# Solvability of a Class of Integral Inclusions

Authors: Ying Chen; Shihuang Hong
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper presents sufficient conditions for the existence of positive solutions for a class of integral inclusions. Our results are obtained via a new fixed point theorem for multivalued operators developed in the paper, in which some nonnegative function is used to describe the cone expansion and compression instead of the classical norm-type, and lead to new existence principles.


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## Body

## 1. Introduction

Let(E,∥·∥) be a Banach space. A nonempty convex closed set  P⊂E is called a cone of E if the following conditions hold:(1.1)x∈P,λ≥0,impliesλx∈P,x∈P,-x∈P,impliesx=θ,whereθ stands for the zero element of E. A cone P is said to be normal if there exists a positive constant N, which is called the normal constant of P, such that θ≤x≤y (x,y∈E) implies that ∥x∥≤N∥y∥. Here, the partially order “≤” in E is introduced as follows: x≤y if and only if y-x∈P for any x, y∈E, x<y if and only if x≤y and x≠y.Given a coneP of E, denote that  P+=P∖{θ}. For u0∈P+, denote that(1.2)P(u0)={x∈P:λu0≤x,forsomeλ>0}.For notational purposes forη>0, let (1.3)Ωη={y∈E:‖y‖<η},∂Ωη={y∈E:‖y‖=η},Ω¯η={y∈E:‖y‖≤η},∂ΩdenotetheboundaryofsetΩ.This paper is concerned with the existence of solutions for the following multivalued integral inclusion:(1.4)x(t)=f(t,x)∫0+∞ux(t,s)ds,
where f:ℝ+×ℝ→ℝ  is a single-valued map, U:H×ℝ⊸2ℝ is a multivalued map, and ux∈SU,x. Here, ℝ+=[0,+∞), H=ℝ+×ℝ+, and the set of L1-selections SU,x of the multivalued map U is defined by(1.5)SU,x∶={fx∈L1(R+,R):fx(t,s)∈U(t,s,x(s))a.e.fort≥0}.Some problems considered in the vehicular traffic theory, biology, and queuing theory lead to the following nonlinear functional-integral equation:(1.6)x(t)=f(t,x(t))∫01u(t,s,x(s))ds,t∈[0,1].(cf. [1]). The Volterra counterpart of the above equation on unbounded interval was studied by [2]. Namely, in [2], the existence of solutions of the following integral equation:(1.7)x(t)=f(t,x(t))∫0tu(t,s,x(s))ds,t≥0was proved by using the technique associated with measures of noncompactness, and the functions were assumed continuous and bounded onℝ+. The sufficient conditions for the existence of solutions to this equation, under the assumption of u being a multivalued map, was presented by [3] via a fixed-point theorem due to Martelli [4] on ordered Banach spaces, [5] via expansion and compression fixed point theorems for multivalued mapping due to Agarwal and O’Regan [6]. When f(t,x)=1, also, [7] established the existence of solutions to the multivalued problem (1.4) in Fréchet spaces. In this paper, we give existence results of positive solutions for system (1.4).The fundamental tool used in the proof of our main results is essentially the fixed point theorem (see Theorem2.3) based on expansion and compression fixed point theorems for multivalued mappings. However, the hypotheses imposed on functions on the right-hand side of (1.4) and methods of the proof in this paper are different from the above-cited works.Cone compression and expansion fixed point theorems are frequently used tools for studying the existence of positive solutions for boundary value problems of integral and differential equations. For instance, in [8–10], authors considered the existence of positive solutions for singular second-order m-point boundary value problem, in [11] Leggett and Williams discussed the nonlinear equation modelling certain infectious diseases. In [12] Zima discussed a three-point boundary value problem for second-order ordinary differential equations. In [13, 14] the authors proved multiplicity of positive radial solutions for an elliptic system on an annulus and so on. The original result of Krasnoselskii fixed point theorem concerning cone compression and expansion was obtained by Krasnoselskii [15]. Afterward, a lot of generalization of this theorem has appeared (see, e.g., [8, 11, 12, 16, 17]). For instance, in [16] Guo and Lashmikantham gave the result of the norm type, and in [17] Anderson and Avery obtained a generalization of the norm type by applying conditions formulated in the terms of two functionals replacing the norm type assumptions. In [8] Zhang and Sun obtained an extension, in which the norm is replayed with some uniformly continuous convex function (see [8], Corollary 2.1). On the other hand, in [11], Leggett and Williams obtained another generalization of Krasnoselskiis original result. In [18] one can find some refinements of [11]. In [12] Zima proved another result via replacing Leggett and Williams type-ordering conditions by the conditions of the norm type (see [12], Theorem 2.1). In addition, Agarwal and O’Regan [6] extended Krasnoselskii's fixed point theorem of norm type to multivalued operator problems and obtained fixed point theorems for k-set contractive multivalued operators (see [6], Theorems 2.4 and 2.8). In general, while the expansion may be easily verified for a large class of nonlinear integral operators, the compression is a rather stringent condition and is usually not easily verified. By improving the compression of the cone theorem via replacing the cone P with the set P(u0), the result of Leggett and Williams [11] has the advantage which consists in its usually being easier to apply even when the compression of the cone theorem is also applicable to a large class of operators. In this paper we will extend Leggett and Williams fixed point theorem to multivalued operator problems and obtain a fixed point theorem for k-set-contractive multivalued operators, in which the norm of [11] will be replayed with some nonnegative function. Our result is not only the fundamental tool to prove our main theorem, but also a generalization of corresponding results in [6, 8, 11, 12].


## 2. Preliminaries

We begin this section with gathering together some definitions and known facts. For two subsetsC, D of E, we write C≤D (or D≥C) if(2.1)∀p∈D,∃q∈Csuchthatq≤p.A multivalued operatorA is called upper semicontinuous (u.s.c.) on E if for each x∈E the set A(x) is a nonempty closed subset of E, and if for each open set B of E containing A(x), there exists an open neighborhood V of x such that A(V)⊆B.A is called a k-set contraction if γ(A(D))≤kγ(D) for all bounded sets D of E and A(D) is bounded, where γ denotes the Kuratowskii measure of noncompactness.Throughout this paper, we denote byCK(C) the family of nonempty, compact, and convex subsets of set C and denote by K∂U(U¯,C) the set of all u.s.c., k-set-contractive maps A:U¯→CK(C) with x∉A(x) for x∈∂U.The nonzero fixed point theorems of multivalued operators (see [6], Theorems 2.3 and 2.7) will play an important role in this section. It is not hard to extend these results on open sets, so we have the following.Lemma 2.1.
LetE be an ordered Banach space and  P a cone in E, and let Ω1 and Ω2 be bounded open sets in E such that θ∈Ω1 and Ω¯1⊂Ω2. Assume that A:Ω¯2→CK(P) is a u.s.c., k-set contractive (here 0≤k<1) map and assume one of the following conditions hold:
(2.2)x∉λAx,∀λ∈[0,1),x∈∂Ω2∩P,(2.3)thereexistsav∈P+withx∉Ax+δvforx∈∂Ω1∩P,δ≥0.
Or
(2.4)x∉λAx,∀λ∈[0,1),x∈∂Ω1∩P,(2.5)thereexistsav∈P+withx∉Ax+δvforx∈∂Ω2∩P,δ≥0.
Then A has at least one fixed point y with y∈(Ω¯2∖Ω1)∩P.Lemma 2.2 (see [19]).
LetE be a Banach space, D a closed convex subset of E, and U an open subset of D with θ∈U. Suppose that A:U¯⊸CK(D) is u.s.c, k-set-contractive (here 0≤k<1). Then either  (h1) there existsx∈U¯ with x∈Ax, or(h2) there existsu∈∂U and λ∈(0,1) with u∈λAx.The proof of the following theorem is not complicated but it is essential to prove our main results.Theorem 2.3.
Assume thatΩ1 and Ω2 are bounded open sets in E such that θ∈Ω1 and Ω¯1⊂Ω2. Let A:P∩Ω¯2→CK(P) be a u.s.c, k-set-contractive (here 0≤k<1) operator, u0∈P+, and ρ:P→[0,+∞) a nondecreasing function with ρ(θ)=0 and ρ(x)>0 for x∈P+. Moreover,  (h)ρ(λx)≤λρ(x), for all x∈P and λ∈[0,1].If one of the following two conditions holds:  (H1)  (i)ρ(y)>ρ(x) for all y∈A(x) and x∈P(u0)∩∂Ω1,(ii)ρ(y)≤ρ(x) for all y∈A(x) and x∈P∩∂Ω2;(H2)  (i)ρ(y)≤ρ(x) for all y∈A(x) and x∈P∩∂Ω1,(ii)ρ(y)>ρ(x) for all y∈A(x) and x∈P(u0)∩∂Ω2, then A has a positive fixed point in the set P∩(Ω¯2∖Ω1).Proof.
We seek to apply Lemma2.1. It is sufficient to check that A satisfies the conditions (2.2) and (2.3) in Ω1 and in Ω2, respectively, provided that the condition (H1) holds. First, (H1)(ii) with x∈P∩∂Ω2 implies that (2.2) is true. To see this suppose that there exist x∈P∩∂Ω2 and λ∈[0,1) with x∈λA(x). Then there exists y∈A(x) with x=λy. Therefore, by the condition (h), we have
(2.6)0<ρ(x)=ρ(λy)≤λρ(y)<ρ(y)≤ρ(x),
a contradiction. Next, we will prove that for any x∈P∩∂Ω1 and any δ≥0,
(2.7)x∉A(x)+δu0.
Suppose, on the contrary, that there exist x0∈P∩∂Ω1 and t≥0 such that x0∈A(x0)+tu0, that is, there exists y0∈A(x0) such that x0=y0+tu0. Hence,
(2.8)x0-tu0=y0∈A(x0).
Clearly, t≠0 (otherwise, this proof is completed). Noting that A(x0)⊂P, we conclude that
(2.9)tu0≤tu0+y
for all y∈A(x0). Then, combining (2.8), we get that x0∈P(u0). Since x0-tu0=y0, we have
(2.10)θ≤y0≤x0,x0≠y0.
In virtue of the monotonicity of ρ, we have
(2.11)ρ(y0)≤ρ(x0).
Since x0∈P(u0)∩∂Ω1, (2.11) contradicts (H1)(i). Hence, (2.7) is true. This implies that (2.3) is true. The result of Theorem 2.3 now follows from Lemma 2.1.
Similarly, we can prove that the result of Theorem2.3 follows if (H2) holds. This proof is completed.Corollary 2.4.
Assume thatΩ1, Ω2 and the multivalued mapping A are given as in Theorem 2.3, u0∈P+, and a function ρ:P→[0,+∞) satisfies the condition (h), ρ(θ)=0 and ρ(x)>0 for x∈P+. Moreover, there exists a constant N>0 such that  (h′)θ≤x≤y with x, y∈E implies that ρ(x)≤Nρ(y). If either  (H′1)  (i)ρ(y)>Nρ(x) for all y∈A(x) and x∈P(u0)∩∂Ω1,(ii)ρ(y)≤ρ(x) for all y∈A(x) and x∈P∩∂Ω2, or(H′2)  (i)ρ(y)≤ρ(x) for all y∈A(x) and x∈P∩∂Ω1,(ii)ρ(y)>Nρ(x) for all y∈A(x) and x∈P(u0)∩∂Ω2 is satisfied, then A has a positive fixed point in P∩(Ω¯2∖Ω1).Proof.
We seek to apply Lemma2.1. The hypothesis (2.2) is true, the proof of which is the same as Theorem 2.3. Next, we will prove that (2.7) is satisfied for any x∈P∩∂Ω1 and any δ≥0. Suppose, on the contrary, that there exist x0∈P∩∂Ω1 and t≥0 such that x0∈A(x0)+tu0, that is, (2.8) holds. Similarly, we have x0∈P(u0) and θ≤y0≤x0 with x0-tu0=y0∈A(x0). In virtue of the condition (h′), we have
(2.12)ρ(y0)≤Nρ(x0).
Since x0∈P∩∂Ω1, (2.12) contradicts (H′1)(i). Hence, (2.7) is true. This shows that the conditions of Lemma 2.1 are satisfied.
Similarly, we can prove that the result of Corollary2.4 follows if (H′2 ) holds. This proof is completed.Remark 2.5.
If the functionρ is convex on P, namely, ρ(tx+(1-t)y)≤tρ(x)+(1-t)ρ(y) for all x, y∈P and t∈[0,1], then the condition (h) holds provided that ρ(θ)=0. From this point of view, we extend the corresponding result of [8]. Let ρ(x)=∥x∥. Then ρ(x) is a convex function with ρ(θ)=0, ρ(x)>0 for x≠0, and the condition (h) is satisfied. Obviously, ρ is nondecreasing if ∥·∥ be increasing with respect to P. This shows that Theorem 2.3 contains the corresponding result of [6]. In addition, the condition (h′) holds if P is a normal cone. Hence, Corollary 2.4 extends and improves the corresponding result of [11].Remark 2.6.
LetE=C[0,1] with the norm ∥φ∥=max0≤t≤1|φ(t)| and the cone
(2.13)P0={φ∈C[0,1]:φ(t)≥0,∀t∈[0,1]}.
Define φ1≤φ2 if and only if φ1(t)≤φ2(t) for every t∈[0,1]. Then the function ρ:P0→[0,+∞) defined by
(2.14)ρ(φ)=(∫01φp(t)dt)1/p,(p≥1)
is nondecreasing convex and ρ(θ)=0, ρ(φ)>0 for φ≠0, and ρ yields the condition (h).
In what follows, we combine Lemma2.2 and Theorem 2.3 to establish existence of multiple fixed points.Theorem 2.7.
Assume that the conditions of Theorem2.3 hold and
(2.15)x∉A(x),∀x∈∂Ω1∩P.
Then A has at least two fixed points x1 and x2 with x1∈Ω1∩P and x2∈P∩(Ω¯2∖Ω1).Proof.
Theorem2.3 guarantees that A has at least one fixed point x2 with x2∈P∩(Ω¯2∖Ω1). In addition, we obtain in the proof of Theorem 2.3 that x∉λA(x) for all λ∈[0,1) and x∈∂Ω1∩P. Hence, we combine (2.15) and Lemma 2.2 to conclude that A has a fixed point x1∈Ω1∩P. This completes the proof of Theorem 2.7.For constantsL, r, R with 0<r<L<R, let us suppose that  (H3)x∉A(x) for all  x∈∂ΩL∩P;(H4) (i)ρ(x)<ρ(y) for all y∈A(x) and x∈P(u0)∩∂Ωr,(ii)ρ(y)≤ρ(x) for all y∈A(x) and x∈P∩∂ΩL,(iii)ρ(x)<ρ(y) for all y∈A(x) and x∈P(u0)∩∂ΩR;(H5) (i)ρ(y)≤ρ(x) for all y∈A(x) and x∈P∩∂Ωr,(ii)ρ(x)<ρ(y) for all y∈A(x) and x∈P(u0)∩∂ΩL,(iii)ρ(y)≤ρ(x) for all y∈A(x) and x∈P∩∂ΩR.Theorem 2.8.
LetA:Ω¯R∩P→CK(P) be a u.s.c., k-set-contractive (here 0≤k<1) operator and the function ρ be given as in Theorem 2.3. If either the conditions (H3) and (H4) or the conditions (H3) and (H5) hold, then A has at least two positive fixed points x1 and x2 with x1∈P∩(ΩL∖Ωr) and x2∈P∩(Ω¯R∖Ω¯L).Proof.
Theorem2.3 implies that A has a fixed point x1∈P∩(Ω¯L∖Ωr). (H3) shows that x1∉∂ΩL. Hence, x1∈P∩(Ω¯L∖Ωr). Again, Theorem 2.3 guarantees the existence of x2. This proof is completed.


## 3. Main Results

In this section, we shall discuss the existence of solutions of integral inclusion (1.4) by using fixed point theorems involved in Section 2. Let us start by defining that a function x∈C(ℝ+,ℝ) is said to be a solution of (1.4) if it satisfies (1.4).ByBC∶=BC(ℝ+,ℝ),  we mean the Banach algebra consisting of all functions defined, bounded, and continuous on ℝ+ with the norm(3.1)‖x‖=sup⁡{|x(t)|:t≥0}.For anyx, y∈BC, define that x≤y if and only if x(t)≤y(t) for each t≥0, x<y, if and only if x≤y and there exists some t≥0 such that x(t)≠y(t).In following Theorem3.1, we need impose the following hypotheses on the single valued map f and the multivalued map U.  (S1)
f:ℝ+×ℝ+→ℝ+ is a continuous function.(S2)
There exists a bounded continuous functiong:ℝ+→ℝ+, such that(3.2)|f(t,x)|≤g(t),foranyt≥0,x≥0.(S3)
There exist positive constants𝒞, δ, ξ with 𝒞>0 and 0<δ<ξ<+∞ such that |f(t,x)|≥𝒞 for t∈[δ,ξ], x≥0.(S4)
U:H×ℝ+→CK(ℝ+) is L1-Carathéodory, that is,(t,s)→U(t,s,x) is measurable for every x∈ℝ+;x→U(t,s,x) is u.s.c. for a.e. (t,s)∈H.In addition, the setSU,x is nonempty for each fixed x∈BC.(S5)
There exist a bounded, continuous, and nondecreasing functionβ:ℝ+→ℝ+, a function α∈L1(ℝ+,ℝ+) with ∫δξα(t)dt>0, and a continuous function γ:[δ,ξ]→(0,+∞) such that(3.3)|ux(t,s)|≤α(s)β(x(s)),fora.e.(t,s)∈H,x≥0,ux∈SU,x;|ux(t,s)|≥γ(t)α(s)β(x(s)),fora.e.(t,s)∈[δ,ξ]×[0,+∞),x≥0,ux∈SU,x;lim⁡ɛ→0+β(ɛ)ɛ=+∞.(S6)
There exists a positive numberR>0 such that β(R)>0 and(3.4)Rβ(R)∫δξg(s)ds∫0+∞α(τ)dτ≥MCM,whereM=supt≥0g(t), ℳ=minδ≤t≤ξγ(t).Theorem 3.1.
If the conditions (S1)–(S6) hold, then (1.4) has at least one (positive) solution x∈BC(ℝ+,ℝ) with x≥0 on ℝ+ and with r<∥x∥≤R for given 0<r<R.Proof.
Let us define the multivalued mapA on the space BC by the following:
(3.5)(A(x))(t)={f(t,x(t))}∫0+∞ux(t,s)ds,ux∈SU,x,t≥0.We will show that A has a fixed point recurring to Theorem 2.3. Define the function ρ:BC(ℝ+,ℝ)→[0,+∞) by
(3.6)ρ(x)=max⁡δ≤t≤ξ|x(t)|
and the set X+ by
(3.7)X+={x∈BC:x(t)≥0,fort≥0,x(t)≥CMM‖x‖,fort∈[δ,ξ]}.
It is easy to see that X+ is a cone of BC and ρ:X+→[0,+∞) given in (3.6) is a nondecreasing convex function with ρ(θ)=0 and ρ(φ)>0 for φ≠θ.
First we point out thatA(x)∈CK(X+) for each fixed x∈Ω¯p∩X+ with p>0. In fact, for any y∈A(x), there exists ux∈SU,x such that y(t)=f(t,x)∫0∞ux(t,s)ds for t≥0. (S1) and (S4) imply that  y(t)≥0 for t≥0 and αβ(x)∈L1(ℝ+,ℝ+), where β(x) is defined by (β(x))(t)=β(x(t)). Applying our assumptions we have the following estimate:
(3.8)|y(t)|≤|f(t,x)|∫0+∞|ux(t,s)|ds≤g(t)∫0+∞α(s)β(x(s))ds≤sup⁡t≥0g(t)∫0∞α(s)β(x(s))ds,
and so(3.9)‖y‖≤M∫0+∞α(s)β(x(s))ds.
In addition (S2) and (S5), together with (3.9), guarantee that
(3.10)y(t)=f(t,s)∫0+∞ux(t,s)ds≥C∫0+∞γ(t)α(s)β(x(s))ds≥Cmin⁡δ≤t≤ξγ(t)∫0+∞α(s)β(x(s))ds≥CMM‖y‖.
This implies that, by the arbitrariness of y, A(x) is bounded and A(x)⊂X+ for each x∈Ω¯p∩X+. Similar to [20] we can infer that A(x) is convex for each x∈Ω¯p∩X+. In the light of our assumptions and the Lebesgue dominated convergence theorem, we can see that A(x) is compact for each x∈Ω¯p∩X+. Hence, A maps Ω¯p∩X+ into CK(X+).
Next, we prove thatA has closed graph. Take xk→x*, hk∈A(xk) and hk→h* as k→∞. We shall prove that h*∈A(x*). hk∈A(xk) means that there exists uk∈SU,xk such that for each t≥0,
(3.11)hk(t)=f(t,xk)∫0+∞uk(t,s)ds,k=1,2,….
Let h*(t)=f(t,x*)h(t). From the continuity of f, it follows that f(t,xk)→f(t,x*). From (S4) it follows that uk→ux*∈SU,x* as k→∞. From (S5) and the Lebesgue dominated convergence theorem it follows that ∫0+∞uk(t,s)ds→∫0+∞ux*(t,s)ds. It is easy to see that ∫0+∞ux*(t,s)ds=h(t), that is, for each t≥0,
(3.12)h*(t)=f(t,x*)∫0+∞ux*(t,s)ds.
This implies thath*∈A(x*). We want to point out that u.s.c. is equivalent to the condition of being a closed graph multivalued map when the map has nonempty compact values; that is, we have shown that A is u.s.c. It is clear that A is a k-set-contractive multivalued map with k=0.
It remains to prove (in virtue of Theorem2.3) that the condition (H1) holds to conclude that A has a fixed point in X+, that is, that (1.4) has a positive solution. Given x∈∂ΩR∩X+ with R satisfying the condition (S6), for any y∈A(x), there exists ux∈SU,x such that y(t)=f(t,x(t))∫0∞ux(t,s)ds, t≥0. Hence,
(3.13)ρ(y)=max⁡δ≤t≤ξ|y(t)|=max⁡δ≤t≤ξ|f(t,x)∫0+∞ux(t,s)ds|≤max⁡δ≤t≤ξg(t)∫0+∞α(s)β(x(s))ds≤β(R)max⁡δ≤t≤ξg(t)∫0+∞α(s)ds.
In addition (S6) shows that
(3.14)β(x(τ))≤β(R)≤CMRMmax⁡δ≤t≤ξ⁡g(t)∫0+∞α(s)ds,
and this together with (3.13) gives the following:
(3.15)ρ(y)≤CMRMmax⁡δ≤t≤ξ⁡g(t)∫0+∞α(s)dsmax⁡δ≤t≤ξg(t)∫0+∞α(s)ds=CMRM≤ρ(x).
Thus, A satisfies condition (H1)(ii).
Take𝒦∶=M/𝒞2ℳminδ≤t≤ξγ(t)∫δξα(s)ds. (S5) shows that there exists a positive number r<R small enough such that
(3.16)β(ɛ)>Kɛ,0<ɛ≤r.
Let u0(t)≡1. To prove that (H1)(i) is true, let y∈Ax with y≠x and x∈∂Ωr∩X+(1). Then there exists ux∈SU,x with y(t)=f(t,x)∫0+∞ux(t,s)ds. In virtue of the definition of X+(1), there exists 0<λ<1 such that
(3.17)λ≤x(t)≤r,t≥0.
Note that there exists η∈[δ,ξ] such that ρ(y)=y(η). Now our assumptions imply that
(3.18)ρ(y)=y(η)=f(η,x)∫0+∞ux(η,s)ds≥C∫0+∞γ(η)α(s)β(x(s))ds≥Cmin⁡δ≤t≤ξγ(t)∫δξα(s)β(x(s))ds>CMmin⁡δ≤t≤ξ⁡γ(t)C2Mmin⁡δ≤t≤ξ⁡γ(t)∫δξα(s)ds∫δξα(s)x(s)ds≥‖x‖≥ρ(x).
So ρ(y)>ρ(x) for all y∈A(x). This shows that (H1)(i) is satisfied. Conclusively, Theorem 2.3 guarantees that A has a fixed point y with r≤∥y∥≤R. This proof is completed.Theorem 3.2.
Suppose that conditions (S1)–(S6) hold. Then (1.4) has at least two positive solutions if the following conditions are satisfied:
(3.19)lim⁡η→+∞β(η)η=+∞.Example 3.3.
Letδ=1/4, ξ=3/4, γ(t)=1/2 for t∈[δ,ξ],
(3.20)α(s)={1,s∈[0,1],s,s=2,3,…,s-2,others,β(x)=x and U(t,s,x)=[(1/2)α(s)x,α(s)x]. Let f(t,x)=e-t(sinx+1), g(t)=2e-t, ℒ=(1/2e), R=32e3/4. It is clear that conditions (S1)–(S6) are satisfied. Hence, Theorem 3.1 guarantees the problem
(3.21)x(t)=e-t(sin⁡x+1)∫0∞ux(t,s)ds
with ux∈[(1/2)α(s)x,α(s)x] having at least a positive solution x with ∥x∥≤32e3/4.



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