On Semilinear Integro-Differential Equations with Nonlocal Conditions in Banach Spaces

Abstract

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set.

1. Introduction

In this paper, we investigate the following problem:𝑥^′(𝑡)=𝐴𝑥(𝑡)+^𝑡₀𝐹()𝑥(𝑠)((𝑡[]𝑥(0)(𝑥)𝑡−𝑠𝑑𝑠+𝑔𝑡,𝑥)),𝑡∈𝐽∶=0,𝑇,+ℎ=𝑥₀. (1.1)(1.2) Here 𝑥(𝑡) takes values in a Banach space 𝑋; 𝐹(𝑡), for each 𝑡∈𝐽, is a linear operator on 𝑋; maps 𝑔∶𝐽×𝑋→𝑋 and ℎ∶𝐶(𝐽;𝑋)→𝑋 are given. In this model, 𝐴 is the generator of a strongly continuous semigroup 𝑆(⋅) on 𝑋.

It is known that (1.1) with 𝑔=𝑔(𝑡) arises from some real applications. For example, the classical heat equation for medium with memory can be written as𝑥_𝑡=𝜕𝑡,𝑦²𝜕𝑦²𝑥𝑡,𝑦+^𝑡₀𝑏(𝑡−𝑠)𝑥𝑠,𝑦𝑑𝑠+𝑔𝑡,𝑦,𝑥0,𝑦=𝑥₀, (1.3) where 𝑡∈ℝ⁺ and 𝑦∈[0,𝑎]⊂ℝ⁺ (for more details, see [1, 2]). In addition, if we replace the initial condition 𝑥(0,𝑦)=𝑥₀ by the nonlocal condition (1.2), it allows to describe the model more effectively. As an example of ℎ, the following function can be considered:ℎ(𝑥)=^𝑝_𝑖=1𝑐_𝑖𝑥𝑡_𝑖, (1.4) where 𝑐_𝑖(𝑖=1,…,𝑝) are given constants and 0≤𝑡₁<⋯<𝑡_𝑝≤𝑇. As another example, one can takeℎ(𝑥)=^𝑝_𝑖=1𝐾_𝑖𝑥𝑡_𝑖, (1.5) where 𝐾_𝑖∶𝑋→𝑋 are given linear operators. Regarding to (1.3), in the case 𝑋=𝐿²(0,𝑎), the operators 𝐾_𝑖 can be given by𝐾_𝑖𝑥𝑡_𝑖,𝑦=^𝑎₀𝑘_𝑖𝜉,𝑦𝑥𝑡_𝑖,𝜉𝑑𝜉, (1.6) where 𝑘_𝑖(𝑖=1,…,𝑝) are continuous kernel functions.

Semilinear problem (1.1)-(1.2) with 𝐹=0 was studied extensively. In [3–5], the existence and uniqueness results were obtained by using the contraction mapping principle, under the Lipschitz conditions imposed on 𝑔 and ℎ. Supposing Carathéodory-type conditions on 𝑔, the authors in [6] proved the global existence result with the assumption that the semigroup 𝑆(𝑡) is compact. However, as it was indicated in [7], if the Lipschitz condition is relaxed, one may get difficulties in proving the compactness of the solution map since the map 𝑡↦𝑆(𝑡), in general, is not uniformly continuous in [0,𝑇], even in case when 𝑆(𝑡) is compact. Recently, Fan and Li [8] gave an asymptotical method to solve this problem for the case when 𝑆(𝑡) is a compact strongly continuous semigroup and the nonlocal function ℎ is supposed to be continuous only.

It is known that, in the case 𝐹=0, the mild solution of (1.1)-(1.2) on 𝐽 is defined via the integral equation𝑥(𝑡)=𝑆(𝑡)𝑥₀−ℎ(𝑥)+^𝑡₀𝑆(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝐽. (1.7) Problem (1.1)-(1.2) involving integro-differential equations was introduced in [2]. The complete references to integro-differential equations can be found in [1, 9, 10]. For some additional problems on solvability and controllability of integro-differential equations, we refer the reader to [11–13]. In order to represent the mild solutions via the variation of constants formula for this case, the notion of so-called resolvent for the corresponding linear equation𝑥^′(𝑡)=𝐴𝑥(𝑡)+^𝑡₀𝐹(𝑡−𝑠)𝑥(𝑠)𝑑𝑠,𝑡∈𝐽 (1.8) can be applied. More precisely, an operator-valued function 𝑅(⋅)∶𝐽↦𝐿(𝑋) is called the resolvent of (7) if it satisfies the following:(1)𝑅(0)=𝐼, the identity operator on 𝑋,(2)for each 𝑣∈𝑋, the map 𝑡↦𝑅(𝑡)𝑣 is continuous on 𝐽,(3)if 𝑌 is the Banach space formed from 𝐷(𝐴), the domain of 𝐴, endowed with the graph norm, then 𝑅(𝑡)∈𝐿(𝑌),𝑅(⋅)𝑦∈𝐶¹(𝐽;𝑋)∩𝐶(𝐽;𝑌) for 𝑦∈𝑌 and𝑑𝑅(𝑡)𝑅(𝑡)𝑑𝑡𝑦=𝐴𝑦+^𝑡₀𝐹()𝑅(𝑠)(𝑡)𝑡−𝑠𝑦𝑑𝑠=𝑅𝐴𝑦+^𝑡₀𝑅(𝑡−𝑠)𝐴𝐹(𝑠)𝑑𝑠,𝑡∈𝐽. (1.9) For the existence of resolvent operators, we refer the reader to [14].

It is worth noting that, from definition of resolvent operator and the uniform boundedness principle, there exists 𝐶_𝑅<+∞ such thatsup_𝑡∈𝐽‖𝑅(𝑡)‖_𝐿(𝑋)≤𝐶_𝑅. (1.10) Then the mild solution on 𝐽 can be represented as𝑥(𝑡)=𝑅(𝑡)𝑥₀−ℎ(𝑥)+^𝑡₀𝑅(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝐽. (1.11) By a similar approach as in [3], the authors in [2] obtained the existence and uniqueness of solutions for (1.11) with the assumptions of the Lipschitz conditions on 𝑔 and ℎ.

In this work, instead of the Lipschitz conditions posed on 𝑔 and ℎ, we assume the regularity of 𝑔 and ℎ expressed in terms of the measure of noncompactness. The mentioned regularity can be considered as a generalization of the Lipschitz condition. We first prove the existence of solutions for (1.1)-(1.2) in Section 2. Our method is to find fixed points of a corresponding condensing map, which yields the existence but does not provide the uniqueness of solutions. The arguments in this work are mainly based on the estimates with measure of noncompactness (MNC estimates). It should be noted that this technique was developed in [15], and it has been employed widely for differential inclusions. In Section 3, we prove that the solution set of our problem is continuously dependent on initial data. Section 4 is devoted to a special case when ℎ is a Lipschitz function and 𝑅(𝑡) is compact for 𝑡>0. We show that, in this case, the solution set to (1.1)-(1.2) has the so-called 𝑅_𝛿-set structure. We end this paper with an example in Section 5.

2. Existence Results

We start with the recalling of some notions and facts (see, e.g. [15, 16]).

Definition 2.1. Let ℰ be a Banach space with power set 𝒫(ℰ), and (𝒜,≥) a partially ordered set. A function 𝛽∶𝒫(ℰ)→𝒜 is called a measure of noncompactness (MNC) in ℰ if 𝛽(coΩ)=𝛽(Ω)foreveryΩ∈𝒫(ℰ), (2.1) where coΩ is the closure of convex hull of Ω. An MNC 𝛽 is called (i)monotone, if Ω₀,Ω₁∈𝒫(ℰ) such that Ω₀⊂Ω₁, then 𝛽(Ω₀)≤𝛽(Ω₁);(ii)nonsingular, if 𝛽({𝑎}∪Ω)=𝛽(Ω) for any 𝑎∈ℰ,Ω∈𝒫(ℰ);(iii)invariant with respect to union with compact sets, if 𝛽(𝐾∪Ω)=𝛽(Ω) for every relatively compact set 𝐾⊂ℰ and Ω∈𝒫(ℰ). If, in addition, 𝒜 is a cone in a normed space, we say that 𝛽 is (iv)algebraically semiadditive, if 𝛽(Ω₀+Ω₁)≤𝛽(Ω₀)+𝛽(Ω₁) for any Ω₀,Ω₁∈𝒫(ℰ);(v)regular, if 𝛽(Ω)=0 is equivalent to the relative compactness of Ω.
An important example of MNC is the Hausdorff MNC, which satisfies all properties given in the previous definition:𝜒(Ω)=inf{𝜀∶Ωhasafinite𝜀-net}. (2.2) Other examples of MNC defined on the space 𝐶(𝐽;𝑋) of continuous functions on an interval 𝐽=[0,𝑇] with values in a Banach space 𝑋 are the following: (i)the modulus of fiber noncompactness:𝛾(Ω)=sup_𝑡∈𝐽𝜒(Ω(𝑡)), (2.3) where 𝜒 is the Hausdorff MNC on 𝑋 and Ω(𝑡)={𝑦(𝑡)∶𝑦∈Ω};(ii)the modulus of equicontinuity:mod_𝐶(Ω)=lim_𝛿→0sup_𝑦∈Ωmax_{𝑡1−𝑡2<𝛿}𝑦𝑡₁−𝑦𝑡₂. (2.4) As indicated in [15], these MNCs satisfy all properties mentioned in Definition 2.1 except the regularity.
Let 𝒯∈𝐿(ℰ), that is, 𝒯 is a bounded linear operator from ℰ into ℰ. We recall the notion of 𝜒-norm (see e.g., [16]) as follows: ‖𝒯‖_𝜒∶=inf𝑀∶𝜒(𝒯Ω)≤𝑀𝜒(Ω),Ω⊂ℰisaboundedset. (2.5) The 𝜒-norm of 𝒯 can be evaluated as ‖𝒯‖_𝜒=𝜒𝒯𝐒₁=𝜒𝒯𝐁₁, (2.6) where 𝐒₁ and 𝐁₁ are the unit sphere and the unit ball in ℰ, respectively. It is easy to see that ‖𝒯‖_𝜒≤‖𝒯‖_𝐿(𝑋). (2.7)

Definition 2.2. A continuous map ℱ∶𝑍⊆ℰ→ℰ is said to be condensing with respect to a MNC 𝛽 (𝛽-condensing) if for every bounded set Ω⊂Z that is not relatively compact, we have 𝛽(ℱ(Ω))_╱≧𝛽(Ω). (2.8)
Let 𝛽 be a monotone nonsingular MNC in ℰ. The application of the topological degree theory for condensing maps (see, e.g., [15, 16]) yields the following fixed point principles.

Theorem 2.3 (cf. [15, Corollary 3.3.1]). Let ℳ be a bounded convex closed subset of ℰ and ℱ∶ℳ→ℳ a 𝛽- condensing map. Then Fixℱ={𝑥=ℱ(𝑥)} is a nonempty compact set.

Theorem 2.4 (cf. [15, Corollary 3.3.3]). Let 𝒱⊂ℰ be a bounded open neighborhood of zero, and ℱ∶𝒱→ℰ a 𝛽- condensing map satisfying the following boundary condition: 𝑥≠𝜆ℱ(𝑥) (2.9) for all x∈𝜕𝒱 and 0<𝜆≤1. Then the fixed point set Fix(ℱ)={𝑥=ℱ(𝑥)}⊂𝒱 is nonempty and compact.

Now, returning to problem (1.1)-(1.2), we impose the following assumptions for 𝑔 and ℎ: (G1)the map 𝑔∶𝐽×𝑋→𝑋 is continuous;(G2)there exist function 𝜇∈𝐿¹(𝐽) and nondecreasing function Υ∶ℝ⁺→ℝ⁺ such that𝑔𝑡,𝜂_𝑋≤𝜇(𝑡)Υ𝜂_𝑋 (2.10)

for a.e. 𝑡∈𝐽 and for all 𝜂∈𝑋;(G3)there exists a function 𝑘∈𝐿¹(𝐽) such that for each nonempty, bounded set Ω⊂𝑋 we have𝜒𝑔(𝑡,Ω)≤𝑘(𝑡)𝜒(Ω) (2.11)

for a.e. 𝑡∈𝐽, where 𝜒 is the Hausdorff MNC in 𝑋;(H1)ℎ∶𝐶(𝐽;𝑋)→𝑋 is a continuous function and there is a nondecreasing function Θ∶ℝ⁺→ℝ⁺ such that‖ℎ(𝑥)‖_𝑋≤Θ‖𝑥‖_𝐶, (2.12)

for all 𝑥∈𝐶(𝐽;𝑋), where ‖𝑥‖_𝐶=‖𝑥‖_{𝐶(𝐽;𝑋)};(H2)there is a constant C_ℎ such that𝜒(ℎ(Ω))≤𝐶_ℎ𝛾(Ω) (2.13)

for any bounded subset Ω⊂𝐶(𝐽;𝑋), where 𝛾 is defined in (2.3).(H3)if Ω⊂𝐶(𝐽;𝑋) is a bounded set, thenmod_𝐶(𝑅(⋅)ℎ(Ω))=0. (2.14)

Remark 2.5. (1) If 𝑋 is a finite dimensional space, one can exclude the hypothesis (G3) since it can be deduced from (G2).
(2) It is known (see, e.g, [15, 16]) that condition (G3) is fulfilled if 𝑔𝑡,𝜂=𝑔₁𝑡,𝜂+𝑔₂𝑡,𝜂, (2.15) where 𝑔₁ is Lipschitz with respect to the second argument: 𝑔₁(𝑡,𝜉)−𝑔₁𝑡,𝜂_𝑋≤𝑘(𝑡)𝜉−𝜂_𝑋 (2.16) for a.e. 𝑡∈𝐽 and 𝜉,𝜂∈𝑋 with 𝑘∈𝐿¹(𝐽) and 𝑔₂ is compact in second argument; that is, for each 𝑡∈𝐽 and bounded Ω⊂𝑋, the set 𝑔₂(𝑡,Ω) is relatively compact in 𝑋.
(3) If we assume that ℎ is completely continuous, that is, it is continuous and compact on bounded sets, then (H2)-(H3) will be satisfied. It is obvious that if the function ℎ in (1.4) obeys (H1)-(H2) and function 𝑡↦𝑅(𝑡) is uniformly continuous, (H3) is also satisfied. It is worth noting that the function ℎ given by (1.5)-(1.6) obeys (H1)–(H3).
As in [2], we assume in the sequel that(F1)𝐹(𝑡)∈𝐿(𝑋) for 𝑡∈𝐽 and for 𝑥(⋅) continuous with values in 𝑌=𝐷(𝐴),𝐴𝐹(⋅)𝑥(⋅)∈𝐿¹(𝐽;𝑋);(F2) for each 𝑥∈𝑋, the function 𝑡↦𝐹(𝑡)𝑥 is continuously differentiable on 𝐽.It is known that under conditions (F1)-(F2), the resolvent operator for (1.8) exists. We assume, in addition, that(HA)𝑡↦𝑅(𝑡) is uniformly norm continuous for 𝑡>0. We define the following operator: Φ∶𝐿¹()(),Φ𝑓𝑡)𝐽;𝑋⟶𝐶𝐽;𝑋(=^𝑡₀𝑅(𝑡−𝑠)𝑓(𝑠)𝑑𝑠. (2.17) Before collecting some properties of Φ, we recall the following definitions.

Definition 2.6. A subset 𝒬 of 𝐿¹(𝐽;𝑋) is said to be integrably bounded if there exists a function 𝜇∈𝐿¹(𝐽) such that 𝑓(𝑡)_𝑋≤𝜇(t)fora.e.𝑡∈𝐽, (2.18) for all 𝑓∈𝒬.

Definition 2.7. The sequence {𝜉_𝑛}⊂𝐿¹(𝐽;𝑋) is called semicompact if it is integrably bounded and the set {𝜉_𝑛(𝑡)} is relatively compact in 𝑋 for a.e. 𝑡∈𝐽.
By using hypothesis (HA) and the same arguments as those in [15, Lemma 4.2.1, Theorem 4.2.2, Proposition 4.2.1, and Theorem 5.1.1], one can verify the following properties for Φ:
(Φ1) the operator Φ sends any integrably bounded set in 𝐿¹(𝐽;𝑋) to equicontinuous set in 𝐶(𝐽;𝑋);(Φ2)the following inequality holds: Φ(𝜉)(𝑡)−Φ𝜂(𝑡)_𝑋≤𝐶_𝑅^𝑡₀𝜉(𝑠)−𝜂(𝑠)_𝑋𝑑𝑠 (2.19) for every 𝜉,𝜂∈𝐿¹(𝐽;𝑋),𝑡∈𝐽;(Φ3)for any compact 𝐾⊂𝑋 and sequence {𝜉_n}⊂𝐿¹(𝐽;𝑋) such that {𝜉_𝑛(𝑡)}⊂𝐾 for a.e. 𝑡∈𝐽, the weak convergence 𝜉_𝑛⇀𝜉 implies the uniform convergence Φ(𝜉_𝑛)→Φ(𝜉);(Φ4)if {𝜉_n}⊂𝐿¹(𝐽;𝑋) is an integrably bounded sequence and 𝑞∈𝐿¹(𝐽) is a nonnegative function such that 𝜒({𝜉_𝑛(𝑡)})≤𝑞(𝑡),fora.e.𝑡∈𝐽, then𝜒Φ𝜉_𝑛(𝑡)≤2𝐶_𝑅^𝑡₀𝑞(𝑠)𝑑𝑠,𝑡∈𝐽; (2.20)(Φ5)if {𝜉_𝑛}⊂𝐿¹(𝐽;𝑋) is a semicompact sequence, then {𝜉_𝑛} is weakly compact in 𝐿¹(𝐽;𝑋) and {Φ(𝜉_𝑛)} is relatively compact in 𝐶(𝐽;𝑋). Moreover, if 𝜉_𝑛⇀𝜉₀, then Φ(𝜉_𝑛)→Φ(𝜉₀). Denote Φ^∗(𝑥)(𝑡)=𝑅(𝑡)𝑥₀−ℎ(𝑥) (2.21) for 𝑡∈𝐽 and 𝑥∈𝐶(𝐽;𝑋). By 𝑁_𝑔 we denote the Nemytskii operator corresponding to the nonlinearity 𝑔, that is, 𝑁_𝑔(𝑥)(𝑡)=𝑔(𝑡,𝑥(𝑡))for𝑡∈𝐽,𝑥∈𝐶(𝐽;𝑋). (2.22) We see that 𝑥 is a solution of (1.1)-(1.2) if and only if 𝑥=Φ^∗(𝑥)+Φ𝑁_𝑔(𝑥). (2.23) Let Ψ(𝑥)=Φ^∗(𝑥)+Φ𝑁_𝑔(𝑥). (2.24) Then the solutions of (1.1)-(1.2) can be considered as the fixed points of Ψ, the operator defined on 𝐶(𝐽;𝑋).
It follows from (G1) and (H1) that Ψ is continuous on 𝐶(𝐽;𝑋). Consider the function 𝜈∶𝒫(𝐶(𝐽;𝑋))⟶ℝ²₊,𝜈(Ω)=max_{(Ω)𝐷∈Δ}𝛾(𝐷),mod_𝐶(𝐷), (2.25) where 𝛾 and mod_C are defined in (2.3) and (2.4), respectively, Δ(Ω) denotes the collection of all countable subsets of Ω, and the maximum is taken in the sense of the ordering in the cone ℝ²₊. By applying the same arguments as in [15], we have that 𝜈 is well defined. That is, the maximum is archiving in Δ(Ω) and so 𝜈 is an MNC in the space 𝐶(𝐽;𝑋), which satisfies all properties in Definition 2.1 (see [15, Example 2.1.3] for details).

Theorem 2.8. Let 𝐹 satisfy (F1)-(F2). Assume that conditions (G1)–(G3) and (H1)–(H3) are fulfilled. If ℓ∶=𝐶_𝑅𝐶_ℎ+2^𝑇₀𝑘(𝑠)𝑑𝑠<1, (2.26) then Ψ is 𝜈-condensing.

Proof. Let Ω⊂𝐶(𝐽;𝑋) be such that 𝜈(Ψ(Ω))≥𝜈(Ω). (2.27) We will show that Ω is relatively compact in 𝐶(𝐽;𝑋). By the definition of 𝜈, there exists a sequence {𝑧_𝑛}⊂Ψ(Ω) such that 𝜈(Ψ(Ω))=𝛾𝑧_𝑛,mod_𝐶𝑧_𝑛. (2.28) Following the construction of Ψ, one can take a sequence {𝑥_𝑛}⊂Ω such that 𝑧_𝑛=Φ^∗𝑥_𝑛+Φ𝑔_𝑛, (2.29) where 𝑔_𝑛(𝑡)=𝑔𝑡,𝑥_𝑛(𝑡)Φ,𝑡∈𝐽,_∗𝑥_𝑛(𝑡)=𝑅(𝑡)𝑥₀−ℎ𝑥_𝑛,Φ𝑔_𝑛(𝑡)=^𝑡₀𝑅(𝑡−𝑠)𝑔_𝑛(𝑠)𝑑𝑠. (2.30) Using assumption (G3), we have 𝜒𝑔_𝑛(𝑠)=𝜒𝑔𝑠,𝑥_𝑛(𝑠)≤𝑘(𝑠)𝜒𝑥_𝑛(𝑠(𝑠)𝛾𝑥)≤𝑘_𝑛, (2.31) for all 𝑠∈𝐽. Then by (Φ4), we obtain 𝜒Φ𝑔_𝑛(𝑡)≤2𝐶_𝑅^𝑡₀𝑘(𝑠)𝑑𝑠𝛾𝑥_𝑛. (2.32) Noting that Φ^∗𝑥_𝑛(𝑡)=𝑅(𝑡)𝑥₀−𝑅(𝑡)ℎ𝑥_𝑛, (2.33) we have 𝜒Φ^∗𝑥_𝑛(𝑡)=𝜒𝑅(𝑡)ℎ𝑥_𝑛≤𝐶_𝑅𝐶_ℎ𝛾𝑥_𝑛 (2.34) due to (2.5)-(2.7) and (H2). Combining (2.29), (2.31), and (2.32), we get 𝛾𝑧_𝑛≤ℓ𝛾𝑥_𝑛. (2.35) Combining the last inequality with (2.27), we have 𝛾𝑥_𝑛≤ℓ𝛾𝑥_𝑛, (2.36) and therefore 𝛾𝑥_𝑛=0. (2.37) But then (2.35) implies 𝛾𝑧_𝑛=0. (2.38) Putting (2.37) together with (2.31), we obtain that {𝑔_𝑛} is semicompact. Hence, by (Φ5) one that has {Φ(𝑔_𝑛)} is relatively compact. This yields mod_𝐶Φ𝑔_𝑛=0. (2.39) By (H3), we have mod_𝐶Φ^∗𝑥_𝑛=0. (2.40) Taking (2.29) into account again, we obtain mod_𝐶𝑧_𝑛=0. (2.41) Now it follows from (2.38)-(2.41) that 𝜈(Ω)=(0,0). (2.42) By regularity of 𝜈, we come to the conclusion that Ω is relatively compact.

Remark 2.9. If 𝑅(𝑡) is compact for 𝑡>0, we can drop assumption (G3) in the foregoing theorem. Indeed, for any bounded sequence {𝑥_𝑛}⊂𝐶(𝐽;𝑋), by setting 𝜉_𝑛(𝑡,𝑠)=𝑅(𝑡−𝑠)𝑔(𝑠,𝑥_𝑛(𝑠)), one sees that under hypothesis (G2), {𝜉_𝑛(𝑡,⋅)} is an integrably bounded sequence in 𝐿¹(0,𝑡;𝑋). In addition, since 𝑅(𝑡),𝑡>0, is compact, we have 𝜒𝜉_𝑛(𝑡,𝑠)=0,fora.e.𝑠∈[0,𝑡]. (2.43) Then by [15, Corollary 4.2.5], we obtain 𝜒^𝑡₀𝜉_𝑛(𝑡,𝑠)𝑑𝑠=0, (2.44) for each 𝑡∈𝐽. By this reason, inequality (2.32) becomes 𝜒Φ𝑔_𝑛(𝑡)=0, (2.45) without the reference to (G3).

We now can formulate the existence result in the following way.

Theorem 2.10. Under assumptions of Theorem 2.8, if one has liminf_𝑟→∞𝐶_𝑅𝑟Θ(𝑟)+Υ(𝑟)^𝑇₀𝜇(𝑠)𝑑𝑠<1, (2.46) then the solution set to problem (1.1)-(1.2) is nonempty and compact.

Proof. We will use Theorem 2.3. Applying the results of Theorem 2.8, we only need to verify the existence of a number 𝑟>0 such that Ψ𝐵_𝑟⊆𝐵_𝑟, (2.47) where 𝐵_𝑟 is the closed ball in 𝐶(𝐽;𝑋) centered at origin with radius 𝑟. Indeed, assume to the contrary that for each 𝑛∈ℕ⧵{0}, there is 𝑥_𝑛∈𝐶(𝐽;𝑋) such that 𝑥_𝑛𝐶≤𝑛,butΨ𝑥_𝑛𝑐>𝑛. (2.48) Recalling that Ψ𝑥_𝑛(𝑡)=𝑅(𝑡)𝑥₀−ℎ𝑥_𝑛+^𝑡₀𝑅(𝑡−𝑠)𝑔𝑠,𝑥_𝑛(𝑠)𝑑𝑠, (2.49) we have Ψ𝑥_𝑛(𝑡)_𝑋≤𝐶_𝑅𝑥_0𝑋+Θ𝑥_𝑛𝐶+𝐶_𝑅^𝑡₀𝜇(𝑠)Υ𝑥_𝑛(𝑠)_𝑋𝑑𝑠, (2.50) due to (H1) and (G2). Then 𝑛<Ψ𝑥_𝑛𝐶≤𝐶_𝑅𝑥_0𝑋+Θ(𝑛)+𝐶_𝑅Υ(𝑛)^𝑇₀𝜇(𝑠)𝑑𝑠. (2.51) Equivalently, Ψ𝑥1<^𝑛𝐶𝑛≤1𝑛𝐶_𝑅𝑥_0𝑋+Θ(𝑛)+𝐶_𝑅Υ(𝑛)^𝑇₀𝜇(𝑠)𝑑𝑠. (2.52) Passing in the last inequality to the limit as 𝑛→+∞, one gets a contradiction due to assumption (2.46). Thus the proof is completed.

We have some special cases related to the growth of Υ and Θ.

Corollary 2.11. Assume hypotheses of Theorem 2.8, in which (G2) and (H1) are replaced by(G2′)‖𝑔(𝑡,𝜂)‖_𝑋≤𝜇(𝑡)(1+‖𝜂‖^𝑝),𝜇∈𝐿¹(𝐽),0≤𝑝<1, for all (𝑡,𝜂)∈𝐽×𝑋;(H1′) ℎ∶C(J;X)→X is continuous and ‖ℎ(𝑥)‖_𝑋≤ℎ₀+ℎ₁‖𝑥‖^𝑞_𝑐,ℎ₀,ℎ₁>0,0≤𝑞<1, (2.53) for all 𝑥∈𝐶(𝐽;𝑋), respectively.Then the solution set to problem (1.1)-(1.2) is nonempty and compact.

Proof. Since 𝑝<1 and 𝑞<1, condition (2.46) in Theorem 2.10 is testified obviously. Then we get the conclusion.

Corollary 2.12. Assume hypotheses of Theorem 2.8, in which (G2) and (H1) are replaced by(G2′′)‖𝑔(𝑡,𝜂)‖_𝑋≤𝜇(𝑡)(1+‖𝜂‖),𝜇∈𝐿¹(𝐽), for all (𝑡,𝜂)∈𝐽×𝑋;(H1′′)ℎ∶𝐶(𝐽;𝑋)→𝑋 is continuous and ‖ℎ(x)‖_X≤ℎ₀+ℎ₁‖x‖_C,forsomeℎ₀,ℎ₁>0, (2.54) for all 𝑥∈𝐶(𝐽;𝑋), respectively. If one has C_Rℎ₀+^T₀𝜇(s)ds<1, (2.55) then the solution set to problem (1)-(2) is nonempty and compact.

Proof. Under (G2′′) and (H1′′), condition (2.55) is equivalent to (2.46) and the conclusion of Theorem 2.10 holds.

It should be mentioned that if 𝑞=0 in (H1′), that is, the nonlocal function ℎ is uniformly bounded, then one can relax the growth of Υ, by the arguments similar to those in [17].

Theorem 2.13. Assume the hypotheses of Theorem 2.8, in which (H1) is replaced by(H1b)ℎ is a continuous function and ||ℎ(𝑥)||_𝑋≤𝑀_ℎ for all 𝑥∈𝐶(𝐽;𝑋), where 𝑀_ℎ is a positive constant.If one has 𝐶_𝑅^𝑇₀𝜇(𝑠)𝑑𝑠<^∞_𝑀𝑑𝑧Υ(𝑧), (2.56) where 𝑀=𝐶_𝑅(||𝑥₀||_𝑋+𝑀_ℎ), then the solution set to problem (1.1)-(1.2) is nonempty and compact.

Proof. In this case we employ Theorem 2.4. It suffices to verify the boundary condition in Theorem 2.4. We show that if 𝑥=𝜆Ψ(𝑥) for 𝜆∈(0,1], then 𝑥 must belong to a bounded set. Indeed, suppose 𝑥(𝑡)=𝜆𝑅(𝑡)𝑥₀−ℎ(𝑥)+𝜆^𝑡₀𝑅(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠. (2.57) It follows that ‖𝑥(𝑡)‖_𝑋≤𝐶_𝑅𝑥_0𝑋+𝑀_ℎ+𝐶_𝑅^𝑡₀𝜇(𝑠)Υ‖𝑥(𝑠)‖_𝑋𝑑𝑠. (2.58) Putting 𝑣(𝑡)=𝐶_𝑅𝑥_0𝑋+𝑀_ℎ+𝐶_𝑅^𝑡₀𝜇(𝑠)Υ‖𝑥(𝑠)‖_𝑋𝑑𝑠, (2.59) we have ‖𝑥(𝑡)‖_𝑋≤𝑣(𝑡), for all 𝑡∈𝐽, and 𝑣^′(𝑡)=𝐶_𝑅𝜇(𝑡)Υ‖𝑥(𝑡)‖_𝑋≤𝐶_𝑅𝜇(𝑡)Υ(𝑣(𝑡)), (2.60) due to the fact that Υ is nondecreasing. Then, by using (2.56), we have _𝑀^𝑣(𝑡)𝑑𝑧Υ(𝑧)≤𝐶_𝑅^𝑇₀𝜇(𝑠)𝑑𝑠<^∞_𝑀𝑑𝑧Υ(𝑧), (2.61) for all 𝑡∈𝐽. The last inequalities imply that sup_𝑡∈𝐽𝑣(𝑡) is bounded, so is ‖𝑥‖_𝐶.

3. Continuous Dependence Result

We start with some notions from the theory of multivalued maps (see, e.g. [15] for details).

Let (𝑌,𝜚_𝑌) and (𝑍,𝜚_𝑍) be metric spaces; 𝒦(𝑍) denotes the collection of all nonempty compact subsets of 𝑍. A multivalued map (multimap) 𝐺∶𝑌→𝒦(𝑍) is said to be (i)  upper semicontinuous (u.s.c.) if for each 𝑦∈𝑌 and 𝜖>0 there exists 𝛿=𝛿(𝑦,𝜖)>0 such that condition 𝜚_𝑌(𝑦,𝑦^′)<𝛿 implies 𝐺(𝑦^′)⊂𝑈_𝜖(𝐺(𝑦)), where 𝑈_𝜖(𝐺(𝑦)) denotes the 𝜖-neighborhood of the set 𝐺(𝑦) induced by the metric 𝜚_𝑍;(ii)  closed if its graph {(𝑦,𝑧)∈𝑌×𝑍∶𝑧∈𝐺(𝑦)} is a closed subset of 𝑌×𝑍;(iii)  compact if 𝐺(𝑌) is relatively compact in 𝑍;(iv)  quasicompact if its restriction to any compact set is compact.

The following assertion gives a sufficient condition for upper semicontinuity.

Lemma 3.1 (see[15]). Let 𝐺∶𝑌→𝒦(𝑍) be a closed quasicompact multimap. Then 𝐺 is u.s.c.

Consider the solution multimap(),𝑊(𝑣)=()-()𝑊∶𝑋⊸𝐶𝐽;𝑋𝑥∶𝑥isasolutionof1.11.2withinitialvalue𝑥₀=𝑣. (3.1) Notice that, as we demonstrated previously, under conditions of our existence theorems, the solution multimap 𝑊 has compact values. We will study the continuity properties of 𝑊.

Theorem 3.2. Under the assumptions of Theorem 2.10, the solution map 𝑊 defined in (3.1) is u.s.c.

Proof. We first prove that 𝑊 is a quasicompact multimap. Let 𝒬⊂𝑋 be a compact set. We will show that 𝑊(𝒬)  is relatively compact in 𝐶(𝐽;𝑋). Suppose that {𝑥_𝑛}⊂W(𝒬). Then there exists a sequence {𝑣_𝑛}⊂𝒬 such that 𝑥_𝑛(𝑡)=𝑅(𝑡)𝑣_𝑛−𝑅(𝑡)ℎ𝑥_𝑛+Φ𝑔_𝑛(𝑡), (3.2) where 𝑔_𝑛(𝑡)=𝑔(𝑡,𝑥_𝑛(𝑡)).
Notice that the sequence {𝑥_𝑛} is bounded. In fact, from (3.2) we have the estimate 𝑥_𝑛𝑐≤𝐶_𝑅𝑣_𝑛𝑥+Θ𝑥_𝑛𝑐+𝐶_𝑅Υ𝑥_𝑛𝑐^𝑇₀𝜇(𝑠)𝑑𝑠. (3.3) Supposing to the contrary that the sequence ‖𝑥_𝑛‖_𝐶 is unbounded, by dividing the last inequality over ‖𝑥_𝑛‖_𝐶 and using condition (2.46) and the boundedness of the sequence {𝑣_𝑛}, we get a contradiction.
Since {𝑣_𝑛} is relatively compact, we obtain from (3.2) that 𝜒𝑥_𝑛(𝑡)≤𝜒𝑅(𝑡)ℎ𝑥_𝑛+𝜒Φ𝑔_𝑛(𝑡). (3.4) Using (G3) we have 𝜒𝑔_𝑛(𝑠)≤𝑘(𝑠)𝜒𝑥_𝑛(𝑠)≤𝑘(𝑠)𝛾𝑥_𝑛 (3.5) for all 𝑠∈𝐽. Referring to (Φ4), one gets 𝜒Φ𝑔_𝑛(𝑠)≤2𝐶_𝑅^𝑡₀𝑘(𝑠)𝑑𝑠𝛾𝑥_𝑛, (3.6) and then 𝛾Φ𝑔_𝑛≤2𝐶_𝑅^𝑇₀𝑘(𝑠)𝑑𝑠𝛾𝑥_𝑛. (3.7) On the other hand, by (H2) one has 𝜒𝑅(𝑡)ℎ𝑥_𝑛≤𝐶_𝑅𝐶_ℎ𝛾𝑥_𝑛. (3.8) Combining the last inequality with (3.4)-(3.7), we have 𝛾𝑥_𝑛≤ℓ𝛾𝑥_𝑛. (3.9) This leads to the conclusion that 𝛾({𝑥_𝑛})=0.
Now, condition (G2) implies that {𝑔_𝑛} is integrably bounded in 𝐿¹(𝐽;𝑋). Thus (Φ1) ensures that {Φ(𝑔_𝑛)} is equicontinuous. Then applying condition (H3), we obtain mod_𝐶𝑥_𝑛≤mod_𝐶Φ^∗𝑥_𝑛+mod_𝐶Φ𝑔_𝑛=0. (3.10) So we have 𝜈({𝑥_𝑛})=(0,0) and therefore {𝑥_𝑛} is relatively compact in 𝐶(𝐽;𝑋).
In order to prove that 𝑊 is u.s.c., it remains, according to Lemma 3.1, to show that 𝑊 is closed. Let 𝑣_𝑛→𝑣 in 𝑋 and 𝑥_𝑛∈𝑊(𝑣_𝑛),𝑥_𝑛→𝑥 in 𝐶(𝐽;𝑋). We claim that 𝑥∈𝑊(𝑣). Indeed, one has 𝑥_𝑛(𝑡)=Φ^∗𝑥_𝑛(𝑡)+^𝑡₀𝑅(𝑡−𝑠)𝑔𝑠,𝑥_𝑛(𝑠)𝑑𝑠. (3.11) We first observe that Φ^∗𝑥_𝑛=𝑅(⋅)𝑣_𝑛−ℎ𝑥_𝑛⟶𝑅(⋅)[𝑣−ℎ(𝑥)]=Φ^∗(𝑥) (3.12) in 𝐶(𝐽;𝑋) in accordance with (H1). In addition, since 𝑔 is a continuous function, we have 𝑔(𝑠,𝑥_𝑛(𝑠))→𝑔(𝑠,𝑥(𝑠)) a.e. 𝑠∈𝐽. The Lebesgue dominated convergence theorem implies that 𝑔⋅,𝑥_𝑛(⋅)−𝑔(⋅,𝑥(⋅))⟶0in𝐿¹(𝐽;𝑋) (3.13) due to the fact that {𝑔(⋅,𝑥_𝑛(⋅))} is integrably bounded. Therefore, taking (3.11) into account, we arrive at 𝑥(𝑡)=Φ^∗(𝑥)(𝑡)+^𝑡₀𝑅(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝐽. (3.14) The proof is completed.

4. Lipschitz Assumption for the Function from Nonlocal Condition

4.1. Existence Result

In this section, we assume that ℎ is a Lipschitz function.(H2’) There exists a constant ℎ₀>0 such thatℎ(𝑥)−ℎ𝑦_𝑋≤ℎ₀𝑥−𝑦_𝐶. (4.1)

This implies the growth of ℎ:‖ℎ(𝑥)‖_𝑋≤ℎ₀‖𝑥‖_𝐶+‖ℎ(0)‖_𝑋, (4.2) and the last inequality covers (H1).

Let 𝜒_𝐶 be the Hausdorff MNC in 𝐶(𝐽;𝑋). We haveΦ^∗(𝑥)(𝑡)−Φ^∗𝑦(𝑡)_𝑋=𝑅(𝑡)ℎ(𝑥)−ℎ𝑦_𝑋≤𝐶_𝑅ℎ₀𝑥−𝑦_𝐶, (4.3) for all 𝑡∈𝐽, where Φ^∗ is given in (2.21). ThusΦ^∗(𝑥)−Φ^∗𝑦_𝐶≤𝐶_𝑅ℎ₀𝑥−𝑦_𝐶. (4.4) Then we know that (see [15]) condition (H2’) implies𝜒_𝐶Φ^∗(Ω)≤𝐶_𝑅ℎ₀𝜒_𝐶(Ω), (4.5) for any bounded set Ω⊂𝐶(𝐽;𝑋). We recall the following facts, which will be used in the sequel: for each bounded set Ω⊂𝐶(𝐽;𝑋), one has the following:(i)𝜒(Ω(𝑡))≤𝜒_𝐶(Ω), for all 𝑡∈𝐽;(ii)if Ω is an equicontinuous set (mod_𝐶(Ω)=0), then𝜒_𝐶(Ω)=sup_𝑡∈𝐽𝜒(Ω(𝑡))=𝛾(Ω). (4.6)

Theorem 4.1. Assume that 𝑔 satisfies (G1)–(G3) and ℎ obeys (H2’). If the following relations ℓ∶=𝐶_𝑅ℎ₀+2^𝑇₀𝑘(𝑠)𝐶𝑑𝑠<1,_𝑅ℎ₀+𝐶_𝑅^𝑇₀𝜇(𝑠)𝑑𝑠liminf_𝑟→∞Ψ(𝑟)𝑟<1, (4.7)(4.8) hold true, then problem (1.1)-(1.2) has at least one solution.

Proof. As we know from the proof of Theorem 2.10, condition (4.8) implies that there exists a ball 𝐵_𝑟⊂𝐶(𝐽,𝑋), 𝑟>0, such that Ψ𝐵_𝑟⊂𝐵_𝑟. (4.9) To apply Theorem 2.3, we verify that Ψ is 𝜒_𝐶-condensing. Let Ω⊂𝐶(𝐽;𝑋) be a bounded set satisfying the inequality 𝜒_𝐶(Ψ(Ω))≥𝜒_𝐶(Ω). (4.10) We will show that Ω is relatively compact. Notice that Ψ(Ω)⊂Φ^∗(Ω)+Φ∘𝑁_𝑔(Ω), (4.11) where Φ^∗(Ω)(𝑡)=𝑅(𝑡)𝑥₀−ℎ(Ω),Φ∘𝑁_𝑔(Ω)(𝑡)=^𝑡₀𝑅(𝑡−𝑠)𝑔𝑠,𝑦(𝑠)𝑑𝑠∶𝑦∈Ω. (4.12) Then we have 𝜒_𝐶(Ψ(Ω))≤𝜒_𝐶Φ^∗(Ω)+𝜒_𝐶Φ∘𝑁_𝑔(Ω). (4.13)
The boundedness of Ω in 𝐶(𝐽;𝑋) implies that 𝑁_𝑔(Ω) is a bounded set in 𝐿¹(𝐽;𝑋). By (Φ1), the set Φ∘𝑁_𝑔(Ω) is equicontinuous and therefore we have𝜒_𝐶Φ∘𝑁_𝑔(Ω)=sup_𝑡∈𝐽𝜒Φ∘𝑁_𝑔(Ω)(𝑡)≤sup_𝑡∈𝐽2𝐶_𝑅^𝑡₀𝑘(𝑠)𝜒(Ω(𝑠))𝑑𝑠, (4.14) due to (G3) and (Φ4). Thus 𝜒_𝐶Φ∘𝑁_𝑔(Ω)≤2𝐶_𝑅𝜒_𝐶(Ω)^𝑇₀𝑘(𝑠)𝑑𝑠. (4.15) Combining (4.5), (4.13), and (4.15), we obtain 𝜒_𝐶(Ψ(Ω))≤𝐶_𝑅ℎ₀+2^𝑇₀𝑘(𝑠)𝑑𝑠𝜒_𝐶(Ω). (4.16) Relations (4.7) and (4.10) yield 𝜒_𝐶(Ω)≤ℓ𝜒_𝐶(Ω). (4.17) Since ℓ<1, we have 𝜒_𝐶(Ω)=0. The regularity of 𝜒_𝐶 ensures that Ω is relatively compact.

Remark 4.2. (1) Assumption (H2^’) allows us to drop (H1)–(H3).(2) As indicated in Remark 2.9, in the case when 𝑅(𝑡) is compact for 𝑡>0, condition (G3) can be dropped and condition (4.7) is reduced to 𝐶_𝑅ℎ₀<1, (4.18) which is covered by (4.8). Recall that in this case we have 𝜒Φ∘𝑁_𝑔𝑥_𝑛(𝑡)=0 (4.19) for any bounded sequence {𝑥_𝑛}⊂𝐶(𝐽;𝑋) and for all 𝑡∈𝐽.

4.2. The Structure of the Solution Set

We are in a position to study the structure of the solution set to (1.1)-(1.2) under the hypotheses of Theorem 4.1 and the assumption that 𝑅(𝑡),𝑡>0, is compact. At first, let us recall some notions.

Definition 4.3. A subset 𝐵 of a metric space 𝑌 is said to be contractible in 𝑌 if the inclusion map 𝑖_𝐵∶𝐵→𝑌 is null-homotopic; that is, there exist 𝑦₀∈𝑌 and a continuous map ℎ∶𝐵×[0,1]→𝑌 such that ℎ(𝑦,0)=𝑦 and ℎ(𝑦,1)=𝑦₀ for every 𝑦∈𝐵.

The following notion [18] is important for our purposes.

Definition 4.4. Let 𝑌 be a metric space; a subset 𝐵⊂𝑌 is called an 𝑅_𝛿-set if 𝐵 can be represented as the intersection of a decreasing sequence of compact contractible sets.

The next lemma gives us a tool for the recognition of 𝑅_𝛿-set.

Lemma 4.5 (see [19]). Let 𝑋 be a metric space, 𝐸 a Banach space, and 𝑉∶𝑋→𝐸 a proper map; that is, 𝑉 is continuous and 𝑉⁻¹(𝐾) is compact for each compact set 𝐾⊂𝐸. Suppose that there exists a sequence {𝑉_𝑛} of mappings from 𝑋 into 𝐸 such that (1)𝑉_𝑛 is proper and {𝑉_𝑛} converges to 𝑉 uniformly on 𝑋;(2)for a given point 𝑦₀∈𝐸 and for all 𝑦 in a neighborhood 𝒩(𝑦₀) of 𝑦₀ in 𝐸, there exists exactly one solution 𝑥_𝑛 of the equation 𝑉_𝑛(𝑥)=𝑦. Then 𝑉⁻¹(𝑦₀) is an 𝑅_𝛿-set.

In order to use this lemma, we need the following result, which is called Lasota-Yorke Approximation Theorem (see, e.g. [20]).

Lemma 4.6. Let 𝐸 be a normed space, 𝑋 a metric space, and 𝑓∶𝑋→𝐸 a continuous map. Then for each 𝜖>0, there is a locally Lipschitz map 𝑓_𝜖∶𝑋→𝐸 such that 𝑓_𝜖(𝑥)−𝑓(𝑥)_𝐸<𝜖,forevery𝑥∈𝑋. (4.20)

We now can formulate the main result of this section.

Theorem 4.7. Assume that 𝑔 satisfies (G1)-(G2) and ℎ obeys (H2′). If 𝑅(𝑡) is compact for 𝑡>0 and 𝐶_𝑅ℎ₀+𝐶_𝑅^𝑇₀𝜇(𝑠)𝑑𝑠liminf_𝑟→∞Υ(𝑟)𝑟<1, (4.21) then the solution set of problem (1.1)-(1.2) is an 𝑅_𝛿-set.

Proof. By Theorem 4.1 and Remark 4.2, the hypotheses of Theorem 4.7 provide the existence result, that is, Fix(Ψ)≠∅. (4.22) We will show that Fix(Ψ) is an 𝑅_𝛿-set.
Consider the nonlinearity 𝑔. By Lemma 4.6, there exists a sequence of functions {𝑔_𝑛} such that(i)𝑔_𝑛∶𝐽×𝑋→𝑋 is continuous and locally Lipschitz map;(ii)‖𝑔_𝑛(𝑡,𝜂)−𝑔(𝑡,𝜂)‖_𝑋<𝜖_𝑛 for all (𝑡,𝜂)∈𝐽×𝑋, where 𝜖_𝑛→0 as 𝑛→∞. One can assume, without loss of generality, that 𝑔_𝑛𝑡,𝜂_𝑋≤𝜇(𝑡)Υ𝜂_𝑋+1 (4.23) for all (𝑡,𝜂)∈𝐽×𝑋 and 𝑛≥1. Let us consider the following equation: 𝑥(𝑡)=𝑦(𝑡)+𝑅(𝑡)𝑥₀−ℎ(𝑥)+^𝑡₀𝑅(𝑡−𝑠)𝑔_𝑛(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝐽, (4.24) where 𝑦∈𝐶(𝐽;𝑋) is a given function. Define Ψ_𝑛∶𝐶(𝐽;𝑋)→𝐶(𝐽;𝑋) by Ψ_𝑛(𝑥)(𝑡)=𝑦(𝑡)+𝑅(𝑡)𝑥₀−ℎ(𝑥)+^𝑡₀𝑅(𝑡−𝑠)𝑔_𝑛(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝐽. (4.25)
By applying the same arguments as in the proof of Theorem 4.1 and in Remark 4.2, we can see that Ψ_𝑛 is 𝜒_𝐶-condensing. In addition, using the similar estimates as in Theorem 2.10, one can find a ball 𝐵_𝑟, 𝑟>0, such that Ψ_𝑛𝐵_𝑟⊂𝐵_𝑟, (4.26) due to (4.23). Therefore, Ψ_𝑛 has a fixed point due to Theorem 2.3 and then (4.24) has at least one solution. Moreover, since ℎ(⋅) is Lipschitzian and 𝑔_𝑛(𝑡,⋅) is a locally Lipschitz function, the solution to (4.24) is unique.
Now by setting 𝑉_𝑛(𝑥)(𝑡)=𝑥(𝑡)−𝑅(𝑡)𝑥₀−ℎ(𝑥)+^𝑡₀𝑅(𝑡−𝑠)𝑔_𝑛((𝑠,𝑉(𝑥𝑡)(𝑡)−𝑅(𝑡)𝑥𝑠,𝑥))𝑑𝑠)(=𝑥₀−ℎ(𝑥)+^𝑡₀𝑅(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝐽, (4.27) we see that {𝑉_𝑛} converges to 𝑉 uniformly on 𝐶(𝐽;𝑋). In addition, for a given 𝑦∈𝐶(𝐽;𝑋) the equation 𝑉_𝑛(𝑥)=𝑦 (4.28) has a unique solution, which is the fixed point of Ψ_𝑛 mentioned previously.
We now show that 𝑉 and 𝑉_𝑛 are proper. We proceed with 𝑉_𝑛; the proof for 𝑉 is similar. Obviously, 𝑉_𝑛 is continuous. Let 𝐾⊂𝐶(𝐽;𝑋) be a compact set and 𝑉_𝑛(Ω)=𝐾. We claim that Ω is a compact set in 𝐶(𝐽;𝑋). Since 𝑉_𝑛 is continuous and 𝐾 is closed, we deduce that Ω is closed. Assume that {𝑥_𝑗} is a sequence in Ω, then one can take a sequence {𝑦_𝑗}⊂𝐾 such that 𝑉_𝑛𝑥_𝑗=𝑦_𝑗. (4.29) Equivalently, 𝑥_𝑗(𝑡)=𝑦_𝑗(𝑡)+𝑅(𝑡)𝑥₀−ℎ𝑥_𝑗+^𝑡₀𝑅(𝑡−𝑠)𝑔_𝑛𝑠,𝑥_𝑗(𝑠)𝑑𝑠,𝑡∈𝐽. (4.30) We first show that the sequence {𝑥_𝑗} is bounded. We have 𝑥_𝑗(𝑡)_𝑋≤𝑦_𝑗(𝑡)_𝑋+𝐶_𝑅𝑥_0𝑥+ℎ₀𝑥_𝑗𝐶+‖ℎ(0)‖_𝑋+𝐶_𝑅^𝑡₀𝜇(𝑠)Υ𝑥_𝑗(𝑠)_𝑋+1𝑑𝑠, (4.31) due to (H2^’) and (4.23). Thus 𝑥_𝑗𝐶≤𝑦_𝑗𝐶+𝐶_𝑅𝑥_0𝑋+‖ℎ(0)‖_𝑋+𝑇𝐶_𝑅+𝐶_𝑅ℎ₀𝑥_𝑗𝐶+𝐶_𝑅Υ𝑥_𝑗𝐶^𝑇₀𝜇(𝑠)𝑑𝑠. (4.32) If {𝑥_𝑗} is unbounded, then there is a subsequence (still denoted by {𝑥_𝑗}) such that ||𝑥_𝑗||_𝐶→+∞ as 𝑗→+∞. Now from the last inequality, it follows that 11≤𝑥_𝑗𝐶𝑦_𝑗𝐶+𝐶_𝑅𝑥_0𝑋+‖ℎ(0)‖_𝑋+𝑇𝐶_𝑅+𝐶_𝑅ℎ₀+𝐶_𝑅^𝑇₀𝜇(𝑠)Υ𝑥𝑑𝑠^𝑗𝐶𝑥_𝑗𝐶. (4.33) Passing in the last inequality to limits as 𝑗→+∞, one gets a contradiction due to the hypotheses of the Theorem. Now from (4.30), we have 𝜒_𝐶𝑥_𝑗≤𝜒_𝐶𝑦_𝑗+𝜒_𝐶𝑅(⋅)ℎ𝑥_𝑗+𝜒_𝐶Φ∘𝑁_𝑔𝑛𝑥_𝑗. (4.34) Using the same arguments as in the proof of Theorem 4.1 and Remark 4.2, we obtain that 𝜒_𝐶Φ∘𝑁_𝑔𝑛𝑥_𝑗=sup_𝑡∈𝐽𝜒Φ∘𝑁_𝑔𝑛𝑥_𝑗𝑡𝜒()=0,_𝐶𝑅(⋅)ℎ𝑥_𝑗=sup_𝑡∈𝐽𝜒𝑅(𝑡)ℎ𝑥_𝑗≤𝐶_𝑅ℎ₀𝜒_𝐶𝑥_𝑗. (4.35) Substituting the last inequalities into (4.34) and using the fact that {𝑦_𝑗} is a compact sequence, we obtain 𝜒_𝐶𝑥_𝑗≤𝐶_𝑅ℎ₀𝜒_𝐶𝑥_𝑗. (4.36) Noting that 𝐶_𝑅ℎ₀<1, we deduce 𝜒_𝐶({𝑥_𝑗})=0. Therefore, {𝑥_𝑗} is a relatively compact sequence in 𝐶(𝐽;𝑋) and we arrive at the conclusion that Ω is compact and then 𝑉_𝑛 is proper.
Finally, by the observation that FixΨ=𝑉⁻¹(0), from Lemma 4.5, we obtain that FixΨ is an 𝑅_𝛿-set.

Remark 4.8. The topological structure of the solution set of problem (1.1)-(1.2) for the case of a non-compact resolvent 𝑅(𝑡) is an open problem.

Further Remarks
Some additional remarks can be given in the case when 𝑅(𝑡),𝑡>0, is compact. Following the technique presented in [8], we can consider the following problem: 𝑥′(𝑡)=𝐴𝑥(𝑡)+^𝑡₀𝐹()𝑥(𝑠)((𝑡[]𝑥(0)𝑡−𝑠𝑑𝑠+𝑔𝑡,𝑥)),𝑡∈𝐽∶=0,𝑇,+ℎ_𝑛(𝑥)=𝑥₀, (4.37)(4.38) where ℎ_𝑛(𝑥)=𝑅(1/𝑛)ℎ(𝑥),𝑛∈ℕ⧵{0}. Since ℎ is continuous, ℎ_𝑛 is a completely continuous function. Then it satisfies (H2)-(H3). Therefore, under the assumptions (G1)–(G3), and (H1) and (2.46), problem (4.37)-(4.38) has at least one mild solution 𝑥_𝑛∈𝐶(𝐽;𝑋). Furthermore, 𝜒Φ𝑁_𝑔(Ω)(𝑡)=𝜒^𝑡₀𝑅(𝑡−𝑠)𝑔(𝑠,Ω(𝑠))𝑑𝑠≤2^𝑡₀𝜒𝑅(𝑡−𝑠)𝑔(𝑠,Ω(𝑠))𝑑𝑠=0 (4.39) for all bounded subset Ω⊂𝐶(𝐽;𝑋). Thus, one can drop assumption (G3), and then the solution operator for (4.37)-(4.38) Ψ_𝑛(𝑥)(𝑡)=𝑅(𝑡)𝑥₀−ℎ_𝑛(𝑥)+^𝑡₀𝑅(𝑡−𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠 (4.40) is 𝜈-condensing without assuming the condition 𝐶_𝑅𝐶_ℎ+2^𝑇₀𝑘(𝑠)𝑑𝑠<1. (4.41) In fact, we have the following assertion.

Theorem 4.9. Let 𝑅(𝑡) be compact for 𝑡>0. Under assumptions (G1)-(G2) and (H1) and (2.46), the solution set of problem (4.37)-(4.38) is a nonempty compact set.

Using the same arguments as in [8], one can prove that the sequence {𝑥_𝑛} of solutions to (4.37)-(4.38) is relatively compact. Finally, passing to the limit as 𝑛→+∞ in the equation𝑥_𝑛(𝑡)=𝑅(𝑡)𝑥₀(𝑡)𝑅1−𝑅𝑛ℎ𝑥_𝑛+^𝑡₀𝑅(𝑡−𝑠)𝑔𝑠,𝑥_𝑛(𝑠)𝑑𝑠, (4.42) we obtain the solution of problem (1.1)-(1.2).

5. Example

We conclude this note with an example, in which we find a representation for the resolvent operator generated by the linear part and impose suitable conditions to demonstrate the existence result and the structure of the solution set. Precisely, consider the following system:𝜕𝑢=𝜕𝜕𝑡𝜕𝑥²²𝑢(𝑥,𝑡)+^𝑡₀𝑎(𝑡−𝑠)𝑢(𝑥,𝑠)𝑑𝑠+^𝑥₀𝑓()(],𝑢()𝑡,𝑦,𝑢𝑦,𝑡𝑑𝑦,𝑥∈0,𝜋,𝑡∈0,𝑏𝑥,0+^𝑚_𝑖=1𝛼_𝑖𝑢𝑥,𝑡_𝑖=𝑢₀,𝑡_𝑖∈()𝑢()()0,𝑏,0,𝑡=𝑢𝜋,𝑡=0. (5.1)(5.2)(5.3) Let 𝑋=𝐿²(0;𝜋),𝐷(𝐴)=𝑧∈𝑋∶𝑧and𝑧^′areabsolutelycontinuous,𝑧(0)=𝑧(𝜋)=0, (5.4) and 𝐴∶𝐷(𝐴)→𝑋, 𝐴𝑧=𝑧^′′. Then it is known that 𝐴 generates a strongly continuous semigroup {𝑆(𝑡)}_𝑡≥0 on 𝑋. Recall that the functions𝜙_𝑛(𝑥)=2𝜋sin𝑛𝑥∶𝑛=1,2,… (5.5) form an orthonormal basis in 𝑋 and they are the eigenfunctions corresponding to the eigenvalues {𝜆_𝑛=𝑛²∶𝑛=1,2,…} of −𝐴.

We are in a position to consider the linear part of (5.1)–(5.3):𝜕𝑢=𝜕𝜕𝑡𝜕𝑥²²𝑢(𝑥,𝑡)+^𝑡₀𝑎()𝑢()[][],𝑢()𝑡−𝑠𝑥,𝑠𝑑𝑠,𝑥∈0,𝜋,𝑡∈0,𝑏𝑥,0=𝑢₀𝑢()(),0,𝑡=𝑢𝜋,𝑡=0. (5.6)(5.7)(5.8) Assume that 𝑎∈𝐶¹(0;𝑏) and 𝑢₀=∑^∞_𝑛=1𝛾_𝑛𝜙_𝑛. We are searching for the resolvent operator generated by (5.6)–(5.8) in the following form:𝑅(𝑡)𝑢₀(𝑥)=𝑢(𝑥,𝑡)=^∞_𝑛=1𝛾_𝑛𝑇_𝑛(𝑡)𝜙_𝑛(𝑥), (5.9) where 𝑇_𝑛 is the solution of the equation𝑇^′_𝑛(𝑡)=−𝑛²𝑇_𝑛(𝑡)+^𝑡₀𝑎(𝑡−𝑠)𝑇_𝑛(𝑠)𝑑𝑠 (5.10) subject to 𝑇_𝑛(0)=1. We know that𝑇_𝑛(𝜆)=1𝜆+𝑛²(1+𝑎(𝜆)), (5.11) where 𝑇_𝑛 and 𝑎 are the Laplace transforms of 𝑇_𝑛 and 𝑎, respectively. For the simple case, when 𝑎 is constant, 𝑎<0, we have𝑇_𝑛(𝜆)=𝜆𝜆²+𝑛²𝜆+𝑛²𝑎. (5.12) By some computations, one gets𝑇_𝑛(𝑡)=𝑝(𝑛)2𝑝(𝑛)+𝑛²𝑒^{𝑝(𝑛)𝑡}+𝑞(𝑛)2𝑞(𝑛)+𝑛²𝑒^{𝑞(𝑛)𝑡}, (5.13) where 𝑝(𝑛)=(1/2)(−𝑛²+𝑛√𝑛²−4𝑎) and 𝑞(𝑛)=(1/2)(−𝑛²−𝑛√𝑛²−4𝑎).

Taking assumption (HA) into account, we conclude that 𝑇_𝑛 needs to satisfy the following condition:𝑇_𝑛(𝑡)−𝑇_𝑛(𝑠)≤𝑐_𝑛|𝑡−𝑠|,for𝑡,𝑠∈(0,𝑏],^∞_𝑛=1𝑐²_𝑛<+∞. (5.14) This condition is obviously fulfilled for 𝑇_𝑛 given by (5.13).

We now verify the compactness of 𝑅(𝑡) for 𝑡>0. Since the embedding 𝐻¹₀(0;𝜋)⊂𝑋 is compact, it is sufficient to find a condition providing that the set𝜕𝑅(𝑡)𝐵𝜕𝑥^𝑋()=𝜕0,𝑟𝑅(𝑡)‖𝑣‖𝜕𝑥𝑣∶_𝑋≤𝑟 (5.15) is bounded in 𝑋. It is easy to verify that this condition follows from^∞_𝑛=1𝑛𝑇_𝑛(𝑡)²<+∞,for𝑡>0. (5.16) The last inequality also holds for 𝑇_𝑛 given by (5.13).

As far as nonlinear problem (5.1)–(5.3) is concerned, we see that the nonlocal function ℎ(𝑢)(𝑥)=∑^𝑚_𝑖=1𝛼_𝑖𝑢(𝑥,𝑡_𝑖) is a Lipschitz function with the constant ℎ₀=∑^𝑚_𝑖=1𝛼_𝑖. Let 𝑔(𝑡,𝑣)(𝑥)=∫^𝑥₀𝑓(𝑡,𝑦,𝑣(𝑦))𝑑𝑦. Then the nonlinearity 𝑔 satisfies (G1)-(G2) if we assume that 𝑓∶[0,𝑏]×[0,𝜋]×ℝ→ℝ is continuous and there exists a function 𝜇∈𝐿¹(0;𝑏) such that𝑓𝑡,𝑦,𝜂≤𝜇(𝑡)1+𝜂,∀𝑡∈[0,𝑏],𝑦∈[0,𝜋],𝜂∈ℝ. (5.17) As indicated in [16], the Hausdorff MNC of a bounded set Ω⊂𝐿²(0;𝜋) can be expressed by𝜒(Ω)=12lim_𝛿→0sup_𝑣∈Ωmax_{0≤𝜔≤𝛿}𝑣−𝑣_{𝜔𝐿2(0;𝜋)}, (5.18) where 𝑣_𝜔 denote the 𝜔-translation of 𝑣:𝑣_𝜔(𝑥)=𝑣()𝑣(𝜋)𝜔+𝑥,0≤𝑥≤𝜋−𝜔,,𝜋−𝜔≤𝑥≤𝜋, (5.19) or, alternatively, the Steklov function:𝑣_𝜔(𝑥)=12𝜔^𝑥+𝜔_𝑥−𝜔𝑣(𝑧)𝑑𝑧 (5.20) (𝑣 is extended outside of [0,𝜋] by zero). Therefore, condition (G3) is provided by the following inequality:_𝑥^𝑥+𝜔𝑓𝑡,𝑦,𝑣𝑦𝑑𝑦≤𝑘(𝑡)𝑣(𝑥)−𝑣_𝜔(𝑥),𝑡∈[0,𝑏];𝑥,𝜔∈[0,𝜋], (5.21) for a nonnegative function 𝑘∈𝐿¹(0;𝑏).

Applying Theorem 4.1, we obtain that problem (5.1)-(5.2) has at least one solution if (5.14), (5.17), and (5.21) take place together with the following estimates:𝐶_𝑅ℎ₀+2^𝑏₀𝑘(𝑠)𝐶𝑑𝑠<1,_𝑅ℎ₀+^𝑏₀𝜇(𝑠)𝑑𝑠<1. (5.22) If we assume that (5.16) holds and hence 𝑅(𝑡) is compact for 𝑡>0, then the solution set of (5.1)–(5.3) is an 𝑅_𝛿-set if (5.17) holds and𝐶_𝑅ℎ₀+^𝑏₀𝜇(𝑠)𝑑𝑠<1, (5.23) due to Theorem 4.7.

Acknowledgment

The authors would like to express their appreciation to the referees for their valuable comments and suggestions. The work is partially supported by the Taiwan-Russian Grant 09-01-92003-NNS, the Taiwan NSC Grants 98-2923-E-037-001-MY3, 98-2923-E-110-003-MY3, 99-2115-M-110-007-MY3 and Russian FBR Grants 11-01-00328 and 10-01-00143.