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.
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→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<t1<t2,…,tm<T. Moreover, h>0 denotes the system delay. Specifically, we will consider abstract second order systems
(1)x′′(t)-Ax(t)∈F(t,x(t),x′(t),xt),t∈J=[0,T],t≠tk,k=1,…,m,(2)Δx(tk)=Ik1(x(tk),x′(tk),xtk),k=1,…,m,(3)Δx′(tk)=Ik2(x(tk),x′(tk),xtk),k=1,…,m,(4)x(θ)+g(x)(θ)=φ(θ),θ∈[-h,0],x′(0)=z∈X,
where x(t)∈X, xt, t≥0 denotes the function defined by xt(θ)=x(t+θ) for θ∈[-h,0], Δy(t)=y(t+)-y(t-) indicates the gap of a piecewise continuous function y(·) at t, φ:[-h,0]→X is an appropriate function, and F, Ik1, Ik2, 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(5)∂2u(t,ξ)∂t2-∂2u(t,ξ)∂ξ2∈∫0ξf0(t,η,u(t,η)∂u(t,η)∂t∫-h0u(t+θ,η)dθ)dη,∈∫0ξf0(t,η,u(t,η)∂u(t,η)∂tu(t+θ,η)55.5511t≠tk,(6)u(t,0)=u(t,π)=0,t∈[0,T],(7)u(tk+,ξ)=u(tk,ξ)+ak1(ξ)∫0πqk1(η)u(tk,η)dη+bk1(ξ),(8)∂∂tu(tk+,ξ)=∂∂tu(tk,ξ)+ak2(ξ)∫0πqk2(η)∂∂tu(tk,η)dη+bk2(ξ),(9)u(θ,ξ)+σ(ξ)∫0T∫0ξu(t+θ,η)dηdt=φ(θ,ξ),θ∈[-h,0],∂∂tu(0,ξ)=z(ξ),
for t∈J=[0,T], ξ∈(0,π), and k=1,…,m. In this system we assume that f0 is a multivalued map, and the inclusion indicated in (5) will be explained in Section 4. Moreover, aki, bki, qki, i=1,2, φ 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 ℒ(Y,Z) the Banach space of the bounded linear operators from Y into Z and we abbreviate this notation to 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
(10)S(t)x=∫0tC(s)xds,x∈X,t∈ℝ.
We denote by M, M1 some positive constants such that ∥C(t)∥≤M and ∥S(t)∥≤M1 for t∈J. The function S:ℝ→ℒ(X) is continuous for the norm of operators and ∥S(t)∥≤Mt 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 C1 on ℝ. We know from Kisyński [35] that E endowed with the norm
(11)∥x∥E=∥x∥+sup0≤t≤1∥AS(t)x∥,x∈E,
is a Banach space.
The operator valued function G(t)=[C(t)S(t)AS(t)C(t)] is a strongly continuous group of bounded linear operators on the space E×X, generated by the operator 𝒜=[0IA0] defined on D(A)×E. It follows from this property that AS(t):E→X is a bounded linear operator and that AS(·):ℝ→ℒ(E;X) is a strongly continuous operator valued map. We denote by M2 a positive constant such that ∥AS(t)∥ℒ(E;X)≤M2 for all t∈J. In addition ([34])
(12)S(t+s)=C(t)S(s)+C(s)S(t),s,t∈ℝ,
which implies that
(13)AS(t+s)x=C(t)AS(s)x+C(s)AS(t)x,AS(t+s)x=C(t)AS(s)xx∈E,s,t∈ℝ.
Furthermore, if x:[0,∞)→X is a locally integrable function, then
(14)y(t)=∫0tS(t-s)x(s)ds
defines an E-valued continuous function.
The existence of solutions of the second order abstract Cauchy problem
(15)x′′(t)=Ax(t)+f(t),t∈J,x(0)=z1,x′(0)=z2,
where f:[0,T]→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
(16)x(t)=C(t)z1+S(t)z2+∫0tS(t-s)f(s)ds,t∈J,
is called mild solution of (15), and that when z1∈E the function x(·) is continuously differentiable and
(17)x′(t)=AS(t)z1+C(t)z2+∫0tC(t-s)f(s)ds.
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 𝒫(Ω) denotes the collection of all nonempty subsets of Ω and 𝒫b(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
(18)η(B)=inf{ɛ>0:Bhasafinitecoverbyclosedballsofradius<ɛη(B)=inf5byclosedballsofradius<ɛ}.
Remark 2.
Let B1,B2⊆Ω be bounded sets. The Hausdorff measure of noncompactness has the following properties.
If B1⊆B2, then η(B1)⩽η(B2).
η(B)=η(B¯).
η(B)=0 if and only if B is totally bounded.
η(B1∪B2)=max{η(B1),η(B2)}.
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 B1,B2⊆Y be bounded sets. The following properties hold.
For λ∈ℝ, η(λB)=|λ|η(B).
η(B1+B2)⩽η(B1)+η(B2), where B1+B2={b1+b2:b1∈B1,b2∈B2}.
η(B)=η(co¯(B)).
Henceforth we use the notations υ(Y) and 𝒦υ(Y) to denote the following sets:
υ(Y)={D∈𝒫(Y):Disconvex},
𝒦υ(Y)={D∈υ(Y):Discompact}.
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 ℱ:Ω→𝒫(Y) is said to be
upper semicontinuous (u.s.c. for short) if ℱ-1(V)={w∈Ω:ℱ(w)⊆V} is an open subset of Ω for all open set V⊆Y;
closed if its graph Gℱ={(w,y);y∈ℱ(w)} is a closed subset of Ω×Y;
compact if its range ℱ(Ω) is relatively compact in Y;
quasicompact if ℱ(K) is relatively compact in Y for any compact subset K⊂Ω.
Definition 5.
A multivalued map ℱ:Ω→𝒫(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(ℱ(D))<η(D).
The next result is essential for the development of the rest of our work. We point out that if ℱ:Ω→Kυ(Y) is u.s.c., then ℱ 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 ℱ:M→Kυ(M) be a u.s.c. β-condensing multivalued map. Then Fix(ℱ)={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→X satisfying the following conditions:
the function x(·) is continuous on I∖{t1,…,tm}, and
there exist limt→tj+x(t) and limt→tj- and x(tj)=limt→tj-x(t) for all 1≤j≤m.
We consider PC(I;X) endowed with the norm of the uniform convergence
(19)∥x∥PC=supt∈I∥x(t)∥.
It is well known that PC(I;X) is a Banach space. Furthermore, let Πj:PC(I;X)→C([tj,tj+1];X), -1≤j≤m, be the map defined by
(20)Πj(x)(t)={x(t)ift∈(tj,tj+1],x(tj+)ift=tj,
where we set t-1=-h, t0=0, and tm+1=T. For each j∈{0,…,m} and D⊆PC([-h,T];X), we denote by Dj the range of D under the operator Πj; that is, Dj=Πj(D).
We define the subspace PC1([-h,T];X) of PC([-h,T];X) consisting of all functions which are continuously differentiable at [0,T]∖{t1,…,tm} and there exist x′(tk+) and x′(tk-) for all 1≤k≤m. It is straightforward to show that the space PC1([-h,T];X) endowed with the norm
(21)∥x∥PC1=∥x∥PC+sup0≤t≤T∥x′(t)∥
is a Banach space.
From now on we denote by Chp, 1≤p<∞, the space of piecewise continuous functions v:[-h,0]→X endowed with the norm
(22)∥v∥Chp=1h1/p(∫-h0∥v(θ)∥pdθ)1/p.
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]→ℒ(X) be a strongly continuous operator valued map. Let D⊂X be a bounded set. Then β({G(·)x:x∈D})≤sup0≤t≤T∥G(t)∥χ(D).
Lemma 8.
Let D⊂PC([-h,T];X) be a bounded set. Then β(D)=maxk=-1,…,mβ(Dk).
Lemma 9 (see [39]).
Let W⊆C(J;X) be a bounded set. Then χ(W(t))⩽β(W) for all t∈J. Furthermore, if W is equicontinuous on J, then χ(W(t)) is continuous on J, and
(23)β(W)=sup{χ(W(t)):t∈J}.
Lemma 10.
Let D⊆C(J;X) be a bounded set. Then there exists a countable set D0⊆D such that β(D0)=β(D).
A set W⊆L1(J;X) is said to be uniformly integrable if there exists a positive function μ∈L1(J) such that ∥w(t)∥≤μ(t) a.e. for t∈J and all w∈W.
Lemma 11.
Let G:[0,T]→ℒ(X) be a strongly continuous operator valued map and Λ:L1([0,T];X)→C([0,T];X) be the map defined by
(24)Λ(u)(t)=∫0tG(t-s)u(s)ds.
Let W⊂L1([0,T];X). Assume that there is a compact set K⊂X and a positive function q∈L1([0,T]) such that W(t)⊆K for all t∈[0,T] and χ(W(t))≤q(t). Then
(25)β(Λ(W))≤2sup0≤t≤T∥G(t)∥∫0Tq(t)dt.
Proof.
It is clear that W is uniformly integrable. Applying Lemma 10 and [5, Theorem 4.2.2], we can affirm that
(26)χ(Λ(W)(t))≤2sup0≤t≤T∥G(t)∥∫0tq(s)ds.
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
(27)|||(x,y)|||=max{∥x∥PC([-h,T];X),∥y∥PC([0,T];X)}.
The following property is immediate.
Lemma 12.
Let W⊂PC([-h,T];X)×PC([0,T];X) be a bounded set.
Assume that W=W1×W2, where W1⊂PC([-h,T];X) and W2⊂PC([0,T];X) are bounded sets. Then β(W)=max{β(W1),β(W2)}.
Let
(28)W1={x∈PC([-h,T];X):(x,y)∈W,W1=5forsomey∈PC([0,T];X)},W2={y∈PC([0,T];X):(x,y)∈W,W115y∈PC([0,T];X):(x,y)∈Wforsomex∈PC([-h,T];X)}.
Then max{β(W1),β(W2)}≤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 φ∈C([-h,0];X). Moreover, in what follows we assume that F is a multivalued map from J×X×X×Chp into 𝒦υ(X) that satisfies the following properties.
The function F(·,y1,y2,ψ):[0,T]→𝒦υ(X) admits a strongly measurable selection for each yi∈X, i=1,2, and ψ∈Chp.
For each t∈[0,T], the function F(t,·,·,·):X×X×Chp→𝒦υ(X) is u.s.c.
For each r>0, there is a function μr∈L1([0,T]) such that
(29)∥F(t,y1,y2,ψ)∥:=sup{∥v∥:v∈F(t,y1,y2,ψ)}≤μr(t),∥F(t,y1,y2,ψ)∥:=sup{∥v∥:v∈F(t,11ψ)}a.e.t∈[0,T],
for all yi∈X, i=1,2, and ψ∈Chp such that
(30)∥y1∥+∥y2∥+∥ψ∥Chp≤r.
There exists a positive integrable function k(·) on [0,T] such that
(31)χ(F(t,Ω1,Ω2,Q))≤k(t)(χ(Ω1)+χ(Ω2))+supθ∈[-h,0]χ(Q(θ)),a.e.t∈[0,T],
for all bounded sets Ωi⊆X, i=1,2, and Q⊆Chp such that supθ∈[-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]→Chp, t↦xt, is continuous. Hence, the function [0,T]→X×X×Chp, t↦(x(t),y(t),xt), 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]→𝒦υ(X), t↦F(t,x(t),y(t),xt) admits a Bochner integrable selection. As a consequence, the set
(32)𝒮F,x,y={f∈L1(J;X):f(t)∈F(t,x(t),y(t),xt),t∈J}≠∅,
and 𝒮F,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.
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)→E is continuous and takes bounded sets in PC([-h,T];X) into bounded subsets of E.
There is a continuous function ℓ:[-h,0]→[0,∞) and a constant ℓ1≥0 such that
(33)χ(g(W)(θ))≤ℓ(θ)supt∈[-h,T]χ(W(t)),θ∈[-h,0],χE(g(W)(0))≤ℓ1supt∈[-h,T]χ(W(t)),
for all bounded set W⊂PC([-h,T],X).
For each bounded set W⊂PC([-h,T];X) the set g(W) is equicontinuous.
Next we establish the conditions on maps Iki, i=1,2, k=1,…,m.
We assume that Ik1:X×X×Chp→E and Ik2:X×X×Chp→X satisfy the following conditions.
The maps Iki, i=1,2, k=1,…,m are continuous and takes bounded sets into bounded sets.
There are positive constants dki,j, i=1,2, j=0,1,2, k=1,…,m, such that
(34)χE(Ik1(D0×D1×W))≤dk1,0χ(D0)+dk1,1χ(D1)+dk1,2sup-h≤θ≤0χ(W(θ)),χ(Ik2(D0×D1×W))≤dk2,0χ(D0)+dk2,1χ(D1)+dk2,2sup-h≤θ≤0χ(W(θ)),
for all bounded subsets D0,D1 of X, and W⊆Chp such that supθ∈[-h,0]{∥ψ(θ)∥:ψ∈W}<∞.
Remark 14.
Let W⊂PC([-h,T];X) be a bounded set. Then for all t∈[0,T], Wt={wt:w∈W} is a bounded subset of Chp and sup-h≤θ≤0χ(Wt(θ))≤β(W) for all 0≤t≤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(·)∈PC1([-h,T];X) is said to be a mild solution of (1)–(4) if conditions (2)–(4) are satisfied, and the integral equation
(35)x(t)=C(t)(φ(0)-g(x)(0))+S(t)z+∫0tS(t-s)f(s)ds+∑tk<tC(t-tk)Ik1(x(tk),x′(tk),x(tk))+∑tk<tS(t-tk)Ik2(x(tk),x′(tk),x(tk)),
is verified for f∈𝒮F,x,x′ and all t∈[0,T].
To establish our results, we need to study two integral operators defined on the set 𝒮F,x,y for functions x∈PC([-h,T];X) and y∈PC([0,T];X). Initially we mention some properties of 𝒮F,x,y. A first result establishes that 𝒮F,x,y is closed. Specifically we have the following property ([5, Lemma 5.1.1]).
Lemma 16.
Let {xn}n=1∞⊂PC([-h,T];X) and {yn}n=1∞⊂PC([0,T];X) be sequences that converge to x0∈PC([-h,T];X) and y0∈PC([0,T];X), respectively. Suppose that {fn}n=1∞⊂L1([0,T];X), fn∈𝒮F,xn,yn, is a sequence that converges weakly to f0∈L1([0,T];X). Then f0∈𝒮F,x0,y0.
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≤t≤tF(t,x(t),y(t),xt) is compact in X. In addition, as a consequence of (F3), the set 𝒮F,x,y is uniformly integrable over J; that is to say, there exists a positive function μx,y∈L1(J) such that ∥f(t)∥≤μx,y(t) a.e. for t∈J and all f∈𝒮F,x,y.
We introduce now the operators Λ1,Λ2:L1([0,T];X)→C([0,T];X) given by
(36)Λ1f(t)=∫0tS(t-s)f(s)ds,Λ2f(t)=∫0tC(t-s)f(s)ds.
It is clear that Λ1, Λ2 are bounded linear operators. Using Λ1, Λ2 we can construct the multivalued maps Λ~1,Λ~2:PC([-h,T];X)×PC([0,T];X)→υ(C([0,T];X)) given by
(37)Λ~1(x,y)=Λ1(𝒮F,x,y),Λ~2(x,y)=Λ2(𝒮F,x,y).
Since C(·) and S(·) are strongly continuous operator valued functions, the assertion in [5, Lemma 4.2.1] remains valid for Λ1 and Λ2. 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×Chp→𝒦υ(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 φ(0)∈E and let x∈PC1([-h,T];X). We define Γ(x) to be the set formed by all functions u given by
(38)u(t)={φ(t)-g(x)(t),t∈[-h,0],C(t)(φ(0)-g(x)(0))+S(t)z+∫0tS(t-s)f(s)ds+∑tk<tC(t-tk)Ik1(x(tk),x′(tk),xtk)+∑tk<tS(t-tk)Ik2(x(tk),x′(tk),xtk),t∈[0,T],
for f∈𝒮F,x,x′. It follows from our hypotheses that u∈PC1([-h,T];X). Hence, Γ:PC1([-h,T];X)→𝒫(PC1([-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 ℱ:PC([-h,T];X)×PC([0,T];X)→𝒫(PC([-h,T];X)×PC([0,T];X)) defined as follows. For (x,y)∈PC([-h,T];X)×PC([0,T];X), ℱ(x,y) is the set consisting of all functions (u,v) given by
(39)u(t)={φ(t)-g(x)(t),t∈[-h,0],C(t)(φ(0)-g(x)(0))+S(t)z+Λ1(f)(t)+∑tk<tC(t-tk)Ik1(x(tk),y(tk),xtk)+∑tk<tS(t-tk)Ik2(x(tk),y(tk),xtk),t∈[0,T],(40)v(t)={AS(t)(φ(0)-g(x)(0))+C(t)z+Λ2(f)(t)+∑tk<tAS(t-tk)Ik1(x(tk),y(tk),xtk)+∑tk<tC(t-tk)Ik2(x(tk),y(tk),xtk),t∈[0,T],
for f∈𝒮F,x,y. It follows from (g1) and (I1) that ℱ is well defined.
We use the following notations:
(41)Ni=∑k=1m(dki,0+dki,2),i=1,2,Ni+2=∑k=1mdki,1,i=1,2,N5=max{∫0Tmax-h≤θ≤0ℓ(θ),Mℓ(0)+6M1∫0Tk(t)dtN5=max5+M(N1+N3)+M1(N2+N4)6M1∫0Tk(t)dt},N6=M2ℓ1+6M∫0Tk(t)dt+M2(N1+N3)+M(N2+N4),N7=max{N5,N6}.
Theorem 18.
Assume that φ(0)∈E, and conditions (F1)–(F4), (g1)–(g3) and (I1)-(I2) are fulfilled. If N7<1, then the map ℱ:PC([-h,T];X)×PC([0,T];X)→Kυ(PC([-h,T];X)×PC([0,T];X)) is u.s.c. and β-condensing.
Proof.
It follows from our hypotheses and Lemma 17 that ℱ is a u.s.c. multivalued map with convex compact values. It remains to prove that ℱ is β-condensing. Let Ω⊂PC([-h,T];X)×PC([0,T];X) be a bounded set such that β(ℱ(Ω))≥β(Ω). It follows from Lemma 10 that there exists a sequence (wn)n in ℱ(Ω) such that β(ℱ(Ω))=β({wn:n∈ℕ}). We can write wn=(un,vn)∈ℱ(xn,yn) for some (xn,yn)∈Ω. It follows from Lemma 12 that
(42)β({wn:n∈ℕ})≤max{β({un:n∈ℕ}),β({vn:n∈ℕ})}.
Here we will estimate separately the values β({un:n∈ℕ}) and β({vn:n∈ℕ}). To estimate β({un:n∈ℕ}), using (39), we can write
(43)un(t)={φ(t)-g(xn)(t),t∈[-h,0],C(t)(φ(0)-g(xn)(0))+S(t)z+Λ1(fn)(t)+∑tk<tC(t-tk)Ik1(xn(tk),yn(tk),(xn)tk)+∑tk<tS(t-tk)Ik2(xn(tk),yn(tk),(xn)tk),t∈[0,T],
for fn∈𝒮F,xn,yn.
For θ∈[-h,0], applying (g2), we get
(44)χ({un(θ):n∈ℕ})=χ({g(xn)(θ):n∈ℕ})≤ℓ(θ)supt∈[-h,T]χ({xn(t):n∈ℕ}).
Using now condition (g3) and Lemma 9 we infer that
(45)β({φ-g(xn):n∈ℕ})≤max-h≤θ≤0ℓ(θ)β({xn:n∈ℕ}).
Now we consider functions un defined on [0,T]. From (43) and using Lemma 7, we get
(46)β({un(·):n∈ℕ})≤Mχ({g(xn)(0):n∈ℕ})+β({Λ1(fn)(·):n∈ℕ})+M∑k=1mχ({Ik1(xn(tk),yn(tk),(xn)tk):n∈ℕ})+M1∑k=1mχ({Ik2(xn(tk),yn(tk),(xn)tk):n∈ℕ}).
Using now conditions (g2), (I2), and Remark 14, we have
(47)χ({g(xn)(0):n∈ℕ})≤ℓ(0)β({xn:n∈ℕ}),χ({Iki(xn(tk),yn(tk),(xn)tk):n∈ℕ})≤dki,0χ({xn(tk):n∈ℕ})+dki,1χ({yn(tk):n∈ℕ})+dki,2β({xn:n∈ℕ})≤(dki,0+dki,2)β({xn:n∈ℕ})+dki,1β({yn:n∈ℕ}).
On the other hand, since fn∈𝒮F,xn,yn, for t∈[0,T] we have that fn(t)∈F(t,xn(t),yn(t),(xn)t). This implies that {fn:n∈ℕ} is uniformly integrable and, applying condition (F4),
(48)χ({fn(t):n∈ℕ})≤k(t)[sup-h≤θ≤0χ({xn(t):n∈ℕ})+χ({yn(t):n∈ℕ})+sup-h≤θ≤0χ({xn(t+θ):n∈ℕ})]≤k(t)[2β({xn:n∈ℕ})+β({yn:n∈ℕ})].
Combining this estimate with Lemma 11 we infer that
(49)β({Λ1(fn)(·):n∈ℕ})≤2M1[2β({xn:n∈ℕ})+β({yn:n∈ℕ})]∫0Tk(t)dt.
Substituting in (46), we obtain
(50)β({un(·):n∈ℕ})≤Mℓ(0)β({xn:n∈ℕ})+2M1[2β({xn:n∈ℕ})+β({yn:n∈ℕ})]∫0Tk(t)dt+M∑k=1m[(dk1,0+dk1,2)β({xn:n∈ℕ})+dk1,1β({yn:n∈ℕ})]+M1∑k=1m[(dk2,0+dk2,2)β({xn:n∈ℕ})+dk2,1β({yn:n∈ℕ})]≤[Mℓ(0)+4M1∫0Tk(t)dt+MN1+M1N2]×β({xn:n∈ℕ})+[2M1∫0Tk(t)dt+MN3+M1N4]×β({yn:n∈ℕ})≤[Mℓ(0)+6M1∫0Tk(t)dt+M(N1+N3)+M1(N2+N4)∫0Tk(t)dt+M(N1+N3)]β({(xn,yn):n∈ℕ}).
Combining with (45), and using Lemma 12, it yields
(51)β({un(·):n∈ℕ})≤max{max-h≤θ≤0ℓ(θ)β({xn:n∈ℕ}),∫0Tk(t)dt+M(N1+N3)+M1(N2+N4)]≤max5.[Mℓ(0)+6M1∫0Tk(t)dt+M(N1+N3)≤max55.∫0Tk(t)dt+M(N1+N3)+M1(N2+N4)]β({(xn,yn):n∈ℕ})max-h≤θ≤0}≤N5β({(xn,yn):n∈ℕ}).
We next estimate β({vn:n∈ℕ}). Using (40) we can write
(52)vn(t)={AS(t)(φ(0)-g(xn)(0))+C(t)z+Λ2(fn)(t)+∑tk<tAS(t-tk)Ik1(xn(tk),yn(tk),(xn)tk)+∑tk<tC(t-tk)Ik2(xn(tk),yn(tk),(xn)tk),t[0,T],
for fn∈𝒮F,xn,yn.
From (52) and using Lemma 7, we get
(53)β({vn(·):n∈ℕ})≤M2χE({g(xn)(0):n∈ℕ})+β({Λ2(fn)(·):n∈ℕ})+M2∑k=1mχE({Ik1(xn(tk),yn(tk),(xn)tk):n∈ℕ})+M∑k=1mχ({Ik2(xn(tk),yn(tk),(xn)tk):n∈ℕ}).
Using again conditions (g2) and (I2), Lemma 12, and also our previous estimates, we obtain
(54)β({vn(·):n∈ℕ})≤M2ℓ1β({xn:n∈ℕ})+2M[2β({xn:n∈ℕ})+β({yn:n∈ℕ})]×∫0Tk(t)dt+M2∑k=1m[(dk1,0+dk1,2)β({xn:n∈ℕ})×∫0Tk(t)dt+M2∑k=1m5+dk1,1β({yn:n∈ℕ})]+M∑k=1m[(dk2,0+dk2,2)β({xn:n∈ℕ})+M∑k=1m5+dk2,1β({yn:n∈ℕ})]≤[M2ℓ1+4M∫0Tk(t)dt+M2N1+MN2]×β({xn:n∈ℕ})+[2M∫0Tk(t)dt+M2N3+MN4]β({yn:n∈ℕ})≤N6β({(xn,yn):n∈ℕ}).
Finally, collecting these estimates, we get
(55)β({(xn,yn):n∈ℕ})≤β(Ω)≤β({wn:n∈ℕ})≤max{β({un:n∈ℕ}),β({vn:n∈ℕ})}≤N7β({(xn,yn):n∈ℕ}).
This implies that β(Ω)=β({(xn,yn):n∈ℕ})=0, which in turn implies that ℱ 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 ℱ. It is follows from (17), (39), and (40) that x(·)∈PC1([-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∈ℝ. 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∈ℝ such that t+s∈[0,T] we can decompose
(56)S(t+s)x-S(t)x=S(t)C(s)x+S(s)C(t)x-S(t)x=(C(s)-I)S(t)x+S(s)C(t)x.If we restrict us to consider x∈D, using that S(t)(D) is relatively compact, C(t)(D) is bounded, and ∥S(s)∥≤Ms, we obtain that (C(s)-I)S(t)x→0 and S(s)C(t)x→0 when s→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∈ℝ. Then the map Λ1 is compact.
Proof.
Let W⊂L1([0,T];X) be a bounded set. It follows from [40, Theorem 5] that the set {Λ1(u)(t):u∈W} is relatively compact in X for every t∈[0,T]. On the other hand, using again (56) we can write
(57)∥Λ1(u)(t+s)-Λ1(u)(t)∥=∥∫0t+sS(t+s-ξ)u(ξ)dξ-∫0tS(t-ξ)u(ξ)dξ∥≤∥(C(s)-I)∫0tS(t-ξ)u(ξ)dξ∥+∥S(s)∫0tC(t-ξ)u(ξ)dξ∥+∥∫tt+sS(t+s-ξ)u(ξ)dξ∥≤∥(C(s)-I)∫0tS(t-ξ)u(ξ)dξ∥+M2|s|∫0T∥u(ξ)∥dξ∥+M|s|∥∫0T∥u(ξ)∥dξ.
Since {Λ1(u)(t):u∈W} is relatively compact in X, ∥(C(s)-I)∫0tS(t-ξ)u(ξ)dξ∥→0 as s→0 uniformly for u∈W. Combining with the above estimate, it follows that Λ1(u)(t+s)-Λ1(u)(t)→0 as s→0 uniformly for u∈W. Therefore, the set Λ1(W) is equicontinuous. The Ascoli-Arzelá theorem shows that Λ1 is a compact operator.
We define the constants
(58)N5′=max{max-h≤θ≤0ℓ(θ),M(ℓ(0)+N1+N3)},N7′=max{N5′,N6}.
Corollary 22.
Assume that the operator S(t) is compact for all t∈ℝ. Assume further that φ(0)∈E and that conditions (F1)–(F4), (g1)–(g3), and (I1)-(I2) hold. If N7′<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({un(·):n∈ℕ}) for un defined on [0,T]. Using Lemmas 20 and 21 we can see that
(59)β({un(·):n∈ℕ})≤Mℓ(0)β({xn:n∈ℕ})+M∑k=1m[(dk1,0+dk1,2)β({xn:n∈ℕ})+dk1,1β({yn:n∈ℕ})]≤M[ℓ(0)+N1]β({xn:n∈ℕ})+MN3β({yn:n∈ℕ})≤M[ℓ(0)+N1+N3]β({(xn,yn):n∈ℕ}).
Combining with (45), for un defined on [-h,T], we obtain
(60)β({un(·):n∈ℕ})≤N5′β({(xn,yn):n∈ℕ}).
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):
(61)x′′(t)-Ax(t)∈F(t,x(t),xt),t∈J=[0,T],t≠tk,k=1,…,m,(62)Δx(tk)=Ik(x(tk),xtk),k=1,…,n,(63)x(θ)+g(x)(θ)=φ(θ),θ∈[-h,0],x′(0)=z.
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 Ik as Ik1 with Ik2=0 and modifying slightly the conditions about F, Ik, and g. We assume that F is a multivalued map from J×X×Chp into 𝒦υ(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]→𝒦υ(X), t↦F(t,x(t),xt) admits a Bochner integrable selection. As a consequence, the set
(64)𝒮F,x={f∈L1(J;X):f(t)∈F(t,x(t),xt),t∈J}≠∅,
and 𝒮F,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.
The function g is continuous and takes bounded sets in PC([-h,T];X) into bounded subsets of C([-h,0];X).
There is a continuous function ℓ:[-h,0]→[0,∞) such that
(65)χ(g(W)(θ))≤ℓ(θ)supt∈[-h,T]χ(W(t)),θ∈[-h,0],
for all bounded sets W⊂PC([-h,T],X).
For each bounded set W⊂PC([-h,T];X) the set g(W) is equicontinuous.
Next we establish the conditions on maps Ik, k=1,…,m. We assume that Ik:X×Chp→X satisfy the following conditions.
The maps Ik, k=1,…,m are continuous and takes bounded sets into bounded sets.
There are positive constants dkj, j=1,2, k=1,…,m, such that
(66)χ(Ik(D1×W))≤dk1χ(D1)+dk2sup-h≤θ≤0χ(W(θ)),
for all bounded sets D1⊂X and W⊂Chp such that sup-h≤θ≤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
(67)x(t)=C(t)(φ(0)-g(x)(0))+S(t)z+∫0tS(t-s)f(s)ds+∑tk<tC(t-tk)Ik(x(tk),x(tk))
is verified for f∈𝒮F,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
(68)u(t)={φ(t)-g(x)(t),t∈[-h,0],C(t)(φ(0)-g(x)(0))+S(t)z+∫0tS(t-s)f(s)ds+∑tk<tC(t-tk)Ik1(x(tk),x′(tk),xtk),t∈[0,T],
for f∈𝒮F,x. It follows from our hypotheses that u∈PC([-h,T];X). Hence, Γ:PC([-h,T];X)→𝒫(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
(69)N7′′=max{max-h≤θ≤0ℓ(θ),[ℓ(0)+∑k=1m(dk1+dk2)]4M1∫0Tk(t)dt)N7′′=max5.(M[ℓ(0)+∑k=1m(dk1+dk2)]N7′′=max5.88+[ℓ(0)+∑k=1m(dk1+dk2)]4M1∫0Tk(t)dt)}.
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 N7′′<1, then the map Γ:PC([-h,T];X)→Kυ(PC([-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))≥β(Ω). It follows from Lemma 10 that there exists a sequence (un)n in Γ(Ω) such that β(Γ(Ω))=β({un:n∈ℕ}). We can write un∈Γ(xn) for some xn∈Ω.
To estimate β({un:n∈ℕ}), using (68) we can write
(70)un(t)={φ(t)-g(xn)(t),t∈[-h,0],C(t)(φ(0)-g(xn)(0))+S(t)z+Λ1(fn)(t)+∑tk<tC(t-tk)Ik(xn(tk),(xn)tk),t∈[0,T],
for fn∈𝒮F,xn.
From (45), we have
(71)β({φ-g(xn):n∈ℕ})≤max-h≤θ≤0ℓ(θ)β({xn:n∈ℕ}).
Now we consider functions un defined on [0,T]. From (70) and using Lemma 7, we get
(72)β({un(·):n∈ℕ})≤Mχ({g(xn)(0):n∈ℕ})+β({Λ1(fn)(·):n∈ℕ})+M∑k=1mχ({Ik(xn(tk),(xn)tk):n∈ℕ}).
Using now conditions (g2), (I2), and Remark 14, we have
(73)χ({g(xn)(0):n∈ℕ})≤ℓ(0)β({xn:n∈ℕ}),χ({Ik(xn(tk),(xn)tk):n∈ℕ})≤dk1χ({xn(tk):n∈ℕ})+dk2β({xn:n∈ℕ})≤(dk1+dk2)β({xn:n∈ℕ}).
On the other hand, since fn∈𝒮F,xn, for t∈[0,T], we have that fn(t)∈F(t,xn(t),(xn)t). This implies that {fn:n∈ℕ} is uniformly integrable and, applying condition (F4),
(74)χ({fn(t):n∈ℕ})≤k(t)[sup-h≤θ≤0χ({xn(t):n∈ℕ})+sup-h≤θ≤0χ({xn(t+θ):n∈ℕ})]≤2k(t)β({xn:n∈ℕ}).
Combining this estimate with Lemma 11 we infer that
(75)β({Λ1(fn)(·):n∈ℕ})≤4M1β({xn:n∈ℕ})∫0Tk(t)dt.
Substituting this estimate in (72), we obtain
(76)β({un(·):n∈ℕ})≤(M[ℓ(0)+∑k=1m(dk1+dk2)]+4M1∫0Tk(t)dt)×β({xn:n∈ℕ})=N7′′β({xn:n∈ℕ}).
Collecting these assertions, we get
(77)β({xn:n∈ℕ})≤β(Γ(Ω))≤N7′′β({xn:n∈ℕ}),
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∈ℝ. Assume further that conditions (F1)–(F4), (g1)–(g3), and (I1)-(I2) hold. If
(78)max{max-h≤θ≤0ℓ(θ),M[ℓ(0)+∑k=1m(dk1+dk2)]}<1,
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=L2([0,π]) and A:D(A)⊆X→X is the map defined by Ax=(d2/dξ2)x(ξ) with domain D(A)={x∈X:x′′∈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∈ℝ on X. Furthermore, A has a discrete spectrum and the eigenvalues are -n2, n∈ℕ, with corresponding eigenvectors zn(ξ)=(2/π)1/2sin(nξ). Furthermore, the set {zn:n∈ℕ} is an orthonormal basis of X and the following properties hold.
For x∈D(A), Ax=-∑n=1∞n2〈x,zn〉zn.
For x∈X,
(79)C(t)x=∑n=1∞cos(nt)〈x,zn〉zn,S(t)x=∑n=1∞sin(nt)n〈x,zn〉zn.
Consequently, ∥C(t)∥=∥S(t)∥≤1 for all t∈ℝ and S(t) is a compact operator for every t∈ℝ.
The space E={x∈H1(0,π):x(0)=x(π)=0} (see [30] for details) and ∥x∥E≤∥x∥+∥x′∥. In particular, we observe that the inclusion ι:E→X is compact. Moreover, the function S(·) is 2π-periodic. Using this property and (13) we can show that
(80)∥AS(t)∥ℒ(E;X)≤2,∀t∈ℝ.
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 ∥AS(t)∥ℒ(E;X)≤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
(81)AS(t)=AS(1+s)=C(1)AS(s)+C(s)AS(1).
Since ∥C(τ)∥≤1 for all τ∈ℝ, we obtain
(82)∥AS(t)∥ℒ(E;X)≤∥C(1)∥∥AS(s)∥ℒ(E;X)+∥C(s)∥∥AS(1)∥ℒ(E;X)≤2.
Similarly, for t∈[π/2,π], we can write t=π-s with s∈[0,π/2], and using again (13), we can write
(83)AS(t)=AS(π-s)=C(s)AS(π)-C(π)AS(s)=-C(π)AS(s),
which implies
(84)∥AS(t)∥ℒ(E;X)≤∥C(π)∥∥AS(s)∥ℒ(E;X)≤2.
In view of that AS(-t)=-AS(t), this completes the proof of the assertion.
In what follows we assume that z∈X and that φ∈C([-h,0];X), where we have identified φ(θ)(ξ)=φ(θ,ξ) for θ∈[-h,0] and ξ∈[0,π].
Initially we construct the multivalued function F. We assume that f0:J×[0,π]×ℝ3→𝒦υ(ℝ) is a bounded multivalued map that satisfies the following conditions.
There exist positive constants L, L1, and L2 such that
(85)dH(f0(t1,ξ1,ϕ1),f0(t2,ξ2,ϕ2))2≤L2|t1-t2|2+L12|ξ1-ξ2|2+L22∥ϕ1-ϕ2∥2,
for all (t,ξ1,ϕ1),(t,ξ2,ϕ2)∈J×[0,π]×ℝ3, where dH denotes the Hausdorff metric and ∥·∥ denotes the Euclidean norm in ℝ3.
There exists a positive function μ∈L1([0,T]) such that |s|≤μ(t) for all s∈f0(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 B1,B2⊆Ω be bounded sets. Then, for every u,v∈Ω,
(86)ρ(υ,B2)≤d(υ,u)+ρ(u,B1)+dH(B1,B2).
We have the following consequences.
Proposition 28.
Under the previous conditions, the following properties hold.
For each t∈J and ϕ∈ℝ3, the function f0(t,·,ϕ) is measurable.
For each t∈J and ξ∈[0,π], the function f0(t,ξ,·) is upper semicontinuous.
For each t∈J and ϕ∈L2([0,π];ℝ3) the set(87)𝒯f0,ϕ(t)={w∈L2([0,π]):w(ξ)∈f0(t,ξ,ϕ)}≠∅
is closed convex in L2([0,π]).
Proof.
Consider the following.
It is an immediate consequence of the fact that the multivalued map f0(t,·,ϕ) is dH-continuous.
We know that f0(t,ξ,ℝ3) is bounded and, therefore, relatively compact. We will show that the graphf(t,ξ,·) is closed. Assume that ϕn,ϕ∈ℝ3 and ϕn→ϕ, sn∈f0(t,ξ,ϕn), sn→s as n→∞. Using Lemma 27 we can write
(88)ρ(s,f0(t,ξ,ϕ))≤|s-sn|+dH(f0(t,ξ,ϕn),f0(t,ξ,ϕ)).
Since f0(t,ξ,·) is dH-continuous, it follows that ρ(s,f0(t,ξ,ϕ))=0. In view of that the set f0(t,ξ,ϕ) is closed, we conclude that s∈f0(t,ξ,ϕ). Applying [3, Proposition 1.2] we obtain that f0(t,ξ,·) is upper semicontinuous.
For t∈J and ϕ∈L2([0,π];ℝ3) the map f0(t,·,ϕ):[0,π]→𝒦υ(ℝ), ξ↦f0(t,ξ,ϕ(ξ)), is measurable with closed values. It follows from [3, Proposition 3.2] that there exists a measurable selection w such that w(ξ)∈f0(t,ξ,ϕ(ξ)). Using that f is bounded it follows that w∈L2([0,π]). This shows that 𝒯f0,ϕ(t)≠∅. Since the values of f0 are convex, it follows that 𝒯f0,ϕ(t) is also convex.
To establish that 𝒯f0,ϕ(t) is closed, we consider a sequence wn,w∈L2([0,π]), wn∈𝒯f0,ϕ(t) such that wn→w, n→∞, for the norm in L2([0,π]). By passing to a subsequence if necessary, we can assume that wn(ξ)→w(ξ), n→∞, a.e. ξ∈[0,π]. Since f0(t,ξ,ϕ(ξ)) is closed, it follows that w(ξ)∈f0(t,ξ,ϕ(ξ)), which in turn implies that w∈𝒯f0,ϕ(t).
For ϕ∈L2([0,π];ℝ3) we define
(89)𝒯f0,ϕ1(t)={∫0ξw(η)dη:w∈𝒯f0,ϕ(t)},
and F1(ϕ):J→𝒫(X) is given by F1(ϕ)(t)=𝒯f0,ϕ1(t).
Proposition 29.
Under the previous conditions, F1 is a measurable and upper semicontinuous map with convex compact values.
Proof.
Initially we show that 𝒯f0,ϕ1(t) is closed. Let υn(ξ)=∫0ξwn(η)dη be a sequence in 𝒯f0,ϕ1(t) that converges to υ as n→∞. Since 𝒯f0,ϕ(t) is sequentially weakly compact, there is w∈L2([0,π]) and a subsequence wnk such that wnk→w as k→∞ in the weak topology. Since 𝒯f0,ϕ(t) is a closed convex set, it follows that w∈𝒯f0,ϕ(t). In view of that
(90)∫0ξwnk(η)dη-∫0ξw(η)dη=∫0π(wnk-w)χ[0,ξ](η)dη⟶0,k⟶∞,
where χ[0,ξ] denotes the characteristic function of the interval [0,ξ], we obtain that υ(ξ)=∫0ξw(η)dη and υ∈𝒯f0,ϕ1(t).
Since the functions in 𝒯f0,ϕ(t) are uniformly bounded,
(91)∫0π|v(ξ+δ)-v(ξ)|2dξ=∫0π|∫ξξ+δw(η)dη|2dξ≤πsup0≤ξ≤π|w(ξ)|2δ2
converge to zero as δ→0 uniformly for v∈𝒯f0,ϕ1(t). From [41, Theorem IV.8.20] we conclude that 𝒯f0,ϕ1(t) is relatively compact.
On the other hand, proceeding as in the proof of Proposition 28(ii), we get that F1(ϕ) is upper semicontinuous. Finally, as a consequence of a remark in [5, page 21], we can affirm that F1(ϕ) is measurable.
The following consequence is essential for our construction.
Corollary 30.
Under the above conditions, there exists a measurable selection for F1(ϕ).
Proof.
It follows from [3, Proposition 3.2].
We now consider the map F:J×X×X×Ch2→𝒦υ(X) defined by
(92)F(t,x,y,ψ)=𝒯f0,ϕ1(t),
for ϕ=(x,y,∫-h0ψ(θ)dθ). Since F(·,x,y,ψ)=F1(ϕ)(·), 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 υ∈F(t,x,y,ψ), then there exists w∈𝒯f0,ϕ(t) such that υ(ξ)=∫0ξw(η)dη with w(ξ)∈f0(t,ξ,x(ξ),y(ξ),∫-h0ψ(θ,ξ)dθ). Therefore, there exists w1(ξ)∈f0(t,0) such that
(93)|w(ξ)|≤μ(t)+1+dH(f0(t,ξ,x(ξ),y(ξ),∫-h0ψ(θ,ξ)dθ)+dH(f0(t,ξ(∫-h0ψ(θ,ξ)dθ)),f0(t,0))≤μ(t)+1+(L12ξ2+L22(x(ξ)2+y(ξ)2(∫-h0ψ(θ,ξ)dθ)2≤μ(t)+1+55555555..5+(∫-h0ψ(θ,ξ)dθ)2))1/2.
Hence
(94)∫0π|v(ξ)|2dξ≤π2∫0π|w(ξ)|2dξ≤2π2∫0π[(μ(t)+1)2+L12ξ2+L22(x(ξ)2+y(ξ)2+(∫-h0ψ(θ,ξ)dθ)2)×(x(ξ)2+y(ξ)2+(∫-h0ψ(θ,ξ)dθ)2)]dξ≤2π2[π(μ(t)+1)2+L12π33+L22(∥x∥2+∥y∥2+∫0π(∫-h0ψ(θ,ξ)dθ)2)×L12π33(∥x∥2+∥y∥2+∫0π(∫-h0ψ(θ,ξ)dθ)2)dξ]≤2π2[(∥x∥2+∥y∥2+h2∥ψ∥Ch22)π(μ(t)+1)2+L12π33+L12π33L22(∥x∥2+∥y∥2+h2∥ψ∥Ch22)],
which shows that F satisfies the condition (F3).
Proposition 31.
Let x,x1,y,y1∈X, ψ,ψ1∈Ch2, and v∈F(t,x,y,ψ). Then
(95)ρ(υ,F(t,x1,y1,ψ1))≤πL2(∥x-x1∥2+∥y-y1∥2+h2∥ψ-ψ1∥Ch22)1/2.
Proof.
To abbreviate the text we introduce the notations ϕ(ξ)=(x(ξ),y(ξ),∫-h0ψ(θ,ξ)dθ) and ϕ1(ξ)=(x1(ξ),y1(ξ),∫-h0ψ1(θ,ξ)dθ). Since v∈𝒯f0,ϕ1(t), there is w∈𝒯f0,ϕ(t) such that v(ξ)=∫0ξw(η)dη. Moreover, in view of that 𝒯f0,ϕ11(t) is convex compact in L2([0,π]), there is v0∈𝒯f0,ϕ11(t) the nearest point of v in 𝒯f0,ϕ11(t). Consequently, there is w0∈𝒯f0,ϕ1(t) such that v0(ξ)=∫0ξw0(η)dη. Therefore,
(96)ρ(υ,F(t,x1,y1,ψ1))2=∥υ-υ0∥2=∫0π(∫0ξ(w(η)-w0(η))dη)2dξ≤π2∫0π|w(η)-w0(η)|2dη≤π2L22∫0π[|∫-h0(ψ(θ,ξ)-ψ1(θ,ξ))dθ|2|x(η)-x1(η)|2+|y(η)-y1(η)|2+|∫-h0(ψ(θ,ξ)-ψ1(θ,ξ))dθ|2]dη≤π2L22[∥x-x1∥2+∥y-y1∥2+h2∥ψ-ψ1∥Ch22],
which completes the proof.
Corollary 32.
Under the above conditions, let Ωi⊂L2([0,π]), i=1,2, be bounded sets, and let Q⊂Ch2 be a set uniformly bounded. Then
(97)χ(F(t,Ω1,Ω2,Q))≤πL2(χ(Ω1)2+χ(Ω2)2+h2γ(Q)2)1/2,
where γ(Q) denotes the Hausdorff measure of noncompactness for the norm of uniform convergence.
Proof.
Let ɛ>0. We abbreviate the notation by writing si=χ(Ωi) and s=γ(Q). There exist x1,…,xi1∈L2([0,π]), y1,…,yj1∈L2([0,π]), and ψ1,…,ψk1∈Ch2 having the following property: given x∈Ω1, y∈Ω2, and ψ∈Q there are xi, yj, and ψk such that ∥x-xi∥≤s1+ɛ, ∥y-yj∥≤s2+ɛ and ∥ψ-ψk∥∞≤s+ɛ. Hence, if v∈F(t,x,y,ψ), using Proposition 31, we obtain that
(98)ρ(v,F(t,xi,yj,ψk))≤πL2((s1+ɛ)2+(s2+ɛ)2+h2(s+ɛ)2)1/2.
Since F(t,xi,yj,ψk) is a compact set in L2([0,π]), and ɛ>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)→C([-h,0];X) by
(99)g(x)(θ,ξ)=σ(ξ)∫0T∫0ξx(t+θ,η)dηdt,(x)(θ,ξ)=σ(ξ)5551x∈PC([-h,T];X),
for θ∈[-h,0] and ξ∈[0,π]. We assume that σ(·) is a function of class C1 such that σ(π)=0. It is clear that
(100)g(x)(0,ξ)=σ(ξ)∫0T∫0ξx(t,η)dηdt∈H1(0,π)
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)→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 ℝ 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,
(101)|g(x)(θ,ξ)|≤∥σ∥∞|∫0T∫0ξx(t,η)dηdt||g(x)(θ,ξ)|≤∥σ∥∞∫0T∫0π|x(t,η)|dηdt|g(x)(θ,ξ)|≤∥σ∥∞∫0Tπ1/2(∫0π|x(t,η)|2dη)1/2dt|g(x)(θ,ξ)|≤∥σ∥∞π1/2Tsup-h≤t≤T∥x(t)∥,
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 L2([0,π]). Hence χ(g(W)(θ))=0, and we can take ℓ(θ)=0. This shows that g satisfies the first part of condition (g2). Furthermore, it follows from (c) that
(102)χE(g(W)(0))≤χ(g(W)(0))+χ(ddξg(W)(0))=χ(ddξg(W)(0)).
From the definition of g we obtain
(103)ddξg(x)(0,ξ)=ddξσ(ξ)∫0T∫0ξx(t,η)dηdt+σ(ξ)∫0Tx(t,ξ)dt.
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)∫0Tx(t,ξ)dt=σ∫0Tx(t)dt in the space L2([0,π]), using [42, Theorem 3.1], we conclude that
(104)χ(ddξg(W)(0))≤∥σ∥∞∫0Tχ(W(t))dt≤∥σ∥∞Tsupt∈[-h,T]χ(W(t)),
which shows that g also satisfies the second part of condition (g2) with ℓ1=∥σ∥∞T.
On the other hand,
(105)∥g(x)(θ+δ)-g(x)(θ)∥2=∫0π|g(x)(θ+δ,ξ)-g(x)(θ,ξ)|2dξ=∫0πσ(ξ)2|∫0ξ∫0T(x(t+θ+δ,η)=∫0πσ(ξ)2|∫0ξ∫0T(.-x(t+θ,η))dtdη∫0T|2dξ≤∥σ∥∞2∫0π|∫0ξ[∫θ+δT+θ+δx(s,η)ds-∫θT+θx(s,η)ds]dη|2dξ=∥σ∥∞2∫0π|∫0ξ[∫T+θT+θ+δx(s,η)ds-∫θθ+δx(s,η)ds]dη|2dξ≤∥σ∥∞2∫0π(∫0ξ[∫T+θT+θ+δ|x(s,η)|ds+∫θθ+δ|x(s,η)|ds]dη)2dξ≤∥σ∥∞2∫0π(∫T+θT+θ+δ∫0π|x(s,η)|dηds+∫θθ+δ∫0π|x(s,η)|dηds)2dξ≤π∥σ∥∞2∫0π(∫T+θT+θ+δ∥x(s)∥ds+∫θθ+δ∥x(s)∥ds)2dξ≤4π2∥σ∥∞2sup-h≤s≤T∥x(s)∥2δ2
converges to zero as δ→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 Ik1,Ik2:X×X×Ch2→X by
(106)Ik1(x,y,ψ)(ξ)=ak1(ξ)∫0πqk1(η)x(η)dη+bk1(ξ),Ik2(x,y,ψ)(ξ)=ak2(ξ)∫0πqk2(η)y(η)dη+bk2(ξ),
for x,y∈L2([0,π]), and ξ∈[0,π]. We assume that qki(·),aki(·),bki(·)∈L2([0,π]) for i=1,2, and that ak1(·),bk1(·)∈C1([0,π]) are functions such that ak1(0)=ak1(π)=bk1(0)=bk1(π)=0. Proceeding as above, it is easy to see that Ik1:X×X×Ch2→E and Ik2:X×X×Ch2→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→ℝ, y↦∫0πqk2(η)y(η)dη, is a bounded linear functional with norm ∥qk2∥, we deduce that
(107)χ(Ik2(D0×D1×W))≤∥ak2∥∥qk2∥χ(D1).
Using this argument together with condition (c), we get
(108)χ(Ik1(D0×D1×W))≤(∥ak1∥+∥ddξak1∥)∥qk1∥χ(D0),
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 Ni introduced in Section 3 are the following:
(109)N1=∑k=1m(dk1,0+dk1,2)N1=∑k=1mdk1,0=∑k=1m(∥ak1∥+∥ddξak1∥)∥qk1∥,N2=∑k=1m(dk2,0+dk2,2)=0,N3=∑k=1mdk1,1=0,N4=∑k=1mdk2,1=∑k=1m∥ak2∥∥qk2∥,N5′=M(ℓ(0)+N1+N3)=N1,N6=M2ℓ1+6M∫0Tk(t)dt+M2(N1+N3)N1=+M(N2+N4)N1=(2∥σ∥∞+6πL2)T+2N1+N4,N7′=max{N5′,N6}=N6.
Combining with Corollary 22, we have established the following result.
Theorem 33.
Assume that φ(0,·)∈E, and
(110)(2∥σ∥∞+6πL2)T+∑k=1m[2(∥ak1∥+∥ddξak1∥)∥qk1∥+∥ak2∥∥qk2∥]<1.
Then there exists a mild solution of systems (5)–(9).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
Hernán R. Henríquez was partially supported by CONICYT under Grants FONDECYT 1130144 and DICYT-USACH.
AbbasS.BenchohraM.N'GuérékataG. M.2012New York, NY, USASpringer10.1007/978-1-4614-4036-9MR2962045BenchohraM.HendersonJ.NtouyasS.2006New York, NY, USAHindawi Publishing Corporation10.1155/9789775945501MR2322133DeimlingK.19921Berlin, GermanyWalter de Gruyter & Co.De Gruyter Series in Nonlinear Analysis and Applications10.1515/9783110874228MR1189795GórniewiczL.20062ndDordrecht, The NetherlandsSpringerMR2238622KamenskiiM.ObukhovskiiV.ZeccaP.20017Berlin, GermanyWalter de Gruyter & Co.De Gruyter Series in Nonlinear Analysis and Applications10.1515/9783110870893MR1831201SamoĭlenkoA. M.PerestyukN. A.199514SingaporeWorld Scientific10.1142/9789812798664MR1355787AbadaN.BenchohraM.HammoucheH.Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions2009246103834386310.1016/j.jde.2009.03.004MR2514728ZBL1171.34052BenchohraM.GatsoriE. P.GórniewiczL.NtouyasS. K.Nondensely defined evolution impulsive differential equations with nonlocal conditions200342185204MR2031389ZBL1060.34027ChuongN. M.KeT. D.Generalized Cauchy problems involving nonlocal and impulsive conditions201212236739210.1007/s00028-012-0136-4MR2923939ZBL1258.35131HernándezE.RabelloM.HenríquezH. R.Existence of solutions for impulsive partial neutral functional differential equations200733121135115810.1016/j.jmaa.2006.09.043MR2313705ZBL1123.34062HernándezE.HenríquezH. R.McKibbenM. A.Existence results for abstract impulsive second-order neutral functional differential equations20097072736275110.1016/j.na.2008.03.062MR2499742ZBL1173.34049HuJ.LiuX.Existence results of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces2009493-451652610.1016/j.mcm.2008.02.005MR2483655ZBL1165.45305KavithaV.ArjunanM. M.RavichandranC.Existence results for a second order impulsive neutral functional integrodifferential inclusions in Banach spaces with infinite delay201255321333MR2968473KavithaV.ArjunanM. M.RavichandranC.Existence results for non-densely defined impulsive neutral functional differential inclusions with state-dependent delay2012134422437MR2948808LakribM.OumansourA.YadiK.Existence results for second order impulsive functional differential equations with infinite delay201291111MR2923208ZBL1246.34075LiuX.LiY.Positive solutions for neumann boundary value problems of second-order impulsive differential equations in Banach spaces2012201214401923MR289805310.1155/2012/401923ZBL1244.34044ObukhovskiiV.YaoJ.-C.On impulsive functional differential inclusions with Hille-Yosida operators in Banach spaces20107361715172810.1016/j.na.2010.05.009MR2661354ZBL1214.34052ByszewskiL.Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem1991162249450510.1016/0022-247X(91)90164-UMR1137634ZBL0748.34040ByszewskiL.LakshmikanthamV.Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space1991401111910.1080/00036819008839989MR1121321ZBL0694.34001ByszewskiL.Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation1991402-317318010.1080/00036819108840001MR1095412ZBL0725.35060ByszewskiL.Uniqueness criterion for solution of abstract nonlocal Cauchy problem199361495410.1155/S104895339300005XMR1214305ZBL0776.34050ByszewskiL.AkcaH.On a mild solution of a semilinear functional-differential evolution nonlocal problem199710326527110.1155/S1048953397000336MR1468121ZBL1043.34504ByszewskiL.WiniarskaT.An abstract nonlocal second order evolution problem2012321758210.7494/OpMath.2012.32.1.75MR2852470ZBL1254.34078FuX.On solutions of neutral nonlocal evolution equations with nondense domain2004299239241010.1016/j.jmaa.2004.02.062MR2098250ZBL1064.34065Hernández M.E.HenríquezH. R.Existence results for second order differential equations with nonlocal conditions in Banach spaces200952111313710.1619/fesi.52.113MR2538282ZBL1177.34078JacksonD.Existence and uniqueness of solutions to semilinear nonlocal parabolic equations1993172125626510.1006/jmaa.1993.1022MR1199510ZBL0814.35060WuJ.1996119New York, NY, USASpringer10.1007/978-1-4612-4050-1MR1415838GatsoriE. P.GórniewiczL.NtouyasS. K.SficasG. Y.Existence results for semilinear functional differential inclusions with infinite delay2005614758MR2133104ZBL1079.34061ArendtW.BattyC. J. K.HieberM.NeubranderF.2001BaselBirkhäuserMR1886588FattoriniH. O.1985108Amsterdam, The NetherlandsNorth-HollandNorth-Holland Mathematics StudiesMR797071HaaseM.2006169Basel, SwitzerlandBirkhäuser10.1007/3-7643-7698-8MR2244037PiskarevS. I.Evolution equations in Banach spaces. Theory of cosine operator functions2004, http://www.icmc.usp.br/andcarva/minicurso.pdfVasilV. V.PiskarevS. I.Differential equations in Banach spaces. II: theory of cosine operator functions200412223055317410.1023/B:JOTH.0000029697.92324.47MR2084186TravisC. C.WebbG. F.Second order differential equations in Banach spaceProceedings of the International Symposium on Nonlinear Equations in Abstract Spaces1987New York, NY, USAAcademic Press331361KisyńskiJ.On cosine operator functions and one-parameter groups of operators19724993105MR0312328ZBL0232.47045TravisC. C.WebbG. F.Compactness, regularity, and uniform continuity properties of strongly continuous cosine families197734555567MR0500288ZBL0386.47024TravisC. C.WebbG. F.Cosine families and abstract nonlinear second order differential equations1978321-27696MR049958110.1007/BF01902205ZBL0388.34039AkhmerovR. R.KamenskiĭM. I.PotapovA. S.RodkinaA. E.SadovskiĭB. N.1992Basel, SwitzerlandBirkhäuserMR1153247BanaśJ.GoebelK.198060New York, NY, USAMarcel DekkerLecture Notes in Pure and Applied MathematicsMR591679HenríquezH. R.On non-exact controllable systems1985421718310.1080/00207178508933347ZBL0569.93008DunfordN.SchwartzJ. T.1988New York, NY, USAJohn Wiley & SonsMR1009162HeinzH.-P.On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions19837121351137110.1016/0362-546X(83)90006-8MR726478ZBL0528.47046