# An Existence Result for Neutral Delay Integrodifferential Equations with Fractional Order and Nonlocal Conditions

Authors: Fang Li; Gaston M. N'Guérékata
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We study the existence of mild solutions of a class of neutral delay integrodifferential equations with fractional order and nonlocal conditions in a Banach spaceX. An existence result on the mild solution is obtained by using the theory of the measures of noncompactness and the theory of condensing maps. Two examples are given to illustrate the existence theorem.


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## Body

## 1. Introduction

Differential and integrodifferential equations of the fractional order are playing an increasingly important role in engineering, physics, and other fields of science, such as the fractal theory and the diffusion in porous media, electrolysis chemical, fractional biological neurons, condensate physics, statistical mechanics, so they attract the attention of many researchers (see, e.g., [1–12] and the references therein).Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (see, e.g., [3, 4, 6, 7, 9, 13, 14]).This paper is concerned with an existence result for nonlocal neutral delay fractional integrodifferential equations in a separable Banach spaceX:
(1.1)Dq(x(t)-h(t,xt))=A(x(t)-h(t,xt))+∫0tK(t,s)f(s,x(s),xs)ds,t∈[0,T],x(t)=g(x)(t)+ϕ(t),t∈[-r,0],
where T>0,0<q<1, 0<r<∞. The fractional derivative is understood here in the Caputo sense. A is the infinitesimal generator of an analytic semigroup S(·) of uniformly bounded linear operators on X, that is, there exists M≥1 such that ∥S(t)∥≤M. Here h:[0,T]×C([-r,0],X)→X, f:[0,T]×X×C([-r,0],X)→X, K:D→R(D={(t,s)∈[0,T]×[0,T]:t≥s}), g:C([-r,0],X)→C([-r,0],X), ϕ∈C([-r,0],X), where C([a,b],X) denotes the space of all continuous functions from [a,b] to X.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].The nonlocal condition can be applied in physics with better effect than that of the classical initial condition. There have been many significant developments in the study of nonlocal Cauchy problems (see, e.g., [10, 15–19] and references cited there in). To the authors’ knowledge, few papers can be found in the literature for the solvability of the fractional order delay integrodifferential equations of neutral type with nonlocal conditions.In this paper, motivated by above works, we study the neutral delay fractional integrodifferential equations with nonlocal condition (1.1) in a separable Banach space X and obtain the existence theorem based on a special measure of noncompactness without the assumptions that the nonlinearity f satisfies a Lipschitz type condition and the semigroup {S(t)}t≥0 generated by A is compact. Two examples are given to show the applications of the abstract result.


## 2. Preliminaries

Throughout this paper, we denote byX a separable Banach space with norm ∥·∥, by L(X) the Banach space of all linear and bounded operators on X, and by C([a,b],X) the space of all X-valued continuous functions on [a,b] with the supremum norm as follows:(2.1)‖x‖[a,b]=‖x‖C([a,b],X)=sup⁡{‖x(t)‖:t∈[a,b]},foranyx∈C([a,b],X).
Moreover, we abbreviate ∥u∥L1([0,T],R+) with ∥u∥L1, for any u∈L1([0,T],R+).We will need the following facts from the theory of measures of noncompactness and condensing maps (see, e.g., [20, 21]).Definition 2.1.
LetE be a Banach space and (𝒜,≥) a partially ordered set. A function β:P(E)→𝒜 is called a measure of noncompactness (MNC) in E if
(2.2)β(co⁡¯(Ω))=β(Ω)foreveryΩ∈P(E),
where P(E) denotes the class of all nonempty subsets of E.
A MNCβ is called:
(i)
monotone, ifΩ0,Ω1∈P(E),Ω0⊂Ω1 implies β(Ω0)≤β(Ω1);(ii)
nonsingular, ifβ({a}∪Ω)=β(Ω) for every a∈E,Ω∈P(E);(iii)
invariant with respect to union with compact sets, ifβ({D}∪Ω)=β(Ω) for every relatively compact set D⊂E,Ω∈P(E).If𝒜 is a cone in a normed space, we say that the MNC β is(iv)
algebraically semiadditive, ifβ(Ω0+Ω1)≤β(Ω0)+β(Ω1) for each Ω0,Ω1∈P(E);(v)
regular, ifβ(Ω)=0 is equivalent to the relative compactness of Ω;(vi)
real, if𝒜 is [0,+∞) with the natural order.As an example of the MNC possessing all these properties, we may consider the Hausdorff MNC(2.3)χ(Ω)=inf⁡{ɛ>0:Ωhasafiniteɛ-net}.Now, letG:[0,h]→P(E) be a multifunction. It is called: (i)
integrable, if it admits a Bochner integrable selectiong:[0,h]→E,g(t)∈G(t) for a.e. t∈[0,h];(ii)
integrably bounded, if there exists a functionϑ∈L1([0,h],E) such that(2.4)‖G(t)‖:=sup⁡{‖g‖:g∈G(t)}≤ϑ(t),a.e.t∈[0,h].We present the following assertion aboutχ-estimates for a multivalued integral [21, Theorem 4.2.3].Proposition 2.2.
For an integrable, integrably bounded multifunctionG:[0,h]→P(X), where X is a separable Banach space, let
(2.5)χ(G(t))≤q(t),fora.e.t∈[0,h],
where q∈L+1([0,h]). Then, χ(∫0tG(s)ds)≤∫0tq(s)ds for all t∈[0,h].LetE be a Banach space, and β a monotone nonsingular MNC in E.Definition 2.3.
A continuous map𝔉:Y⊆E→E is called condensing with respect to a MNC β (or β-condensing) if, for every bounded set Ω⊆Y which is not relatively compact, we have
(2.6)β(F(Ω))≱β(Ω).The following fixed point principle (see, e.g., [20, 21]) will be used later.Theorem 2.4.
Let𝔐 be a bounded convex closed subset of E and 𝔉:𝔐→𝔐 a β-condensing map. Then, Fix𝔉={x:x=𝔉(x)} is nonempty.Theorem 2.5.
LetV⊂E be a bounded open neighborhood of zero and 𝔉:V¯→E a β-condensing map satisfying the boundary condition
(2.7)x≠λF(x)
for all x∈∂V and 0<λ≤1. Then, Fix𝔉 is a nonempty compact set.We state a generalization of Gronwall’s lemma for singular kernels [22, Lemma 7.1.1].Lemma 2.6.
Letv,w:[0,T]→[0,+∞) be continuous functions. If w(·) is nondecreasing and there are constants a>0 and 0<α<1 such that
(2.8)v(t)≤w(t)+a∫0t(t-s)-αv(s)ds,
then there exists a constant k=k(α) such that
(2.9)v(t)≤w(t)+ka∫0t(t-s)-αw(s)ds,foreacht∈[0,T].Based on the work in [1, 2, 23], we set (2.10)Q(t)=∫0∞ξq(σ)S(tqσ)dσ,R(t)=q∫0∞σtq-1ξq(σ)S(tqσ)dσ,
and ξq is a probability density function defined on (0,∞) such that(2.11)ξq(σ)=1qσ-1-(1/q)ϖq(σ-1/q)≥0,
where(2.12)ϖq(σ)=1π∑n=1∞(-1)n-1σ-qn-1Γ(nq+1)n!sin⁡(nπq),σ∈(0,∞).Remark 2.7 (see [23]).
It is not difficult to verify that forv∈[0,1],
(2.13)∫0∞σvξq(σ)dσ=∫0∞σ-qvϖq(σ)dσ=Γ(1+v)Γ(1+qv).Then, we can see(2.14)‖R(t)‖≤Cq,Mtq-1,t>0,
where Cq,M=qM/Γ(1+q).We define the mild solution for problem (1.1) as follows.Definition 2.8.
A functionx∈C([-r,T],X) satisfying the equation
(2.15)x(t)={g(x)(t)+ϕ(t),t∈[-r,0],Q(t)(g(x)(0)+ϕ(0)-h(0,ϕ+g(x)))+h(t,xt)+∫0t∫0sR(t-s)K(s,τ)f(τ,x(τ),xτ)dτds,t∈[0,T],
is called a mild solution of problem (1.1).


## 3. Main Result

We will require the following assumptions.(H1)
f:[0,T]×X×C([-r,0],X)→X satisfies f(·,v,w):[0,T]→X is measurable for all (v,w)∈X×C([-r,0],X) and f(t,·,·):X×C([-r,0],X)→X is continuous for a.e. t∈[0,T], and there exist two functions μi(·)∈L1([0,T],R+)(i=1,2) such that(3.1)‖f(t,v,w)‖≤μ1(t)‖v‖+μ2(t)‖w‖[-r,0],for almost allt∈[0,T]. (H2)
There exists a functionη∈L1([0,T],R+) such that for any bounded sets D1⊂X, D2⊂C([-r,0],X),(3.2)χ(f(τ,D1,D2))≤η(τ)(χ(D1)+sup⁡θ∈[-r,0]χ(D2(θ))),a.e.τ∈[0,T].(H3)
(i) There exists a continuous functionLg:[-r,0]→R+ such that(3.3)‖g(x)(t)-g(y)(t)‖≤Lg(t)‖x(t)-y(t)‖,t∈[-r,0].(ii)
The functiong(x)(·):[-r,0]→C([-r,0],X) is equicontinuous and uniformly bounded, that is, there exists a constant N>0 such that
(3.4)‖g(x)‖[-r,0]≤N∀x∈C([-r,0],X).(H4)
(i) There exists a constantLh>0 such that(3.5)‖h(t1,φ)-h(t2,φ̃)‖≤Lh(|t1-t2|+‖φ-φ̃‖[-r,0]),t1,t2∈[0,T],φ,φ̃∈C([-r,0],X).(ii)
For every bounded setΩ⊂C([-r,0],X) and t∈[0,T], there exists a constant 0<ω<1 such that(3.6)χ(h(t,Ω))≤ωsup⁡s∈[-r,0]χ(Ω(s)).(H5)
For eacht∈[0,T], K(t,·) is measurable on [0,t] and K(t)=esssup{|K(t,s)|,0≤s≤t} is bounded on [0,T]. The map t→Kt is continuous from [0,T] to L∞([0,T],R), here, Kt(s)=K(t,s).(H6)
There existsM*∈(0,1) such that(3.7)L0+MKTqΓ(q+1)max⁡{2‖η‖L1,‖μ1‖L1+‖μ2‖L1}<M*,whereL0=MLg*(1+ω)+max{Lh,ω}, Lg*=supt∈[-r,0]Lg(t), K=supt∈[0,T]K(t).Theorem 3.1.
Assume that (H1)–(H6) are satisfied. Then, the mild solutions set of problem (1.1) is a nonempty compact subset of the space C([-r,T],X).Proof.
Define the operatorΛ:C([-r,T],X)→C([-r,T],X) in the following way:
(3.8)(Λx)(t)={g(x)(t)+ϕ(t),t∈[-r,0],Q(t)(g(x)(0)+ϕ(0)-h(0,ϕ+g(x)))+h(t,xt)+∫0t∫0sR(t-s)K(s,τ)f(τ,x(τ),xτ)dτds,t∈[0,T].
It is clear that the operator Λ is well defined, and the fixed point of Λ is the mild solution of problem (1.1).
The operatorΛ can be written in the form Λ=∑i=13Λi, where the operators Λi,i=1,2,3 are defined as follows:(3.9)(Λ1x)(t)={g(x)(t)+ϕ(t),t∈[-r,0],Q(t)(g(x)(0)+ϕ(0)),t∈[0,T],(Λ2x)(t)={0,t∈[-r,0],-Q(t)h(0,ϕ+g(x))+h(t,xt),t∈[0,T],(Λ3x)(t)={0,t∈[-r,0],∫0t∫0sR(t-s)K(s,τ)f(τ,x(τ),xτ)dτds,t∈[0,T].
Obviously, under the assumptions of g and h, Λ1 and Λ2 are continuous, respectively. For t∈[0,T], we can prove that Λ3 is continuous.
Indeed, let{xk}k∈N be a sequence such that xk→x in C([-r,T],X) as k→∞. Since f satisfies (H1), for almost every t∈[0,T], we get(3.10)f(t,xk(t),xtk)⟶f(t,x(t),xt),ask⟶∞.
Noting that xk→x in C([-r,T],X), we can see that there exists ɛ>0 such that ∥xk-x∥[-r,T]≤ɛ for k sufficiently large. Therefore, we have
(3.11)‖f(t,xk(t),xtk)-f(t,x(t),xt)‖≤μ1(t)‖xk(t)‖+μ2(t)‖xtk‖[-r,0]+μ1(t)‖x(t)‖+μ2(t)‖xt‖[-r,0]≤μ1(t)‖xk(t)-x(t)‖+μ1(t)‖x(t)‖+μ2(t)‖xtk-xt‖[-r,0]+μ2(t)‖xt‖[-r,0]+μ1(t)‖x(t)‖+μ2(t)‖xt‖[-r,0]=μ1(t)‖xk(t)-x(t)‖+2μ1(t)‖x(t)‖+μ2(t)‖xtk-xt‖[-r,0]+2μ2(t)‖xt‖[-r,0]≤μ1(t)ɛ+2μ1(t)‖x‖[0,T]+μ2(t)ɛ+2μ2(t)‖x‖[-r,T]≤(μ1(t)+μ2(t))ɛ+2(μ1(t)+μ2(t))‖x‖[-r,T].
It follows from the Lebesgue’s Dominated Convergence Theorem that
(3.12)‖(Λ3xk)(t)-(Λ3x)(t)‖≤∫0t∫0s‖R(t-s)K(s,τ)[f(τ,xk(τ),xτk)-f(τ,x(τ),xτ)]‖dτds≤KCq,M∫0t∫0s(t-s)q-1‖f(τ,xk(τ),xτk)-f(τ,x(τ),xτ)‖dτds→0,ask⟶∞.
Therefore, we obtain that
(3.13)lim⁡k→∞‖Λ3xk-Λ3x‖[-r,T]=0.
This shows that Λ3 is continuous. Therefore, Λ is continuous.
Consider the set(3.14)Bρ={x∈C([-r,T],X):‖x‖[-r,T]≤ρ}.
In view of (H4), for x∈Bρ,
(3.15)‖h(t,xt)‖≤‖h(t,xt)-h(t,0)‖+‖h(t,0)‖≤Lhρ+M1,
where M1=supt∈[0,T]∥h(t,0)∥.
Next, we show that there exists someρ>0 such that ΛBρ⊂Bρ. Suppose, on the contrary, that for each ρ>0, there exist xρ(·)∈Bρ and some t∈[-r,T] such that ∥(Λxρ)(t)∥>ρ. Now, if t∈[-r,0], then(3.16)ρ<‖(Λxρ)(t)‖=‖g(xρ)(t)+ϕ(t)‖≤N+‖ϕ‖[-r,0],
and if t∈[0,T], then
(3.17)ρ<‖(Λxρ)(t)‖≤‖(Λ1xρ)(t)‖+‖(Λ2xρ)(t)‖+‖(Λ3xρ)(t)‖≤‖Q(t)(g(xρ)(0)+ϕ(0))‖+‖-Q(t)h(0,ϕ+g(xρ))+h(t,xtρ)‖+∫0t∫0s‖R(t-s)K(s,τ)f(τ,xρ(τ),xτρ)‖dτds≤M(N+‖ϕ(0)‖)+MLh(‖ϕ‖[-r,0]+N)+ρLh+M1(M+1)+ρCq,MK∫0t(t-s)q-1∫0s(μ1(τ)+μ2(τ))dτds≤M2+ρLh+TqKMρΓ(q+1)(‖μ1‖L1+‖μ2‖L1),
where M2:=M(N+∥ϕ(0)∥)+MLh(∥ϕ∥[-r,0]+N)+M1(M+1).
Denote byLρ the right-hand side of (3.17), then we have(3.18)ρ≤max⁡{‖ϕ‖[-r,0]+N,Lρ}.
Dividing both sides of (3.18) by ρ and taking ρ→∞, we have
(3.19)Lh+TqKMΓ(q+1)(‖μ1‖L1+‖μ2‖L1)≥1.
This contradicts (H6). Hence, for some positive number, ρ,ΛBρ⊂Bρ.
Letχ be a Hausdorff MNC in X; we consider the measure of noncompactness β in the space C([-r,T],X) with values in the cone R+2 of the following way: for every bounded subset Ω⊂C([-r,T],X),(3.20)β(Ω)=(Ψ(Ω),modc(Ω)),
where
(3.21)Ψ(Ω)=sup⁡t∈[-r,T]χ(Ω(t)),
and modc(Ω) is the module of equicontinuity of Ω given by
(3.22)modc(Ω)=lim⁡δ→0sup⁡x∈Ωmax⁡|t1-t2|≤δ‖x(t1)-x(t2)‖.
Next, we show that the operatorΛ is β-condensing on every bounded subset of C([-r,T],X).
LetΩ⊂C([-r,T],X) be a nonempty, bounded set such that
(3.23)β(Λ(Ω))≥β(Ω).
Firstly, we estimateΨ(Ω). For t∈[-r,0], x,y∈Ω, we have(3.24)‖g(x)(t)-g(y)(t)‖≤Lg*‖x(t)-y(t)‖,
if we denote χ(Ω[a,b]):=supt∈[a,b]χ(Ω(t)), then
(3.25)χ(g(Ω)[-r,0])≤Lg*χ(Ω[-r,0]).
Then,
(3.26)χ((ΛΩ)([-r,0]))=χ({ϕ([-r,0])}+g(Ω)[-r,0])=χ(g(Ω)[-r,0])≤Lg*χ(Ω[-r,0]),
by (3.23), we can see that
(3.27)χ(Ω[-r,0])=0.
Furthermore, χ((ΛΩ)([-r,0]))=0.
Fort∈[0,T], one gets(3.28)‖(Λ1x)(t)-(Λ1y)(t)‖≤‖Q(t)‖‖g(x)(0)-g(y)(0)‖≤MLg*‖x(0)-y(0)‖,
thus χ((Λ1Ω)([0,T]))≤MLg*Ψ(Ω).
Moreover, we see that(3.29)χ(-Q(t)h(0,ϕ+g(Ω)))≤Mωsup⁡s∈[-r,0]χ(g(Ω)(s))≤MωLg*Ψ(Ω),χ(h(t,Ωt))≤ωsup⁡-r≤θ≤0χ(Ω(t+θ))≤ωΨ(Ω),
where Ωt={xt:x∈Ω}. Now, we can see χ((Λ2Ω)([0,T]))≤ω(MLg*+1)Ψ(Ω).
For anyt∈[0,T], we set(3.30)Φ(Ω)(t)={∫0t∫0sR(t-s)K(s,τ)f(τ,x(τ),xτ)dτds:x∈Ω}.
We consider the multifunction s∈[0,t]⊸G(s),
(3.31)G(s)={R(t-s)∫0sK(s,τ)f(τ,x(τ),xτ)dτ:x∈Ω}.
Obviously, G is integrable, and, from (H1), it follows that it is integrably bounded. Moreover, noting that (H2) and Proposition 2.2, we have the following estimate for a.e. s∈[0,t]:
(3.32)χ(G(s))≤Cq,M(t-s)q-1K⋅χ({∫0sf(τ,x(τ),xτ)dτ:x∈Ω})=Cq,M(t-s)q-1K⋅χ(∫0sf(τ,Ω(τ),Ωτ)dτ)≤Cq,M(t-s)q-1K⋅∫0s[η(τ)(χ(Ω(τ))+sup⁡θ∈[-r,0]χ(Ω(τ+θ)))]dτ.
Moreover, the equality (3.27) implies that
(3.33)sup⁡-r≤σ≤0χ(Ω(σ))=0.
Therefore, for τ∈[0,T], we have
(3.34)sup⁡τ-r≤σ≤τχ(Ω(σ))≤sup⁡-r≤σ≤0χ(Ω(σ))+sup⁡0≤σ≤τχ(Ω(σ))=sup⁡0≤σ≤τχ(Ω(σ)),
then we can see
(3.35)χ(G(s))≤2Cq,M(t-s)q-1K⋅‖η‖L1sup⁡0≤θ≤tχ(Ω(θ))≤2Cq,M(t-s)q-1K⋅‖η‖L1⋅Ψ(Ω).
Applying Proposition 2.2, we have
(3.36)χ(Φ(Ω)(t))=χ(∫0tG(s)ds)≤2MKTqΓ(q+1)⋅‖η‖L1⋅Ψ(Ω),
that is,
(3.37)χ((Λ3Ω)([0,T]))≤2MKTqΓ(q+1)‖η‖L1Ψ(Ω).
Now, we can see
(3.38)χ((ΛΩ)([0,T]))≤χ((Λ1Ω)([0,T]))+χ((Λ2Ω)([0,T]))+χ((Λ3Ω)([0,T]))≤(MLg*(1+ω)+ω+2MKTqΓ(q+1)‖η‖L1)Ψ(Ω),
furthermore
(3.39)Ψ(ΛΩ)≤(MLg*(1+ω)+ω+2MKTqΓ(q+1)‖η‖L1)Ψ(Ω),
which implies, by (H6) and (3.23), Ψ(Ω)=0. Next, we will prove modc(Ω)=0.
Noting (H3)(ii) and the continuity of{S(t)}t≥0 in the uniform operator topology for t>0, we can see(3.40)modc(Λ1Ω)=0.
Letδ>0, t1,t2∈[0,T] such that 0<|t1-t2|≤δ and x∈Ω, we obtain(3.41)‖h(t1,xt1)-h(t2,xt2)‖≤Lh(|t1-t2|+‖xt1-xt2‖[-r,0])=Lh(|t1-t2|+sup⁡θ∈[-r,0],|t1-t2|≤δ‖x(t1+θ)-x(t2+θ)‖)≤Lh(|t1-t2|+sup⁡s1,s2∈[-r,T],|s1-s2|≤δ‖x(s1)-x(s2)‖).
Moreover, noting the continuity of {S(t)}t≥0 in the uniform operator topology for t>0, then we have
(3.42)modc(Λ2Ω)≤Lhmodc(Ω).
For0<t2<t1<T and x∈Ω, we have(3.43)‖(Λ3x)(t1)-(Λ3x)(t2)‖≤I1+I2,
where
(3.44)I1=K∫0t2∫0s‖[R(t1-s)-R(t2-s)]f(τ,x(τ),xτ)‖dτds,I2=K∫t2t1∫0s‖R(t1-s)‖‖f(τ,x(τ),xτ)‖dτds.
ForI1, we have(3.45)I1≤qK∫0t2∫0∞σ‖[(t1-s)q-1-(t2-s)q-1]ξq(σ)S((t1-s)qσ)∫0sf(τ,x(τ),xτ)dτ‖dσds+qK∫0t2∫0∞σ(t2-s)q-1ξq(σ)‖S((t1-s)qσ)-S((t2-s)qσ)‖∫0s‖f(τ,x(τ),xτ)‖dτdσds≤Cq,MK∫0t2|(t1-s)q-1-(t2-s)q-1|∫0s(μ1(τ)‖x(τ)‖+μ2(τ)‖xτ‖[-r,0])dτds+qK∫0t2∫0∞σ(t2-s)q-1ξq(σ)‖S((t1-s)qσ)-S((t2-s)qσ)‖∫0s‖f(τ,x(τ),xτ)‖dτdσds≤K(‖μ1‖L1+‖μ2‖L1)‖x‖[-r,T]⋅[Cq,M∫0t2|(t1-s)q-1-(t2-s)q-1|ds+q∫0t2∫0∞σ(t2-s)q-1×ξq(σ)‖S((t1-s)qσ)-S((t2-s)qσ)‖dσds∫0t2].
Clearly, the first term on the right-hand side of (3.45) tends to 0 as t2→t1. The second term on the right-hand side of (3.45) tends to 0 as t2→t1 as a consequence of the continuity of S(t) in the uniform operator topology for t>0.
In view of the assumption ofμi(s)(i=1,2), we see that(3.46)I2=K∫t2t1∫0s‖R(t1-s)‖‖f(τ,x(τ),xτ)‖dτds≤KCq,M(‖μ1‖L1+‖μ2‖L1)‖x‖[-r,T]∫t2t1(t1-s)q-1ds→0,ast2⟶t1.
Thus, the set {(Λ3x)(·):x∈Ω} is equicontinuous, then modc(Λ3Ω)=0.
Since(3.47)modc(ΛΩ)≤∑i=13modc(ΛiΩ)≤L0modc(Ω),
then modc(Ω)=0, which yields from (3.23), hence
(3.48)β(Ω)=(0,0).
The regularity property of β implies the relative compactness of Ω.
Now, it follows from Definition2.3 that Λ is β-condensing.
According to Theorem2.4, problem (1.1) has at least one mild solution.
Next, forδ∈(0,1], we consider the following one-parameter family of maps:(3.49)Π:[0,1]×C([-r,T],X)⟶C([-r,T],X),(δ,x)⟶Π(δ,x)=δΛ(x).
We will prove that the fixed point set of the family Π,
(3.50)Fix⁡Π={x∈Π(δ,x)forsomeδ∈(0,1]}
is a priori bounded.
Letx∈FixΠ, for t∈[0,T], we have(3.51)‖xt‖[-r,0]=sup⁡-r≤θ≤0‖x(t+θ)‖≤sup⁡-r≤τ≤0‖x(τ)‖+sup⁡0≤τ≤t‖x(τ)‖≤N+‖ϕ‖[-r,0]+sup⁡0≤τ≤t‖x(τ)‖.
Then,(3.52)‖x(t)‖≤‖Q(t)(g(x)(0)+ϕ(0))‖+‖-Q(t)h(0,ϕ+g(x))‖+‖h(t,xt)‖+∫0t∫0s‖R(t-s)K(s,τ)f(τ,x(τ),xτ)‖dτds≤M(N+‖ϕ(0)‖)+Lh(M+1)(‖ϕ‖[-r,0]+N)+M1(M+1)+Lhsup⁡0≤τ≤t‖x(τ)‖+Cq,MK∫0t(t-s)q-1∫0s[μ1(τ)‖x(τ)‖+μ2(τ)(‖ϕ‖[-r,0]+N+sup⁡0≤θ≤τ‖x(θ)‖)]dτds≤θ1+Lhsup⁡0≤τ≤t‖x(τ)‖+Cq,MK(‖μ1‖L1+‖μ2‖L1)∫0t(t-s)q-1sup⁡0≤τ≤s‖x(τ)‖ds=θ1+Lhsup⁡0≤τ≤t‖x(τ)‖+θ2∫0t(t-s)q-1sup⁡0≤τ≤s‖x(τ)‖ds,
where
(3.53)θ1:=L+MKTq‖μ2‖L1Γ(q+1)(‖ϕ‖[-r,0]+N),L:=M(N+‖ϕ(0)‖)+Lh(M+1)(‖ϕ‖[-r,0]+N)+M1(M+1),θ2:=Cq,MK(‖μ1‖L1+‖μ2‖L1).
We denote κ(t):=sup0≤s≤t∥x(s)∥. Let t̃∈[0,t] such that κ(t)=∥x(t̃)∥. Then, by (3.52), we can see
(3.54)κ(t)≤θ1+Lhκ(t)+θ2∫0t(t-s)q-1κ(s)ds.
By Lemma 2.6, there exists a constant k̃=k̃(q) such that
(3.55)κ(t)≤θ11-Lh+k̃θ1θ2(1-Lh)2∫0t(t-s)q-1ds≤θ11-Lh+k̃θ1θ2Tqq(1-Lh)2:=ζ.
Therefore,
(3.56)sup⁡t∈[-r,T]‖x(t)‖≤sup⁡t∈[-r,0]‖x(t)‖+sup⁡t∈[0,T]‖x(t)‖=sup⁡t∈[-r,0]‖g(x)(t)+ϕ(t)‖+sup⁡t∈[0,T]‖x(t)‖≤N+‖ϕ‖[-r,0]+ζ.
Now, we consider a closed ball(3.57)BR={x∈C([-r,T],X):‖x‖[-r,T]≤R}⊂C([-r,T],X).
We take the radius R>0 large enough to contain the set FixΠ inside itself. Moreover, from the proof above, Λ:BR→C([-r,T],X) is β-condensing and it remains to apply Theorem 2.5.


## 4. Applications

Example 4.1.
In this section, we consider the following integrodifferential model:(4.1)∂q∂tq[v(t,ξ)-e-t∫-r0γ1(θ)1+|v(t+θ,ξ)|dθ]=∂2∂ξ2[v(t,ξ)-e-t∫-r0γ1(θ)1+|v(t+θ,ξ)|dθ]+∫0t(t-s)sksin⁡|v(s,ξ)|ds+∫0t(t-s)∫-r0γ2(θ)s2/3⋅sin⁡(|v(s+θ,ξ)|s)dθds,v(t,0)-e-t∫-r0γ1(θ)1+|v(t+θ,0)|dθ=0,v(t,1)-e-t∫-r0γ1(θ)1+|v(t+θ,1)|dθ=0,v(θ,ξ)=v0(θ,ξ)+eμθk2⋅|v(θ,ξ)|1+|v(θ,ξ)|,-r≤θ≤0,
where 0≤t≤1, ξ∈[0,1], k∈N, r>0, μ>0, γ1:[-r,0]→R, γ2:[-r,0]→R, v0:[-r,0]×[0,1]→R are continuous functions, and ∫-r0|γ1(θ)|dθ<1, ∫-r0|γ2(θ)|dθ<∞.
SetX=L2([0,1],R) and define A by(4.2)D(A)=H2(0,1)∩H01(0,1),Au=u′′.
Then, A generates a compact, analytic semigroup S(·) of uniformly bounded linear operators, and ∥S(t)∥≤1.
Forξ∈[0,1] and φ∈C([-r,0],X), we set(4.3)x(t)(ξ)=v(t,ξ),ϕ(θ)(ξ)=v0(θ,ξ),θ∈[-r,0],h(t,φ)(ξ)=e-t∫-r0γ1(θ)1+|φ(θ)(ξ)|dθ,g(φ(θ))(ξ)=eμθk2⋅|φ(θ)(ξ)|1+|φ(θ)(ξ)|,K(t,s)=t-s,f(t,x(t),φ)(ξ)=tksin⁡|x(t)(ξ)|+∫-r0γ2(θ)t2/3⋅sin⁡(|φ(θ)(ξ)|t)dθ.
Then, (4.1) can be reformulated as the abstract (1.1).
Moreover, fort∈(0,1], we can see(4.4)‖f(t,x(t),φ)‖≤tk‖x(t)‖+1t3‖φ‖[-r,0]∫-r0|γ2(θ)|dθ=μ1(t)‖x(t)‖+μ2(t)‖φ‖[-r,0],
where μ1(t):=tk,μ2(t):=(1/t3)∫-r0|γ2(θ)|dθ.
For anyx1,x2∈X, φ,φ̃∈C([-r,0],X),(4.5)‖f(t,x1(t),φ)(ξ)-f(t,x2(t),φ̃)(ξ)‖≤tk‖x1(t)-x2(t)‖+1t3∫-r0|γ2(θ)|‖φ(θ)(ξ)-φ̃(θ)(ξ)‖dθ.
Therefore, for any bounded setsD1⊂X, D2⊂C([-r,0],X), we have(4.6)χ(f(t,D1,D2))≤tk⋅χ(D1)+1t3∫-r0|γ2(θ)|χ(D2(θ))dθ≤tk⋅χ(D1)+1t3sup⁡-r≤θ≤0χ(D2(θ))∫-r0|γ2(θ)|dθ≤η(t)(χ(D1)+sup⁡-r≤θ≤0χ(D2(θ))),a.e.t∈[0,1],
where η(t):=max{tk,(1/t3)∫-r0|γ2(θ)|dθ}.
Forφ,φ̃∈C([-r,0],X), θ∈[-r,0], we can see(4.7)‖g(φ)(ξ)-g(φ̃)(ξ)‖≤eμθk2⋅‖φ-φ̃‖≤1k2⋅‖φ-φ̃‖.
We denote Lg:=1/k2. Moreover,
(4.8)‖g(φ)(ξ)‖≤1k2:=N.
Fort1,t2∈[0,1], φ,φ̃∈C([-r,0],X), we have(4.9)‖h(t1,φ)-h(t2,φ̃)‖≤|t1-t2|∫-r0‖γ1(θ)1+|φ(θ)(ξ)|‖dθ+e-t2∫-r0‖γ1(θ)(11+|φ(θ)(ξ)|-11+|φ̃(θ)(ξ)|)‖dθ≤|t1-t2|∫-r0|γ1(θ)|dθ+∫-r0|γ1(θ)|‖φ(θ)(ξ)-φ̃(θ)(ξ)‖dθ≤|t1-t2|∫-r0|γ1(θ)|dθ+∫-r0|γ1(θ)|dθ⋅‖φ-φ̃‖[-r,0]=Lh(|t1-t2|+‖φ-φ̃‖[-r,0]),
where Lh=∫-r0|γ1(θ)|dθ.
Moreover, from above inequality, we can see that for any bounded setD̃⊂C([-r,0],X) and t∈[0,1],(4.10)χ(h(t,D̃))≤∫-r0|γ1(θ)|χ(D̃(θ))dθ≤∫-r0|γ1(θ)|dθ⋅sup⁡θ∈[-r,0]χ(D̃(θ)).
Suppose further that there exists a constantM*∈(0,1) such that(4.11)L0+MKΓ(q+1)max⁡{2‖η‖L1,‖μ1‖L1+‖μ2‖L1}<M*,
then (4.1) has a mild solution by Theorem 3.1.
For example, if we put(4.12)γ1(θ)=γ2(θ)=ekθ,q=0.5,k=7,r=1,
then Lh=(e7-1)/(7e7)≈0.143, Lg*=1/49≈0.020, max{2∥η∥L1,∥μ1∥L1+∥μ2∥L1}=2∥η∥L1≈0.579,Γ(3/2)≈0.886, L0=MLg*(1+ω)+Lh=(1/49)(1+(e7-1)/(7e7))+(e7-1)/(7e7)≈0.166, thus, we see
(4.13)L0+MKΓ(q+1)max⁡{2‖η‖L1,‖μ1‖L1+‖μ2‖L1}<0.9<1.Example 4.2.
LetΩ be a bounded domain with smooth boundary ∂Ω in Rn and let A(ξ,D) be the symmetric second-order differential operator given by
(4.14)A(ξ,D)u=-∑k,l=1n∂∂ξk(ak,l(ξ)∂u∂ξl),
where ξ=(ξ1,…,ξn)∈Rn.
We assume that the coefficientsak,l(ξ)=al,k(ξ) are real valued and continuously differentiable in Ω¯ and A(ξ,D) is strongly elliptic, that is, there is a constant C0>0 such that(4.15)∑k,l=1nak,l(ξ)ηiηj≥C0∑k=1nηk2,
for any η=(η1,…,ηn)∈Rn.
SetX=L2(Ω) and define A by(4.16)D(A)=H2(Ω)∩H01(Ω),Au=A(ξ,D)u.
Then, -A generates an analytic semigroup S̃(·) of uniformly bounded linear operators, and ∥S̃(t)∥≤1 [24].
We consider the following integrodifferential problem:(4.17)∂q∂tq[v(t,ξ)-e-t∫t-rta(s-t)sin⁡v(s,ξ)ds]+A(ξ,D)[v(t,ξ)-e-t∫t-rta(s-t)sin⁡v(s,ξ)ds]=∫0tf1(s)v(s,ξ)ds+∫0t∫s-rsb(s,τ-s)f2(s,v(τ,ξ))dτds,(t,ξ)∈[0,1]×Ω,v(t,ξ)-e-t∫t-rta(s-t)sin⁡v(s,ξ)ds=0,(t,ξ)∈[0,1]×∂Ω,v(θ,ξ)=v0(θ,ξ)+∫Ωc(ξ,z)1+|v(θ,z)|dz,-r≤θ≤0,
where r>0, a:[-r,0]→R, b:[0,1]×[-r,0]→R, v0:[-r,0]×Ω→R, f1:[0,1]→R are continuous functions, c(ξ,z)∈L2(Ω×Ω,R) and ∫-r0|a(θ)|dθ<1.
Forξ∈Ω and φ∈C([-r,0],X), we set(4.18)x(t)(ξ)=v(t,ξ),ϕ(θ)(ξ)=v0(θ,ξ),θ∈[-r,0],h(t,φ)(ξ)=e-t∫-r0a(θ)sin⁡φ(θ)(ξ)dθ,g(φ(θ))(ξ)=∫Ωc(ξ,z)1+|φ(θ)(z)|dz,f(t,x(t),φ)(ξ)=f1(t)x(t)(ξ)+∫-r0b(t,θ)f2(t,φ(θ)(ξ))dθ.
Then (4.17) can be reformulated as the abstract (1.1) (K(t,s)≡1).
Furthermore, we assume the following.
(1)
The functionf2:[0,1]×C([-r,0],X)→R is continuous, and there exist continuous functions l(t) and Lf(t) such that(4.19)‖f2(t,φ)(ξ)‖≤l(t)‖φ‖[-r,0],‖f2(t,φ)-f2(t,φ̃)‖≤Lf(t)‖φ-φ̃‖.(2)
The functionb(t,θ) is continuous in [0,1]×[-r,0] and(4.20)∫-r0|b(t,θ)|dθ=p(t)<∞.(3)
The functionc(ξ,z),ξ,z∈Ω is measurable, and there exists a constant N¯ such that(4.21)∫Ω‖c(ξ,z)‖dz≤N¯.
Thus, fort∈[0,1], we can see(4.22)‖f(t,x(t),φ)‖≤μ1(t)‖x(t)‖+μ2(t)‖φ‖[-r,0],
where μ1(t):=|f1(t)|,μ2(t):=l(t)p(t).
For any bounded setsD1⊂X, D2⊂C([-r,0],X), we have(4.23)χ(f(t,D1,D2))≤η(t)(χ(D1)+sup⁡-r≤θ≤0χ(D2(θ))),a.e.t∈[0,1],
where η(t):=max{|f1(t)|,Lf(t)p(t)}.
Forφ,φ̃∈C([-r,0],X), θ∈[-r,0], we can see(4.24)‖g(φ)(ξ)-g(φ̃)(ξ)‖≤(∫Ω∫Ωc2(ξ,z)dzdξ)1/2⋅‖φ-φ̃‖:=L̃g⋅‖φ-φ̃‖.
Moreover,
(4.25)‖g(φ)(ξ)‖≤∫Ω‖c(ξ,z)‖dz≤N¯.
It is clear that(4.26)‖h(t1,φ)-h(t2,φ̃)‖≤L¯h⋅(|t1-t2|+‖φ-φ̃‖[-r,0]),
where L¯h:=max{1,(meas·(Ω))1/2}·∫-r0|a(θ)|dθ, and for any bounded set D̃⊂C([-r,0],X) and t∈[0,1],
(4.27)χ(h(t,D̃))≤∫-r0|a(θ)|dθ⋅sup⁡θ∈[-r,0]χ(D̃(θ)).
Suppose further that there exists a constant M̃*∈(0,1) such that
(4.28)L0+1Γ(q+1)max⁡{2‖η‖L1,‖μ1‖L1+‖μ2‖L1}<M̃*,
where L0=L̃g(1+∫-r0|a(θ)|dθ)+L¯h, then (4.17) has a mild solution by Theorem 3.1.



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*Source: 952782-2011-11-03.xml*