Open access peer-reviewed chapter

Existence, Regularity, and Compactness Properties in the α-Norm for Some Partial Functional Integrodifferential Equations with Finite Delay

By Boubacar Diao, Khalil Ezzinbi and Mamadou Sy

Submitted: April 1st 2019Reviewed: June 17th 2019Published: May 13th 2020

DOI: 10.5772/intechopen.88090

Downloaded: 58

Abstract

The objective, in this work, is to study the alpha-norm, the existence, the continuity dependence in initial data, the regularity, and the compactness of solutions of mild solution for some semi-linear partial functional integrodifferential equations in abstract Banach space. Our main tools are the fractional power of linear operator theory and the operator resolvent theory. We suppose that the linear part has a resolvent operator in the sense of Grimmer. The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. An application is provided to illustrate our results.

Keywords

  • integrodifferential
  • mild solution
  • resolvent operator
  • fractional power operator

1. Introduction

We consider, in this manuscript, partial functional equations of retarded type with deviating arguments in terms of involving spatial partial derivatives in the following form [1]:

dutdt=Aut+0tBtsusds+Ftut for t0,u0=φCα=Cr0DAα],E1

where Ais the infinitesimal generator of an analytic semigroup Ttt0on a Banach space X. Btis a closed linear operator with domain DBtDAtime-independent. For 0<α<1, Aαis the fractional power of Awhich will be precise in the sequel. The domain DAαis endowed with the norm xα=Aαxcalled αnorm. Cαis the Banach space Cr0DAαof continuous functions from r0to DAαendowed with the following norm:

ϕα=suprθ0ϕθα for ϕCα.

F:R+×CαXis a continuous function, and as usual, the history function utCαis defined by

utθ=ut+θ for θr0.

As a model for this class, one may take the following Lotka-Volterra equation:

utxt=2utxx2+0thts2usxx2ds+r0gtut+θxx for t0andx0π,ut0=utπ=0 for t0,uθx=u0θx for θr0andx0π.E2

Here u0:r0×0πR,g:R+×RRand h:R+Rare appropriate functions.

In the particular case where α=0, many results are obtained in the literature under various hypotheses concerning A, B, and F(see, for instance, [2, 3, 4, 5, 6] and the references therein). For example, in [7], Ezzinbi et al. investigated the existence and regularity of solutions of the following equation:

dutdt=Aut+0tBtsusds+Ftut for t0,u0=φCr0X,E3

The authors obtained also the uniqueness and the representation of solutions via a variation of constant formula, and other properties of the resolvent operator were studied. In [8], Ezzinbi et al. studied a local existence and regularity of Eq. (3). To achieve their goal, the authors used the variation of constant formula, the theory of resolvent operator, and the principle contraction method. Ezzinbi et al. in [9] studied the local existence and global continuation for Eq. (3). Recall that the resolvent operator plays an important role in solving Eq. (3); in the weak and strict sense, it replaces the role of the c0semigroup theory. For more details in this topic, here are the papers of Chen and Grimmer [2], Hannsgen [10], Smart [11], Miller [12, 13], and Miller and Wheeler [14, 15]. In the case where the nonlinear part involves spatial derivative, the above obtained results become invalid. To overcome this difficulty, we shall restrict our problem in a Banach space YαX, to obtain our main results for Eq. (1).

Considering the case where B=0, Travis and Webb in [16] obtained results on the existence, stability, regularity, and compactness of Eq. (1). To achieve their goal, the authors assumed that Ais the infinitesimal generator of a compact analytic semigroup and Fis only continuous with respect to a fractional power of A in the second variable. The present paper is motivated by the paper of Travis and Webb in [16].

The paper is organized as follows. In Section 2, we recall some fundamental properties of the resolvent operator and fractional powers of closed operators. The global existence, uniqueness, and continuous dependence with respect to the initial data are studied in Section 3. In Section 4, we study the local existence and bowing up phenomena. In Section 5 we prove, under some conditions, the regularity of the mild solutions. And finally, we illustrate our main results in Section 6 by examining an example.

2. Fractional power of closed operators and resolvent operator for integrodifferential equations

We shall write Yfor DAendowed with the graph norm xY=x+Ax,Yαfor DAαand LYαXwill denote the space of bounded linear operators from Yαto X, and for Y0=X, we write LXwith norm .LX. We also frequently use the Laplace transform of fwhich is denoted by f. If we assume that Agenerates an analytic semigroup and, without loss of generality, that 0ϱA, then one can define the fractional power Aαfor 0<α<1, as a closed linear operator on its domain Yαwith its inverse Aαgiven by

Aα=1Γα0tα1Ttdt,

where Γis the gamma function

Γα=0tα1etdt.

We have the following known results.

Theorem 2.1. [17] The following properties are true.

  1. Yα=DAαis a Banach space with the norm xα=Aαxfor xYα.

  2. Aαis a closed linear operator with domain Yα=ImAαand Aα=Aα1.

  3. Aαis a bounded linear operator in X.

  4. If 0<αβthen DAβDAα. Moreover the injection is compact if Ttis compact for t>0.

Definition 2.2. [18] A family of bounded linear operators Rtt0in Xis called resolvent operator for the homogeneous equation of Eq. (3) if:

  1. R0=Iand RtM1expσtfor some M11and σR.

  2. For all xX, tRtxis continuous for t0.

  3. RtLYfor t0. For xY, R.xC1R+XCR+Y, and for t0we have

Rtx=ARtx+0tBtsRsxds=RtAx+0tRtsBsxds.E4

What follows is we assume the hypothesis taken from [1] which implies the existence of an analytic resolvent operator Rtt0.

(V1) Agenerates an analytic semigroup on X. Btt0is a closed operator on Xwith domain at least DAa.e t0with Btxstrongly measurable for each xYand BtxbtxY,forbLloc10with bλabsolutely convergent for Reλ>0.

(V2) ρλ=λI+ABλ1exists as a bounded operator on Xwhich is analytic for λΛ=λC:argλ<π/2+δ,where 0<δ<π/2. In Λif λϵ>0, there exists M=Mϵ>0so that ρλM/λ.

(V3) λLXfor λΛand is analytic from Λto LX. BλLYXand BλρλLYXfor λΛ. Given ϵ>0, there exists a positive constant M=Mεso that λx+BλρλxM/λxYfor xYand λΛwith λεand Bλ0as λin Λ. In addition, λxMλnxfor some n>0,λΛwith λε. Further, there exists DDA2which is dense in Ysuch that ADand BλDare contained in Yand BλxYis bounded for each xDand λΛwith λϵ.

Theorem 2.3. [1] Assume that conditions (V1)–(V3) are satisfied. Then there exists an analytic resolvent operator Rtt0. Moreover, there exist positive constants N,Nαsuch that RtNandAαRtNαtαfor t>0and 0α<1.

We take the following hypothesis.

(H0) The semigroup Ttt0is compact for t>0.

Theorem 2.4. [19] Under the conditions (V1)–(V3) and (H0), the corresponding resolvent operator Rtt0is compact for t>0.

3. Global existence, uniqueness, and continuous dependence with respect to the initial data

Definition 3.1. A function u:0bYαis called a strict solution of Eq. (1), if:

  1. tutis continuously differentiable on 0b.

  2. utYfor t0b.

  3. usatisfies Eq. (1) on 0b.

Definition 3.2. A continuous function u:0bYαis called a mild solution of Eq. (1) if

ut=Rtφ0+0tRtsFsusds for t0b,u0=φCα.E5

Now to obtain our first result, we take the following assumption.

(H1) There exists a constant LF>0such that

Ftφ1F(tφ2)LFφ1φ2α for t0andφ1,φ2Cα.

Theorem 3.3. Assume that (V1)–(V3) and (H1) hold. Then for φCα, Eq. (1) has a unique mild solution which is defined for all t0.

Proof. Let a>0. For φCα, we define the set by

=yC0aYα:y0=φ0.

The set is a closed subset of C0aYαwhere C0aYαis the space of continuous functions from 0ato Yαequipped with the uniform norm topology

yα=sup0taytα for yC0aYα.

For y, we introduce the extension y¯of yon radefined by y¯t=ytfor t0aand y¯t=φtfor tr0. We consider the operator Γdefined on by

Γyt=Rtφ0+0tRtsFsy¯sds for t0a.

We claim that Γ.In fact for y, we have Γy0=φ0, and by continuity of Fand Rtxfor xX, we deduce that Γy.In order to obtain our result, we apply the strict contraction principle. In fact, let u,vand t0a. Then

ΓuΓvt=0tRtsFsu¯sFsv¯sds.

Using the αnorm, we have

AαΓuΓvtLFNα0t1tsαu¯sv¯sαdsLFNα0t1tsαsup0τauτvταdsLFNα0adssαuvα.

Now we choose asuch that

LFNα0adssα<1.

Then Γis a strict contraction on , and it has a unique fixed point ywhich is the unique mild solution of Eq. (1) on 0a. To extend the solution of Eq. (1) in a2a, we show that the following equation has a unique mild solution:

ddtzt=Azt+atBtszsds+Ftzt for ta2a,za=yaCraYα.E6

Notice that the solution of Eq. (6) is given by

zt=Rtaza+atRtsFszsds for ta2a.

Let z¯be the function defined by z¯t=ztfor ta2aand z¯t=ytfor tra. Consider now again the set defined by

=zCa2aYα:za=ya,

provided with the induced topological norm. We define the operator Γaon by

Γazt=Rtaza+atRtsFsz¯sds for ta2a.

We have Γaza=yaand Γazis continuous. Then it follows that Γa.Moreover, for u,v, one has

AαΓauΓavtLFNαat1tsαu¯sv¯sαds.

Since u¯=v¯=φin r0,we deduce that

AαΓauΓavLFNα0adssαuvα.

Then we deduce that Γahas a unique fixed point in which extends the solution yin a2a. Proceeding inductively, yis uniquely and continuously extended to nan+1afor all n1, and this ends the proof.

Now we show the continuous dependence of the mild solutions with respect to the initial data.

Theorem 3.4. Assume that (V1)–(V3) and (H1) hold. Then the mild solution u.φof Eq. (1) defines a continuous Lipschitz operator Ut,t0in Cαby Utφ=ut.φ. That is, Utφis continuous from 0to Cαfor each fixed φCα. Moreover there exist a real number δand a scalar function Psuch that for t0and φ1,φ2Cαwe have

Utφ1Utφ2Pδeδtφ1φ2α.E7

Proof. We use the gamma formula

Γ1αkα1=0ekssαds,

where k>0(see [20], p. 265). The continuity is obvious that the map tut.φis continuous. Now, let φ1,φ2Cα.If we pose wt=utφ1utφ2,then we have

wtαM1eσtφ1φ2α+LFNα0teσtstsαwsαds.E8

Let δa real number be such that

σδ<0andM1maxeδr1LFΓ1αδσα1<1.

We define the function Pby

Pδ=M1M3M41M1M4LFΓ1αδσα11

where

M3=maxeδr1,M4=maxeδr1.

Fix t¯>0and let E=sup0st¯eδsws. If 0τt¯, then from Eq. (8), we have

eδτwταM1eσδτφ1φ2α+LFNα0τeσδτsτsαeδswsαdsM1φ1φ2α+LFM1EΓ1αδσα1.E9

If rτ0, we have

eδτwταφ1φ2αM3.E10

Therefore, Eqs. (9) and (10) imply that

suprτt¯eδτwταM1M3φ1φ2Cα+LFM1EΓ1αδσα1.E11

For 0tt¯, we have

eδtwtα=suprθ0eδθeδt+θwt+θαM4suprθ0eδt+θwt+θαM4suprτt¯eδτwτα.E12

Then from Eqs. (11) and (12), we deduce that for 0tt¯

eδtwtαM1M3M4φ1φ2α+LFM1M3EΓ1αδσα1,

which implies that

EM1M3M4φ1φ2α+LFM1M4EΓ1αδσα1.

Then the result follows.

4. Local existence, blowing up phenomena, and the compactness of the flow

We start by generalizing a result, obtained in [19] in the case of the usual norm on Xα=0, in the case where α0. We take the following assumption.

(H2) BtLXβXfor some 0<β<1, a.e t0and Btxbtxβfor xXβ,with bLlocq0where q>1/1β.

Theorem 4.1. Assume that (V1)–(V3) and (H2) hold. Then for any a>0,there exists a positive constant M=Masuch that for xXwe have

AαRt+hxRhRtxM0hdssαx for 0h<ta.

Proof. Let a>0and xX. Then

ddtRt+hx=ARt+hx+0t+hBt+hsRsxds=ARt+hx+0tBtsRs+hxds+tt+hBt+hsRsxds.

We deduce that Rt+hxsatisfies the equation of the form

ddtyt=Ayt+0tBtsysds+ft.

Then by the variation o constante formula, it follows that

Rt+hx=RtRhx+0tRtsss+hBs+huRuxduds=RhRtx+0hRhs0tBuRs+tuxduds.

Which yields that

Rt+hxRhRtx=0hRhs0tBuRs+tuxduds.

Taking the αnorm, we obtain that

AαRt+hxRhRtxNα0h1hsα0tBuRs+tuxdudsNα0h1hsα0tbuAβRt+suxdudsNαNβ0hdshsα0tbutuβxdu.

Let pbe such that 1/q+1/p=1, so p<1/β.Then it follows that

AαRt+hxRhRtxNαNβbLq0auβLp0a0hdssαx.

And the proof is complete.

The local existence result is given by the following Theorem.

Theorem 4.2. Suppose that (V1)–(V3), (H0), and (H2) hold. Moreover, assume that Fdefined from J×Ωinto Xis continuous where J×Ωis an open set in R+×Cα. Then for each φΩ, Eq. (1) has at least one mild solution which is defined on some interval 0b.

Proof. Let φΩ. For any real ζJand p>0,we define the following sets:

Iζ=t:0tζandHp=ϕCα:ϕαp.

For ϕHp, we choose ζand psuch that tϕ+φIζ×Hpand HpΩ.By continuity of F, there exists N10such that Ftϕ+φN1for tϕin Iζ×Hp. We consider φ¯CrζYαas the function defined by φ¯t=Rtφ0for tIζand φ¯0=φ. Suppose that p¯<pand choose 0<b<ζsuch that

NαN10bdssα<p¯andφ¯tφαpp¯ for tIb.

Let K0=ηCrbYα:η0=0and ηtαp¯ for 0tb.Then we have Ftφ¯t+ηtN1for 0tband ηK0,since ηt+φ¯tφαp.Consider the mapping Sfrom K0to CrbYαdefined by 0=0

t=0tRtsFsηs+φ¯sds for 0tb.E13

Notice that finding a fixed point of Sin K0is equivalent to finding a mild solution of Eq. (1) in K0. Furthermore, Sis a mapping from K0to K0, since if ηK0we have 0=0and

tα0tAαRtsF(sηs+φ¯s)ds.

Then

tαNαN10tdstsαNαN10bdssα<p¯

which implies that SK0K0. We claim that t:ηK0}is compact in Yαfor fixed trb.In fact, let βbe such that 0<αβ<1. The above estimate show that Aβt:ηK0is bounded in X. Since Aαβis compact operator, we infer that AαβAβt:ηK0is compact in X,hence t:ηK0is compact in Yα. Next, we show that t:ηK0is equicontinuous. The equicontinuity of t:ηK0at t=0follows from the above estimation of t. Now let 0<t0<tbwith t0be fixed. Then we have

Aαtt00t0AαRtsRtt0Rt0sFsηs+φ¯sds+AαRtt0I0t0Rt0sF(sηs+φ¯s)ds+t0tAαRtsF(sηs+φ¯s)ds.E14

Using Theorem 4.1, it follows that

Aαtt0t0N1M0tt0dssα+Rtt0IAα0t0Rt0sF(sηs+φ¯s)ds+ NαN10tt01sαds.

As the set t0:ηK0is compact in Yα, we have

limtt0+tt0α=0uniformlyinηK0.

We obtain the same results by taking t0be fixed with 0<t<t0b.Then we claim that limtt0tt0α=0uniformly in ηK0which means that t:ηK0is equicontinuous. Then by Ascoli-Arzela theorem, :ηK0is relatively compact in K0. Finally, we prove that Sis continuous. Since Fis continuous, given ε>0, there exists δ>0, such that

sup0sbηsη̂sα<δ implies that Fsηs+φ¯sF(sη̂s+φ¯s)<ε.

Then for 0tb, we have

tSη̂tαNα0t1tsαFsηs+φ¯sF(sη̂s+φ¯s)dsNαε0tdssα.

This yields the continuity of S, and using Schauder’s fixed point theorem, we deduce that Shas a fixed point. Then the proof of the theorem is complete.

The following result gives the blowing up phenomena of the mild solution in finite times.

Theorem 4.3. Assume that (V1)–(V3), (H0), and (H2) hold and Fis a continuous and bounded mapping. Then for each φCα, Eq. (1) has a mild solution u.φon a maximal interval of existence rbφ. Moreover if bφ<, then lim¯tbφutφα=+.

Proof. Let u.φbe the mild solution of Eq. (1) defined on 0b. Similar arguments used in the local existence result can be used for the existence of b1>band a function u.ubdefined from bb1to Yαsatisfying

utub.φ=Rtubφ+btRtsFsusds for tbb1.

By a similar proceeding, we show that the mild solution u.φcan be extended to a maximal interval of existence rbφ. Assume that bφ<+and lim¯tbφutφα<+. There exists N2>0such that FsusN2,for s0bφ. We claim that u.φis uniformly continuous. In fact, let 0<htt+h<bφ. Then

ut+hut=Rt+hRtφ0+0tRt+hsRtsFsusds+tt+hRt+hsFsusds.

By continuity of AαRt, we claim that AαRt+hRtφ0is uniformly continuous on each compact set. Moreover, Theorem 4.1 implies that AαRt+hsRtsFsus0uniformly in twhen h0.In fact we have

0tRt+hsRtsFsusαds0tRt+hsRhRtsFsusαds+RhIAα0tRtsF(sus)ds

Then using Theorem 4.1, we obtain that

0tRt+hsRtsFsusαdsbφN2M0hdssα+RhIAα0tRtsF(sus)ds.

We claim that the set {Aα0tRtsFsusds:t[0,bφ)}is relatively compact. In fact, let tnn0be a sequence of 0bφ. Then there exist a subsequence tnkkand a real number t0such that tnkt0. Using the dominated convergence theorem, we deduce that

0tnkAαRtnksFsusds0t0AαRt0sFsusds.

This implies that {Aα0tRtsFsusds:t[0,bφ)}is relatively compact. Now using Banach-Steinhaus’ theorem, we deduce that

RhIAα0tRtsFsusds0

uniformly when h0with respect to t0bφ. Moreover we have

tt+hRt+hsFsusdsαN2Nα0hdssα.

Consequently ut+hutα0ash0uniformly in t0bφ. If h0, that is, for tt0, we have

utut0=RtRt0φ00tRt0sRt0tRtsFsusdsRt0tI0tRtsFsusdstt0Rt0sFsusds,

one can show similar results by using the same reasoning. This implies that u.φis uniformly continuous. Therefore limtbφutφexistsinYα.And consequently, u.φcan be extended to bφwhich contradicts the maximality of 0bφ.

The next result gives the global existence of the mild solutions under weak conditions of F. To achieve our goal, we introduce a following necessary result which is a consequence of Lemma 7.1.1 given in ([21], p. 197, Exo 4).

Lemma 4.4. [21] Let α,a,b0,β<1and 0<d<.Also assume that vis nonnegative and locally integrable on 0dwith

vtatα+b0tvstsβds for t0d.

Then there exists a constant M2=M2abαβd<such that vtM2/tαon 0d.

Theorem 4.5. Assume that (V1)–(V3), (H0), and (H2) hold and Fis a completely continuous function on R+×Cα. Moreover suppose that there exist continuous nonnegative functions f1and f2such that Ftφf1tφα+f2tfor φCαand t0.Then Eq. (1) has a mild solution which is defined for t0.

Proof. Let 0bφbe the maximal interval of existence of a mild solution u.φ. Assume that bφ<+.By Theorem 4.3 we have lim¯ttφutφα=+. Recall that the solution of Eq. (1) is given by u0=φand

utφ=Rtφ0+0tRtsFsus.φds for t0bφ.

Then taking the αnorm, we obtain

utφαRtφ0α+k2Nα0bφdssαds+k1Nα0t1tsαus.φαds,

where k1=max0tbφf1tand k2=max0tbφf2t. Then we deduce that

utφαNφ0α+k1Nα0t1tsαsuprτsuτφαds+k2Nα0bφdssα.E15

Now we claim that the function

t0t1tsαsuprτsuτφαds,

is nondecreasing. In fact, let 0t1t2. Then

0t11t1sαsuprτsuτφαds=0t11sαsuprτt1suτφαds0t21sαsuprτt2suτφαds=0t21t2sαsuprτsuτφαds

which yields the result. Then it follows from Eq. (15) that

suprstusφαNφ0α+k2Nα0bφdssαds+k1Nα0t1tsαsuprτsuτφαds.

Then using Lemma 4.4, we deduce that u.φis bounded in 0bφ. Then we obtain that lim¯tbφutφα<,which contradicts our hypothesis. Then the mild solution is global.

We focus now to the compactness of the flow defined by the mild solutions.

Theorem 4.6. Assume that (V1)–(V3) and (H0)–(H2) hold. Then the flow Utdefined from Cαto Cαby Utφ=ut.φis compact for t>r, where ut.φdenotes the mild solution starting from φ.

Proof. We use Ascoli-Arzela’s theorem. Let E=φγ:γΓbe a bounded subset of Cαand let t>rbe fixed, but arbitrary. We will prove that UtE¯is compact. It follows from (H1) and inequality Eq. (7) that there exists N5such that

F(tutφγ)N2utφγ)+Ft0=N5 for γΓ.

For each γΓ,we define fγCαby fγ=ut.φγ. We show now that for fixed θr0,the set fγθ:γΓis precompact in Yα. For any γΓ, we have

fγθ=Rt+θφγ0+0t+θRt+θsFsus.φds.

As Rtis compact for t>0, we need only to prove that the set

0t+θRt+θsF(sus(.φγ))ds:γΓ

is compact. Also we have

μRε0t+θεRt+θεsF(sus(.φγ))ds:γΓ=0,

where μis the measure of non-compactness. Moreover, using Theorem 4.1, we have

Aα0t+θεRt+θεsRεRt+θεsF(sus(.φγ))ds0t+θεRt+θsRεRt+θεsF(sus(.φ))αdsN5M0εdssα0asε0.

We deduce that

μ0t+θεRt+θsF(sus(.φγ))ds:γΓ=0.

On the other hand, for 0<αβ<1, we have

Aβt+θεt+θRt+θsF(sus(.φγ))dst+θεt+θRt+θsF(sus(.φγ))βdsNβN5t+θεt+θdst+θsβ=NβN50εdssβ0asε0.

Thus Aβt+θεt+θRt+θsF(sus(.φγ))ds:γΓis a bounded subset of X. The precompactness in Yαnow follows from the compactness of Aβ:XYα. Then the set UtEθ:rθ0is precompacted in Yα. We prove that the family fγ:γΓis equicontinuous. Let γin Γ,0<ε<tr,and rθ̂θ0with θ̂be fixed and h=θθ̂. Then

Aαfγh+θ̂fγθ̂Rt+θ̂+hRt+θ̂φγ0α+0t+θ̂AαRt+θ̂+hsRhRt+θ̂sF(sus.φγds+RhIAα0t+θ̂Rt+θ̂sF(sus(.φγ))ds+t+θ̂t+θ̂+hAαRt+θ̂+hsF(sus(.φγ))ds.

Then it follows that

Aαfγh+θ̂fγθ̂Rt+θ̂+hRt+θ̂Aαφγ0+MN5t+θ̂0hdssα+RhIAα0t+θ̂Rt+θ̂sF(sus(.φγ))ds+N5Nα0hdssα.

Using the compactness of the set Aα0t+θRt+θsF(sus(.φγ))ds:γΓand the continuity of tRtxfor xX, the right side of the above inequality can be made sufficiently small for h>0small enough. Then we conclude that fγ:γΓis equicontinuous. Consequently, by Ascoli-Arzela’s theorem, we conclude that the set Utφ:φEis compact, which means that the operator Utis compact for t>r.

5. Regularity of the mild solutions

We define the set Cα1by Cα1=C1r0Yαas the set of continuously differentiable functions from r0to Yα. We assume the following hypothesis.

(H3) Fis continuously differentiable, and the partial derivatives DtFand DφFare locally Lipschitz in the classical sense with respect to the second argument.

Theorem 5.1. Assume that (V1)–(V3), (H1), and (H3) hold. Let φin Cα1be such that φ0Yand φ̇0=0+F0φ.Then the corresponding mild solution ubecomes a strict solution of Eq. (1).

Proof. Let a>0. Take φCα1such that φ0Yand φ̇0=0+F0φ, and let ube the mild solution of Eq. (1) which is defined on [0,+[. Consider the following equation:

vt=Rtφ̇0+0tRtsDtFsus+DφF(sus)vsds+0tRtsBsφ0ds for t0,v0=φ̇.E16

Using the strict contraction principle, we can show that there exists a unique continuous function vsolution in 0aof Eq. (16). We introduce the function wdefined by

wt=φ0+0tvsdsift0,φtifrt0.

Then it follows

wt=φ+0tvs ds for t0a.

Consequently, the maps twtand t0tRtsFswsdsare continuously differentiable, and the following formula holds

ddt0tRtsFswsds=RtF0w0+0tRtsDtFsws+DφFswsvsds=RtF0φ+0tRtsDtFsws+DφFswsvsds.

This implies that

0tRsF0φds=0tRtsFswsds0t0sRsτDtFτwτ+DφFτwτvτds.

On the other hand, from equality Eq. (4), we have

0tRs0ds=Rtφ0φ00t0sRsτBτφ0ds.

We rewrite was follows:

wt= φ00tRs0ds+0tRsF0φds+0t0sRsτDtFτuτ+DφFτuτvτds+0t0sRsτBτφ0ds.

Then it follows that

wt= Rtφ0+0tRtsFswsds+0t0sRsτ(DτFτuτDτF(τwτ))ds+0t0sDφFτuτvτDφFτwτvτds.

We deduce, for t0a,that

utwtα0tAαRts(F(sus)F(sws))ds+0t0sAαRsτ(DτF(τuτ)DτF(τwτ))ds+0t0sAαRsτ(DφF(τuτ)DφF(τwτ))vτds.E17

The set H=usws:s0ais compact in Cα. Since the partial derivatives of Fare locally Lipschitz with respect to the second argument, it is well-known that they are globally Lipschitz on H. Then we deduce that

utwtαNαha0t1tsαuswsαdsNαha0t1tsαsup0τauτwταds,

where ha=LFNα+aNαLipDtF+aNαLipDφF,with LipDφFand LipDtFthe Lipschitz constant of DφFand DtF, respectively, which implies that

uwαNαha0adssαuwα.

If we choose asuch that

Nαha0adssα<1,

then u=win 0a. Now we will prove that u=win 0+.Assume that there exists t0>0such that ut0wt0. Let t1=inft>0:utwt>0.By continuity, one has ut=wtfor tt1, and there exists ε>0such that utwt>0for tt1t1+ε. Then it follows that for tt1t1+ε,

utwtαNαhε0εdssαsupετt1+εuτwτα.

Now choosing εsuch that

Nαhε0εdssα<1,

then u=win t1t1+εwhich gives a contradiction. Consequently, ut=wtfor t0. We conclude that tutfrom 0+to Yαand tFtutfrom 0+×Cαto Xare continuously differentiable. Thus, we claim that uis a strict solution of Eq. (1) on 0+[22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

6. Application

For illustration, we propose to study the model Eq. (2) given in the Introduction. We recall that this is defined by

twtx=2x2wtx+0thts2x2wsxds+r0gtxwt+θx for t0andx0π,wt0=wtπ=0 for t0,wθx=w0θx for θr0andx0π,E18

where w0:r0×0πR, g:R+×RRand h:R+R+are appropriate functions. To study this equation, we choose X=L20π, with its usual norm .. We define the operator A:Y=DAXXby

Aw=w with  domainDA=H20πH010π,

and Btx=htAxX,for0,xY.For α=1/2, we define Y1/2=DA1/21/2where x1/2=A1/2xfor each xY1/2. We define C1/2 = Cr0Y1/2equipped with norm and the functions uand φand Fby ut=wtx,φθx=w0θxfor a.e x0πand θr0,t0, and finally

Ftφx=r0gtxφθx for a.ex0πandφC1/2.

Then Eq. (18) takes the abstract form

dutdt=Aut+0tBtsusds+Ftut for t0,u0=φC1/2=Cr0DA1/2],E19

The Ais a closed operator and generates an analytic compact semigroup Ttt0on X. Thus, there exists δin 0π/2and M0such that Λ=λC:argλ<π2+δ0is contained in ρA, the resolvent set of A, and RλA<M/λfor λΛ. The operator Btis closed and for xY, BtxhtxY. The operator Ahas a discrete spectrum, the eigenvalues are n2, and the corresponding normalized eigenvectors are enx=2πsinnx,n=1,2,. Moreover the following formula holds:

  1. Au=n=1n2uenenuDA.

  2. A1/2u=n=11nuenenfor uX.

  3. A1/2u=n=1nuenenfor uDA1/2=uX:n=11nuenenX.

One also has the following result.

Lemma 6.1 [16] LetφY1/2.Thenφis absolutely continuous,φ'Xand

φ'=A12φ.

We assume the following assumptions.

(H4) The scalar function h.L10and satisfies g1λ=1+hλ0(hthe Laplace transform of h) and λg11λΛfor λΛ. Further, hλ0as λ,for λΛand hλ1=λn.

(H5) The function g:R+×RRis continuous and Lipschitz with respect to the second variable.

By assumption (H4), the operator ρλ=λI+g1λA1=g11λλg11λI+A1exists as a bounded operator on X, which is analytic in Λand satisfies ρλ<M/λ.On the other hand, for xX, we have

λx=AλI+g1λA1x=A+λg11λIλg11λIλI+g1λA1x=g11λIλg11λλg11λI+A1x.

Since λg11λλg11λI+A1is bounded because g11λΛ, then λxhas the growth properties of g11λwhich tends to 1 if λgoes to infinity. Then we deduce that λLX. Moreover, it is analytic from Λto LX. Now, for xY, one has

λx=g11λλg11λI+A1AxandBλρλx=hλρλAx.

Then it follows that

λxM/λxYandBλρλM/λxY.

We deduce that λLYX, Bλ=hλALYX, and BλρλLYX.Considering D=C00π, we see that the conditions (V1)–(V3) and (H0) are verified. Hence the homogeneous linear equation of Eq. (18) has an analytic compact resolvent operator Rtt0. The function Fis continuous in the first variable from the fact that gis continuous in the first variable. Moreover from Lemma 6.1 and the continuity of g, we deduce that Fis continuous with respect to the second argument. This yields the continuity of Fin R+×C1/2. In addition, by assumption (H5) we deduce that

Ftφ1Ftφ2rLfφ1φ2C1/2.

Then Fis a continuous globally Lipschitz function with respect to the second argument. We obtain the following important result.

Proposition 6.2. Suppose that the assumptions (H4)–(H5) hold. Then Eq. (19) has a mild solution which is defined for t0.

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Boubacar Diao, Khalil Ezzinbi and Mamadou Sy (May 13th 2020). Existence, Regularity, and Compactness Properties in the <em>α</em>-Norm for Some Partial Functional Integrodifferential Equations with Finite Delay, Nonlinear Systems -Theoretical Aspects and Recent Applications, Walter Legnani and Terry E. Moschandreou, IntechOpen, DOI: 10.5772/intechopen.88090. Available from:

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