solutions of volterra integral equations with infinite delay

Math. Nachr. 281, No. 3, 325 – 336 (2008) / DOI 10.1002/mana.200510605

Solutions of Volterra integral equations with infinite delay

Daniel Franco∗1 and Donal O’Regan∗∗2

1 Departamento de Matematica Aplicada, Universidad Nacional de Educacion a Distancia, Apartado de Correos 60149,Madrid, 28080, Spain

2 Department of Mathematics, National University of Ireland, Galway, Ireland

Received 11 March 2005, revised 20 October 2005, accepted 27 January 2006Published online 6 February 2008

Key words Volterra integral equation, infinite delay, existence results, periodic solutions, positive solutionsMSC (2000) 45D05, 45M15, 45M20, 45G10

We present several new existence results for a Volterra integral equation with infinite delay. We discuss periodicand bounded solutions. Sufficient conditions for the existence of positive periodic solutions are also provided.The techniques we employ have not been used for this equation before. Our results generalize and complementthose in the literature and several examples are presented to show their applicability.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

In this paper we are interested in the existence of solutions to the nonlinear equation

f(t) = y(t) −∫ t

−∞k(t, s)g(s, y(s)) ds, t ∈ R. (1.1)

Thinking of y as an input and f as the output we observe that (1.1) is an example of a hereditary law wherethe output in a fixed time t not only depends on the value of the input at t but also on its past history. Suchlaws appear in modelling situations in physics, engineering, biology, etc. For example, in [16] Equation (1.1)is considered in the study of hereditary response in continuum physics for a material with large memory, and in[20] it is employed in the study of the response of nonlinear feedback systems to periodic input signals. We referthe reader to [1] for a brief description of the meaning of (1.1) in the context of epidemics. In [4] the relationbetween (1.1) and the common variation of parameters formula is discussed. A more complete list of referencesabout (1.1) can be found in [10]. Finally, we refer the reader to [2], [19] for general results on Volterra equationson finite intervals.

Although the literature concerning existence results for Equation (1.1) is small, some authors have consideredit from different points of view. In the next paragraphs we describe roughly the literature to date.

In relation to the existence of periodic solutions using topological methods most of the results appear in [4],[5], [9], [11], [12], [13]. The techniques made use of Banach’s contraction principle [4], [11], [13], Schauder’sfixed point theorem [11], topological transversality method [9], resolvent equations [11], [12], [13], and Liapunovfunctionals [5]. A common hypothesis in all the above papers is the continuity of g which sometimes is assumedto be independent of its first variable (g(s, x) = h(x) or g(s, x) = x) together with a local Lipschitz conditionwith respect to the second variable. In [5] the regularity of the kernel

(k ∈ C2

)is an essential hypothesis whereas

in [11]–[13] it is the integrability of the resolvent which plays an important role. However, as it is pointed outin [13], this last condition can be difficult to achieve. For example, if the kernel is of convolution type, i.e., thefunction k satisfies k(t, s) = a(t−s) for certain function a ∈ L1(R+), and b is the resolvent of the equation then∫ ∞

0

|a(t)| dt < 1 ⇐⇒ b ∈ L1(R+).

∗ Corresponding author: e-mail: [email protected], Phone: +34 913988134, Fax: +34 913986912∗∗ e-mail: [email protected], Phone: +353 91 493091, Fax: +353 91 750542

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326 Franco and O’Regan: Integral equations with infinite delay

The existence of bounded solutions has also been considered in [5], [6], [11]–[13]. In [6], [11] the resolventis employed to study (1.1). The Banach contraction principle is the main tool used in Theorem 5 of [13] wheresufficient conditions are presented for the existence of a bounded solution. In [12], firstly the linear case g(s, x) =x is considered and using the resolvent equation the author proves the existence of a unique solution x(t). Nextthe nonlinear case is considered in a perturbed situation and assuming that the resolvent r satisfies for each ω > 0

limt→∞

∫ ω

−∞|r(t, s)| ds = 0,

the author shows that if y(t) is a bounded solution of the perturbed equation then |x(t) − y(t)| → 0 as t → ∞.On the other hand, assuming respectively a “strong” Lipschitz condition and a monotone condition of the secondvariable of the nonlinearity g, Banach’s contraction principle and the monotone iterative technique are employedin [11] to show the existence of bounded solutions. Again, in the above papers the resolvent is assumed to beintegrable.

The existence of almost periodic solutions for (1.1), and as a consequence of bounded continuous solutions(see [7] for general theory on almost periodic functions), has been considered in [1], [5], [14]. In [14] the authorsprove that if there exists a bounded solution with suitable stability properties then that solution is in fact an almostperiodic solution. In [5] Equation (1.1) is considered in the convolution case and existence is established usingLiapunov functionals and a theorem of Schaefer type. In [1] Equation (1.1) is also considered in the convolutioncase with f ≡ 0 and using the Hilbert projective metric the authors show that the equation has exactly onepseudo-almost-periodic positive solution.

To the best of our knowledge the techniques we are going to employ here have not been used for (1.1) before.Our results generalize and complement those in [1], [4], [5], [9], [11]–[14], [16], [17], [20]. We include severalexamples to show the applicability of our results.

We finish this introduction by explaining briefly what we are going to do in the following sections. Also wepresent some definitions and notation. It is clear that if we define an operatorN : E → E from a certain space offunctions E into itself by

[Ny](t) = f(t) +∫ t

−∞k(t, s)g(s, y(s)) ds, t ∈ R, (1.2)

then finding a solution u ∈ E for (1.1) is equivalent to find a fixed point of N . Now, since we will be interestedin bounded and periodic solutions, the space E will be one of the following.BC(R) will be the space of bounded continuous functions on R with values in R. Let 0 < ω < ∞. We

define Cω(R) to be the subspace of BC(R) consisting of all ω-periodic mappings, that is, if y ∈ Cω(R), then yis continuous on R and

y(t+ ω) = y(t) for all t ∈ R.

The norm on Cω(R) will be the same as the norm | · |0 on BC(R),

|y|0 = supt∈R

|y(t)|

and for y ∈ Cω(R) it is clear that |y|0 = supt∈Iω|y(t)|, where Iω is a compact subinterval of R of length ω.

Sufficient conditions for the existence of at least one solution y ∈ Cω(R) or y ∈ BC(R) of (1.1) willbe presented in Section 3, while in Section 4 we will deal with the existence of at least one positive solutiony ∈ Cω(R). In each case a Nonlinear Alternative and Krasnosel’skii’s fixed point theorem [15] are instrumentalin obtaining our results.

Theorem 1.1 (Nonlinear Alternative) Let C be a convex subset of a normed linear space E, and let U be anbounded open subset of C, with p ∈ U. Then every continuous and completely continuous map N : U → C hasat least one of the following two properties:

(i) N has a fixed point,(ii) there is an x ∈ ∂U with x = (1 − δ)p + δNx for some 0 < δ < 1.

c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.mn-journal.com

Math. Nachr. 281, No. 3 (2008) 327

Let E be normed linear space, we say that K ⊂ E is a cone if is closed, nonempty, K = 0 and wheneverx, y ∈ K and λ, µ ∈ R with λ ≥ 0, µ ≥ 0 then λx+ µy ∈ K . If D is a subset of E we write DK = D ∩K and∂KD = (∂D) ∩K .

Theorem 1.2 (Krasnosel’skii’s Fixed Point Theorem) Let (E, ‖ · ‖) be a Banach space, and let K ⊂ E be acone in E. Assume Ω1, Ω2 are open bounded subsets of E with 0 ∈ Ω1, Ω1 ⊂ Ω2, and let

N : (Ω2 \ Ω1)K −→ K

be a continuous and completely continuous operator such that, either

‖Nu‖ ≤ ‖u‖ for u ∈ ∂KΩ1, and ‖Nu‖ ≥ ‖u‖ for u ∈ ∂KΩ2,

or

‖Nu‖ ≥ ‖u‖ for u ∈ ∂KΩ1, and ‖Nu‖ ≤ ‖u‖ for u ∈ ∂KΩ2.

Then N has a fixed point in (Ω2 \ Ω1)K .

Recall that a completely continuous operator means an operator which transforms every bounded set into arelatively compact set.

In order to apply the above results we will study in Section 2 conditions on k, f and g which guarantee thatthe integral operatorN defined in (1.2) satisfies

N : E −→ E is well-defined, continuous and completely continuous,

where E is Cω(R) or BC(R). Of course, to prove the completely continuity of the operator we shall needa compactness criteria in those spaces. The Ascoli–Arzela Theorem is suitable for Cω(R) but in the case ofBC(R) we use the following result, which was proven in [22], recently employed by M. Zima in the study ofboundary value problems on the half line [21].

Theorem 1.3 Let Ω ⊂ BC(R). Suppose that the functions y ∈ Ω are equicontinuous in each compactinterval I ⊂ R and uniformly bounded in the sense of the norm

|y|ξ = supt∈R

ξ(|t|)|y(t)|,

where ξ : [0,∞) → (0,∞) is positive, continuous and

limt→∞ ξ(t) = ∞.

Then Ω is relatively compact in BC(R).

2 Admissibility results

We have not yet established the basic framework for the functions that define Equation (1.1). Since we arelooking for bounded continuous functions it is natural to assume f ∈ BC(R). On the other hand, since theintegral

∫ t

−∞ k(t, s)g(s, y(s)) ds has to be defined for every t ∈ R we shall assume

• k : R × R → R is continuous in (t, s) for s ≤ t and there exists p ≥ 1 with

supt∈R

(∫ t

−∞|k(t, s)|p ds

) 1p

≤ β <∞. (2.1)

• g : R × R → R is a Caratheodory function, i.e., for each u ∈ R, t → g(t, u) is measurable, and for a.e.t ∈ R, u → g(t, u) is continuous. Moreover, for each R > 0 there exists µR ∈ Lq(R) (here q is theconjugate to p) such that |g(t, x)| ≤ µR(t) for a.e. t ∈ R

1.

We begin by presenting a condition on k which guarantees thatN defined in (1.2) transforms bounded contin-uous functions to bounded continuous functions and that N is continuous.

1 As usual for u ∈ Lq(R) we consider the norm ‖u‖q =

R|u(s)|q ds

1q if 1 < q < ∞ and ‖u‖∞ = ess supt∈R |u(t)| if q = ∞.

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Math. Nachr. 281, No. 3 (2008) 329

Theorem 2.3 Suppose that condition (A) holds. In addition, assume that there exists ω > 0 such that

(B1) k(t+ ω, s+ ω) = k(t, s), −∞ < s ≤ t <∞,

(B2) g(t+ ω, x) = g(t, x) for almost every t ∈ R and all x ∈ R,

(B3) f ∈ Cω(R).

Then N : Cω(R) → Cω(R) is well-defined.

P r o o f. From Theorem 2.1 we know that N : Cω(R) ⊂ BC(R) → BC(R) is well-defined. Moreover, usingthat y ∈ Cω(R) and (B1)–(B3) we have

Ny(t+ ω) = f(t+ ω) +∫ t+ω

−∞k(t+ ω, s)g(s, y(s)) ds

= f(t) +∫ t

−∞k(t+ ω, s+ ω)g(s+ ω, y(s+ ω)) ds

= f(t) +∫ t


= Ny(t).

Hence Ny ∈ Cω(R).

In the following results, we look for conditions which guarantee the completely continuity of N . We shallconsider first the periodic case.

Theorem 2.4 Assume conditions (B1)–(B3) hold. In addition assume that

(B4) For each ε > 0 there exists δ > 0 such that if |σ| < δ, then∫ t

−∞|k(t+ σ, s) − k(t, s)|p ds < ε

uniformly on compact sets of R.

Then N : Cω(R) → Cω(R) is a completely continuous operator.

P r o o f. Notice that (B4) implies (A) and therefore N is well-defined. Let Ω ⊂ Cω(R) a bounded set. Then,there exists R > 0 such that |y|0 < R for each y ∈ Ω. Essentially the same reasoning that we have employed in(2.3) assures that NΩ is a uniformly bounded set.

Now, we will show that NΩ is equicontinuous. First, we note that since k is continuous and satisfies (B1) wehave

limδ→0

∫ t+δ

t


∫ δ

0

|k(t+ δ, s+ t)|p ds = 0 (2.4)

uniformly for t ∈ R.On the other hand, from Holder’s inequality we have for each y ∈ Ω and t ∈ [0, ω]∣∣∣∣

∫ t

−∞k(t, s)g(s, y(s)) ds−

∫ t+δ

−∞k(t+ δ, s)g(s, y(s)) ds

∣∣∣∣≤((∫ t

−∞|k(t, s) − k(t+ δ, s)|p ds

) 1p

+(∫ t+δ

t

|k(t+ δ, s)|p ds) 1

p

)‖µR‖q.

The equicontinuity of NΩ follows from (B4) and (2.4).Finally, since NΩ ⊂ Cω(R) we can use the Ascoli–Arzela Theorem to obtain that NΩ is relatively compact.



Next, we present a condition which guarantees the completely continuity and the continuity of N : BC(R) →BC(R).

(C) There exist constants γ, η > 0 and a continuous function ξ : [0,∞) → (0,∞) satisfying

limt→∞ ξ(t) = ∞ such that sup

t∈R

(∫ t

−∞(ξ(|t|)|k(t, s)|)p ds

) 1p

≤ γ and |f |ξ ≤ η.

Theorem 2.5 Assume that (B4) and (C) hold. Then N : BC(R) → BC(R) is completely continuous.

P r o o f. We shall apply Theorem 1.3. Let Ω ⊂ BC(R) be a bounded set and let C be a compact set in R.Without loss of generality assume C = [a, b]. Now, since k([a, b+ 1] × [a, b+ 1]) is bounded, we have

limδ→0

∫ t+δ

t


∫ δ

0

|k(t+ δ, s+ t)|p ds = 0 (2.5)

uniformly for t ∈ [a, b].Therefore, the equicontinuity of NΩ on [a, b] follows from (B4), (2.5) and Holder’s inequality.Next, we have to show that NΩ is uniformly bounded in the sense of the norm | · |ξ where ξ was introduced in

(C). Let y ∈ Ω, then we have

ξ(|t|)|Ny(t)| = ξ(|t|)∣∣∣∣f(t) +

∫ t


∣∣∣∣≤ |f |ξ +

∫ t

−∞ξ(|t|)|k(t, s)||g(s, y(s))| ds

≤ η + γ‖µR‖q.

Thus |Ny|ξ ≤ η + γ‖µR‖q and we are finished.

3 Existence theory

Using the Nonlinear Alternative we obtain our first existence principle, which generalizes Theorem 2.3 in [9].

Theorem 3.1 Assume that (B1)–(B4) hold. In addition, suppose that there exists a constantM > 0, indepen-dent of λ, with |y|0 = M for any solution y ∈ Cω(R) of

y(t) = λ

(f(t) +

∫ t


), t ∈ R, (3.1)

for each λ ∈ (0, 1). Then (1.1) has a solution y ∈ Cω(R).

P r o o f. Apply Theorem 1.1 with E = C = Cω(R), p = 0, U = y ∈ Cω(R) : |y|0 < M and N : E → Egiven by

Ny(t) = f(t) +∫ t

−∞k(t, s)g(s, y(s)) ds.

We now present a condition which guarantees that (3.1) holds.

Corollary 3.2 Let p = 1. Assume that (B1)–(B4) hold and

|g(s, y)| ≤ φ(s)ψ(|y|)with φ ∈ L∞(R), ψ : R

+ → R continuous and nondecreasing, and

supx∈[0,∞)

x

|f |0 + β‖φ‖∞ψ(x)> 1. (3.2)

Then (1.1) has at least one solution y ∈ Cω(R).


Math. Nachr. 281, No. 3 (2008) 331

P r o o f. First, we note that in the above hypotheses g satisfies the assumption at the beginning of Section 2with q = ∞. Next, we will show that there exists M > 0 such that |y|0 = M for any solution y ∈ Cω(R) of(3.1) with λ ∈ (0, 1). Let y be a solution of (3.1) for some λ ∈ (0, 1). Then we have

|y(t)| =∣∣∣∣λ(f(t) +

∫ t


)∣∣∣∣ ≤ |f |0 + β‖φ‖∞ψ(|y|0),

and so

|y|0|f |0 + β‖φ‖∞ψ(|y|0) ≤ 1.

From (3.2), if we choose M such that

M

|f |0 + β‖φ‖∞ψ(M)> 1

then our result follows.

Example 3.3 Consider the equation

y(t) = cos t+∫ t

−∞e−(t−s)g(s, y(s)) ds

where g is a Caratheodory function, 2π-periodic in the first variable and there exists φ ∈ L∞(R) such that|g(s, x)| ≤ φ(s)|x| for a.e. s ∈ R and all x ∈ R. Then (1.1) has a periodic solution provided that

‖φ‖∞ < 1. (3.3)

To see the above notice that (B1)–(B3) hold for ω = 2π. Let p = 1 and note (2.1) holds with β = 1 and (B4)holds uniformly in R since∫ t

−∞|k(t+ δ, s) − k(t, s)| ds =

∫ t

−∞

∣∣∣e−(t+δ−s) − e−(t−s)∣∣∣ ds

=∣∣e−δ − 1

∣∣ ∫ t

−∞e−(t−s) ds

=∣∣e−δ − 1

∣∣ .Then, we get the result as a direct application of Corollary 3.2 since (3.3) guarantees that condition (3.2) holds.

Similar existence principles can be obtained for BC(R). We only have to replace conditions (B1)–(B3) by(C) and the space Cω(R) by BC(R) in the above results. We establish one for completeness.

Theorem 3.4 Let p = 1. Assume that (C) and (B4) hold and

|g(s, y)| ≤ φ(s)ψ(|y|)with φ ∈ L∞(R), ψ : R

+ → R continuous and nondecreasing, and

supx∈[0,∞)

x

|f |0 + β‖φ‖∞ψ(x)> 1. (3.4)

Then (1.1) has at least one solution y ∈ BC(R).Example 3.5 Consider the equation

y(t) =sin tet2

+∫ t

−∞

e−(t−s)

|t| + 1g(s, y(s)) ds



where g is a Caratheodory function such that there exists φ ∈ L∞(R) with |g(s, x)| ≤ φ(s)|x| for a.e. s and allx. Then (1.1) has a bounded solution provided that

‖φ‖∞ < 1.

Let p = 1. Condition (C) holds with γ = 2, η = 1 and ξ(t) = t+ 1 because

(|t| + 1)sin tet2

≤ |t| + 1et2

≤ 2

and

supt∈R

(|t| + 1)∫ t

−∞

e−(t−s)

|t| + 1ds =

∫ t

−∞e−(t−s) ds = 1.

Note since

supt∈R

∫ t

−∞

e−(t−s)

|t| + 1ds ≤

∫ t

−∞e−(t−s) ds = 1

then (2.1) holds with β = 1.Finally, we get (B4) from

∫ t

−∞|k(t+ δ, s) − k(t, s)| ds =

∫ t

−∞

∣∣∣∣ e−(t+δ−s)

|t+ δ| + 1− e−(t−s)

|t| + 1

∣∣∣∣ ds=∣∣∣∣ e−δ

|t+ δ| + 1− 1

|t| + 1

∣∣∣∣∫ t

−∞e−(t−s) ds

=∣∣∣∣ e−δ

|t+ δ| + 1− 1

|t| + 1

∣∣∣∣ .4 Existence theory of positive periodic solutions

In this section for clarity and simplicity we shall consider Equation (1.1) with f ≡ 0 and g(s, x) = g(x), i.e.,

y(t) = Ny(t), t ∈ R, (4.1)

with

Ny(t) =∫ t

−∞k(t, s)g(y(s)) ds, t ∈ R.

We assume p = 1, since in the other case g cannot satisfy the conditions at the beginning of Section 2.We look for positive periodic solutions. In this section we assume (B1), (B4) and the following hypotheses

hold:

(D1) g : R → R is continuous, nondecreasing, g(x) ≥ 0 for all x ∈ R, and g(x) > 0 for x > 0.

(D2) k(t, s) > 0 for s ≤ t.

When considering the existence of solution for an integral equation on a compact interval via fixed points inconical shells a common hypothesis in the literature is to assume that the kernel satisfies cφ(s) ≤ k(t, s) ≤ φ(s)with φ a positive function and c ∈ (0, 1]. Roughly speaking, such a condition guarantees that the associatedoperator maps the conical shell into the cone. In our case, such a condition is impossible to achieve so we needto take a different approach. We obtain our result via integral inequalities adapting the ideas of [18].


Math. Nachr. 281, No. 3 (2008) 333

Theorem 4.1 Assume that (B1), (B4) and (D1)–(D2) hold. In addition, suppose that there exist R1, R2 > 0,R1 = R2 such that

(E1) There exists a ∈ Cω(R) with 0 < a(t) ≤ 1, t ∈ R and for any constantR ∈ [minR1, R2,maxR1, R2][N(Ra)](t) ≥ |NR|0 a(t), t ∈ R; (4.2)

(E2) supr∈[0,ω]

∫ r

−∞ k(r, s)g(R1) ds < R1;

(E3)∫ t

−∞ k(t, s)g(R2a(s)) ds > R2 for some t ∈ R.

Then (4.1) has at least one positive periodic solution and either

0 < R1 < |y|0 < R2 and y(t) ≥ a(t)R1 ≥ 0, t ∈ R, if R1 < R2

or

0 < R2 < |y|0 < R1 and y(t) ≥ a(t)R2 ≥ 0, t ∈ R if R2 < R1

holds.

P r o o f. We shall apply the Krasnosel’skii’s fixed point theorem with the following cone

Ka = y ∈ Cω(R) : y(t) ≥ a(t)|y|0where a is the function described in (E1).

Assume in what follows that 0 < R2 < R1 (a similar argument holds if 0 < R1 < R2). Let

ΩR = y ∈ Cω(R) : |y|0 ≤ R.

Now take y ∈ Ka ∩ (ΩR1 \ ΩR2). Then y satisfies 0 < R2 ≤ |y|0 ≤ R1 and R2a(t) ≤ y(t) ≤ R1. Now letR ∈ [R2, R1] be such that |y|0 = R and 0 < Ra(t) ≤ y(t) ≤ R, t ∈ R. Condition (D1) yields

Ny(t) ≥∫ t

−∞k(t, s)g(Ra(s)) ds, t ∈ R,

and since Ny ∈ Cω(R) and g is nondecreasing

|Ny|0 ≤ supr∈[0,ω]

∫ r

−∞k(r, s)g(R) ds.

Combining both of the above inequalities and using (E1) we have that

Ny(t) ≥

∫ t

−∞k(t, s)g(Ra(s)) ds

supr∈[0,ω]

∫ r

−∞k(r, s)g(R) ds

|Ny|0 ≥ a(t)|Ny|0.

Thus Ny ∈ Ka.Recall that by the results in Section 2 we know that N is continuous and completely continuous from Cω into

Cω , so N : Ka ∩ (ΩR1 \ ΩR2) → Ka is continuous and completely continuous.Now, we show that

|Ny|0 < |y|0 for y ∈ Ka ∩ ∂ΩR1

and

|Ny|0 > |y|0 for y ∈ Ka ∩ ∂ΩR2 .



Let y ∈ Ka ∩ ∂ΩR1 . Then |y|0 = R1 and

0 ≤ a(t)R1 ≤ y(t) ≤ R1, t ∈ R.

By (D1) and (E2) we obtain

|Ny|0 = supr∈[0,ω]

|Ny(r)| ≤ supr∈[0,ω]

∫ r

−∞k(r, s)g(R1) ds < R1 = |y|0.

Now let y ∈ Ka ∩ ∂ΩR2 . Then |y|0 = R2 and

0 ≤ a(t)R2 ≤ y(t) ≤ R2, t ∈ R.

From (D1) and (E3) we know there exists t ∈ R with

|Ny|0 ≥∫ t

−∞k(t, s)g(R2a(s)) ds > R2 = |y|0.

Now apply Krasnosel’skii’s fixed point theorem.

Condition (E1) is not easy to check in practice, so we will next discuss sufficient conditions for (E1) to hold.

Corollary 4.2 Suppose that (B1), (B4) and (D1)–(D2) hold. In addition assume that k satisfies for someA > 0

k(t, s) ≥ Ae−(t−s). (4.3)

Also suppose the function g satisfies g(ab) = g(a)g(b) and assume that the differential equation

a′(t) + a(t) − A

kωg(a(t)) = 0 (4.4)

has a periodic solution a(t) with 0 < a(t) ≤ 1; here kω = supr∈[0,ω]

∫ r

−∞ k(r, s) ds. Then (E1) holds.

P r o o f. From (4.3)∫ t

−∞ k(t, s)g(a(s)) dskω

≥∫ t

−∞Ae−(t−s)g(a(s)) dskω

.

Therefore, if we can guarantee that∫ t

−∞Ae−(t−s)g(a(s)) dskω

= a(t), t ∈ R, (4.5)

has a periodic solution such that 0 < a(t) ≤ 1 then we are finished.Differentiating (4.5) we obtain

A

kω

[−et

∫ t

−∞esg(a(s)) ds+ g(a(t))

]= a′(t)

and using (4.5) we arrive at the differential equation

a′(t) + a(t) − A

kωg(a(t)) = 0.

Now, condition (4.4) guarantees the existence of a ∈ Cω(R) such that 0 < a(t) ≤ 1 for t ∈ R.

Corollary 4.3 Suppose that (B1), (B4) and (D1)–(D2) hold. In addition assume that k satisfies (4.3), thefunction g satisfies g(ab) = g(a)g(b) and A

kωg has a fixed point in (0, 1].

Then (E1) holds.


Math. Nachr. 281, No. 3 (2008) 335

P r o o f. It is easy to check that Equation (4.4) has a constant periodic solution provided that the function Akωg

has a fixed point in (0, 1].

Example 4.4 Consider the equation

y(t) =∫ t

−∞(sin t+ 2)e−(t−s)

√y(s) ds. (4.6)

Let k(t, s) = (sin t + 2)e−(t−s) for s ≤ t and g(x) =√x. We easily see that (B1) and (D1)–(D2) hold with

ω = 2π. The function k satisfies condition (B4) since∫ t

−∞|k(t+ δ, s) − k(t, s)| ds =

∫ t

−∞

∣∣(2 + sin(t+ δ))es−t−δ − (2 + sin t)es−t∣∣ ds

≤∫ t

−∞|2 + sin(t+ δ)| ∣∣es−t − es−t−δ

∣∣ ds+∫ t

−∞| sin(t+ δ) − sin(t)| es−t ds

≤ 3 |1 − e−δ| + | sin(t+ δ) − sin t|.Moreover k satisfies (4.3) with A = 1

k(t, s) ≥ e−(t−s) for s ≤ t

and

k2π = supr∈[0,2π]

∫ r

−∞(sin r + 2)e−(r−s) ds = sup

r∈[0,2π]

[sin r + 2] = 3.

Therefore, Akωg(x) = 1

3

√x and 1

9 ∈ (0, 1] is a fixed point of Akωg. Hence, by Corollary 4.3 condition (E1)

holds.Next, taking R1 > 9 we have

supt∈R

∫ t

−∞(2 + sin t)e−(t−s)g(R1) ds ≤ 3g(R1) = 3

√R1 < R1

and (E2) is satisfied.Finally, for t = π

2 we obtain

∫ π2

−∞

(2 + sin π

2

)e−(π

2 −s)g(

19R2

)ds = 3g

(19R2

)=√R2.

Thus taking R2 < 1 we guarantee that (E3) holds and the existence of a 2π-positive solution y for (4.1) with1 ≤ |y|0 ≤ 9 follows from Theorem 4.1.

Remark 4.5 In this section if we want to consider a kernel k satisfying (2.1) with p = 1 we would need toassume g(t, x) = φ(t)h(x) with h satisfying (D1) and φ ∈ Lq(R).

Acknowledgements This research was partially done during a visit of Daniel Franco to the National University of Irelandat Galway. That visit was supported by Secretarıa de Estado de Universidades e Investigacion (Spain).

This research has been supported in part by Ministerio de Educacion y Ciencia (Spain), project MTM2004-06652-C03-03.

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solutions of volterra integral equations with infinite delay

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