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 Applied Analysis 9 Following the construction of Ψ, one can take a sequence {xn} ⊂ Ω such that zn Φ∗ xn Φ ( gn ) , 2.29and Applied Analysis 9 Following the construction of Ψ, one can take a sequence {xn} ⊂ Ω such that zn Φ∗ xn Φ ( gn ) , 2.29 where gn t g t, xn t , t ∈ J, Φ∗ xn t R t x0 − h xn , Φ ( gn ) t ∫ t 0 R t − s gn s ds. 2.30 Using assumption G3 , we have χ ({ gn s }) χ ({ g s, xn s }) ≤ k s χ {xn s } ≤ k s γ {xn} , 2.31 for all s ∈ J . Then by Φ4 , we obtain χ ({ Φ ( gn ) t }) ≤ 2CR (∫ t 0 k s ds ) γ {xn} . 2.32 Noting that Φ∗ xn t R t x0 − R t h xn , 2.33 we have χ {Φ∗ xn t } χ {R t h xn } ≤ CRChγ {xn} 2.34 due to 2.5 2.7 and H2 . Combining 2.29 , 2.31 , and 2.32 , we get γ {zn} ≤ γ {xn} . 2.35 Combining the last inequality with 2.27 , we have γ {xn} ≤ γ {xn} , 2.36