# An Existence Result for Nonlocal Impulsive Second-Order Cauchy Problems with Finite Delay

Authors: Fang Li; Huiwen Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We deal with the existence of mild solutions of a class of nonlocal impulsive second-order functional differential equations with finite delay in a real Banach spaceX. An existence result on the mild solution is obtained by using the theory of the measures of noncompactness. An example is presented.


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## Body

## 1. Introduction

The Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (see, e.g., [1–5]).The literature concerning second- and higher-order ordinary functional differential equations is very extensive. We only mention the works [1, 6–15], which are directly related to this work.On the other hand, the impulsive conditions have advantages over traditional initial value problems because they can be used to model phenomena that cannot be modeled by traditional initial value problems, such as the dynamics of populations subject to abrupt changes (harvesting, diseases, etc.) (see [16–27] and references therein). For this reason, the theory of impulsive differential equations has become an important area of investigation in recent years. Partial differential equations of first and second order with impulses have been studied by Rogovchenko [26], Liu [25], Cardinali and Rubbioni [19], Liang et al. [24],  Henríquez and Vásquez [1], Hernández et al., [21–23], Arthi and Balachandran [17], and so forth.Moreover, we consider the nonlocal conditionx(0)=g(x)+x0, where g is a mapping from some space of functions so that it constitutes a nonlocal condition (see [24, 28–30] and the references therein), where it is demonstrated that nonlocal conditions have better effects in applications than traditional initial value problems.In this paper, we pay our attention to the investigation of the existence of mild solutions to the following impulsive second-order functional differential equations with finite delay in a real Banach spaceX:
(1)d2dt2x(t)=Ax(t)+f(t,xt,x(t)),t∈(0,T],t≠tk,k=1,2,…,p,(2)x(t)=g(x)(t)+ϕ(t),t∈[-r,0],(3)x′(0)=ξ∈X,(4)Δx(tk)=Ik(x(tk)),k=1,2,…,p,
where A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators {C(t)}t∈R on X. f, g are given functions to be specified later. ϕ∈C([-r,0],X), where C([a,b],X) denotes the space of all continuous functions from [a,b] to X.The impulsive moments{tk} are given such that 0=t0<t1<⋯<tp<tp+1=T, Ik:X→X(k=1,2,…,p) are appropriate functions, Δx(tk) represents the jump of a function x at tk, which is defined by Δx(tk)=x(tk+)-x(tk-), where x(tk+) and x(tk-) are, respectively, the right and the left limits of x at tk.For any continuous functionx defined on the interval [-r,T] and any t∈[0,T], we denote by xt the element of C([-r,0],X) defined by xt(θ)=x(t+θ) for θ∈[-r,0].In this paper, motivated by above works, we study (1)–(4) in X and obtain the existence theorem based on theory on measures of noncompactness without the assumptions that the nonlinearity f satisfies a Lipschitz type condition and the cosine family of bounded linear operators {C(t)}t∈R generated by A is compact.


## 2. Preliminaries

Throughout this paper, we setJ=[0,T], a compact interval in R. We denote by X a Banach space with norm ∥·∥, by L(X) the Banach space of all linear and bounded operators on X. We abbreviate ∥u∥L1(J,R+) with ∥u∥L1, for any u∈L1(J,R+).Let(5)PC(J,X):={x:J⟶X;x(t)iscontinuousatt≠tk,left  continuousatt=tk,andtherightlimitx(tk+)existsfork=1,2,…,p}.It is easy to check thatPC(J,X) is a Banach space with the norm
(6)∥x∥PC=supt∈J∥x(t)∥,foranyx∈PC(J,X).We letJ0=(t0,t1], J1=(t1,t2],…,Jp=(tp,tp+1].For𝔅⊆PC(J,X), we denote by 𝔅|J-i the set
(7)𝔅|J-i={x∈C([ti,ti+1],X);x(ti)=v(ti+),x(t)=v(t),t∈Ji,v∈𝔅}i=0,1,2,…,p.A family{C(t)}t∈R in L(X) is called a cosine function on X if (i)
C(0)=I is the identity operator in X;(ii)
C(t+s)+C(t-s)=2C(t)C(s) for all s,t∈R;(iii)
The mapt→C(t)x is strongly continuous for each x∈X. The associated sine function is the family {S(t)}t∈R of operators defined by
(8)S(t)x=∫0t‍C(s)xds,forx∈X,t∈R.One can define the infinitesimal generatorA of C(·) by
(9)D(A)={x∈X;limt→02t-2(C(t)x-x)∈X}Ax=limt→02t-2(C(t)x-x),x∈D(A).In this paper, we assume there exist positive constantsM and N such that
(10)∥C(t)∥≤M,∥S(t)∥≤Nforeveryt∈J.
The following properties are well known [6, 7, 11, 12]:
(11)(i)C(t)x∈D(A),C(t)Ax=AC(t)xforx∈D(A),t∈R;(ii)S(t)x∈D(A),S(t)Ax=AS(t)xforx∈D(A),t∈R;(iii)∫0tS(s)xds∈D(A),(iii)A∫0t‍S(s)xds=C(t)x-xforx∈X,t∈R;(iv)C(t)x-x=∫0t‍S(s)Axdsforx∈D(A),t∈R.For more details on strongly continuous cosine and sine families, we refer the reader to [6, 7, 11, 12].Next, we recall that the Hausdorff measure of noncompactnessχ(·) on each bounded subset Ω of Banach space Y is defined by
(12)χ(Ω)=inf{ε>0;Ωhasafiniteε-netinX}.This measure of noncompactness satisfies some basic properties as follows.Lemma 1 (see [31]).
LetY be a real Banach space, and let B,C⊆Y be bounded. Then (1)
χ(B)=0 if and only if B is precompact;(2)
χ(B)=χ(B¯)=χ(convB), where B¯ and convB mean the closure and convex hull of B, respectively;(3)
χ(B)≤χ(C) if B⊆C;(4)
χ(B∪C)≤max{χ(B),χ(C)};(5)
χ(B+C)≤χ(B)+χ(C), where B+C={x+y;x∈B,y∈C};(6)
χ(αB)=|α|χ(B), for any α∈R;(7)
letZ be a Banach space and Q:D(Q)⊆Y→Z Lipschitz continuous with constant ν. Then χ(QB)≤ν·χ(B) for all B⊆D(Q) being bounded.Proposition 2 (see [32], Page 125).
LetΩ be a bounded set for a real Banach space X. Then, for every ε>0 there exists a sequence {xn}n=1∞ in Ω such that
(13)χ(Ω)≤2χ({xn}n=1∞)+ε.In the sequel, we make use of the following formulation of Theorem 4.2.2 of [33] obtained by using Theorem 2 of [34].Proposition 3.
Let{fn}n=1∞ be a sequence in L1(J,X) such that there exist v,q∈L+1([0,T]) with the properties:(i)
supn∈N∥fn(t)∥≤v(t),a.e.t∈J;(ii)
χ({fn}n=1∞)≤q(t),a.e.t∈J. Then, for every t∈J, we have
(14)χ({∫0t‍S(t-s)fn(s)ds}n=1∞)≤2N∫0t‍q(s)ds,
where N is from (10).A continuous mapQ:W⊆Y→Y is said to be a χ-contraction if there exists a positive constant ν<1 such that χ(QC)≤ν·χ(C) for any bounded closed subset C⊆W.Theorem 4 (see [31] (Darbo-Sadovskii)).
IfU⊆Y is bounded closed and convex, the continuous map ℱ:U→U is a χ-contraction, then the map ℱ has at least one fixed point in U.Definition 5.
A functionx:[-r,T]→X is called a mild solution of system (1)–(4) if x0=g(x)+ϕ, x|J∈PC(J,X) and
(15)x(t)=C(t)(ϕ(0)+g(x)(0))+S(t)ξ+∫0t‍S(t-s)f(s,xs,x(s))ds+∑0<tk<t‍C(t-tk)Ik(x(tk)),t∈J.Remark 6.
A mild solution of (1)–(4) satisfies (2) and (4). However, a mild solution may be not differentiable at zero.


## 3. Existence Result and Proof

In this section, we study the existence of mild solutions for the system (1)–(4).Let𝔉(T) stand for the space
(16)𝔉(T)={x:[-r,T]→X;x|J∈PC(J,X),x0∈C([-r,0],X)}
endowed with norm
(17)∥x∥𝔉(T)=supt∈[-r,0]∥x(t)∥+supt∈J∥x(t)∥.We will require the following assumptions.(H1)(i)  f:J×C([-r,0],X)×X→X satisfies f(·,v,w):J→X is measurable for all (v,w)∈C([-r,0],X)×X and f(t,·,·):C([-r,0],X)×X→X is continuous for a.e. t∈J, and there exists a function μ(·)∈L1(J,R+) such that
(18)∥f(t,v,w)∥≤μ(t)(1+∥w∥)
for almost all t∈J;(ii)   there exists a function η∈L1(J,R+) such that for any bounded sets D1⊂C([-r,0],X), D2⊂X(19)χ(f(t,D1,D2))≤η(t)(supθ∈[-r,0]χ(D1(θ))+χ(D2)),a.e.t∈J.(H2)Ik:X→X are compact operators and there exist positive constants M1,M2 such that
(20)∥Ik(x)∥≤M1∥x∥+M2,foranyx∈X,k=1,2,…,p.(H3)g:C([-r,0],X)→X is a compact operator and there exists a constant N1>0 such that
(21)∥g(x)∥[-r,0]≤N1forallx∈C([-r,0],X),where∥g(x)∥[-r,0]=supt∈[-r,0]∥g(x)(t)∥.(H4)
There existsM*∈(0,1) such that 8N∫0T‍η(s)ds<M*.Theorem 7.
Assume that (H1)–(H4) are satisfied, then there exists a mild solution of (1)–(4) on [-r,T] provided that pMM1<1.Proof.
Define the operatorΛ:𝔉(T)→𝔉(T) in the following way:
(22)(Λx)(t)={g(x)(t)+ϕ(t),t∈[-r,0],C(t)(g(x)(0)+ϕ(0))+S(t)ξ+∫0t‍S(t-s)f(s,xs,x(s))ds+∑0<tk<tC(t-tk)Ik(x(tk)),t∈J.
It is clear that the operator Λ is well defined, and the fixed point of Λ is the mild solution of problems (1)–(4).
The operatorΛ can be written in the form Λ=Λ1+Λ2, where the operators Λ1,Λ2 are defined as follows:
(23)(Λ1x)(t)={g(x)(t)+ϕ(t),t∈[-r,0],C(t)(g(x)(0)+ϕ(0))+S(t)ξ,t∈J,(Λ2x)(t)={0,t∈[-r,0],∫0t‍S(t-s)f(s,xs,x(s))ds+∑0<tk<tC(t-tk)Ik(x(tk)),t∈J.
Obviously, under the assumptions ofg, Λ1 is continuous. For t∈J, we can prove that Λ2 is continuous.
Indeed, let{xn}n∈N be a sequence such that xn→x in 𝔉(T) as n→∞. Since f satisfies (H1)(i), for almost every t∈J, we get
(24)f(t,xtn,xn(t))→f(t,xt,x(t)),asn→∞.
Noting thatxn→x in 𝔉(T), we can see that there exists ε>0 such that ∥xn-x∥𝔉(T)≤ε for n sufficiently large. Therefore, we have
(25)∥f(t,xtn,xn(t))-f(t,xt,x(t))∥≤μ(t)(1+∥xn(t)∥)+μ(t)(1+∥x(t)∥)≤2μ(t)+μ(t)∥xn(t)-x(t)∥+2μ(t)∥x(t)∥≤2μ(t)+μ(t)ε+2μ(t)∥x∥𝔉(T).
It follows from the Lebesgue’s dominated convergence theorem that
(26)∫0t‍∥S(t-s)[f(s,xsn,xn(s))-f(s,xs,x(s))]∥ds≤N∫0t‍∥f(s,xsn,xn(s))-f(s,xs,x(s))∥ds→0,asn→∞.
Moreover, noting that (H2), we obtain that
(27)limn→∞∥Λ2xn-Λ2x∥𝔉(T)=0.
This shows that Λ2 is continuous. Therefore, Λ is continuous.
Let us introduce in the space𝔉(T) the equivalent norm defined as
(28)∥x∥*=supt∈[-r,0]∥x(t)∥+supt∈J(e-Lt∥x(t)∥),
where L>0 is a constant chosen so that
(29)Nsupt∈J∫0t‍μ(s)e-L(t-s)ds<1.
Noting that for any ψ∈L1(J,X), we have
(30)limL→+∞supt∈J∫0t‍e-L(t-s)ψ(s)ds=0,
so, we can take the appropriate L to satisfy (29).
Consider the set(31)Bρ={x∈𝔉(T);∥x∥*≤ρ},
where ρ is a constant chosen so that
(32)ρ≥N1+∥ϕ∥[-r,0]+ℓ+pMM21-pMM1>0,
where ℓ:=M(N1+∥ϕ(0)∥)+N(∥ξ∥+∥μ∥L1) and ∥ϕ∥[-r,0]=supt∈[-r,0]∥ϕ(t)∥.
Now, ift∈[-r,0], x∈Bρ, then
(33)∥(Λx)(t)∥=∥g(x)(t)+ϕ(t)∥≤N1+∥ϕ∥[-r,0].
For t∈J, x∈Bρ, we have
(34)∥(Λx)(t)∥≤∥C(t)(g(x)(0)+ϕ(0))∥+∥S(t)ξ∥+∫0t‍∥S(t-s)f(s,xs,x(s))∥ds+∑0<tk<t‍∥C(t-tk)Ik(x(tk))∥≤M(N1+∥ϕ(0)∥+∑0<tk<t‍∥Ik(x(tk))∥)+N(∥ξ∥+∫0t‍μ(s)(1+eLse-Ls∥x(s)∥)ds)=ℓ+M∑0<tk<t‍∥Ik(x(tk))∥+N∫0t‍μ(s)eLse-Ls∥x(s)∥ds,
then
(35)e-Lt∥(Λx)(t)∥≤e-Lt[ℓ+M∑0<tk<t‍∥Ik(x(tk))∥+N∫0t‍μ(s)eLse-Ls∥x(s)∥ds∑0<tk<t‍]≤ℓ+ρpMM1+pMM2+N∫0t‍μ(s)e-L(t-s)ds·∥x∥*,
therefore,
(36)supt∈J(e-Lt∥(Λx)(t)∥)≤ℓ+pMM2+[pMM1+supt∈J(N∫0t‍μ(s)e-L(t-s)ds)]ρ.
It results that
(37)∥Λx∥*=supt∈[-r,0]∥(Λx)(t)∥+supt∈J(e-Lt∥(Λx)(t)∥)≤N1+∥ϕ∥[-r,0]+ℓ+pMM2+(pMM1+Nsupt∈J∫0t‍μ(s)e-L(t-s)ds)ρ.
LetL→+∞, we obtain
(38)∥Λx∥*≤N1+∥ϕ∥[-r,0]+ℓ+pMM2+pMM1ρ≤ρ.
Hence for some positive number ρ, ΛBρ⊂Bρ.
Using the strong continuity of{C(t)}t∈R and the compactness condition on the operators Ik, for ε>0, there exists δ>0 such that
(39)∥(C(t+h)-C(t))Ik(x)∥≤ε,x∈Bρ,t∈J,k=1,2,…,p,
when |h|<δ. If t∈[tk,tk+1] and h<δ such that t+h∈[tk,tk+1], then
(40)∥∑0<tk<t(C(t+h-tk)-C(t-tk))Ik(x(tk))‍∥≤∑k=1p‍∥(C(t+h-tk)-C(t-tk))Ik(x(tk))∥≤pε.
Forx∈Bρ, by the hypothesis (H1)(i) and (40), we get
(41)∥(Λ2x)(t+h)-(Λ2x)(t)∥≤∥∫0t+h‍S(t+h-s)f(s,xs,x(s))ds-∫0t‍S(t-s)f(s,xs,x(s))ds∥+pε≤pε+∫0t‍∥(S(t+h-s)-S(t-s))f(s,xs,x(s))∥ds+∫tt+h‍∥S(t+h-s)f(s,xs,x(s))∥ds≤pε+[Mh∫0t‍μ(s)ds+N∫tt+h‍μ(s)ds]·(1+ρ).
As h→0 and ε→0, the right-hand side of the inequality above tends to zero independent of x, so Λ2 maps bounded sets into equicontinuous sets.
For bounded setB⊂PC(J,X), we consider the map
(42)χpc(B)=maxi=0,1,…,pχi(B|J-i),
where χi is the Hausdorff measure of noncompactness on the Banach space C(Ji,X) and B|J-i is defined in (7).
Furthermore, we define the Hausdorff measure of noncompactnessχ𝔉 on 𝔉(T) as follows:
(43)χ𝔉(𝕐):=χpc(𝕐|PC(J,X))+supt∈[-r,0]χ(𝕐(t)),𝕐⊂𝔉(T).
For every bounded subsetΩ~⊂PC(J,X), by applying Proposition 2, for any ε>0 there exists a sequence {xn}n=1∞⊂Ω~ such that
(44)χpc(Λ2Ω~)≤2χpc(Λ2{xn})+ε,
noting that the definition of χpc, we have
(45)χpc(Λ2Ω~)≤2maxi=0,1,…,pχi(Λ2{xn}|J-i)+ε.
Then, noting the equicontinuity of Λ2|J-i,  i=0,1,…,p, we can apply Lemmas 2.1 and 2.2 of [35] to obtain
(46)χi(Λ2{xn}|J-i)=supt∈J-iχ(Λ2{xn}(t)).
Then from (45) and (46), we have
(47)χpc(Λ2Ω~)≤2maxi=0,1,…,p(supt∈J-iχ(Λ2{xn}(t)))+ε=2supt∈Jχ(Λ2{xn}(t))+ε.
For every bounded subsetΩ⊂𝔉(T), we have
(48)χ𝔉(Λ2Ω)=χpc((Λ2Ω)|PC(J,X))+supt∈[-r,0]χ((Λ2Ω)(t))=χpc((Λ2Ω)|PC(J,X)),
moreover, by applying Proposition 2, for any ε>0 there exists a sequence {yn}n=1∞⊂Ω such that
(49)χ𝔉(Λ2Ω)≤2χ𝔉(Λ2{yn})+ε=2χpc((Λ2{yn})|PC(J,X))+ε.
Combining with (48) and (49), we have
(50)χ𝔉(Λ2Ω)=χpc((Λ2Ω)|PC(J,X))≤2χpc((Λ2{yn})|PC(J,X))+ε.
Using the induction of (45)–(47) above, we can see
(51)χ𝔉(Λ2Ω)=χpc((Λ2Ω)|PC(J,X))≤2supt∈Jχ(Λ2{yn}(t)|t∈J)+ε.
Thus, from (51), (H2) and Proposition 3 and (3) in Lemma 1, we can see
(52)χ𝔉(Λ2Ω)≤2supt∈Jχ(Λ2{yn}(t)|t∈J)+ε=2supt∈J[∑0<tk<t‍χ({∑0<tk<t‍∫0t‍S(t-s)f(s,y~ns,y~n(s))ds+∑0<tk<t‍C(t-tk)Ik(y~n(tk))})]+ε≤2supt∈J[2N∫0t‍η(s)(supθ∈[-r,0]χ({yn(s+θ)})+χ({y~n(s)})supθ∈[-r,0])dssupθ∈[-r,0]]+ε,
where y~n(t):=yn(t)|t∈J.
Noting that(53)supθ∈[-r,0]χ({yn(s+θ)})≤supθ∈[-r,0]χ({yn(θ)})+sups∈Jχ({y~n(s)})≤supθ∈[-r,0]χ(Ω(θ))+sups∈Jχ(Ω(s))≤supθ∈[-r,0]χ(Ω(θ))+χpc(Ω)=χ𝔉(Ω),(54)χ({y~n(s)})≤χ(Ω(s))≤χpc(Ω).
Thus, by (52), we see
(55)χ𝔉(Λ2Ω)≤2supt∈J[2N∫0t‍η(s)(supθ∈[-r,0]χ({yn(s+θ)})+χ({y~n(s)})supθ∈[-r,0]∫0t‍)ds]+ε≤2supt∈J(4N∫0t‍η(s)ds)·χ𝔉(Ω)+ε=8N∫0T‍η(s)ds·χ𝔉(Ω)+ε.
Sinceε is arbitrary, we can obtain
(56)χ𝔉(Λ2Ω)≤8N∫0T‍η(s)ds·χ𝔉(Ω).
Combining with (H3), we have(57)χ𝔉(ΛΩ)≤χ𝔉(Λ1Ω)+χ𝔉(Λ2Ω)≤8N∫0T‍η(s)ds·χ𝔉(Ω)<M*χ𝔉(Ω),
hence Λ is a χ𝔉-contraction on 𝔉(T). According to Theorem 4, the operator Λ has at least one fixed point x in Bρ. This completes the proof.Next, we establish a condition that guarantee that a mild solution satisfies (3).Proposition 8.
Assume that the hypotheses of Theorem7 are fulfilled and that ϕ(0)+g(x)(0)∈D(A). If x(·) is a mild solution of (1)–(4), then condition (3) holds.Proof.
Clearly,(1/t)∫0t‍S(t-s)f(s,xs,x(s))ds→0 as t→0. Moreover, noting that ϕ(0)+g(x)(0)∈D(A) and (11), we have C(·)(ϕ(0)+g(x)(0)) is of class C1. Therefore, we can see that
(58)limt→0+x(t)-x(0)t=limt→0+1t[∫0t‍(C(t)-I)(ϕ(0)+g(x)(0))+S(t)ξ+∫0t‍S(t-s)f(s,xs,x(s))ds]=ξ,
which shows the assertion.


## 4.  Application

In this section, we consider an application of the theory developed in Section3 to the study of an impulsive partial differential equation with unbounded delay.Example 9.
X
=
L
2
(
[
0
,
π
]
), A:D(A)⊆X→X is the map defined by Aϑ=ϑ′′ with domain D(A)={ϑ∈X:ϑ′′∈X,ϑ(0)=ϑ(π)=0}.
We consider the following integrodifferential model:(59)∂2∂t2v(t,ξ)=∂2∂ξ2v(t,ξ)+sin|v(t,ξ)|+t2∫t-rt‍γ(θ-t)·cos(|v(θ,ξ)|t)dθ,v(t,π)=v(t,0)=0,v(θ,ξ)=v0(θ,ξ)+∫0π‍c(ξ,s)sin(1+v(θ,s))ds,-r≤θ≤0,∂∂tv(0,ξ)=ω(ξ),Δv(tk,ξ)=∫0π‍ρk(ξ,y)dy·cos2(v(tk,ξ)),1≤k≤p,
where t∈[0,T], r>0, ξ∈[0,π], 0<t1<t2<⋯<tp<T, ω∈X and vt(θ,ξ)=v(t+θ,ξ). γ:[-r,0]→R and c(ξ,s),ρk(ξ,z)∈L2([0,π]×[0,π],R) satisfy the following assumptions. (1)
The functionγ:[-r,0]→R is a continuous function and ∫-r0‍|γ(θ)|dθ<∞.(2)
The functionc(ξ,s),ξ,s∈[0,π] is measurable and there exists a constant N1 such that (π∫0π∫0πc2(ξ,s)dsdξ)1/2≤N1.(3)
For everyk=1,2,…,p, the function ρk(ξ,z),z∈[0,π], is measurable and there exists a constant N¯ such that
(60)(∫0π‍(∫0π‍ρk(ξ,z)dz)2dξ)1/2≤N¯.
To treat the above problem, we define(61)D(A)=H2([0,π])∩H01([0,π]),Au=u′′.A is the infinitesimal generator of a strongly continuous cosine function {C(t)}t∈R on X. Moreover, A has a discrete spectrum, the eigenvalues are -n2,n∈N, with the corresponding normalized eigenvectors ωn(x)=(2/π)sin(nx); the set {ωn;n∈N} is an orthonormal basis of X and the following properties hold. (a)
Ifω∈D(A), then Aω=-∑n=1∞n2〈ω,ωn〉ωn.(b)
For eachω∈X, C(t)ω=∑n=1∞cos(nt)〈ω,ωn〉ωn and S(t)ω=∑n=1∞(sin(nt)/n)〈ω,ωn〉ωn. Consequently, ∥C(t)∥=∥S(t)∥≤1 for t∈R, and {S(t)} is compact for every t∈R.
Forξ∈[0,π] and φ∈C([-r,0],X), we set
(62)x(t)(ξ)=v(t,ξ),ϕ(θ)(ξ)=v0(θ,ξ),θ∈[-r,0],g(φ(θ))(ξ)=∫0π‍c(ξ,s)sin(1+φ(θ)(s))ds,f(t,φ,x(t))(ξ)=sin|x(t)(ξ)|+t2∫-r0‍γ(θ)·cos(|φ(θ)(ξ)|t)dθ,Ik(x(tk))(ξ)=∫0π‍ρk(ξ,y)dy·cos2(x(tk)(ξ)).
Then the above equation (59) can be reformulated as the abstract (1)–(4).
Fort∈[0,T], we can see
(63)∥f(t,φ,x(t))∥≤∥x(t)∥+t2∫-r0‍|γ(θ)|dθ≤μ(t)(1+∥x(t)∥),
where μ(t):=max{1,t2∫-r0‍|γ(θ)|dθ}.
For anyx1,x2∈X,  φ,φ~∈C([-r,0],X),
(64)∥f(t,φ,x1(t))(ξ)-f(t,φ~,x2(t))(ξ)∥≤∥x1(t)-x2(t)∥+t∫-r0‍|γ(θ)|∥φ(θ)(ξ)-φ~(θ)(ξ)∥dθ.
Therefore, for any bounded setsD1⊂C([-r,0],X), D2⊂X, we have
(65)χ(f(t,D1,D2))≤χ(D2)+t∫-r0‍|γ(θ)|χ(D1(θ))dθ≤χ(D2)+tsup-r≤θ≤0χ(D1(θ))∫-r0‍|γ(θ)|dθ≤η(t)(sup-r≤θ≤0χ(D1(θ))+χ(D2)),a.e.t∈[0,T],
where η(t):=max{1,t∫-r0‍|γ(θ)|dθ}.
Forx∈X,
(66)∥Ik(x)∥≤N¯(1+∥x∥),k=1,2,…,p.
Suppose further that there exists a constantM~*∈(0,1) such that 8∫0T‍η(s)ds<M~* and pN¯<1, then (59) has at least a mild solution by Theorem 7.



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*Source: 724854-2013-02-27.xml*