A Note on Impulsive Fractional Evolution Equations with Nondense Domain

Abstract

This paper is concerned with the existence of integral solutions for nondensely defined fractional functional differential equations with impulse effects. Some errors in the existing paper concerned with nondensely defined fractional differential equations are pointed out, and correct formula of integral solutions is established by using integrated semigroup and some probability densities. Sufficient conditions for the existence are obtained by applying the Banach contraction mapping principle. An example is also given to illustrate our results.

1. Introduction

The aim in this paper is to study the existence of the integral solutions for the fractional semilinear differential equations of the form 𝐷^𝑞𝑦(𝑡)=𝐴𝑦(𝑡)+𝑓𝑡,𝑦_𝑡,𝑡∈𝐽∶=[0,𝑏],𝑡≠𝑡_𝑘,𝑘=1,…,𝑚,Δ𝑦_{𝑡=𝑡𝑘}=𝐼_𝑘𝑦𝑡⁻_𝑘𝑦(𝑡)(𝑡)[],,𝑘=1,…,𝑚,=𝜙,𝑡∈−𝜏,0 (1.1) where 0<𝑞<1, 𝐷^𝑞 is the Caputo fractional derivative. 𝑓∶𝐽×𝒟→𝐸 is a given function, 𝒟={𝜓∶[−𝜏,0]→𝐸,𝜓 is continuous everywhere except for a finite number of points 𝑠 at which 𝜓(𝑠⁻),𝜓(𝑠⁺) exist and 𝜓(𝑠⁻)=𝜓(𝑠)}, and 𝐸 is a real Banach space with the norm |⋅|. Denoting the domain of 𝐴 by 𝐷(𝐴), 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 is nondensely closed linear operator on 𝐸, 𝜙∈𝒟. 𝐼_𝑘∶𝐸→𝐸, 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑏, 𝑦(𝑡⁺_𝑘) and 𝑦(𝑡⁻_𝑘) represent the right and left limits at 𝑡_𝑘 of 𝑦(𝑡) as usual; we assume 𝑦(𝑡⁻_𝑘)=𝑦(𝑡_𝑘). Δ𝑦|_{𝑡=𝑡𝑘}=𝑦(𝑡⁺_𝑘)−𝑦(𝑡⁻_𝑘) represents the jump in the state 𝑦 at time 𝑡_𝑘. Moreover, for any 𝑡∈𝐽, the histories 𝑦_𝑡 belong to 𝒟 defined by 𝑦_𝑡(𝜍)=𝑦(𝑡+𝜍),𝜍∈[−𝜏,0].

In the past decades, the theory of fractional differential equations has become an important area of investigation because of its wide applicability in many branches of physics, economics, and technical sciences [1–10]. In recent years, many authors were devoted to mild solutions to fractional evolution equations, and there have been a lot of interesting works. For instance, in [11], El-Borai discussed the following equation in Banach space 𝑋: 𝐷^𝑞𝑢(𝑡)(𝑡)(𝑡)𝑢(𝑡),𝑢(0)=𝐴𝑢+𝐵=𝑢₀, (1.2) where 𝐴 generates an analytic semigroup, and the solution was given in terms of some probability densities. In [12], Zhou and Jiao concerned the existence and uniqueness of mild solutions for fractional evolution equations by some fixed point theorems. Cao et al. [13] studied the 𝛼-mild solutions for a class of fractional evolution equations and optimal controls in fractional powder space. For more information on this subject, the readers may refer to [14–16] and the references therein.

Research on integer order differential evolution equations including a nondensely defined operator was initialed by Da Prato and Sinestrari [17] and has been extensively investigated by many authors [18–25]. The main methods used in their work are based on integrated semigroup theory. Recently, existence results for integral solutions of nondensely defined fractional evolution equations were established in some papers [9, 26]. But there are some errors in transforming integral solution into an available form. For example, definition of integral solution [9] is given by 𝑥(𝑡)=𝑆(𝑡)𝑥₀−𝑔(𝑥)+lim_𝜆→∞1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓(𝑠,𝑥(𝑠))𝑑𝑠,𝑡≥0. (1.3) Here 𝐷(𝐴)⊂𝐸 and 𝐷(𝐴)≠𝐸. 𝐵(𝜆,𝐴)∶=𝜆(𝜆𝐼−𝐴)⁻¹ will be introduced in next section. If we let 𝑓 take values in 𝐷(𝐴), then (1.3) becomes 𝑥(𝑡)=𝑆(𝑡)𝑥₀(𝑥)+1−𝑔Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑆(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠. (1.4) According to [19], integral solution should be mild solution in this case. But as pointed in [14], (1.4) is not the mild solution.

Motivated by these papers and the fact that impulse effects exist widely in the realistic situations, we give the definition of integral solution and prove the existence results for impulsive semilinear fractional differential equations with nondensely defined operators. The rest of the paper will be organized as follows. In Section 2, we will recall some basic definitions and preliminary facts from integrated semigroups and fractional derivation and integration which would be used later. Section 3 is devoted to the existence of integral solutions of problem (1.1). We present an example to illustrate our results in Section 4. At last, we end the paper with a conclusion.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary results which would be used in the rest of the paper.

We denote by 𝐶([0,𝑏];𝐸) the Banach space of all continuous functions from [0,𝑏] into 𝐸 with the norm ‖𝑥‖_∞=sup𝑦(𝑡)∶𝑡∈[0,𝑏]. (2.1) For 𝜙∈𝒟 the norm of 𝜙 is defined by 𝜙_𝒟=sup𝜙(𝜍)∶𝜍∈[−𝜏,0]. (2.2)𝐵(𝐸) denotes the Banach space of bounded linear operators from 𝐸 into 𝐸, with the norm ‖𝑁‖=sup𝑁𝑦∶𝑦=1, (2.3) where 𝑁∈𝐵(𝐸) and 𝑦∈𝐸. Let 𝐿^𝑝([0,𝑏];𝐸) be the space of 𝐸-valued Bochner function on [0,𝑏] with the norm ‖𝑥‖_𝐿𝑝=^𝑏₀𝑦(𝑠)^𝑝𝑑𝑠^1/𝑝,1≤𝑝<∞. (2.4) In order to define an integral solution of problem (1.1), we will introduce the set of functions PC=𝑦∶𝐽⟶𝐷(𝐴),iscontinuousexceptfor𝑡=𝑡_𝑘𝑡,𝑘=1,2,…,𝑚,thereexist𝑦⁻_𝑘and𝑦𝑡⁺_𝑘suchthat𝑦𝑡⁻_𝑘=𝑦𝑡_𝑘. (2.5) Endowed with the norm ‖𝑦‖_PC=sup_𝑡∈𝐽|𝑦(𝑡)|, (PC,‖⋅‖_PC) is a Banach space.

Seting Ω=𝑦∶[−𝜏,𝑏]⟶𝐷(𝐴)∶𝑦∈𝒟∩PC, (2.6) then Ω is a Banach space with the norm 𝑦_Ω=max𝑦_𝒟,𝑦_PC. (2.7)

Definition 2.1 (see [27]). Letting 𝐸 be a Banach space, an integrated semigroup is a family of operators (𝑆(𝑡))_𝑡≥0 of bounded linear operators 𝑆(𝑡) on 𝐸 with the following properties:(i)𝑆(0)=0;(ii)𝑡→𝑆(𝑡) is strongly continuous;(iii)𝑆(𝑠)𝑆(𝑡)=∫^𝑠₀(𝑆(𝑡+𝑟)−𝑆(𝑟))𝑑𝑟 for all 𝑡,𝑠≥0.

Definition 2.2 (see [28]). An operator 𝐴 is called a generator of an integrated semigroup, if there exists 𝜔∈𝐑 such that (𝜔,+∞)⊂𝜌(𝐴), and there exists a strongly continuous exponentially bounded family (𝑆(𝑡))_𝑡≥0 of linear bounded operators such that 𝑆(0)=0 and (𝜆𝐼−𝐴)⁻¹=𝜆∫^∞₀𝑒^−𝜆𝑡𝑆(𝑡)𝑑𝑡 for all 𝜆>𝜔.

Proposition 2.3 (see [27]). Let 𝐴 be the generator of an integrated semigroup (𝑆(𝑡))_𝑡≥0. Then for all 𝑥∈𝐸 and 𝑡≥0, ^𝑡₀𝑆(𝑠)𝑥𝑑𝑠∈𝐷(𝐴),𝑆(𝑡)𝑥=𝐴^𝑡₀𝑆(𝑠)𝑥𝑑𝑠+𝑡𝑥. (2.8)

Definition 2.4     (see [29]). We say that a linear operator 𝐴 satisfies the Hille-Yosida condition if there exists 𝑀≥0 and 𝜔∈𝐑 such that (𝜔,+∞)⊂𝜌(𝐴) and sup(𝜆−𝜔)^𝑛(𝜆𝐼−𝐴)^−𝑛,𝑛∈𝐍,𝜆>𝜔≤𝑀. (2.9)
Here and hereafter, we assume that 𝐴 satisfies the Hille-Yosida condition. Let us introduce the part 𝐴₀ of 𝐴 in 𝐷(𝐴)∶𝐴₀=𝐴 on𝐷(𝐴₀)={𝑥∈𝐷(𝐴);𝐴𝑥∈𝐷(𝐴)}. Let (𝑆(𝑡))_𝑡≥0 be the integrated semigroup generated by 𝐴. We note that (𝑆′(𝑡))_𝑡≥0 is a 𝐶₀-semigroup on 𝐷(𝐴) generated by 𝐴₀ and ‖𝑆′(𝑡)‖≤𝑀𝑒^𝜔𝑡, 𝑡≥0, where 𝑀 and 𝜔 are the constants considered in the Hille-Yosida condition [28, 30].
Let 𝐵(𝜆,𝐴)∶=𝜆(𝜆𝐼−𝐴)⁻¹; then for all 𝑥∈𝐷(𝐴), 𝐵(𝜆,𝐴)𝑥→𝑥 as 𝜆→∞. Also from the Hille-Yosida condition it is easy to see that lim_𝜆→∞|𝐵(𝜆,𝐴)𝑥|≤𝑀|𝑥|.
For more properties on integral semigroup theory the interested reader may refer to [18, 30].

Definition 2.5 (see [3]). The Riemann-Liouville fractional integral of order 𝑞∈𝐑⁺ of a function ℎ∶(0,𝑏]→𝐸 is defined by 𝐼^𝑞₀ℎ(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1ℎ(𝑠)𝑑𝑠, (2.10) provided the right-hand side is pointwise defined on (0,𝑏] and where Γ is the gamma function.

Remark 2.6. According to [10], 𝐼^𝑞₀𝐼^𝛽₀=𝐼₀^𝑞+𝛽 holds for all 𝑞, 𝛽≥0.

Definition 2.7 (see [3]). The Caputo fractional derivative of order 0<𝑞<1 of a function 𝑓∈𝐶¹([0,∞),𝐸) is defined by 𝐷^𝑞𝑓(𝑡)=1Γ1−𝑞^𝑡₀(𝑡−𝑠)^−𝑞𝑓′(𝑠)𝑑𝑠,𝑡>0. (2.11)

3. Main Results

In this section we will establish the existence and uniqueness of integral solution for problem (1.1).

Definition 3.1. A function 𝑦∈Ω is said to be an integral solutions of (1.1) if(i)∫^𝑡_𝑡𝑘(𝑡−𝑠)^𝑞−1𝑦(𝑠)𝑑𝑠∈𝐷(𝐴) for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1],𝑘=0,1,…,𝑚,(ii)𝑦(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0],(iii) 𝑦(𝑡)=𝜙(0)+1Γ𝑞𝐴^𝑡₀(𝑡−𝑠)^𝑞−1𝑦(𝑠)1𝑑𝑠+Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈0,𝑡₁,𝑦𝑡⁻₁+𝐼₁𝑦𝑡⁻₁+1Γ𝑞𝐴^𝑡_𝑡1(𝑡−𝑠)^𝑞−1𝑦(𝑠)+1𝑑𝑠Γ𝑞^𝑡_𝑡1(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑦𝑡⁻_𝑚+𝐼_𝑚𝑦𝑡⁻_𝑚+1Γ𝑞𝐴^𝑡_𝑡𝑚(𝑡−𝑠)^𝑞−1𝑦(𝑠)+1𝑑𝑠Γ𝑞^𝑡_𝑡𝑚(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (3.1)

Lemma 3.2. If 𝑦 is an integral solution of (1.1), then for all 𝑡∈[0,𝑏], 𝑦(𝑡)∈𝐷(𝐴). In particular, 𝜙(0), 𝑦(𝑡⁻₁)+𝐼₁(𝑦(𝑡⁻₁)),…,𝑦(𝑡⁻_𝑚)+𝐼₁(𝑦(𝑡⁻_𝑚)) belong to 𝐷(𝐴).

Proof. Using Remark 2.6, for each 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝐼¹_𝑡𝑘𝑦(𝑡)=𝐼_𝑡^1−𝑞_𝑘𝐼^𝑞_𝑡𝑘𝑦(𝑡)∈𝐷(𝐴) since 𝐼^𝑞_𝑡𝑘𝑦(𝑡)∈𝐷(𝐴). Consequently, for ℎ>0 such that 𝑡+ℎ∈(𝑡_𝑘,𝑡_𝑘+1], (1/ℎ)∫_𝑡^𝑡+ℎ𝑦(𝑠)𝑑𝑠∈𝐷(𝐴) because 𝐼¹_𝑡𝑘𝑦(𝑡)=∫^𝑡_𝑡𝑘𝑦(𝑠)𝑑𝑠∈𝐷(𝐴). Hence, we deduce that 𝑦(𝑡)=lim_ℎ→0(1/ℎ)∫_𝑡^𝑡+ℎ𝑦(𝑠)𝑑𝑠∈𝐷(𝐴). The proof is completed.

Lemma 3.3 (see [31]). Let Ψ_𝑞(𝜃)=(1/𝜋)∑^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑞𝑛−1}(Γ(𝑛𝑞+1)/𝑛!)sin(𝑛𝜋𝑞), 𝜃∈𝐑⁺; then Ψ_𝑞(𝜃) is a one-sided stable probability density function, and its Laplace transform is given by ^∞₀𝑒^−𝑝𝜃Ψ_𝑞(𝜃)𝑑𝜃=𝑒^−𝑝𝑞,𝑞∈(0,1),𝑝>0. (3.2)

Lemma 3.4. For 𝑡∈(0,𝑏], the integral solution in Definition 3.1 is given by 𝑦(𝑡)=𝔖(𝑡)𝜙(0)+lim_𝜆→∞^𝑡₀(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈0,𝑡₁,𝔖𝑡−𝑡₁𝑦𝑡⁻₁+𝐼₁𝑦𝑡⁻₁+lim_𝜆→∞^𝑡_𝑡1(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝔖𝑡−𝑡_𝑚𝑦𝑡⁻_𝑚+𝐼_𝑚𝑦𝑡⁻_𝑚+lim_𝜆→∞^𝑡_𝑡𝑚(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈𝑡_𝑚,𝑏, (3.3) where 𝔖(𝑡)=^∞₀ℎ_𝑞(𝜃)𝑆^′𝑡^𝑞𝜃𝑑𝜃,𝔗(𝑡)=𝑞^∞₀𝜃ℎ_𝑞(𝜃)𝑆′𝑡^𝑞𝜃𝑑𝜃, (3.4) where ℎ_𝑞(𝜃)=(1/𝑞)𝜃^{−1−1/𝑞}Ψ_𝑞(𝜃^−1/𝑞) is the probability density function defined on 𝐑⁺.

Proof. From the definition, for 𝑡∈(0,𝑡₁] we have 𝑦(𝑡)(0)+1=𝜙Γ𝑞𝐴^𝑡₀(𝑡−𝑠)^𝑞−1𝑦(𝑠)1𝑑𝑠+Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈[0,𝑏]. (3.5) Consider the Laplace transform 𝑣𝑝=^∞₀𝑒^−𝑝𝑡𝐵(𝜆,𝐴)𝑦(𝑡)𝑑𝑡,𝑤𝑝=^∞₀𝑒^−𝑝𝑡𝐵(𝜆,𝐴)𝑓𝑡,𝑦_𝑡𝑑𝑡,𝑝>0. (3.6) Note that for each 0<𝑡≤𝑡₁, 𝐵_𝜆𝑦(𝑡),𝐵(𝜆,𝐴)𝑓(𝑡,𝑦_𝑡)∈𝐷(𝐴), then we have 𝑣(𝑝),𝑤(𝑝)∈𝐷(𝐴). Applying (3.6) to (3.5) yields 𝑣𝑝=1𝑝𝐵()𝜙(0)+1𝜆,𝐴𝑝^𝑞𝑝+1𝐴𝑣𝑝^𝑞𝑤𝑝=𝑝^𝑞−1𝑝^𝑞𝐼−𝐴⁻¹𝐵(𝜆,𝐴)𝜙(0)+𝑝^𝑞𝐼−𝐴⁻¹𝑤𝑝=𝑝^𝑞−1∞₀𝑒^{−𝑝𝑞𝑠}𝑆′(𝑠)𝐵(𝜆,𝐴)𝜙(0)𝑑𝑠+^∞₀𝑒^{−𝑝𝑞𝑠}𝑆′(𝑠)𝑤𝑝𝑑𝑠, (3.7) where 𝐼 is the identity operator defined on 𝐸.
From (3.2), we get 𝑝^𝑞−1∞₀𝑒^{−𝑝𝑞𝑠}𝑆^′(𝑠)𝐵()𝜙(0)=𝜆,𝐴𝑑𝑠_∞0𝑝^𝑞−1𝑒^{−(𝑝𝑡)𝑞}𝑆′𝑡^𝑞𝐵(𝜆,𝐴)𝜙(0)𝑞𝑡^𝑞−1=𝑑𝑡_∞0−1𝑝𝑑𝑒𝑑𝑡^{−(𝑝𝑡)𝑞}𝑆′𝑡^𝑞𝐵(𝜆,𝐴)𝜙(0)𝑑𝑡=^∞₀𝜃Ψ_𝑞(𝜃)𝑒^{−𝑝𝑡𝜃}𝑆^′𝑡^𝑞𝐵()𝜙(0)=𝜆,𝐴𝑑𝜃𝑑𝑡_∞0Ψ_𝑞(𝜃)𝑒^−𝑝𝑠𝑆^′𝑠𝜃^𝑞𝐵()𝜙(0)=𝜆,𝐴𝑑𝜃𝑑𝑠_∞0𝑒^{−𝑝𝑡∞}₀Ψ_𝑞(𝜃)𝑆^′𝑡𝜃^𝑞𝐵(𝜆,𝐴)𝜙(0)𝑑𝜃𝑑𝑡,_∞0𝑒^{−𝑝𝑞𝑠}(𝑠)𝑤𝑝=𝑆′𝑑𝑠_∞0𝑒^{−𝑝𝑞𝑠}𝑒^−𝑝𝑡𝑆^′(𝑠)𝐵(𝜆,𝐴)𝑓𝑡,𝑦_𝑡=𝑑𝑡𝑑𝑠_∞0𝑞𝑠^𝑞−1𝑒^{−(𝑝𝑠)𝑞}𝑒^−𝑝𝑡𝑆^′𝑠^𝑞𝐵(𝜆,𝐴)𝑓𝑡,𝑦_𝑡=𝑑𝑡𝑑𝑠_∞0𝑞Ψ_𝑞(𝜃)𝑒^{−𝑝𝑠𝜃}𝑒^−𝑝𝑡𝑆^′𝑠^𝑞𝐵(𝜆,𝐴)𝑓𝑡,𝑦_𝑡=𝑑𝜃𝑑𝑡𝑑𝑠_∞0𝑞Ψ_𝑞(𝜃)𝑒^{−𝑝(𝑠+𝑡)}𝑠^𝑞−1𝜃^𝑞𝑆^′𝑠𝜃^𝑞𝐵(𝜆,𝐴)𝑓𝑡,𝑦_𝑡𝑑𝜃𝑑𝑡𝑑𝑠=^∞₀𝑒^−𝑝𝑠𝑞^𝑠₀^∞₀Ψ_𝑞(𝜃)(𝑠−𝑡)^𝑞−1𝜃^𝑞𝑆^′(𝑠−𝑡)^𝑞𝜃^𝑞𝐵(𝜆,𝐴)𝑓𝑡,𝑦_𝑡=𝑑𝜃𝑑𝑡𝑑𝑠_∞0𝑒^−𝑝𝑡𝑞^𝑡₀^∞₀Ψ_𝑞(𝜃)(𝑡−𝑠)^𝑞−1𝜃^𝑞𝑆^′(𝑡−𝑠)^𝑞𝜃^𝑞𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝜃𝑑𝑠𝑑𝑡. (3.8) According to (3.7) and (3.8), we have 𝑣𝑝=^∞₀𝑒^{−𝑝𝑡∞}₀Ψ_𝑞(𝜃)𝑆^′𝑡𝜃^𝑞𝐵(𝜆,𝐴)𝜙(0)𝑑𝜃𝑑𝑡+^∞₀𝑒^−𝑝𝑡𝑞^𝑡₀^∞₀Ψ_𝑞(𝜃)(𝑡−𝑠)^𝑞−1𝜃^𝑞𝑆^′(𝑡−𝑠)^𝑞𝜃^𝑞𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝜃𝑑𝑠𝑑𝑡. (3.9) Inverting the last Laplace transform, we obtain 𝐵(𝜆,𝐴)𝑦(𝑡)=^∞₀Ψ_𝑞(𝜃)𝑆^′𝑡𝜃^𝑞𝐵(𝜆,𝐴)𝜙(0)𝑑𝜃+𝑞^𝑡₀^∞₀Ψ_𝑞(𝜃)(𝑡−𝑠)^𝑞−1𝜃^𝑞𝑆^′(𝑡−𝑠)^𝑞𝜃^𝑞𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠=𝑑𝜃𝑑𝑠_∞0ℎ_𝑞(𝜃)𝑆′𝑡^𝑞𝜃𝐵(𝜆,𝐴)𝜙(0)𝑑𝜃+𝑞^𝑡₀^∞₀𝜃(𝑡−𝑠)^𝑞−1ℎ_𝑞(𝜃)𝑆^′(𝑡−𝑠)^𝑞𝜃𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝜃𝑑𝑠. (3.10) In view of lim_𝜆→∞𝐵(𝜆,𝐴)𝑥=𝑥 for 𝑥∈𝐷(𝐴) and Lemma 3.2, we have 𝑦(𝑡)=^∞₀ℎ_𝑞(𝜃)𝑆′𝑡^𝑞𝜃𝜙(0)𝑑𝜃+lim_𝜆→∞𝑞^𝑡₀^∞₀𝜃(𝑡−𝑠)^𝑞−1ℎ_𝑞(𝜃)𝑆^′(𝑡−𝑠)^𝑞𝜃𝐵(𝜆,𝐴)×𝑓𝑠,𝑦_𝑠(𝑡)𝜙(0)𝑑𝜃𝑑𝑠=𝔖+lim_𝜆→∞^𝑡₀(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠. (3.11)
For 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘=1,2,…,𝑚, we can prove the results by the similar methods used previously. The proof is completed.

Remark 3.5. According to [31], one can easily check that ^∞₀𝜃ℎ_𝑞(𝜃)𝑑𝜃=^∞₀1𝜃^𝑞Ψ_𝑞(𝜃)1𝑑𝜃=Γ.1+𝑞 (3.12)
We are now in a position to state and prove our main results for the existence and uniqueness of solutions of problem (1.1).

Let us list the following hypotheses.(H1)𝐴 satisfies the Hille-Yosida condition, and assume that 𝑀∶=sup{‖𝑆′(𝑡)‖∶𝑡∈[0,+∞]}<∞.(H2) For 𝑢∈𝒟,𝑓(⋅,𝑢)∶[0,𝑏]→𝐸 is strongly measurable.(H3) There exists a constant 𝑞₁∈(0,𝑞) and 𝑙∈𝐿^1/𝑞1([0,𝑏];𝐑⁺) such that 𝑓(𝑡,𝑢)≤𝑙(𝑡),a.e.𝑡∈𝐽,andeach𝑢∈𝒟. (3.13)(H4) There exists 𝜌>0 such that 𝐼_𝑘(𝑢)−𝐼_𝑘(𝑣)≤𝜌|𝑢−𝑣|∀𝑢,𝑣∈𝐸,𝑘=1,…,𝑚. (3.14)(H5) There exists a constant 𝜅 such that 𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)≤𝜅‖𝑢−𝑣‖_𝒟,for𝑡∈𝐽andevery𝑢,𝑣∈𝒟. (3.15)

Theorem 3.6. Assuming that hypotheses (H1)–(H5) hold, then problem (1.1) has a unique integral solution 𝑦∈Ω provided that 𝑀(1+𝜌)+(𝑀𝑀𝜅𝑏^𝑞/Γ(1+𝑞))<1.

Proof. Define 𝑄∶Ω→Ω by 𝑄𝑦(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0], (3.16) and for 𝑡∈𝐽, (𝑡)=𝔖(𝑡)𝜙(0)𝑄𝑦+lim_𝜆→∞^𝑡₀(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)×𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈0,𝑡₁,𝔖𝑡−𝑡₁𝑦𝑡⁻₁+𝐼₁𝑦𝑡⁻₁+lim_𝜆→∞^𝑡_𝑡1(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝔖𝑡−𝑡_𝑚𝑦𝑡⁻_𝑚+𝐼_𝑚𝑦𝑡⁻_𝑚+lim_𝜆→∞^𝑡_𝑡𝑚(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (3.17)
Firstly we check that 𝑄 is well defined on Ω.
For each 𝑦∈Ω, take 𝑡∈(0,𝑡₁]. It is obvious that 𝔖(𝑡)𝜙(0) is well defined. Direct calculation shows that (𝑡−𝑠)^𝑞−1∈𝐿^{(1/(1−𝑞1))}[0,𝑡], for 𝑡∈[0,𝑡₁] and 𝑞₁∈(0,𝑞). Let 𝑎=𝑞−11−𝑞₁∈(−1,0),𝑀₁=‖𝑙‖_{𝐿11/𝑞[0,𝑏]}. (3.18) Then for 𝑡∈[0,𝑡₁], we have (𝑡−𝑠)^𝑡₀^𝑞−1𝑓𝑠,𝑦_𝑠𝑑𝑠≤^𝑡₀(𝑡−𝑠)^{((𝑞−1)/(1−𝑞1))}𝑑𝑠^1−𝑞1‖𝑙‖_{𝐿11/𝑞[0,𝑡]}≤𝑀₁(1+𝑎)^1−𝑞1𝑏^{(1+𝑎)(1−𝑞1)}. (3.19) From (H1), (3.12), (3.19), and the fact that ‖𝐵(𝜆,𝐴)‖≤𝑀, we get ^𝑡₀^∞₀𝜃(𝑡−𝑠)^𝑞−1ℎ_𝑞(𝜃)𝑆^′(𝑡−𝑠)^𝑞𝜃𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝜃𝑑𝑠≤𝑀𝑀^𝑡₀^∞₀𝜃ℎ_𝑞(𝜃)(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦_𝑠≤𝑑𝜃𝑑𝑠𝑀𝑀₀Γ1+𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦_𝑠≤𝑑𝑠𝑀𝑀₀𝑀₁Γ1+𝑞(1+𝑎)^1−𝑞1𝑏^{(1+𝑎)(1−𝑞1)},for𝑡∈0,𝑡₁. (3.20) It means that |∫^∞₀𝜃(𝑡−𝑠)^𝑞−1ℎ_𝑞(𝜃)𝑆′((𝑡−𝑠)^𝑞𝜃)𝐵(𝜆,𝐴)𝑓(𝑠,𝑦_𝑠)𝑑𝜃| is Lebesgue integrable with respect to 𝑠∈[0,𝑡] for all 𝑡∈[0,𝑡₁]. Therefore ∫^∞₀𝜃(𝑡−𝑠)^𝑞−1ℎ_𝑞(𝜃)𝑆′((𝑡−𝑠)^𝑞𝜃)𝐵(𝜆,𝐴)𝑓(𝑠,𝑦_𝑠)𝑑𝜃 is Bochner integrable with respect to 𝑠∈[0,𝑡] for all 𝑡∈[0,𝑡₁].
From [19], we know lim_𝜆→∞∫^𝑡₀(𝑡−𝑠)^𝑞−1𝑆′((𝑡−𝑠)^𝑞𝜃)𝐵(𝜆,𝐴)𝑓(𝑠,𝑦_𝑠)𝑑𝑠 exists; then lim_𝜆→∞^𝑡₀(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠=lim_𝜆→∞𝑞^𝑡₀^∞₀𝜃(𝑡−𝑠)^𝑞−1ℎ_𝑞(𝜃)𝑆^′(𝑡−𝑠)^𝑞𝜃𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝜃𝑑𝑠=lim_𝜆→∞𝑞^∞₀𝜃ℎ_𝑞(𝜃)^𝑡₀(𝑡−𝑠)^𝑞−1𝑆^′(𝑡−𝑠)^𝑞𝜃𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠𝑑𝜃=𝑞^∞₀𝜃ℎ_𝑞(𝜃)lim_𝜆→∞^𝑡₀(𝑡−𝑠)^𝑞−1𝑆^′(𝑡−𝑠)^𝑞𝜃𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠𝑑𝑠𝑑𝜃 (3.21) exists. Therefore we get (𝑄𝑦)(⋅) which is well defined on [0,𝑡₁].
For 𝑡∈(𝑡_𝑘,𝑡_𝑘+1],𝑘=1,2,…,𝑚, similar discussion could obtain (𝑄𝑦)(⋅) is well defined. Hence, 𝑄 is well defined on Ω.
Secondly, we will prove operator 𝑄 is a contraction.
For 𝑡∈(0,𝑡₁] and 𝑦,𝑧∈Ω, by the hypotheses and ‖𝐵(𝜆,𝐴)‖≤𝑀, we get (𝑡)−(𝑡=𝑄𝑦𝑄𝑧)()lim_𝜆→∞^𝑡₀(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠−𝑓𝑠,𝑧_𝑠=𝑑𝑠lim_𝜆→∞𝑞^𝑡₀^∞₀𝜃(𝑡−𝑠)^𝑞−1ℎ_𝑞(𝜃)𝑆^′(𝑡−𝑠)^𝑞𝜃𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠−𝑓𝑠,𝑧_𝑠≤𝑑𝜃𝑑𝑠𝑀𝑀Γ1+𝑞^𝑡₀𝑞(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑦_𝑠−𝑓𝑠,𝑧_𝑠≤𝑑𝑠𝑀𝑀𝜅Γ1+𝑞^𝑡₀𝑞(𝑡−𝑠)^𝑞−1𝑦_𝑠−𝑧_𝑠𝒟≤𝑑𝑠𝑀𝑀𝜅𝑏^𝑞Γ1+𝑞𝑦−𝑧_Ω. (3.22) Now take 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘=1,2,…,𝑚 and 𝑦,𝑧∈Ω: (𝑡)−(𝑡≤𝔖𝑄𝑦𝑄𝑧)()𝑡−𝑡_𝑘𝑦𝑡⁻_𝑘+𝐼_𝑘𝑦𝑡⁻_𝑘−𝑧𝑡⁻_𝑘−𝐼_𝑘𝑧𝑡⁻_𝑘+lim_𝜆→∞^𝑡_𝑡𝑘(𝑡−𝑠)^𝑞−1𝔗(𝑡−𝑠)𝐵(𝜆,𝐴)𝑓𝑠,𝑦_𝑠−𝑓𝑠,𝑧_𝑠≤𝑑𝑠𝑀1+𝜌𝑦−𝑧_Ω+𝑀𝑀𝜅𝑏^𝑞Γ1+𝑞𝑦−𝑧_Ω≤𝑀1+𝜌+𝑀𝑀𝜅𝑏^𝑞Γ1+𝑞𝑦−𝑧_Ω. (3.23) In view of 𝑀(1+𝜌)+(𝑀𝑀𝜅𝑏^𝑞/Γ(1+𝑞))<1, we have that the operator 𝑄 is a contraction. By the Banach contraction principle we have that 𝑄 has a unique fixed point 𝑦∈Ω, which gives rise to a unique integral solution to the problem (1.1). The proof is finished.

Remark 3.7. For impulsive Caputo fractional differential equations, its integral solutions (or mild solutions; see [14]) can be expressed only by using piecewise functions. Thus Definition 2.3 given in [15] is unsuitable.

4. An Example

As an application of our results we consider the following fractional differential equations of the form 𝐷^𝑞𝑢()=𝜕𝑡,𝑧²𝜕𝑧²𝑢(𝑡,𝑧)+𝐹𝑡,𝑢_𝑡()[][]⧵1𝜍,𝑧,𝑧∈0,𝜋,𝑡∈0,12,𝑢12⁺1,𝑧−𝑢2⁻1,𝑧=𝜌𝑢2⁻[],𝑢()()[],𝑢()()[][].,𝑧,𝑧∈0,𝜋𝑡,0=𝑢𝑡,𝜋=0,𝑡∈0,1𝜍,𝑧=𝜙𝜍,𝑧,𝜍∈−1,0,𝑧∈0,𝜋 (4.1)

Consider 𝐸=𝐶([0,𝜋];𝐑) endowed with the supnorm and the operator 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 defined by 𝐷(𝐴)=𝑢∈𝐶²])()()𝜕([0,𝜋;𝐑∶𝑢𝑡,0=𝑢𝑡,𝜋=0,𝐴𝑢=²𝜕𝑧²𝑢(𝑡,𝑧). (4.2) Now, we have 𝐷(𝐴)={𝑢∈𝐸∶𝑢(𝑡,0)=𝑢(𝑡,𝜋)=0}≠𝐸. As we know from [17] that 𝐴 satisfies the Hille-Yosida condition with (0,+∞)⊆𝜌(𝐴) and 𝜆>0, |𝑅(𝜆,𝐴)|≤1/𝜆. Hence, operator 𝐴 satisfies (H1) and 𝑀=𝑀=1/2.

Then the system (4.1) can be reformulated as 𝐷^𝑞𝑦(𝑡)=𝐴𝑦(𝑡)+𝑓𝑡,𝑦_𝑡[]1,𝑡∈𝐽∶=0,𝑏,𝑡≠2,Δ𝑦_𝑡=1/2𝑦1=𝐼2⁻𝑦(𝑡)(𝑡)[],,𝑘=1,…,𝑚,=𝜙,𝑡∈−𝜏,0 (4.3) where 𝑦(𝑡)(𝑧)=𝑢(𝑡,𝑧), 𝑓(𝑡,𝑦_𝑡)(𝑧)=𝐹(𝑡,𝑢_𝑡(𝜍,𝑧)), 𝐼(𝑥)=𝜌𝑥, 𝜙(𝑡)(𝑧)=𝜙(𝑡,𝑧).

If we take 𝑞=1/3,𝜌=1/10, 𝑓(𝑡,𝑦_𝑡)=(1/(𝑡+1)(𝑡+2))sin𝑦_𝑡. We easily get that 𝑓()(≤1𝑡,𝑢−𝑡𝑡,𝑣)3‖𝑢−𝑣‖_𝒟,for𝑡∈𝐽andevery𝑢,𝑣∈𝒟. (4.4) Then all conditions of Theorem 3.6 are satisfied and we deduce (4.1) has a unique integral solution.

5. Conclusions

An essence error of the formula of solutions which appeared in the recent work on the nondensely defined fractional evolution differential equations is reported in this work. A correct formula of integral solutions for nondensely defined fractional evolution equations could be obtained from the results in this paper.

In view of the complicated definitions for integral or mild solutions for impulsive fractional evolution equations, many fixed point theorems related to completely continuous operators are hard to be used to establish the existence results. As far as we know, only [14] applied Leray Schauder Alternative theorem to the existence of mild solutions of impulsive fractional differential equations. But there is a mistake in proving that Γ(𝐵_𝑟) is equicontinuous (page 2009, Step 3). How to overcome this difficulty is our next work.

Acknowledgments

This work was partially supported by National Natural Science Foundation of China (11171122) and University Science Foundation of Anhui Province (KJ2012B187).