Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay

Abstract

We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach space 𝑋. The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.

1. Introduction

In recent years, fractional calculus has becomes an active area of research due to its demonstrated applications in widespread fields of science and engineering such as mechanics, electrical engineering, medicine, biology, ecology, and many others. The memory and hereditary properties of various materials and processes can be described by a differential equation with fractional order. The fractional differential equation also describes the efficiency of nonlinear oscillations of the earthquake. The details on the theory and its applications can be found in [1–4] and references given therein.

On the other hand, many real world processes and phenomena which are subjected during their development to short-term external influences can be modeled as impulsive differential equation with fractional order which have been used efficiently in modelling many practical problems. Their duration is negligible compared with the total duration of the entire process and phenomena. Such process is investigated in various fields such as biology, physics, control theory, population dynamics, economics, chemical technology, and medicine. In addition, the improvement of the hypothesis of the functional differential equation with infinite delay relies on a choice of phase space. There are various phase spaces which have been studied. Hale and Kato in [5] introduced a common phase space 𝒫. For more details on phase space, we refer to books by Hale and Kato [5], Hino et al. [6] and papers [7–10]. For the study of impulsive differential equation, we refer to papers [7, 8, 11–18] and references given therein.

The purpose of this work is to establish the existence of mild solution for impulsive fractional differential equation with infinite delay: 𝐃^𝑞_𝑡𝑢(𝑡)+𝑔𝑡,𝑢_𝑡=𝐴𝑢(𝑡)+𝑔𝑡,𝑢_𝑡+ℐ_𝑡^1−𝑞𝑓𝑡,𝑢_𝑡,𝑡∈𝐽=[0,𝑇],𝑡≠𝑡_𝑖𝑢₀𝑡=𝜑∈𝒫,Δ𝑢_𝑖=𝐼_𝑖𝑢_𝑡𝑖,𝑖=1,2,…,𝑛,𝑛∈ℕ, (1) where 0<𝑇<∞, 0<𝑞<1, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a closed and densely defined linear operator and infinitesimal generator of a solution (resolvent) operator {𝑆_𝑞(𝑡)}_𝑡≥0 on Banach space 𝑋, and 𝐃^𝑞_𝑡 denotes the fractional derivative in Caputo sense and ℐ_𝑡^1−𝑞 denotes the Riemann-Liouville fractional integral operator. The history 𝑢_𝑡∶(−∞,0]→𝑋 defined by 𝑢_𝑡(𝑠)=𝑢(𝑡+𝑠) for 𝑠∈(−∞,0] belongs to some abstract phase space 𝒫 defined axiomatically and 𝐼_𝑖∈𝐶(𝑋,𝑋)(𝑖=1,…,𝑛); 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑛≤𝑡_𝑛+1=𝑇 are fixed numbers and Δ𝑢(𝑡) denotes the jump of the function 𝑢 at the point 𝑡, given by Δ𝑢(𝑡)=𝑢(𝑡⁺)−𝑢(𝑡⁻). The functions 𝑓,𝑔∶𝐽×𝒫→𝑋 are appropriate functions and satisfy some conditions to be specified later.

In [13], authors have considered the following impulsive fractional differential equation in a Banach space of the form 𝐃^𝑞_𝑡𝑢(𝑡)(𝑡)((𝑡,(][]=𝐴𝑢+𝑓𝑡,𝑢))𝑞∈0,1,𝑡∈𝐽=0,𝑇,𝑡≠𝑡_𝑖,𝑢(0)=𝑢₀,𝑢𝑡⁺_𝑖=𝑢𝑡⁻_𝑖+𝑦_𝑖,𝑖=1,2,…,𝑛,𝑛∈ℕ, (2) where 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of a 𝐶₀-semigroup {𝑇(𝑡)∶𝑡≥0} on a Banach space, 𝑓∶𝐽×𝑋→𝑋 is continuous, and 𝑢₀, 𝑦_𝑖 are the element of 𝑋. Authors have established some existence and uniqueness results for system (2) under the different assumptions on initial conditions.

In this work, we adopt the idea of Wang et al. [13] and establish the existence of a mild solution for the problem (1) by using the measure of noncompactness and solution operator. The tool of measure of noncompactness has been used in linear operator theory, theory of differential and integral equations, the fixed point theory, and many others. For an initial study of theory of the measure of noncompactness, we refer to book of Banaś and Goebel [19] and Akhmerov et al. [20] and papers [9, 21–25] and references given therein.

This paper is organized as follows: In Section 2 we recall some basic definitions, lemmas, and theorems. We will prove the existence of a mild solution for the system (1) in Section 3. In the last section, we shall discuss an example to illustrate the application of the abstract results.

2. Preliminaries

Now we provide some basic definitions, notations, theorems, lemmas, and preliminary facts which will be used throughout this paper.

Let 𝑋 be a Banach space and let 𝐶([0,𝑇];𝑋) be the Banach space of continuous functions 𝑢(𝑡) from [0,𝑇] to 𝑋 equipped with the norm ‖𝑦‖_𝐶=sup_{𝑡∈[0,𝑇]}‖𝑦(𝑡)‖_𝑋 and 𝐿^𝑝((0,𝑇);𝑋) denotes the Banach space of all Bochner-measurable functions from (0,𝑇) to 𝑋 with the norm ‖𝑢‖_𝐿^𝑝=_(0,𝑇)‖𝑢(𝑠)‖^𝑝_𝑋𝑑𝑠^1/𝑝. (3)

Assume that 0∈𝜌(𝐴), that is, 𝐴 is invertible. Then, this permits us to define the positive fractional power 𝐴^𝛼 as closed linear operator with domain 𝐷(𝐴^𝛼)⊆𝐻 for 𝛼∈(0,1]. Moreover, 𝐷(𝐴^𝛼) is dense in 𝐻 with the norm 𝑦_𝛼=𝐴^𝛼𝑦. (4) It is easy to see that 𝐷(𝐴^𝛼) which is dense in 𝑋 is a Banach space. Henceforth, we use 𝑋_𝛼 as notation of 𝐷(𝐴^𝛼). Also, we have that 𝑋_𝜅↪𝑋_𝛼 for 0<𝛼<𝜅 and, therefore, the embedding is continuous. Then, we define 𝑋_−𝛼=(𝑋_𝛼)^∗, for each 𝛼>0. The space 𝑋_−𝛼, standing for the dual space of 𝑋_𝛼, is a Banach space with the norm ‖𝑧‖_−𝛼=‖𝐴^−𝛼𝑧‖ for 𝑧∈𝑋_−𝛼. For more details on the fractional powers of closed linear operators, we refer to book by Pazy [26].

To consider the mild solution for the impulsive problem, we propose that the set 𝒫𝒞([0,𝑇];𝑋)={𝑢∶[0,𝑇]→𝑋∶𝑢iscontinuousat𝑡≠𝑡_𝑖andleftcontinuousat𝑡 = 𝑡_𝑖and𝑢(𝑡⁺_𝑖)exists,forall𝑖=1,…,𝑚}. Clearly, 𝒫𝒞([0,𝑇];𝑋) is a Banach space endowing the norm ‖𝑢‖_𝒫𝒞=sup_{𝑡∈[0,𝑇]}‖𝑢(𝑠)‖. For a function 𝑢∈𝒫𝒞([0,𝑇];𝑋) and 𝑖∈{0,1,…,𝑚}, we define the function 𝑢_𝑖∈𝐶([𝑡_𝑖,𝑡_𝑖+1],𝑋) such that 𝑢_𝑖(𝑡)=𝑢(𝑡),for𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑢𝑡⁺_𝑖,for𝑡=𝑡_𝑖. (5) For 𝑊⊂𝒫𝒞([0,𝑇];𝑋) and 𝑖∈{0,1,…,𝑚}, we have 𝑊_𝑖={𝑢_𝑖∶𝑢∈𝑊} and following Accoli-Arzelà type criteria.

Lemma 1. A set 𝐵⊂𝒫𝒞([0,𝑇];𝑋) is relatively compact in 𝒫𝒞([0,𝑇];𝑋) if and only if each set 𝐵_𝑗(𝑗=1,2,…,𝑚) is relatively compact in 𝐶([𝑡_𝑗,𝑡_𝑗+1],𝑋)(𝑗=0,1,…,𝑚).

For the differential equation with infinite delay, Hale and Kato [5] proposed the phase space 𝒫 satisfying certain fundamental axioms.

Definition 2 (see [6]). A phase space 𝒫 is a linear space which contains all the functions mapping (−∞,0] into Banach space 𝑋 with a seminorm ‖⋅‖_𝒫. The fundamental axioms assumed on 𝒫 are the following,(A)If 𝑢∶(−∞,𝑎+𝑇]→𝑋, 𝑇>0 is a continuous function on [𝑎,𝑎+𝑇] such that 𝑢_𝑎∈𝒫 and 𝑢|_{[𝑎,𝑎+𝑇]}∈𝒫∈𝒫𝒞([𝑎,𝑎+𝑇];𝑋), then for every 𝑡∈[𝑎,𝑎+𝑇), the following conditions hold:(i)𝑢_𝑡∈𝒫,(ii)𝐻‖𝑢_𝑡‖_𝒫≥‖𝑢(𝑡)‖,(iii)‖𝑢_𝑡‖_𝒫≤𝑁(𝑡+𝑎)‖𝑢_𝑎‖_𝒫+𝐾(𝑡−𝑎)sup{‖𝑢(𝑠)‖∶𝑎≤𝑠≤𝑡}. Where 𝐻 is a positive constant, 𝑁,𝐾∶[0,∞)→[0,∞), 𝑁 is a locally bounded, 𝐾 is continuous, and 𝐻, 𝑁, 𝐾 are independent of 𝑢(⋅).(A1)For the function 𝑢 in (A1), 𝑢_𝑡 is a 𝒫-valued continuous function for 𝑡∈[𝑎,𝑎+𝑇].(B)The space 𝒫 is complete.

Now, we state some basic definitions and properties of fractional calculus.

Mittag-Leffler. The definition of one parameter Mittag-Leffler function is given as 𝐸_𝛼(𝑧)=^∞_𝑘=0𝑧^𝑘Γ(),𝛼𝑘+1 (6) and two-parameter Mittag-Leffler function is defined as 𝐸_𝛼,𝛽(𝑧)=^∞_𝑘=0𝑧^𝑘Γ=1𝛼𝑧+𝛽2𝜋𝑖_𝐶𝜇^𝛼−𝛽𝑒^𝜇𝜇^𝑎−𝑧𝑑𝜇,𝛼,𝛽>0,𝑧∈ℂ, (7) where 𝐶 is a contour which starts and ends at −∞ and encircles the disc |𝜇|≤|𝑧|^1/2 counter clockwise. The Laplace transform of the Mittag-leffler is defined as 𝐿𝑡^𝛽−1𝐸_𝛼,𝛽−𝜌^𝛼𝑡^𝛼=𝜆^𝛼−𝛽𝜆^𝛼+𝜌^𝛼,Re𝜆>𝜌^1/𝛼,𝜌>0. (8) For more details we refer to [1].

Laplace transform of integer order derivatives is defined as 𝐿𝐹^𝑛(𝑡);𝜆=𝜆^𝑛𝐿[𝐹(𝑡)]−^𝑛−1_𝑘=0𝜆^{𝑛−𝑘−1}𝐹^𝑘(0)=𝜆^𝑛𝐿[𝐹(𝑡)]−^𝑛−1_𝑘=0𝜆^𝑘𝐹^{𝑛−𝑘−1}(0). (9)

Definition 3. The Riemann-Liouville fractional integral operator ℐ is defined as ℐ^𝑞_𝑡𝐹(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝐹(𝑠)𝑑𝑠, (10) where 𝐹∈𝐿¹((0,𝑇);𝑋) and 𝑞>0 is the order of the fractional integration.

Definition 4. The Riemann-Liouville fractional derivative is given as 𝐃^𝑞_𝑡𝐹(𝑡)=𝐷^𝑛_𝑡ℐ_𝑡^𝑛−𝑞𝐹(𝑡),𝑛−1<𝑞<𝑛,𝑛∈ℕ, (11) where 𝐷^𝑛_𝑡=𝑑^𝑛/𝑑𝑡^𝑛, 𝐹∈𝐿¹((0,𝑇);𝑋), and ℐ_𝑡^𝑛−𝑞𝐹∈𝑊^𝑛,1((0,𝑇);𝑋). Here, the notation 𝑊^𝑛,1((0,𝑇);𝑋) stands for the Sobolev space defined as 𝑊^𝑛,1))=((0,𝑇;𝑋𝑦∈𝑋∶∃𝑧∈𝐿¹))∶𝑦(𝑡)=((0,𝑇;𝑋_{𝑛−1𝑘=0}𝑑_𝑘𝑡^𝑘+𝑡𝑘!(^𝑛−1)!(𝑡)(.𝑛−1∗𝑧,𝑡∈0,𝑇) (12) Note that 𝑧(𝑡)=𝑦^𝑛(𝑡) and 𝑑_𝑘=𝑦^𝑘(0).

Definition 5. The Caputo fractional derivative is given as 𝐃^𝛼_𝑡𝐹(𝑡)=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝐹^𝑛(𝑠)𝑑𝑠,𝑛−1<𝛼<𝑛, (13) where 𝐹∈𝐶^𝑛−1((0,𝑇);𝑋)∩𝐿¹((0,𝑇);𝑋) and the following holds ℐ^𝑞_𝑡𝐃^𝑞_𝑡𝐹(𝑡)=𝐹(𝑡)−^𝑛−1_𝑘=0𝑡^𝑘𝐹𝑘!^𝑘(0). (14)

Definition 6 (see [27]). A family {𝑆_𝑞(𝑡)}_𝑡≥0⊂ℒ(𝑋) of bounded linear operators in 𝑋 is called a resolvent (or solution operator) generating by 𝐴 if the following conditions are fulfilled:(1)𝑆_𝑞(𝑡) is strongly continuous on ℝ⁺ and 𝑆_𝑞(0)=𝐼;(2)for 𝑥∈𝐷(𝐴) and 𝑡≥0, 𝑆_𝑞(𝑡)𝐷(𝐴)⊂𝐷(𝐴) and 𝐴𝑆_𝑞(𝑡)𝑥=𝑆_𝑞(𝑡)𝐴𝑥;(3)𝑆_𝑞(𝑡)𝑥 is the solution of the equation 𝑢(𝑡)1=𝑥+Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1(𝑠)(𝐴)𝐴𝑢𝑑𝑠,∀𝑥∈𝐷,𝑡≥0, (15)where ℒ(𝑋) denotes the space of all bounded linear operators from 𝑋 into 𝑋 endowed with the norm of operators.

Also, the solution operator 𝑆_𝑞(𝑡) for (15) is defined as (see [27]) 𝜆^𝑞−1𝜆^𝑞𝐼−𝐴⁻¹𝑥=^∞₀𝑒^−𝜆𝑡𝑆_𝑞(𝑡)𝑥𝑑𝑡,Re𝜆>𝜔,𝑥∈𝑋, (16) where 𝜔≥0 and {𝜆^𝑞∶Re𝜆>𝜔}⊂𝜌(𝐴).

Let (𝜔,𝜃)=𝜆∈ℂ∶arg(𝜆−𝜔)<𝜃. (17)

Definition 7 (see [27]). A solution operator 𝑆_𝑞(𝑡) is said to be analytic if 𝑆_𝑞(⋅)∶ℝ⁺→ℒ(𝑋) admits analytic extension to a sector ∑(0,𝜃₀) for some 0<𝜃₀≤𝜋/2. Furthermore, An analytic resolvent 𝑆_𝑞(𝑡) is said to be of analyticity type (𝜔₀,𝜃₀) if, for 𝜃₀>𝜃 and 𝜔₀<𝜔, there exists 𝑀=𝑀(𝜔,𝜃) such that ‖𝑆_𝑞(𝑡)‖≤𝑀𝑒^𝜔Re𝑧 for 𝑧∈∑(0,𝜃); here Re𝑧 means the real part of 𝑧.

In this work, we assume that solution operator {𝑆_𝑞(𝑡)}_𝑡≥0 is analytic; that is, {𝑆_𝑞(𝑡)}_𝑡≥0 satisfy the following property.(HA)The map 𝑡↦𝑆_𝑞(𝑡) is continuous from [0,𝑇] to ℒ(𝑋) endowed with the uniform operator norm ‖⋅‖_ℒ(𝑋).

Without loss of generality, we have that there exist a positive constant 𝑀 such that ‖𝑆_𝑞(𝑡)‖≤𝑀, for 𝑡≥0.

Definition 8 (see [19]). The Hausdorff measure of noncompactness 𝜒_𝑍 is defined as 𝜒_𝑍(𝐹),=inf𝜖>0∶𝐹canbecoveredbyfinitenumberofballswithradius𝜖 (18) for bounded set 𝐹⊂𝑍, where 𝑍 is a Banach space.

Lemma 9 (see [19]). For any bounded set 𝑈,𝑉⊂𝑌, where 𝑌 is a Banach space. Then, the following properties are fulfilled:(i)𝜒_𝑌(𝑈)=0 if and only if 𝑈 is pre-compact;(ii)𝜒_𝑌(𝑈)=𝜒_𝑌(conv𝑈)=𝜒_𝑌(𝑈), where conv𝑈 and 𝑈 denotes the convex hull and closure of 𝑈, respectively;(iii)𝜒_𝑌(𝑈)⊂𝜒_𝑌(𝑉), when 𝑈⊂𝑉;(iv)𝜒_𝑌(𝑈+𝑉)≤𝜒_𝑌(𝑈)+𝜒_𝑌(𝑉), where 𝑈+𝑉={𝑢+𝑣∶𝑢∈𝑈,𝑣∈𝑉};(v)𝜒_𝑌(𝑈∪𝑉)≤max{𝜒_𝑌(𝑈),𝜒_𝑌(𝑉)};(vi)𝜒_𝑌(𝜆𝑈)=𝜆⋅𝜒_𝑌(𝑈), for any 𝜆∈ℝ;(vii)if the map 𝑃∶𝐷(𝑃)⊂𝑌→𝒵 is continuous and satisfies the Lipschitsz condition with constant 𝜅, then we have that 𝜒_𝒵(𝑃𝑈)≤𝜅𝜒_𝑌(𝑈) for any bounded subset 𝑈⊂𝐷(𝑃), where 𝑌 and 𝒵 are Banach space.

The details on the measure of noncompactness and its applications can be found in a book by Banaś and Goebel [19] and papers [9, 10, 21, 23, 24].

Lemma 10 (see [19]). A bounded and continuous map 𝑄∶𝐷⊂𝑍→𝑍 is a 𝜒_𝑍-contraction if there exists a constant 0<𝜅<1 such that 𝜒_𝑍(𝑄(𝑈))≤𝜅𝜒_𝑍(𝑈), for any bounded closed subset 𝑈⊂𝐷, where 𝑍 is a Banach space.

Lemma 11 (see [28]). Let 𝐷⊂𝑍 be closed and convex with 0∈𝐷 and let the continuous map 𝑄∶𝐷→𝐷 be a 𝜒_𝑍-contraction. If the set {𝑢∈𝐷∶𝑢=𝜆𝑄𝑢,for0<𝜆<1} is bounded, then the map 𝑄 has a fixed point in 𝐷.

Lemma 12 ((Darbo-Sadovskii) [19]). Let 𝐷⊂𝑍 be bounded, closed, and convex. If the continuous map 𝑄∶𝐷→𝐷 is a 𝜒_𝑍-contraction, then the map 𝑄 has a fixed point in 𝐷.

In this work, we consider that 𝜒 denotes the Hausdorff measure of noncompactness in 𝑋, 𝜒_𝐶 denotes the Hausdorff measure in noncompactness of 𝐶([0,𝑇];𝑋) and 𝜒_𝒫𝒞 denotes the Hausdorff measure of noncompactness in 𝒫𝒞([0,𝑇];𝑋).

Lemma 13 (see [19, 21]). If 𝑈 is bounded subset of 𝐶([0,𝑇];𝑋). Then, one has that 𝜒(𝑈(𝑡))≤𝜒_𝐶(𝑈), forall𝑡∈[0,𝑇], where 𝑈(𝑡)={𝑢(𝑡);𝑢∈𝑈}⊆𝑋. Furthermore, if 𝑈 is equicontinuous on [0,𝑇], then 𝜒(𝑈(𝑡)) is continuous on the interval [0,𝑇] and 𝜒_𝐶(𝑈)=sup_{[]𝑡∈0,𝑇}𝜒(𝑈(𝑡)). (19)

Lemma 14 (see [19]). If 𝑈⊂𝐶([0,𝑇];𝑋) is bounded and equicontinuous set, then 𝜒(𝑈(𝑡)) is continuous and 𝜒^𝑡₀𝑈(𝑠)𝑑𝑠≤^𝑡₀𝜒(𝑈(𝑠))𝑑𝑠,∀𝑡∈[0,𝑇], (20) where ∫^𝑡₀𝑈(𝑠)𝑑𝑠={∫^𝑡₀𝑢(𝑠)𝑑𝑠,𝑢∈𝑈}.

Lemma 15 (see [29]). (1) If 𝑈⊂𝒫𝒞([0,𝑇];𝑋) is bounded, then 𝜒(𝑈(𝑡))≤𝜒_𝒫𝒞(𝑈), forall𝑡∈[0,𝑇], where 𝑈(𝑡)={𝑢(𝑡)∶𝑢∈𝑈}⊂𝑋.
(2) If  𝑈 is piecewise equicontinuous on [0,𝑇], then 𝜒(𝑈(𝑡)) is piecewise continuous for 𝑡∈[0,𝑇] and 𝜒_𝒫𝒞(𝑈)=sup𝜒(𝑈(𝑡))∶𝑡∈[0,𝑇]. (21)
(3) If 𝑈⊂𝒫𝒞([0,𝑇];𝑋) is bounded and equicontinuous, then 𝜒(𝑈(𝑡)) is piecewise continuous for 𝑡∈[0,𝑇] and 𝜒^𝑡₀𝑈(𝑠)𝑑𝑠≤^𝑡₀𝜒(𝑈(𝑠))𝑑𝑠,∀𝑡∈[0,𝑇], (22) where ∫^𝑡₀𝑈(𝑠)𝑑𝑠={∫^𝑡₀𝑢(𝑠)𝑑𝑠∶𝑢∈𝑈}.

3. Main Results

In this section, we will establish the existence results of solution for (1) by using solution operator and Hausdorff's measure of noncompactness.

From [13], we adopt the following concept of solution for impulsive differential problem (1).

Definition 16. A piecewise continuous function 𝑢∶(−∞,𝑇]→𝑋 is said to be a mild solution for impulsive problem (1) if 𝑢₀=𝜑, 𝑢(⋅)|_𝐽∈𝒫𝒞 and 𝑢(𝑡)=𝑆_𝑞(𝑡)𝜑(0)+𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑢_𝑠𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝜑(0)+𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑢_𝑡1+𝑆_𝑞𝑡−𝑡₁×𝑔𝑡₁,𝑢_𝑡1+𝐼₁𝑢_𝑡1−𝑔𝑡₁,𝑢_𝑡1+^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑢_𝑠𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡)𝜑(0)+𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+^𝑛_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝐼₁𝑢_𝑡𝑖+^𝑛_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖×𝑔𝑡_𝑖,𝑢_𝑡𝑖+𝐼_𝑖𝑢_𝑡𝑖−𝑔𝑡_𝑖,𝑢_𝑡𝑖+^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑢_𝑠𝑑𝑠,𝑡∈𝑡_𝑛,𝑇, (23) where 𝑆_𝑞(𝑡)=12𝜋𝑖_Γ𝑒^𝜆𝑡𝜆^𝑞−1𝑅𝜆^𝑞,𝐴𝑑𝜆. (24)

Now we list the following assumptions which are required to establish main results.(Hf)The function 𝑓∶𝐽×𝒫→𝑋 satisfies the following conditions:(1)the function 𝑓(⋅,𝑢)∶𝐽→𝑋 is strongly measurable for every 𝑢∈𝒫 and 𝑢₀∈𝒫, 𝑢|_𝐽∈𝒫𝒞;(2)𝑓(𝑡,⋅)∶𝒫→𝑋 is a continuous function for each 𝑡∈𝐽;(3)there exists an integrable function 𝑚_𝑓∶[0,𝑇]→[0,∞) and a nondecreasing continuous function Ω∶[0,∞)→(0,∞) such that 𝑓(𝜏,𝑥)≤𝑚_𝑓(𝜏)Ω‖𝑥‖_𝒫,𝜏∈𝐽,𝑥∈𝒫; (25)(4)there exists an integrable function 𝜂∶[0,𝑇]→[0,∞) such that, for any bounded set 𝐵⊂𝒫, we have 𝜒𝑆_𝑞(𝜏)𝑓(𝜏,𝐵)≤𝜂(𝜏)sup_{−∞≤𝜃≤0}𝜒(𝐵(𝜃)); (26) for almost everywhere 𝜏∈𝐽, where 𝐵(𝜃)={𝑢(𝜃)∶𝑢∈𝐵}.(Hg)(1) For 0<𝛽<1, 𝐴^𝛽𝑔(⋅,⋅) is Lipschitz continuous function for all (𝑡,𝑣)∈𝐽×𝒫 and there exist positive constants 𝐶₁ and 𝐶₂ such that 𝐴^𝛽𝑔(𝑡,𝑢)≤𝐶₁‖𝑢‖_𝒫+𝐶₂; (27)(2)there exists a constant 𝐿_𝑔>0 such that 𝐴^𝛽𝑔(𝑡,𝑢)−𝐴^𝛽𝑔(𝑡,𝑣)≤𝐿_𝑔‖𝑢−𝑣‖_𝒫, (28) for all 𝑢,𝑣∈𝒫.(HI)The functions 𝐼_𝑖∶𝑋→𝑋,(𝑖=1,…,𝑛) are continuous functions and satisfy the following conditions:(1)There is a constant 𝐿_𝐼>0 such that 𝐼_𝑖(𝑢)−𝐼_𝑖(𝑣)≤𝐿_𝐼‖𝑢−𝑣‖_𝒫,∀𝑢,𝑣∈𝒫. (29)(2)There exist positive constants 𝐿_𝑗(𝑗=1,2) such that 𝐼_𝑖(𝑢)=𝐿₁‖𝑢‖_𝒫+𝐿₂, (30) for all 𝑢∈𝒫.(H1) 𝐾_𝑇𝑀1−𝜁₂^𝑡₀𝑚_𝑓(𝑠)𝑑𝑠<^∞_𝑑𝑑𝑠Ω(𝑠), (31) where 𝜁₁=𝑀𝐾_𝑇𝐻+𝑁_𝑇+𝑀𝐶₁𝐴^−𝛽𝐾_𝑇𝜑_𝒫+𝐾_𝑇𝐶₂𝐴^−𝛽(𝑀+1)+𝐾_𝑇𝑛𝑀𝐿₂1+𝐿_𝑔,𝜁₂=𝐴^−𝛽𝐶₁+𝑛𝑀𝐿₁1+𝐿_𝑔𝐾_𝑇𝜁<1,𝑑=¹1−𝜁₂; (32)(H2)𝐾_𝑇𝐴^−𝛽𝐿_𝑔+𝑛𝑀𝐿_𝐼+𝑀𝑛𝐿_𝑔2+𝐿_𝐼+^𝑡₀𝜂(𝑠)𝑑𝑠<1. (33)

Now, let 𝑧∶(−∞,𝑇]→𝑋 be a function given by 𝑧₀=𝜑 and 𝑧(𝑡)=𝑆_𝑞(𝑡)𝜑(0) on 𝐽. It is easy to see that ‖𝑧_𝑡‖_𝒫≤(𝐾_𝑇𝑀𝐻+𝑁_𝑇)‖𝜑‖_𝒫, where 𝐾_𝑇=sup_{0≤𝑡≤𝑇}𝐾(𝑡), 𝑁_𝑇=sup_{0≤𝑡≤𝑇}𝑁(𝑡).

Theorem 17. Suppose that hypotheses (𝐻𝐴), (𝐻𝑓), (𝐻𝑔), (𝐻𝐼), and (𝐻1)−(𝐻2) are satisfied. Then, there exists a mild solution for the impulsive problem (1).

Proof. Consider the space 𝑆(𝑇)={𝑢∶(−∞,𝑇]→𝑋∶𝑢₀=0,𝑢|_𝐽∈𝒫} endowed with supremum norm ‖⋅‖_𝒫. Define the operator 𝑄∶𝑆(𝑇)→𝑆(𝑇) by(𝑡)=(]𝑆𝑄𝑥)(0,𝑡∈−∞,0_𝑞(𝑡)𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+𝑧_𝑡+^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑢_𝑠+𝑧_𝑠𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+𝑧_𝑡+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑢_𝑡1+𝑧_𝑡1+𝑆_𝑞𝑡−𝑡₁×𝑔𝑡₁,𝑢_𝑡1+𝑧_𝑡1+𝐼₁𝑢_𝑡1+𝑧_𝑡1−𝑔𝑡₁,𝑢_𝑡1+𝑧_𝑡1+^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑢_𝑠+𝑧_𝑠𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡)𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+𝑧_𝑡+^𝑛_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝐼₁𝑢_𝑡𝑖+𝑧_𝑡𝑖+^𝑛_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖×𝑔𝑡_𝑖,𝑢_𝑡𝑖+𝑧_𝑡𝑖+𝐼_𝑖𝑢_𝑡𝑖+𝑧_𝑡𝑖−𝑔𝑡_𝑖,𝑢_𝑡𝑖+𝑧_𝑡𝑖+^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑢_𝑠+𝑧_𝑠𝑑𝑠,𝑡∈𝑡_𝑛,𝑇, (34)and we have that 𝑢_𝑡+𝑧_𝑡𝒫≤𝑢_𝑡𝒫+𝑧_𝑡𝒫,≤𝐾(𝑡)sup{‖𝑢(𝑠)‖∶0≤𝑠≤𝑡}+𝑁(𝑡)𝑢_0𝒫+𝐾(𝑡)sup{‖𝑧(𝑠)‖∶0≤𝑠≤𝑡}+𝑁(𝑡)𝑧_0𝒫,≤𝐾(𝑡)‖𝑢‖_𝑡+𝐾(𝑡)𝑀𝐻𝜑_𝒫+𝑁(𝑡)𝜑_𝒫,≤𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐾_𝑇‖𝑢‖_𝑡, (35) where ‖𝑢‖_𝑡=sup_{0≤𝑠≤𝑡}‖𝑢(𝑠)‖. Thus, 𝑄 is well defined and with the values in 𝑆(𝑇) by our assumptions. By Lebesgue dominated convergence theorem, axioms of phase space and assumptions (Hf), (Hg), and (HI), it is clear that 𝑄 is continuous map. Furthermore, by uniformly continuity of the map 𝑡↦𝑆_𝑞(𝑡) on (0,𝑇], we obtain that set 𝑄(𝑆(𝑇)) is equicontinuous. We prove the result in following steps.
Step 1. The set {𝑢∈𝒫𝒞∶𝑢=𝜆𝑄𝑢,for0<𝜆<1} is bounded.
Let 𝑢_𝜆 be a solution of 𝑢=𝜆𝑄𝑢 for 0<𝜆<1. Therefore, we have that 𝑢_𝜆𝑡+𝑧_𝑡𝒫≤𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐾_𝑇𝑢_𝜆𝑡. (36) Take 𝑣_𝜆(𝑡)=(𝐾_𝑇𝑀𝐻+𝑁_𝑇)‖𝜑‖_𝒫+𝐾_𝑇‖𝑢_𝜆‖_𝑡, for 𝑡∈𝐽. Then, we get that for 𝑡∈[0,𝑡₁]𝑢_𝜆(𝑡)=𝜆𝑄𝑢_𝜆(𝑡)≤𝑄𝑢_𝜆(𝑡,𝐴)≤𝑀^−𝛽𝐶₁𝜑_𝒫+𝐶₂+𝐴^−𝛽𝐶₁𝑣_𝜆(𝑡)+𝐶₂+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝑑𝑠. (37) For 𝑡∈(𝑡₁,𝑡₂], we have 𝑢_𝜆(𝑡)≤𝑀𝐴^−𝛽𝐶₁𝜑_𝒫+𝐶₂+𝐴^−𝛽𝐶₁𝑣_𝜆(𝑡)+𝐶₂+𝑀𝐼₁𝑣_𝜆𝑡₁+𝑀𝑔𝑡₁,𝑣_𝜆𝑡₁+𝐼₁𝑣_𝜆𝑡₁−𝑔𝑡₁,𝑣_𝜆𝑡₁+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠𝐴)𝑑𝑠,≤𝑀^−𝛽𝐶₁𝜑_𝒫+𝐶₂+𝐴^−𝛽𝐶₁𝑣_𝜆(𝑡)+𝐶₂+𝑀𝐿₁𝑣_𝜆(𝑡)+𝐿₂+𝑀𝐿_𝑔𝐿₁𝑣_𝜆(𝑡)+𝐿₂+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝑑𝑠. (38) For 𝑡∈(𝑡_𝑛,𝑇], 𝑢_𝜆(𝑡)≤𝑀𝐴^−𝛽𝐶₁𝜑_𝒫+𝐶₂+𝐴^−𝛽𝐶₁𝑣_𝜆(𝑡)+𝐶₂+𝑛𝑀𝐿₁𝑣_𝜆(𝑡)+𝐿₂+𝑛𝑀𝐿_𝑔𝐿₁𝑣_𝜆(𝑡)+𝐿₂+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝑑𝑠. (39) Therefore, for all 𝑡∈[0,𝑇]=𝐽, we have 𝑢_𝜆(𝑡)≤𝑀𝐴^−𝛽𝐶₁𝜑_𝒫+𝐶₂+𝐴^−𝛽𝐶₁𝑣_𝜆(𝑡)+𝐶₂+𝑛𝑀𝐿₁1+𝐿_𝑔𝑣_𝜆(𝑡)+𝑛𝑀𝐿₂1+𝐿_𝑔+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝐴𝑑𝑠,≤𝑀^−𝛽𝐶₁𝜑_𝒫+𝐶₂+𝐴^−𝛽𝐶₂+𝑛𝑀𝐿₂1+𝐿_𝑔+𝐴^−𝛽𝐶₁+𝑛𝑀𝐿₁1+𝐿_𝑔𝑣_𝜆(𝑡)+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝑑𝑠. (40) From 𝑣_𝜆(𝑡)=(𝐾_𝑇𝑀𝐻+𝑁_𝑇)‖𝜑‖_𝒫+𝐾_𝑇‖𝑢_𝜆‖_𝑡, it implies that 𝑣_𝜆(𝑡)≤𝑀𝐾_𝑇𝐻+𝑁_𝑇+𝑀𝐴^−𝛽𝐶₁𝐾_𝑇𝜑_𝒫+𝐾_𝑇𝐴^−𝛽𝐶₂(𝑀+1)+𝐾_𝑇𝑛𝑀𝐿₂1+𝐿_𝑔+𝐾_𝑇𝐴^−𝛽𝐶₁+𝑛𝑀𝐿₁1+𝐿_𝑔𝑣_𝜆(𝑡)+𝑀𝐾_𝑇^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝑑𝑠, (41) and consequently, 𝑣_𝜆(𝑡)≤𝜁₁1−𝜁₂+𝑀𝐾_𝑇1−𝜁₂^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝑑𝑠,=𝑑+𝑀𝐾_𝑇1−𝜁₂^𝑡₀𝑚_𝑓(𝑠)Ω𝑣_𝜆(𝑠)𝑑𝑠, (42) where 𝜁₁=𝑀𝐾_𝑇𝐻+𝑁_𝑇+𝑀𝐶₁𝐴^−𝛽𝐾_𝑇𝜑_𝒫+𝐾_𝑇𝐴^−𝛽𝐶₂(𝑀+1)+𝑛𝐾_𝑇𝑀𝐿₂1+𝐿_𝑔,𝜁₂=𝐾_𝑇𝐴^−𝛽𝐶₁+𝑛𝑀𝐿₁1+𝐿_𝑔. (43) Let 𝜉_𝜆(𝑡)=𝑑+(𝑀𝐾_𝑇/(1−𝜁₂))∫^𝑡₀𝑚_𝑓(𝑠)Ω(𝜉_𝜆(𝑠))𝑑𝑠. Thus 𝜉_𝜆(0)=𝑑. Therefore, we get 𝜉^′_𝜆(𝑡)Ω𝜉_𝜆(𝑡)≤𝐾_𝑇𝑀1−𝜁₂𝑚_𝑓(𝑡). (44) Integrating above inequality we have that ^𝜉𝜆_𝑑^(𝑡)𝑑𝑠Ω(𝑠)≤𝐾_𝑇𝑀1−𝜁₂^𝑡₀𝑚_𝑓(𝑠)𝑑𝑠<^∞_𝑑𝑑𝑠Ω(𝑠). (45) It gives that the functions 𝜉_𝜆(𝑡) are bounded on interval 𝐽. Therefore, the functions 𝑣_𝜆(𝑡) are bounded and 𝑢_𝜆(⋅) are also bounded on 𝐽.
Step 2. The map 𝑄 is a 𝜒-contraction.
Firstly, we introduce decomposition of 𝑄 into 𝑄=𝑄₁+𝑄₂, for 𝑡≥0 such that 𝑄₁𝑢(𝑡)=𝑆_𝑞(𝑡)𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+𝑧_𝑡,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝑔0,𝜑−𝑔𝑡,𝑢_𝑡+𝑧_𝑡+^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝐼_𝑖𝑢_𝑡𝑖+𝑧_𝑡𝑖+^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖×𝑔𝑡_𝑖,𝑢_𝑡𝑖+𝑧_𝑡𝑖+𝐼_𝑖𝑢_𝑡𝑖+𝑧_𝑡𝑖−𝑔𝑡_𝑖,𝑢_𝑡𝑖+𝑧_𝑡𝑖,𝑡∈𝑡_𝑚,𝑡_𝑚+1𝑄,𝑚=1,2,3,…,𝑛;₂𝑢(𝑡)=^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑢_𝑠+𝑧_𝑠𝑑𝑠,𝑡∈[0,𝑇]. (46) To prove the result, we firstly show that 𝑄₁ is Lipschitz continuous. For 𝑥₁,𝑥₂∈𝑆(𝑇) and 𝑡∈[0,𝑡₁], we have that 𝑄₁𝑥₁(𝑡)−𝑄₁𝑥₂(𝑡)=𝑔𝑡,𝑥_1𝑡+𝑧_𝑡−𝑔𝑡,𝑥_2𝑡+𝑧_𝑡,≤𝐴^−𝛽𝐿_𝑔𝑥_1𝑡−𝑥_2𝑡. (47) For 𝑡∈(𝑡_𝑚,𝑡_𝑚+1], 𝑚=1,2,…,𝑛 we have 𝑄₁𝑥₁(𝑡)−𝑄₁𝑥₂(𝑡≤𝐴_−𝛽)𝐿_𝑔𝑥_1𝑡−𝑥_2𝑡𝒫+𝑚𝑀𝐿_𝐼𝑥_1𝑡−𝑥_2𝑡𝒫+𝑀𝑚𝐿_𝑔2+𝐿_𝐼𝑥_1𝑡−𝑥_2𝑡𝒫,=𝐾_𝑇𝐴^−𝛽𝐿_𝑔+𝑚𝑀𝐿_𝐼+𝑀𝑚𝐿_𝑔2+𝐿_𝐼×𝑥₁−𝑥_2𝑇. (48) Thus for 𝑡∈[0,𝑇] we have 𝑄₁𝑥₁(𝑡)−𝑄₁𝑥₂(𝑡)≤𝐾_𝑇𝐴^−𝛽𝐿_𝑔+𝑛𝑀𝐿_𝐼+𝑀𝑛𝐿_𝑔2+𝐿_𝐼×𝑥₁−𝑥_2𝑇. (49) Taking supremum on [0,𝑇], we get 𝑄₁𝑥₁−𝑄₁𝑥_2𝑇≤𝐾_𝑇𝐴^−𝛽𝐿_𝑔+𝑛𝑀𝐿_𝐼+𝑀𝑛𝐿_𝑔2+𝐿_𝐼×𝑥₁−𝑥_2𝑇. (50) Hence, it implies that 𝑄₁ satisfies the Lipschitz condition with Lipschitz constant 𝐿, where 𝐿=𝐾_𝑇[‖𝐴^−𝛽‖𝐿_𝑔+𝑛𝑀𝐿_𝐼+𝑀𝑛𝐿_𝑔(2+𝐿_𝐼)].
Therefore, from Lemma 9 (vii), we have that for any bounded set 𝐵∈𝒫𝒞𝜒_𝒫𝒞𝑄₁𝐵≤𝐿𝜒_𝒫𝒞(𝐵). (51)
Next, we show that 𝑄₂ is a 𝜒-contraction. Let 𝐵 be an arbitrary bounded subset 𝑆(𝑇). Since 𝑆_𝑞(𝑡) is equicontinuous solution operator, therefore 𝑆_𝑞(𝑡−𝑠)𝑓(𝑠,𝑢_𝑠+𝑧_𝑠) is piecewise continuous. From Lemma 9, we have that for any bounded set 𝐵∈𝒫𝒞, 𝜒𝑄₂𝐵(𝑡)=𝜒^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝐵_𝑠+𝑧_𝑠≤𝑑𝑠_𝑡0𝜂(𝑠)sup_{−∞<𝜔≤0}𝜒(𝐵()(≤𝑠+𝜔+𝑧𝑠+𝜔))𝑑𝑠_𝑡0𝜂(𝑠)sup_{0≤𝜏≤𝑠}𝜒(𝐵(𝜏))𝑑𝑠≤𝜒_𝒫𝒞(𝐵)^𝑡₀𝜂(𝑠)𝑑𝑠,𝑡∈[0,𝑇]. (52) Thus for any bounded set 𝐵∈𝒫𝒞𝜒_𝒫𝒞(𝑄𝐵)=𝜒_𝒫𝒞𝑄₁𝐵+𝑄₂𝐵≤𝜒_𝒫𝒞𝑄₁𝐵+𝜒_𝒫𝒞𝑄₂𝐵≤𝐿+^𝑡₀𝜂(𝑠)𝑑𝑠𝜒_𝒫𝒞(𝐵). (53) By the assumption (H2), we obtain that 𝜒_𝒫𝒞(𝑄𝐵)<𝜒_𝒫𝒞(𝐵); that is, 𝑄 is a contraction. Therefore, 𝑄 has at least one fixed point in 𝐵 by Darbo fixed point theorem. Let 𝑢 be a fixed point of 𝑄 on 𝑆(𝑇), then 𝑦=𝑢+𝑧 is a mild solution for (1).

Theorem 18. Suppose that (Hf), (Hg), and (HI) are satisfied and 𝐾_𝑇𝐴^−𝛽𝐶₁+𝑛𝑀𝐿₁1+𝐿_𝑔+^𝑇₀𝑚_𝑓(𝑠)𝑑𝑠lim_𝜏→∞Ω(𝜏)sup𝜏<1. (54) Then, there exists a mild solution for the impulsive problem (1).

Proof. Proceeding as in the proof of Theorem 17, we infer that 𝑄 defined by (34) is continuous from 𝑆(𝑇) into 𝑆(𝑇). Next we indicate that there exists 𝑟>0 such that 𝑄(𝐵_𝑟)⊂𝐵_𝑟, where 𝐵 is defined by 𝐵_𝑟={𝑢∈𝑆(𝑇)∶‖𝑢‖_𝑇≤𝑟}. To this end, let us assume that assertion is false, then for any 𝑟>0 there exists 𝑢_𝑟∈𝐵_𝑟 and 𝑡_𝑟∈𝐽 such that 𝑟<‖𝑄𝑢_𝑟(𝑡_𝑟)‖. Therefore, for 𝑡_𝑟∈[0,𝑡₁] and 𝑢_𝑟∈𝐵_𝑟, 𝑟<𝑄𝑢_𝑟𝑡_𝑟≤𝑀𝑔𝑡,𝜑+𝑔𝑡,𝑢_𝑟𝑡𝑟+𝑧_𝑡𝑟+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑢_𝑟𝑠+𝑧_𝑠𝑔+𝐴𝑑𝑠,≤𝑀𝑡,𝜑^−𝛽𝐶₁𝑢_𝑟𝑡𝑟+𝑧_𝑡𝑟𝒫+𝐶₂+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑢_𝑟𝑠+𝑧_𝑠𝑔+𝐴𝑑𝑠,≤𝑀𝑡,𝜑^−𝛽×𝐶₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐶₁𝐾_𝑇𝑟+𝐶₂+𝑀^𝑡₀𝑚_𝑓(𝑠)𝑑𝑠Ω𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐾_𝑇𝑟=𝑟₀. (55) For 𝑡_𝑟∈(𝑡₁,𝑡₂], we have 𝑟<𝑄𝑢_𝑟𝑡_𝑟𝑔+𝐴≤𝑀𝑡,𝜑_−𝛽𝐶₁𝑢_𝑟𝑡𝑟+𝑧_𝑡𝑟𝒫+𝐶₂+𝑀𝐿₁𝑢_𝑟𝑡𝑟+𝑧_𝑡𝑟𝒫+𝐿₂+𝑀𝐿_𝑔𝐿₁𝑢_𝑟𝑡𝑟+𝑧_𝑡𝑟𝒫+𝐿₂+𝑀^𝑡₀𝑚_𝑓(𝑠)Ω𝑢_𝑟𝑠+𝑧_𝑠𝑔+𝐴𝑑𝑠,≤𝑀𝑡,𝜑^−𝛽×𝐶₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐶₁𝐾_𝑇𝑟+𝐶₂+𝑀𝐿₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐿₁𝐾_𝑇𝑟+𝐿₂+𝑀𝐿_𝑔𝐿₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐿₁𝐾_𝑇𝑟+𝐿₂+𝑀^𝑡₀𝑚_𝑓(𝑠)𝑑𝑠Ω𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐾_𝑇𝑟,=𝑟₁. (56) For 𝑡_𝑟∈(𝑡_𝑛,𝑇], we have 𝑟<𝑄𝑢_𝑟𝑡_𝑟𝑔+𝐴≤𝑀𝑡,𝜑_−𝛽𝐶₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐶₁𝐾_𝑇𝑟+𝐶₂+𝑛𝑀𝐿₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐿₁𝐾_𝑇𝑟+𝐿₂+𝑛𝑀𝐿_𝑔𝐿₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐿₁𝐾_𝑇𝑟+𝐿₂+𝑀^𝑡₀𝑚_𝑓(𝑠)𝑑𝑠Ω𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐾_𝑇𝑟,=𝑟_𝑛, (57) which implies that 𝑟<max{𝑟₀,𝑟₁,…,𝑟_𝑛}. Therefore, we conclude that 𝑟<𝑀𝑔𝑡,𝜑+𝐴^−𝛽×𝐶₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐶₁𝐾_𝑇𝑟+𝐶₂+𝑛𝑀𝐿₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐿₁𝐾_𝑇𝑟+𝐿₂+𝑛𝑀𝐿_𝑔𝐿₁𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐿₁𝐾_𝑇𝑟+𝐿₂+𝑀^𝑡₀𝑚_𝑓(𝑠)𝑑𝑠Ω𝐾_𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐾_𝑇𝑟. (58) We divide both the sides of (58) by 𝑟 and letting 𝑟→∞, we obtain that 1<𝐾_𝑇𝐴^−𝛽𝐶₁+𝑛𝑀𝐿₁+𝑛𝑀𝐿_𝑔𝐿₁+𝑀^𝑇₀𝑚_𝑓(𝑠)𝑑𝑠×lim_𝑟→∞Ω𝐾sup^𝑇𝑀𝐻+𝑁_𝑇𝜑_𝒫+𝐾_𝑇𝑟𝑟,<𝐾_𝑇𝐴^−𝛽𝐶₁+𝑛𝑀𝐿₁1+𝐿_𝑔+^𝑇₀𝑚_𝑓(𝑠)𝑑𝑠lim_𝜏→∞Ω(𝜏)sup𝜏. (59) This contradicts the inequality (54). Hence, there exists a positive constant 𝑟>0 such that 𝑄(𝐵_𝑟)⊂𝐵_𝑟. Moreover, by uniform continuity of the map 𝑡↦𝑆_𝑞(𝑡) on (0,𝑇], we have that set 𝑄(𝐵_𝑟) is equicontinuous. As the proof of Theorem 17, we infer that system (1) has a mild solution.

4. Example

In this section, we consider an example to illustrate the application of the theory. Here we take the space 𝐶₀×𝐿²(ℎ,𝑋) as the phase space 𝒫 (see, [7]).

Now we study the following fractional differential equation with infinite delay: 𝑑^𝑞𝑑𝑡^𝑞𝑢(𝑡,𝑥)+^𝑡_−∞^𝜋₀𝑏𝑢=𝜕𝑡−𝑠,𝜇,𝑥𝑠,𝜇𝑑𝜇𝑑𝑠₂𝑑𝑥²𝑢(𝑡,𝑥)+^𝑡_−∞^𝜋₀𝑏𝑡−𝑠,𝜇,𝑥𝑢𝑠,𝜇𝑑𝜇𝑑𝑠+ℐ_𝑡^1−𝑞𝑡_−∞𝑓(([][],𝑢()()𝑢()()[],𝑡𝑡,𝑡−𝑠,𝑥,𝑢𝑠,𝑥))𝑑𝑠,𝑡∈0,𝑇,𝑥∈0,𝜋𝑡,0=𝑢𝑡,𝜋=0,0≤𝑡≤𝑇,𝜏,𝑥=𝜑𝜏,𝑥,0≥𝜏,𝑥∈0,𝜋Δ𝑢_𝑘,𝑥=^𝑡𝑘_−∞𝑎_𝑘𝑡_𝑘−𝑠𝑢(𝑠,𝑥)𝑑𝑠,𝑘=1,2,…,𝑛, (60) where 𝜑∈𝐶₀×𝐿²(ℎ;𝑋) and 0<𝑡₁<⋯<𝑡_𝑛<𝑡_𝑛+1=𝑇 are prefixed numbers. Let 𝑋=𝐿²[0,𝜋] and consider the operator 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 as 𝐴𝑣=𝑣^′′ with the domain 𝐷(𝐴)=𝑣∈𝑋∶𝑣,𝑣^′areabsolutelycontinuousand𝑣^′′𝑣(0)(𝜋.∈𝑋,=0=𝑣) (61) Then, 𝐴=^∞_𝑛=1𝑛²𝑣,𝑣_𝑛𝑣_𝑛,𝑣∈𝐷(𝐴), (62) Here 𝑣_𝑛(𝑥)=(2/𝜋)sin(𝑛𝑥), 𝑛∈ℕ is the orthogonal set of eigenvectors of 𝐴. It is clear that 𝐴 is the infinitesimal generator of an analytic semigroup {𝑆(𝑡)}_𝑡≥0 in 𝑋 which is given by 𝑆(𝑡)𝑣=^∞_𝑛=1𝑒^−𝑛2𝑡𝑣,𝑣_𝑛𝑣_𝑛,∀𝑣∈𝑋,𝑡>0. (63) By Theorem 3.1 in [27], we get that 𝐴 is the infinitesimal generator of solution operator {𝑆_𝑞(𝑡)}_𝑡≥0 and there exists a positive constant 𝑀 such that ‖𝑆_𝑞(𝑡)‖_ℒ(𝑋)≤𝑀 for 𝑡∈[0,𝑇]. The functions 𝑏,𝑓,𝑎_𝑘 satisfy the following conditions:(1)the functions 𝑏(𝑠,𝜇,𝑥), (𝜕𝑏/𝜕𝑥)(𝑠,𝜇,𝑥) are measurable such that 𝑏(𝑠,𝜇,𝜋)=𝑏(𝑠,𝜇,0)=0 and 𝐿_𝑔=max^𝜋₀⁰_−∞^𝜋₀1ℎ(𝑠)𝜕^𝑖𝑏𝑠,𝜇,𝑥𝜕𝑥^𝑖2𝑑𝜇𝑑𝑠𝑑𝑥^1/2∶𝑖=0,1<∞; (64)(2)𝑓∶ℝ⁴→ℝ is continuous and there exist a 𝜂∶𝐶(ℝ²,ℝ) such that 𝑓𝑡₁,𝑡₂,𝑥,𝑢≤𝜂𝑡₁,𝑡₂‖𝑢‖,𝑡₁,𝑡₂,𝑥,𝑢∈ℝ⁴; (65)(3)the functions 𝑎_𝑘∶[0,∞)→ℝ, 𝑘=1,2,…,𝑛 are continuous and 𝐿_𝐼=⁰_−∞𝑎_𝑘(𝑠)²ℎ(𝑠)𝑑𝑠^1/2<∞. (66)The impulsive system (60) might be reformulated as the abstract impulsive Cauchy problem (1) where 𝑔𝑡,𝜙(𝑥)=⁰_−∞^𝜋₀𝑏𝜙𝐹(𝑥)=𝑠,𝑦,𝑥𝑠,𝑦𝑑𝑦𝑑𝑠,𝑡,𝜙⁰_−∞𝑓(𝐼𝑡,𝑠,𝑥,𝜙𝑠,𝑥)𝑑𝑠,_𝑘𝜙(𝑥)=⁰_−∞𝑎_𝑘(𝑠)𝜙(𝑠,𝑥)𝑑𝑠,𝑘=1,…,𝑛. (67) It may be verified that 𝑔, 𝐹 satisfy the assumptions (Hf), (Hg), and (HI); that is, 𝑔(𝑡,⋅) and 𝐼_𝑘, 𝑘=1,…,𝑛 are bounded linear operators and the range of 𝑔(⋅) is contained in 𝑋_1/2, ‖𝐴^1/2𝑔(𝑡,⋅)‖≤𝐿_𝑔, ‖𝐼_𝑘‖≤𝐿_𝐼, 𝑘=1,…,𝑛, and ‖𝐹(𝑡,𝜙)‖≤𝑚_𝐹(𝑡)‖𝜙‖_𝒫 for every 𝑡∈[0,𝑇], where 𝑚_𝐹(𝑡) is defined as 𝑚_𝐹(𝑡)=(∫⁰_−∞𝜂(𝑡,𝑠)²/ℎ(𝑠)𝑑𝑠)^1/2. Applying Theorem 17, we obtain that problem (60) has a mild solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the UGC (University Grants Commission, India) under Grant no. (6405-11-061).nt.