# Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay

Authors: Jaydev Dabas; Archana Chauhan; Mukesh Kumar
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

This paper is concerned with the existence and uniqueness of a
mild solution of a semilinear fractional-order functional evolution differential equation with the infinite delay and impulsive effects. The existence and uniqueness of a mild solution is established using a solution operator and the classical fixed-point theorems.


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## Body

## 1. Introduction

This paper is concerned with the existence and uniqueness of a mild solution of an impulsive fractional-order functional differential equation with the infinite delay of the form(1.1)Dtαx(t)=Ax(t)+f(t,xt,Bx(t)),t∈J=[0,T],t≠tk,Δx(tk)=Ik(x(tk-)),k=1,2,…,m,x(t)=ϕ(t),ϕ(t)∈Bh,
where T>0,0<α<1,A:D(A)⊂X→X is the infinitesimal generator of an α-resolvent family Sα(t)t≥0, the solution operator Tα(t)t≥0 is defined on a complex Banach space X, Dα is the Caputo fractional derivative, f:J×𝔅h×X→X is a given function, and 𝔅h is a phase space defined in Section 2. Here, 0=t0<t1<⋯<tm<tm+1=T, Ik∈C(X,X), (k=1,2,…,m), are bounded functions, Δx(tk)=x(tk+)-x(tk-),x(tk+)=limh→0x(tk+h) and x(tk-)=limh→0x(tk-h) represent the right and left limits of x(t) at t=tk, respectively.We assume thatxt:(-∞,0]→X, xt(s)=x(t+s), s≤0, belongs to an abstract phase space 𝔅h. The term Bx(t) is given by Bx(t)=∫0tK(t,s)x(s)ds, where K∈C(D,ℝ+) is the set of all positive continuous functions on D={(t,s)∈ℝ2:0≤s≤t≤T}.Differential equations with impulsive conditions constitute an important field of research due to their numerous applications in ecology, medicine biology, electrical engineering, and other areas of science. Many physical phenomena in evolution processes are modelled as impulsive differential equations and have been studied extensively by several authors, for instance, see [1–3], for more information on these topics. Impulsive integro-differential equations with delays represent mathematical models for problems in the areas such as population dynamics, biology, ecology, and epidemic and have been studied by many authors [2–7]. The study of fractional differential equations has emerged as a new branch of applied mathematics, which has been used for construction and analysis of mathematical models in science and engineering. In fact, the fractional differential equations are considered as models alternative to nonlinear differential equations. Many physical systems can be represented more accurately through fractional derivative formulation. For more detail, see, for instance, the papers [1, 3–5, 7–12] and references therein.Recently, in [4], the author has established sufficient conditions for the existence of a mild solution for a fractional integro-differential equation with a state-dependent delay. Mophou and N’Guérékata [7] have investigated the existence and uniqueness of a mild solution for the fractional differential equation (1.1) without impulsive conditions. Authors of [7] have established the results assuming that A generates an α-resolvent family (Sα(t))t≥0 on a complex Banach space X by means of classical fixed-point methods.In [5], Benchohra et al. have considered the following nonlinear functional differential equation with infinite delay(1.2)Dqx(t)=f(t,xt),t∈[0,T],0<q<1,x(t)=ϕ(t),t∈]-∞,0],
where Dq is Riemann-Liouville fractional derivative, ϕ∈𝔅h, with ϕ(0)=0, and established the existence of a mild solution for the considered problem using the Banach fixed-point and the nonlinear alternative of Leray-Schauder theorems.Motivated by the above-mentioned works, we consider the problem (1.1) to study the existence and uniqueness of a mild solution using the solution operator and fixed-point theorems. The paper is organized as follows: in Section 2, we introduce some function spaces and notations and present some necessary definitions and preliminary results that will be used to prove our main results. The proof of our main results is given in Section 3. In the last section one example is presented.


## 2. Preliminaries

In this section, we mention some definitions and properties required for establishing our results. LetX be a complex Banach space with its norm denoted as ∥·∥X, and L(X) represents the Banach space of all bounded linear operators from X into X, and the corresponding norm is denoted by ∥·∥L(X). Let C(J,X) denote the space of all continuous functions from J into X with supremum norm denoted by ∥·∥C(J,X). In addition, Br(x,X) represents the closed ball in X with the center at x and the radius r.To describe a fractional-order functional differential equation with the infinite delay, we need to discuss the abstract phase space𝔅h in a convenient way (for details see [3]). Let h:(-∞,0]→(0,∞) be a continuous function with l=∫-∞0h(t)dt<∞. For any a>0, we define𝔅={ψ:[-a,0]→X such that ψ(t) is bounded and measurable} and equip the space 𝔅 with the norm
(2.1)‖ψ‖[-a,0]=sup⁡s∈[-a,0]|ψ(s)|,∀ψ∈B.
Let us define by
(2.2)Bh={∫-∞0ψ:(-∞,0]⟶X,suchthatforanyc>0,ψ∣[-c,0]∈Bwithψ(0)=0and∫-∞0h(s)‖ψ‖[s,0]ds<∞}.If𝔅h is endowed with the norm (2.3)‖ψ‖Bh=∫-∞0h(s)‖ψ‖[s,0]ds,∀ψ∈Bh,
then it is known that (𝔅h,∥·∥𝔅h) is a Banach space.Now, we consider the space(2.4)Bh′={x:(-∞,T]⟶Xsuchthatx∣Jk∈C(Jk,X)andthereexistx(tk+)andx(tk-)withx(tk)=x(tk-),x0=ϕ∈Bh,k=1,…,m(-∞,T]0(-∞,T](-∞,T]},
where x∣Jk is the restriction of x to Jk=(tk,tk+1],k=0,1,2,…,m. The function ∥·∥𝔅h′ to be a seminorm in 𝔅h′, it is defined by (2.5)‖x‖Bh′=sup⁡{|x(s)|:s∈[0,T]}+‖ϕ‖Bh,x∈Bh′.
If x:]-∞,T]→X,T>0, is such that x0∈𝔅h, then for all t∈J, the following conditions hold:(1)
xt∈𝔅h,(2)
∥xt∥𝔅h≤C1(t)sup0<s<t∥x(s)∥+C2(t)∥x0∥𝔅h,(3)
∥x(t)∥≤H∥xt∥𝔅h, where H>0 is a constant and C1:[0,∞)→[0,∞) is continuous, C2:[0,∞)→[0,∞) is locally bounded, and C1,C2 are independent of x(·). For more details, see [6].A two parameter function of the Mittag-Lefller type is defined by the series expansion(2.6)Eα,β(z)=∑k=0∞zkΓ(αk+β)=12πi∫Cμα-βeμμα-zdμ,α,β>0,z∈C,
where C is a contour which starts and ends at -∞ and encircles the disc |μ|≤|z|1/2 counter clockwise. For short, Eα(z)=Eα,1(z). It is an entire function which provides a simple generalization of the exponent function: E1(z)=ez and the cosine function: E2(z2)=cosh(z),E2(-z2)=cos(z), and plays an important role in the theory of fractional differential equations. The most interesting properties of the Mittag-Lefller functions are associated with their Laplace integral(2.7)∫0∞e-λttβ-1Eα,β(ωtα)dt=λα-βλα-ω,Reλ>ω1/α,ω>0,
see [12] for more details.Definition 2.1.
A closed and linear operatorA is said to be sectorial if there are constants ω∈R,θ∈[π/2,π],M>0, such that the following two conditions are satisfied:
(2.8)(1)ρ(A)⊂∑(θ,ω)={λ∈C:λ≠ω,|arg(λ-ω)|<θ},(2)‖R(λ,A)‖L(X)≤M|λ-ω|,λ∈∑(θ,ω).
Sectorial operators are well studied in the literature. For details see [13].Definition 2.2 (see Definition  2.3 in [10]).
LetA be a closed and linear operator with the domain D(A) defined in a Banach space X. Let ρ(A) be the resolvent set of A. We say that A is the generator of an α-resolvent family if there exist ω≥0 and a strongly continuous function Sα:R+→L(X) such that {λα:Reλ>ω}⊂ρ(A) and
(2.9)(λαI-A)-1x=∫0∞e-λtSα(t)xdt,Reλ>ω,x∈X,
in this case, Sα(t) is called the α-resolvent family generated by A.Definition 2.3 (see Definition  2.1 in [4]).
LetA be a closed linear operator with the domain D(A) defined in a Banach space X and α>0. We say that A is the generator of a solution operator if there exist ω≥0 and a strongly continuous function Sα:R+→L(X) such that {λα:Reλ>ω}⊂ρ(A) and
(2.10)λα-1(λαI-A)-1x=∫0∞e-λtSα(t)xdt,Reλ>ω,x∈X,
in this case, Sα(t) is called the solution operator generated by A.The concept of the solution operator is closely related to the concept of a resolvent family (see [14] Chapter 1). For more details on α-resolvent family and solution operators, we refer to [14, 15] and the references therein.Definition 2.4.
The Riemann-Liouville fractional integral operator for orderα>0, of a function f:ℝ+→ℝ and f∈L1(ℝ+,X), is defined by
(2.11)I0f(t)=f(t),Iαf(t)=1Γ(α)∫0t(t-s)α-1f(s)ds,α>0,t>0,
where Γ(·) is the Euler gamma function. The Laplace transform of a function f∈L1(ℝ+,X) is defined by
(2.12)f̂(λ)=∫0∞e-λtf(t)dt,Re(λ)>ω,
provided the integral is absolutely convergent for Re(λ)>ω.Definition 2.5.
Caputo’s derivative of orderα for a function f:[0,∞)→ℝ is defined as
(2.13)Dtαf(t)=1Γ(n-α)∫0t(t-s)n-α-1f(n)(s)ds=In-αf(n)(t),
for n-1≤α<n,n∈N. If 0<α≤1, then
(2.14)Dtαf(t)=1Γ(1-α)∫0t(t-s)-αf(1)(s)ds.
Obviously, Caputo’s derivative of a constant is equal to zero. The Laplace transform of the Caputo derivative of order α>0 is given as
(2.15)L{Dtαf(t);λ}=λαf̂(λ)-∑k=0n-1λα-k-1f(k)(0);n-1≤α<n.Lemma 2.6.
Iff satisfies the uniform Holder condition with the exponent β∈(0,1] and A is a sectorial operator, then the unique solution of the Cauchy problem
(2.16)Dtαx(t)=Ax(t)+f(t,xt,Bx(t)),t>t0,t0∈R,0<α<1,x(t)=ϕ(t),t≤t0,
is given by
(2.17)x(t)=Tα(t-t0)(x(t0+))+∫t0tSα(t-s)f(s,xs,Bx(s))ds,
where
(2.18)Tα(t)=Eα,1(Atα)=12πi∫B̂reλtλα-1λα-Adλ,Sα(t)=tα-1Eα,α(Atα)=12πi∫B̂reλt1λα-Adλ,B̂r denotes the Bromwich path. Sα(t) is called the α-resolvent family, and Tα(t) is the solution operator, generated by A.Proof.
Lett-t0=u, then we get
(2.19)Duαx(u+t0)=Ax(u+t0)+f(u+t0,xu+t0,Bx(u+t0)),u>0.
Taking the Laplace transform of (2.19), we have
(2.20)λαL{x(u+t0)}-λα-1x(t0+)=AL{x(u+t0)}+L{f(u+t0,xu+t0,Bx(u+t0))}.
Since (λαI-A)-1 exists, that is, λα∈ρ(A), from (2.20), we obtain
(2.21)L{x(u+t0)}=λα-1(λαI-A)-1x(t0+)+(λαI-A)-1L{f(u+t0,xu+t0,Bx(u+t0))}.
By the inverse Laplace transform of (2.21), we get
(2.22)x(u+t0)=Eα,1(Auα)x(t0+)+∫0u(u-s)α-1Eα,α(A(u-s)α)f(s+t0,xs+t0,Bx(s+t0))ds.
Set u+t0=t, in (2.22), we have
(2.23)x(t)=Eα,1(A(t-t0)α)x(t0+)+∫0t-t0(t-t0-s)α-1Eα,α(A(t-t0-s)α)f(s+t0,xs+t0,Bx(s+t0))ds.
On simplification, we obtain
(2.24)x(t)=Eα,1(A(t-t0)α)x(t0+)+∫t0t(t-θ)α-1Eα,α(A(t-θ)α)f(θ,xθ,Bx(θ))dθ.
Set Tα(t)=Eα,1(Atα) and Sα(t)=tα-1Eα,α(Atα) in (2.24). We have
(2.25)x(t)=Tα(t-t0)x(t0+)+∫t0tSα(t-θ)f(θ,xθ,Bx(θ))dθ.
This completes the proof of the lemma.Now, we give the definition of a mild solution of the system (1.1) by investigating the classical solution of the system (1.1).Definition 2.7.
A functionx:(-∞,T]→X is called a mild solution of (1.1) if the following holds: x0=ϕ∈𝔅h on (-∞,0] with ϕ(0)=0;Δx∣t=tk=Ik(x(tk-)),k=1,…,m, the restriction of x(·) to the interval [0,T)∖{t1,…,tm} is continuous and satisfies the following integral equation:
(2.26)x(t)={ϕ(t),t∈(-∞,0],∫0tSα(t-s)f(s,xs,Bx(s))ds,t∈[0,t1],Tα(t-t1)(x(t1-)+I1(x(t1-)))+∫t1tSα(t-s)f(s,xs,Bx(s))ds,t∈(t1,t2],⋮Tα(t-tm)(x(tm-)+Im(x(tm-)))+∫tmtSα(t-s)f(s,xs,Bx(s))ds,t∈(tm,T].Now, we introduce the following assumptions:(H1)
there existμ1,μ2>0 such that(2.27)‖f(t,φ,x)-f(t,ψ,y)‖X≤μ1‖φ-ψ‖Bh+μ2‖x-y‖X,t∈I,(φ,ψ)∈Bh2,x,y∈X.(H2)
for eachk=1,…,m, there exists ρk>0 such that(2.28)‖Ik(x)-Ik(y)‖X≤ρk‖x-y‖X,∀x,y∈X.(H3)
(2.29)max⁡1≤i≤m{M̃T(1+ρi)+M̃STαα(μ1C1*+μ2B*)}<1,
where C1*=sup0<τ<TC1(τ) and B*=supt∈[0,t]∫0tK(t,s)ds<∞ and(2.30)M̃T=sup⁡0≤t≤T‖Tα(t)‖L(X),M̃S=sup⁡0≤t≤TCeωt(1+t1-α).Ifα∈(0,1) and A∈Aα(θ0,ω0), then for any x∈X and t>0, we have ∥Tα(t)∥L(X)≤Meωt and ∥Sα(t)∥L(X)≤Ceωt(1+tα-1),t>0,ω>ω0. Hence, we have ∥Tα(t)∥L(X)≤M̃T,∥Sα(t)∥L(X)≤tα-1M̃S. See [1] for details.


## 3. The Main Results

Our first result is based on the Banach contraction principle.Theorem 3.1.
Assume that the assumptions (H1)–(H3) are satisfied. IfA∈𝔸α(θ0,ω0), then the system (1.1) has a unique mild solution.Proof.
Consider the operatorN:𝔅h′→𝔅h′ defined by
(3.1)(Nx)(t)={ϕ(t),t∈(-∞,0],∫0tSα(t-s)f(s,xs,Bx(s))ds,t∈[0,t1],Tα(t-t1)(x(t1-)+I1(x(t1-)))+∫t1tSα(t-s)f(s,xs,Bx(s))ds,t∈(t1,t2],⋮Tα(t-tm)(x(tm-)+Im(x(tm-)))+∫tmtSα(t-s)f(s,xs,Bx(s))ds,t∈(tm,T].
Let y(·):(-∞,T]→X be the function defined by
(3.2)y(t)={ϕ(t),t∈(-∞,0]0,t∈J,
then y0=ϕ. For each z∈C(J,ℝ) with z(0)=0, we denote by z¯ the function defined by
(3.3)z¯(t)={0,t∈(-∞,0];z(t),t∈J.
If x(·) satisfies (2.26), then we can decompose x(·) as x(t)=y(t)+z¯(t) for t∈J, which implies xt=yt+z¯t for t∈J, and the function z(·) satisfies
(3.4)z(t)={∫0tSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈[0,t1],Tα(t-t1)[y(t1-)+z¯(t1-)+I1((y(t1-))+z¯(t1-))]+∫t1tSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈(t1,t2],⋮Tα(t-tm)[y(tm-)+z¯(tm-)+Im((y(tm-))+z¯(tm-))]+∫tmtSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈(tm,T].
Set 𝔅h′′={z∈𝔅h′ such that z0=0} and let ∥·∥𝔅h′′ be the seminorm in 𝔅h′′ defined by
(3.5)‖z‖Bh′′=sup⁡t∈J‖z(t)‖X+‖z0‖Bh=sup⁡t∈J‖z(t)‖X,z∈Bh′′,
thus (𝔅h′′,∥·∥𝔅h′′) is a Banach space. We define the operator P:𝔅h′′→𝔅h′′ by
(3.6)(Pz)(t)={∫0tSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈[0,t1],Tα(t-t1)[z(t1-)+I1(z(t1-))]+∫t1tSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈(t1,t2],⋮Tα(t-tm)[z(tm-)+Im(z(tm-))]+∫tmtSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈(tm,T].
It is clear that the operator N has a unique fixed-point if and only if P has a unique fixed-point. To prove that P has a unique fixed-point, let z,z*∈𝔅h′′, then for all t∈[0,t1]. We have
(3.7)‖P(z)(t)-P(z*)(t)‖X≤∫0t‖Sα(t-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))-f(s,ys+z¯s*,B(y(s)+z¯*(s)))‖Xds≤M̃S∫0t(t-s)α-1[μ1‖z¯s-z¯s*‖Bh+μ2‖B(y(s)+z¯(s))-B(y(s)+z¯*)‖X]ds≤M̃Sα(μ1C1*+μ2B*)Tα‖z-z*‖Bh′′.
For t∈(t1,t2], we have
(3.8)‖P(z)(t)-P(z*)(t)‖X≤‖Tα(t-t1)‖L(X)[‖z(t1-)-z*(t1-)‖X+‖I1(z(t1-))-I1(z*(t1-))‖X]+∫t1t‖Sα(t-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))-f(s,ys+z¯s*,B(y(s)+z¯*(s)))‖X≤M̃T[‖z(t1-)-z*(t1-)‖X+ρ1‖z(t1-)-z*(t1-)‖X]+M̃S∫t1t(t-s)α-1[μ1‖z¯s-z¯s*‖Bh+μ2‖B(y(s)+z¯(s))-B(y(s)+z¯*)‖X]ds≤M̃T(1+ρ1)‖z-z*‖Bh′′+M̃Sα(μ1C1*+μ2B*)Tα‖z-z*‖Bh′′.
Similarly, when t∈(ti,ti+1],i=2,…,m, we get
(3.9)‖P(z)(t)-P(z*)(t)‖X≤M̃T(1+ρi)‖z-z*‖Bh′′+M̃Sα(μ1C1*+μ2B*)Tα‖z-z*‖Bh′′.
Thus, for all t∈[0,T], we have
(3.10)‖P(z)-P(z*)‖Bh′′≤max⁡1≤i≤m{M̃T(1+ρi)+M̃Sα(μ1C1*+μ2B*)Tα}‖z-z*‖Bh′′.
Hence, P is a contraction map, and therefore it has an unique fixed-point z∈𝔅h′′, which is a mild solution of (1.1) on (-∞,T]. This completes the proof of the theorem.The second result is established using the following Krasnoselkii’s fixed-point theorem.Theorem 3.2.
LetB be a closed-convex and nonempty subset of a Banach space X. Let P and Q be two operators such that (i)Px+Qy∈B whenever x,y∈B, (ii)P is compact and continuous; (iii)Q is a contraction mapping, then there exists z∈B such that z=Pz+Qz.Now, we make the following assumptions:(H4)
f:J×𝔅h×X→X is continuous, and there exist two continuous functions μ1,μ2:J→(0,∞) such that(3.11)‖f(t,ψ,x)‖X≤μ1(t)‖ψ‖Bh+μ2(t)‖x‖X,(t,ψ,x)∈J×B×X.(H5)
the functionIk:X→X is continuous, and there exists Ω>0 such that
(3.12)Ω=max⁡1≤k≤m,x∈Br{‖Ik(x)‖X}.Before going further, we need the following lemma.Lemma 3.3 (see Lemma  3.2 in [7]).
Let(3.13)C1*=sup⁡0<τ<TC1(τ),C2*=sup⁡0<τ<TC2(τ),μ1*=sup⁡0<τ<Tμ1(τ),μ2*=sup⁡0<τ<Tμ2(τ)
then for any s∈J,
(3.14)μ1(s)‖ys+z¯s‖Bh+μ2(s)‖B(y(s)+z¯(s))‖X≤μ1*[C2*‖ϕ‖Bh+C1*sup⁡0≤τ≤s‖z(τ)‖X]+μ2*∫0sK(s,τ)‖z(τ)‖Xdτ.
If ∥z∥X<r,r>0, then
(3.15)μ1(s)‖ys+z¯s‖Bh+μ2(s)‖B(y(s)+z¯(s))‖X≤μ1*[C2*‖ϕ‖Bh+C1*r]+μ2*rB*=λ.Theorem 3.4.
Suppose that the assumptions (H1), (H4), (H5) are satisfied with(3.16)[M̃Sα(μ1C1*+μ2B*)Tα]<1,
then the impulsive problem (1.1) has at least one mild solution on (-∞,T].Proof.
Chooser≥[M̃T(r+Ω)+(M̃STαλ/α)] and consider Br={z∈𝔅h′′:∥z∥𝔅h′′≤r}, then Br is a bounded, closed-convex subset in 𝔅h′′.
LetΓ1:Br→Br and Γ2:Br→Br be defined as
(3.17)(Γ1z)(t)={0,t∈[0,t1],Tα(t-t1)[z(t1-)+I1(z(t1-))],t∈(t1,t2],⋮Tα(t-tm)[z(tm-)+Im(z(tm-))],t∈(tm,T],(3.18)(Γ2z)(t)={∫0tSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈[0,t1],∫t1tSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈(t1,t2],⋮∫tmtSα(t-s)f(s,ys+z¯s,B(y(s)+z¯(s)))ds,t∈(tm,T].
Step 1.
Letz,z*∈Br, then show that Γ1z+Γ2z*∈Br, for t∈[0,t1], we have
(3.19)‖(Γ1z)(t)+(Γ2z*)(t)‖X≤∫0t‖Sα(t-s)‖L(X)‖f(s,ys+z¯s*,B(y(s)+z¯*(s)))‖Xds≤M̃S∫0t(t-s)α-1[μ1(s)‖ys+z¯s*‖Bh′′+μ2(s)‖B(y(s)+z¯*(s))‖X]ds,
and by using Lemma 3.3, we conclude that
(3.20)‖(Γ1z)+(Γ2z*)‖Bh′′≤M̃SλTαα<r.
Similarly, when t∈(ti,ti+1], i=1,…,m, we have the estimate
(3.21)‖(Γ1z)(t)+(Γ2z*)(t)‖X≤‖Tα(t-ti)[z(ti-)+Ii(z(ti-))]‖X+∫tit‖Sα(t-s)‖L(X)‖f(s,ys+z¯s*,B(y(s)+z¯*(s)))‖Xds≤M̃T(‖z‖Bh′′+‖Ii(z(ti-))‖X)+∫tit‖Sα(t-s)‖L(X)[μ1(s)‖ys+z¯s*‖Bh′′+μ2(s)‖B(y(s)+z¯*(s))‖X]ds≤M̃T(r+Ω)+M̃STαλα<r,
which implies that ∥Γ1z+Γ2z∥𝔅h′′≤r.Step 2.
We will show that the mapping(Γ1z)(t) is continuous on Br. For this purpose, let {zn}n=1∞ be a sequence in Br with limzn→z∈Br, then for t∈(ti,ti+1], i=0,1,…,m, we have
(3.22)‖(Γ1zn)(t)-(Γ1z)(t)‖X≤‖Tα(t-ti)‖L(X)[‖zn(ti-)-z(ti-)‖X+‖Ii(zn(ti-))-Ii(z(ti-))‖X].
Since the functions Ii,i=1,2,…,m are continuous, hence limn→∞Γ1zn=Γ1z in Br which implies that the mapping Γ1 is continuous on Br.Step 3.
Uniform boundedness of the map(Γ1z)(t) is an implication of the following inequality: for t∈(ti,ti+1], i=0,1,…,m, we have
(3.23)‖(Γ1z)(t)‖X≤‖Tα(t-ti)‖L(X)[‖z(ti-)‖X+‖Ii(z(ti-))‖X]≤M̃T(r+Ω).Step 4.
To show that the map (3.17) is equicontinuous, we proceed as follows. Let u,v∈(ti,ti+1], ti≤u<v≤ti+1, i=0,1,…,m, z∈Br, then we obtain
(3.24)‖(Γ1z)(v)-(Γ1z)(u)‖X≤‖Tα(v-ti)-Tα(u-ti)‖L(X)‖z(ti-)+Ii(z(ti-))‖X≤(r+Ω)‖Tα(v-ti)-Tα(u-ti)‖L(X).
Since Tα is strongly continuous, the continuity of the function t↦∥T(t)∥ allows us to conclude that limu→v∥Tα(v-ti)-Tα(u-ti)∥L(X)=0, which implies that Γ1(Br) is equicontinuous. Finally, combining Step 1 to Step 4 together with Ascoli’s theorem, we conclude that the operator Γ1 is compact.
Now, it only remains to show that the mapΓ2 is a contraction mapping. Let z,z*∈Br and t∈(ti,ti+1], i=0,1,…,m, then we have
(3.25)‖(Γ2z)(t)-(Γ2z*)(t)‖X≤∫tit‖Sα(t-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))-f(s,ys+z¯s*,B(y(s)+z¯*(s)))‖Xds≤M̃S∫tit(t-s)α-1[μ1‖z¯s-z¯s*‖Bh′′+μ2‖B(y(s)+z¯(s))-B(y(s)+z¯*(s))‖X]ds≤M̃Sα(μ1C1*+μ2B*)Tα‖z-z*‖Bh′′,
since (M̃S/α)(μ1C1*+μ2B*)Tα<1, which implies that Γ2 is a contraction mapping. Hence, by the Krasnoselkii fixed-point theorem, we can conclude that the problem (1.1) has at least one solution on (-∞,T]. This completes the proof of the theorem.Our last result is based on the following Schaefer’s fixed-point theorem.Theorem 3.5.
LetP be a continuous and compact mapping on a Banach space X into itself, such that the set {x∈X:x=νPxforsome0≤ν≤1} is bounded, then P has a fixed-point.Lemma 3.6 (see [5]).
Letv:[0,T]→[0,∞) be a real function, w(·) is nonnegative and locally integrable function on [0,T], and there are constants a>0 and 0<α<1 such that
(3.26)v(t)≤w(t)+a∫0tv(s)(t-s)αds.
Then there exists a constant K(α) such that
(3.27)v(t)≤w(t)+aK(α)∫0tw(s)(t-s)αds,foreveryt∈[0,T].Theorem 3.7.
Assume that the assumptions (H4)-(H5) are satisfied, and ifA∈Aα(θ0,ω0) and M̃T<1, then the impulsive problem (1.1) has at least one mild solution on (-∞,T].Proof.
We define the operatorP:𝔅h′′→𝔅h′′ as in Theorem  3.3. Note that P is well defined in 𝔅h′′.We complete the proof in the following steps.Step 1.
For the continuity of the mapP, let {zn} be a sequence in 𝔅h′′ such that zn→z in 𝔅h′′. Since the function f is continuous on J×𝔅h×X, This implies that
(3.28)f(s,ys+z¯sn,B(y(s)+z¯n(s)))⟶f(s,ys+z¯s,B(y(s)+z¯(s)))asn⟶∞.
Now, for every t∈[0,t1], we get
(3.29)‖Pzn(t)-Pz(t)‖X≤∫0t‖Sα(t-s)‖L(X)‖f(s,ys+z¯sn,B(y(s)+z¯n(s)))-f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds≤M̃STααɛ,
where ɛ>0,ɛ→0 as n→∞. Moreover, we have
(3.30)‖Pzn(t)-Pz(t)‖X≤M̃T[‖zn(ti-)-z(ti-)‖X+‖Ii(zn(ti-))-Ii(z(ti-))‖X]+∫tit‖Sα(t-s)‖L(X)‖f(s,ys+z¯sn,B(y(s)+z¯n(s)))-f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds≤M̃T[‖zn(ti-)-z(ti-)‖X+‖Ii(zn(ti-))-Ii(z(ti-))‖X]+M̃STααɛ,
where ɛ>0,ɛ→0 as n→∞, for all t∈(ti,ti+1],i=1,…,m. The impulsive functions Ik,k=1,…,m are continuous, then we get
(3.31)lim⁡n→∞‖Pzn-Pz‖Bh′′=0.
This implies that P is continuous.Step 2.
P maps bounded sets into bounded sets in 𝔅h′′. To prove that for any r>0, there exists a γ>0 such that for each z∈Br={z∈𝔅h′′:∥z∥𝔅h′′≤r}, then we have ∥Pz∥𝔅h′′≤γ, then for any z∈Br,t∈[0,t1], we have
(3.32)‖Pz(t)‖X≤∫0t‖Sα(t-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds≤M̃S∫0t(t-s)α-1[μ1(s)‖ys+z¯s‖Bh+μ2(s)‖B(y(s)+z¯(s))‖X]ds.
Using Lemma 3.3, we obtain ∥Pz(t)∥X≤M̃S(Tα/α)λ. Similarly, we have
(3.33)‖Pz(t)‖X≤M̃T(r+Ω)+M̃STααλ,t∈(ti,ti+1],i=1,…,m.
This implies that
(3.34)‖Pz‖Bh′′≤M̃T(r+Ω)+M̃STααλ=γ,t∈[0,T].Step 3.
We will prove thatP(Br) is equicontinuous. Let u,v∈[0,t1], with u<v, we have
(3.35)‖Pz(v)-Pz(u)‖X≤∫0u‖Sα(v-s)-Sα(u-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds+∫uv‖Sα(v-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds≤Q1+Q2,
where
(3.36)Q1=∫0u‖Sα(v-s)-Sα(u-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds≤λ∫0u‖Sα(v-s)-Sα(u-s)‖L(X)ds.
Since ∥Sα(v-s)-Sα(u-s)∥L(X)≤2M̃S(t1-s)α-1∈L1(I,ℝ+) for s∈[0,t1] and Sα(v-s)-Sα(u-s)→0 as u→v, Sα is strongly continuous. This implies that limu→vQ1=0,
(3.37)Q2=∫uv‖Sα(v-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds≤λM̃S(v-u)αα.
Hence, limu→vQ2=0. Similarly, for u,v∈(ti,ti+1], with u<v,i=1,…,m, we have
(3.38)‖Pz(v)-Pz(u)‖X≤‖Tα(v-ti)-Tα(u-ti)‖L(X)[‖z(ti-)‖X+‖Ii(z(ti-))‖X]+Q1+Q2.
Since Tα is also strongly continuous, so Tα(v-ti)-Tα(u-ti)→0 as u→v. Thus, from the above inequalities, we have limu→v∥Pz(v)-Pz(u)∥X=0. So, P(Br) is equicontinuous. Finally, combining Step 1 to Step 3 with Ascoli’s theorem, we conclude that the operator P is compact.Step 4.
We show that the set(3.39)E={z∈Bh′′suchthatz=νPzforsome0<ν<1}
is bounded. Let z∈E, then z(t)=νPz(t) for some 0<ν<1. Then for each t∈[0,t1], we have
(3.40)‖z(t)‖X≤ν∫0t‖Sα(t-s)‖L(X)‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds≤νM̃S∫0t(t-s)α-1‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds,
for t∈(ti,ti+1],i=1,…,m, we get
(3.41)‖z(t)‖X≤ν[‖Tα(t-ti)‖L(X)(‖z(ti-)‖X+‖Ii(z(ti-))‖X)+∫tit‖Sα(t-s)‖L(X)×‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds]≤[M̃T‖z(ti-)‖X+M̃TΩ+M̃S∫tit(t-s)α-1‖f(s,ys+z¯s,B(y(s)+z¯(s)))‖Xds].
then for all t∈[0,T], we have
(3.42)‖z(t)‖X≤M̃TΩ1-M̃T+M̃S1-M̃T∫0t(t-s)α-1[μ1(s)‖ys+z¯s‖Bh+μ2(s)‖B(y(s)+z¯(s))‖X]ds≤M̃TΩ1-M̃T+M̃Sμ1*C2*‖ϕ‖BhTαα(1-M̃T)+M̃S1-M̃T(μ1*C1*+μ2*B*)∫0t(t-s)α-1sup⁡0≤τ≤s‖z(τ)‖Xds≤ω1+ω2∫0t(t-s)α-1sup⁡0≤τ≤s‖z(τ)‖Xds,
where ω1=M̃TΩ/(1-M̃T)+M̃Sμ1*C2*∥ϕ∥𝔅hTα/α(1-M̃T) and ω2=(M̃S/1-M̃T)(μ1*C1*+μ2*B*). Let τ*∈[0,s] be such that sup0≤τ≤s∥z(τ)∥X=∥z(τ*)∥X,0≤s≤t. If τ*∈[0,t], then (3.43) can be written as
(3.43)‖z(t)‖X≤ω1+ω2∫0t(t-s)α-1‖z(s)‖Xds.
Using Lemma 3.6, there exists a constant K(α), and (3.43) becomes
(3.44)‖z(t)‖X≤ω1[1+∫0tK(α)(t-s)α-1ds]≤ω1[1+K(α)Tαα].
As a consequence of Schaefer’s fixed-point theorem, we deduce that P has a fixed-point on (-∞,T]. This completes the proof of the theorem.


## 4. Applications

To illustrate the application of the theory, we consider the following partial integro-differential equation with fractional derivative of the form(4.1)Dtqu(t,x)=∂2∂x2u(t,x)+∫-∞tH(t,x,s-t)Q(u(s,x))ds+∫0tk(s,t)e-u(s,x)ds,x∈[0,π],t∈[0,b],t≠tk,u(t,0)=0=u(t,π),t≥0,u(t,x)=ϕ(t,x),t∈(-∞,0],x∈[0,π],Δu(ti)(x)=∫-∞tiqi(ti-s)u(s,x)ds,x∈[0,π],
where Dtq is Caputo’s fractional derivative of order 0<q<1, 0<t1<t2<⋯<tn<b are prefixed numbers, and ϕ∈𝔅h. Let X=L2[0,π], and define the operator A:D(A)⊂X→X by Aw=w′′ with the domain D(A):={w∈X:w, w′ are absolutely continuous, w′′∈X,w(0)=0=w(π)}, then(4.2)Aw=∑n=1∞n2(w,wn)wn,w∈D(A),
where wn(x)=2/πsin(nx),n∈ℕ is the orthogonal set of eigenvectors of A. It is well known that A is the infinitesimal generator of an analytic semigroup (T(t))t≥0 in X and is given by(4.3)T(t)w=∑n=1∞e-n2t(w,wn)wn,∀w∈X,andeveryt>0.
From these expressions, it follows that (T(t))t≥0 is a uniformly bounded compact semigroup, so that R(λ,A)=(λ-A)-1 is a compact operator for all λ∈ρ(A),thatis,A∈𝔸α(θ0,ω0). Let h(s)=e2s,s<0, then l=∫-∞0h(s)ds=1/2, and define(4.4)‖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 u(t)(x)=u(t,x),(4.5)f(t,ϕ,Bu(t))(x)=∫-∞0H(t,x,θ)Q(ϕ(θ)(x))dθ+Bu(t)(x),
where Bu(t)(x)=∫0tk(s,t)e-u(s,x)ds. Then with these settings, the above equation (4.1) can be written in the abstract form of the equations (1.1). The functions H, k, and Q are satisfying some conditions, and qi:ℝ→ℝ are continuous and di=∫-∞0h(s)qi2(s)ds<∞ for i=1,2,…,n. Suppose further that (1)
the functionH(t,x,θ) is continuous in [0,b]×[0,π]×(-∞,0] and H(t,x,θ)≥0, ∫-∞0H(t,x,θ)dθ=p1(t,x)<∞,(2)
the functionQ(·) is continuous and for each (θ,y)∈(-∞,0]×[0,π], 0≤Q(u(θ)(x))≤(∫-∞0e2s∥u(s,·)∥L2ds).Now, we can see that(4.6)‖f(t,ϕ,Bu(t))‖L2=[∫0π(∫-∞0h(t,x,θ)Q(ϕ(θ)(x))dθ+Bu(t)(x)dx)2]1/2≤[∫0π(∫-∞0h(t,x,θ)(∫-∞0e2s‖ϕ(s)(⋅)‖L2ds)dθ)2dx]1/2+[∫0π(∫0tk(s,t)e-u(s,x)ds)2dx]1/2≤[∫0π(∫-∞0h(t,x,θ)(∫-∞0e2ssup⁡s∈[θ,0]‖ϕ(s)‖L2ds)dθ)2dx]1/2+[∫0π(∫0tk(s,t)e-u(s,x)ds)2dx]1/2≤[∫0π(∫-∞0h(t,x,θ)dθ)2dx]1/2‖ϕ‖Bh+‖Bu(t)‖L2≤[∫0π(p(t,x))2dx]1/2‖ϕ‖Bh+‖Bu(t)‖L2≤p¯(t)‖ϕ‖Bh+‖Bu(t)‖L2.If we takeμ1(t)=p¯(t) and μ2(t)=1, hence f satisfies (H4), and similarly we can show that Ik satisfy (H5). All conditions of Theorem 3.7 are now fulfilled, so we deduce that the system (4.1) has a mild solution on (-∞,T].



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*Source: 793023-2011-10-15.xml*