Research ArticlePositive Solutions for a Fourth-Order RiemannndashStieltjes IntegralBoundary Value Problem

Yujun Cui 1 Donal OrsquoRegan2 and Jiafa Xu 3

1State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Provinceand the Ministry of Science and Technology Shandong University of Science and Technology QingdaoShandong 266590 China2School of Mathematics Statistics and Applied Mathematics National University of Ireland Galway Ireland3School of Mathematical Sciences Qufu Normal University Qufu 273165 China

Correspondence should be addressed to Yujun Cui cyj720201163com

Received 4 December 2018 Accepted 3 December 2019 Published 18 December 2019

Academic Editor Higinio Ramos

Copyright copy 2019 YujunCui et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper we use the fixed point index to study the existence of positive solutions for the fourth-order RiemannndashStieltjes

integral boundary value problem minus x(4)(t) f(t x(t) xprime(t) xPrime(t) xPrimeprime(t)) t isin (0 1)

x(0) xprime(0) xPrimeprime(1) 0 xPrime(0) α[xPrime(t)]1113896 where f [0 1] times R+ times R+ times R+ times

R+⟶ R+ is a continuous function and α[xPrime] denotes a linear function Two existence theorems are obtained with someappropriate inequality conditions on the nonlinearity f which involve the spectral radius of related linear operators eseconditions allow f(t z1 z2 z3 z4) to have superlinear or sublinear growth in zi i 1 2 3 4

1 Introduction

In this paper we investigate the existence of positive so-lutions for the following fourth-order RiemannndashStieltjesintegral boundary value problem

minus x(4)(t) f t x(t) xprime(t) xPrime(t) xPrimeprime(t)( 1113857 t isin (0 1)

x(0) xprime(0) xPrimeprime(1) 0 xPrime(0) α xPrime(t)1113858 1113859

⎧⎨

⎩

(1)

where α[xPrime(t)] 111393810 xPrime(t)dβ(t) denotes the Riemannndash

Stieltjes integral with a suitable function β of bounded variationand

(H0)α[1] isin [0 1) (2)

e deformation of an elastic beam in an equilibriumstate can be described by a fourth-order ordinary equationboundary value problem [1] and there are a large number ofpapers in the literature in this direction for example see[1ndash30] and the references therein In [1] the author usedKrasnoselrsquoskiirsquos fixed point theorem to establish one or twopositive solutions for the fourth-order boundary valueproblem

u(4)(t) + βuPrime(t) λf t u(t) uPrime(t)( 1113857 t isin (0 1)

u(0) u(1) 11139461

0p(s)u(s)ds

uPrime(0) uPrime(1) 11139461

0q(s)uPrime(s)ds

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(3)

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3748631 12 pageshttpsdoiorg10115520193748631

When f isin C([0 1] times R+ times Rminus R+) and in [2] the au-thors studied the existence of positive solutions for thefourth-order m-point boundary value problem

u(4) + αuPrime minus βu f(t u) t isin (0 1)

u(0) 1113944mminus 2

i1aiu ξi( 1113857 u(1) 1113944

mminus 2

i1biu ξi( 1113857

uPrime(0) 1113944mminus 2

i1aiuPrime ξi( 1113857

uPrime(1) 1113944mminus 2

i1biuPrime ξi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

where f isin C([0 1] times R+R+) satisfies superlinear andsublinear growth conditions

superlinear lim infu⟶+infin

mintisin[01]

f(t u)

ugt λlowast

lim supu⟶0+

maxtisin[01]

f(t u)

ult λlowast

sublinear lim infu⟶0+

mintisin[01]

f(t u)

ugt λlowast

lim supu⟶+infin

maxtisin[01]

f(t u)

ult λlowast

(5)

where λlowast is the first eigenvalue of the relevant linearoperator

In [3] the authors studied the existence of an iterativesolution for the fourth-order boundary value problem

u(4)(t) f t u(t) uprime(t)( 1113857 t isin (0 1)

u(0) uprime(0) uprime(1) uPrime(1) 0

⎧⎨

⎩ (6)

where f [0 1] times R2⟶ R is continuous and satisfies someappropriate Lipschitz condition and in [4] the authors usedthe method of upper and lower solution to establish exis-tence results for the fourth-order four-point boundary valueproblem on time scales

uΔΔΔΔ(t) f t u(σ(t)) uΔΔ(t)( 1113857 t isin [0 1]T

u(0) u σ4(1)( 1113857 0

αuΔΔ ξ1( 1113857 minus βuΔΔΔ ξ1( 1113857 0

cuΔΔ ξ2( 1113857 + ηuΔΔΔ ξ2( 1113857 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(7)

where f [0 1] times R times R⟶ R is a continuous functionere are only a few papers in the literature which

consider general nonlinearities for fourth-order boundaryvalue problems e difficulty lies in a priori estimates forthird-order derivatives so some authors adopted a Nagumo-type growth condition (see (H7) in Section 3) to overcomethis difficulty for example see [15 16 31ndash34] and thereferences therein In [15] the author studied the existenceof positive solutions for the fourth-order boundary valueproblem

u(4) f t u uprime uPrime uPrimeprime( 1113857 t isin (0 1)

u(0) uprime(0) uPrime(1) uPrimeprime(1) 0

⎧⎨

⎩ (8)

where f isin C([0 1] times R+ times R+ times R+ times Rminus R+) satisfiessome inequality conditions where f grows both super-linearly and sublinearly about its variables u uprime uPrime and uPrimeprimeWhen f is superlinear a Nagumo-type condition is used torestrict the growth of f on uPrime and uPrimeprime

Integral boundary conditions arise in thermal conduc-tion problems [35] semiconductor problems [36] andhydrodynamic problems [37] and there are some papers inthe literature devoted to this direction (see [1 9 1934 38ndash48]) In [19] the authors studied p-Laplacian fourth-order differential equations with RiemannndashStieltjes integralboundary conditions

ϕp1uPrime(t)( 11138571113872 1113873Prime λp1minus 1a1(t)f1(t u(t) v(t)) 0lt tlt 1

ϕp1vPrime(t)( 11138571113872 1113873Prime μp2minus 1a2(t)f2(t u(t) v(t))

u(0) u(1) 11139461

0u(s)dξ1(s)

ϕp1uPrime(0)( 1113857 ϕp1

uPrime(1)( 1113857 11139461

0uPrime(s)( 1113857dη1(s)ϕp1

v(0) v(1) 11139461

0v(s)dξ2(s)

ϕp2vPrime(0)( 1113857 ϕp2

vPrime(1)( 1113857 11139461

0ϕp2

vPrime(s)( 1113857dη2(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

e authors used fixed point theory in cones to obtainthe existence of positive solutions for the above problem andprovided the interval ranges of the parameters λ and μ forthese solutions

In [38] the authors studied the fractional differentialequation with a singular decreasing nonlinearity and a p-Laplacian operator

minus Dα0+ φp minus D

c0+z( 11138571113872 1113873(x) f(x z(x)) 0ltxlt 1

z(0) 0 Dc0+z(0) D

c0+z(1) 0

z(1) 11139461

0z(x)dχ(x)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(10)

Using a double iterative technique they showed that theabove problem has a unique positive solution and from aniterative technique they established an appropriate se-quence which converges uniformly to the unique positivesolution

Motivated by the aforementioned works the aim of thispaper is to study the existence of positive solutions for thefourth-order RiemannndashStieltjes integral boundary valueproblem (1) e novelty is of two folds (1) we provide someuseful inequality conditions on f involving the first eigen-value of the relevant linear operator (these conditions implythat f grows superlinearly and sublinearly) and (2) for the

2 Mathematical Problems in Engineering

superlinear case an appropriate Nagumo-type condition isused to restrict the growth of f on xPrimeprime in (1)

2 Preliminaries

In this section we first transform (1) into an equivalentHammerstein-type integral equation For this letxPrime(t) y(t) for t isin [0 1] en from the conditionsx(0) xprime(0) 0 we have

xprime(t) 1113946t

0xPrime(s)ds 1113946

t

0y(s)ds

x(t) 1113946t

0xprime(s)ds 1113946

t

01113946

s

0y(τ)dτ ds

(11)

erefore substituting (11) into (1) gives

minus yPrime(t) f t 1113946t

01113946

s

0y(τ)dτ ds 1113946

t

0y(s)ds y(t) yprime(t)1113888 1113889 t isin (0 1)

y(0) α[y(t)]

yprime(1) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)

Lemma 1 lte problem (12) can be transformed into theHammerstein-type integral equation

y(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

(13)

where G1(t s) (11 minus α[1]) 111393810 K1(t s)dβ(t) + K1(t s)

and K1(t s) min t s for t s isin [0 1]

Proof Using the function g on [0 1] to replacef(t 1113938t

0 1113938s

0 y

(τ)dτ ds 1113938t

0 y(s)ds y(t) yprime(t)) in (12) we consider thefollowing problem

minus yPrime(t) g(t) t isin (0 1)

y(0) α[y(t)]

yprime(1) 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(14)

From the differential equation in (14) we obtain

y(t) 1113946t

0(s minus t)g(s)ds + c1t + c2 for some ci isin R i 1 2

(15)

and then

yprime(t) minus 1113946t

0g(s)ds + c1

yPrime(t) minus g(t)

(16)

e condition yprime(1) 0 implies that

c1 11139461

0g(s)ds (17)

Using the condition y(0) α[y(t)] it enables us toobtain

c2 11139461

01113946

t

0(s minus t)g(s)ds + 1113946

1

0tg(s)ds + c21113890 1113891dβ(t)

11139461

011139461

0K1(t s)g(s)ds + c21113890 1113891dβ(t)

(18)

Hence we have

c2 1

1 minus α[1]11139461

011139461

0K1(t s)g(s)ds dβ(t) (19)

As a result substituting c1 and c2 into (15) gives

y(t) 1113946t

0(s minus t)g(s)ds + 1113946

1

0tg(s)ds +

11 minus α[1]

11139461

011139461

0K1(t s)g(s)ds dβ(t)

11139461

0G1(t s)g(s)ds

(20)

is completes the proofLet E C1[0 1] y max ||y||C ||yprime||C1113864 1113865 with ||y||C

maxtisin[01]|y(t)| and P y isin E y(t)ge 0 yprime(t)ge 0 forallt isin1113864

[0 1] en (E middot) is a Banach space and P is a cone onE From Lemma 1 we can define an operator A P⟶ P asfollows

(Ay)(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y(r)dr dτ1113874

1113946s

0y(τ)dτ y(s) yprime(s)1113875ds fory isin E

(21)

en A is a completely continuous operator from theArzelandashAscoli theorem (this argument is standard)

Mathematical Problems in Engineering 3

Remark 1

(i) In our work we need the nonnegativity of Greenrsquosfunction G1 so we have the following assumption

(H1) 11139461

0K1(t s)dβ(t)ge 0 for s isin [0 1] (22)

(ii) We need some inequality conditions on the non-linearity f(t z1 z2 z3 z4) with respect to the vari-ables zi i 1 2 3 4 We consider some useful linearoperators

L1y( 1113857(t) 11139461

0G1(t s) 1113946

s

01113946τ

0y(r)dr dτ ds ≔ 1113946

1

0G2(t s)y(s)ds

L2y( 1113857(t) 11139461

0G1(t s) 1113946

1

01113946

s

0y(τ)dτ ds ≔ 1113946

1

0G3(t s)y(s)ds fory isin E t isin [0 1]

(23)

If we know the function β we can obtain the functionsG2 and G3

Example 1 Let β(t) 0 for t isin [0 1] enG1(t s) K1(t s) for t s isin [0 1] Let

h(s τ) 1 0le τ le sle 1

0 0le sle τ le 11113896 (24)

and then from (22) we find

G3(t s) 11139461

0G1(t τ)h(τ s)dτ for t s isin [0 1] (25)

We consider two cases

(i) Case 1 when 0le tle sle 1 we have

G3(t s) 1113946t

00 middot τdτ + 1113946

s

t0 middot tdτ + 1113946

1

s1 middot tdτ t(1 minus s)

(26)

(ii) Case 2 when 0le sle tle 1 we have

G3(t s) 1113946s

00 middot τdτ + 1113946

t

s1 middot τdτ + 1113946

1

t1 middot tdτ t minus

12t2

minus12s2

(27)

We now calculate G2 For this let 1113938τ0 y(r)dr j(τ) and

k(τ r) 0 0le τ le rle 1

1 0le rle τ le 11113896 en we have

11139461

0G1(t s) 1113946

s

0j(τ)dτ ds 1113946

1

0G3(t τ)j(τ)dτ

11139461

0G3(t τ) 1113946

τ

0y(r)dr dτ

11139461

0G3(t τ) 1113946

1

0k(τ r)y(r)dr dτ

(28)

erefore from (22) we have

G2(t s) 11139461

0G3(t τ)k(τ s)dτ for t s isin [0 1] (29)

We consider two cases

(i) Case 1 when 0le tle sle 1 we have

G2(t s) 1113946t

00 middot t minus

12t2

minus12τ21113874 1113875dτ

+ 1113946s

t0 middot t(1 minus τ)dτ + 1113946

1

s1 middot t(1 minus τ)dτ

t12

+12s2

minus s1113874 1113875

(30)

(ii) Case 2 when 0le sle tle 1 we have

G2(t s) 1113946s

00 middot t minus

12t2

minus12τ21113874 1113875dτ

+ 1113946t

s1 middot t minus

12t2

minus12τ21113874 1113875dτ + 1113946

1

t1 middot t(1 minus τ)dτ

12

t minus ts +12t2s minus

16t3

+16s3

(31)

Example 2 Let β(t) (t2) for t isin [0 1] en we have

11 minus α[1]

11139461

0K1(t s)dβ(t) 1113946

1

0K1(t s)dt 1113946

s

0t dt

+ 11139461

ssdt s minus

12s2 for s isin [0 1]

(32)

Hence G1(t s) s minus (12)s2 + K1(t s) for t s isin [0 1]Note (22) and Example 1 so we only need to calculate

11139461

0τ minus

12τ21113874 1113875h(τ s)dτ 1113946

1

sτ minus

12τ21113874 1113875dτ

13

minus12s2

+16s3

11139461

0

13

minus12τ2 +

16τ31113874 1113875k(τ s)dτ 1113946

1

s

13

minus12τ2 +

16τ31113874 1113875dτ

524

minus13

s +16s3

minus124

s4 for s isin [0 1]

(33)

erefore we obtain

4 Mathematical Problems in Engineering

G3(t s) 13

minus12s2

+16s3

+

t(1 minus s) 0le tle sle 1

t minus12t2

minus12s2 0le sle tle 1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

G2(t s) 524

minus13

s +16s3

minus124

s4

+

t12

+12s2

minus s1113874 1113875 0le tle sle 1

12

t minus ts +12t2s minus

16t3

+16s3 0le sle tle 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(34)

Lemma 2 (KreinndashRutman see([49] theorem 193)) Let Pbe a reproducing cone in a real Banach space E and letL E⟶ E be a compact linear operator with L(P) sub P If

r(L)gt 0 then there exists φ isin P 0 such that Lφ r(L)φwhere r(L) is the spectral radius of L

Lemma 3 For not all zero numbers a b c and dge 0 we let

Labcdy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds + d 1113946

1

0G1(t s)yprime(s)ds

Labcy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds

Llowastabcy1113872 1113873(s) a 1113946

1

0G2(t s)y(t)dt + b 1113946

1

0G3(t s)y(t)dt + c 1113946

1

0G1(t s)y(t)dt fory isin P

(35)

en130

a +18

b +13

c +13

d1113874 1113875κ1 le r Labcd1113872 1113873le18

a +13

b +12

c +12

d1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Labc1113872 1113873le18

a +13

b +12

c1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Llowastabc1113872 1113873le

18

a +13

b +12

c1113874 1113875κ2

(36)

where κ1 ≔ (α[t] minus α[1] + 1)(1 minus α[1]) and κ2 ≔ 1(1 minus

α[1])

Proof We first give some inequalities for Gi i 1 2 3 Notethat tsleK1(t s)le s for t s isin [0 1] and from the definitionof G1 we see that

κ1ts α[t] minus α[1] + 1

1 minus α[1]tsleG1(t s)le

11 minus α[1]

s

κ2s for t s isin [0 1]

(37)

With h and k as before note

12κ1t 1 minus s

21113872 1113873 1113946

1

sκ1tτdτ leG3(t s) 1113946

1

0G1(t τ)h(τ s)dτ le 1113946

1

sκ2τdτ

12κ2 1 minus s

21113872 1113873

for t s isin [0 1]

16κ1t 2 + s

3minus 3s1113872 1113873 1113946

1

s

12κ1t 1 minus τ21113872 1113873dτ leG2(t s) 1113946

1

0G3(t τ)k(τ s)dτ le 1113946

1

s

12κ2 1 minus τ21113872 1113873dτ

16κ2 2 + s

3minus 3s1113872 1113873 for t s isin [0 1]

(38)

Mathematical Problems in Engineering 5

For convenience let 1113957ψ0(t) t 1113957ψ1(t) (16)(2 + t3 minus

3t) and 1113957ψ2(t) (12)(1 minus t2) for t isin [0 1]We only prove the inequalities in (35) about the spectral

radius of Labc For convenience let Gabc(t s) aG2(t s) +

bG3(t s) + cG1(t s) for t s isin [0 1] en we have

Labcy1113872 1113873(t) 11139461

0Gabc(t s)y(s)ds fory isin P (39)

us we obtain

Labc

sup

||y||1Labcy

sup

tisin[01]y111139461

0Gabc(t s)y(s)ds

le maxtisin[01]

11139461

0Gabc(t s)dsle 1113946

1

0κ2 a1113957ψ1(s)1113858

+ b1113957ψ2(s) + c1113957ψ0(s)1113859ds

le18

a +13

b +12

c1113874 1113875κ2

(40)For all n isin N+ we note that

Lnabcy1113872 1113873(t) L L

nminus 1abcy1113872 1113873(t) 1113946

1

0Gabc t snminus 1( 1113857 L

nminus 1abcy1113872 1113873 snminus 1( 1113857dsnminus 1

middot middot middot

1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857y(s)dsnminus 1dsnminus 2 middot middot middot ds fory isin P

(41)

Hence we can obtain

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868le max

tisin[01]1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

le κn2 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)( 1113857ds1113890 1113891

n

le18

a +13

b +12

c1113874 1113875n

κn2

(42)

Gelfandrsquos theorem implies that

r Labc1113872 1113873 limn⟶infin

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

n

1113969

le18

a +13

b +12

c1113874 1113875κ2 (43)

Next we introduce a conclusion in ([50] problem 214)Let y isin C[0 1] and a functional J on C[0 1] be as

J(x) 11139461

0x(t)y(t)dt forallx isin C[0 1] (44)

en we have

J 11139461

0|y(t)|dt (45)

We note that there exists t0 isin [0 1] such thatmaxtisin[01] 1113938

10 Gabc(t s)ds 1113938

10 Gabc(t0 s)ds en in (38)

for fixed t we define a linear function

Labcty 11139461

0Gabc(t s)y(s)ds fory isin C[0 1] (46)

and thus

Labct

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 1113946

1

0Gabc(t s)ds (47)

en by the definition of the norm of linear function weknow that for all εgt 0 there exists yεt isin C[0 1] with ||yεt||

1 such that

11139461

0Gabc(t s)ds Labct

sup

y1Labcty|ge |Labctyεt

11138681113868111386811138681113868111386811138681113868

ge 11139461

0Gabc(t s)ds minus ε

(48)

On the contrary note from the definition of our normwehave

Labcy

suptisin[01]

Labcty|ge |Labct0y

11138681113868111386811138681113868

11138681113868111386811138681113868 (49)

Consequently we have

Labc

sup

y1Labcy

ge Labct0

yεt0

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868ge 11139461

0Gabc t0 s( 1113857ds minus ε

(50)

For the arbitrariness of ε we have

Labc

1113946

1

0Gabc t0 s( 1113857ds max

tisin[01]11139461

0Gabc(t s)ds (51)

Also for all n isin N+ we obtain

6 Mathematical Problems in Engineering

Lnabc

max

tisin[01]1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857 middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

ge κn1 max

tisin[01]t1113888 1113889 1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

a1113957ψ1 snminus 1( 1113857 + b1113957ψ2 snminus 1( 1113857 + c1113957ψ0 snminus 1( 11138571113858 1113859

middot snminus 1 a1113957ψ1 snminus 2( 1113857 + b1113957ψ2 snminus 2( 1113857 + c1113957ψ0 snminus 2( 11138571113858 1113859 middot middot middot s1 a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859dsnminus 1dsnminus 2 middot middot middot ds

κn1 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859ds 1113946

1

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds1113888 1113889

nminus 1

(52)

From Gelfandrsquos theorem we have

r Labc1113872 1113873 limn⟶infin

Lnabc

n

1113969

ge κ1 11139461

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds

ge130

a +18

b +13

c1113874 1113875κ1

(53)

is completes the proof

Lemma 4 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set and thatA Ω cap P⟶ P is a continuous compact operator If thereexists a ω0 isin P 0 such that

ω minus Aωne λω0 forallλge 0 ω isin zΩ cap P (54)

then i(AΩ cap P P) 0 where i denotes the fixed point indexon P

Lemma 5 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set with0 isin Ω and that A ΩcapP⟶ P is a continuous compactoperator If

ω minus λAωne 0 forallλ isin [0 1] ω isin zΩ capP (55)

then i(AΩcapP P) 1

3 Main Results

In our paper we let Bρ y isin P ylt ρ1113864 1113865 for ρgt 0 NowzBρ y isin P y ρ1113864 1113865 and Bρ y isin P yle ρ1113864 1113865 Now welist our assumptions on the nonlinearity f

(H2)f isin C [0 1] times R+

times R+

times R+

times R+R

+( 1113857 (56)

(H3) ere exist not all zero numbersa1 b1 c1 andd1 ge 0 and e1 gt 0 such that r(La1 b1 c1 d1

)lt 1and f(t z1 z2 z3 z4)le a1z1 + b1z2 + c1z3 + d1z4 + e1

for (t z1 z2 z3 z4) isin [0 1] times R+ times R + times R+ times R+(H4) ere exist not all zero numbers

a2 b2 c2 andd2 ge 0 and ρ1 gt 0 such that r(La2 b2 c2 d2)ge 1

and f(t z1 z2 z3 z4)ge a2z1 + b2z2 + c2z3 + d2z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ1] times [0 ρ1]times [0 ρ1] times [0 ρ1](H5) ere exist not all zero numbers

a3 b3 c3 andd3 ge 0 and ρ2 gt 0 such that r(La3 b3 c3 d3)lt 1

and f(t z1 z2 z3 z4)le a3z1 + b3z2 + c3z3 + d3z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ2] times [0 ρ 2] times [0 ρ2] times [0 ρ2](H6) ere exist not all zero numbers a4 b4 and c4 ge 0

and e2 gt 0 such that r(Llowasta4 b4 c4)gt 1 and f(t z1 z2

z3 z4)ge a4z1 + b4z2 + c4z3 minus e2 for (t z1 z2 z3 z4) isin[0 1] times R+ times R+ times R+ times R+

(H7) For any Mgt 0 there exists a positive continuousfunction HM(9) on R+ such that 1113938

+infin0 (9d9HM(9) + δ0)

+infin forallδ0 gt 0 and f(t z1 z2 z3 z4)leHM(z4) for (t z1 z2

z3 z4) isin [0 1] times [0 M]times [0 M] times [0 M] times R+

Remark 2 Considering Lemma 3 one can adjust the co-efficients ai bi ci anddi (i 1 2 3 4) such that the spectralradii r(Laibicidi

) r(La4 b4 c4) r(Llowasta4 b4 c4

)(i 1 2 3) satisfy

their respective conditions in (H3)ndash(H6)

Theorem 1 Suppose that (H0ndashH4) hold lten (1) has atleast one positive solution

Proof Let W y isin P y λAy λ isin [0 1]1113864 1113865 Now weprove thatW is a bounded set in P If y isinW then from (H3)we have

y(t) λ(Ay)(t)le (Ay)(t)le 11139461

0G1(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0a1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0G1(t s)yprime(s)ds + e1η(t)

La1 b1 c1 d1y1113872 1113873(t) + e1η(t)

(57)

Mathematical Problems in Engineering 7

where

η(t) 11139461

0

11 minus α[1]

11139461

0K1(t s)dβ(t) + K1(t s)1113890 1113891ds

1

1 minus α[1]11139461

011139461

0K1(t s)dβ(t)ds + t minus

12

t2

(58)

for t isin [0 1] is implies that

I minus La1 b1 c1d11113872 1113873y1113872 1113873(t)le e1η(t) (59)

Note that η isin P(ηprime(t) 1 minus tge 0 forallt isin [0 1]) y isinWand we obtain

yprime(t) λ(Ay)prime(t) λ11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0

z

zta1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0

z

ztG1(t s)yprime(s)ds + e1ηprime(t)

La1 b1 c1 d1y1113872 1113873prime(t) + e1ηprime(t)

(60)

us

I minus La1 b1 c1 d11113872 1113873y1113872 1113873prime(t)le e1ηprime(t) (61)

Since r(La1 b1 c1 d1)lt 1 we know that I minus La1 b1 c1 d1

has abounded inverse operator (I minus La1 b1 c1 d1

)minus 1 with

I minus La1 b1 c1 d11113872 1113873

minus 1 I + La1 b1 c1 d1

+ L2a1 b1 c1 d1

+ middot middot middot

+ Lna1 b1 c1 d1

+ middot middot middot (62)

Note that La1 b1 c1 d1(P) sub P and we obtain (Iminus

La1 b1 c1 d1)minus 1(P) sub P erefore

y(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875(t)

yprime(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875prime(t)

for t isin [0 1]

(63)

is implies that

yC le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

C

yprime

Cle I minus La1 b1 c1 d1

1113872 1113873minus 1

e1η1113874 1113875prime

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868C

(64)

erefore we have

||y||le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

(65)

at is W is bounded Now we can selectR1 gtmax supW ρ11113864 1113865 (ρ1 is defined in (H4)) such that

yne λAy fory isin zBR1capP and λ isin [0 1] (66)

From Lemma 5 we have

i A BR1capP P1113872 1113873 1 (67)

On the contrary since La2 b2 c2 d2(P) sub P and

r(La2 b2 c2 d2)ge 1 it follows from Lemma 2 that there exists

φ0 isin P | 0 such that La2b2 c2 d2φ0 r(La2 b2 c2 d2

)φ0 andφ0 r(La2 b2 c2 d2

)minus 1La2 b2 c2 d2φ0 isin P Now we show that

y minus Ayne λφ0 fory isin zBρ1 capP λge 0 (68)

If this claim is false then there exist y0 isin zBρ1 capP andλ0 ge 0 such that y0 minus Ay0 λ0φ0 Note that λ0 gt 0 (other-wise the theorem is proved) en from (H4) we have

Ay0( 1113857(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y0(r)drdτ 1113946

s

0y0(τ)dτ y0(s) y0prime(s)1113874 1113875ds

ge 11139461

0G1(t s) a2 1113946

s

01113946τ

0y0(r)drdτ + b2 1113946

s

0y0(τ)dτ + c2y0(s) + d2y0prime(s)1113874 1113875ds

11139461

0a2G2(t s) + b2G3(t s) + c2G1(t s)( 1113857y0(s)ds + d2 1113946

1

0G1(t s)y0prime(s)ds

La2 b2 c2d2y01113872 1113873(t)

(69)

8 Mathematical Problems in Engineering

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

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superlinear case an appropriate Nagumo-type condition isused to restrict the growth of f on xPrimeprime in (1)

2 Preliminaries

In this section we first transform (1) into an equivalentHammerstein-type integral equation For this letxPrime(t) y(t) for t isin [0 1] en from the conditionsx(0) xprime(0) 0 we have

xprime(t) 1113946t

0xPrime(s)ds 1113946

t

0y(s)ds

x(t) 1113946t

0xprime(s)ds 1113946

t

01113946

s

0y(τ)dτ ds

(11)

erefore substituting (11) into (1) gives

minus yPrime(t) f t 1113946t

01113946

s

0y(τ)dτ ds 1113946

t

0y(s)ds y(t) yprime(t)1113888 1113889 t isin (0 1)

y(0) α[y(t)]

yprime(1) 0

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)

Lemma 1 lte problem (12) can be transformed into theHammerstein-type integral equation

y(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

(13)

where G1(t s) (11 minus α[1]) 111393810 K1(t s)dβ(t) + K1(t s)

and K1(t s) min t s for t s isin [0 1]

Proof Using the function g on [0 1] to replacef(t 1113938t

0 1113938s

0 y

(τ)dτ ds 1113938t

0 y(s)ds y(t) yprime(t)) in (12) we consider thefollowing problem

minus yPrime(t) g(t) t isin (0 1)

y(0) α[y(t)]

yprime(1) 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(14)

From the differential equation in (14) we obtain

y(t) 1113946t

0(s minus t)g(s)ds + c1t + c2 for some ci isin R i 1 2

(15)

and then

yprime(t) minus 1113946t

0g(s)ds + c1

yPrime(t) minus g(t)

(16)

e condition yprime(1) 0 implies that

c1 11139461

0g(s)ds (17)

Using the condition y(0) α[y(t)] it enables us toobtain

c2 11139461

01113946

t

0(s minus t)g(s)ds + 1113946

1

0tg(s)ds + c21113890 1113891dβ(t)

11139461

011139461

0K1(t s)g(s)ds + c21113890 1113891dβ(t)

(18)

Hence we have

c2 1

1 minus α[1]11139461

011139461

0K1(t s)g(s)ds dβ(t) (19)

As a result substituting c1 and c2 into (15) gives

y(t) 1113946t

0(s minus t)g(s)ds + 1113946

1

0tg(s)ds +

11 minus α[1]

11139461

011139461

0K1(t s)g(s)ds dβ(t)

11139461

0G1(t s)g(s)ds

(20)

is completes the proofLet E C1[0 1] y max ||y||C ||yprime||C1113864 1113865 with ||y||C

maxtisin[01]|y(t)| and P y isin E y(t)ge 0 yprime(t)ge 0 forallt isin1113864

[0 1] en (E middot) is a Banach space and P is a cone onE From Lemma 1 we can define an operator A P⟶ P asfollows

(Ay)(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y(r)dr dτ1113874

1113946s

0y(τ)dτ y(s) yprime(s)1113875ds fory isin E

(21)

en A is a completely continuous operator from theArzelandashAscoli theorem (this argument is standard)

Mathematical Problems in Engineering 3

Remark 1

(i) In our work we need the nonnegativity of Greenrsquosfunction G1 so we have the following assumption

(H1) 11139461

0K1(t s)dβ(t)ge 0 for s isin [0 1] (22)

(ii) We need some inequality conditions on the non-linearity f(t z1 z2 z3 z4) with respect to the vari-ables zi i 1 2 3 4 We consider some useful linearoperators

L1y( 1113857(t) 11139461

0G1(t s) 1113946

s

01113946τ

0y(r)dr dτ ds ≔ 1113946

1

0G2(t s)y(s)ds

L2y( 1113857(t) 11139461

0G1(t s) 1113946

1

01113946

s

0y(τ)dτ ds ≔ 1113946

1

0G3(t s)y(s)ds fory isin E t isin [0 1]

(23)

If we know the function β we can obtain the functionsG2 and G3

Example 1 Let β(t) 0 for t isin [0 1] enG1(t s) K1(t s) for t s isin [0 1] Let

h(s τ) 1 0le τ le sle 1

0 0le sle τ le 11113896 (24)

and then from (22) we find

G3(t s) 11139461

0G1(t τ)h(τ s)dτ for t s isin [0 1] (25)

We consider two cases

(i) Case 1 when 0le tle sle 1 we have

G3(t s) 1113946t

00 middot τdτ + 1113946

s

t0 middot tdτ + 1113946

1

s1 middot tdτ t(1 minus s)

(26)

(ii) Case 2 when 0le sle tle 1 we have

G3(t s) 1113946s

00 middot τdτ + 1113946

t

s1 middot τdτ + 1113946

1

t1 middot tdτ t minus

12t2

minus12s2

(27)

We now calculate G2 For this let 1113938τ0 y(r)dr j(τ) and

k(τ r) 0 0le τ le rle 1

1 0le rle τ le 11113896 en we have

11139461

0G1(t s) 1113946

s

0j(τ)dτ ds 1113946

1

0G3(t τ)j(τ)dτ

11139461

0G3(t τ) 1113946

τ

0y(r)dr dτ

11139461

0G3(t τ) 1113946

1

0k(τ r)y(r)dr dτ

(28)

erefore from (22) we have

G2(t s) 11139461

0G3(t τ)k(τ s)dτ for t s isin [0 1] (29)

We consider two cases

(i) Case 1 when 0le tle sle 1 we have

G2(t s) 1113946t

00 middot t minus

12t2

minus12τ21113874 1113875dτ

+ 1113946s

t0 middot t(1 minus τ)dτ + 1113946

1

s1 middot t(1 minus τ)dτ

t12

+12s2

minus s1113874 1113875

(30)

(ii) Case 2 when 0le sle tle 1 we have

G2(t s) 1113946s

00 middot t minus

12t2

minus12τ21113874 1113875dτ

+ 1113946t

s1 middot t minus

12t2

minus12τ21113874 1113875dτ + 1113946

1

t1 middot t(1 minus τ)dτ

12

t minus ts +12t2s minus

16t3

+16s3

(31)

Example 2 Let β(t) (t2) for t isin [0 1] en we have

11 minus α[1]

11139461

0K1(t s)dβ(t) 1113946

1

0K1(t s)dt 1113946

s

0t dt

+ 11139461

ssdt s minus

12s2 for s isin [0 1]

(32)

Hence G1(t s) s minus (12)s2 + K1(t s) for t s isin [0 1]Note (22) and Example 1 so we only need to calculate

11139461

0τ minus

12τ21113874 1113875h(τ s)dτ 1113946

1

sτ minus

12τ21113874 1113875dτ

13

minus12s2

+16s3

11139461

0

13

minus12τ2 +

16τ31113874 1113875k(τ s)dτ 1113946

1

s

13

minus12τ2 +

16τ31113874 1113875dτ

524

minus13

s +16s3

minus124

s4 for s isin [0 1]

(33)

erefore we obtain

4 Mathematical Problems in Engineering

G3(t s) 13

minus12s2

+16s3

+

t(1 minus s) 0le tle sle 1

t minus12t2

minus12s2 0le sle tle 1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

G2(t s) 524

minus13

s +16s3

minus124

s4

+

t12

+12s2

minus s1113874 1113875 0le tle sle 1

12

t minus ts +12t2s minus

16t3

+16s3 0le sle tle 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(34)

Lemma 2 (KreinndashRutman see([49] theorem 193)) Let Pbe a reproducing cone in a real Banach space E and letL E⟶ E be a compact linear operator with L(P) sub P If

r(L)gt 0 then there exists φ isin P 0 such that Lφ r(L)φwhere r(L) is the spectral radius of L

Lemma 3 For not all zero numbers a b c and dge 0 we let

Labcdy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds + d 1113946

1

0G1(t s)yprime(s)ds

Labcy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds

Llowastabcy1113872 1113873(s) a 1113946

1

0G2(t s)y(t)dt + b 1113946

1

0G3(t s)y(t)dt + c 1113946

1

0G1(t s)y(t)dt fory isin P

(35)

en130

a +18

b +13

c +13

d1113874 1113875κ1 le r Labcd1113872 1113873le18

a +13

b +12

c +12

d1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Labc1113872 1113873le18

a +13

b +12

c1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Llowastabc1113872 1113873le

18

a +13

b +12

c1113874 1113875κ2

(36)

where κ1 ≔ (α[t] minus α[1] + 1)(1 minus α[1]) and κ2 ≔ 1(1 minus

α[1])

Proof We first give some inequalities for Gi i 1 2 3 Notethat tsleK1(t s)le s for t s isin [0 1] and from the definitionof G1 we see that

κ1ts α[t] minus α[1] + 1

1 minus α[1]tsleG1(t s)le

11 minus α[1]

s

κ2s for t s isin [0 1]

(37)

With h and k as before note

12κ1t 1 minus s

21113872 1113873 1113946

1

sκ1tτdτ leG3(t s) 1113946

1

0G1(t τ)h(τ s)dτ le 1113946

1

sκ2τdτ

12κ2 1 minus s

21113872 1113873

for t s isin [0 1]

16κ1t 2 + s

3minus 3s1113872 1113873 1113946

1

s

12κ1t 1 minus τ21113872 1113873dτ leG2(t s) 1113946

1

0G3(t τ)k(τ s)dτ le 1113946

1

s

12κ2 1 minus τ21113872 1113873dτ

16κ2 2 + s

3minus 3s1113872 1113873 for t s isin [0 1]

(38)

Mathematical Problems in Engineering 5

For convenience let 1113957ψ0(t) t 1113957ψ1(t) (16)(2 + t3 minus

3t) and 1113957ψ2(t) (12)(1 minus t2) for t isin [0 1]We only prove the inequalities in (35) about the spectral

radius of Labc For convenience let Gabc(t s) aG2(t s) +

bG3(t s) + cG1(t s) for t s isin [0 1] en we have

Labcy1113872 1113873(t) 11139461

0Gabc(t s)y(s)ds fory isin P (39)

us we obtain

Labc

sup

||y||1Labcy

sup

tisin[01]y111139461

0Gabc(t s)y(s)ds

le maxtisin[01]

11139461

0Gabc(t s)dsle 1113946

1

0κ2 a1113957ψ1(s)1113858

+ b1113957ψ2(s) + c1113957ψ0(s)1113859ds

le18

a +13

b +12

c1113874 1113875κ2

(40)For all n isin N+ we note that

Lnabcy1113872 1113873(t) L L

nminus 1abcy1113872 1113873(t) 1113946

1

0Gabc t snminus 1( 1113857 L

nminus 1abcy1113872 1113873 snminus 1( 1113857dsnminus 1

middot middot middot

1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857y(s)dsnminus 1dsnminus 2 middot middot middot ds fory isin P

(41)

Hence we can obtain

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868le max

tisin[01]1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

le κn2 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)( 1113857ds1113890 1113891

n

le18

a +13

b +12

c1113874 1113875n

κn2

(42)

Gelfandrsquos theorem implies that

r Labc1113872 1113873 limn⟶infin

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

n

1113969

le18

a +13

b +12

c1113874 1113875κ2 (43)

Next we introduce a conclusion in ([50] problem 214)Let y isin C[0 1] and a functional J on C[0 1] be as

J(x) 11139461

0x(t)y(t)dt forallx isin C[0 1] (44)

en we have

J 11139461

0|y(t)|dt (45)

We note that there exists t0 isin [0 1] such thatmaxtisin[01] 1113938

10 Gabc(t s)ds 1113938

10 Gabc(t0 s)ds en in (38)

for fixed t we define a linear function

Labcty 11139461

0Gabc(t s)y(s)ds fory isin C[0 1] (46)

and thus

Labct

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 1113946

1

0Gabc(t s)ds (47)

en by the definition of the norm of linear function weknow that for all εgt 0 there exists yεt isin C[0 1] with ||yεt||

1 such that

11139461

0Gabc(t s)ds Labct

sup

y1Labcty|ge |Labctyεt

11138681113868111386811138681113868111386811138681113868

ge 11139461

0Gabc(t s)ds minus ε

(48)

On the contrary note from the definition of our normwehave

Labcy

suptisin[01]

Labcty|ge |Labct0y

11138681113868111386811138681113868

11138681113868111386811138681113868 (49)

Consequently we have

Labc

sup

y1Labcy

ge Labct0

yεt0

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868ge 11139461

0Gabc t0 s( 1113857ds minus ε

(50)

For the arbitrariness of ε we have

Labc

1113946

1

0Gabc t0 s( 1113857ds max

tisin[01]11139461

0Gabc(t s)ds (51)

Also for all n isin N+ we obtain

6 Mathematical Problems in Engineering

Lnabc

max

tisin[01]1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857 middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

ge κn1 max

tisin[01]t1113888 1113889 1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

a1113957ψ1 snminus 1( 1113857 + b1113957ψ2 snminus 1( 1113857 + c1113957ψ0 snminus 1( 11138571113858 1113859

middot snminus 1 a1113957ψ1 snminus 2( 1113857 + b1113957ψ2 snminus 2( 1113857 + c1113957ψ0 snminus 2( 11138571113858 1113859 middot middot middot s1 a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859dsnminus 1dsnminus 2 middot middot middot ds

κn1 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859ds 1113946

1

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds1113888 1113889

nminus 1

(52)

From Gelfandrsquos theorem we have

r Labc1113872 1113873 limn⟶infin

Lnabc

n

1113969

ge κ1 11139461

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds

ge130

a +18

b +13

c1113874 1113875κ1

(53)

is completes the proof

Lemma 4 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set and thatA Ω cap P⟶ P is a continuous compact operator If thereexists a ω0 isin P 0 such that

ω minus Aωne λω0 forallλge 0 ω isin zΩ cap P (54)

then i(AΩ cap P P) 0 where i denotes the fixed point indexon P

Lemma 5 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set with0 isin Ω and that A ΩcapP⟶ P is a continuous compactoperator If

ω minus λAωne 0 forallλ isin [0 1] ω isin zΩ capP (55)

then i(AΩcapP P) 1

3 Main Results

In our paper we let Bρ y isin P ylt ρ1113864 1113865 for ρgt 0 NowzBρ y isin P y ρ1113864 1113865 and Bρ y isin P yle ρ1113864 1113865 Now welist our assumptions on the nonlinearity f

(H2)f isin C [0 1] times R+

times R+

times R+

times R+R

+( 1113857 (56)

(H3) ere exist not all zero numbersa1 b1 c1 andd1 ge 0 and e1 gt 0 such that r(La1 b1 c1 d1

)lt 1and f(t z1 z2 z3 z4)le a1z1 + b1z2 + c1z3 + d1z4 + e1

for (t z1 z2 z3 z4) isin [0 1] times R+ times R + times R+ times R+(H4) ere exist not all zero numbers

a2 b2 c2 andd2 ge 0 and ρ1 gt 0 such that r(La2 b2 c2 d2)ge 1

and f(t z1 z2 z3 z4)ge a2z1 + b2z2 + c2z3 + d2z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ1] times [0 ρ1]times [0 ρ1] times [0 ρ1](H5) ere exist not all zero numbers

a3 b3 c3 andd3 ge 0 and ρ2 gt 0 such that r(La3 b3 c3 d3)lt 1

and f(t z1 z2 z3 z4)le a3z1 + b3z2 + c3z3 + d3z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ2] times [0 ρ 2] times [0 ρ2] times [0 ρ2](H6) ere exist not all zero numbers a4 b4 and c4 ge 0

and e2 gt 0 such that r(Llowasta4 b4 c4)gt 1 and f(t z1 z2

z3 z4)ge a4z1 + b4z2 + c4z3 minus e2 for (t z1 z2 z3 z4) isin[0 1] times R+ times R+ times R+ times R+

(H7) For any Mgt 0 there exists a positive continuousfunction HM(9) on R+ such that 1113938

+infin0 (9d9HM(9) + δ0)

+infin forallδ0 gt 0 and f(t z1 z2 z3 z4)leHM(z4) for (t z1 z2

z3 z4) isin [0 1] times [0 M]times [0 M] times [0 M] times R+

Remark 2 Considering Lemma 3 one can adjust the co-efficients ai bi ci anddi (i 1 2 3 4) such that the spectralradii r(Laibicidi

) r(La4 b4 c4) r(Llowasta4 b4 c4

)(i 1 2 3) satisfy

their respective conditions in (H3)ndash(H6)

Theorem 1 Suppose that (H0ndashH4) hold lten (1) has atleast one positive solution

Proof Let W y isin P y λAy λ isin [0 1]1113864 1113865 Now weprove thatW is a bounded set in P If y isinW then from (H3)we have

y(t) λ(Ay)(t)le (Ay)(t)le 11139461

0G1(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0a1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0G1(t s)yprime(s)ds + e1η(t)

La1 b1 c1 d1y1113872 1113873(t) + e1η(t)

(57)

Mathematical Problems in Engineering 7

where

η(t) 11139461

0

11 minus α[1]

11139461

0K1(t s)dβ(t) + K1(t s)1113890 1113891ds

1

1 minus α[1]11139461

011139461

0K1(t s)dβ(t)ds + t minus

12

t2

(58)

for t isin [0 1] is implies that

I minus La1 b1 c1d11113872 1113873y1113872 1113873(t)le e1η(t) (59)

Note that η isin P(ηprime(t) 1 minus tge 0 forallt isin [0 1]) y isinWand we obtain

yprime(t) λ(Ay)prime(t) λ11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0

z

zta1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0

z

ztG1(t s)yprime(s)ds + e1ηprime(t)

La1 b1 c1 d1y1113872 1113873prime(t) + e1ηprime(t)

(60)

us

I minus La1 b1 c1 d11113872 1113873y1113872 1113873prime(t)le e1ηprime(t) (61)

Since r(La1 b1 c1 d1)lt 1 we know that I minus La1 b1 c1 d1

has abounded inverse operator (I minus La1 b1 c1 d1

)minus 1 with

I minus La1 b1 c1 d11113872 1113873

minus 1 I + La1 b1 c1 d1

+ L2a1 b1 c1 d1

+ middot middot middot

+ Lna1 b1 c1 d1

+ middot middot middot (62)

Note that La1 b1 c1 d1(P) sub P and we obtain (Iminus

La1 b1 c1 d1)minus 1(P) sub P erefore

y(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875(t)

yprime(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875prime(t)

for t isin [0 1]

(63)

is implies that

yC le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

C

yprime

Cle I minus La1 b1 c1 d1

1113872 1113873minus 1

e1η1113874 1113875prime

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868C

(64)

erefore we have

||y||le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

(65)

at is W is bounded Now we can selectR1 gtmax supW ρ11113864 1113865 (ρ1 is defined in (H4)) such that

yne λAy fory isin zBR1capP and λ isin [0 1] (66)

From Lemma 5 we have

i A BR1capP P1113872 1113873 1 (67)

On the contrary since La2 b2 c2 d2(P) sub P and

r(La2 b2 c2 d2)ge 1 it follows from Lemma 2 that there exists

φ0 isin P | 0 such that La2b2 c2 d2φ0 r(La2 b2 c2 d2

)φ0 andφ0 r(La2 b2 c2 d2

)minus 1La2 b2 c2 d2φ0 isin P Now we show that

y minus Ayne λφ0 fory isin zBρ1 capP λge 0 (68)

If this claim is false then there exist y0 isin zBρ1 capP andλ0 ge 0 such that y0 minus Ay0 λ0φ0 Note that λ0 gt 0 (other-wise the theorem is proved) en from (H4) we have

Ay0( 1113857(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y0(r)drdτ 1113946

s

0y0(τ)dτ y0(s) y0prime(s)1113874 1113875ds

ge 11139461

0G1(t s) a2 1113946

s

01113946τ

0y0(r)drdτ + b2 1113946

s

0y0(τ)dτ + c2y0(s) + d2y0prime(s)1113874 1113875ds

11139461

0a2G2(t s) + b2G3(t s) + c2G1(t s)( 1113857y0(s)ds + d2 1113946

1

0G1(t s)y0prime(s)ds

La2 b2 c2d2y01113872 1113873(t)

(69)

8 Mathematical Problems in Engineering

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

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Remark 1

(i) In our work we need the nonnegativity of Greenrsquosfunction G1 so we have the following assumption

(H1) 11139461

0K1(t s)dβ(t)ge 0 for s isin [0 1] (22)

(ii) We need some inequality conditions on the non-linearity f(t z1 z2 z3 z4) with respect to the vari-ables zi i 1 2 3 4 We consider some useful linearoperators

L1y( 1113857(t) 11139461

0G1(t s) 1113946

s

01113946τ

0y(r)dr dτ ds ≔ 1113946

1

0G2(t s)y(s)ds

L2y( 1113857(t) 11139461

0G1(t s) 1113946

1

01113946

s

0y(τ)dτ ds ≔ 1113946

1

0G3(t s)y(s)ds fory isin E t isin [0 1]

(23)

If we know the function β we can obtain the functionsG2 and G3

Example 1 Let β(t) 0 for t isin [0 1] enG1(t s) K1(t s) for t s isin [0 1] Let

h(s τ) 1 0le τ le sle 1

0 0le sle τ le 11113896 (24)

and then from (22) we find

G3(t s) 11139461

0G1(t τ)h(τ s)dτ for t s isin [0 1] (25)

We consider two cases

(i) Case 1 when 0le tle sle 1 we have

G3(t s) 1113946t

00 middot τdτ + 1113946

s

t0 middot tdτ + 1113946

1

s1 middot tdτ t(1 minus s)

(26)

(ii) Case 2 when 0le sle tle 1 we have

G3(t s) 1113946s

00 middot τdτ + 1113946

t

s1 middot τdτ + 1113946

1

t1 middot tdτ t minus

12t2

minus12s2

(27)

We now calculate G2 For this let 1113938τ0 y(r)dr j(τ) and

k(τ r) 0 0le τ le rle 1

1 0le rle τ le 11113896 en we have

11139461

0G1(t s) 1113946

s

0j(τ)dτ ds 1113946

1

0G3(t τ)j(τ)dτ

11139461

0G3(t τ) 1113946

τ

0y(r)dr dτ

11139461

0G3(t τ) 1113946

1

0k(τ r)y(r)dr dτ

(28)

erefore from (22) we have

G2(t s) 11139461

0G3(t τ)k(τ s)dτ for t s isin [0 1] (29)

We consider two cases

(i) Case 1 when 0le tle sle 1 we have

G2(t s) 1113946t

00 middot t minus

12t2

minus12τ21113874 1113875dτ

+ 1113946s

t0 middot t(1 minus τ)dτ + 1113946

1

s1 middot t(1 minus τ)dτ

t12

+12s2

minus s1113874 1113875

(30)

(ii) Case 2 when 0le sle tle 1 we have

G2(t s) 1113946s

00 middot t minus

12t2

minus12τ21113874 1113875dτ

+ 1113946t

s1 middot t minus

12t2

minus12τ21113874 1113875dτ + 1113946

1

t1 middot t(1 minus τ)dτ

12

t minus ts +12t2s minus

16t3

+16s3

(31)

Example 2 Let β(t) (t2) for t isin [0 1] en we have

11 minus α[1]

11139461

0K1(t s)dβ(t) 1113946

1

0K1(t s)dt 1113946

s

0t dt

+ 11139461

ssdt s minus

12s2 for s isin [0 1]

(32)

Hence G1(t s) s minus (12)s2 + K1(t s) for t s isin [0 1]Note (22) and Example 1 so we only need to calculate

11139461

0τ minus

12τ21113874 1113875h(τ s)dτ 1113946

1

sτ minus

12τ21113874 1113875dτ

13

minus12s2

+16s3

11139461

0

13

minus12τ2 +

16τ31113874 1113875k(τ s)dτ 1113946

1

s

13

minus12τ2 +

16τ31113874 1113875dτ

524

minus13

s +16s3

minus124

s4 for s isin [0 1]

(33)

erefore we obtain

4 Mathematical Problems in Engineering

G3(t s) 13

minus12s2

+16s3

+

t(1 minus s) 0le tle sle 1

t minus12t2

minus12s2 0le sle tle 1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

G2(t s) 524

minus13

s +16s3

minus124

s4

+

t12

+12s2

minus s1113874 1113875 0le tle sle 1

12

t minus ts +12t2s minus

16t3

+16s3 0le sle tle 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(34)

Lemma 2 (KreinndashRutman see([49] theorem 193)) Let Pbe a reproducing cone in a real Banach space E and letL E⟶ E be a compact linear operator with L(P) sub P If

r(L)gt 0 then there exists φ isin P 0 such that Lφ r(L)φwhere r(L) is the spectral radius of L

Lemma 3 For not all zero numbers a b c and dge 0 we let

Labcdy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds + d 1113946

1

0G1(t s)yprime(s)ds

Labcy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds

Llowastabcy1113872 1113873(s) a 1113946

1

0G2(t s)y(t)dt + b 1113946

1

0G3(t s)y(t)dt + c 1113946

1

0G1(t s)y(t)dt fory isin P

(35)

en130

a +18

b +13

c +13

d1113874 1113875κ1 le r Labcd1113872 1113873le18

a +13

b +12

c +12

d1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Labc1113872 1113873le18

a +13

b +12

c1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Llowastabc1113872 1113873le

18

a +13

b +12

c1113874 1113875κ2

(36)

where κ1 ≔ (α[t] minus α[1] + 1)(1 minus α[1]) and κ2 ≔ 1(1 minus

α[1])

Proof We first give some inequalities for Gi i 1 2 3 Notethat tsleK1(t s)le s for t s isin [0 1] and from the definitionof G1 we see that

κ1ts α[t] minus α[1] + 1

1 minus α[1]tsleG1(t s)le

11 minus α[1]

s

κ2s for t s isin [0 1]

(37)

With h and k as before note

12κ1t 1 minus s

21113872 1113873 1113946

1

sκ1tτdτ leG3(t s) 1113946

1

0G1(t τ)h(τ s)dτ le 1113946

1

sκ2τdτ

12κ2 1 minus s

21113872 1113873

for t s isin [0 1]

16κ1t 2 + s

3minus 3s1113872 1113873 1113946

1

s

12κ1t 1 minus τ21113872 1113873dτ leG2(t s) 1113946

1

0G3(t τ)k(τ s)dτ le 1113946

1

s

12κ2 1 minus τ21113872 1113873dτ

16κ2 2 + s

3minus 3s1113872 1113873 for t s isin [0 1]

(38)

Mathematical Problems in Engineering 5

For convenience let 1113957ψ0(t) t 1113957ψ1(t) (16)(2 + t3 minus

3t) and 1113957ψ2(t) (12)(1 minus t2) for t isin [0 1]We only prove the inequalities in (35) about the spectral

radius of Labc For convenience let Gabc(t s) aG2(t s) +

bG3(t s) + cG1(t s) for t s isin [0 1] en we have

Labcy1113872 1113873(t) 11139461

0Gabc(t s)y(s)ds fory isin P (39)

us we obtain

Labc

sup

||y||1Labcy

sup

tisin[01]y111139461

0Gabc(t s)y(s)ds

le maxtisin[01]

11139461

0Gabc(t s)dsle 1113946

1

0κ2 a1113957ψ1(s)1113858

+ b1113957ψ2(s) + c1113957ψ0(s)1113859ds

le18

a +13

b +12

c1113874 1113875κ2

(40)For all n isin N+ we note that

Lnabcy1113872 1113873(t) L L

nminus 1abcy1113872 1113873(t) 1113946

1

0Gabc t snminus 1( 1113857 L

nminus 1abcy1113872 1113873 snminus 1( 1113857dsnminus 1

middot middot middot

1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857y(s)dsnminus 1dsnminus 2 middot middot middot ds fory isin P

(41)

Hence we can obtain

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868le max

tisin[01]1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

le κn2 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)( 1113857ds1113890 1113891

n

le18

a +13

b +12

c1113874 1113875n

κn2

(42)

Gelfandrsquos theorem implies that

r Labc1113872 1113873 limn⟶infin

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

n

1113969

le18

a +13

b +12

c1113874 1113875κ2 (43)

Next we introduce a conclusion in ([50] problem 214)Let y isin C[0 1] and a functional J on C[0 1] be as

J(x) 11139461

0x(t)y(t)dt forallx isin C[0 1] (44)

en we have

J 11139461

0|y(t)|dt (45)

We note that there exists t0 isin [0 1] such thatmaxtisin[01] 1113938

10 Gabc(t s)ds 1113938

10 Gabc(t0 s)ds en in (38)

for fixed t we define a linear function

Labcty 11139461

0Gabc(t s)y(s)ds fory isin C[0 1] (46)

and thus

Labct

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 1113946

1

0Gabc(t s)ds (47)

en by the definition of the norm of linear function weknow that for all εgt 0 there exists yεt isin C[0 1] with ||yεt||

1 such that

11139461

0Gabc(t s)ds Labct

sup

y1Labcty|ge |Labctyεt

11138681113868111386811138681113868111386811138681113868

ge 11139461

0Gabc(t s)ds minus ε

(48)

On the contrary note from the definition of our normwehave

Labcy

suptisin[01]

Labcty|ge |Labct0y

11138681113868111386811138681113868

11138681113868111386811138681113868 (49)

Consequently we have

Labc

sup

y1Labcy

ge Labct0

yεt0

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868ge 11139461

0Gabc t0 s( 1113857ds minus ε

(50)

For the arbitrariness of ε we have

Labc

1113946

1

0Gabc t0 s( 1113857ds max

tisin[01]11139461

0Gabc(t s)ds (51)

Also for all n isin N+ we obtain

6 Mathematical Problems in Engineering

Lnabc

max

tisin[01]1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857 middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

ge κn1 max

tisin[01]t1113888 1113889 1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

a1113957ψ1 snminus 1( 1113857 + b1113957ψ2 snminus 1( 1113857 + c1113957ψ0 snminus 1( 11138571113858 1113859

middot snminus 1 a1113957ψ1 snminus 2( 1113857 + b1113957ψ2 snminus 2( 1113857 + c1113957ψ0 snminus 2( 11138571113858 1113859 middot middot middot s1 a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859dsnminus 1dsnminus 2 middot middot middot ds

κn1 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859ds 1113946

1

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds1113888 1113889

nminus 1

(52)

From Gelfandrsquos theorem we have

r Labc1113872 1113873 limn⟶infin

Lnabc

n

1113969

ge κ1 11139461

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds

ge130

a +18

b +13

c1113874 1113875κ1

(53)

is completes the proof

Lemma 4 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set and thatA Ω cap P⟶ P is a continuous compact operator If thereexists a ω0 isin P 0 such that

ω minus Aωne λω0 forallλge 0 ω isin zΩ cap P (54)

then i(AΩ cap P P) 0 where i denotes the fixed point indexon P

Lemma 5 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set with0 isin Ω and that A ΩcapP⟶ P is a continuous compactoperator If

ω minus λAωne 0 forallλ isin [0 1] ω isin zΩ capP (55)

then i(AΩcapP P) 1

3 Main Results

In our paper we let Bρ y isin P ylt ρ1113864 1113865 for ρgt 0 NowzBρ y isin P y ρ1113864 1113865 and Bρ y isin P yle ρ1113864 1113865 Now welist our assumptions on the nonlinearity f

(H2)f isin C [0 1] times R+

times R+

times R+

times R+R

+( 1113857 (56)

(H3) ere exist not all zero numbersa1 b1 c1 andd1 ge 0 and e1 gt 0 such that r(La1 b1 c1 d1

)lt 1and f(t z1 z2 z3 z4)le a1z1 + b1z2 + c1z3 + d1z4 + e1

for (t z1 z2 z3 z4) isin [0 1] times R+ times R + times R+ times R+(H4) ere exist not all zero numbers

a2 b2 c2 andd2 ge 0 and ρ1 gt 0 such that r(La2 b2 c2 d2)ge 1

and f(t z1 z2 z3 z4)ge a2z1 + b2z2 + c2z3 + d2z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ1] times [0 ρ1]times [0 ρ1] times [0 ρ1](H5) ere exist not all zero numbers

a3 b3 c3 andd3 ge 0 and ρ2 gt 0 such that r(La3 b3 c3 d3)lt 1

and f(t z1 z2 z3 z4)le a3z1 + b3z2 + c3z3 + d3z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ2] times [0 ρ 2] times [0 ρ2] times [0 ρ2](H6) ere exist not all zero numbers a4 b4 and c4 ge 0

and e2 gt 0 such that r(Llowasta4 b4 c4)gt 1 and f(t z1 z2

z3 z4)ge a4z1 + b4z2 + c4z3 minus e2 for (t z1 z2 z3 z4) isin[0 1] times R+ times R+ times R+ times R+

(H7) For any Mgt 0 there exists a positive continuousfunction HM(9) on R+ such that 1113938

+infin0 (9d9HM(9) + δ0)

+infin forallδ0 gt 0 and f(t z1 z2 z3 z4)leHM(z4) for (t z1 z2

z3 z4) isin [0 1] times [0 M]times [0 M] times [0 M] times R+

Remark 2 Considering Lemma 3 one can adjust the co-efficients ai bi ci anddi (i 1 2 3 4) such that the spectralradii r(Laibicidi

) r(La4 b4 c4) r(Llowasta4 b4 c4

)(i 1 2 3) satisfy

their respective conditions in (H3)ndash(H6)

Theorem 1 Suppose that (H0ndashH4) hold lten (1) has atleast one positive solution

Proof Let W y isin P y λAy λ isin [0 1]1113864 1113865 Now weprove thatW is a bounded set in P If y isinW then from (H3)we have

y(t) λ(Ay)(t)le (Ay)(t)le 11139461

0G1(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0a1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0G1(t s)yprime(s)ds + e1η(t)

La1 b1 c1 d1y1113872 1113873(t) + e1η(t)

(57)

Mathematical Problems in Engineering 7

where

η(t) 11139461

0

11 minus α[1]

11139461

0K1(t s)dβ(t) + K1(t s)1113890 1113891ds

1

1 minus α[1]11139461

011139461

0K1(t s)dβ(t)ds + t minus

12

t2

(58)

for t isin [0 1] is implies that

I minus La1 b1 c1d11113872 1113873y1113872 1113873(t)le e1η(t) (59)

Note that η isin P(ηprime(t) 1 minus tge 0 forallt isin [0 1]) y isinWand we obtain

yprime(t) λ(Ay)prime(t) λ11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0

z

zta1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0

z

ztG1(t s)yprime(s)ds + e1ηprime(t)

La1 b1 c1 d1y1113872 1113873prime(t) + e1ηprime(t)

(60)

us

I minus La1 b1 c1 d11113872 1113873y1113872 1113873prime(t)le e1ηprime(t) (61)

Since r(La1 b1 c1 d1)lt 1 we know that I minus La1 b1 c1 d1

has abounded inverse operator (I minus La1 b1 c1 d1

)minus 1 with

I minus La1 b1 c1 d11113872 1113873

minus 1 I + La1 b1 c1 d1

+ L2a1 b1 c1 d1

+ middot middot middot

+ Lna1 b1 c1 d1

+ middot middot middot (62)

Note that La1 b1 c1 d1(P) sub P and we obtain (Iminus

La1 b1 c1 d1)minus 1(P) sub P erefore

y(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875(t)

yprime(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875prime(t)

for t isin [0 1]

(63)

is implies that

yC le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

C

yprime

Cle I minus La1 b1 c1 d1

1113872 1113873minus 1

e1η1113874 1113875prime

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868C

(64)

erefore we have

||y||le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

(65)

at is W is bounded Now we can selectR1 gtmax supW ρ11113864 1113865 (ρ1 is defined in (H4)) such that

yne λAy fory isin zBR1capP and λ isin [0 1] (66)

From Lemma 5 we have

i A BR1capP P1113872 1113873 1 (67)

On the contrary since La2 b2 c2 d2(P) sub P and

r(La2 b2 c2 d2)ge 1 it follows from Lemma 2 that there exists

φ0 isin P | 0 such that La2b2 c2 d2φ0 r(La2 b2 c2 d2

)φ0 andφ0 r(La2 b2 c2 d2

)minus 1La2 b2 c2 d2φ0 isin P Now we show that

y minus Ayne λφ0 fory isin zBρ1 capP λge 0 (68)

If this claim is false then there exist y0 isin zBρ1 capP andλ0 ge 0 such that y0 minus Ay0 λ0φ0 Note that λ0 gt 0 (other-wise the theorem is proved) en from (H4) we have

Ay0( 1113857(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y0(r)drdτ 1113946

s

0y0(τ)dτ y0(s) y0prime(s)1113874 1113875ds

ge 11139461

0G1(t s) a2 1113946

s

01113946τ

0y0(r)drdτ + b2 1113946

s

0y0(τ)dτ + c2y0(s) + d2y0prime(s)1113874 1113875ds

11139461

0a2G2(t s) + b2G3(t s) + c2G1(t s)( 1113857y0(s)ds + d2 1113946

1

0G1(t s)y0prime(s)ds

La2 b2 c2d2y01113872 1113873(t)

(69)

8 Mathematical Problems in Engineering

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

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MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

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OptimizationJournal of

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Engineering Mathematics

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Operations ResearchAdvances in

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

G3(t s) 13

minus12s2

+16s3

+

t(1 minus s) 0le tle sle 1

t minus12t2

minus12s2 0le sle tle 1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

G2(t s) 524

minus13

s +16s3

minus124

s4

+

t12

+12s2

minus s1113874 1113875 0le tle sle 1

12

t minus ts +12t2s minus

16t3

+16s3 0le sle tle 1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(34)

Lemma 2 (KreinndashRutman see([49] theorem 193)) Let Pbe a reproducing cone in a real Banach space E and letL E⟶ E be a compact linear operator with L(P) sub P If

r(L)gt 0 then there exists φ isin P 0 such that Lφ r(L)φwhere r(L) is the spectral radius of L

Lemma 3 For not all zero numbers a b c and dge 0 we let

Labcdy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds + d 1113946

1

0G1(t s)yprime(s)ds

Labcy1113872 1113873(t) a 11139461

0G2(t s)y(s)ds + b 1113946

1

0G3(t s)y(s)ds + c 1113946

1

0G1(t s)y(s)ds

Llowastabcy1113872 1113873(s) a 1113946

1

0G2(t s)y(t)dt + b 1113946

1

0G3(t s)y(t)dt + c 1113946

1

0G1(t s)y(t)dt fory isin P

(35)

en130

a +18

b +13

c +13

d1113874 1113875κ1 le r Labcd1113872 1113873le18

a +13

b +12

c +12

d1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Labc1113872 1113873le18

a +13

b +12

c1113874 1113875κ2

130

a +18

b +13

c1113874 1113875κ1 le r Llowastabc1113872 1113873le

18

a +13

b +12

c1113874 1113875κ2

(36)

where κ1 ≔ (α[t] minus α[1] + 1)(1 minus α[1]) and κ2 ≔ 1(1 minus

α[1])

Proof We first give some inequalities for Gi i 1 2 3 Notethat tsleK1(t s)le s for t s isin [0 1] and from the definitionof G1 we see that

κ1ts α[t] minus α[1] + 1

1 minus α[1]tsleG1(t s)le

11 minus α[1]

s

κ2s for t s isin [0 1]

(37)

With h and k as before note

12κ1t 1 minus s

21113872 1113873 1113946

1

sκ1tτdτ leG3(t s) 1113946

1

0G1(t τ)h(τ s)dτ le 1113946

1

sκ2τdτ

12κ2 1 minus s

21113872 1113873

for t s isin [0 1]

16κ1t 2 + s

3minus 3s1113872 1113873 1113946

1

s

12κ1t 1 minus τ21113872 1113873dτ leG2(t s) 1113946

1

0G3(t τ)k(τ s)dτ le 1113946

1

s

12κ2 1 minus τ21113872 1113873dτ

16κ2 2 + s

3minus 3s1113872 1113873 for t s isin [0 1]

(38)

Mathematical Problems in Engineering 5

For convenience let 1113957ψ0(t) t 1113957ψ1(t) (16)(2 + t3 minus

3t) and 1113957ψ2(t) (12)(1 minus t2) for t isin [0 1]We only prove the inequalities in (35) about the spectral

radius of Labc For convenience let Gabc(t s) aG2(t s) +

bG3(t s) + cG1(t s) for t s isin [0 1] en we have

Labcy1113872 1113873(t) 11139461

0Gabc(t s)y(s)ds fory isin P (39)

us we obtain

Labc

sup

||y||1Labcy

sup

tisin[01]y111139461

0Gabc(t s)y(s)ds

le maxtisin[01]

11139461

0Gabc(t s)dsle 1113946

1

0κ2 a1113957ψ1(s)1113858

+ b1113957ψ2(s) + c1113957ψ0(s)1113859ds

le18

a +13

b +12

c1113874 1113875κ2

(40)For all n isin N+ we note that

Lnabcy1113872 1113873(t) L L

nminus 1abcy1113872 1113873(t) 1113946

1

0Gabc t snminus 1( 1113857 L

nminus 1abcy1113872 1113873 snminus 1( 1113857dsnminus 1

middot middot middot

1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857y(s)dsnminus 1dsnminus 2 middot middot middot ds fory isin P

(41)

Hence we can obtain

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868le max

tisin[01]1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

le κn2 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)( 1113857ds1113890 1113891

n

le18

a +13

b +12

c1113874 1113875n

κn2

(42)

Gelfandrsquos theorem implies that

r Labc1113872 1113873 limn⟶infin

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

n

1113969

le18

a +13

b +12

c1113874 1113875κ2 (43)

Next we introduce a conclusion in ([50] problem 214)Let y isin C[0 1] and a functional J on C[0 1] be as

J(x) 11139461

0x(t)y(t)dt forallx isin C[0 1] (44)

en we have

J 11139461

0|y(t)|dt (45)

We note that there exists t0 isin [0 1] such thatmaxtisin[01] 1113938

10 Gabc(t s)ds 1113938

10 Gabc(t0 s)ds en in (38)

for fixed t we define a linear function

Labcty 11139461

0Gabc(t s)y(s)ds fory isin C[0 1] (46)

and thus

Labct

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 1113946

1

0Gabc(t s)ds (47)

en by the definition of the norm of linear function weknow that for all εgt 0 there exists yεt isin C[0 1] with ||yεt||

1 such that

11139461

0Gabc(t s)ds Labct

sup

y1Labcty|ge |Labctyεt

11138681113868111386811138681113868111386811138681113868

ge 11139461

0Gabc(t s)ds minus ε

(48)

On the contrary note from the definition of our normwehave

Labcy

suptisin[01]

Labcty|ge |Labct0y

11138681113868111386811138681113868

11138681113868111386811138681113868 (49)

Consequently we have

Labc

sup

y1Labcy

ge Labct0

yεt0

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868ge 11139461

0Gabc t0 s( 1113857ds minus ε

(50)

For the arbitrariness of ε we have

Labc

1113946

1

0Gabc t0 s( 1113857ds max

tisin[01]11139461

0Gabc(t s)ds (51)

Also for all n isin N+ we obtain

6 Mathematical Problems in Engineering

Lnabc

max

tisin[01]1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857 middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

ge κn1 max

tisin[01]t1113888 1113889 1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

a1113957ψ1 snminus 1( 1113857 + b1113957ψ2 snminus 1( 1113857 + c1113957ψ0 snminus 1( 11138571113858 1113859

middot snminus 1 a1113957ψ1 snminus 2( 1113857 + b1113957ψ2 snminus 2( 1113857 + c1113957ψ0 snminus 2( 11138571113858 1113859 middot middot middot s1 a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859dsnminus 1dsnminus 2 middot middot middot ds

κn1 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859ds 1113946

1

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds1113888 1113889

nminus 1

(52)

From Gelfandrsquos theorem we have

r Labc1113872 1113873 limn⟶infin

Lnabc

n

1113969

ge κ1 11139461

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds

ge130

a +18

b +13

c1113874 1113875κ1

(53)

is completes the proof

Lemma 4 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set and thatA Ω cap P⟶ P is a continuous compact operator If thereexists a ω0 isin P 0 such that

ω minus Aωne λω0 forallλge 0 ω isin zΩ cap P (54)

then i(AΩ cap P P) 0 where i denotes the fixed point indexon P

Lemma 5 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set with0 isin Ω and that A ΩcapP⟶ P is a continuous compactoperator If

ω minus λAωne 0 forallλ isin [0 1] ω isin zΩ capP (55)

then i(AΩcapP P) 1

3 Main Results

In our paper we let Bρ y isin P ylt ρ1113864 1113865 for ρgt 0 NowzBρ y isin P y ρ1113864 1113865 and Bρ y isin P yle ρ1113864 1113865 Now welist our assumptions on the nonlinearity f

(H2)f isin C [0 1] times R+

times R+

times R+

times R+R

+( 1113857 (56)

(H3) ere exist not all zero numbersa1 b1 c1 andd1 ge 0 and e1 gt 0 such that r(La1 b1 c1 d1

)lt 1and f(t z1 z2 z3 z4)le a1z1 + b1z2 + c1z3 + d1z4 + e1

for (t z1 z2 z3 z4) isin [0 1] times R+ times R + times R+ times R+(H4) ere exist not all zero numbers

a2 b2 c2 andd2 ge 0 and ρ1 gt 0 such that r(La2 b2 c2 d2)ge 1

and f(t z1 z2 z3 z4)ge a2z1 + b2z2 + c2z3 + d2z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ1] times [0 ρ1]times [0 ρ1] times [0 ρ1](H5) ere exist not all zero numbers

a3 b3 c3 andd3 ge 0 and ρ2 gt 0 such that r(La3 b3 c3 d3)lt 1

and f(t z1 z2 z3 z4)le a3z1 + b3z2 + c3z3 + d3z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ2] times [0 ρ 2] times [0 ρ2] times [0 ρ2](H6) ere exist not all zero numbers a4 b4 and c4 ge 0

and e2 gt 0 such that r(Llowasta4 b4 c4)gt 1 and f(t z1 z2

z3 z4)ge a4z1 + b4z2 + c4z3 minus e2 for (t z1 z2 z3 z4) isin[0 1] times R+ times R+ times R+ times R+

(H7) For any Mgt 0 there exists a positive continuousfunction HM(9) on R+ such that 1113938

+infin0 (9d9HM(9) + δ0)

+infin forallδ0 gt 0 and f(t z1 z2 z3 z4)leHM(z4) for (t z1 z2

z3 z4) isin [0 1] times [0 M]times [0 M] times [0 M] times R+

Remark 2 Considering Lemma 3 one can adjust the co-efficients ai bi ci anddi (i 1 2 3 4) such that the spectralradii r(Laibicidi

) r(La4 b4 c4) r(Llowasta4 b4 c4

)(i 1 2 3) satisfy

their respective conditions in (H3)ndash(H6)

Theorem 1 Suppose that (H0ndashH4) hold lten (1) has atleast one positive solution

Proof Let W y isin P y λAy λ isin [0 1]1113864 1113865 Now weprove thatW is a bounded set in P If y isinW then from (H3)we have

y(t) λ(Ay)(t)le (Ay)(t)le 11139461

0G1(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0a1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0G1(t s)yprime(s)ds + e1η(t)

La1 b1 c1 d1y1113872 1113873(t) + e1η(t)

(57)

Mathematical Problems in Engineering 7

where

η(t) 11139461

0

11 minus α[1]

11139461

0K1(t s)dβ(t) + K1(t s)1113890 1113891ds

1

1 minus α[1]11139461

011139461

0K1(t s)dβ(t)ds + t minus

12

t2

(58)

for t isin [0 1] is implies that

I minus La1 b1 c1d11113872 1113873y1113872 1113873(t)le e1η(t) (59)

Note that η isin P(ηprime(t) 1 minus tge 0 forallt isin [0 1]) y isinWand we obtain

yprime(t) λ(Ay)prime(t) λ11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0

z

zta1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0

z

ztG1(t s)yprime(s)ds + e1ηprime(t)

La1 b1 c1 d1y1113872 1113873prime(t) + e1ηprime(t)

(60)

us

I minus La1 b1 c1 d11113872 1113873y1113872 1113873prime(t)le e1ηprime(t) (61)

Since r(La1 b1 c1 d1)lt 1 we know that I minus La1 b1 c1 d1

has abounded inverse operator (I minus La1 b1 c1 d1

)minus 1 with

I minus La1 b1 c1 d11113872 1113873

minus 1 I + La1 b1 c1 d1

+ L2a1 b1 c1 d1

+ middot middot middot

+ Lna1 b1 c1 d1

+ middot middot middot (62)

Note that La1 b1 c1 d1(P) sub P and we obtain (Iminus

La1 b1 c1 d1)minus 1(P) sub P erefore

y(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875(t)

yprime(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875prime(t)

for t isin [0 1]

(63)

is implies that

yC le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

C

yprime

Cle I minus La1 b1 c1 d1

1113872 1113873minus 1

e1η1113874 1113875prime

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868C

(64)

erefore we have

||y||le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

(65)

at is W is bounded Now we can selectR1 gtmax supW ρ11113864 1113865 (ρ1 is defined in (H4)) such that

yne λAy fory isin zBR1capP and λ isin [0 1] (66)

From Lemma 5 we have

i A BR1capP P1113872 1113873 1 (67)

On the contrary since La2 b2 c2 d2(P) sub P and

r(La2 b2 c2 d2)ge 1 it follows from Lemma 2 that there exists

φ0 isin P | 0 such that La2b2 c2 d2φ0 r(La2 b2 c2 d2

)φ0 andφ0 r(La2 b2 c2 d2

)minus 1La2 b2 c2 d2φ0 isin P Now we show that

y minus Ayne λφ0 fory isin zBρ1 capP λge 0 (68)

If this claim is false then there exist y0 isin zBρ1 capP andλ0 ge 0 such that y0 minus Ay0 λ0φ0 Note that λ0 gt 0 (other-wise the theorem is proved) en from (H4) we have

Ay0( 1113857(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y0(r)drdτ 1113946

s

0y0(τ)dτ y0(s) y0prime(s)1113874 1113875ds

ge 11139461

0G1(t s) a2 1113946

s

01113946τ

0y0(r)drdτ + b2 1113946

s

0y0(τ)dτ + c2y0(s) + d2y0prime(s)1113874 1113875ds

11139461

0a2G2(t s) + b2G3(t s) + c2G1(t s)( 1113857y0(s)ds + d2 1113946

1

0G1(t s)y0prime(s)ds

La2 b2 c2d2y01113872 1113873(t)

(69)

8 Mathematical Problems in Engineering

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

For convenience let 1113957ψ0(t) t 1113957ψ1(t) (16)(2 + t3 minus

3t) and 1113957ψ2(t) (12)(1 minus t2) for t isin [0 1]We only prove the inequalities in (35) about the spectral

radius of Labc For convenience let Gabc(t s) aG2(t s) +

bG3(t s) + cG1(t s) for t s isin [0 1] en we have

Labcy1113872 1113873(t) 11139461

0Gabc(t s)y(s)ds fory isin P (39)

us we obtain

Labc

sup

||y||1Labcy

sup

tisin[01]y111139461

0Gabc(t s)y(s)ds

le maxtisin[01]

11139461

0Gabc(t s)dsle 1113946

1

0κ2 a1113957ψ1(s)1113858

+ b1113957ψ2(s) + c1113957ψ0(s)1113859ds

le18

a +13

b +12

c1113874 1113875κ2

(40)For all n isin N+ we note that

Lnabcy1113872 1113873(t) L L

nminus 1abcy1113872 1113873(t) 1113946

1

0Gabc t snminus 1( 1113857 L

nminus 1abcy1113872 1113873 snminus 1( 1113857dsnminus 1

middot middot middot

1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857y(s)dsnminus 1dsnminus 2 middot middot middot ds fory isin P

(41)

Hence we can obtain

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868le max

tisin[01]1113946 1

01113946 1

0middot middot middot 1113946

1

01113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857

middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

le κn2 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)( 1113857ds1113890 1113891

n

le18

a +13

b +12

c1113874 1113875n

κn2

(42)

Gelfandrsquos theorem implies that

r Labc1113872 1113873 limn⟶infin

Lnabc

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

n

1113969

le18

a +13

b +12

c1113874 1113875κ2 (43)

Next we introduce a conclusion in ([50] problem 214)Let y isin C[0 1] and a functional J on C[0 1] be as

J(x) 11139461

0x(t)y(t)dt forallx isin C[0 1] (44)

en we have

J 11139461

0|y(t)|dt (45)

We note that there exists t0 isin [0 1] such thatmaxtisin[01] 1113938

10 Gabc(t s)ds 1113938

10 Gabc(t0 s)ds en in (38)

for fixed t we define a linear function

Labcty 11139461

0Gabc(t s)y(s)ds fory isin C[0 1] (46)

and thus

Labct

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 1113946

1

0Gabc(t s)ds (47)

en by the definition of the norm of linear function weknow that for all εgt 0 there exists yεt isin C[0 1] with ||yεt||

1 such that

11139461

0Gabc(t s)ds Labct

sup

y1Labcty|ge |Labctyεt

11138681113868111386811138681113868111386811138681113868

ge 11139461

0Gabc(t s)ds minus ε

(48)

On the contrary note from the definition of our normwehave

Labcy

suptisin[01]

Labcty|ge |Labct0y

11138681113868111386811138681113868

11138681113868111386811138681113868 (49)

Consequently we have

Labc

sup

y1Labcy

ge Labct0

yεt0

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868

11138681113868111386811138681113868ge 11139461

0Gabc t0 s( 1113857ds minus ε

(50)

For the arbitrariness of ε we have

Labc

1113946

1

0Gabc t0 s( 1113857ds max

tisin[01]11139461

0Gabc(t s)ds (51)

Also for all n isin N+ we obtain

6 Mathematical Problems in Engineering

Lnabc

max

tisin[01]1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857 middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

ge κn1 max

tisin[01]t1113888 1113889 1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

a1113957ψ1 snminus 1( 1113857 + b1113957ψ2 snminus 1( 1113857 + c1113957ψ0 snminus 1( 11138571113858 1113859

middot snminus 1 a1113957ψ1 snminus 2( 1113857 + b1113957ψ2 snminus 2( 1113857 + c1113957ψ0 snminus 2( 11138571113858 1113859 middot middot middot s1 a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859dsnminus 1dsnminus 2 middot middot middot ds

κn1 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859ds 1113946

1

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds1113888 1113889

nminus 1

(52)

From Gelfandrsquos theorem we have

r Labc1113872 1113873 limn⟶infin

Lnabc

n

1113969

ge κ1 11139461

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds

ge130

a +18

b +13

c1113874 1113875κ1

(53)

is completes the proof

Lemma 4 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set and thatA Ω cap P⟶ P is a continuous compact operator If thereexists a ω0 isin P 0 such that

ω minus Aωne λω0 forallλge 0 ω isin zΩ cap P (54)

then i(AΩ cap P P) 0 where i denotes the fixed point indexon P

Lemma 5 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set with0 isin Ω and that A ΩcapP⟶ P is a continuous compactoperator If

ω minus λAωne 0 forallλ isin [0 1] ω isin zΩ capP (55)

then i(AΩcapP P) 1

3 Main Results

In our paper we let Bρ y isin P ylt ρ1113864 1113865 for ρgt 0 NowzBρ y isin P y ρ1113864 1113865 and Bρ y isin P yle ρ1113864 1113865 Now welist our assumptions on the nonlinearity f

(H2)f isin C [0 1] times R+

times R+

times R+

times R+R

+( 1113857 (56)

(H3) ere exist not all zero numbersa1 b1 c1 andd1 ge 0 and e1 gt 0 such that r(La1 b1 c1 d1

)lt 1and f(t z1 z2 z3 z4)le a1z1 + b1z2 + c1z3 + d1z4 + e1

for (t z1 z2 z3 z4) isin [0 1] times R+ times R + times R+ times R+(H4) ere exist not all zero numbers

a2 b2 c2 andd2 ge 0 and ρ1 gt 0 such that r(La2 b2 c2 d2)ge 1

and f(t z1 z2 z3 z4)ge a2z1 + b2z2 + c2z3 + d2z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ1] times [0 ρ1]times [0 ρ1] times [0 ρ1](H5) ere exist not all zero numbers

a3 b3 c3 andd3 ge 0 and ρ2 gt 0 such that r(La3 b3 c3 d3)lt 1

and f(t z1 z2 z3 z4)le a3z1 + b3z2 + c3z3 + d3z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ2] times [0 ρ 2] times [0 ρ2] times [0 ρ2](H6) ere exist not all zero numbers a4 b4 and c4 ge 0

and e2 gt 0 such that r(Llowasta4 b4 c4)gt 1 and f(t z1 z2

z3 z4)ge a4z1 + b4z2 + c4z3 minus e2 for (t z1 z2 z3 z4) isin[0 1] times R+ times R+ times R+ times R+

(H7) For any Mgt 0 there exists a positive continuousfunction HM(9) on R+ such that 1113938

+infin0 (9d9HM(9) + δ0)

+infin forallδ0 gt 0 and f(t z1 z2 z3 z4)leHM(z4) for (t z1 z2

z3 z4) isin [0 1] times [0 M]times [0 M] times [0 M] times R+

Remark 2 Considering Lemma 3 one can adjust the co-efficients ai bi ci anddi (i 1 2 3 4) such that the spectralradii r(Laibicidi

) r(La4 b4 c4) r(Llowasta4 b4 c4

)(i 1 2 3) satisfy

their respective conditions in (H3)ndash(H6)

Theorem 1 Suppose that (H0ndashH4) hold lten (1) has atleast one positive solution

Proof Let W y isin P y λAy λ isin [0 1]1113864 1113865 Now weprove thatW is a bounded set in P If y isinW then from (H3)we have

y(t) λ(Ay)(t)le (Ay)(t)le 11139461

0G1(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0a1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0G1(t s)yprime(s)ds + e1η(t)

La1 b1 c1 d1y1113872 1113873(t) + e1η(t)

(57)

Mathematical Problems in Engineering 7

where

η(t) 11139461

0

11 minus α[1]

11139461

0K1(t s)dβ(t) + K1(t s)1113890 1113891ds

1

1 minus α[1]11139461

011139461

0K1(t s)dβ(t)ds + t minus

12

t2

(58)

for t isin [0 1] is implies that

I minus La1 b1 c1d11113872 1113873y1113872 1113873(t)le e1η(t) (59)

Note that η isin P(ηprime(t) 1 minus tge 0 forallt isin [0 1]) y isinWand we obtain

yprime(t) λ(Ay)prime(t) λ11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0

z

zta1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0

z

ztG1(t s)yprime(s)ds + e1ηprime(t)

La1 b1 c1 d1y1113872 1113873prime(t) + e1ηprime(t)

(60)

us

I minus La1 b1 c1 d11113872 1113873y1113872 1113873prime(t)le e1ηprime(t) (61)

Since r(La1 b1 c1 d1)lt 1 we know that I minus La1 b1 c1 d1

has abounded inverse operator (I minus La1 b1 c1 d1

)minus 1 with

I minus La1 b1 c1 d11113872 1113873

minus 1 I + La1 b1 c1 d1

+ L2a1 b1 c1 d1

+ middot middot middot

+ Lna1 b1 c1 d1

+ middot middot middot (62)

Note that La1 b1 c1 d1(P) sub P and we obtain (Iminus

La1 b1 c1 d1)minus 1(P) sub P erefore

y(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875(t)

yprime(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875prime(t)

for t isin [0 1]

(63)

is implies that

yC le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

C

yprime

Cle I minus La1 b1 c1 d1

1113872 1113873minus 1

e1η1113874 1113875prime

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868C

(64)

erefore we have

||y||le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

(65)

at is W is bounded Now we can selectR1 gtmax supW ρ11113864 1113865 (ρ1 is defined in (H4)) such that

yne λAy fory isin zBR1capP and λ isin [0 1] (66)

From Lemma 5 we have

i A BR1capP P1113872 1113873 1 (67)

On the contrary since La2 b2 c2 d2(P) sub P and

r(La2 b2 c2 d2)ge 1 it follows from Lemma 2 that there exists

φ0 isin P | 0 such that La2b2 c2 d2φ0 r(La2 b2 c2 d2

)φ0 andφ0 r(La2 b2 c2 d2

)minus 1La2 b2 c2 d2φ0 isin P Now we show that

y minus Ayne λφ0 fory isin zBρ1 capP λge 0 (68)

If this claim is false then there exist y0 isin zBρ1 capP andλ0 ge 0 such that y0 minus Ay0 λ0φ0 Note that λ0 gt 0 (other-wise the theorem is proved) en from (H4) we have

Ay0( 1113857(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y0(r)drdτ 1113946

s

0y0(τ)dτ y0(s) y0prime(s)1113874 1113875ds

ge 11139461

0G1(t s) a2 1113946

s

01113946τ

0y0(r)drdτ + b2 1113946

s

0y0(τ)dτ + c2y0(s) + d2y0prime(s)1113874 1113875ds

11139461

0a2G2(t s) + b2G3(t s) + c2G1(t s)( 1113857y0(s)ds + d2 1113946

1

0G1(t s)y0prime(s)ds

La2 b2 c2d2y01113872 1113873(t)

(69)

8 Mathematical Problems in Engineering

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

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Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Lnabc

max

tisin[01]1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

Gabc t snminus 1( 1113857Gabc snminus 1 snminus 2( 1113857 middot middot middot Gabc s1 s( 1113857dsnminus 1dsnminus 2 middot middot middot ds

ge κn1 max

tisin[01]t1113888 1113889 1113946

1

0

1113946

1

0

middot middot middot 1113946

1

01113980radicradicradic11139791113978radicradicradic1113981n

a1113957ψ1 snminus 1( 1113857 + b1113957ψ2 snminus 1( 1113857 + c1113957ψ0 snminus 1( 11138571113858 1113859

middot snminus 1 a1113957ψ1 snminus 2( 1113857 + b1113957ψ2 snminus 2( 1113857 + c1113957ψ0 snminus 2( 11138571113858 1113859 middot middot middot s1 a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859dsnminus 1dsnminus 2 middot middot middot ds

κn1 1113946

1

0a1113957ψ1(s) + b1113957ψ2(s) + c1113957ψ0(s)1113858 1113859ds 1113946

1

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds1113888 1113889

nminus 1

(52)

From Gelfandrsquos theorem we have

r Labc1113872 1113873 limn⟶infin

Lnabc

n

1113969

ge κ1 11139461

0as1113957ψ1(s) + bs1113957ψ2(s) + cs1113957ψ0(s)1113858 1113859ds

ge130

a +18

b +13

c1113874 1113875κ1

(53)

is completes the proof

Lemma 4 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set and thatA Ω cap P⟶ P is a continuous compact operator If thereexists a ω0 isin P 0 such that

ω minus Aωne λω0 forallλge 0 ω isin zΩ cap P (54)

then i(AΩ cap P P) 0 where i denotes the fixed point indexon P

Lemma 5 (see [51]) Let E be a real Banach space and P be acone on E Suppose that Ω sub E is a bounded open set with0 isin Ω and that A ΩcapP⟶ P is a continuous compactoperator If

ω minus λAωne 0 forallλ isin [0 1] ω isin zΩ capP (55)

then i(AΩcapP P) 1

3 Main Results

In our paper we let Bρ y isin P ylt ρ1113864 1113865 for ρgt 0 NowzBρ y isin P y ρ1113864 1113865 and Bρ y isin P yle ρ1113864 1113865 Now welist our assumptions on the nonlinearity f

(H2)f isin C [0 1] times R+

times R+

times R+

times R+R

+( 1113857 (56)

(H3) ere exist not all zero numbersa1 b1 c1 andd1 ge 0 and e1 gt 0 such that r(La1 b1 c1 d1

)lt 1and f(t z1 z2 z3 z4)le a1z1 + b1z2 + c1z3 + d1z4 + e1

for (t z1 z2 z3 z4) isin [0 1] times R+ times R + times R+ times R+(H4) ere exist not all zero numbers

a2 b2 c2 andd2 ge 0 and ρ1 gt 0 such that r(La2 b2 c2 d2)ge 1

and f(t z1 z2 z3 z4)ge a2z1 + b2z2 + c2z3 + d2z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ1] times [0 ρ1]times [0 ρ1] times [0 ρ1](H5) ere exist not all zero numbers

a3 b3 c3 andd3 ge 0 and ρ2 gt 0 such that r(La3 b3 c3 d3)lt 1

and f(t z1 z2 z3 z4)le a3z1 + b3z2 + c3z3 + d3z4 for (t

z1 z2 z3 z4) isin [0 1] times [0 ρ2] times [0 ρ 2] times [0 ρ2] times [0 ρ2](H6) ere exist not all zero numbers a4 b4 and c4 ge 0

and e2 gt 0 such that r(Llowasta4 b4 c4)gt 1 and f(t z1 z2

z3 z4)ge a4z1 + b4z2 + c4z3 minus e2 for (t z1 z2 z3 z4) isin[0 1] times R+ times R+ times R+ times R+

(H7) For any Mgt 0 there exists a positive continuousfunction HM(9) on R+ such that 1113938

+infin0 (9d9HM(9) + δ0)

+infin forallδ0 gt 0 and f(t z1 z2 z3 z4)leHM(z4) for (t z1 z2

z3 z4) isin [0 1] times [0 M]times [0 M] times [0 M] times R+

Remark 2 Considering Lemma 3 one can adjust the co-efficients ai bi ci anddi (i 1 2 3 4) such that the spectralradii r(Laibicidi

) r(La4 b4 c4) r(Llowasta4 b4 c4

)(i 1 2 3) satisfy

their respective conditions in (H3)ndash(H6)

Theorem 1 Suppose that (H0ndashH4) hold lten (1) has atleast one positive solution

Proof Let W y isin P y λAy λ isin [0 1]1113864 1113865 Now weprove thatW is a bounded set in P If y isinW then from (H3)we have

y(t) λ(Ay)(t)le (Ay)(t)le 11139461

0G1(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0a1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0G1(t s)yprime(s)ds + e1η(t)

La1 b1 c1 d1y1113872 1113873(t) + e1η(t)

(57)

Mathematical Problems in Engineering 7

where

η(t) 11139461

0

11 minus α[1]

11139461

0K1(t s)dβ(t) + K1(t s)1113890 1113891ds

1

1 minus α[1]11139461

011139461

0K1(t s)dβ(t)ds + t minus

12

t2

(58)

for t isin [0 1] is implies that

I minus La1 b1 c1d11113872 1113873y1113872 1113873(t)le e1η(t) (59)

Note that η isin P(ηprime(t) 1 minus tge 0 forallt isin [0 1]) y isinWand we obtain

yprime(t) λ(Ay)prime(t) λ11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0

z

zta1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0

z

ztG1(t s)yprime(s)ds + e1ηprime(t)

La1 b1 c1 d1y1113872 1113873prime(t) + e1ηprime(t)

(60)

us

I minus La1 b1 c1 d11113872 1113873y1113872 1113873prime(t)le e1ηprime(t) (61)

Since r(La1 b1 c1 d1)lt 1 we know that I minus La1 b1 c1 d1

has abounded inverse operator (I minus La1 b1 c1 d1

)minus 1 with

I minus La1 b1 c1 d11113872 1113873

minus 1 I + La1 b1 c1 d1

+ L2a1 b1 c1 d1

+ middot middot middot

+ Lna1 b1 c1 d1

+ middot middot middot (62)

Note that La1 b1 c1 d1(P) sub P and we obtain (Iminus

La1 b1 c1 d1)minus 1(P) sub P erefore

y(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875(t)

yprime(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875prime(t)

for t isin [0 1]

(63)

is implies that

yC le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

C

yprime

Cle I minus La1 b1 c1 d1

1113872 1113873minus 1

e1η1113874 1113875prime

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868C

(64)

erefore we have

||y||le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

(65)

at is W is bounded Now we can selectR1 gtmax supW ρ11113864 1113865 (ρ1 is defined in (H4)) such that

yne λAy fory isin zBR1capP and λ isin [0 1] (66)

From Lemma 5 we have

i A BR1capP P1113872 1113873 1 (67)

On the contrary since La2 b2 c2 d2(P) sub P and

r(La2 b2 c2 d2)ge 1 it follows from Lemma 2 that there exists

φ0 isin P | 0 such that La2b2 c2 d2φ0 r(La2 b2 c2 d2

)φ0 andφ0 r(La2 b2 c2 d2

)minus 1La2 b2 c2 d2φ0 isin P Now we show that

y minus Ayne λφ0 fory isin zBρ1 capP λge 0 (68)

If this claim is false then there exist y0 isin zBρ1 capP andλ0 ge 0 such that y0 minus Ay0 λ0φ0 Note that λ0 gt 0 (other-wise the theorem is proved) en from (H4) we have

Ay0( 1113857(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y0(r)drdτ 1113946

s

0y0(τ)dτ y0(s) y0prime(s)1113874 1113875ds

ge 11139461

0G1(t s) a2 1113946

s

01113946τ

0y0(r)drdτ + b2 1113946

s

0y0(τ)dτ + c2y0(s) + d2y0prime(s)1113874 1113875ds

11139461

0a2G2(t s) + b2G3(t s) + c2G1(t s)( 1113857y0(s)ds + d2 1113946

1

0G1(t s)y0prime(s)ds

La2 b2 c2d2y01113872 1113873(t)

(69)

8 Mathematical Problems in Engineering

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

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Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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AnalysisInternational Journal of

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Submit your manuscripts atwwwhindawicom

where

η(t) 11139461

0

11 minus α[1]

11139461

0K1(t s)dβ(t) + K1(t s)1113890 1113891ds

1

1 minus α[1]11139461

011139461

0K1(t s)dβ(t)ds + t minus

12

t2

(58)

for t isin [0 1] is implies that

I minus La1 b1 c1d11113872 1113873y1113872 1113873(t)le e1η(t) (59)

Note that η isin P(ηprime(t) 1 minus tge 0 forallt isin [0 1]) y isinWand we obtain

yprime(t) λ(Ay)prime(t) λ11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a1 1113946

s

01113946τ

0y(r)drdτ + b1 1113946

s

0y(τ)dτ + c1y(s) + d1yprime(s) + e11113874 1113875ds

11139461

0

z

zta1G2(t s) + b1G3(t s) + c1G1(t s)( 1113857y(s)ds + d1 1113946

1

0

z

ztG1(t s)yprime(s)ds + e1ηprime(t)

La1 b1 c1 d1y1113872 1113873prime(t) + e1ηprime(t)

(60)

us

I minus La1 b1 c1 d11113872 1113873y1113872 1113873prime(t)le e1ηprime(t) (61)

Since r(La1 b1 c1 d1)lt 1 we know that I minus La1 b1 c1 d1

has abounded inverse operator (I minus La1 b1 c1 d1

)minus 1 with

I minus La1 b1 c1 d11113872 1113873

minus 1 I + La1 b1 c1 d1

+ L2a1 b1 c1 d1

+ middot middot middot

+ Lna1 b1 c1 d1

+ middot middot middot (62)

Note that La1 b1 c1 d1(P) sub P and we obtain (Iminus

La1 b1 c1 d1)minus 1(P) sub P erefore

y(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875(t)

yprime(t)le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875prime(t)

for t isin [0 1]

(63)

is implies that

yC le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

C

yprime

Cle I minus La1 b1 c1 d1

1113872 1113873minus 1

e1η1113874 1113875prime

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868C

(64)

erefore we have

||y||le I minus La1 b1 c1 d11113872 1113873

minus 1e1η1113874 1113875

(65)

at is W is bounded Now we can selectR1 gtmax supW ρ11113864 1113865 (ρ1 is defined in (H4)) such that

yne λAy fory isin zBR1capP and λ isin [0 1] (66)

From Lemma 5 we have

i A BR1capP P1113872 1113873 1 (67)

On the contrary since La2 b2 c2 d2(P) sub P and

r(La2 b2 c2 d2)ge 1 it follows from Lemma 2 that there exists

φ0 isin P | 0 such that La2b2 c2 d2φ0 r(La2 b2 c2 d2

)φ0 andφ0 r(La2 b2 c2 d2

)minus 1La2 b2 c2 d2φ0 isin P Now we show that

y minus Ayne λφ0 fory isin zBρ1 capP λge 0 (68)

If this claim is false then there exist y0 isin zBρ1 capP andλ0 ge 0 such that y0 minus Ay0 λ0φ0 Note that λ0 gt 0 (other-wise the theorem is proved) en from (H4) we have

Ay0( 1113857(t) 11139461

0G1(t s)f s 1113946

s

01113946τ

0y0(r)drdτ 1113946

s

0y0(τ)dτ y0(s) y0prime(s)1113874 1113875ds

ge 11139461

0G1(t s) a2 1113946

s

01113946τ

0y0(r)drdτ + b2 1113946

s

0y0(τ)dτ + c2y0(s) + d2y0prime(s)1113874 1113875ds

11139461

0a2G2(t s) + b2G3(t s) + c2G1(t s)( 1113857y0(s)ds + d2 1113946

1

0G1(t s)y0prime(s)ds

La2 b2 c2d2y01113872 1113873(t)

(69)

8 Mathematical Problems in Engineering

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

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Mathematical PhysicsAdvances in

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

which implies that

y0 Ay0 + λ0φ0 geLa2 b2 c2 d2y0 + λ0φ0 ge λ0φ0 (70)

Let λlowast sup λgt 0 y0 ge λφ01113864 1113865 en λ0 isin λgt 0 y0 ge1113864

λφ0and y0 ge λlowastφ0 However we note that y0 ge La2 b2 c2 d2

y0 +

λ0φ0 ge La2 b2 c2 d2λlowastφ0 + λ0φ0 (λlowastr(La2 b2 c2 d 2) + λ0)φ0 and

this contradicts the definition of λlowast for r(La2 b2 c2 d2)ge 1

erefore (66) holds as required From Lemma 4 we have

i A Bρ1 capP P1113872 1113873 0 (71)

From (65) and (69) we have

i A BR1∖Bρ11113872 1113873capP P1113872 1113873 i A BR1

capP P1113872 1113873 minus i A Bρ1 capP P1113872 1113873 1

(72)

and hence A has at least one fixed point in (BR1 Bρ1)capP ie

(1) has at least one positive solution is completes theproof

Theorem 2 Suppose that (H0ndashH2) and (H5ndashH7) holdlten(1) has at least one positive solution

Proof We show that

yne λAy fory isin zBρ2 capP λ isin [0 1] (73)

If the claim is false then there exist y1 isin zBρ2 capP andλ1 isin [0 1] such that y1(t) λ1(Ay1)(t) for t isin [0 1] Fort isin [0 1] from (H5) we have

y1(t)le 11139461

0G1(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0G1(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3d3y11113872 1113873(t)

(74)

Also y1prime(t) λ1(Ay1)prime(t) for t isin [0 1] implies that

y1prime(t) λ11139461

tf s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s)f s 1113946

s

01113946τ

0y1(r)drdτ 1113946

s

0y1(τ)dτ y1(s) y1prime(s)1113874 1113875ds

le 11139461

0

zG1

zt(t s) a3 1113946

s

01113946τ

0y1(r)drdτ + b3 1113946

s

0y1(τ)dτ + c3y1(s) + d3y1prime(s)1113874 1113875ds

La3 b3 c3 d3y11113872 1113873prime(t)

(75)

Note that La3 b3 c3 d3(P) sub P and r(La3 b3 c3 d3

)lt 1 and wehave

I minus La3 b3 c3 d31113872 1113873y11113872 1113873(t)le 0

I minus La3 b3 c3 d31113872 1113873y11113872 1113873prime(t)le 0

for t isin [0 1]

(76)

ereforey1(t) 0

yprime(t) 0

for t isin [0 1]

(77)

is contradicts the fact that y1 isin zBρ2 capP Hence (71) istrue as required From Lemma 5 we have

i A Bρ2 capP P1113872 1113873 1 (78)

On the contrary from Lemma 2 there exists ψ1 isin P | 0

such that (Llowasta4 b4 c4ψ1)(s) r(Llowasta4 b4 c4

)ψ1(s) for s isin [0 1] LetU y isin P y minus Ay λφ2 for λge 01113864 1113865 where φ2(t) t minus

(12)t2 isin P for t isin [0 1] Note that λgt 0 (otherwise thetheorem is proved) We shall show thatU is a bounded set inP If y isin P then from (H6) we have

Mathematical Problems in Engineering 9

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

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Decision SciencesAdvances in

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AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

y(t) (Ay)(t) + λφ2(t)ge (Ay)(t)

ge 11139461

0G1(t s) a4 1113946

s

01113946τ

0y(r)drdτ + b4 1113946

s

0y(τ)dτ + c4y(s) minus e21113874 1113875ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus e2 1113946

1

0G1(t s)ds

ge 11139461

0a4G2(t s) + b4G3(t s) + c4G1(t s)( 1113857y(s)ds minus

12κ2e2

(79)

Multiplying both sides of the above inequality by ψ1(t)

and integrating from 0 to 1 yields

11139461

0y(t)ψ1(t)dtge 1113946

1

0ψ1(t) 1113946

1

0a4G2(t s) + b4G3(t s)(

+ c4G1(t s)1113857y(s)dsdt minus12κ2e2 1113946

1

0ψ1(t)dt

ge r Llowasta4 b4 c4

1113872 1113873 11139461

0y(t)ψ1(t)dt

minus12κ2e2 1113946

1

0ψ1(t)dt

(80)

is together with r(Llowasta4 b4 c4)gt 1 implies that

11139461

0y(t)ψ1(t)dt le

κ2e2 111393810 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

(81)

Note that y isin U and we have

yprime(t) (Ay)prime(t) + λφ2prime(t) 11139461

tf s 1113946

s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875ds + λ(1 minus t)ge 0

yPrime(t) (Ay)Prime(t) + λφPrime2(t) minus f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889 minus λle 0

(82)

en y is a concave and increasing function on [0 1]Hence

11139461

0y(t)ψ1(t)dt 1113946

1

0y

t

1middot 1 +

1 minus t

1middot 01113874 1113875ψ1(t)dt

ge 11139461

0(ty(1) +(1 minus t)y(0))ψ1(t)dt

ge 11139461

0ty(1)ψ1(t)dt

(83)

is enables us to obtain

||y||C y(1)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

11139461

0tψ1(t)dt1113890 1113891

minus 1

(84)

Now note (82) and we see there is an Mgt 0 such that

1113946s

01113946τ

0y(r)drdτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

1113946s

0y(τ)dτ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868 |y(s)| leM for s isin [0 1]

(85)

is together with (H7) implies that

f s 1113946s

01113946τ

0y(r)drdτ 1113946

s

0y(τ)dτ y(s) yprime(s)1113874 1113875leHM yprime(s)( 1113857

for s isin [0 1]

(86)

Note that y isin U and we obtain

minus yPrime(t) minus (Ay)Prime(t) minus λφPrime2 (t)

f t 1113946t

01113946τ

0y(r)drdτ 1113946

t

0y(τ)dτ y(t) yprime(t)1113888 1113889

+ λleHM yprime(t)( 1113857 + λ

(87)

10 Mathematical Problems in Engineering

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

is implies thatminus yPrime(t)yprime(t)

HM yprime(t)( 1113857 + λleyprime(t) (88)

and then if we let 9 yprime we have

1113946yprime C

0

9d9

HM(9) + λle 1113946

1

011139461

0yprime(t)dt

y(1) minus y(0)leκ2e2 1113938

10 ψ1(t)dt

2r Llowasta4 b4 c41113872 1113873 minus 2

middot 11139461

0tψ1(t)dt1113890 1113891

minus 1

(89)

erefore combining this and (H7) there exists Nlowast gt 0such that

yprime1113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868C leNlowast (90)

us U is bounded (see (82) and (88)) TakingR2 gtmax supU ρ21113864 1113865 we have

y minus Ayne λφ2 fory isin zBR2capP λge 0 (91)

From Lemma 4 we have

i A BR2capP P1113872 1113873 0 (92)

From (76) and (90) we have

i A BR2∖Bρ21113872 1113873capP P1113872 1113873 i A BR2

capP P1113872 1113873

minus i A Bρ2 capP P1113872 1113873 minus 1(93)

and hence A has at least one fixed point in (BR2 Bρ2)capP ie

(1) has at least one positive solution is completes theproof

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the China Postdoctoral ScienceFoundation (grant no 2019M652348) Technology ResearchFoundation of Chongqing Educational Committee (grantno KJQN201900539) Natural Science Foundation ofChongqing Normal University (grant no 16XYY24) andShandong Natural Science Foundation (grant noZR2018MA011)

References

[1] Z Bai ldquoPositive solutions of some nonlocal fourth-orderboundary value problemrdquo Applied Mathematics and Com-putation vol 215 no 12 pp 4191ndash4197 2010

[2] X Hao N Xu and L Liu ldquoExistence and uniqueness ofpositive solutions for fourth-order m-point boundary valueproblems with two parametersrdquo Rocky Mountain Journal ofMathematics vol 43 no 4 pp 1161ndash1180 2013

[3] Y Wei Q Song and Z Bai ldquoExistence and iterative methodfor some fourth order nonlinear boundary value problemsrdquoApplied Mathematics Letters vol 87 pp 101ndash107 2019

[4] Y Pang and Z Bai ldquoUpper and lower solution method for afourth-order four-point boundary value problem on timescalesrdquo Applied Mathematics and Computation vol 215no 6 pp 2243ndash2247 2009

[5] F Zhu L Liu and Y Wu ldquoPositive solutions for systems of anonlinear fourth-order singular semipositone boundary valueproblemsrdquo Applied Mathematics and Computation vol 216no 2 pp 448ndash457 2010

[6] W Fan X Hao L Liu and Y Wu ldquoNontrivial solutions ofsingular fourth-order Sturm-Liouville boundary value prob-lems with a sign-changing nonlinear termrdquo Applied Mathe-matics and Computation vol 217 no 15 pp 6700ndash67082011

[7] K Zhang ldquoNontrivial solutions of fourth-order singularboundary value problems with sign-changing nonlineartermsrdquo Topological Methods in Nonlinear Analysis vol 40no 1 pp 53ndash70 2012

[8] Y Zou ldquoOn the existence of positive solutions for a fourth-order boundary value problemrdquo Journal of Function Spacesvol 2017 p 5 2017

[9] X Zhang and Y Cui ldquoPositive solutions for fourth-ordersingular p-Laplacian differential equations with integralboundary conditionsrdquo Boundary Value Problems vol 2010p 23 2010

[10] Y Cui and J Sun ldquoExistence of multiple positive solutions forfourth-order boundary value problems in Banach spacesrdquoBoundary Value Problems vol 2012 no 1 p 13 2012

[11] Y Cui and Y Zou ldquoExistence and uniqueness theorems forfourth-order singular boundary value problemsrdquo Computersamp Mathematics with Applications vol 58 no 7 pp 1449ndash1456 2009

[12] OA Arqub ldquoAn iterative method for solving fourth-orderboundary value problems of mixed type integro-differentialequationsrdquo Journal of Computational and Applied Mathe-matics vol 18 no 5 pp 857ndash874 2015

[13] A Cabada and S Tersian ldquoMultiplicity of solutions of a twopoint boundary value problem for a fourth-order equationrdquoApplied Mathematics and Computation vol 219 no 10pp 5261ndash5267 2013

[14] G Bonanno and B Di Bella ldquoInfinitely many solutions for afourth-order elastic beam equationrdquo Nonlinear DifferentialEquations and Applications NoDEA vol 18 no 3 pp 357ndash368 2011

[15] Y Li ldquoExistence of positive solutions for the cantilever beamequations with fully nonlinear termsrdquo Nonlinear AnalysisReal World Applications vol 27 pp 221ndash237 2016

[16] Z Yang and J Sun ldquoPositive solutions of a fourth-orderboundary value problem involving derivatives of all ordersrdquoCommunications on Pure and Applied Analysis vol 11 no 5pp 1615ndash1628 2012

[17] K Zhang D OrsquoRegan and Z Fu ldquoNontrivial solutions forboundary value problems of a fourth order differenceequation with sign-changing nonlinearityrdquo Advances inDifference Equations vol 2018 no 1 p 13 2018

[18] J Liu and Z Zhao ldquoOn the nonhomogeneous fourth-order p-Laplacian generalized Sturm-Liouville nonlocal boundary

Mathematical Problems in Engineering 11

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

value problemsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 12 2012

[19] J Jiang L Liu and Y Wu ldquoPositive solutions for p-Laplacianfourth-order differential system with integral boundaryconditionsrdquo Discrete Dynamics in Nature and Societyvol 2012 p 19 2012

[20] Y Cui and Y Zou ldquoExistence and uniqueness of solutions forfourth-order boundary-value problems in Banach spaceselectronrdquo Journal of Differential Equations vol 2009 no 33pp 1ndash8 2009

[21] M R H Tavani ldquoExistence results for fourth-order elasticbeam equations on the real linerdquo Dynamic Systems andApplications vol 27 no 1 pp 149ndash163 2018

[22] M Tuz ldquoe existence of symmetric positive solutions offourth-order elastic beam equationsrdquo Symmetry vol 11 no 1p 121 2019

[23] N Ghawadri N Senu F Adel Fawzi F Ismail andZ Ibrahim ldquoDiagonally implicit Runge-Kutta type methodfor directly solving special fourth-order ordinary differentialequations with III-posed problem of a beam on elasticfoundationrdquo Algorithms vol 12 no 1 p 10 2019

[24] Y Tian S Shang and Q Huo ldquoAntiperiodic solutions offourth-order impulsive differential equationrdquo MathematicalMethods in the Applied Sciences vol 41 no 2 pp 769ndash7802017

[25] B Azarnavid K Parand and S Abbasbandy ldquoAn iterativekernel based method for fourth order nonlinear equation withnonlinear boundary conditionrdquo Communications in Non-linear Science and Numerical Simulation vol 59 pp 544ndash5522018

[26] R Jiang and C Zhai ldquoCombined effects of concave andconvex nonlinearities in nonperiodic fourth-order equationselectronrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 30 no 30 pp 1ndash14 2018

[27] Y Zhang J-P Sun and J Zhao ldquoPositive solutions for afourth-order three-point BVP with sign-changing Greenrsquosfunctionrdquo Electronic Journal of Qualitative lteory of Differ-ential Equations vol 5 no 5 pp 1ndash11 2018

[28] Y Han ldquoA class of fourth-order parabolic equation witharbitrary initial energyrdquo Nonlinear Analysis Real WorldApplications vol 43 pp 451ndash466 2018

[29] R Behl A Cordero S S Motsa and J R TorregrosaldquoMultiplicity anomalies of an optimal fourth-order class ofiterative methods for solving nonlinear equationsrdquo NonlinearDynamics vol 91 no 1 pp 81ndash112 2018

[30] Z Bai Z Du and S Zhang ldquoIterative method for a class offourth-order p-Laplacian beam equationrdquo Journal of AppliedAnalysis and Computation vol 9 no 4 pp 1443ndash1453 2019

[31] Y Li Y Ding and E Ibrahim ldquoPositive radial solutions forelliptic equations with nonlinear gradient terms on an exteriordomainrdquo Mediterranean Journal of Mathematics vol 15no 3 p 83 2018

[32] Y Li and Y Li ldquoPositive solutions of a third-order boundaryvalue problem with full nonlinearityrdquo Mediterranean Journalof Mathematics vol 14 no 3 p 128 2017

[33] Y Li ldquoPositive solutions for second-order boundary valueproblems with derivative termsrdquoMathematische Nachrichtenvol 289 no 16 pp 2058ndash2068 2016

[34] J Zhang G Zhang and H Li ldquoPositive solutions of second-order problem with dependence on derivative in nonlinearityunder Stieltjes integral boundary conditionrdquo ElectronicJournal of Qualitative lteory of Differential Equations vol 4no 4 pp 1ndash13 2018

[35] J R Cannon ldquoe solution of the heat equation subject to thespecifcation of energyrdquo Quarterly of Applied Mathematicsvol 21 no 2 pp 155ndash160 1963

[36] N I Ionkin ldquoe solution of a certain boundary valueproblem of the theory of heat conduction with a nonclassicalboundary conditionrdquo Journal of Differential Equationsvol 13 no 2 pp 294ndash304 1977

[37] R Y Chegis ldquoNumerical solution of a heat conductionproblem with an integral conditionrdquo LitovskiıMatematicheskiı Sbornik vol 24 no 4 pp 209ndash215 1984

[38] J Wu X Zhang L Liu Y Wu and Y Cui ldquoe convergenceanalysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular de-creasing nonlinearityrdquo Boundary Value Problems vol 2018no 1 p 15 2018

[39] X Zhang J Wu L Liu Y Wu and Y Cui ldquoConvergenceanalysis of iterative scheme and error estimation of positivesolution for a fractional differential equationrdquo MathematicalModelling and Analysis vol 23 no 4 pp 611ndash626 2018

[40] J Webb ldquoPositive solutions of nonlinear differential equa-tions with Riemann-Stieltjes boundary conditionsrdquo ElectronicJournal of Qualitativelteory of Differential Equations vol 86no 86 pp 1ndash13 2016

[41] B Ahmad Y Alruwaily A Alsaedi and S K NtouyasldquoExistence and stability results for a fractional order differ-ential equation with non-conjugate Riemann-Stieltjes inte-gro-multipoint boundary conditionsrdquo Mathematics vol 7no 3 p 249 2019

[42] F Wang L Liu Y Wu and Y Zou ldquoIterative analysis of theunique positive solution for a class of singular nonlinearboundary value problems involving two types of fractionalderivatives with p-Laplacian operatorrdquo Complexity vol 2019p 21 2019

[43] Q Song and Z Bai ldquoPositive solutions of fractional differ-ential equations involving the Riemann-Stieltjes integralboundary conditionrdquo Advances in Difference Equationsvol 2018 no 1 p 7 2018

[44] W Ma and Y Cui ldquoe eigenvalue problem for Caputo typefractional differential equation with Riemann-Stieltjes integralboundary conditionsrdquo Journal of Function Spaces vol 2018p 9 2018

[45] W Ma S Meng and Y Cui ldquoResonant integral boundaryvalue problems for Caputo fractional differential equationsrdquoMathematical Problems in Engineering vol 2018 p 8 2018

[46] S Meng and Y Cui ldquoMultiplicity results to a conformablefractional differential equations involving integral boundaryconditionrdquo Complexity vol 2019 p 8 2019

[47] S Meng and Y Cui ldquoe uniqueness theorem of the solutionfor a class of differential systems with coupled integralboundary conditionsrdquo Discrete Dynamics in Nature and So-ciety vol 2018 p 7 2018

[48] X Zhang L Liu Y Wu and Y Zou ldquoExistence anduniqueness of solutions for systems of fractional differentialequations with Riemann-Stieltjes integral boundary condi-tionrdquo Advances in Difference Equations vol 2018 no 1 p 152018

[49] K Deimling Nonlinear Functional Analysis Springer BerlinGermany 1985

[50] K Chang and Q Lin lte Lecture of Functional Analysis ePress of Beijing University Beijing China 2001

[51] D Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones Academic Press Orlando FL USA 1988

12 Mathematical Problems in Engineering

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom