Second Order Impulsive Retarded Differential Inclusions with Nonlocal Conditions

Abstract

In this work we establish some existence results for abstract second order Cauchy problems modeled by a retarded differential inclusion involving nonlocal and impulsive conditions. Our results are obtained by using fixed point theory for the measure of noncompactness.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Hernán R. Henríquez 1 and Marcos Rabelo 2 and Luciana Vale 2

Academic Editor:Geraldo Botelho

1, Departamento de Matemática, Universidad de Santiago (USACH), Casilla 307, Correo 2, Santiago, Chile
2, Departamento de Matemática, Universidade Federal de Goiás, Campus Catalão, 75704-020 Catalão, GO, Brazil

Received 27 August 2013; Accepted 3 December 2013; 23 January 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper we are interested in studying the existence of solutions to evolution systems that can be described by equations that suffer abrupt changes in their trajectories and simultaneously depend on nonlocal initial conditions. More specifically, the aim of this paper is to establish existence results for abstract second order evolution problems with delay whose equations can be written as differential inclusions with nonlocal initial conditions and subjected to impulses.

To describe the problem, throughout this work we denote by X a Banach space provided with a norm || · || . We assume that A : D ( A ) ⊆ X [arrow right] X is the infinitesimal generator of a cosine functions of operators C ( t ) on X . We study the system on an interval J = [ 0 , T ] , for some T > 0 , and we assume that the impulses occur at fixed moments 0 < _{t 1} < _{t 2} , ... , _{t m} < T . Moreover, h > 0 denotes the system delay. Specifically, we will consider abstract second order systems [figure omitted; refer to PDF] where x ( t ) ∈ X , _{x t} , t ...5; 0 denotes the function defined by _{x t} ( θ ) = x ( t + θ ) for θ ∈ [ - h , 0 ] , Δ y ( t ) = y ( ^{t +} ) - y ( ^{t -} ) indicates the gap of a piecewise continuous function y ( · ) at t , [straight phi] : [ - h , 0 ] [arrow right] X is an appropriate function, and F , ^{I k 1} , ^{I k 2} , and g are maps that will be specified later.

As a model, we consider a general wave equation described by a second order differential inclusion with impulses and nonlocal initial conditions [figure omitted; refer to PDF] for t ∈ J = [ 0 , T ] , ξ ∈ ( 0 , π ) , and k = 1 , ... , m . In this system we assume that _{f 0} is a multivalued map, and the inclusion indicated in (5) will be explained in Section 4. Moreover, ^{a k i} , ^{b k i} , ^{q k i} , i = 1,2 , [straight phi] and z are appropriate functions.

Here we briefly discuss the context in which our work is inserted. We do not intend to make an exhaustive list of references but just mention those most recent and directly related to the topic of this paper. Differential inclusions and impulsive differential inclusions are used to describe many phenomena arising from different fields as physics, chemistry, population dynamics, and so forth. For this reason, last years several researchers have studied various aspects of the theory. We mention here to [1-6] and references in these texts for the motivations of the theory.

In particular, there are phenomena in nature that experiment abrupt changes at fixed moments of time. Such kind of systems are well described by impulsive systems. In the study of ordinary and partial differential equations with impulsive action, interesting questions appear such as local and global existence, stability, controllability, and so forth. For this reason this topic has attracted the attention of many authors in the last time. We only mention here the papers [7-17] which are directly related with the objective of this paper.

The concept of nonlocal initial condition was introduced by Byszewski and Lakshmikantham to extend the classical theory of initial value problems ([18-22]). This notion is more appropriate than the classical theory to describe natural phenomena because it allows us to consider additional information. Thenceforth, the study of differential equations with nonlocal initial conditions has been an active topic of research. The interested reader can consult [23-26] and the references therein for recent developments on issues similar to those addressed in this paper.

On the other hand, it is well known that retarded functional differential equations are used to model important concrete phenomena. For general aspects of the theory of partial differential equations with delay we refer to [27], and for functional differential inclusions we refer to [7, 9, 12-14, 28]. In similar way, there exists an extensive literature concerning abstract second order problems. In the autonomous case, the existence of solutions to the second order abstract Cauchy problem is strongly related with the concept of cosine functions.

In this paper, we combine the theory of cosine functions with the properties of the measure of noncompactness and some properties of function spaces introduced in [9] to establish the existence of solutions to the problems (1)-(4).

This paper has four sections. In Section 2 we develop some properties about the abstract Cauchy problem of second order, the measure of noncompactness, and multivalued analysis which are needed to establish our results. In Section 3 we discuss the existence of mild solutions to problems (1)-(4). Finally, in Section 4 we apply our results to establish the existence of solutions to problems (5)-(9).

The terminology and notations are those generally used in functional analysis. In particular, if ( Y , _{|| · || Y} ) and ( Z , _{|| · || Z} ) are Banach spaces, we denote by [Lagrangian (script capital L)] ( Y , Z ) the Banach space of the bounded linear operators from Y into Z and we abbreviate this notation to [Lagrangian (script capital L)] ( Y ) whenever Z = Y .

2. Preliminaries

2.1. The Second Order Abstract Cauchy Problem

In this section we collect the main facts concerning the existence of solutions for second order abstract differential equations. For the theory of cosine functions of operators we refer to [29-34]. We next only mention a few concepts and properties relative to the second order abstract Cauchy problem. Throughout this paper, A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators C ( t ) on the Banach space X . We denote by S ( t ) the sine function associated with C ( t ) which is defined by [figure omitted; refer to PDF] We denote by M , _{M 1} some positive constants such that || C ( t ) || ...4; M and || S ( t ) || ...4; _{M 1} for t ∈ J . The function S : ... [arrow right] [Lagrangian (script capital L)] ( X ) is continuous for the norm of operators and || S ( t ) || ...4; M t for every t ∈ J . The notation E stands for the space formed by the vectors x ∈ X for which C ( · ) x is a function of class ^{C 1} on ... . We know from Kisynski [35] that E endowed with the norm [figure omitted; refer to PDF] is a Banach space.

The operator valued function G ( t ) = [ C ( t ) S ( t ) A S ( t ) C ( t ) ] is a strongly continuous group of bounded linear operators on the space E × X , generated by the operator ...9C; = [ 0 I A 0 ] defined on D ( A ) × E . It follows from this property that A S ( t ) : E [arrow right] X is a bounded linear operator and that A S ( · ) : ... [arrow right] [Lagrangian (script capital L)] ( E ; X ) is a strongly continuous operator valued map. We denote by _{M 2} a positive constant such that _{|| A S ( t ) || [Lagrangian (script capital L)] ( E ; X )} ...4; _{M 2} for all t ∈ J . In addition ([34]) [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] Furthermore, if x : [ 0 , ∞ ) [arrow right] X is a locally integrable function, then [figure omitted; refer to PDF] defines an E -valued continuous function.

The existence of solutions of the second order abstract Cauchy problem [figure omitted; refer to PDF] where f : [ 0 , T ] [arrow right] X is an integrable function has been discussed in [30, 32-34, 36]. Similarly, the existence of solutions for the semilinear second order abstract Cauchy problem has been treated in [37]. We only mention here that the function x ( · ) given by [figure omitted; refer to PDF] is called mild solution of (15), and that when ^{z 1} ∈ E the function x ( · ) is continuously differentiable and [figure omitted; refer to PDF]

2.2. Measure of Noncompactness and Multivalued Maps

In this subsection we recall some facts concerning multivalued analysis, which will be used later. Let Ω be a metric space. Throughout this paper ...AB; ( Ω ) denotes the collection of all nonempty subsets of Ω and _{...AB; b} ( Ω ) denotes the collection of all bounded nonempty subsets of Ω .

Some of our results are based on the concept of measure of noncompactness. For this reason, we next recall a few properties of this concept. For general information the reader can see [5, 9, 38, 39]. In this paper, we use the notion of Hausdorff measure of noncompactness.

Definition 1.

Let B be a bounded subset of a metric space Ω . The Hausdorff measure of noncompactness of B is defined by [figure omitted; refer to PDF]

Remark 2.

Let _{B 1} , _{B 2} ⊆ Ω be bounded sets. The Hausdorff measure of noncompactness has the following properties.

(a) If _{B 1} ⊆ _{B 2} , then η ( _{B 1} ) [= or <, slanted] η ( _{B 2} ) .

(b) η ( B ) = η ( B ¯ ) .

(c) η ( B ) = 0 if and only if B is totally bounded.

(d) η ( _{B 1} ∪ _{B 2} ) = max ... ... { η ( _{B 1} ) , η ( _{B 2} ) } .

In what follows, we assume that Y is a normed space. For a bounded set B ⊆ Y , we denote by co ... ¯ ( B ) the closed convex hull of the set B .

Remark 3.

Let _{B 1} , _{B 2} ⊆ Y be bounded sets. The following properties hold.

(a) For λ ∈ ... , η ( λ B ) = | λ | η ( B ) .

(b) η ( _{B 1} + _{B 2} ) [= or <, slanted] η ( _{B 1} ) + η ( _{B 2} ) , where _{B 1} + _{B 2} = { _{b 1} + _{b 2} : _{b 1} ∈ _{B 1} , _{b 2} ∈ _{B 2} } .

(c) η ( B ) = η ( co ... ¯ ( B ) ) .

Henceforth we use the notations [upsilon] ( Y ) and ...A6; [upsilon] ( Y ) to denote the following sets:

(s1) [upsilon] ( Y ) = { D ∈ ...AB; ( Y ) : D is convex } ,

(s2) ...A6; [upsilon] ( Y ) = { D ∈ [upsilon] ( Y ) : D is compact } .

We refer the reader to the already mentioned references to abstract concepts of measure of noncompactness and for many examples of measure of noncompactness.

Definition 4.

Let Ω be metric space. We said that a multivalued map ... : Ω [arrow right] ...AB; ( Y ) is said to be

(i) upper semicontinuous (u.s.c. for short) if ^{... - 1} ( V ) = { w ∈ Ω : ... ( w ) ⊆ V } is an open subset of Ω for all open set V ⊆ Y ;

(ii) closed if its graph _{G ...} = { ( w , y ) ; y ∈ ... ( w ) } is a closed subset of Ω × Y ;

(iii): compact if its range ... ( Ω ) is relatively compact in Y ;

(iv) quasicompact if ... ( K ) is relatively compact in Y for any compact subset K ⊂ Ω .

Definition 5.

A multivalued map ... : Ω [arrow right] ...AB; ( Y ) is said to be a condensing map with respect to η (abbreviated, η -condensing) if for every bounded set D ⊂ Ω , η ( D ) > 0 , η ( ... ( D ) ) < η ( D ) .

The next result is essential for the development of the rest of our work. We point out that if ... : Ω [arrow right] K [upsilon] ( Y ) is u.s.c., then ... is closed. This allows us to establish the following version of the fixed point theorem [5, Corollary 3.3.1].

Theorem 6.

Let M be a convex closed subset of Y , and let ... : M [arrow right] K [upsilon] ( M ) be a u.s.c. β -condensing multivalued map. Then F i x ( ... ) = { y ∈ F ( y ) } is a nonempty compact set.

2.3. Function Spaces

Let I be any of the intervals [ 0 , T ] or [ - h , T ] . The space PC ( I ; X ) is formed by all piecewise continuous functions x : I [arrow right] X satisfying the following conditions:

(c1) the function x ( · ) is continuous on I \ { _{t 1} , ... , _{t m} } , and

(c2) there exist _{lim ... t [arrow right] ^{t j +}} x ( t ) and _{lim ... t [arrow right] ^{t j -}} and x ( _{t j} ) = _{lim ... t [arrow right] ^{t j -}} x ( t ) for all 1 ...4; j ...4; m .

We consider PC ( I ; X ) endowed with the norm of the uniform convergence [figure omitted; refer to PDF] It is well known that PC ( I ; X ) is a Banach space. Furthermore, let _{Π j} : PC ( I ; X ) [arrow right] C ( [ _{t j} , _{t j + 1} ] ; X ) , - 1 ...4; j ...4; m , be the map defined by [figure omitted; refer to PDF] where we set _{t - 1} = - h , _{t 0} = 0 , and _{t m + 1} = T . For each j ∈ { 0 , ... , m } and D ⊆ PC ( [ - h , T ] ; X ) , we denote by _{D j} the range of D under the operator _{Π j} ; that is, _{D j} = _{Π j} ( D ) .

We define the subspace P ^{C 1} ( [ - h , T ] ; X ) of PC ( [ - h , T ] ; X ) consisting of all functions which are continuously differentiable at [ 0 , T ] \ { _{t 1} , ... , _{t m} } and there exist ^{x [variant prime]} ( ^{t k +} ) and ^{x [variant prime]} ( ^{t k -} ) for all 1 ...4; k ...4; m . It is straightforward to show that the space P ^{C 1} ( [ - h , T ] ; X ) endowed with the norm [figure omitted; refer to PDF] is a Banach space.

From now on we denote by ^{C h p} , 1 ...4; p < ∞ , the space of piecewise continuous functions v : [ - h , 0 ] [arrow right] X endowed with the norm [figure omitted; refer to PDF] In what follows we denote by χ the Hausdorff measure of noncompactness in X and by β the Hausdorff measure of noncompactness in a space of continuous (or piecewise continuous) functions with values in X . We next collect some properties of measure β which are needed to establish our results.

Lemma 7.

Let G : [ 0 , T ] [arrow right] [Lagrangian (script capital L)] ( X ) be a strongly continuous operator valued map. Let D ⊂ X be a bounded set. Then β ( { G ( · ) x : x ∈ D } ) ...4; _{sup ... 0 ...4; t ...4; T} || G ( t ) || χ ( D ) .

Lemma 8.

Let D ⊂ P C ( [ - h , T ] ; X ) be a bounded set. Then β ( D ) = _{max ... k = - 1 , ... , m} β ( _{D k} ) .

Lemma 9 (see [39]).

Let W ⊆ C ( J ; X ) be a bounded set. Then χ ( W ( t ) ) [= or <, slanted] β ( W ) for all t ∈ J . Furthermore, if W is equicontinuous on J , then χ ( W ( t ) ) is continuous on J , and [figure omitted; refer to PDF]

Lemma 10.

Let D ⊆ C ( J ; X ) be a bounded set. Then there exists a countable set _{D 0} ⊆ D such that β ( _{D 0} ) = β ( D ) .

A set W ⊆ ^{L 1} ( J ; X ) is said to be uniformly integrable if there exists a positive function μ ∈ ^{L 1} ( J ) such that || w ( t ) || ...4; μ ( t ) a.e. for t ∈ J and all w ∈ W .

Lemma 11.

Let G : [ 0 , T ] [arrow right] [Lagrangian (script capital L)] ( X ) be a strongly continuous operator valued map and Λ : ^{L 1} ( [ 0 , T ] ; X ) [arrow right] C ( [ 0 , T ] ; X ) be the map defined by [figure omitted; refer to PDF] Let W ⊂ ^{L 1} ( [ 0 , T ] ; X ) . Assume that there is a compact set K ⊂ X and a positive function q ∈ ^{L 1} ( [ 0 , T ] ) such that W ( t ) ⊆ K for all t ∈ [ 0 , T ] and χ ( W ( t ) ) ...4; q ( t ) . Then [figure omitted; refer to PDF]

Proof.

It is clear that W is uniformly integrable. Applying Lemma 10 and [5, Theorem 4.2.2], we can affirm that [figure omitted; refer to PDF] Since the set Λ ( W ) is equicontinuous, using Lemma 11, we obtain the assertion.

We also need to consider the product space PC ( [ - h , T ] ; X ) × PC ( [ 0 , T ] ; X ) provided with the norm [figure omitted; refer to PDF] The following property is immediate.

Lemma 12.

Let W ⊂ P C ( [ - h , T ] ; X ) × P C ( [ 0 , T ] ; X ) be a bounded set.

(a) Assume that W = _{W 1} × _{W 2} , where _{W 1} ⊂ P C ( [ - h , T ] ; X ) and _{W 2} ⊂ P C ( [ 0 , T ] ; X ) are bounded sets. Then β ( W ) = max ... { β ( _{W 1} ) , β ( _{W 2} ) } .

(b) Let [figure omitted; refer to PDF]

: Then max ... { β ( _{W 1} ) , β ( _{W 2} ) } ...4; β ( W ) .

3. Existence Results

In this section we establish some results of existence of mild solutions of problems (1)-(4). Initially we will establish the general framework of conditions under which we will study this problem. Throughout this section, χ denotes the Hausdorff measure of noncompactness in X . We assume that [straight phi] ∈ C ( [ - h , 0 ] ; X ) . Moreover, in what follows we assume that F is a multivalued map from J × X × X × ^{C h p} into ...A6; [upsilon] ( X ) that satisfies the following properties.

(F1) The function F ( · , ^{y 1} , ^{y 2} , ψ ) : [ 0 , T ] [arrow right] ...A6; [upsilon] ( X ) admits a strongly measurable selection for each ^{y i} ∈ X , i = 1,2 , and ψ ∈ ^{C h p} .

(F2) For each t ∈ [ 0 , T ] , the function F ( t , · , · , · ) : X × X × ^{C h p} [arrow right] ...A6; [upsilon] ( X ) is u.s.c.

(F3) For each r > 0 , there is a function _{μ r} ∈ ^{L 1} ( [ 0 , T ] ) such that [figure omitted; refer to PDF]

: for all ^{y i} ∈ X , i = 1,2 , and ψ ∈ ^{C h p} such that [figure omitted; refer to PDF]

(F4) There exists a positive integrable function k ( · ) on [ 0 , T ] such that [figure omitted; refer to PDF]

: for all bounded sets _{Ω i} ⊆ X , i = 1,2 , and Q ⊆ ^{C h p} such that _{sup ... θ ∈ [ - h , 0 ]} { || ψ ( θ ) || : ψ ∈ Q } < ∞ .

Remark 13.

Let x ( · ) ∈ PC ( [ - h , T ] ; X ) and y ( · ) ∈ PC ( [ 0 , T ] ; X ) . Then the function [ 0 , T ] [arrow right] ^{C h p} , t ... _{x t} , is continuous. Hence, the function [ 0 , T ] [arrow right] X × X × ^{C h p} , t ... ( x ( t ) , y ( t ) , _{x t} ) , is strongly measurable. Combining this assertion with conditions (F1) and (F2) and applying [5, Theorem 1.3.5] we infer that the function [ 0 , T ] [arrow right] ...A6; [upsilon] ( X ) , t ... F ( t , x ( t ) , y ( t ) , _{x t} ) admits a Bochner integrable selection. As a consequence, the set [figure omitted; refer to PDF] and _{...AE; F , x , y} is convex.

Next we introduce the conditions on the function g . We assume that g is a map from PC ( [ - h , T ] ; X ) into C ( [ - h , 0 ] ; X ) such that the values g ( x ) ( 0 ) ∈ E for all x ( · ) ∈ PC ( [ - h , T ] ; X ) and that the following conditions are fulfilled.

(g1) The function g is continuous and takes bounded sets in PC ( [ - h , T ] ; X ) into bounded subsets of C ( [ - h , 0 ] ; X ) . Moreover, the map g ( · ) ( 0 ) : PC ( [ - h , T ] ; X ) [arrow right] E is continuous and takes bounded sets in PC ( [ - h , T ] ; X ) into bounded subsets of E .

(g2) There is a continuous function [cursive l] : [ - h , 0 ] [arrow right] [ 0 , ∞ ) and a constant _{[cursive l] 1} ...5; 0 such that [figure omitted; refer to PDF]

: for all bounded set W ⊂ PC ( [ - h , T ] , X ) .

(g3) For each bounded set W ⊂ PC ( [ - h , T ] ; X ) the set g ( W ) is equicontinuous.

Next we establish the conditions on maps ^{I k i} , i = 1,2 , k = 1 , ... , m .

We assume that ^{I k 1} : X × X × ^{C h p} [arrow right] E and ^{I k 2} : X × X × ^{C h p} [arrow right] X satisfy the following conditions.

(I1) The maps ^{I k i} , i = 1,2 , k = 1 , ... , m are continuous and takes bounded sets into bounded sets.

(I2) There are positive constants ^{d k i , j} , i = 1,2 , j = 0,1 , 2 , k = 1 , ... , m , such that [figure omitted; refer to PDF]

: for all bounded subsets _{D 0} , _{D 1} of X , and W ⊆ ^{C h p} such that _{sup ... θ ∈ [ - h , 0 ]} { || ψ ( θ ) || : ψ ∈ W } < ∞ .

Remark 14.

Let W ⊂ PC ( [ - h , T ] ; X ) be a bounded set. Then for all t ∈ [ 0 , T ] , _{W t} = { _{w t} : w ∈ W } is a bounded subset of ^{C h p} and _{sup ... - h ...4; θ ...4; 0} χ ( _{W t} ( θ ) ) ...4; β ( W ) for all 0 ...4; t ...4; T .

Motivated by expressions (16) and (17) (see also [12]), we introduce the following concept of mild solution to problems (1)-(4).

Definition 15.

A function x ( · ) ∈ P ^{C 1} ( [ - h , T ] ; X ) is said to be a mild solution of (1)-(4) if conditions (2)-(4) are satisfied, and the integral equation [figure omitted; refer to PDF] is verified for f ∈ _{...AE; F , x , ^{x [variant prime]}} and all t ∈ [ 0 , T ] .

To establish our results, we need to study two integral operators defined on the set _{...AE; F , x , y} for functions x ∈ PC ( [ - h , T ] ; X ) and y ∈ PC ( [ 0 , T ] ; X ) . Initially we mention some properties of _{...AE; F , x , y} . A first result establishes that _{...AE; F , x , y} is closed. Specifically we have the following property ([5, Lemma 5.1.1]).

Lemma 16.

Let { _{x n} ^{} n = 1 ∞} ⊂ P C ( [ - h , T ] ; X ) and { _{y n} ^{} n = 1 ∞} ⊂ P C ( [ 0 , T ] ; X ) be sequences that converge to _{x 0} ∈ P C ( [ - h , T ] ; X ) and _{y 0} ∈ P C ( [ 0 , T ] ; X ) , respectively. Suppose that { _{f n} ^{} n = 1 ∞} ⊂ ^{L 1} ( [ 0 , T ] ; X ) , _{f n} ∈ _{...AE; F , x n , y n} , is a sequence that converges weakly to _{f 0} ∈ ^{L 1} ( [ 0 , T ] ; X ) . Then _{f 0} ∈ _{...AE; F , x 0 , y 0} .

On the other hand, since the values of F are convex compact sets, and, as already mentioned, the graph of F is closed, we can assert that for functions x ( · ) ∈ PC ( [ - h , T ] ; X ) and y ( · ) ∈ PC ( [ 0 , T ] ; X ) , the set _{∪ 0 ...4; t ...4; t} F ( t , x ( t ) , y ( t ) , _{x t} ) is compact in X . In addition, as a consequence of (F3), the set _{...AE; F , x , y} is uniformly integrable over J ; that is to say, there exists a positive function _{μ x , y} ∈ ^{L 1} ( J ) such that || f ( t ) || ...4; _{μ x , y} ( t ) a.e. for t ∈ J and all f ∈ _{...AE; F , x , y} .

We introduce now the operators _{Λ 1} , _{Λ 2} : ^{L 1} ( [ 0 , T ] ; X ) [arrow right] C ( [ 0 , T ] ; X ) given by [figure omitted; refer to PDF] It is clear that _{Λ 1} , _{Λ 2} are bounded linear operators. Using _{Λ 1} , _{Λ 2} we can construct the multivalued maps _{Λ ~ 1} , _{Λ ~ 2} : PC ( [ - h , T ] ; X ) × PC ( [ 0 , T ] ; X ) [arrow right] [upsilon] ( C ( [ 0 , T ] ; X ) ) given by [figure omitted; refer to PDF] Since C ( · ) and S ( · ) are strongly continuous operator valued functions, the assertion in [5, Lemma 4.2.1] remains valid for _{Λ 1} and _{Λ 2} . Hence, combining our previous remarks with [5, Lemma 4.2.1, Corollary 5.1.2] we can establish the following property.

Lemma 17.

Let F : [ 0 , T ] × X × X × ^{C h p} [arrow right] ...A6; [upsilon] ( X ) be a multivalued map satisfying conditions (F1)-(F4). Then _{Λ ~ 1} and _{Λ ~ 2} are u.s.c. maps with convex compact values.

We next define the solution map for problems (1)-(4) as follows. Assume that [straight phi] ( 0 ) ∈ E and let x ∈ P ^{C 1} ( [ - h , T ] ; X ) . We define Γ ( x ) to be the set formed by all functions u given by [figure omitted; refer to PDF] for f ∈ _{...AE; F , x , ^{x [variant prime]}} . It follows from our hypotheses that u ∈ P ^{C 1} ( [ - h , T ] ; X ) . Hence, Γ : P ^{C 1} ( [ - h , T ] ; X ) [arrow right] ...AB; ( P ^{C 1} ( [ - h , T ] ; X ) ) . Furthermore, it is clear that x ( · ) is a mild solution of problems (1)-(4) if and only if x ( · ) is a fixed point of Γ .

We are now in a position to prove the main result of this section. We introduce the map ... : PC ( [ - h , T ] ; X ) × PC ( [ 0 , T ] ; X ) [arrow right] ...AB; ( PC ( [ - h , T ] ; X ) × PC ( [ 0 , T ] ; X ) ) defined as follows. For ( x , y ) ∈ PC ( [ - h , T ] ; X ) × PC ( [ 0 , T ] ; X ) , ... ( x , y ) is the set consisting of all functions ( u , v ) given by [figure omitted; refer to PDF] for f ∈ _{...AE; F , x , y} . It follows from (g1) and (I1) that ... is well defined.

We use the following notations: [figure omitted; refer to PDF]

Theorem 18.

Assume that [straight phi] ( 0 ) ∈ E , and conditions (F1)-(F4), (g1)-(g3) and (I1)-(I2) are fulfilled. If _{N 7} < 1 , then the map ... : P C ( [ - h , T ] ; X ) × P C ( [ 0 , T ] ; X ) [arrow right] K [upsilon] ( P C ( [ - h , T ] ; X ) × P C ( [ 0 , T ] ; X ) ) is u.s.c. and β -condensing.

Proof.

It follows from our hypotheses and Lemma 17 that ... is a u.s.c. multivalued map with convex compact values. It remains to prove that ... is β -condensing. Let Ω ⊂ PC ( [ - h , T ] ; X ) × PC ( [ 0 , T ] ; X ) be a bounded set such that β ( ... ( Ω ) ) ...5; β ( Ω ) . It follows from Lemma 10 that there exists a sequence ( _{w n ) n} in ... ( Ω ) such that β ( ... ( Ω ) ) = β ( { _{w n} : n ∈ ... } ) . We can write _{w n} = ( _{u n} , _{v n} ) ∈ ... ( _{x n} , _{y n} ) for some ( _{x n} , _{y n} ) ∈ Ω . It follows from Lemma 12 that [figure omitted; refer to PDF] Here we will estimate separately the values β ( { _{u n} : n ∈ ... } ) and β ( { _{v n} : n ∈ ... } ) . To estimate β ( { _{u n} : n ∈ ... } ) , using (39), we can write [figure omitted; refer to PDF] for _{f n} ∈ _{...AE; F , x n , y n} .

For θ ∈ [ - h , 0 ] , applying (g2), we get [figure omitted; refer to PDF] Using now condition (g3) and Lemma 9 we infer that [figure omitted; refer to PDF] Now we consider functions _{u n} defined on [ 0 , T ] . From (43) and using Lemma 7, we get [figure omitted; refer to PDF] Using now conditions (g2), (I2), and Remark 14, we have [figure omitted; refer to PDF] On the other hand, since _{f n} ∈ _{...AE; F , x n , y n} , for t ∈ [ 0 , T ] we have that _{f n} ( t ) ∈ F ( t , _{x n} ( t ) , _{y n} ( t ) , ( _{x n ) t} ) . This implies that { _{f n} : n ∈ ... } is uniformly integrable and, applying condition (F4), [figure omitted; refer to PDF] Combining this estimate with Lemma 11 we infer that [figure omitted; refer to PDF] Substituting in (46), we obtain [figure omitted; refer to PDF] Combining with (45), and using Lemma 12, it yields [figure omitted; refer to PDF] We next estimate β ( { _{v n} : n ∈ ... } ) . Using (40) we can write [figure omitted; refer to PDF] for _{f n} ∈ _{...AE; F , x n , y n} .

From (52) and using Lemma 7, we get [figure omitted; refer to PDF] Using again conditions (g2) and (I2), Lemma 12, and also our previous estimates, we obtain [figure omitted; refer to PDF] Finally, collecting these estimates, we get [figure omitted; refer to PDF] This implies that β ( Ω ) = β ( { ( _{x n} , _{y n} ) : n ∈ ... } ) = 0 , which in turn implies that ... is a β -condensing map.

Corollary 19.

Under the hypotheses of Theorem 18, there exists a mild solution of problems (1)-(4).

Proof.

It follows from Theorem 18 and Theorem 6 that there is a fixed point ( x , y ) of ... . It is follows from (17), (39), and (40) that x ( · ) ∈ P ^{C 1} ( [ - h , T ] ; X ) and that x ( · ) is a fixed point of Γ .

The sine functions S ( t ) involved in concrete problems are frequently compact. This allows us to reduce the conditions to obtain the existence of mild solutions to problems (1)-(4). To establish this result some previous properties about sine operators are needed.

Lemma 20.

Assume that S ( t ) is a compact operator for all t ∈ ... . If D ⊂ X is a bounded set, then the set { S ( · ) x : x ∈ D } is relatively compact in C ( [ 0 , T ] ; X ) .

Proof.

The set S ( t ) ( D ) is relatively compact in X for all t ∈ [ 0 , T ] . Moreover, for fixed t ∈ [ 0 , T ] and s ∈ ... such that t + s ∈ [ 0 , T ] we can decompose [figure omitted; refer to PDF] If we restrict us to consider x ∈ D , using that S ( t ) ( D ) is relatively compact, C ( t ) ( D ) is bounded, and || S ( s ) || ...4; M s , we obtain that ( C ( s ) - I ) S ( t ) x [arrow right] 0 and S ( s ) C ( t ) x [arrow right] 0 when s [arrow right] 0 uniformly for x ∈ D . Consequently, the set { S ( · ) x : x ∈ D } is equicontinuous, and the Ascoli-Arzelá theorem implies that { S ( · ) x : x ∈ D } is relatively compact in C ( [ 0 , T ] ; X ) .

Lemma 21.

Assume that S ( t ) is a compact operator for all t ∈ ... . Then the map _{Λ 1} is compact.

Proof.

Let W ⊂ ^{L 1} ( [ 0 , T ] ; X ) be a bounded set. It follows from [40, Theorem 5] that the set { _{Λ 1} ( u ) ( t ) : u ∈ W } is relatively compact in X for every t ∈ [ 0 , T ] . On the other hand, using again (56) we can write [figure omitted; refer to PDF] Since { _{Λ 1} ( u ) ( t ) : u ∈ W } is relatively compact in X , || ( C ( s ) - I ) ^{∫ 0 t} ... S ( t - ξ ) u ( ξ ) d ξ || [arrow right] 0 as s [arrow right] 0 uniformly for u ∈ W . Combining with the above estimate, it follows that _{Λ 1} ( u ) ( t + s ) - _{Λ 1} ( u ) ( t ) [arrow right] 0 as s [arrow right] 0 uniformly for u ∈ W . Therefore, the set _{Λ 1} ( W ) is equicontinuous. The Ascoli-Arzelá theorem shows that _{Λ 1} is a compact operator.

We define the constants [figure omitted; refer to PDF]

Corollary 22.

Assume that the operator S ( t ) is compact for all t ∈ ... . Assume further that [straight phi] ( 0 ) ∈ E and that conditions (F1)-(F4), (g1)-(g3), and (I1)-(I2) hold. If ^{N 7 [variant prime]} < 1 , then there exists a mild solution of problems (1)-(4).

Proof.

We repeat the construction carried out in the proof of Theorem 18. The only modification is related with the estimate of β ( { _{u n} ( · ) : n ∈ ... } ) for _{u n} defined on [ 0 , T ] . Using Lemmas 20 and 21 we can see that [figure omitted; refer to PDF] Combining with (45), for _{u n} defined on [ - h , T ] , we obtain [figure omitted; refer to PDF] Proceeding as in the proof of Theorem 18 and Corollary 19, we get that Γ has a fixed point x , which is a mild solution of problems (1)-(4).

We now are concerned with the following particular case of problems (1)-(4): [figure omitted; refer to PDF] From an intuitive viewpoint this model corresponds to an incomplete second order equation in which the impulses on the path do not lead to changes in the velocity.

We can reduce this problem to a particular case of problems (1)-(4) taking _{I k} as ^{I k 1} with ^{I k 2} = 0 and modifying slightly the conditions about F , _{I k} , and g . We assume that F is a multivalued map from J × X × ^{C h p} into ...A6; [upsilon] ( X ) that satisfies conditions (F1)-(F4) (now we omit the variable y in these conditions). Proceeding as in Remark 13, for x ( · ) ∈ PC ( [ - h , T ] ; X ) the function [ 0 , T ] [arrow right] ...A6; [upsilon] ( X ) , t ... F ( t , x ( t ) , _{x t} ) admits a Bochner integrable selection. As a consequence, the set [figure omitted; refer to PDF] and _{...AE; F , x} is convex.

Next we describe the conditions on the function g . We assume that g is a map from PC ( [ - h , T ] ; X ) into C ( [ - h , 0 ] ; X ) that satisfies the following.

(g1) The function g is continuous and takes bounded sets in PC ( [ - h , T ] ; X ) into bounded subsets of C ( [ - h , 0 ] ; X ) .

(g2) There is a continuous function [cursive l] : [ - h , 0 ] [arrow right] [ 0 , ∞ ) such that [figure omitted; refer to PDF]

: for all bounded sets W ⊂ PC ( [ - h , T ] , X ) .

(g3) For each bounded set W ⊂ PC ( [ - h , T ] ; X ) the set g ( W ) is equicontinuous.

Next we establish the conditions on maps _{I k} , k = 1 , ... , m . We assume that _{I k} : X × ^{C h p} [arrow right] X satisfy the following conditions.

(I1) The maps _{I k} , k = 1 , ... , m are continuous and takes bounded sets into bounded sets.

(I2) There are positive constants ^{d k j} , j = 1,2 , k = 1 , ... , m , such that [figure omitted; refer to PDF]

: for all bounded sets _{D 1} ⊂ X and W ⊂ ^{C h p} such that _{sup ... - h ...4; θ ...4; 0} { || ψ ( θ ) || : ψ ∈ W } < ∞ .

We now establish our concept of mild solution.

Definition 23.

A function x ( · ) ∈ PC ( [ - h , T ] ; X ) is said to be a mild solution of (61)-(63) if conditions (62)-(63) are satisfied, and the integral equation [figure omitted; refer to PDF] is verified for f ∈ _{...AE; F , x} and all t ∈ [ 0 , T ] .

We next define the solution map associated with our concept of mild solution for problems (61)-(63) as follows. Let x ∈ PC ( [ - h , T ] ; X ) . We define Γ ( x ) to be the set formed by all functions u given by [figure omitted; refer to PDF] for f ∈ _{...AE; F , x} . It follows from our hypotheses that u ∈ PC ( [ - h , T ] ; X ) . Hence, Γ : PC ( [ - h , T ] ; X ) [arrow right] ...AB; ( PC ( [ - h , T ] ; X ) ) . Furthermore, it is clear that x ( · ) is a mild solution of problems (61)-(63) if and only if x ( · ) is a fixed point of Γ .

We define [figure omitted; refer to PDF] We are now in a position to prove the following result.

Theorem 24.

Assume that conditions (F1)-(F4), (g1)-(g3), and (I1)-(I2) hold. If ^{N 7 [variant prime][variant prime]} < 1 , then the map Γ : P C ( [ - h , T ] ; X ) [arrow right] K [upsilon] ( P C ( [ - h , T ] ; X ) ) is u.s.c. and β -condensing.

Proof.

We proceed as in the proof of Theorem 18. We only include here a sketch of the proof. To prove that Γ is β -condensing. Let Ω ⊂ PC ( [ - h , T ] ; X ) be a bounded set such that β ( Γ ( Ω ) ) ...5; β ( Ω ) . It follows from Lemma 10 that there exists a sequence ( _{u n ) n} in Γ ( Ω ) such that β ( Γ ( Ω ) ) = β ( { _{u n} : n ∈ ... } ) . We can write _{u n} ∈ Γ ( _{x n} ) for some _{x n} ∈ Ω .

To estimate β ( { _{u n} : n ∈ ... } ) , using (68) we can write [figure omitted; refer to PDF] for _{f n} ∈ _{...AE; F , x n} .

From (45), we have [figure omitted; refer to PDF]

Now we consider functions _{u n} defined on [ 0 , T ] . From (70) and using Lemma 7, we get [figure omitted; refer to PDF]

Using now conditions (g2), (I2), and Remark 14, we have [figure omitted; refer to PDF]

On the other hand, since _{f n} ∈ _{...AE; F , x n} , for t ∈ [ 0 , T ] , we have that _{f n} ( t ) ∈ F ( t , _{x n} ( t ) , ( _{x n ) t} ) . This implies that { _{f n} : n ∈ ... } is uniformly integrable and, applying condition (F4), [figure omitted; refer to PDF]

Combining this estimate with Lemma 11 we infer that [figure omitted; refer to PDF]

Substituting this estimate in (72), we obtain [figure omitted; refer to PDF] Collecting these assertions, we get [figure omitted; refer to PDF] which implies that β ( Γ ( Ω ) ) = 0 , which in turn shows that Γ is β -condensing and completes the proof.

The following assertions are immediate consequences of Theorem 24.

Corollary 25.

Under the hypotheses of Theorem 24, there exists a mild solution of problems (61)-(63).

Corollary 26.

Assume that the operator S ( t ) is compact for all t ∈ ... . Assume further that conditions (F1)-(F4), (g1)-(g3), and (I1)-(I2) hold. If [figure omitted; refer to PDF] then there exists a mild solution of problems (61)-(63).

4. Applications

In this section we apply our abstract results to study the existence of solutions to the impulsive retarded wave equation described by (5)-(9). To model this problem in abstract form, in what follows we consider the space X = ^{L 2} ( [ 0 , π ] ) and A : D ( A ) ⊆ X [arrow right] X is the map defined by A x = ( ^{d 2} / d ^{ξ 2} ) x ( ξ ) with domain D ( A ) = { x ∈ X : ^{x [variant prime][variant prime]} ∈ X , x ( 0 ) = x ( π ) = 0 } . It is well known that A is the infinitesimal generator of a strongly continuous cosine function ( C ( t ) _{) t ∈ ...} on X . Furthermore, A has a discrete spectrum and the eigenvalues are - ^{n 2} , n ∈ ... , with corresponding eigenvectors _{z n} ( ξ ) = ^{( 2 / π ) 1 / 2} sin ( n ξ ) . Furthermore, the set { _{z n} : n ∈ ... } is an orthonormal basis of X and the following properties hold.

(a) For x ∈ D ( A ) , A x = - ^{∑ n = 1 ∞} ... ^{n 2} Y9; x , _{z n} YA; _{z n} .

(b) For x ∈ X , [figure omitted; refer to PDF]

: Consequently, || C ( t ) || = || S ( t ) || ...4; 1 for all t ∈ ... and S ( t ) is a compact operator for every t ∈ ... .

(c) The space E = { x ∈ ^{H 1} ( 0 , π ) : x ( 0 ) = x ( π ) = 0 } (see [30] for details) and _{|| x || E} ...4; || x || + || ^{x [variant prime]} || . In particular, we observe that the inclusion ι : E [arrow right] X is compact. Moreover, the function S ( · ) is 2 π -periodic. Using this property and (13) we can show that [figure omitted; refer to PDF]

In fact, using the periodicity of S ( · ) is sufficient to establish the property for t ∈ [ - π , π ] . It is an immediate consequence of the definition of the norm in E that _{|| A S ( t ) || [Lagrangian (script capital L)] ( E ; X )} ...4; 1 , for t ∈ [ 0,1 ] . For t ∈ [ 1 , π / 2 ] , we can write t = 1 + s with s ∈ [ 0,1 ] , and using (13) we have [figure omitted; refer to PDF] Since || C ( τ ) || ...4; 1 for all τ ∈ ... , we obtain [figure omitted; refer to PDF] Similarly, for t ∈ [ π / 2 , π ] , we can write t = π - s with s ∈ [ 0 , π / 2 ] , and using again (13), we can write [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] In view of that A S ( - t ) = - A S ( t ) , this completes the proof of the assertion.

In what follows we assume that z ∈ X and that [straight phi] ∈ C ( [ - h , 0 ] ; X ) , where we have identified [straight phi] ( θ ) ( ξ ) = [straight phi] ( θ , ξ ) for θ ∈ [ - h , 0 ] and ξ ∈ [ 0 , π ] .

Initially we construct the multivalued function F . We assume that _{f 0} : J × [ 0 , π ] × ^{... 3} [arrow right] ...A6; [upsilon] ( ... ) is a bounded multivalued map that satisfies the following conditions.

(f1) There exist positive constants L , _{L 1} , and _{L 2} such that [figure omitted; refer to PDF]

: for all ( t , _{ξ 1} , _{[varphi] 1} ) , ( t , _{ξ 2} , _{[varphi] 2} ) ∈ J × [ 0 , π ] × ^{... 3} , where _{d H} denotes the Hausdorff metric and || · || denotes the Euclidean norm in ^{... 3} .

(f2) There exists a positive function μ ∈ ^{L 1} ( [ 0 , T ] ) such that | s | ...4; μ ( t ) for all s ∈ _{f 0} ( t , 0 ) .

In a metric space ( Ω , d ) , we denote ρ ( w , B ) = inf ... ... { d ( w , b ) : b ∈ B } . We will use the following property of the Hausdorff metric.

Lemma 27.

Let ( Ω , d ) be a metric space, and let _{B 1} , _{B 2} ⊆ Ω be bounded sets. Then, for every u , v ∈ Ω , [figure omitted; refer to PDF]

We have the following consequences.

Proposition 28.

Under the previous conditions, the following properties hold.

(i) For each t ∈ J and [varphi] ∈ ^{... 3} , the function _{f 0} ( t , · , [varphi] ) is measurable.

(ii) For each t ∈ J and ξ ∈ [ 0 , π ] , the function _{f 0} ( t , ξ , · ) is upper semicontinuous.

(iii): For each t ∈ J and [varphi] ∈ ^{L 2} ( [ 0 , π ] ; ^{... 3} ) the set [figure omitted; refer to PDF]

: is closed convex in ^{L 2} ( [ 0 , π ] ) .

Proof.

Consider the following.

(i) It is an immediate consequence of the fact that the multivalued map _{f 0} ( t , · , [varphi] ) is _{d H} -continuous.

(ii) We know that _{f 0} ( t , ξ , ^{... 3} ) is bounded and, therefore, relatively compact. We will show that the graph f ( t , ξ , · ) is closed. Assume that _{[varphi] n} , [varphi] ∈ ^{... 3} and _{[varphi] n} [arrow right] [varphi] , _{s n} ∈ _{f 0} ( t , ξ , _{[varphi] n} ) , _{s n} [arrow right] s as n [arrow right] ∞ . Using Lemma 27 we can write [figure omitted; refer to PDF]

: Since _{f 0} ( t , ξ , · ) is _{d H} -continuous, it follows that ρ ( s , _{f 0} ( t , ξ , [varphi] ) ) = 0 . In view of that the set _{f 0} ( t , ξ , [varphi] ) is closed, we conclude that s ∈ _{f 0} ( t , ξ , [varphi] ) . Applying [3, Proposition 1.2] we obtain that _{f 0} ( t , ξ , · ) is upper semicontinuous.

(iii): For t ∈ J and [varphi] ∈ ^{L 2} ( [ 0 , π ] ; ^{... 3} ) the map _{f 0} ( t , · , [varphi] ) : [ 0 , π ] [arrow right] ...A6; [upsilon] ( ... ) , ξ ... _{f 0} ( t , ξ , [varphi] ( ξ ) ) , is measurable with closed values. It follows from [3, Proposition 3.2] that there exists a measurable selection w such that w ( ξ ) ∈ _{f 0} ( t , ξ , [varphi] ( ξ ) ) . Using that f is bounded it follows that w ∈ ^{L 2} ( [ 0 , π ] ) . This shows that _{...AF; f 0 , [varphi]} ( t ) ...0; ∅ . Since the values of _{f 0} are convex, it follows that _{...AF; f 0 , [varphi]} ( t ) is also convex.

To establish that _{...AF; f 0 , [varphi]} ( t ) is closed, we consider a sequence _{w n} , w ∈ ^{L 2} ( [ 0 , π ] ) , _{w n} ∈ _{...AF; f 0 , [varphi]} ( t ) such that _{w n} [arrow right] w , n [arrow right] ∞ , for the norm in ^{L 2} ( [ 0 , π ] ) . By passing to a subsequence if necessary, we can assume that _{w n} ( ξ ) [arrow right] w ( ξ ) , n [arrow right] ∞ , a.e. ξ ∈ [ 0 , π ] . Since _{f 0} ( t , ξ , [varphi] ( ξ ) ) is closed, it follows that w ( ξ ) ∈ _{f 0} ( t , ξ , [varphi] ( ξ ) ) , which in turn implies that w ∈ _{...AF; f 0 , [varphi]} ( t ) .

For [varphi] ∈ ^{L 2} ( [ 0 , π ] ; ^{... 3} ) we define [figure omitted; refer to PDF] and _{F 1} ( [varphi] ) : J [arrow right] ...AB; ( X ) is given by _{F 1} ( [varphi] ) ( t ) = ^{...AF; _{f 0} , [varphi] 1} ( t ) .

Proposition 29.

Under the previous conditions, _{F 1} is a measurable and upper semicontinuous map with convex compact values.

Proof.

Initially we show that ^{...AF; _{f 0} , [varphi] 1} ( t ) is closed. Let _{[upsilon] n} ( ξ ) = ^{∫ 0 ξ} ... _{w n} ( η ) d η be a sequence in ^{...AF; _{f 0} , [varphi] 1} ( t ) that converges to [upsilon] as n [arrow right] ∞ . Since _{...AF; f 0 , [varphi]} ( t ) is sequentially weakly compact, there is w ∈ ^{L 2} ( [ 0 , π ] ) and a subsequence _{w n k} such that _{w n k} [arrow right] w as k [arrow right] ∞ in the weak topology. Since _{...AF; f 0 , [varphi]} ( t ) is a closed convex set, it follows that w ∈ _{...AF; f 0 , [varphi]} ( t ) . In view of that [figure omitted; refer to PDF] where _{χ [ 0 , ξ ]} denotes the characteristic function of the interval [ 0 , ξ ] , we obtain that [upsilon] ( ξ ) = ^{∫ 0 ξ} ... w ( η ) d η and [upsilon] ∈ ^{...AF; _{f 0} , [varphi] 1} ( t ) .

Since the functions in _{...AF; f 0 , [varphi]} ( t ) are uniformly bounded, [figure omitted; refer to PDF] converge to zero as δ [arrow right] 0 uniformly for v ∈ ^{...AF; _{f 0} , [varphi] 1} ( t ) . From [41, Theorem IV.8.20] we conclude that ^{...AF; _{f 0} , [varphi] 1} ( t ) is relatively compact.

On the other hand, proceeding as in the proof of Proposition 28(ii), we get that _{F 1} ( [varphi] ) is upper semicontinuous. Finally, as a consequence of a remark in [5, page 21], we can affirm that _{F 1} ( [varphi] ) is measurable.

The following consequence is essential for our construction.

Corollary 30.

Under the above conditions, there exists a measurable selection for _{F 1} ( [varphi] ) .

Proof.

It follows from [3, Proposition 3.2].

We now consider the map F : J × X × X × ^{C h 2} [arrow right] ...A6; [upsilon] ( X ) defined by [figure omitted; refer to PDF] for [varphi] = ( x , y , ^{∫ - h 0} ... ψ ( θ ) d θ ) . Since F ( · , x , y , ψ ) = _{F 1} ( [varphi] ) ( · ) , it follows from our construction that F satisfies condition (F1). Moreover, proceeding as in the proof of Proposition 28(ii) we conclude that F is upper semicontinuous, which shows that F satisfies condition (F2). On the other hand, if [upsilon] ∈ F ( t , x , y , ψ ) , then there exists w ∈ _{...AF; f 0 , [varphi]} ( t ) such that [upsilon] ( ξ ) = ^{∫ 0 ξ} ... w ( η ) d η with w ( ξ ) ∈ _{f 0} ( t , ξ , x ( ξ ) , y ( ξ ) , ^{∫ - h 0} ... ψ ( θ , ξ ) d θ ) . Therefore, there exists ^{w 1} ( ξ ) ∈ _{f 0} ( t , 0 ) such that [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] which shows that F satisfies the condition (F3).

Proposition 31.

Let x , ^{x 1} , y , ^{y 1} ∈ X , ψ , ^{ψ 1} ∈ ^{C h 2} , and v ∈ F ( t , x , y , ψ ) . Then [figure omitted; refer to PDF]

Proof.

To abbreviate the text we introduce the notations [varphi] ( ξ ) = ( x ( ξ ) , y ( ξ ) , ^{∫ - h 0} ... ψ ( θ , ξ ) d θ ) and ^{[varphi] 1} ( ξ ) = ( ^{x 1} ( ξ ) , ^{y 1} ( ξ ) , ^{∫ - h 0} ... ^{ψ 1} ( θ , ξ ) d θ ) . Since v ∈ ^{...AF; _{f 0} , [varphi] 1} ( t ) , there is w ∈ _{...AF; f 0 , [varphi]} ( t ) such that v ( ξ ) = ^{∫ 0 ξ} ... w ( η ) d η . Moreover, in view of that ^{...AF; _{f 0} , [varphi] 1 1} ( t ) is convex compact in ^{L 2} ( [ 0 , π ] ) , there is _{v 0} ∈ ^{...AF; _{f 0} , [varphi] 1 1} ( t ) the nearest point of v in ^{...AF; _{f 0} , [varphi] 1 1} ( t ) . Consequently, there is _{w 0} ∈ _{...AF; f 0 , ^{[varphi] 1}} ( t ) such that _{v 0} ( ξ ) = ^{∫ 0 ξ} ... _{w 0} ( η ) d η . Therefore, [figure omitted; refer to PDF] which completes the proof.

Corollary 32.

Under the above conditions, let _{Ω i} ⊂ ^{L 2} ( [ 0 , π ] ) , i = 1,2 , be bounded sets, and let Q ⊂ ^{C h 2} be a set uniformly bounded. Then [figure omitted; refer to PDF] where γ ( Q ) denotes the Hausdorff measure of noncompactness for the norm of uniform convergence.

Proof.

Let [varepsilon] > 0 . We abbreviate the notation by writing _{s i} = χ ( _{Ω i} ) and s = γ ( Q ) . There exist ^{x 1} , ... , ^{x _{i 1}} ∈ ^{L 2} ( [ 0 , π ] ) , ^{y 1} , ... , ^{y _{j 1}} ∈ ^{L 2} ( [ 0 , π ] ) , and ^{ψ 1} , ... , ^{ψ _{k 1}} ∈ ^{C h 2} having the following property: given x ∈ _{Ω 1} , y ∈ _{Ω 2} , and ψ ∈ Q there are ^{x i} , ^{y j} , and ^{ψ k} such that || x - ^{x i} || ...4; _{s 1} + [varepsilon] , || y - ^{y j} || ...4; _{s 2} + [varepsilon] and _{|| ψ - ^{ψ k} || ∞} ...4; s + [varepsilon] . Hence, if v ∈ F ( t , x , y , ψ ) , using Proposition 31, we obtain that [figure omitted; refer to PDF] Since F ( t , ^{x i} , ^{y j} , ^{ψ k} ) is a compact set in ^{L 2} ( [ 0 , π ] ) , and [varepsilon] > 0 was chosen arbitrarily, this completes the proof of the assertion.

Corollary 32 shows that F satisfies the condition (F4).

On the other hand, we define g : PC ( [ - h , T ] ; X ) [arrow right] C ( [ - h , 0 ] ; X ) by [figure omitted; refer to PDF] for θ ∈ [ - h , 0 ] and ξ ∈ [ 0 , π ] . We assume that σ ( · ) is a function of class ^{C 1} such that σ ( π ) = 0 . It is clear that [figure omitted; refer to PDF] and g ( x ) ( 0,0 ) = g ( x ) ( 0 , π ) = 0 . This implies that g ( x ) ( 0 ) ∈ E for all x ∈ PC ( [ - h , T ] ; X ) . Moreover, g is a continuous map that takes bounded sets into bounded sets, and g ( x ) ( 0 ) : PC ( [ - h , T ] ; X ) [arrow right] E is also continuous and takes bounded sets into bounded sets in E . This shows that g satisfies condition (g1).

It is clear that g ( x ) ( θ ) is a continuous function from [ 0 , π ] into ... for each x ∈ PC ( [ - h , T ] , X ) . Let W ⊂ PC ( [ - h , T ] , X ) be a bounded set. It is not difficult to see that the set { g ( x ) ( θ ) : x ∈ W } is equicontinuous. Moreover, [figure omitted; refer to PDF] which shows that { g ( x ) ( θ , ξ ) : x ∈ W } is bounded. The Ascoli-Arzelá theorem implies that { g ( x ) ( θ ) : x ∈ W } is relatively compact in C ( [ 0 , π ] ) . Therefore, { g ( x ) ( θ ) : x ∈ W } is also relatively compact in ^{L 2} ( [ 0 , π ] ) . Hence χ ( g ( W ) ( θ ) ) = 0 , and we can take [cursive l] ( θ ) = 0 . This shows that g satisfies the first part of condition (g2). Furthermore, it follows from (c) that [figure omitted; refer to PDF] From the definition of g we obtain [figure omitted; refer to PDF] Arguing as above, we can affirm that the first term on the right hand side of (103) defines a relatively compact set in X . Since σ ( ξ ) ^{∫ 0 T} ... x ( t , ξ ) d t = σ ^{∫ 0 T} ... x ( t ) d t in the space ^{L 2} ( [ 0 , π ] ) , using [42, Theorem 3.1], we conclude that [figure omitted; refer to PDF] which shows that g also satisfies the second part of condition (g2) with _{[cursive l] 1} = _{|| σ || ∞} T .

On the other hand, [figure omitted; refer to PDF] converges to zero as δ [arrow right] 0 uniformly for x ∈ W . This shows that the set g ( W ) is equicontinuous. Consequently, we can affirm that g satisfies condition (g3).

We define ^{I k 1} , ^{I k 2} : X × X × ^{C h 2} [arrow right] X by [figure omitted; refer to PDF] for x , y ∈ ^{L 2} ( [ 0 , π ] ) , and ξ ∈ [ 0 , π ] . We assume that ^{q k i} ( · ) , ^{a k i} ( · ) , ^{b k i} ( · ) ∈ ^{L 2} ( [ 0 , π ] ) for i = 1,2 , and that ^{a k 1} ( · ) , ^{b k 1} ( · ) ∈ ^{C 1} ( [ 0 , π ] ) are functions such that ^{a k 1} ( 0 ) = ^{a k 1} ( π ) = ^{b k 1} ( 0 ) = ^{b k 1} ( π ) = 0 . Proceeding as above, it is easy to see that ^{I k 1} : X × X × ^{C h 2} [arrow right] E and ^{I k 2} : X × X × ^{C h 2} [arrow right] X are continuous maps that take bounded sets into bounded sets. This shows that condition (I1) is verified. In addition, using that the map X [arrow right] ... , y ... ^{∫ 0 π} ... ^{q k 2} ( η ) y ( η ) d η , is a bounded linear functional with norm || ^{q k 2} || , we deduce that [figure omitted; refer to PDF] Using this argument together with condition (c), we get [figure omitted; refer to PDF] which shows that condition (I2) is also verified.

We complete our model by defining x ( t ) = u ( t , · ) . It is not difficult to see that under the conditions specified previously, systems (5)-(9) are described by the abstract models (1)-(4). The constants _{N i} introduced in Section 3 are the following: [figure omitted; refer to PDF]

Combining with Corollary 22, we have established the following result.

Theorem 33.

Assume that [straight phi] ( 0 , · ) ∈ E , and [figure omitted; refer to PDF] Then there exists a mild solution of systems (5)-(9).

Acknowledgment

Hernán R. Henríquez was partially supported by CONICYT under Grants FONDECYT 1130144 and DICYT-USACH.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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