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8CIENCE~DIRECT e Applied Mathematics Letters 17 (2004) 969-976

Applied Mathematics Letters

www.elsevier.com/locate/aml

A Second-Order Singular Three-Point Boundary Value Problem

P . SINGH Department of Mathematics, Baylor University

Waco, TX76798, U.S.A. pcobb©gulfcoast.edu

(Received April 2003; accepted August PO03)

A b s t r a c t - - T h e existence of a positive solution is obtained for the second-order three-point bound- ary value problem y l l+ f ( x , y ) = O, 0 < x <_ 1, y(O) = O, y (p ) -y (1 ) = 0, where 0 < p < 1 is fixed and where f (x , y) is singular at x --- 0, y = 0, and possibly at y = c~. The method applies a fixed-point theorem for mappings that are decreasing with respect to a cone. (~) 2004 Elsevier Ltd. All rights reserved.

K e y w o r d s - - F i x e d - p o i n t theorem, Boundary value problem.

1. I N T R O D U C T I O N

In this paper, we consider positive solutions for a three-point boundary value problem for the second-order ordinary differential equation

y" + f(x, y) = O, 0 < x <_ 1, (1)

and satisfying the three-point boundary conditions

~(o)=o, (2) y(p)-y(1) =o, (3)

where f (x,y) is singular at x = 0, y = 0, and possibly at y = c~. The value p C (0,1) is fixed throughout this paper. In particular, we assume

(i) f : (0, 1] × (0, c~) --~ (0, c~) is continuous and decreasing in y for every x E (0, 1] and for

each y e (0, ~ ) , f~ f(x, y) dx < ~ , (ii) limu_~0+ f(x, y) = c~ and l imy_~ f(x, y) = 0, uniformly on compact subsets of (0, 1].

Note that the function f(x, y) = 1/v/~-~ satisfies such conditions. The study of singular boundary value problems for ordinary differential equations enjoys sub-

stantial history since the paper by Gatica, 01iker and Waltman [1]. They studied singularities of the type in (i) and (ii) for second-order Sturm-Louiville boundary value problems. The key for the

0893-9659/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by .4.A~-TEX doi:10.1016/j.aml.2003.08.011

970 P. SINGH

Catica, Oliker, Wal tman results hinged on an application of a fixed-point theorem for operators tha t are decreasing with respect to a cone. Subsequent works for similar singularities were done by Henderson and Yin [2,3], Eloe and Henderson [4], and O 'Regan [5] in which right focal, focal, and higher-order boundary value problems were considered. Furthermore, Lomtat idze [6] proved the existence of at least one positive solution of a singular three-point boundary value problem.

By the same token, some at tent ion has been devoted to three-point boundary value problems. Included among these works are papers by Gup ta [7,8], Anderson [9], Ma [10], and Yang [11] in which positive solutions and multiple positive solutions are obtained. In the present work, the Gatica, Oliker, Wal tman fixed-point theorem is applied to obtain solutions for (1)-(3).

Our intent is to t ransform the boundary value problem into an integral equation by use of an appropriate Green 's function, C(x, t), which will play the role of the kernel of operators , Tn, for which we seek fixed points. These fixed points will form a sequence of i terates converging to a solution of the boundary value problem.

In the next section, we give definitions and some properties of cones in a Banach space. We then s tate the fixed-point theorem due to Gatica, Oliker and Wal tman [1].

2. A F I X E D - P O I N T T H E O R E M

We begin by giving definitions and some properties of cones in a Banach space. For references, see [12,13].

Let B be a real Banach space. A nonempty set K: C B is called a cone if the following conditions are satisfied:

(a) the set K: is closed; (b) if u, v E/C, then au + fly E ~ for all a , fl >_ 0; (c) u , - u E/C imply u = 0.

Given a cone, K:, a partial order, ~, is induced on B by x <_ y, for x , y E B iff y - x C K:. (For clarity, we sometimes write x _ y (w.r.t. K:).) If x, y E B with x _< y, let (x, y) denote the closed-order interval between x and y given by (x, y) = {z C B ] x < z < y}. A cone K: is normal

in B provided there exists 5 > 0 such tha t ]Iel +e2I] _> 5, for all el ,e2 E K: with [lelll = ]]e2II = 1.

REMARK. If K: is a normal cone in B, then closed-order intervals are norm bounded (see [9, p. 24]).

We now state the fixed-point theorem due to Gatica, Oliker and Wal tman [1].

THEOREM 1. Let B be a Banach space, }C a normal cone in B, D a subset of ]C such that if x, y are elements of D, x <_ y, then (x, y) is contained in D, and let T : D --+ ]C be a continuous decreasing mapping which is compact on any closed-order interval contained in D. Suppose there

exists an xo E D such tha t T2xo is defined (where T2xo = T(Txo ) ), and furthermore, Txo,T2xo

are (order) comparable to xo. Then, T has a fixed point in D provided that either

(I) Txo <_ xo and T2xo < xo or Txo >_ xo and T2xo >_ xo, or (II) the complete sequence of iterates {Tnxo}~_o is defined and there exists Yo E D such that

Tyo C D and yo <_ Tnxo, for every n.

3. T H E S E C O N D - O R D E R S I N G U L A R P R O B L E M

We look for positive solutions of (1)-(3) on (0, 1] of class C[0, 1] N C(2)(0, 1] using Gatica- Oliker-Waltman methods. Our intent is to t ransform (1)-(3) into an integral equation by use of a Green 's function, G(x, t), then find fixed points of operators, Tn. These fixed points will form a sequence of i terates converging to a solution of (1)-(3). We observe tha t positive solutions of (1)-(3) are also concave-down on [0, 1].

Three-Point Boundary Value Problem 971

We shall consider the following Banach space, /3, with associated norm:

/3 = {u: [0, 1] -~ ~ I u is continuous},

Ilull = sup lu(x)l . ~e[0,1]

We define a normal cone, K:, in /3 as

:= {u e / 3 I u(x) > 0 on [0, 1]}.

Moreover, we define the function gl : [0,1] ~ [0, oc) by

J" x, i f 0 < x < p , gl (x) l p, i f p < x < l .

And for t9 > 0, let go(z) = O. gl.

We observe at this point, for each positive (and concave-down) solution, y(x), of (1)-(3), for some O > O , go(x) < _ y ( x ) , O < x < l .

For the final hypothesis, we assume

(iii) 0 < f ~ f ( x , go(x))dx < cx~, for all • > O.

We will apply Theorem I to operators whose kernel is the Green 's function for - y " = 0 and satisfies (2),(3). As shown by Yang [11], this Green 's function, G : [0, 1] × [0, 1] ~ [0, c~), is given by

x, x < t < p ,

t, t < x and t < p, a ( z , t ) = 1 - t ~ _ p . X , t > _ p a n d z < t ,

p - t t + T ~ _ p . X , x > t > p .

Notice, if (x, t) • (0, 1) × (0, 1), then G(x, t) > O. We define a subset, D, of the cone as

D := {¢ • / ~ I 30(¢) > 0 such tha t ¢(x) >_ go(x), x • [0, 1]}.

Moreover, let the integral opera tor T : D ~ K: be defined by

/01 Tu(x) := G(z,t)f( t ,u(t))dt .

Notice, it suffices to define D as above, since due to the singularity in f , (limy--,0+ f (x , y) = co), we cannot define our operator T on all of the cone ~ . Fur thermore, it can easily be verified tha t T is well defined. In tha t direction, for ¢ • D, there exists 0(¢) > 0 such tha t go(x) <_ ¢(x), 0 < x < 1. And since f (x ,y) decreases with respect to y, we have

f(x, ¢(x)) _</(×,go(x)), for 0 < x _< 1. Thus,

/0 /0 0 < a(z , t ) / ( t ,¢( t ) )d t < G(z, t) f( t , go(t))dt < ~ .

REMARK. In a similar light, one can see tha t T is decreasing with respect to D. Moreover, ¢ • D is a solution of (1)-(3) iff T ¢ = ¢.

One direction of the remark is obviously true. To see the other direction, we let ¢ • D. Then, (T¢)(z) = f~o a(z , t ) / ( t , ¢(t)) dt and we have tha t (T¢)"(z) = - f ( x , ¢(x)) < 0, x • (0, 1]. Moreover, we see t ha t (T¢)(x) > 0, (T¢)(0) = 0, and (T¢)(p) - (T¢)(1) = 0. Thus, there exists a 0(T¢) such tha t (T¢)(x) _> go(x), which implies tha t T ¢ • D. T h a t is, T : D ~ D, and hence, the remark.

We now present a number of lemmas tha t allow us to apply Theorem 1.

972 P. SINGH

LEMMA 2. I f f satisfies (i)-Oii), then there exists an S > 0 such that I1¢1t ~ s for any solution ¢ in D of (1)-(3).

PROOF. We shall prove the lemma by contradiction. Assume that the conclusion is false. Then, there exists a sequence, {¢n)n°°=l , of solutions to (1)-(3), such tha t ¢n(x) > 0 for z C (0, 1] and

I1¢.11 --- 11¢~+111 and l i r n I1¢~11 = o~,

Now, for any solution ¢ of (1), we have ¢"(x) = - f ( x , ¢(x)) < 0, for 0 < x _< 1. This says that ¢ is concave downward. In particular, the graph of each ¢~(x) is concave.

Furthermore, we claim tha t the boundary conditions (2),(3) and the concavity of ¢,~ give us

¢~(x) > p¢~(x~) = pll¢~ll, p _< x _< 1,

where x~ c (p, 1) is where the maximum value of the solution, ¢n(X), occurs. To see this, we consider the line segment joining (0, O) and (x~, ¢n(Xn)), which gives us y(x) = ([[¢n[]/xn)x, 0 < x < x~. Thus, y(p) = ([[¢,~[[/xn)p > p[]¢~[[. Furthermore, we certainly have y(p) < ¢~(p) and ¢ , ( x ) > ¢ , (p) , x e ~v, 1]. Thus,

which implies tha t

and hence, the claim. Now let us define

Cn(X) ~ Ca(P) > Y(P) > PllCnl], p ~ X ~ 1,

¢~(x) > p ¢ . ( x n ) , ; < ~ < 1,

M : = m a x { G ( x , t ) : (x,t) e [0,1] × [0,1]}.

Then, from Condition (ii), there exists an no such that , if n > no and for x E ~o, 1],

1 f ( x , ¢~(x)) _<

M(1 - p )

Let 0 = P¢no (x~o). Then, the line segment joining (0, 0) with (p, ¢) and the line segment joining (p, ¢) with (1, 0) must lie under the graph of ¢~ for n _> no. Tha t is,

for x E [0, 1]. ~)n(X) ~ gO(X),

< 1, we have

Ten(x )

f = a(x , t ) f ( t , ¢~ (t)) dt

/o f = Pa(z , t ) f ( t ,¢ ,~( t ) )d t+ a ( z , t ) f ( t , ¢~( t ) )d t

// f < a(x , t ) f ( t , go(t)) at + a(x , t ) f ( t , Cn(t)) at

<_ G(x, t ) f ( t , go(t)) dt + M(1 - p) dt

= --.In" G(x, t ) f ( t , go(t)) dt + 1

f < M f( t , go(t)) dt + 1

<00.

Thus, for n _> no and 0 < x

¢~(x) =

This is a contradiction to the assumption that lim~--.oo ][¢~[I = oo. Hence, there exists an S > 0 such that I]¢[] -< S for any solution ¢ E D of (1)-(3). |

Three-Point Boundary Value Problem 973

LEMMA 3. If f satisfies (i)--(iii), then there exists an R > 0 such that I1¢11 >-- R, for any solution ¢ in D of (1)-(3).

PROOF. We assume the conclusion is false. Thus, we may assume tha t

lim I1¢~11 = 0 n---+~

uniformly on [0, 1]. Let

m = inf { G ( x , t ) : (x,t) E ~,1] × ~v, 1]} > 0.

From Condition (ii) on f , we have tha t

lim f ( x , y ) = oo y-*0+

uniformly on compact subsets of (0,1]. Hence, there exists ~ > 0 such tha t for x 6 ~v, 1] and

0 < y < 6, we have 1 :(x' Y) > m(1 - p~'

Now, there exists an no such that n >_ no implies tha t

5 o < ¢~(x) < ~,

for x e (0, 1]. So, for x e (0, 1] and n >_ no,

¢~(x) = T¢~(x)

= fo 1 G(x, t ) f ( t , ¢~(t)) dt

,1

>mj

>mj

f ( t , ¢~(t)) dt

if (t,~)dt ol 1

- - dt m(1 - ; )

This is a contradiction to the assumption tha t lim~--.oo I1¢~11 = 0 uniformly on [0,1]. Hence, there

exists an R > 0 such tha t R _< Nell. |

Thus, altogether, for ¢ E D a solution of (1)-(3), Lemmas 2 and 3 give us

R < II¢ll -< s .

We now state our existence result.

THEOREM 4. If f satisfies (i)-(iii), then (1)-(3) has at least one positive solution.

P R o o f . For all n, let lb,(x) = T(n), where n is the constant function of tha t value on [0, 1]. In

particular,

/o 1 ¢~ (x) = a(x , t ) f ( t , n) dr.

Since f is decreasing, and as we have observed, T is i s o a decreasing mapping, we have

¢ . + 1 ( x ) < Cn(x), ¢ . ( ~ ) > 0, for x e (0,1].

And by (ii), l im, -~o Cn(x) = 0 uniformly on [0, 1].

974 P. SINGH

W e n o w def ine fn: (0, 1] x [0, oo) --* (0, a~) , as

f~(x, t) = f(x, max{t , ¢~(x)}) .

Then, fn is continuous and fn does not have a singularity at y = 0. Moreover, for (x, t) E (0, 1] x (0, ee) we have tha t

fn(x,t) <_

and, in particular,

fn(x, t) = f (x , max{t,¢~(x)}) <_ f (x ,¢~(x)) .

Next, we define a sequence of operators T~ : ]C ~ K:, for ¢ E K and x E [0, 1], by

T**¢(x) := G(x, t)fn(t, ¢(t)) dt.

I t is s tandard tha t each T~ is a compact mapping on/C. Moreover, T~(0) >_ 0, and T~2(0) > 0. Thus, by Theorem 1, each T~ has a fixed point in K:, for each n. So, for all n, there exists a Cn E/C such tha t Then(x) = Cn(x), 0 < x < 1. Hence, for n > 1, Cn satisfies the boundary conditions of the problem.

In addition, for each Cn, we note tha t

1 Then(x) = .In G(x,t)f~(t,¢~(t)) dt

= ./a I a(x, t) f( t , max{¢~ (t), ¢~( t )}) dt

< a(z , t ) f ( t ,¢n( t ) )d t

= T¢~(x) ,

which gives us tha t Cn(x) = T~¢n(x) < T¢~(x) for all n and 0 < x < 1. Now, arguing much along the same lines of L e m m a 2 and using T~¢n(x) _< T¢~(x) , it is fairly

straightforward to show tha t there exists an S > 0 such tha t 1[¢~][ - S, for all n. Similarly, we can follow the proof of L e m m a 3 to show tha t there exists an R > 0 such tha t []¢~I[ > R. Hence, altogether, for all n we have tha t

n < II¢,,ll < s .

As observed before, we know tha t Cn(x) >_ P[]¢n[I, for x E [p, 1] and for all n. Thus, if 0 = pR, it follows tha t go(x) <_ ¢~(x) for all n. Our first observation gives us the previous inequality for x E [p, 1], and the concavity of ¢~(x) gives us the previous inequality for x E [0, 1]. This implies tha t the sequence {¢~}~=1 is contained in the order interval (go, S}, where S is the constant function of tha t value on [0, 1]. Tha t is, the sequence {¢~}~=1 is contained in D. Thus, using the fact tha t T is a compact mapping, we may assume tha t lim~--.o~ T e n , say ¢*, exists.

To conclude the proof of this theorem, we now show tha t

lim ( T e n ( x ) - Cn(x)) = 0.

This will give us ¢* E (go, S) and

0 < x < l .

Three-Point Boundary Value Problem 975

To tha t end, let us fix 0 = pR. Then, as observed above, go(x) < ¢,~(x) for all n, and 0 < x < 1. Let E > 0 be given and choose 6 such t ha t 0 < 5 < 1 and such t ha t

~0 ~ £ f(t, go(t)) dt < 2M'

where M = max{G(x,t) : (x,t) e [0, 11 × [0, 1]}. Then, there exists an no such t ha t for n _> n0,

¢~(t) < g0(t) < ¢=(t), t e [6,1].

Thus, for t C [6, 1],

f n ( t , ¢~( t ) ) = f ( t , m a x { ¢ n ( t ) , Cn( t )}) = f ( t , ¢~( t ) ) ,

and we have tha t , for 0 < x < 1,

T e n ( x ) -- Cn(X) = T e n ( x ) - Then(x)

[z z = G(x, t)f(t, ¢n(t)) dt + G(x, t)f(t, Cn(t)) dt

~ 1 - G(x,t)f~(t,4)~(t)) at + a(z,t)f,~(t, Cn(t)) dt

~ 6

= fo a(z , t ) I ( t ,¢~( t ) )d t - fo G(z,t)I,~(t,¢,~(t))dt.

Thus, we have tha t , for 0 < x < 1,

; ] I T C h ( X ) - Cn(X)l < M f ( t , ¢ n ( t ) ) d t + f ( t , m a x { ¢ n ( t ) , ¢ n ( t ) } ) d t

< M f(t, Cn(t)) dt + f(t , ¢,~(t)) dt

6

<_ 2M fo f(t, go(t)) dt

<~.

Since x E [0,1] was a rb i t ra ry , we have t h a t for n > no, HTCn - ¢~11 < e, and hence, t he resul t ,

T¢* = ¢*.

This completes the proof.

R E F E R E N C E S 1. J.A. Gatica, V. Oliker and P. Waltman, Singular nonlinear boundary value problems for second-order ordinary

differential equations, J. Differential Equations 79, 62-78~ (1989). 2. J. Henderson and W. Yin, Singular (k, n - k) boundary value problems between conjugate and right focal.

Positive solutions of nonlinear problems, J. Comput. Appl. Math. 88, 57-69, (1998). 3. J. Henderson and W. Yin, Focal boundary-value problems for singular ordinary differential equations, In

Advances in Nonlinear Dynamics, Volume 5, Stability Control Theory Methods Appl., pp. 283-295, Gordon and Breach, Amsterdam, (1997).

4. P. Eloe and J. Henderson, Singular nonlinear boundary value problems for higher ordinary differential equa- tions, Nonlinear Anal. 17, 1-10, (1991).

5. D. O'Regan, Existence of solutions to third order boundary value problems, Proc. Royal Irish Acad. Sect. A 90 (2), 173-189, (1990).

976 P. SINGH

6. A.G. Lomtatidze, A singular three-point boundary value problem, (in Russian), Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 17, 122-134, (1986).

7. C.P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinm'y differential equation, J. Math. Anal. Appl. 168, 540-551, (1992).

8. C.P. Gupta, A sharper condition for the solvability of a three-point second order boundary value problem, J. Math. Anal. Appl. 205, 586-579, (1997).

9. D. Anderson, Multiple positive solutions for a three-point boundary value problem, Mathl. Comput. Modelling 27 (6), 40-57, (1998).

10. R. Ma, Positive solutions of a nonlinear three-point boundary value problem, Electron. J. Differential Equa- tions 1998 (34), 1-8, (1998).

11. ]3. Yang, Boundary value problems for ordinary differential equations, Doctoral Dissertation, Mississippi State University, Mississippi State, MS, (August 2002).

12. M.A. Krasnosel'skii, Positive Solutions to Operator Equations, Noordhoff, Croningen, The Netherlands, (1964).

13. H. Amann, Fixed point equations and nonlinear e[genvalue problems in ordered Banach spaces, SIAM Rev. 18, 620-709, (1976).