# Existence of Solution via Integral Inequality of Volterra-Fredholm Neutral Functional Integrodifferential Equations with Infinite Delay

Authors: Kishor D. Kucche; Machindra B. Dhakne
Journal: International Journal of Differential Equations
 (2014)

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## Abstract

In this work we study existence results for mixed Volterra-Fredholm neutral functional integrodifferential equations with infinite delay in Banach spaces. To obtain a priori bounds of solutions required in Krasnoselski-Schaefer type fixed point theorem, we have used an integral inequality established by B. G. Pachpatte. The variants for obtained results are given. An example is considered to illustrate the obtained results.


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## Body

## 1. Introduction

In this paper we establish existence results for the mixed Volterra-Fredholm neutral functional integrodifferential equations with infinite delay of the form(1)ddt[x(t)-g(t,xt,∫0t‍e(t,s,xs)ds)]=Ax(t)+f(t,xt,∫0t‍k(t,s,xs)ds,∫0b‍w(t,s,xs)ds),hhhHHHt∈J=[0,b],x0=ϕ∈Bh,
where A is the infinitesimal generator of a compact analytic semigroup of bounded linear operators T(t), t≥0 in a Banach space X, g:J×Bh×X→X, e, k, h:Δ×Bh→X, and f:J×Bh×X×X→X are given functions, Δ={(t,s):0≤s≤t≤b}, and Bh is a phase space defined later. The histories xt:(-∞,0]→X, xt(s)=x(t+s), s≤0, belong to the abstract phase space Bh.Due to the importance of neutral functional differential and integrodifferential equations with infinite delay in diverse fields of applied mathematics, these equations have generated considerable interest among researchers. Excellent account on the work with infinite delay can be found in [1–5]. The work in partial neutral functional differential equations with unbounded delay was initiated by Hernández and Henríquez [6, 7] and they have investigated the results pertaining to existence of mild, strong, and periodic solutions to the neutral functional differential equations. Recently, several works reported on existence results and controllability problem for various special forms of (1) and their variants with impulse or inclusion. Hernández [8]  proved existence results for special form of (1) with e=0, h=0, by using the Leray-Schauder alternative. Li et al. [9] investigated the controllability problem when e=0 and f=∫0t‍k(s,xs)ds by applying Sadovskii fixed point theorem. Henríquez [10, 11] has studied approximation and regularity of solutions of functional differential equations with unbounded delay.Chang et al. [12] established existence results for neutral functional integrodifferential equations with infinite delay using the resolvent operators and Krasnoselski-Schaefer type fixed point theorem. The work related to existence and controllability results with the impulse effect and infinite delay can be found in [13–15] and some of the references cited therein. The recent investigations on this theme can also be found in the work of Henriquez and dos Santos [16].The authors [17–20] have studied existence, uniqueness, continuous dependence, and other properties of the solution of special forms of (1) with finite delay.In this paper we investigate the existence results for (1) by using Krasnoselski-Schaefer type fixed point theorem via integral inequality by Pachpatte. We further prove existence results for the same equation without using integral inequality with different assumptions on the functions involved in the equation. To study (1), we use an abstract phase space Bh given by Yan [21] instead of seminormed space, introduced by Hale and Kato in [3].The paper is organized as follows. In Section2, we present the preliminaries. Section 3 is concerned with main results and proof. In Section 4, we present an example to illustrate the application of our results.


## 2. Preliminaries

We give some preliminaries from [21, 22] that will be used in our subsequent discussion. Assume that h:(-∞,0]→(0,+∞) is a continuous function with l=∫-∞0‍h(t)dt<+∞. For any a>0, we define
(2)Ba={ψ∶[-a,0]⟶Xsuch thatψ(t)is bounded and measurableψ∶[-a,0]⟶Xsuch thatψ(t)}
and equip the space Ba with the norm
(3)∥ψ∥[-a,0]=sup⁡s∈[-a,0]∥ψ(s)∥,∀ψ∈Ba.
Let us define
(4)Bh={∫-∞0‍ψ∶(-∞,0]⟶Xsuch that for anyc>0,ψ|[-c,0]∈Bc,∫-∞0‍h(s)∥ψ∥[s,0]ds<+∞}.
If Bh is endowed with the norm
(5)∥ψ∥Bh=∫-∞0‍h(s)∥ψ∥[s,0]ds,∀ψ∈Bh,
then it is clear that (Bh,∥·∥Bh) is a Banach space.Now we consider the space(6)Bh′={,and,x0∈Bhx∶(-∞,b]⟶Xsuch thatx|J∈C(J,X),x0∈Bh}.
Set ∥·∥b to be a seminorm in Bh′ defined by
(7)∥x∥b=∥x0∥Bh+sup⁡{∥x(s)∥∶s∈[0,b]},x∈Bh′.LetA:D(A)→X be the infinitesimal generator of a compact analytic semigroup of bounded linear operators T(t), t≥0 on a Banach space X with the norm ∥·∥, and let 0∈ρ(A); then it is possible to define the fractional power (-A)α, for 0<α≤1, as closed linear invertible operator with domain D(-A)α dense in X. The closedness of D(-A)α implies that D(-A)α endowed with the graph norm ∥x∥X=∥x∥+∥(-A)αx∥ is a Banach space. Since (-A)α is invertible, its graph norm ∥x∥X is equivalent to the norm |x|=∥(-A)αx∥. Thus D(-A)α equipped with the norm |·| is a Banach space which we denote by Xα.The following lemmas play an important role in our further discussions.Lemma 1 (see [22]).
The following properties hold.(i)
If0<β<α≤1, then Xα⊂Xβ and the imbedding is compact whenever the resolvent operator of A is compact.(ii)
For every0<α≤1, there exists Cα>0 such that
(8)∥(-A)αT(t)∥≤Cαtα,0<t≤b.Lemma 2 (see [23]).
LetX be a Banach space, and let Φ1, Φ2 be two operators on X such that(a)
Φ1 is contraction, and(b)
Φ2 is completely continuous. Then either(i)
the operator equationΦ1x+Φ2x=x has a solution or(ii)
the setG={x∈X:λΦ1(x/λ)+λΦ2x=x,λ∈(0,1)} is unbounded.Lemma 3 (see [8]).
Letu(·),v(·): [0,b]→[0,∞) be continuous functions. If v(·) is nondecreasing and there are constants θ>0, 0<α<1 such that
(9)u(t)≤v(t)+θ∫0t‍u(s)(t-s)1-αds,t∈J,
then
(10)u(t)≤exp⁡[θn(Γ(α))ntnαΓ(nα)]∑j=0n-1‍(θbαα)jv(t),
for every t∈[0,b] and every n∈N such that nα>1, and Γ(·) is the Gamma function.Lemma 4 (see [24]).
Assumex∈Bh′; then for t∈J, xt∈Bh. Moreover,
(11)l∥x(t)∥≤∥xt∥Bh≤∥x0∥Bh+lsup⁡s∈[0,t]∥x(s)∥,
where l=∫-∞0‍h(t)dt<+∞.Lemma 5 (see [25], p-47).
Letz(t), u(t), v(t), w(t)∈C([α,β],R+), and k≥0 be a real constant and
(12)z(t)≤k+∫αt‍u(s)[+∫αβ‍w(σ)z(σ)dσz(s)+∫αs‍v(σ)z(σ)dσhhhh+∫αβ‍w(σ)z(σ)dσz(s)+∫αs‍v(σ)z(σ)dσ]ds,HHHHHfort∈[α,β].
If
(13)r*=∫αβ‍w(σ)exp⁡(∫ασ‍[u(τ)+v(τ)]dτ)dσ<1,
then
(14)z(t)≤k1-r*exp⁡(∫αt‍[u(s)+v(s)]ds),fort∈[α,β].Definition 6.
A functionx:(-∞,b]→X is called a mild solution of the problem (1) if x0=ϕ∈Bh on (-∞,0], the restriction of x(·) to the interval J is continuous, and for each s∈[0,t) the function AT(t-s)g(s,xs,∫0s‍e(s,τ,xτ)dτ) is integrable and the integral equation
(15)x(t)=T(t)[ϕ(0)-g(0,ϕ,0)]+g(t,xt,∫0t‍e(t,s,xs)ds)+∫0t‍AT(t-s)g(s,xs,∫0s‍e(s,τ,xτ)dτ)ds+∫0t‍T(t-s)f(∫0b‍w(s,τ,xτ)dτs,xs,∫0s‍k(s,τ,xτ)dτ,HHH∫0b‍w(s,τ,xτ)dτs,xs,∫0s‍k(s,τ,xτ)dτ,)ds,t∈J,
is satisfied.Definition 7.
A mapf: J×Bh×X×X→X is said to be an L1 -Caratheodory if (i)
for eacht∈J, the function f(t,·,·,·):Bh×X×X→X is continuous;(ii)
for each(ψ,x,y)∈Bh×X×X; the function f(·,ψ,x,y):J→X is strongly measurable;(iii)
for each positive integerm>0, there exists αm∈L1(J,R+) such thatand for almost all t∈J.
(16)∥f(t,ψ,x,y)∥≤αm(t),∀∥ψ∥Bh≤m,HHHH∥x∥≤m,∥y∥≤m,


## 3. Existence Results

In this section we state and prove our main results. We list the following hypotheses for our convenience.(H1)
A is the infinitesimal generator of a compact analytic semigroup of bounded linear operators T(t), t>0 in X and 0∈ρ(A) such that
(17)∥T(t)∥≤M,∀t≥0,∥(-A)1-αT(t-s)∥≤C1-α(t-s)1-α,0<t≤b,
where 0≤α<1.(H2)
There exist constants0<β<1, L1, L2>0 such that g is Xβ-valued, (-A)βg is continuous, and
(i)
∥e(t,s,ψ)-e(t,s,χ)∥≤L1∥ψ-χ∥Bh, t, s∈J, ψ, χ∈Bh,(ii)
∥(-A)βg(t,ψ,x)-(-A)βg(t,χ,y)∥≤L2[∥ψ-χ∥Bh+∥x-y∥], t∈J, ψ, χ∈Bh with
(18)C0=lL2(1+bL1)[∥(-A)-β∥+(C1-βbβ)β]<1.(H3)
There exist integrable functionsp, q, r: J→[0,∞) such that
(i)
∥k(t,s,ψ)∥≤q(s)∥ψ∥Bh, (t,s)∈Δ, ψ∈Bh,(ii)
∥w(t,s,ψ)∥≤r(s)∥ψ∥Bh, (t,s)∈Δ, ψ∈Bh,(iii)
∥f(t,ψ,x,y)∥≤p(t)(∥ψ∥Bh+∥x∥+∥y∥), for each (t,ψ,x,y)∈J×Bh×X×X.(H4)
For each(t,s)∈Δ, the functions k(t,s,·), w(t,s,·): Bh→X are continuous and for each ψ∈Bh, the functions k(·,·,ψ), w(·,·,ψ): Δ→X are strongly measurable.(H5)
f is an L1-Caratheodory.(H6)
The condition∫0b‍r(σ)exp⁡(∫0σ‍[B0K1p(τ)+q(τ)]dτ)dσ:=R<1 holds, where
(19)B0=exp⁡[K2n(Γ(β))nbnβΓ(nβ)]∑j=0n-1‍(K2bββ)j,HHn∈Nsuchthatnβ>1,K1=lM1-lL2(1+bL1)∥(-A)-β∥,K2=lL2(1+bL1)C1-β1-lL2(1+bL1)∥(-A)-β∥.We set c1=bsup⁡(t,s)∈Δ⁡∥e(t,s,0)∥ and c2=∥(-A)β∥sup⁡t∈J⁡∥g(t,0,0)∥.Using the hypotheses (H1), (H2) and Lemma1, we have the following inequality:
(20)∥AT(t-s)g(s,xs,∫0s‍e(s,τ,xτ)dτ)∥≤∥(-A)1-βT(t-s)∥{∥(-A)βg(s,xs,∫0s‍e(s,τ,xτ)dτ)HHH-(-A)βg(s,0,0)∫0s‍∥+c2}≤C1-β(t-s)1-β{L2[∥xs∥Bh+∥∫0s‍e(s,τ,xτ)dτ∥]+c2}≤C1-β(t-s)1-β{∫0s‍L2[∫0s‍∥xs∥Bh+∫0s∥e(s,τ,xτ)-e(s,τ,0)∥dτ‍HHH+c1∫0s‍]+c2}≤C1-β(t-s)1-β{L2[(1+bL1)∥xs∥Bh+c1∫0s‍]+c2}.
Thus from Bochner theorem, it follows that AT(t-s)g(s,xs,∫0s‍e(s,τ,xτ)dτ) is integrable on [0,t).Throughout this paper, for brevity we set(21)F=M∥ϕ(0)∥+M∥(-A)-β∥[L2∥ϕ∥Bh+c2]+∥(-A)-β∥(L2c1+c2)+C1-β(L2c1+c2)bββ.In the following theorem we establish a priori bound for the mild solution of the following system by using Pachpatte inequality:(22)ddt[x(t)-λg(t,xt,∫0t‍e(t,s,xs)ds)]=Ax(t)+λf(t,xt,∫0t‍k(t,s,xs)ds,∫0b‍w(t,s,xs)ds),HHHhhhht∈J=[0,b],x0=ϕ∈Bh,
where λ∈(0,1). By Definition 6, the mild solution of the system (22) is given by
(23)x(t)=T(t)[ϕ(0)-λg(0,ϕ,0)]+λg(t,xt,∫0t‍e(t,s,xs)ds)+λ∫0t‍AT(t-s)g(s,xs,∫0s‍e(s,τ,xτ)dτ)ds+λ∫0t‍T(t-s)f(s,xs,∫0s‍k(s,τ,xτ)dτ,HHH∫0b‍w(s,τ,xτ)dτ)ds,HHHHHHHt∈J,x0=ϕ∈Bh.Theorem 8.
If hypotheses (H1)–(H6) are satisfied and lettingx(t) be a mild solution of the system (22), then ∥xt∥Bh≤B0K3/(1-R)exp⁡(∫0b‍[B0K1p(s)+q(s)]ds):=K,t∈J,where
(24)K3=∥ϕ∥Bh+lF1-lL2(1+bL1)∥(-A)-β∥.Proof.
Using the hypotheses (H1)–(H3) in (23), we get
(25)∥x(t)∥≤M∥ϕ(0)∥+M∥(-A)-β∥×[∥(-A)βg(0,ϕ,0)-(-A)βg(t,0,0)∥+c2]+∥(-A)-β∥{∥(-A)βg(t,xt,∫0t‍e(t,s,xs)ds)HH-(-A)βg(t,0,0)∫0t‍∥+c2}+∫0t‍∥(-A)1-βT(t-s)∥×{∥(-A)βg(s,xs,∫0s‍e(s,τ,xτ)dτ)-(-A)βg(t,0,0)∫0t‍∥+c2}ds+∫0t‍∥T(t-s)∥∥∫0b‍f(∫0b‍w(s,τ,xτ)dτs,xs,∫0s‍k(s,τ,xτ)dτ,HHHH∫0b‍w(s,τ,xτ)dτs,xs,∫0s‍k(s,τ,xτ)dτ)∥ds≤M∥ϕ(0)∥+M∥(-A)-β∥[L2∥ϕ∥Bh+c2]+∥(-A)-β∥{∫0b‍L2[∫0t‍∥∫0t‍e(t,s,xs)ds∥∥xt∥Bh+∥∫0t‍e(t,s,xs)ds∥]+c2}+∫0t‍C1-β(t-s)1-β×{L2[∥xs∥Bh+∥∫0s‍e(s,τ,xτ)dτ∥]+c2}ds+∫0t‍Mp(s)[+∫0b‍r(τ)∥xτ∥Bhdτ∥xs∥Bh+∫0s‍q(τ)∥xτ∥BhdτHH+∫0b‍r(τ)∥xτ∥Bhdτ∥xs∥Bh+∫0s‍q(τ)∥xτ∥Bhdτ]ds≤M∥ϕ(0)∥+M∥(-A)-β∥[L2∥ϕ∥Bh+c2]+∥(-A)-β∥{L2[(1+bL1)∥xt∥Bh+c1]+c2}+∫0t‍C1-β(t-s)1-β×{L2[(1+bL1)∥xs∥Bh+c1]+c2}ds+∫0t‍Mp(s)[+∫0b‍r(τ)∥xτ∥Bhdτ∥xs∥Bh+∫0s‍q(τ)∥xτ∥Bhdτ+∫0b‍r(τ)∥xτ∥Bhdτ]ds≤M∥ϕ(0)∥+M∥(-A)-β∥[L2∥ϕ∥Bh+c2]+∥(-A)-β∥(L2c1+c2)+C1-β(L2c1+c2)bββ+∥(-A)-β∥L2(1+bL1)∥xt∥Bh+C1-βL2(1+bL1)∫0t‍∥xs∥Bh(t-s)1-βds+∫0t‍Mp(s)[+∫0b‍r(τ)∥xτ∥Bhdτ∥xs∥Bh+∫0s‍q(τ)∥xτ∥Bhdτ+∫0b‍r(τ)∥xτ∥Bhdτ]ds=F+∥(-A)-β∥L2(1+bL1)∥xt∥Bh+C1-βL2(1+bL1)∫0t‍∥xs∥Bh(t-s)1-βds+∫0t‍Mp(s)[+∫0b‍r(τ)∥xτ∥Bhdτ∥xs∥Bh+∫0s‍q(τ)∥xτ∥BhdτHH+∫0b‍r(τ)∥xτ∥Bhdτ]ds.
From inequality (25) and Lemma 4, we have
(26)∥xt∥Bh≤∥ϕ∥Bh+lsup⁡s∈[0,t]|x(s)|≤∥ϕ∥Bh+lF+l∥(-A)-β∥L2(1+bL1)sup⁡s∈[0,t]∥xs∥Bh+lC1-βL2(1+bL1)∫0t‍∥xs∥Bh(t-s)1-βds+∫0t‍lMp(s)[+∫0b‍r(τ)∥xτ∥Bhdτ∥xs∥Bh+∫0s‍q(τ)∥xτ∥BhdτHH+∫0b‍r(τ)∥xτ∥Bhdτ]ds,HHHHHHt∈J.
Define the function μ(t)=sup⁡{∥xs∥Bh: 0≤s≤t}, t∈J; then μ(t) is nondecreasing on J, and we get
(27)μ(t)≤∥ϕ∥Bh+lF+l∥(-A)-β∥L2(1+bL1)μ(t)+lC1-βL2(1+bL1)∫0t‍μ(s)(t-s)1-βds+∫0t‍lMp(s)[+∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτHH+∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτ]ds.
Therefore,
(28)μ(t)≤∥ϕ∥Bh+lF1-lL2(1+bL1)∥(-A)-β∥+lL2(1+bL1)C1-β1-lL2(1+bL1)∥(-A)-β∥∫0t‍μ(s)(t-s)1-βds+lM1-lL2(1+bL1)∥(-A)-β∥×∫0t‍p(s)[+∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτHH+∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτ]ds=K3+K1∫0t‍p(s)[+∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτHH+∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτ]ds+K2∫0t‍μ(s)(t-s)1-βds.
Using Lemma 3, we have
(29)μ(t)≤B0(K3+K1∫0t‍p(s)HHHHHHHHH×[∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτHHHHHHHHHHHH+∫0b‍r(τ)μ(τ)dτμ(s)+∫0s‍q(τ)μ(τ)dτ]ds)=B0K3+∫0t‍B0K1p(s)HHHHHHH×[∫0s‍q(τ)μ(τ)dτμ(s)+∫0b‍r(τ)μ(τ)dτHHHHHHHHHH+∫0s‍q(τ)μ(τ)dτ∫0b‍r(τ)μ(τ)dτ]ds,
where
(30)B0=exp⁡[K2n(Γ(β))nbnβΓ(nβ)]∑j=0n-1‍(K2bββ)j.
Thanks to Pachpatte’s inequality given in Lemma 5 and applying it with z(t)=μ(t) and using hypothesis (H6), we obtain
(31)μ(t)≤B0K31-Rexp⁡(∫0t‍[B0K1p(s)+q(s)]ds),t∈J.
This implies that ∥xt∥Bh≤K, t∈J.Now we define the operatorΨ:Bh′→Bh′ by
(32)Ψx(t)={ϕ(t),t∈(-∞,0]T(t)[ϕ(0)-g(0,ϕ,0)]+g(t,xt,∫0t‍e(t,s,xs)ds)+∫0t‍AT(t-s)×g(s,xs,∫0s‍e(s,τ,xτ)dτ)ds+∫0t‍T(t-s)×f(s,xs,∫0s‍k(s,τ,xτ)dτ,HH∫0b‍w(s,τ,xτ)dτ)ds,t∈J.
For ϕ∈Bh, define ϕ~ by
(33)ϕ~(t)={ϕ(t),t∈(-∞,0],T(t)ϕ(0),t∈J=[0,b];
then ϕ~∈Bh′. Let x(t)=y(t)+ϕ~(t), -∞<t≤b. It is easy to see that x satisfies (15) if and only if y satisfies y0=0 and
(34)y(t)=-T(t)g(0,ϕ,0)+g(t,yt+ϕ~t,∫0t‍e(t,s,ys+ϕ~s)ds)+∫0t‍AT(t-s)g(s,ys+ϕ~s,∫0s‍e(s,τ,yτ+ϕ~τ)dτ)ds+∫0t‍T(t-s)f(∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,HH∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,)ds.
Let Bh′′={y∈Bh′:y0=0∈Bh}; then for any y∈Bh′′ we have (35)∥y∥b=∥y0∥Bh+sup⁡{∥y(s)∥:0≤s≤b}=sup⁡{∥y(s)∥:0≤s≤b};
thus (Bh′′,∥·∥b) is a Banach space. Define Bm={y∈Bh′′:∥y∥b≤m} for some m>0; then Bm⊆Bh′′ is uniformly bounded, and for y∈Bm, from Lemma 4, we have
(36)∥yt+ϕ~t∥Bh≤∥yt∥Bh+∥ϕ~t∥Bh≤∥y0∥Bh+lsup⁡s∈[0,t]∥y(s)∥+∥ϕ~0∥Bh+lsup⁡s∈[0,t]∥ϕ~(s)∥≤lm+∥ϕ~0∥Bh+lsup⁡s∈[0,b]∥T(s)ϕ(0)∥≤l(m+M∥ϕ(0)∥)+∥ϕ∥Bh=m′.
Define the operator Ψ¯:Bh′′→Bh′′ by
(37)Ψ¯y(t)={0,t∈(-∞,0]-T(t)g(0,ϕ,0)+g(∫0t‍t,yt+ϕ~t,∫0t‍e(t,s,ys+ϕ~s)ds)+∫0t‍AT(t-s)×g(∫0s‍s,ys+ϕ~s,∫0s‍e(s,τ,yτ+ϕ~τ)dτ)ds+∫0t‍T(t-s)×f(∫0t‍s,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,∫0b‍w(s,τ,yτ+ϕ~τ)dτ)ds,t∈J.
In the view of Krasnoselski-Schaefer type fixed point theorem, we decompose Ψ¯ as Ψ¯1+Ψ¯2, where Ψ¯1 and Ψ¯2 are defined on Bh′′, respectively, by
(38)Ψ¯1y(t)=-T(t)g(0,ϕ,0)+g(t,yt+ϕ~t,∫0t‍e(t,s,ys+ϕ~s)ds)+∫0t‍AT(t-s)g(∫0s‍s,ys+ϕ~s,HHH∫0s‍∫0s‍e(s,τ,yτ+ϕ~τ)dτ)ds,HHHt∈J,(39)Ψ¯2y(t)=∫0t‍T(t-s)×f(∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,HH∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,)ds,t∈J.
Observe that the operator Ψ having a fixed point is equivalent to Ψ¯ having one. Next, our aim is to prove that the operator Ψ¯1 is a contraction, while Ψ¯2 is a completely continuous operator.Theorem 9.
If the hypotheses (H1) and (H2) are satisfied, thenΨ¯1 is a contraction on Bh′′.Proof.
Let anyu, v∈Bh′′; then by using the hypotheses (H1), (H2) and Lemma 4, from (38) for each t∈J, we have
(40)∥Ψ¯1u(t)-Ψ¯1v(t)∥≤∥(-A)β[(-A)βg(t,ut+ϕ~t,∫0t‍e(t,s,us+ϕ~s)ds)-(-A)βg(∫0t‍t,vt+ϕ~t,∫0t‍e(t,s,vs+ϕ~s)ds)]∥+∫0t‍∥(-A)1-βT(t-s)∥×∥(-A)βg(s,us+ϕ~s,∫0s‍e(s,τ,uτ+ϕ~τ)dτ)-(-A)βg(s,vs+ϕ~s,∫0s‍e(s,τ,vτ+ϕ~τ)dτ)∥ds≤∥(-A)-β∥L2[∫0t‍∥ut-vt∥Bh+∫0t∥e(t,s,us+ϕ~s)‍-e(t,s,vs+ϕ~s)∥ds∫0t‍∥ut-vt∥Bh]+∫0t‍C1-β(t-s)1-βL2[∫0s‍∥us-vs∥Bh+∫0s‍∥e(s,τ,uτ+ϕ~τ)-e(s,τ,vτ+ϕ~τ)∥dτ∫0s‍∥us-vs∥Bh]ds≤∥(-A)-β∥L2(1+bL1)∥ut-vt∥Bh+C1-βL2(1+bL1)∥ut-vt∥Bh∫0t‍ds(t-s)1-β≤L2(1+bL1)[∥(-A)-β∥+C1-βbββ]∥ut-vt∥Bh≤L2(1+bL1)[∥(-A)-β∥+C1-βbββ]×[∥u0∥Bh+∥v0∥Bh+lsup⁡s∈[0,t]∥u(s)-v(s)∥]≤lL2(1+bL1)[∥(-A)-β∥+C1-βbββ]sup⁡s∈[0,b]∥u(s)-v(s)∥,
since ∥u0∥Bh=0, ∥v0∥Bh=0.
This implies that(41)∥Ψ¯1u-Ψ¯1v∥b≤C0∥u-v∥b.
Since C0<1, Ψ¯1 is contraction on Bh′′.Theorem 10.
If the hypotheses (H1), (H3)–(H5) are satisfied, thenΨ¯2:Bh′′→Bh′′ is completely continuous operator.Proof.
We give the proof in the following steps.
Step
1.
Ψ
¯
2 maps bounded sets into bounded sets in Bh′′.
Let anyy∈Bm={y∈Bh′′:∥y∥b≤m}. Then it is enough to prove that ∥Ψ¯2y∥b≤Λ for some constant Λ. By using the hypotheses (H1), (H3) and condition (36) from (39), we have
(42)∥Ψ¯2y(t)∥≤∫0t‍Mp(s)[∫0b‍r(τ)∥yτ+ϕ~τ)∥Bhdτ∥ys+ϕ~s∥Bh+∫0s‍q(τ)∥yτ+ϕ~τ∥BhdτHHH+∫0b‍r(τ)∥yτ+ϕ~τ∥Bhdτ∥ys+ϕ~s∥Bh+∫0s‍q(τ)∥yτ+ϕ~τ)∥Bh]ds≤∫0t‍Mp(s)[m′+∫0s‍m′q(τ)dτ+∫0b‍m′r(τ)dτ]ds≤∫0b‍Mm′p(s)[1+∫0b‍{q(τ)+r(τ)}dτ]ds:=Λ.
Thus for each y∈Bm, we have ∥Ψ¯2y∥b≤Λ.
Step
2.
Ψ
¯
2 maps bounded sets into equicontinuous sets of Bh′′.
Lety∈Bm and t1, t2∈(-∞,b]. Then from (39), using the hypotheses (H1) and (H3) and condition (36), we have the following three cases.
Case
1. Let 0<t1<t2≤b. Then, we have
(43)∥Ψ¯2y(t1)-Ψ¯2y(t2)∥≤∫0t1-ϵ‍∥T(t1-s)-T(t2-s)∥×∥∫0b‍w(s,τ,yτ+ϕ~τ)dτ)f(s,ys+ϕ~s,∫0sk(s,τ,yτ+ϕ~τ)dτ,∫0b‍w(s,τ,yτ+ϕ~τ)dτ)s,ys+ϕ~s,∫0sk(s,τ,yτ+ϕ~τ)dτ,∥ds+∫t1-ϵt1‍∥T(t1-s)-T(t2-s)∥×∥f(s,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,∫0b‍w(s,τ,yτ+ϕ~τ)dτ)∥ds+∫t1t2‍∥T(t2-s)∥∥f(s,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,∫0b‍w(s,τ,yτ+ϕ~τ)dτ)∥ds≤∫0t1-ϵ‍∥T(t1-s)-T(t2-s)∥m′p(s)×[1+∫0b‍{q(τ)+r(τ)}dτ]ds+∫t1-ϵt1‍∥T(t1-s)-T(t2-s)∥m′p(s)×[1+∫0b‍{q(τ)+r(τ)}dτ]ds+∫t1t2‍∥T(t2-s)∥m′p(s)×[1+∫0b‍{q(τ)+r(τ)}dτ]ds.
Case
2. Let t1≤0≤t2≤b. Then, we have
(44)∥Ψ¯2y(t1)-Ψ¯2y(t2)∥≤∫0t2‍∥T(t2-s)∥∥f(s,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,HH∫0b‍w(s,τ,yτ+ϕ~τ)dτ)∥ds≤∫0t2‍Mm′p(s)[1+∫0b‍{q(τ)+r(τ)}dτ]ds.
Case
3.If t1≤t2≤0, then ∥Ψ¯2y(t1)-Ψ¯2y(t2)∥=0.
From Cases 1–3, we deduce that the right-hand side of the above inequality tends to zero ast2-t1 tends to 0 for ϵ sufficiently small, since the compactness of T(t), t>0, implies the continuity in the uniform operator topology. Thus the set {Ψ¯2y:y∈Bm} is equicontinuous.
Step
3.
Ψ
¯
2 maps Bm into a precompact set in Bh′′.
Together with Arzela-Ascoli theorem and Steps 1–2 to proveΨ¯2Bm is precompact in Bh′′, it is sufficient to show that the set {(Ψ¯2y)(t):y∈Bm} is precompact in X. Let 0<t≤b be fixed, and let ϵ be a real number satisfying 0<ϵ<t. For y∈Bm, we define the operators
(45)(Ψ~2ϵy)(t)=∫0t-ϵ‍T(t-s)f(∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,HHH∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,)ds=T(ϵ)∫0t-ϵ‍T(t-s-ϵ)×f(∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,HHH∫0b‍w(s,τ,yτ+ϕ~τ)dτs,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,)ds.
Since T(ϵ) is compact operator, the set V¯ϵ(t)={(Ψ¯2ϵy)(t):y∈Bm} is precompact in X, for every ϵ, 0<ϵ<t. Moreover, for each y∈Bm, we have
(46)∥(Ψ¯2y)(t)-(Ψ~2ϵy)(t)∥≤∫t-ϵt‍m′Mp(s)[1+∫0b‍{q(τ)+r(τ)}dτ]ds⟶0,HHHHHHasϵ→0+.
Therefore there are precompact sets arbitrarily close to the set {(Ψ¯2y)(t):y∈Bm}. Thus the set {(Ψ¯2y)(t):y∈Bm} is precompact in X.
Step
4.
Ψ
¯
2
:
B
h
′′
→
B
h
′′ is continuous.
Let{y(n)}n=0+∞⊆Bh′′, with y(n)→y in Bh′′. Then there is a number m>0 such that ∥y(n)(t)∥≤m for all n and a. e. t∈J, so y(n)∈Bm and y∈Bm.
By using the hypotheses (H4), (H5) and condition (36) we have
(47)f(∫0b‍w(t,s,ys(n)+ϕ~s)dst,yt(n)+ϕ~t,∫0t‍k(t,s,ys(n)+ϕ~s)ds,t,yt(n)+ϕ~t,∫0t‍k(t,s,ys(n)+ϕ~s)ds,∫0b‍w(t,s,ys(n)+ϕ~s)dst,yt(n)+ϕ~t,∫0t‍k(t,s,ys(n)+ϕ~s)ds,)⟶f(∫0b‍w(t,s,ys+ϕ~s)dst,yt+ϕ~t,∫0t‍k(t,s,ys+ϕ~s)ds,t,yt+ϕ~t,∫0t‍k(t,s,ys+ϕ~s)ds,HHH∫0b‍w(t,s,ys+ϕ~s)dst,yt+ϕ~t,∫0t‍k(t,s,ys+ϕ~s)ds,),
for each t∈J, and since
(48)∥f(t,yt(n)+ϕ~t,∫0t‍k(t,s,ys(n)+ϕ~s)ds,∫0b‍w(t,s,ys(n)+ϕ~s)ds)HHH-f(∫0b‍w(t,s,ys+ϕ~s)dst,yt+ϕ~t,∫0t‍k(t,s,ys+ϕ~s)ds,HHHH∫0b‍w(t,s,ys+ϕ~s)dst,yt+ϕ~t,∫0t‍k(t,s,ys+ϕ~s)ds,)∥≤2m′p(s)[1+∫0b‍{q(τ)+r(τ)}dτ],
we have by the dominated convergence theorem that
(49)∥Ψ¯2y(n)-Ψ¯2y∥b=∫0t‍∥T(t-s)∥∥∫0s‍f(∫0s‍s,ys(n)+ϕ~s,HHHH∫0s‍k(s,τ,yτ(n)+ϕ~τ)dτ,HHHH∫0b‍w(s,τ,yτ(n)+ϕ~τ)dτ)HHHH-f(s,ys+ϕ~s,∫0s‍k(s,τ,yτ+ϕ~τ)dτ,HHHHHH∫0b‍w(s,τ,yτ+ϕ~τ)dτ)∥dsHHHHH⟶0asn⟶∞.
Therefore,
(50)∥Ψ¯2y(n)-Ψ¯2y∥b=sup⁡t∈J∥Ψ¯2y(n)(t)-Ψ¯2y(t)∥⟶0.
This implies that Ψ¯2 is continuous.
From Steps 1–4, we can conclude that the operatorΨ¯2 is completely continuous and thus satisfies condition (b) in Lemma 2.Theorem 11.
Assume that the hypotheses (H1)–(H6) hold. Then the problem (1) has at least one mild solution on (-∞,b].Proof.
LetG={y∈Bh′′:y=λΨ¯1(y/λ)+λΨ¯2y for some λ∈(0,1)}. Then for any y∈G, the function x=y+ϕ~ is a mild solution of the system (22) for which we have proved in Theorem 8 that ∥xt∥Bh≤K, t∈J, and hence from Lemma 4, we have
(51)∥y∥b=∥y0∥Bh+sup⁡{∥y(t)∥∶t∈[0,b]}=sup⁡{∥y(t)∥∶t∈[0,b]}≤sup⁡⁡{∥x(t)∥∶t∈[0,b]}+sup⁡⁡{∥ϕ~(t)∥∶t∈[0,b]}≤sup⁡{l-1∥xt∥Bh∶t∈[0,b]}+sup⁡{∥T(t)ϕ(0)∥∶t∈[0,b]}≤l-1K+M∥ϕ(0)∥,
which yields that the set G is bounded.
Consequently, by virtue of Lemma2, Theorem 9, and Theorem 10, the equation Ψ¯1y+Ψ¯2y=y has a solution y*∈Bh′′. Let x(t)=y*(t)+ϕ~(t), t∈(-∞,b]; then x is a fixed point of the operator Ψ which is a mild solution of the problem (1).Theorem 12.
Assume that (H1), (H2), (H4), (H5), and the following hypotheses are satisfied. (H3)′
There exist integrable functionsp, q, r:J→[0,∞) such that
(i)
∥∫0t‍k(t,s,ψ)∥≤q(t)∥ψ∥Bh, (t,s)∈Δ,ψ∈Bh,(ii)
∥∫0b‍w(t,s,ψ)∥≤r(t)∥ψ∥Bh, (t,s)∈Δ,ψ∈Bh,(iii)
∥f(t,ψ,x,y)∥≤p(t)Ω(∥ψ∥Bh+∥x∥+∥y∥), for each (t,ψ,x,y)∈J×Bh×X×X, where Ω:[0,∞)→[0,∞) is continuous nondecreasing function such that
(52)Ω(γ(t)r)≤γ(t)Ω(r),γ(t)=1+p(t)+q(t),
for each t∈J and r≥0.(H6)′
The conditionB0K1∫ob‍p(s)γ(s)ds<∫B0K3∞‍ds/Ω(s) holds, where B0, K1, and K3 are as in (H6) and (24).Then the problem (1) has at least one mild solution on (-∞,b].Proof.
Letx(t) be the solution of (22). By using the hypotheses (H1), (H2), and (H3)′ and (23), we obtain
(53)∥x(t)∥≤F+∥(-A)-β∥L2(1+bL1)∥xt∥Bh+C1-βL2(1+bL1)∫0t‍∥xs∥Bh(t-s)1-βds+∫0t‍Mp(s)Ω(∥xs∥Bh+q(s)∥xs∥Bh+r(s)∥xs∥Bh)ds≤F+∥(-A)-β∥L2(1+bL1)∥xt∥Bh+C1-βL2(1+bL1)∫0t‍∥xs∥Bh(t-s)1-βds+∫0t‍Mp(s)γ(s)Ω(∥xs∥Bh)ds.
By an application of Lemma 4, we get
(54)∥xt∥Bh≤∥ϕ∥Bh+lF+l∥(-A)-β∥L2(1+bL1)∥xt∥Bh+lC1-βL2(1+bL1)∫0t‍∥xs∥Bh(t-s)1-βds+∫0t‍lMp(s)γ(s)Ω(∥xs∥Bh)ds.
Define the function μ as in the proof of Theorem 8 and, proceeding on the same line, we obtain
(55)μ(t)≤K3+K1∫0tp(s)γ(s)Ω(μ(s))ds+K2∫0tμ(s)(t-s)1-βds,t∈J.
Applying Lemma 3 to the above inequality, we obtain
(56)μ(t)≤B0(K3+K1∫0t‍p(s)γ(s)Ω(μ(s))ds),t∈J.
Let
(57)w(t)=B0(K3+K1∫0t‍p(s)γ(s)Ω(μ(s))ds),t∈J.
Then w(0)=B0K3, μ(t)≤w(t), t∈J, and
(58)w′(t)=B0K1p(t)γ(t)Ω(μ(t)),t∈J.
Since Ω is nondecreasing function, we have
(59)w′(t)≤B0K1p(t)γ(t)Ω(w(t)),t∈J.
Therefore,
(60)w′(t)Ω(w(t))≤B0K1p(t)γ(t),t∈J.
Integrating from 0 to t and using the change of variables t→s=w(t) and the hypothesis (H6)′, we obtain
(61)∫B0K3w(t)dsΩ(s)ds≤B0K1∫0b‍p(s)γ(s)ds<∫B0K3∞dsΩ(s)ds,t∈J.
This implies that w(t)<∞. So there is constant K* such that w(t)≤K*, t∈J, and hence ∥xt∥Bh≤μ(t)≤w(t)≤K*, t∈J, where K* depends on the functions p, γ, and Ω.
Define the operatorsΨ¯, Ψ¯1, and Ψ¯2 as discussed above. Note that the set G={y∈Bh′′∶y=λΨ¯1(y/λ)+λΨ¯2y for some λ∈(0,1)} is bounded by l-1K*+M∥ϕ(0)∥. Theorem 9 satisfies condition (a) in Lemma 2. The proof of Ψ¯2 is completely continuous operator which can be completed using the hypotheses (H1), (H3)′, (H4), and (H5) and closely looking at the proof of Theorem 10. Finally by applying Lemma 2, the problem (1) has mild solution on (-∞,b].Theorem 13.
Assume that the hypotheses (H1), (H2), (H4), and (H5) are satisfied. In addition suppose the following.(H3)′′
There exist integrable functionsp, q, r:J→[0,∞) such that
(i)
∥∫0t‍k(t,s,ψ)∥≤q(t)Ω(∥ψ∥Bh),(t,s)∈Δ,ψ∈Bh,(ii)
∥∫0b‍w(t,s,ψ)∥≤r(t)Ω(∥ψ∥Bh),(t,s)∈Δ,ψ∈Bh,(iii)
∥f(t,ψ,x,y)∥≤p(t)Ω(∥ψ∥Bh)+∥x∥+∥y∥, for each (t,ψ,x,y)∈J×Bh×X×X, where Ω: [0,∞)→[0,∞) is continuous nondecreasing function.(H6)′′
The conditionB0K1∫ob‍[p(s)+q(s)+r(s)]ds<∫B0K3∞‍(ds/Ω(s)) holds, where B0, K1, and K3 are as in (H6) and (24).
Then the problem (1) has at least one mild solution on (-∞,b].Proof.
Proceeding as in the proof of Theorem12 with suitable modification, we can complete the proof. Hence we omit the details.


## 4. Application

Consider the following partial neutral mixed integrodifferential equation of the form(62)∂∂t[v(t,x)+G(t,∫-∞0‍P1(s-t)v(s,x)ds),∫0t‍∫-∞s‍P2(s,x,τ-s)Q1(v(τ,x))dτds]=∂2∂x2v(t,x)+H(t,∫-∞0‍P3(s-t)v(s,x)ds,∫0t‍∫-∞s‍P4(s,x,τ-s)Q2(v(τ,x))dτds,∫0b‍∫-∞s‍P5(s,x,τ-s)Q3(v(τ,x))dτds),(x,t)∈[0,π]×[0,b],v(t,0)=v(t,π)=0,t≥0v(t,x)=ϕ(t,x),t∈(-∞,0],x∈[0,π],
where ϕ∈Bh. Consider the space X=L2[0,π] with the norm |·|L2. Define the operator A:X→X by Aw=w′′ with the domain D(A)={w∈X:w,w′ are absolutely continuous, w′′∈X,w(0)=w(π)=0}.ThenAw=∑n=1∞‍-n2<w,wn>wn, and w∈D(A), where wn(x)=2/πsin(nx), n=1,2,…, is the orthogonal set of eigen vectors of A. It is well known that A is the infinitesimal generator of an analytic semigroup T(t), t≥0, and is given by T(t)w=∑n=1∞‍e-n2t<w, wn>wn, and w∈X. Further for every w∈X, (-A)-1/2w=∑n=1∞‍(1/n)<w, wn>wn, and ∥(-A)-1/2∥=1. The operator (-A)1/2 is given by
(63)(-A)1/2w=∑n=1∞‍n<w,wn>wn,
with the domain D((-A)1/2)={w∈X:∑n=1∞‍n<w, wn>wn∈X}. It follows that ∥T(t)∥≤1. Let h(s)=e2s, s<0; then l=∫-∞0‍h(s)ds=1/2, and define
(64)∥ϕ∥h=∫-∞0‍h(s)sup⁡θ∈[s,0]|ϕ(θ)|L2ds.
Hence for (t,ϕ)∈[0,b]×Bh, where ϕ(θ)(x)=ϕ(θ,x), (θ,x)∈(-∞,0]×[0,π]. Set
(65)v(t)(y)=v(t,y)e(t,s,ψ)(y)=∫-∞0‍P2(t,y,θ)Q1(ψ(θ)(y))dθ,k(t,s,ψ)(y)=∫-∞0‍P4(t,y,θ)Q2(ψ(θ)(y))dθ,w(t,s,ψ)(y)=∫-∞0‍P5(t,y,θ)Q3(ψ(θ)(y))dθ,g(t,ψ,∫0t‍e(t,s,ψ)ds)(y)=G(t,∫-∞0‍P1(θ)ψ(θ)(y)dθ,∫0t‍e(t,s,ψ)(y)ds)f(t,ψ,∫0t‍k(t,s,ψ)ds,∫0b‍w(t,s,ψ)ds)(y)=H(t,∫-∞0‍P3(θ)ϕ(θ)(y)dθ,∫0t‍k(t,s,ψ)(y)ds,∫0b‍w(t,s,ψ)(y)ds).
With these choices of functions, the system (1) is the abstract formulation of the system (62). By imposing suitable conditions on the above defined functions to verify the assumptions of Theorem 10, we conclude that system (62) has at least one mild solution on (-∞,b].


## 5. Conclusion

The integral inequality established by B. G. Pachpatte is used to obtain the bound and different existence results are established for more general equations which include the study of Volterra and Fredholm functional integrodifferential equations.



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*Source: 784956-2014-05-14.xml*