# About the Existence Results of Fractional Neutral Integrodifferential Inclusions with State-Dependent Delay in Fréchet Spaces

Authors: Selvaraj Suganya; Dumitru Baleanu; Siva Selvarasu; Mani Mallika Arjunan
Journal: Journal of Function Spaces
 (2016)

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## Abstract

A recent nonlinear alternative for multivalued contractions in Fréchet spaces thanks to Frigon fixed point theorem consolidated with semigroup theory is utilized to examine the existence results for fractional neutral integrodifferential inclusions (FNIDI) with state-dependent delay (SDD). An example is described to represent the hypothesis.


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## Body

## 1. Introduction

We are dealing in this paper with the existence of mild solutions for FNIDI with SDD in Fréchet spaces by making use of the fixed point theorem of Frigon [1, Corollary 3.5]. In Section 3 of this paper, we deliberate the neutral integrodifferential inclusions of fractional-order of the model(1)ddtxt-Gt,xϱt,xt∈∫0tt-sα-2Γα-1Axs-Gs,xϱs,xsds+Ft,xϱt,xt,∫0tet,s,xϱs,xsds,a.e.t∈I=0,+∞,x0=ς∈B,wherein 1<α<2 and A:D(A)⊂E→E is the generator of an integral resolvent family characterized on a complex Banach space (E,·), the convolution integral within the equation is understood because of the Riemann-Liouville fractional integral (see [2]), F:I×B×E→P(E) is a multivalued map, (P(E) is the family of nonempty subsets of E), e:I×I×B→E, G:I×B→E, and ϱ:I×B→(-∞,+∞) are apposite functions, and B is theoretical phase space axioms characterized in Section 2.For almost any continuous functionx characterized on (-∞,b] and any t≥0, we designate by xt the part of B characterized by xt(θ)=x(t+θ) for θ≤0. Now, xt(·) speaks to the historical backdrop of the state from every θ∈(-∞,0] likely the current time t.The notion of a fractional derivative plays an important role in numerous technological innovation and scientific disciplines as the statistical modeling of frameworks and procedures in numerous fields, case in point, physical science, chemical industry, aerodynamics, electrodynamics of complex medium, and so forth. For information, we recommend the readers to refer to the treatise of Abbas et al. [3], Baleanu et al. [4], Podlubny [5], Diethelm [6], Kilbas et al. [7], and Zhou [8] and the papers of fractional differential and integrodifferential systems [9–12] and impulsive fractional differential systems [13–15] and the references cited therein.We recall that the fractional differential inclusions (FDI) occur in the mathematical modeling of specific models in financial aspects, optimal control, and so forth and are usually investigated by numerous writers; see, for instance, [16–18] and the references therein. Fractional equation with delay properties arises in several fields such as biological and physical ones with state-dependent delay or nonconstant delay. Nowadays, the existence results of mild solutions for such problems became very attractive and several researchers are working on it. Recently, several number of papers have been written on the fractional-order problems with state-dependent delay [19–23] and the sources therein. In particular, in [20, 21], the authors analyzed the existence results for neutral differential systems with SDD in Banach spaces, whereas in [19, 22] the authors investigate the same type of problems with SDD and impulsive conditions by utilizing appropriate fixed point theorem. Also, the integrodifferential systems are experienced in numerous ranges of science, the place where it is imperative to deal with aftereffect or delay (e.g., control theory, biology, ecology, and medicine). Particularly, one dependably depicts a model which has inherited qualities by integrodifferential systems in implementation; see, for instance, [23–25].The beginning stage of this work is reflected in [26–30]. Particularly, in [26], Agarwal et al. acquired the existence of mild solutions for FIDE of the structure (2)x′t=∫0tt-sα-2Γα-1Axsds+ft,xϱt,xt,a.e.t∈I=0,b,x0=ς∈B,in which 1<α<2 and A:D(A)⊂X→X is a linear densely described operator of sectorial kind on a complex Banach space (X,·), whereas, in [30], the authors establish the existence results for FIDI of the model (3)x′t-∫0tt-sα-2Γα-1Axsds∈Ft,xϱt,xt,a.e.t∈I=0,∞,x0=ς∈B,where α and A are the same as those mentioned in model (1).The existence of mild solutions for the division of FNIDI with SDD in Fréchet spaces of the structure (1) is by all accounts an unread point. By utilizing a few speculations as a part of [27, 30], our desire here is to yield the existence results for the above model (1) utilizing a nonlinear alternative for multivalued contractions as of late created by Frigon [1, Corollary 3.5].


## 2. Preliminary Notions

Below, we briefly present the mathematical tools required in this paper.LetC([0,n],E), n∈N, be the Banach space of all continuous functions from In=[0,n] into E making use of the standard norm(4)xn=sup⁡xt:0≤t≤n.AllowB(E) to be the space of all bounded linear operators N:E→E, having the common supremum norm(5)NBE=sup⁡Nx:x=1.A measurable functionx:In→E is Bochner integrable if and only if x is Lebesgue integrable (to get extra insights about Bochner integral, see the treatise of Yosida [31]).LetL1(In,E) signify the Banach space of all measurable functions x:In→E which are Bochner integrable making use of the norm (6)xL1=∫0nxtdt∀x∈L1In,E.Recognize the space (7)V+∞=x:-∞,+∞⟶E:xI∈CI,E,x0∈B,where xI is the restraint of x to I.We expect that the phase space(B,·B) is a seminormed linear space of functions mapping (-∞,0] into E and fulfilling the subsequent elementary adages as a result of Hale and Kato (find illustration in [32–34]).(P1)
Ifx:(-∞,n)→E is continuous on In and x0∈B, then for every t∈In the going hand in hand circumstances hold the following:
(i)
xt is in B;(ii)
|x(t)|≤HxtB;(iii)
xtB≤D1tsup⁡{|x(s)|:0≤s≤t}+D2(t)x0B, where H>0 is a constant and D1·:[0,+∞)→[0,+∞) is continuous, D2(·):[0,+∞)→[0,+∞) is locally bounded, and D1, D2 are independent of x(·).(P2)
For functionx(·) in (P1), xt is a B-valued continuous function on In.(P3)
The spaceB is complete.DesignateD1∗=sup⁡{D1(t):t∈In} and D2∗=sup⁡{D2(t):t∈In}.The next step is to review some known results from the fractional calculus.The Laplace transformation of a functionf∈Lloc1(R+,E) is determined by (8)Lfλ=s^λ=∫0∞e-λtftdt,Reλ>ω,if the integral is definitely convergent for Re(λ)>ω. With a specific end goal to give an operator hypothetical methodology, we review the subsequent definition [2].Definition 1.
LetA:D(A)⊂E→E be a closed and linear operator on a Banach space E. One addresses A as the generator of an integral resolvent if one can find ω>0 and a strongly continuous function S:R+→B(E) to ensure that (9)1s^λI-A-1x=∫0∞e-λtStxdt,Reλ>ω,x∈E.For this situation,S(t) is known as the integral resolvent family produced by A. For extra points of interest regarding this, we refer the reader to [35, Proposition 3.1 and Lemma 2.2].Remark 2.
The uniqueness and uniform continuity of the resolvent are long-familiar (see Benchohra and Litimein [30], Pruss [36]).Before we complete this section, we display some long-familiar outcomes from multivalued research.Indicate the following:(10)PX=Y⊂X:Y≠∅,PclX=Y∈PX:Yclosed,PbX=Y∈PX:Ybounded,PcpX=Y∈PX:Ycompact,Pcp,cX=Y∈PX:Ycompact  and  convex.Proposition 3 (see [37, Proposition III.4]).
IfΓ1 and Γ2 are compact valued measurable multifunctions, then the multifunction t→Γ1(t)∩Γ2(t) is measurable. If (Γn) is a sequence of compact valued measurable multifunctions, then t→∩Γn(t) is measurable, and if ∪Γn(t)¯ is compact, then t→∪Γn(t)¯ is measurable.Remark 4.
The definitions of measurable, admissible contraction, metric space, and nonlinear alternative fixed point theorem [1, Corollary 3.5] are classical in multivalued analysis; hence, we keep off it.Remark 5.
For primary and surplus points on Fréchet spaces, we refer the reader to [30].For eachx∈V+∞, specify the set of selections for F by (11)SF,x=v∈L1I,E:vt∈Ft,xϱt,xt,∫0tet,s,xϱs,xsdsfora.e.t∈I.For surplus points of benefit on multivalued maps, think about the treatise of Castaing and Valadier [37] and Graef et al. [38].


## 3. The Main Results

In this part, we prove the existence outcomes for the structure (1). To commence, we delineate the mild solution for the structure (1).Definition 6.
One affirms that the functionx:(-∞,+∞)→E is a mild solution of the model (1) if x(t)=ς(t) for all t≤0, the constraint of x(·) to the period [0,∞) is continuous and one can find v(·)∈L1(J,E), in a way that v(t)∈F(t,xϱ(t,xt),∫0t‍e(t,s,xϱ(s,xs))ds) a.e. t∈[0,∞), and x fulfills the consecutive integral equation:(12)xt=Stς0-G0,ς0+Gt,xϱt,xt+∫0tSt-svsdsfor  everyt∈0,+∞.Set(13)Rϱ-=ϱs,ς:s,ς∈I×B,ϱs,ς≤0.We generally expect thatϱ:I×B→(-∞,b] is continuous. Moreover, we make the subsequent assumption:(Hς)
functiont→ςt is continuous from R(ϱ-) into B and we can find a continuous and bounded function Lς:R(ϱ-)→(0,∞) in a way that (14)ςtB≤LςtςBfor  everyt∈Rϱ-.Lemma 7 (see [21, Lemma 3.1]).
Ifx:(-∞,b]→X is a function to ensure that x0=ς, then (15)xsB≤D2∗+LςςB+D1∗sup⁡xθ:θ∈0,max⁡0,s,s∈Rϱ-∪In,where Lς=supt∈R(ϱ-)Lς(t).The successive hypotheses will be required in whatever is left of this paper.(H1)
The solution operatorS(t)t∈I is compact for t≥0, and we can find M>0 in a way that (16)StBE≤Mfor  everyt∈I.(H2)
(i)
The multivalued mapF:I×B×E→Pcp,c(E) is Carathéodory and there is certainly function ϑ∈Lloc1(I,R+) and a continuous nondecreasing function ☊:R+→(0,∞) in a way that (17)Ft,u,y≤ϑt☊uB+y,for  everyt,u,y∈I×B×E.(ii)
For allR>0, we can find ℘R∈Lloc1(I,R+) to ensure that (18)HdFt,u,y,Ft,u¯,y¯≤℘Rtu-u¯B+y-y¯,in whicht∈I, in addition to, for all u,u¯∈B in conjunction with {uB,u¯B}≤R, y,y¯∈E joined with (19)d0,Ft,0,0≤℘Rta.e.t∈I.(H3)
(i)
There is a functionm∈Lloc1(I,R+) and a continuous nondecreasing function Ω:R+→(0,∞) to ensure that (20)et,s,u≤msΩuB∀t,s,u∈I×I×B.(ii)
There is a constantC1>0 in a way that (21)∫0tet,s,u-et,s,u¯ds≤C1u-u¯B,fort,s∈I,u,u¯∈B.(H4)
(i)
FunctionG(t,·) is continuous on I, and there are certainly positive constants c1, c2 in a way that (22)Gt,u≤c1uB+c2,for  everyt,u∈I×B.(ii)
For everyR>0, there is a function ℘~R(t)∈Lloc1(I,R+) in a way that (23)Gt,u-Gt,u¯≤℘~Rtu-u¯B,t∈I,u,u¯∈BwithuB,u¯B≤R.For each n∈N, we delineate, in V+∞, the family of seminorms by(24)xn=sup⁡e-τLn∗txt:t∈0,n,in which Ln∗(t)=∫0t‍℘n∗(s)ds, ℘n∗(t)=max⁡{℘~n(t)D1∗,℘¯(t)}, ℘¯(t)=M℘n(t)(1+C1)D1∗, and ℘n∗=sup⁡{℘n∗(t):t∈[0,n]} and accept that ℘n∗+1/τ<1 and ℘n, ℘~n are a function from (H2)(ii) and (H4)(ii) appropriately.Theorem 8.
Expect that (H1)–(H4) and (Hς) hold, and believe that μ=1-c1D1∗>0 and (25)∫C+∞ds☊s+Ωs>∫0nνsdsforn∈N,in which C=cn+(D1∗/μ)[M|G(0,ς(0))|+c1cn+c2]. At that point, model (1) has a mild solution on (-∞,+∞).Proof.
We will transmute the structure (1) into a fixed point problem. Recognize the multivalued operator Υ:V+∞→P(V+∞) specified by Υ(h)={h∈V+∞} with (26)ht=ςt,t≤0;Stς0-G0,ς0+Gt,xϱt,xt+∫0tSt-svsds,t∈I,where v∈SF,xϱ(s,xs). For ς∈B, we express function y(·):(-∞,+∞)→E by (27)yt=ςt,t≤0;Stς0,t∈I,and then y0=ς. For every function z∈V+∞ with z0=0, we designate by z~ the function clear by (28)z~t=0,t≤0;zt,t∈I.If x(·) fulfills (12), we are able to decompose it as x(t)=z(t)+y(t), t∈I, which suggests that xt=zt+yt, for each t∈I, and also the function z(·) fulfills (29)zt=Gt,zϱt,zt+yt+yϱt,zt+yt-StG0,ς0+∫0tSt-svsds,t∈I,where v(s)∈SF,zϱ(s,zs+ys)+yϱ(s,zs+ys).
LetV+∞0={z∈V+∞:z(0)=0∈B}. For any z∈V+∞0, we sustain (30)z+∞=sup⁡zs:0≤s<+∞+z0B=sup⁡zs:0≤s<+∞.
Along these lines,V+∞0 is a Banach space with the norm ·+∞. We delimit the operator Υ¯:V+∞0→P(B+∞0) by Υ¯(z)={h∈V+∞0} with (31)ht=Gt,zϱt,zt+yt+yϱt,zt+yt-StG0,ς0+∫0tSt-svsds,t∈I,where v(s)∈SF,zϱ(s,zs+ys)+yϱ(s,zs+ys).
It is vindicated that the operatorΥ has a fixed point if and only if Υ¯ has a fixed point. As a result, let us demonstrate that Υ¯ has a fixed point z∈V+∞0.
Remark 9. (i) By condition (P1) and Lemma 7 in the above discussion, we have the subsequent estimates: (32)zϱs,zs+ys+yϱs,zs+ysB≤zϱs,zs+ysB+yϱs,zs+ysB≤D1∗zs+D2∗+Lςz0B+D1∗ys+D2∗+LςςB≤D1∗zs+D1∗MHςB+D2∗+LςςB≤D1∗zs+D2∗+Lς+D1∗MHςB≤D1∗zs+cn,where cn=(D2∗+Lς+D1∗MH)ςB.
(ii)(33)vt-v¯t≤℘ntzϱt,zt+yt-z¯ϱt,z-t+ytB+C1zϱt,zt+yt-z¯ϱt,z-t+ytB≤℘ntD1∗zt-z¯t+C1D1∗zt-z¯t≤℘nt1+C1D1∗zt-z¯t;
(iii)(34)ht-h¯t≤Gt,zϱt,zt+yt+yϱt,zt+yt-Gt,z¯ϱt,z-t+yt+yϱt,z-t+yt+∫0tSt-sBEvs-v¯sds≤℘~ntD1∗zt-z¯t+M∫0t℘ns1+C1D1∗zs-z¯sds≤℘~ntD1∗zt-z¯t+∫0t℘¯nszs-z¯sds≤℘~ntD1∗eτLn∗te-τLn∗tzt-z¯t+∫0t℘¯nseτLn∗se-τLn∗szs-z¯sds≤℘n∗teτLn∗tz-z¯n+∫0t℘n∗seτLn∗sz-z¯nds≤℘n∗+1τeτLn∗tz-z¯n.
Presentingn∈N, z is ought to be a solution of the inclusion z∈λΥ¯(z) for many λ∈(0,1) and there may be v∈SF,zϱ(s,zs+ys)+yϱ(s,zs+ys) in a way that, for any t∈In, we maintain(35)zt≤StBEG0,ς0+Gt,zϱt,zt+yt+yϱt,zt+yt+∫0tSt-sBEFs,zϱs,zs+ys+yϱs,zs+ys,∫0ses,τ,zϱτ,zτ+yτ+yϱτ,zτ+yτdτds≤MG0,ς0+c1zϱt,zt+yt+yϱt,zt+ytB+c2+M∫0tϑs☊zϱs,zs+ys+yϱs,zs+ysB+∫0smτΩzϱs,zs+ys+yϱs,zs+ysBdτds.From Remark 9(i), we have (36)zt≤MG0,ς0+c1D1∗zt+c1cn+c2+M∫0tϑs☊D1∗zs+cn+∫0smτΩD1∗zs+cndτds≤1μMG0,ς0+c1cn+c2+Mμ∫0tϑs☊D1∗zs+cn+∫0smτΩD1∗zs+cndτds.Thus, (37)D1∗zt+cn≤cn+D1∗μMG0,ς0+c1cn+c2+MD1∗μ∫0tϑs☊D1∗zs+cn+∫0smτΩD1∗zs+cndτds.We conceive function β characterized by (38)βt=sup⁡D1∗zs+cn:0≤s≤b,0≤t<+∞.Permit t∗∈[0,t] in a way that β(t)=D1∗|z(t∗)|+cnςB. Because of the aforementioned inequality, we maintain, for t∈In, (39)βt≤cn+D1∗μMG0,ς0+c1cn+c2+MD1∗μ∫0tϑs☊βs+∫0smτΩβτdτds.Allow us to occupy the right-hand part of the overhead inequality as w(t). Then, we sustain β(t)≤w(t) for all t∈[0,n]. Through the significance of v, we get (40)w0=cn+D1∗μMG0,ς0+c1cn+c2=C.This leads us to the accompanying inequality for t∈In, (41)wt≤cn+D1∗μMG0,ς0+c1cn+c2+MD1∗μ∫0tϑs☊ws+∫0smτΩwτdτds,where (42)w′t≤MD1∗μϑt☊wt+∫0tmsΩwsds.Next, we weigh the function (43)ωt=wt+∫0tmsΩwsds.Then we bring forth ω(0)=w(0)=C and w(t)≤ω(t) for all t∈In. Applying the nondecreasing character of ☊, we receive (44)ω′t=w′t+mtΩwt≤MD1∗μϑt☊ωt+mtΩωt,a.e.t∈In.We characterize the function ν(t)=max⁡(MD1∗/μ)ϑ(t),m(t), t∈In, which suggests that (45)ω′t☊ωt+Ωωt≤νt.From condition (25), we acquire (46)∫ω0=Cωtds☊s+Ωs≤∫0tνsds≤∫0nνsds<∫C+∞ds☊s+Ωs.Subsequently, for every t∈In, we have a constant Λn∗ in a way that ω(t)≤Λn∗ and, consequently, β(t)≤Λn∗. Due to the fact that zn≤β(t), we certainly have zn≤Λn∗. Fix (47)U=z∈V+∞0:sup0≤t≤n⁡zt≤Λn∗+1,∀n∈N.Evidently, U is a closed subset of V+∞0. We should demonstrate that Υ¯:U¯→P(V+∞0) is a contraction and an admissible operator. Initially, we evaluate that Υ¯ is a contraction. In fact, consider z,z¯∈V+∞0 and h∈Υ¯(z). Then, there may be v(t)∈F(t,zϱ(t,zt+yt)+yϱ(t,zt+yt),∫0t‍e(t,s,zϱ(s,zs+ys)+yϱ(s,zs+ys))ds) such that (48)ht=Gt,zϱt,zt+yt+yϱt,zt+yt-StG0,ς0+∫0tSt-svsds,for  everyt∈In.From the hypotheses (H2)(ii) and (H3)(ii), we sustain(49)HdFt,zϱt,zt+yt+yϱt,zt+yt,∫0tet,s,zϱs,zs+ys+yϱs,zs+ysds,Ft,z¯ϱt,z-t+yt+yϱt,z-t+yt,∫0tet,s,z¯ϱs,z-s+ys+yϱs,z-s+ysds≤℘ntzϱt,zt+yt-z¯ϱt,z-t+ytB+C1zϱt,zt+yt-z¯ϱt,z-t+ytB.As a result, there is w∈F(t,z¯ϱ(t,z-t+yt)+yϱ(t,z-t+yt),∫0t‍e(t,s,z¯ϱ(s,z-s+ys)+yϱ(s,z-s+ys))ds) so that (50)vt-w≤℘ntzϱt,zt+yt-z¯ϱt,z-t+ytB+C1zϱt,zt+yt-z¯ϱt,z-t+ytB.Recognize U∗:[0,n]→P(E) specified by (51)U∗t=w∈E:vt-w≤℘ntzϱt,zt+yt-z¯ϱt,z-t+ytB+C1zϱt,zt+yt-z¯ϱt,z-t+ytB.Considering the fact that the multivalued operator V∗(t)=U∗(t)∩F(t,z¯ϱ(t,z-t+yt)+yϱ(t,z-t+yt),∫0t‍e(t,s,z¯ϱ(s,z-s+ys)+yϱ(s,z-s+ys))ds) is measurable (see Proposition 3), there is function v¯(t), which is a measurable choice for V∗. So, v¯(t)∈F(t,z¯ϱ(t,z-t+yt)+yϱ(t,z-t+yt),∫0t‍e(t,s,z¯ϱ(s,z-s+ys)+yϱ(s,z-s+ys))ds), and, from Remark 9(ii), we specify(52)vt-v¯t≤℘nt1+C1D1∗zt-z¯t.For every t∈[0,n], give us a chance to characterize (53)h¯t=Gt,z¯ϱt,z-t+yt+yϱt,z-t+yt-StG0,ς0+∫0tSt-sv¯sds.Again, from Remark 9(iii), we now have (54)ht-h¯t≤Gt,zϱt,zt+yt+yϱt,zt+yt-Gt,z¯ϱt,z-t+yt+yϱt,z-t+yt+∫0tSt-sBEvs-v¯sds≤℘n∗+1τeτLn∗tz-z¯n.As a result (55)h-h¯n≤℘n∗+1τz-z¯n.Being practically equivalent to the connection gotten by exchanging the parts of z and z¯, it takes after that (56)HdΥ¯z,Υ¯z¯≤℘n∗+1τz-z¯n,demonstrating that Υ¯ is a contraction for all n∈N and, from the second aspect of [30, Theorem 3.4], we realize that Υ¯ is likewise admissible contraction operator. With the decision of U, there is no z∈∂U in a way that z=λΥ¯(z) for many λ∈(0,1). From the nonlinear alternative fixed point theorem thanks to Frigon [1, Corollary 3.5], we realize that the operator Υ¯ has a fixed point z∗. This intimates that x∗(t)=z∗(t)+y(t), t∈(-∞,+∞), is a fixed point of the operator Υ, which is a mild solution of the structure (1).


## 4. Illustrative Example

To exemplify our theoretical results, we treat the FNIDI with SDD of the structure(57)∂∂tut,ξ-gt,ut-σut,0,ξ∈∫t0t-sα-2Γα-1∂2∂ξ2-rus,ξ-gs,us-σus,0,ξds+f1t,ut-σut,0,ξ,∫0tηt,s,us-σus,0,ξds,f2t,ut-σut,0,ξ,∫0tηt,s,us-σus,0,ξds,t∈0,∞,ξ∈0,π,ut,0=ut,π=0,t∈0,∞,uθ,ξ=u0θ,ξ,θ∈-∞,0,ξ∈0,π,where 1<α<2, (u0,σ)∈C(R[0,∞)), Lξ=(∂2/∂ξ2-r), r>0 stands for the operator with respect to the special variable ξ,f1,f2:I×B×E→R are measurable in t and continuous in x, g:I×B→R, and η:I×I×B→R are appropriate functions. We expect that, for each t≥0, f1(t,·,·) is lower semicontinuous (i.e., the set {x∈B,y∈E:f1(t,x,y)>ν} is open for all ν∈R) and accept that, for each t≥0, f2(t,·,·) is upper semicontinuous (i.e., the set {x∈B,y∈E:f2(t,x,y)<ν} is open for each ν∈R).RecognizeE=L2([0,π],R) and the operator A:Lξ:D(A)⊂E→E with domain (58)DA=u∈E:u′′∈E,u0=uπ=0.Clearly, A is densely defined in E and is sectorial. Hence, A is a generator of a solution operator on E. For the phase space, we pick B=Cγ={ς∈C((-∞,0]:X):limθ→-∞⁡eγθς(θ) exists in X} invested with the norm (59)ς=sup-∞<θ≤0⁡eγθςθ.Here, we note the phase space Cγ satisfying conditions (P1), (P2), and (P3). Set (60)xtξ=ut,ξ,t≥0,ξ∈0,π,ς0ξ=u0θ,ξ,t≥0,θ≤0,Gt,ςξ=gt,ς0,ξ,t≥0,ξ∈0,π,Ft,ς,xξ=f1t,ς0,ξ,∫0tηt,s,ς0,ξds,f2t,ς0,ξ,∫0tηt,s,ς0,ξds,t≥0,ξ∈0,π,et,s,ςξ=ηt,s,ς0,ξ,ϱt,ς=t-σς0,0.The multivalued map F is u.s.c. with compact convex values. Thus, (H1) and (H2) are fulfilled.Presently, the existence of mild solutions can be reasoned from an immediate utilization of Theorem8. In the perspective of phase space Cγ with the above examination, we have the accompanying results.Corollary 10.
Letς∈B=Cγ be continuous and bounded. Then, there is certainly at least one mild solution of the model (57) on (-∞,+∞).



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*Source: 6165804-2016-02-29.xml*