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

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.

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 formΔ𝐷𝑥^𝛼𝑡𝑥𝑡(_𝑘𝑡)=𝐴𝑥(𝑡)+𝑓𝑡,𝑥_𝑡,𝐵𝑥(𝑡),𝑡∈𝐽=[0,𝑇],𝑡≠𝑡_𝑘,𝑥(𝑡)=𝜙=(𝑡𝐼)_𝑘,𝑥𝜙(𝑡𝑡^−𝑘)∈,𝔅_ℎ𝑘=1,2,…,𝑚,,(1.1) where 𝑇>0,0<𝛼<1,𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of an 𝛼-resolvent family 𝑆_𝛼(𝑡)_𝑡≥0, the solution operator 𝑇_𝛼(𝑡)_𝑡≥0 is defined on a complex Banach space 𝑋, 𝐷^𝛼 is the Caputo fractional derivative, 𝑓∶𝐽×𝔅_ℎ×𝑋→𝑋 is a given function, and 𝔅_ℎ is a phase space defined in Section 2. Here, 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, 𝐼_𝑘∈𝐶(𝑋,𝑋), (𝑘=1,2,…,𝑚), are bounded functions, Δ𝑥(𝑡_𝑘)=𝑥(𝑡^+𝑘)−𝑥(𝑡^−𝑘),𝑥(𝑡^+𝑘)=lim_ℎ→0𝑥(𝑡_𝑘+ℎ) and 𝑥(𝑡^−𝑘)=lim_ℎ→0𝑥(𝑡_𝑘−ℎ) represent the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively.

We assume that 𝑥_𝑡∶(−∞,0]→𝑋, 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠), 𝑠≤0, belongs to an abstract phase space 𝔅_ℎ. The term 𝐵𝑥(𝑡) is given by 𝐵𝑥(𝑡)=∫^𝑡0𝐾(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 (𝑆_𝛼(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 delay𝐷^𝑞𝑥(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡∈[0,𝑇],0<𝑞<1,𝑥(𝑡)=𝜙(𝑡),𝑡∈]−∞,0],(1.2) where 𝐷^𝑞 is Riemann-Liouville fractional derivative, 𝜙∈𝔅_ℎ, 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. Let 𝑋 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, 𝐵_𝑟(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 space 𝔅_ℎ in a convenient way (for details see [3]). Let ℎ∶(−∞,0]→(0,∞) be a continuous function with 𝑙=∫^0−∞ℎ(𝑡)𝑑𝑡<∞. For any 𝑎>0, we define

𝔅={𝜓∶[−𝑎,0]→𝑋 such that 𝜓(𝑡) is bounded and measurable} and equip the space 𝔅 with the norm ‖𝜓‖_[−𝑎,0]=sup_{[]𝑠∈−𝑎,0}||𝜓(𝑠)||,∀𝜓∈𝔅.(2.1) Let us define by 𝔅_ℎ=𝜓𝜓(0∶)=(−0∞a,n0d]⟶_0−∞𝑋,suchthatforany𝑐>0,𝜓∣_[−𝑐,0]∈𝔅withℎ(𝑠)‖𝜓‖_[𝑠,0]𝑑𝑠<∞.(2.2)

If 𝔅_ℎ is endowed with the norm ‖𝜓‖_𝔅ℎ=^0−∞ℎ(𝑠)‖𝜓‖_[𝑠,0]𝑑𝑠,∀𝜓∈𝔅_ℎ,(2.3) then it is known that (𝔅_ℎ,‖⋅‖_𝔅ℎ) is a Banach space.

Now, we consider the space 𝔅^ℎ=𝑥𝑥∶𝑡(−∞,𝑇]⟶𝑋suchthat𝑥∣_𝐽𝑘∈𝐶𝐽_𝑘,𝑋andthereexist_+𝑘and𝑥𝑡^−𝑘with𝑥𝑡_𝑘=𝑥𝑡^−𝑘,𝑥₀=𝜙∈𝔅_ℎ,𝑘=1,…,𝑚,(2.4) where 𝑥∣_𝐽𝑘 is the restriction of 𝑥 to 𝐽_𝑘=(𝑡_𝑘,𝑡_𝑘+1],𝑘=0,1,2,…,𝑚. The function ‖⋅‖_𝔅^′ℎ to be a seminorm in 𝔅^ℎ, it is defined by ‖𝑥‖_𝔅^′ℎ=sup||𝑥(𝑠)||∶𝑠∈[0,𝑇]+‖𝜙‖_𝔅ℎ,𝑥∈𝔅^ℎ.(2.5) If 𝑥∶]−∞,𝑇]→𝑋,𝑇>0, is such that 𝑥₀∈𝔅_ℎ, then for all 𝑡∈𝐽, the following conditions hold:(1)𝑥_𝑡∈𝔅_ℎ, (2)‖𝑥_𝑡‖_𝔅ℎ≤𝐶₁(𝑡)sup_0<𝑠<𝑡‖𝑥(𝑠)‖+𝐶₂(𝑡)‖𝑥₀‖_𝔅ℎ, (3)‖𝑥(𝑡)‖≤𝐻‖𝑥_𝑡‖_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 expansion𝐸_𝛼,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ(𝛼𝑘+𝛽)=21𝜋𝑖_𝐶𝜇^𝛼−𝛽𝑒^𝜇𝜇^𝛼−𝑧𝑑𝜇,𝛼,𝛽>0,𝑧∈ℂ,(2.6) where 𝐶 is a contour which starts and ends at −∞ and encircles the disc |𝜇|≤|𝑧|^1/2 counter clockwise. For short, 𝐸_𝛼(𝑧)=𝐸_𝛼,1(𝑧). It is an entire function which provides a simple generalization of the exponent function: 𝐸₁(𝑧)=𝑒^𝑧 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^∞0𝑒^−𝜆𝑡𝑡^𝛽−1𝐸_𝛼,𝛽(𝜔𝑡^𝛼)𝑑𝑡=𝜆^𝛼−𝛽𝜆^𝛼−𝜔,Re𝜆>𝜔^1/𝛼,𝜔>0,(2.7) see [12] for more details.

Definition 2.1. A closed and linear operator 𝐴 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: ((21))𝜌‖𝑅(𝐴(𝜆),⊂𝐴)‖_(𝜃,𝜔)=_𝐿(𝑋)𝜆∈𝐶∶𝜆≠𝜔,||arg(𝜆−𝜔)||<𝜃,≤𝑀||𝜆−𝜔||,𝜆∈_(𝜃,𝜔).(2.8) Sectorial operators are well studied in the literature. For details see [13].

Definition 2.2 (see Definition  2.3 in [10]). Let 𝐴 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 𝑆_𝛼∶𝑅₊→𝐿(𝑋) such that {𝜆^𝛼∶Re𝜆>𝜔}⊂𝜌(𝐴) and (𝜆^𝛼𝐼−𝐴)⁻¹𝑥=^∞0𝑒^−𝜆𝑡𝑆_𝛼(𝑡)𝑥𝑑𝑡,Re𝜆>𝜔,𝑥∈𝑋,(2.9) in this case, 𝑆_𝛼(𝑡) is called the 𝛼-resolvent family generated by 𝐴.

Definition 2.3 (see Definition  2.1 in [4]). Let 𝐴 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 𝑆_𝛼∶𝑅₊→𝐿(𝑋) such that {𝜆^𝛼∶Re𝜆>𝜔}⊂𝜌(𝐴) and 𝜆^𝛼−1(𝜆^𝛼𝐼−𝐴)⁻¹𝑥=^∞0𝑒^−𝜆𝑡𝑆_𝛼(𝑡)𝑥𝑑𝑡,Re𝜆>𝜔,𝑥∈𝑋,(2.10) in this case, 𝑆_𝛼(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 order 𝛼>0, of a function 𝑓∶ℝ₊→ℝ and 𝑓∈𝐿¹(ℝ₊,𝑋), is defined by 𝐼⁰𝑓(𝑡)=𝑓(𝑡),𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡0(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠,𝛼>0,𝑡>0,(2.11) where Γ(⋅) is the Euler gamma function. The Laplace transform of a function 𝑓∈𝐿¹(ℝ₊,𝑋) is defined by 𝑓(𝜆)=^∞0𝑒^−𝜆𝑡𝑓(𝑡)𝑑𝑡,Re(𝜆)>𝜔,(2.12) provided the integral is absolutely convergent for Re(𝜆)>𝜔.

Definition 2.5. Caputo’s derivative of order 𝛼 for a function 𝑓∶[0,∞)→ℝ is defined as 𝐷^𝛼𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡0(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)𝑑𝑠=𝐼^𝑛−𝛼𝑓^(𝑛)(𝑡),(2.13) for 𝑛−1≤𝛼<𝑛,𝑛∈𝑁. If 0<𝛼≤1, then 𝐷^𝛼𝑡𝑓(𝑡)=1Γ(1−𝛼)^𝑡0(𝑡−𝑠)^−𝛼𝑓⁽¹⁾(𝑠)𝑑𝑠.(2.14) 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 𝐿𝐷^𝛼𝑡𝑓(𝑡);𝜆=𝜆^𝛼𝑓(𝜆)−^𝑛−1_𝑘=0𝜆^{𝛼−𝑘−1}𝑓^(𝑘)(0);𝑛−1≤𝛼<𝑛.(2.15)

Lemma 2.6. If 𝑓 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 𝐷^𝛼𝑡𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓𝑥𝑡,(𝑡𝑥)_𝑡,=𝐵𝜙𝑥((𝑡𝑡)),,𝑡𝑡≤>𝑡₀𝑡₀,𝑡₀∈𝑅,0<𝛼<1,,(2.16) is given by 𝑥(𝑡)=𝑇_𝛼𝑡−𝑡₀𝑥𝑡⁺⁰+^𝑡𝑡₀𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐵𝑥(𝑠)𝑑𝑠,(2.17) where 𝑇_𝛼(𝑡)=𝐸_𝛼,1(𝐴𝑡^𝛼)=12𝜋𝑖𝐵_𝑟𝑒^𝜆𝑡𝜆^𝛼−1𝑆𝜆^𝛼−𝐴𝑑𝜆,_𝛼(𝑡)=𝑡^𝛼−1𝐸_𝛼,𝛼(𝐴𝑡^𝛼)=12𝜋𝑖𝐵_𝑟𝑒^𝜆𝑡1𝜆^𝛼−𝐴𝑑𝜆,(2.18)𝐵_𝑟 denotes the Bromwich path. 𝑆_𝛼(𝑡) is called the 𝛼-resolvent family, and 𝑇_𝛼(𝑡) is the solution operator, generated by 𝐴.

Proof. Let 𝑡−𝑡₀=𝑢, then we get 𝐷^𝛼𝑢𝑥𝑢+𝑡₀=𝐴𝑥𝑢+𝑡₀+𝑓𝑢+𝑡₀,𝑥_𝑢+𝑡0,𝐵𝑥𝑢+𝑡₀,𝑢>0.(2.19) Taking the Laplace transform of (2.19), we have 𝜆^𝛼𝐿𝑥𝑢+𝑡₀−𝜆^𝛼−1𝑥𝑡⁺⁰=𝐴𝐿𝑥𝑢+𝑡₀+𝐿𝑓𝑢+𝑡₀,𝑥_𝑢+𝑡0,𝐵𝑥𝑢+𝑡₀.(2.20) Since (𝜆^𝛼𝐼−𝐴)⁻¹ exists, that is, 𝜆^𝛼∈𝜌(𝐴), from (2.20), we obtain 𝐿𝑥=𝑢𝜆+𝑡_0𝛼−1(𝜆^𝛼𝐼−𝐴)⁻¹𝑥𝑡⁺⁰+(𝜆^𝛼𝐼−𝐴)⁻¹𝐿𝑓𝑢+𝑡₀,𝑥_𝑢+𝑡0,𝐵𝑥𝑢+𝑡₀.(2.21) By the inverse Laplace transform of (2.21), we get 𝑥𝑢+𝑡₀=𝐸_𝛼,1(𝐴𝑢^𝛼)𝑥𝑡⁺⁰+^𝑢0(𝑢−𝑠)^𝛼−1𝐸_𝛼,𝛼(𝐴(𝑢−𝑠)^𝛼)𝑓𝑠+𝑡₀,𝑥_𝑠+𝑡0,𝐵𝑥𝑠+𝑡₀𝑑𝑠.(2.22) Set 𝑢+𝑡₀=𝑡, in (2.22), we have 𝑥(𝑡)=+𝐸∫_{𝛼,1𝑡−𝑡00}𝐴𝑡−𝑡₀^𝛼𝑥𝑡⁺⁰𝑡−𝑡₀−𝑠^𝛼−1𝐸_𝛼,𝛼𝐴𝑡−𝑡₀−𝑠^𝛼𝑓𝑠+𝑡₀,𝑥_𝑠+𝑡0,𝐵𝑥𝑠+𝑡₀𝑑𝑠.(2.23) On simplification, we obtain 𝑥(𝑡)=+𝐸_{𝛼,1𝑡𝑡0}𝐴𝑡−𝑡₀^𝛼𝑥𝑡⁺⁰(𝑡−𝜃)^𝛼−1𝐸_𝛼,𝛼(𝐴(𝑡−𝜃)^𝛼)𝑓𝜃,𝑥_𝜃,𝐵𝑥(𝜃)𝑑𝜃.(2.24) Set 𝑇_𝛼(𝑡)=𝐸_𝛼,1(𝐴𝑡^𝛼) and 𝑆_𝛼(𝑡)=𝑡^𝛼−1𝐸_𝛼,𝛼(𝐴𝑡^𝛼) in (2.24). We have 𝑥(𝑡)=𝑇_𝛼𝑡−𝑡₀𝑥𝑡⁺⁰+^𝑡𝑡₀𝑆_𝛼(𝑡−𝜃)𝑓𝜃,𝑥_𝜃,𝐵𝑥(𝜃)𝑑𝜃.(2.25) 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 function 𝑥∶(−∞,𝑇]→𝑋 is called a mild solution of (1.1) if the following holds: 𝑥₀=𝜙∈𝔅_ℎ on (−∞,0] with 𝜙(0)=0;Δ𝑥∣_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑥(𝑡^−𝑘)),𝑘=1,…,𝑚, the restriction of 𝑥(⋅) to the interval [0,𝑇)⧵{𝑡₁,…,𝑡_𝑚} is continuous and satisfies the following integral equation: 𝑥(𝑡)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝜙(𝑡),𝑡∈(−∞,0^𝑡0],𝑇_𝛼𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,⋮𝑇_𝛼𝑡−𝑡₁𝑥𝑡⁻¹+𝐼₁𝑥𝑡⁻¹+^𝑡𝑡₁𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,𝑡−𝑡_𝑚𝑥𝑡^−𝑚+𝐼_𝑚𝑥𝑡^−𝑚+^𝑡𝑡_𝑚𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑇.(2.26)

Now, we introduce the following assumptions: (H1)there exist 𝜇₁,𝜇₂>0 such that ‖𝑓(𝑡,𝜑,𝑥)−𝑓(𝑡,𝜓,𝑦)‖_𝑋≤𝜇₁‖𝜑−𝜓‖_𝔅ℎ+𝜇₂‖𝑥−𝑦‖_𝑋,𝑡∈𝐼,(𝜑,𝜓)∈𝔅^2ℎ,𝑥,𝑦∈𝑋.(2.27)(H2)for each 𝑘=1,…,𝑚, there exists 𝜌_𝑘>0 such that ‖‖𝐼_𝑘(𝑥)−𝐼_𝑘(𝑦)‖‖_𝑋≤𝜌_𝑘‖𝑥−𝑦‖_𝑋,∀𝑥,𝑦∈𝑋.(2.28)(H3)max_{1≤𝑖≤𝑚}𝑀^𝑇1+𝜌_𝑖+𝑀^𝑆𝑇^𝛼𝛼𝜇¹𝐶^∗1+𝜇₂𝐵^∗<1,(2.29) where 𝐶^∗1=sup_0<𝜏<𝑇𝐶₁(𝜏) and 𝐵^∗=sup_{𝑡∈[0,𝑡]}∫^𝑡0𝐾(𝑡,𝑠)𝑑𝑠<∞ and 𝑀_𝑇=sup_{0≤𝑡≤𝑇}‖‖𝑇_𝛼(𝑡)‖‖_𝐿(𝑋),𝑀_𝑆=sup_{0≤𝑡≤𝑇}𝐶𝑒^𝜔𝑡1+𝑡^1−𝛼.(2.30)

If 𝛼∈(0,1) and 𝐴∈𝐴^𝛼(𝜃₀,𝜔₀), then for any 𝑥∈𝑋 and 𝑡>0, we have ‖𝑇_𝛼(𝑡)‖_𝐿(𝑋)≤𝑀𝑒^𝜔𝑡 and ‖𝑆_𝛼(𝑡)‖_𝐿(𝑋)≤𝐶𝑒^𝜔𝑡(1+𝑡^𝛼−1),𝑡>0,𝜔>𝜔₀. Hence, we have ‖𝑇_𝛼(𝑡)‖_𝐿(𝑋)≤𝑀_𝑇,‖𝑆_𝛼(𝑡)‖_𝐿(𝑋)≤𝑡^𝛼−1𝑀_𝑆. 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. If 𝐴∈𝔸^𝛼(𝜃₀,𝜔₀), then the system (1.1) has a unique mild solution.

Proof. Consider the operator 𝑁∶𝔅^ℎ→𝔅^ℎ defined by (𝑁𝑥)(𝑡)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝜙(𝑡),𝑡∈(−∞,0^𝑡0],𝑇_𝛼𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,+𝑡−𝑡₁𝑥𝑡⁻¹+𝐼₁𝑥𝑡⁻¹_𝑡𝑡1⋮𝑇_𝛼𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,+𝑡−𝑡_𝑚𝑥𝑡^−𝑚+𝐼_𝑚𝑥𝑡^−𝑚_𝑡𝑡𝑚𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐵𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑇.(3.1) Let 𝑦(⋅)∶(−∞,𝑇]→𝑋 be the function defined by 𝑦(𝑡)=𝜙0,(𝑡),𝑡𝑡∈∈𝐽(−,∞,0](3.2) then 𝑦₀=𝜙. For each 𝑧∈𝐶(𝐽,ℝ) with 𝑧(0)=0, we denote by 𝑧 the function defined by 𝑧(𝑡)=0𝑧,(𝑡),𝑡𝑡∈∈𝐽(−.∞,0];(3.3) If 𝑥(⋅) satisfies (2.26), then we can decompose 𝑥(⋅) as 𝑥(𝑡)=𝑦(𝑡)+𝑧(𝑡) for 𝑡∈𝐽, which implies 𝑥_𝑡=𝑦_𝑡+𝑧_𝑡 for 𝑡∈𝐽, and the function 𝑧(⋅) satisfies 𝑧(𝑡)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩^𝑡0𝑇_𝛼𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,+𝑡−𝑡₁𝑦𝑡⁻¹+𝑧𝑡⁻¹+𝐼₁𝑦𝑡⁻¹+𝑧𝑡⁻¹_𝑡𝑡1⋮𝑇_𝛼𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,+𝑡−𝑡_𝑚𝑦𝑡^−𝑚+𝑧𝑡^−𝑚+𝐼_𝑚𝑦𝑡^−𝑚+𝑧𝑡^−𝑚_𝑡𝑡𝑚𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑇.(3.4) Set 𝔅^ℎ={𝑧∈𝔅^ℎ such that 𝑧₀=0} and let ‖⋅‖_𝔅^ℎ′′ be the seminorm in 𝔅^ℎ defined by ‖𝑧‖_𝔅^ℎ′′=sup_𝑡∈𝐽‖𝑧(𝑡)‖_𝑋+‖‖𝑧₀‖‖_𝔅ℎ=sup_𝑡∈𝐽‖𝑧(𝑡)‖_𝑋,𝑧∈𝔅^ℎ,(3.5) thus (𝔅^ℎ,‖⋅‖_𝔅^ℎ′′) is a Banach space. We define the operator 𝑃∶𝔅^ℎ→𝔅^ℎ by (𝑃𝑧)(𝑡)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩^𝑡0𝑇_𝛼𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,+𝑡−𝑡₁𝑧𝑡⁻¹+𝐼₁𝑧𝑡⁻¹_𝑡𝑡1⋮𝑇_𝛼𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,+𝑡−𝑡_𝑚𝑧𝑡^−𝑚+𝐼_𝑚𝑧𝑡^−𝑚_𝑡𝑡𝑚𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑇.(3.6) 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^∗∈𝔅^ℎ, then for all 𝑡∈[0,𝑡₁]. We have ‖𝑃(𝑧)(𝑡)−𝑃(𝑧^∗)(𝑡)‖_𝑋≤≤𝑀^𝑡0‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))−𝑓(𝑠,𝑦_𝑠+𝑧^∗𝑠,𝐵(𝑦(𝑠)+𝑧^∗(𝑠)))‖‖_𝑋𝑑𝑠_𝑆^𝑡0≤𝑀(𝑡−𝑠)^𝛼−1𝜇¹‖‖𝑧_𝑠−𝑧^∗𝑠‖‖_𝔅ℎ+𝜇₂‖‖𝐵𝑦(𝑠)+𝑧(𝑠)−𝐵𝑦(𝑠)+𝑧^∗‖‖_𝑋𝑑𝑠_𝑆𝛼𝜇¹𝐶^∗1+𝜇₂𝐵^∗𝑇^𝛼‖𝑧−𝑧^∗‖_𝔅^ℎ′′.(3.7) For 𝑡∈(𝑡₁,𝑡₂], we have ‖𝑃(𝑧)(𝑡)−𝑃(𝑧^∗)(𝑡)‖_𝑋≤+‖‖𝑇_{𝛼𝑡𝑡1}(𝑡−𝑡₁)‖‖_𝐿(𝑋)‖‖𝑧(𝑡⁻¹)−𝑧^∗(𝑡⁻¹)‖‖_𝑋+‖‖𝐼₁(𝑧(𝑡⁻¹))−𝐼₁(𝑧^∗(𝑡⁻¹))‖‖_𝑋‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))−𝑓(𝑠,𝑦_𝑠+𝑧^∗𝑠,𝐵(𝑦(𝑠)+𝑧^∗(𝑠)))‖‖_𝑋≤+𝑀𝑀_𝑇𝑆‖‖𝑧(𝑡⁻¹)−𝑧^∗(𝑡⁻¹)‖‖_𝑋+𝜌₁‖‖𝑧(𝑡⁻¹)−𝑧^∗(𝑡⁻¹)‖‖_𝑋^𝑡𝑡₁≤𝑀(𝑡−𝑠)^𝛼−1𝜇¹‖‖𝑧_𝑠−𝑧^∗𝑠‖‖_𝔅ℎ+𝜇₂‖‖𝐵𝑦(𝑠)+𝑧(𝑠)−𝐵𝑦(𝑠)+𝑧^∗‖‖_𝑋𝑑𝑠_𝑇1+𝜌₁‖𝑧−𝑧^∗‖_𝔅^ℎ′′+𝑀𝛼^𝑆𝜇¹𝐶^∗1+𝜇₂𝐵^∗𝑇^𝛼‖𝑧−𝑧^∗‖_𝔅^ℎ′′.(3.8) Similarly, when 𝑡∈(𝑡_𝑖,𝑡_𝑖+1],𝑖=2,…,𝑚, we get ‖𝑃(𝑧)(𝑡)−𝑃(𝑧^∗)(𝑡)‖_𝑋≤𝑀_𝑇1+𝜌_𝑖‖𝑧−𝑧^∗‖_𝔅^ℎ′′+𝑀^𝑆𝛼𝜇¹𝐶^∗1+𝜇₂𝐵^∗𝑇^𝛼‖𝑧−𝑧^∗‖_𝔅^ℎ′′.(3.9) Thus, for all 𝑡∈[0,𝑇], we have ‖𝑃(𝑧)−𝑃(𝑧^∗)‖_𝔅^ℎ′′≤max_{1≤𝑖≤𝑚}𝑀^𝑇1+𝜌_𝑖+𝑀𝛼^𝑆𝜇¹𝐶^∗1+𝜇₂𝐵^∗𝑇^𝛼‖𝑧−𝑧^∗‖_𝔅^ℎ′′.(3.10) Hence, 𝑃 is a contraction map, and therefore it has an unique fixed-point 𝑧∈𝔅^ℎ, 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. Let 𝐵 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×𝔅_ℎ×𝑋→𝑋 is continuous, and there exist two continuous functions 𝜇₁,𝜇₂∶𝐽→(0,∞) such that ‖𝑓(𝑡,𝜓,𝑥)‖_𝑋≤𝜇₁(𝑡)‖𝜓‖_𝔅ℎ+𝜇₂(𝑡)‖𝑥‖_𝑋,(𝑡,𝜓,𝑥)∈𝐽×𝔅×𝑋.(3.11)(H5)the function 𝐼_𝑘∶𝑋→𝑋 is continuous, and there exists Ω>0 such that Ω=max_{1≤𝑘≤𝑚,𝑥∈𝐵𝑟}‖‖𝐼_𝑘(𝑥)‖‖_𝑋.(3.12)

Before going further, we need the following lemma.

Lemma 3.3 (see Lemma  3.2 in [7]). Let 𝐶^∗1=sup_0<𝜏<𝑇𝐶₁(𝜏),𝐶^∗2=sup_0<𝜏<𝑇𝐶₂(𝜏),𝜇^∗1=sup_0<𝜏<𝑇𝜇₁(𝜏),𝜇^∗2=sup_0<𝜏<𝑇𝜇₂(𝜏)(3.13) then for any 𝑠∈𝐽, 𝜇₁(𝑠)‖‖𝑦_𝑠+𝑧_𝑠‖‖_𝔅ℎ+𝜇₂(𝑠)‖‖𝐵(𝑦(𝑠)+𝑧(𝑠))‖‖_𝑋≤𝜇^∗1𝐶^∗2‖𝜙‖_𝔅ℎ+𝐶^∗1sup_{0≤𝜏≤𝑠}‖𝑧(𝜏)‖_𝑋+𝜇_∗2^𝑠0𝐾(𝑠,𝜏)‖𝑧(𝜏)‖_𝑋𝑑𝜏.(3.14) If ‖𝑧‖_𝑋<𝑟,𝑟>0, then 𝜇₁(𝑠)‖‖𝑦_𝑠+𝑧_𝑠‖‖_𝔅ℎ+𝜇₂(𝑠)‖‖𝐵(𝑦(𝑠)+𝑧(𝑠))‖‖_𝑋≤𝜇^∗1𝐶^∗2‖𝜙‖_𝔅ℎ+𝐶^∗1𝑟+𝜇^∗2𝑟𝐵^∗=𝜆.(3.15)

Theorem 3.4. Suppose that the assumptions (H1), (H4), (H5) are satisfied with 𝑀𝛼^𝑆𝜇¹𝐶^∗1+𝜇₂𝐵^∗𝑇^𝛼<1,(3.16) then the impulsive problem (1.1) has at least one mild solution on (−∞,𝑇].

Proof. Choose 𝑟≥[𝑀_𝑇(𝑟+Ω)+(𝑀_𝑆𝑇^𝛼𝜆/𝛼)] and consider 𝐵_𝑟={𝑧∈𝔅^ℎ∶‖𝑧‖_𝔅^ℎ′′≤𝑟}, then 𝐵_𝑟 is a bounded, closed-convex subset in 𝔅^ℎ.
Let Γ₁∶𝐵_𝑟→𝐵_𝑟 and Γ₂∶𝐵_𝑟→𝐵_𝑟 be defined as Γ₁𝑧(𝑡)=⎧⎪⎨⎪⎩0,𝑡∈0,𝑡¹𝑇_𝛼,⋮𝑇_𝛼𝑡−𝑡₁𝑧𝑡⁻¹+𝐼₁𝑧𝑡⁻¹,𝑡∈𝑡₁,𝑡₂,Γ𝑡−𝑡_𝑚𝑧𝑡^−𝑚+𝐼_𝑚𝑧𝑡^−𝑚,𝑡∈𝑡_𝑚,𝑇,(3.17)₂𝑧(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩^𝑡0_𝑡𝑡1𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,⋮_𝑡𝑡𝑚𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,𝑆_𝛼(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑇.(3.18)
Step 1. Let 𝑧,𝑧^∗∈𝐵_𝑟, then show that Γ₁𝑧+Γ₂𝑧^∗∈𝐵_𝑟, for 𝑡∈[0,𝑡₁], we have ‖‖(Γ₁𝑧)(𝑡)+(Γ₂𝑧^∗)(𝑡)‖‖_𝑋≤≤𝑀^𝑡0‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧^∗𝑠,𝐵(𝑦(𝑠)+𝑧^∗(𝑠)))‖‖_𝑋𝑑𝑠_𝑆^𝑡0(𝑡−𝑠)^𝛼−1𝜇¹(𝑠)‖‖𝑦_𝑠+𝑧^∗𝑠‖‖_{𝔅ℎ′′}+𝜇₂(𝑠)‖‖𝐵(𝑦(𝑠)+𝑧^∗(𝑠))‖‖_𝑋𝑑𝑠,(3.19) and by using Lemma 3.3, we conclude that ‖‖(Γ₁𝑧)+(Γ₂𝑧^∗)‖‖_𝔅^ℎ′′≤𝑀^𝑆𝜆𝑇^𝛼𝛼<𝑟.(3.20) Similarly, when 𝑡∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=1,…,𝑚, we have the estimate ‖‖(Γ₁𝑧)(𝑡)+(Γ₂𝑧^∗)(𝑡)‖‖_𝑋≤‖‖𝑇_𝛼(𝑡−𝑡_𝑖)[𝑧(𝑡^−𝑖)+𝐼_𝑖(𝑧(𝑡^−𝑖))]‖‖_𝑋≤+𝑀^𝑡𝑡_𝑖‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧^∗𝑠,𝐵(𝑦(𝑠)+𝑧^∗(𝑠)))‖‖_𝑋𝑑𝑠_𝑇+_𝑡𝑡𝑖‖𝑧‖^{𝔅ℎ′′}+‖‖𝐼_𝑖(𝑧(𝑡^−𝑖))‖‖_𝑋≤𝑀‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)𝜇¹(𝑠)‖‖𝑦_𝑠+𝑧^∗𝑠‖‖_{𝔅ℎ′′}+𝜇₂(𝑠)‖‖𝐵(𝑦(𝑠)+𝑧^∗(𝑠))‖‖_𝑋𝑑𝑠_𝑇(𝑟+Ω)+𝑀^𝑆𝑇^𝛼𝜆𝛼<𝑟,(3.21) which implies that ‖Γ₁𝑧+Γ₂𝑧‖_𝔅^ℎ′′≤𝑟.Step 2. We will show that the mapping (Γ₁𝑧)(𝑡) is continuous on 𝐵_𝑟. For this purpose, let {𝑧^𝑛}^∞𝑛=1 be a sequence in 𝐵_𝑟 with lim𝑧^𝑛→𝑧∈𝐵_𝑟, then for 𝑡∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,…,𝑚, we have ‖‖(Γ₁𝑧^𝑛)(𝑡)−(Γ₁𝑧)(𝑡)‖‖_𝑋≤‖‖𝑇_𝛼(𝑡−𝑡_𝑖)‖‖_𝐿(𝑋)‖‖𝑧^𝑛(𝑡^−𝑖)−𝑧(𝑡^−𝑖)‖‖_𝑋+‖‖𝐼_𝑖(𝑧^𝑛(𝑡^−𝑖))−𝐼_𝑖(𝑧(𝑡^−𝑖))‖‖_𝑋.(3.22) Since the functions 𝐼_𝑖,𝑖=1,2,…,𝑚 are continuous, hence lim_𝑛→∞Γ₁𝑧^𝑛=Γ₁𝑧 in 𝐵_𝑟 which implies that the mapping Γ₁ is continuous on 𝐵_𝑟.Step 3. Uniform boundedness of the map (Γ₁𝑧)(𝑡) is an implication of the following inequality: for 𝑡∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,…,𝑚, we have ‖‖(Γ₁𝑧)(𝑡)‖‖_𝑋≤≤𝑀‖‖𝑇_𝛼𝑇(𝑡−𝑡_𝑖)‖‖_𝐿(𝑋)‖‖𝑧(𝑡^−𝑖)‖‖_𝑋+‖‖𝐼_𝑖(𝑧(𝑡^−𝑖))‖‖_𝑋(𝑟+Ω).(3.23)Step 4. To show that the map (3.17) is equicontinuous, we proceed as follows. Let 𝑢,𝑣∈(𝑡_𝑖,𝑡_𝑖+1], 𝑡_𝑖≤𝑢<𝑣≤𝑡_𝑖+1, 𝑖=0,1,…,𝑚, 𝑧∈𝐵_𝑟, then we obtain ‖‖(Γ₁𝑧)(𝑣)−(Γ₁𝑧)(𝑢)‖‖_𝑋≤‖‖𝑇_𝛼(𝑣−𝑡_𝑖)−𝑇_𝛼(𝑢−𝑡_𝑖)‖‖_𝐿(𝑋)‖‖𝑧(𝑡^−𝑖)+𝐼_𝑖(𝑧(𝑡^−𝑖))‖‖_𝑋≤(𝑟+Ω)‖‖𝑇_𝛼(𝑣−𝑡_𝑖)−𝑇_𝛼(𝑢−𝑡_𝑖)‖‖_𝐿(𝑋).(3.24) Since 𝑇_𝛼 is strongly continuous, the continuity of the function 𝑡↦‖𝑇(𝑡)‖ allows us to conclude that lim_𝑢→𝑣‖𝑇_𝛼(𝑣−𝑡_𝑖)−𝑇_𝛼(𝑢−𝑡_𝑖)‖_𝐿(𝑋)=0, which implies that Γ₁(𝐵_𝑟) 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 map Γ₂ is a contraction mapping. Let 𝑧,𝑧^∗∈𝐵_𝑟 and 𝑡∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,…,𝑚, then we have ‖‖(Γ₂𝑧)(𝑡)−(Γ₂𝑧^∗)(𝑡)‖‖_𝑋≤≤𝑀^𝑡𝑡_𝑖‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)−𝑓𝑠,𝑦_𝑠+𝑧^∗𝑠,𝐵𝑦(𝑠)+𝑧^∗(𝑠)‖‖_𝑋𝑑𝑠_𝑆^𝑡𝑡_𝑖≤𝑀(𝑡−𝑠)^𝛼−1𝜇¹‖‖𝑧_𝑠−𝑧^∗𝑠‖‖_{𝔅ℎ′′}+𝜇₂‖‖𝐵𝑦(𝑠)+𝑧(𝑠)−𝐵𝑦(𝑠)+𝑧^∗(𝑠)‖‖_𝑋𝑑𝑠_𝑆𝛼𝜇¹𝐶^∗1+𝜇₂𝐵^∗𝑇^𝛼‖𝑧−𝑧^∗‖_𝔅^ℎ′′,(3.25) since (𝑀_𝑆/𝛼)(𝜇₁𝐶^∗1+𝜇₂𝐵^∗)𝑇^𝛼<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. Let 𝑃 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]). Let 𝑣∶[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 𝑣(𝑡)≤𝑤(𝑡)+𝑎^𝑡0𝑣(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠.(3.26) Then there exists a constant 𝐾(𝛼) such that 𝑣(𝑡)≤𝑤(𝑡)+𝑎𝐾(𝛼)^𝑡0𝑤(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠,forevery𝑡∈[0,𝑇].(3.27)

Theorem 3.7. Assume that the assumptions (H4)-(H5) are satisfied, and if 𝐴∈𝐴^𝛼(𝜃₀,𝜔₀) and 𝑀_𝑇<1, then the impulsive problem (1.1) has at least one mild solution on (−∞,𝑇].

Proof. We define the operator 𝑃∶𝔅^ℎ→𝔅^ℎ as in Theorem  3.3. Note that 𝑃 is well defined in 𝔅^ℎ.We complete the proof in the following steps.Step 1. For the continuity of the map 𝑃, let {𝑧^𝑛} be a sequence in 𝔅^ℎ such that 𝑧^𝑛→𝑧 in 𝔅^ℎ. Since the function 𝑓 is continuous on 𝐽×𝔅_ℎ×𝑋, This implies that 𝑓𝑠,⟶𝑦_𝑠+𝑓𝑧^𝑛𝑠,𝑠𝐵,𝑦_𝑠𝑦(𝑠)+𝑧^𝑛(𝑠)+𝑧_𝑠,𝐵𝑦(𝑠)+𝑧(𝑠)as𝑛⟶∞.(3.28) Now, for every 𝑡∈[0,𝑡₁], we get ‖𝑃𝑧^𝑛(𝑡)−𝑃𝑧(𝑡)‖_𝑋≤≤𝑀^𝑡0‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧^𝑛𝑠,𝐵(𝑦(𝑠)+𝑧^𝑛(𝑠)))−𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠_𝑆𝑇^𝛼𝛼𝜀,(3.29) where 𝜀>0,𝜀→0 as 𝑛→∞. Moreover, we have ‖𝑃𝑧^𝑛(𝑡)−𝑃𝑧(𝑡)‖_𝑋≤+𝑀_{𝑇𝑡𝑡𝑖}‖‖𝑧^𝑛(𝑡^−𝑖)−𝑧(𝑡^−𝑖)‖‖_𝑋+‖‖𝐼_𝑖(𝑧^𝑛(𝑡^−𝑖))−𝐼_𝑖(𝑧(𝑡^−𝑖))‖‖_𝑋≤𝑀‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧^𝑛𝑠,𝐵(𝑦(𝑠)+𝑧^𝑛(𝑠)))−𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠_𝑇‖‖𝑧^𝑛(𝑡^−𝑖)−𝑧(𝑡^−𝑖)‖‖_𝑋+‖‖𝐼_𝑖(𝑧^𝑛(𝑡^−𝑖))−𝐼_𝑖(𝑧(𝑡^−𝑖))‖‖_𝑋+𝑀_𝑆𝑇^𝛼𝛼𝜀,(3.30) where 𝜀>0,𝜀→0 as 𝑛→∞, for all 𝑡∈(𝑡_𝑖,𝑡_𝑖+1],𝑖=1,…,𝑚. The impulsive functions 𝐼_𝑘,𝑘=1,…,𝑚 are continuous, then we get lim_𝑛→∞‖P𝑧^𝑛−𝑃𝑧‖_𝔅^ℎ′′=0.(3.31) This implies that 𝑃 is continuous.Step 2. 𝑃 maps bounded sets into bounded sets in 𝔅^ℎ. To prove that for any 𝑟>0, there exists a 𝛾>0 such that for each 𝑧∈𝐵_𝑟={𝑧∈𝔅^ℎ∶‖𝑧‖_𝔅^ℎ′′≤𝑟}, then we have ‖𝑃𝑧‖_𝔅^ℎ′′≤𝛾, then for any 𝑧∈𝐵_𝑟,𝑡∈[0,𝑡₁], we have ‖𝑃𝑧(𝑡)‖_𝑋≤≤𝑀^𝑡0‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠_𝑆^𝑡0(𝑡−𝑠)^𝛼−1𝜇¹(𝑠)‖‖𝑦_𝑠+𝑧_𝑠‖‖_𝔅ℎ+𝜇₂(𝑠)‖‖𝐵(𝑦(𝑠)+𝑧(𝑠))‖‖_𝑋𝑑𝑠.(3.32) Using Lemma 3.3, we obtain ‖𝑃𝑧(𝑡)‖_𝑋≤𝑀_𝑆(𝑇^𝛼/𝛼)𝜆. Similarly, we have ‖𝑃𝑧(𝑡)‖_𝑋≤𝑀_𝑇(𝑟+Ω)+𝑀_𝑆𝑇^𝛼𝛼𝜆,𝑡∈𝑡^𝑖,𝑡_𝑖+1,𝑖=1,…,𝑚.(3.33) This implies that ‖𝑃𝑧‖_𝔅^ℎ′′≤𝑀_𝑇(𝑟+Ω)+𝑀_𝑆𝑇^𝛼𝛼𝜆=𝛾,𝑡∈[0,𝑇].(3.34)Step 3. We will prove that 𝑃(𝐵_𝑟) is equicontinuous. Let 𝑢,𝑣∈[0,𝑡₁], with 𝑢<𝑣, we have ‖𝑃𝑧(𝑣)−𝑃𝑧(𝑢)‖_𝑋≤+^𝑢0‖‖𝑆_𝛼(𝑣−𝑠)−𝑆_𝛼(𝑢−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠_𝑣𝑢‖‖𝑆_𝛼(𝑣−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠≤𝑄₁+𝑄₂,(3.35) where 𝑄₁≤=𝜆^𝑢0‖‖_𝑢0𝑆_𝛼(𝑣−𝑠)−𝑆_𝛼(𝑢−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠‖‖𝑆_𝛼(𝑣−𝑠)−𝑆_𝛼(𝑢−𝑠)‖‖_𝐿(𝑋)𝑑𝑠.(3.36) Since ‖𝑆_𝛼(𝑣−𝑠)−𝑆_𝛼(𝑢−𝑠)‖_𝐿(𝑋)≤2𝑀_𝑆(𝑡₁−𝑠)^𝛼−1∈𝐿¹(𝐼,ℝ₊) for 𝑠∈[0,𝑡₁] and 𝑆_𝛼(𝑣−𝑠)−𝑆_𝛼(𝑢−𝑠)→0 as 𝑢→𝑣, 𝑆_𝛼 is strongly continuous. This implies that lim_𝑢→𝑣𝑄₁=0, 𝑄₂=^𝑣𝑢‖‖𝑆_𝛼(𝑣−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠≤𝜆𝑀^𝑆(𝑣−𝑢)^𝛼𝛼.(3.37) Hence, lim_𝑢→𝑣𝑄₂=0. Similarly, for 𝑢,𝑣∈(𝑡_𝑖,𝑡_𝑖+1], with 𝑢<𝑣,𝑖=1,…,𝑚, we have ‖𝑃𝑧(𝑣)−𝑃𝑧(𝑢)‖_𝑋≤‖‖𝑇_𝛼(𝑣−𝑡_𝑖)−𝑇_𝛼(𝑢−𝑡_𝑖)‖‖_𝐿(𝑋)‖‖𝑧𝑡^−𝑖‖‖_𝑋+‖‖𝐼_𝑖𝑧𝑡^−𝑖‖‖_𝑋+𝑄₁+𝑄₂.(3.38) Since 𝑇_𝛼 is also strongly continuous, so 𝑇_𝛼(𝑣−𝑡_𝑖)−𝑇_𝛼(𝑢−𝑡_𝑖)→0 as 𝑢→𝑣. Thus, from the above inequalities, we have lim_𝑢→𝑣‖𝑃𝑧(𝑣)−𝑃𝑧(𝑢)‖_𝑋=0. So, 𝑃(𝐵_𝑟) 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 𝐸=𝑧∈𝔅^ℎsuchthat𝑧=𝜈𝑃𝑧forsome0<𝜈<1(3.39) is bounded. Let 𝑧∈𝐸, then 𝑧(𝑡)=𝜈𝑃𝑧(𝑡) for some 0<𝜈<1. Then for each 𝑡∈[0,𝑡₁], we have ‖𝑧(𝑡)‖_𝑋≤≤𝜈𝜈𝑀^𝑡0_𝑆‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠^𝑡0(𝑡−𝑠)^𝛼−1‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠,(3.40) for 𝑡∈(𝑡_𝑖,𝑡_𝑖+1],𝑖=1,…,𝑚, we get ‖𝑧(𝑡)‖_𝑋≤𝜈‖‖𝑇_𝛼(𝑡−𝑡_𝑖)‖‖_𝐿(𝑋)‖‖𝑧(𝑡^−𝑖)‖‖_𝑋+‖‖𝐼_𝑖(𝑧(𝑡^−𝑖))‖‖_𝑋+^𝑡𝑡_𝑖‖‖𝑆_𝛼(𝑡−𝑠)‖‖_𝐿(𝑋)≤×𝑀‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠_𝑇‖‖𝑧(𝑡^−𝑖)‖‖_𝑋+𝑀_𝑇Ω+𝑀_𝑆^𝑡𝑡_𝑖(𝑡−𝑠)^𝛼−1‖‖𝑓(𝑠,𝑦_𝑠+𝑧_𝑠,𝐵(𝑦(𝑠)+𝑧(𝑠)))‖‖_𝑋𝑑𝑠.(3.41) then for all 𝑡∈[0,𝑇], we have ‖𝑧(𝑡)‖_𝑋≤𝑀^𝑇Ω1−𝑀_𝑇+𝑀^𝑆1−𝑀_𝑇^𝑡0≤𝑀(𝑡−𝑠)^𝛼−1𝜇¹(𝑠)‖‖𝑦_𝑠+𝑧_𝑠‖‖_𝔅ℎ+𝜇₂(𝑠)‖‖𝐵(𝑦(𝑠)+𝑧(𝑠))‖‖_𝑋𝑑𝑠_𝑇Ω1−𝑀_𝑇+𝑀^𝑆𝜇^∗1𝐶^∗2‖𝜙‖_𝔅ℎ𝑇^𝛼+𝑀_𝑆𝛼1−𝑀_𝑇1−𝑀_𝑇≤𝜔₁𝜇^∗1𝐶^∗1+𝜇^∗2𝐵^∗^𝑡0(𝑡−𝑠)^𝛼−1sup_{0≤𝜏≤𝑠}‖𝑧(𝜏)‖_𝑋𝑑𝑠+𝜔₂^𝑡0(𝑡−𝑠)^𝛼−1sup_{0≤𝜏≤𝑠}‖𝑧(𝜏)‖_𝑋𝑑𝑠,(3.42) where 𝜔₁=𝑀_𝑇Ω/(1−𝑀_𝑇)+𝑀_𝑆𝜇^∗1𝐶^∗2‖𝜙‖_𝔅ℎ𝑇^𝛼/𝛼(1−𝑀_𝑇) and 𝜔₂=(𝑀_𝑆/1−𝑀_𝑇)(𝜇^∗1𝐶^∗1+𝜇^∗2𝐵^∗). Let 𝜏^∗∈[0,𝑠] be such that sup_{0≤𝜏≤𝑠}‖𝑧(𝜏)‖_𝑋=‖𝑧(𝜏^∗)‖_𝑋,0≤𝑠≤𝑡. If 𝜏^∗∈[0,𝑡], then (3.43) can be written as ‖𝑧(𝑡)‖_𝑋≤𝜔₁+𝜔₂^𝑡0(𝑡−𝑠)^𝛼−1‖𝑧(𝑠)‖_𝑋𝑑𝑠.(3.43) Using Lemma 3.6, there exists a constant 𝐾(𝛼), and (3.43) becomes ‖𝑧(𝑡)‖_𝑋≤𝜔₁1+^𝑡0𝐾(𝛼)(𝑡−𝑠)^𝛼−1𝑑𝑠≤𝜔₁1+𝐾(𝛼𝛼)𝑇^𝛼.(3.44) 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 form𝐷^𝑞𝑡𝑢(𝑡,𝑥)=𝜕²𝜕𝑥²+𝑢(𝑡,𝑥)+^𝑡−∞𝐻(𝑡,𝑥,𝑠−𝑡)𝑄(𝑢(𝑠,𝑥))𝑑𝑠_𝑡0Δ𝑢𝑢𝑢𝑡_𝑖((𝑡𝑡,,𝑥0))==𝜙0=(𝑡,𝑘𝑢𝑥(()𝑡𝑠,,,𝜋𝑡))𝑡𝑒,∈𝑡(−≥^{−𝑢(𝑠,𝑥)}∞𝑑0𝑠,,,0],𝑥𝑥∈∈[0[,0𝜋,]𝜋,],𝑡∈[0,𝑏],𝑡≠𝑡_𝑘,(𝑥)=^𝑡𝑖_−∞𝑞_𝑖𝑡_𝑖−𝑠𝑢(𝑠,𝑥)𝑑𝑠,𝑥∈[0,𝜋],(4.1) where 𝐷^𝑞𝑡 is Caputo’s fractional derivative of order 0<𝑞<1, 0<𝑡₁<𝑡₂<⋯<𝑡_𝑛<𝑏 are prefixed numbers, and 𝜙∈𝔅_ℎ. 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)}, then𝐴𝑤=^∞_𝑛=1𝑛²𝑤,𝑤_𝑛𝑤_𝑛,𝑤∈𝐷(𝐴),(4.2) where 𝑤_𝑛(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 by𝑇(𝑡)𝑤=^∞_𝑛=1𝑒^−𝑛2𝑡𝑤,𝑤_𝑛𝑤_𝑛,∀𝑤∈𝑋,andevery𝑡>0.(4.3) 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∈𝔸^𝛼(𝜃₀,𝜔₀). Let ℎ(𝑠)=𝑒^2𝑠,𝑠<0, then 𝑙=∫^0−∞ℎ(𝑠)𝑑𝑠=1/2, and define‖𝜙‖_ℎ=^0−∞ℎ(𝑠)sup_{[]𝜃∈𝑠,0}‖𝜙(𝜃)‖_𝐿²𝑑𝑠.(4.4) Hence, for (𝑡,𝜙)∈[0,𝑏]×𝔅_ℎ, where 𝜙(𝜃)(𝑥)=𝜙(𝜃,𝑥),(𝜃,𝑥)∈(−∞,0]×[0,𝜋]. Set 𝑢(𝑡)(𝑥)=𝑢(𝑡,𝑥),𝑓(𝑡,𝜙,𝐵𝑢(𝑡))(𝑥)=^0−∞𝐻(𝑡,𝑥,𝜃)𝑄(𝜙(𝜃)(𝑥))𝑑𝜃+𝐵𝑢(𝑡)(𝑥),(4.5) where 𝐵𝑢(𝑡)(𝑥)=∫^𝑡0𝑘(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 𝑞_𝑖∶ℝ→ℝ are continuous and 𝑑_𝑖=∫^0−∞ℎ(𝑠)𝑞^2𝑖(𝑠)𝑑𝑠<∞ for 𝑖=1,2,…,𝑛. Suppose further that (1)the function 𝐻(𝑡,𝑥,𝜃) is continuous in [0,𝑏]×[0,𝜋]×(−∞,0] and 𝐻(𝑡,𝑥,𝜃)≥0, ∫^0−∞𝐻(𝑡,𝑥,𝜃)𝑑𝜃=𝑝₁(𝑡,𝑥)<∞,(2)the function 𝑄(⋅) is continuous and for each (𝜃,𝑦)∈(−∞,0]×[0,𝜋], 0≤𝑄(𝑢(𝜃)(𝑥))≤(∫^0−∞𝑒^2𝑠‖𝑢(𝑠,⋅)‖_𝐿²𝑑𝑠).

Now, we can see that ‖𝑓(𝑡,𝜙,𝐵𝑢(𝑡))‖_𝐿²=^𝜋0^0−∞ℎ(𝑡,𝑥,𝜃)𝑄(𝜙(𝜃)(𝑥))𝑑𝜃+𝐵𝑢(𝑡)(𝑥)𝑑𝑥²^1/2≤^𝜋0^0−∞ℎ(𝑡,𝑥,𝜃)^0−∞𝑒^2𝑠‖𝜙(𝑠)(⋅)‖_𝐿²𝑑𝑠𝑑𝜃²𝑑𝑥^1/2+^𝜋0^𝑡0𝑘(𝑠,𝑡)𝑒^{−𝑢(𝑠,𝑥)}𝑑𝑠²𝑑𝑥^1/2≤⎡⎢⎢⎣^𝜋0^0−∞ℎ(𝑡,𝑥,𝜃)^0−∞𝑒^2𝑠sup_{𝑠∈[𝜃,0]}‖𝜙(𝑠)‖_𝐿²𝑑𝑠𝑑𝜃²𝑑𝑥⎤⎥⎥⎦^1/2+^𝜋0^𝑡0𝑘(𝑠,𝑡)𝑒^{−𝑢(𝑠,𝑥)}𝑑𝑠²𝑑𝑥^1/2≤^𝜋0^0−∞ℎ(𝑡,𝑥,𝜃)𝑑𝜃²𝑑𝑥^1/2‖𝜙‖_𝔅ℎ+‖𝐵𝑢(𝑡)‖_𝐿²≤^𝜋0(𝑝(𝑡,𝑥))²𝑑𝑥^1/2‖𝜙‖_𝔅ℎ+‖𝐵𝑢(𝑡)‖_𝐿²≤𝑝(𝑡)‖𝜙‖_𝔅ℎ+‖𝐵𝑢(𝑡)‖_𝐿².(4.6)

If we take 𝜇₁(𝑡)=𝑝(𝑡) and 𝜇₂(𝑡)=1, hence 𝑓 satisfies (H4), and similarly we can show that 𝐼_𝑘 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].

Acknowledgment

The authors express their deep gratitude to the referees for their valuable suggestions and comments for improvement of the paper.