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

and Applied Analysis 3 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 equation x′ t A [ x t ∫ t 0 F 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 and d dt R t y A [ R t y ∫ t 0 F t − s R s yds ] R t Ay ∫ t 0 R 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 CR < ∞ such that sup t∈J ‖R t ‖L X ≤ CR. 1.10 Then the mild solution on J can be represented as x t R t x0 − h x ∫ t 0 R 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 Rδ-set structure. We end this paper with an example in Section 5. 4 Abstract and Applied Analysis 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 co Ω β Ω for every Ω ∈ P E , 2.1 where co Ω 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 : Ω has a finite ε-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∈J χ Ω t , 2.3 where χ is the Hausdorff MNC on X and Ω t {y t : y ∈ Ω}; ii the modulus of equicontinuity: modC Ω lim δ→ 0 sup y∈Ω max |t1−t2|<δ ∥∥y t1 − y t2 ∥∥. 2.4 As indicated in 15 , these MNCs satisfy all properties mentioned in Definition 2.1 except the regularity. LetT ∈ 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 : inf { M : χ TΩ ≤ Mχ Ω , Ω ⊂ E is a bounded set } . 2.5 Abstract and Applied Analysis 5 The χ-norm of T can be evaluated as ‖T‖χ χ TS1 χ TB1 , 2.6and Applied Analysis 5 The χ-norm of T can be evaluated as ‖T‖χ χ TS1 χ TB1 , 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 setΩ ⊂ 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 βcondensing 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 βcondensing 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 that ∥∥g(t, η)∥∥X ≤ μ t Υ(∥∥η∥∥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 have χ ( g t,Ω ) ≤ k t χ Ω 2.11 for a.e. t ∈ J , where χ is the Hausdorff MNC in X; 6 Abstract and Applied Analysis 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 Ch such that χ h Ω ≤ Chγ 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, then modC 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 g ( t, η ) g1 ( t, η ) g2 ( t, η ) , 2.15 where g1 is Lipschitz with respect to the second argument: ∥∥g1 t, ξ − g1(t, 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. Abstract and Applied Analysis 7 We define the following operator: Φ : L1 J ;X −→ C J ;X , Φ ( f ) t ∫ t 0 R t − s f s ds. 2.17and Applied Analysis 7 We define the following operator: Φ : L1 J ;X −→ C J ;X , Φ ( f ) t ∫ t 0 R t − s f s ds. 2.17 Before collecting some properties of Φ, we recall the following definitions. Definition 2.6. A subsetQ 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 for a.e. t ∈ J, 2.18 for all f ∈ Q. Definition 2.7. The sequence {ξn} ⊂ L1 J ;X is called semicompact if it is integrably bounded and the set {ξn 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 −Φ(η) t ∥∥X ≤ CR ∫ t 0 ∥∥ξ s − η s ∥∥Xds 2.19 for every ξ, η ∈ L1 J ;X , t ∈ J ; Φ3 for any compact K ⊂ X and sequence {ξn} ⊂ L1 J ;X such that {ξn t } ⊂ K for a.e. t ∈ J , theweak convergence ξn ⇀ ξ implies the uniform convergenceΦ ξn → Φ ξ ; Φ4 if {ξn} ⊂ L1 J ;X is an integrably bounded sequence and q ∈ L1 J is a nonnegative function such that χ {ξn t } ≤ q t , for a.e. t ∈ J , then χ {Φ ξn t } ≤ 2CR ∫ t 0 q s ds, t ∈ J ; 2.20 Φ5 if {ξn} ⊂ L1 J ;X is a semicompact sequence, then {ξn} is weakly compact in L1 J ;X and {Φ ξn } is relatively compact in C J ;X . Moreover, if ξn ⇀ ξ0, then Φ ξn → Φ ξ0 . Denote Φ∗ x t R t x0 − h x 2.21 8 Abstract and Applied Analysis for t ∈ J and x ∈ C J ;X . By Ng we denote the Nemytskii operator corresponding to the nonlinearity g, that is, Ng x t g t, x t for t ∈ 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 ΦNg x . 2.23 Let Ψ x Φ∗ x ΦNg 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 D∈Δ Ω ( γ D ,modC D ) , 2.25 where γ and modC are defined in 2.3 and 2.4 , respectively, Δ Ω denotes the collection of all countable subsets ofΩ, and the maximum is taken in the sense of the ordering in the cone R 2 . 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 : CR ( Ch 2 ∫T 0 k 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 {zn} ⊂ Ψ Ω such that ν Ψ Ω ( γ {zn} ,modC {zn} ) . 2.28 Abstract and App


Introduction
In this paper, we investigate the following problem: x t A x t t 0 F t − s x s ds g t, x t , t ∈ J : 0, T , 1.1 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 as x t t, y ∂ 2 ∂y 2 x t, y t 0 b t − s x s, y ds g t, y , x 0, y x 0 , 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 x 0 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: where c i i 1, . . ., p are given constants and 0 ≤ t 1 < • • • < t p ≤ T .As another example, one can take where K i : X → X are given linear operators.Regarding to 1.3 , in the case X L 2 0, a , the operators K i can be given by K i x t i , y a 0 k i ξ, y x t i , ξ dξ, 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 equation x t S t x 0 − h x t 0 S 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 equation 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, 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 1.10 Then the mild solution on J can be represented as 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.

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 where co Ω is the closure of convex hull of Ω.
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 An important example of MNC is the Hausdorff MNC, which satisfies all properties given in the previous definition: 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: where χ is the Hausdorff MNC on X and Ω t {y t : y ∈ Ω}; ii the modulus of equicontinuity: 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: Abstract and Applied Analysis 5 The χ-norm of T can be evaluated as where S 1 and B 1 are the unit sphere and the unit ball in E, respectively.It is easy to see that 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 ∈ L 1 J and nondecreasing function Υ : R → R such that for a.e.t ∈ J and for all η ∈ X; G3 there exists a function k ∈ L 1 J such that for each nonempty, bounded set Ω ⊂ X we have 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 12 for all x ∈ C J; X , where x C x C J;X ; H2 there is a constant C h such that for any bounded subset Ω ⊂ C J; X , where γ is defined in 2.3 .
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 where g 1 is Lipschitz with respect to the second argument: for a.e.t ∈ J and ξ, η ∈ X with k ∈ L 1 J and g 2 is compact in second argument; that is, for each t ∈ J and bounded Ω ⊂ X, the set g 2 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 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: Before collecting some properties of Φ, we recall the following definitions.Φ1 the operator Φ sends any integrably bounded set in L 1 J; X to equicontinuous set in C J; X ; Φ2 the following inequality holds: for every ξ, η ∈ L 1 J; X , t ∈ J; Φ3 for any compact K ⊂ X and sequence {ξ n } ⊂ L 1 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 } ⊂ L 1 J; X is an integrably bounded sequence and q ∈ L 1 J is a nonnegative function such that χ {ξ n t } ≤ q t , for a.e.t ∈ J, then 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 for t ∈ J, x ∈ C J; X .

2.22
We see that x is a solution of 1.1 -1.2 if and only if 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 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 R 2 .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 2.1.3for details .

Theorem 2.8. Let F satisfy (F1)-(F2). Assume that conditions (G1)-(G3) and (H1)-(H3) are fulfilled. If
We will show that Ω is relatively compact in C J; X .By the definition of ν, there exists a sequence Following the construction of Ψ, one can take a sequence {x n } ⊂ Ω such that where

2.30
Using assumption G3 , we have for all s ∈ J. Then by Φ4 , we obtain Noting that we have

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 L 1 0, t; X .In addition, since R t , t > 0, is compact, we have 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 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

2.48
Recalling that we have due to H1 and G2 .Then

2.51
Equivalently, 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 H1 h : C J; X → X is continuous and 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 H1 h : C J; X → X is continuous and 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
where M C R ||x 0 || 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 x 0 − h x λ t 0 R t − s g s, x s ds.

2.57
It follows that we have x t X ≤ v t , for all t ∈ J, and due to the fact that Υ is nondecreasing.Then, by using 2.56 , we have for all t ∈ J.The last inequalities imply that sup t∈J v t is bounded, so is x C .

Continuous Dependence Result
We start with some notions from the theory of multivalued maps see, e.g. 15 for details .Let Y, Y and Z, 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 > 0 there exists δ δ y, > 0 such that condition Y y, y < δ implies G y ⊂ U G y , where U G y denotes the -neighborhood of the set G y induced by the metric The following assertion gives a sufficient condition for upper semicontinuity.Consider the solution multimap x is a solution of 1.1 -1.2 with initial value x 0 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 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 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 χ {x n t } ≤ χ {R t h x n } χ Φg n t .

3.4
Using G3 we have for all s ∈ J. Referring to Φ4 , one gets and then On the other hand, by H2 one has Combining the last inequality with 3.4 -3.7 , we have This leads to the conclusion that γ {x n } 0. Now, condition G2 implies that {g n } is integrably bounded in L 1 J; X .Thus Φ1 ensures that {Φ g n } is equicontinuous.Then applying condition H3 , we obtain 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

3.11
We first observe that 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 due to the fact that {g •, x n • } is integrably bounded.Therefore, taking 3.11 into account, we arrive at The proof is completed.

Existence Result
In this section, we assume that h is a Lipschitz function.H2' There exists a constant h 0 > 0 such that This implies the growth of h: and the last inequality covers H1 .Let χ C be the Hausdorff MNC in C J; X .We have for all t ∈ J, where Φ * is given in 2.21 .Thus Then we know that see 15 condition H2' implies 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:
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 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 We will show that Ω is relatively compact.Notice that where Then we have The boundedness of Ω in C J; X implies that N g Ω is a bounded set in L 1 J; X .By Φ1 , the set Φ • N g Ω is equicontinuous and therefore we have Combining 4.5 , 4.13 , and 4.15 , we obtain Relations 4.7 and 4.10 yield Since < 1, we have χ C Ω 0. The regularity of χ C ensures that Ω is relatively compact.
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 which is covered by 4.8 .Recall that in this case we have for any bounded sequence {x n } ⊂ C J; X and for all t ∈ J.

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 y 0 ∈ Y and a continuous map h : B × 0, 1 → Y such that h y, 0 y and h y, 1 y 0 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 y 0 ∈ E and for all y in a neighborhood N y 0 of y 0 in E, there exists exactly one solution x n of the equation V n x y.
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 > 0, there is a locally Lipschitz map f : X → E such that f x − f x E < , for every x ∈ 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 then the solution set of problem 1.1 -1.2 is an R δ -set.
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 n x j y j .

4.29
Equivalently, x j t y j t R t x 0 − h x j t 0 R t − s g n s, x j s ds, t ∈ J.

4.30
We first show that the sequence {x j } is bounded.We have due to H2 and 4.23 .Thus 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 x j C .

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 Using the same arguments as in the proof of Theorem 4.1 and Remark 4.2, we obtain that

4.35
Abstract and Applied Analysis 21 Substituting the last inequalities into 4.34 and using the fact that {y j } is a compact sequence, we obtain

4.36
Noting that C R h 0 < 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 Fix Ψ V −1 0 , from Lemma 4.5, we obtain that Fix Ψ is an R δ -set.

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: 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 equation we obtain the solution of problem 1.1 -1.2 .

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: 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 have T n λ λ λ 2 n 2 λ n 2 a .

5.12
By some computations, one gets T n t p n 2p n n 2 e p n t q n 2q n n 2 e q n t , 5.13 where p n 1/2 −n 2 n √ n 2 − 4a and q n 1/2 −n 2 − n √ n 2 − 4a .Taking assumption HA into account, we conclude that T n needs to satisfy the following condition:

5.14
This condition is obviously fulfilled for T n given by 5.13 .

Abstract and Applied Analysis
We now verify the compactness of R t for t > 0. Since the embedding H 1 0 0; π ⊂ X is compact, it is sufficient to find a condition providing that the set is bounded in X.It is easy to verify that this condition follows from |nT n t | 2 < ∞, for t > 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 1 α i u x, t i is a Lipschitz function with the constant h 0 m i 1 α i .Let g t, v x x 0 f 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 ∈ L 1 0; b such that f t, y, η ≤ μ t 1 η , ∀t ∈ 0, b , y ∈ 0, π , η ∈ R.
5.17 v ω x 1 2ω x ω x−ω v z dz 5.20 v is extended outside of 0, π by zero .Therefore, condition G3 is provided by the following inequality:

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

Lemma 3 . 1
see 15 .Let G : Y → K Z be a closed quasicompact multimap.Then G is u.s.c.

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.

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 Fix F {x F x } is a nonempty compact set.
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.
We are in a position to consider the linear part of 5.1 -5.3 : t t 0 a t − s u x, s ds x 0 f t, y, u y, t dy, x ∈ 0, π , t ∈ 0, b ,form an orthonormal basis in X and they are the eigenfunctions corresponding to the eigenvalues {λ n n 2 : n 1, 2, . ..} of −A.