Analysis of Caputo Impulsive Fractional Order Differential Equations with Applications

Abstract

We use Sadovskii's fixed point method to investigate the existence and uniqueness of solutions of Caputo impulsive fractional differential equations of order 𝛼∈(0,1) with one example of impulsive logistic model and few other examples as well. We also discuss Caputo impulsive fractional differential equations with finite delay. The results proven are new and compliment the existing one.

1. Introduction

Dynamics of many evolutionary processes from various fields such as population dynamics, control theory, physics, biology, and medicine. undergo abrupt changes at certain moments of time like earthquake, harvesting, shock, and so forth. These perturbations can be well approximated as instantaneous change of states or impulses. These processes are modeled by impulsive differential equations. In 1960, Milman and Myšhkis introduced impulsive differential equations in their paper [1]. Based on their work, several monographs have been published by many authors like Samoilenko and Perestyuk [2], Lakshmikantham et al. [3], Bainov and Simeonov [4, 5], Bainov and Covachev [6], and Benchohra et al. [7]. All the authors mentioned perviously have considered impulsive differential equations as ordinary differential equations coupled with impulsive effects. They considered the impulsive effects as difference equations being satisfied at impulses time. So, the solutions are piecewise continuous with discontinuities at impulses time. In the fields like biology, population dynamics, and so forth, problems with hereditary are best modeled by delay differential equations [8]. Problems associated with impulsive effects and hereditary property are modeled by impulsive delay differential equations.

The origin of fractional calculus (derivatives (𝑑^𝛼/𝑑𝑡^𝛼)𝑓 and integrals 𝐼^𝛼𝑓 of arbitrary order 𝛼>0) goes back to Newton and Leibniz in the 17th century. In a letter correspondence with Leibniz, L’Hospital asked “What if the order of the derivative is 1/2”? Leibniz replied, “Thus it follows that will be equal to 𝑥√𝑑𝑥∶𝑥, an apparent paradox, from which one day useful consequences will be drawn.” This letter of Leibniz was in September 1695. So, 1695 is considered as the birthday of fractional calculus. Fractional order differential equations are generalizations of classical integer order differential equations and are increasingly used to model problems in fluid dynamics, finance, and other areas of application. Recent investigations have shown that sometimes physical systems can be modeled more accurately using fractional derivative formulations [9]. There are several excellent monographs available on this field [10–15]. In [11], the authors give a recent and up-to-date description of the developments of fractional differential and fractional integro-differential equations including applications. The existence and uniqueness of solutions to fractional differential equations has been considered by many authors [16–21]. Impulsive fractional differential equations represent a real framework for mathematical modeling to real world problems. Significant progress has been made in the theory of impulsive fractional differential equations [7, 22–24]. Xu et al. in their paper [25] have described an impulsive delay fishing model.

Fractional derivatives arise naturally in mathematical problems, for 𝛼>0 and a function 𝑓∶[0,𝑇]→ℝ, recall [10, Definitions 3.1, 2.2](a) the Caputo fractional derivative^𝐶𝐷^𝛼𝑓(𝑡)=1Γ(1−𝛼)^𝑡₀(𝑡−𝑠)^−𝛼𝑓^′(𝑠)𝑑𝑠, (1)(b) the Riemann-Liouville fractional derivative𝐷^𝛼𝑓(𝑡)=𝑑^𝛼𝑑𝑡^𝛼𝑓(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡^𝑡₀(𝑡−𝑠)^−𝛼𝑓(𝑠)𝑑𝑠, (2)provided that the right-hand sides exist pointwise on [0,𝑇] (Γ denotes the gamma function). Using the Riemann-Liouville fractional integral [10, Definition 2.1] 𝐼^𝛼₀𝑓(𝑡)=(1/Γ(𝛼))∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠, we have ^𝐶𝐷^𝛼𝑓(𝑡)=𝐼₀^1−𝛼(𝑑/𝑑𝑡)𝑓(𝑡) and 𝐷^𝛼𝑓(𝑡)=(𝑑/𝑑𝑡)𝐼₀^1−𝛼𝑓(𝑡). 𝐼^𝛼₀𝑓 exists, for instance, for all 𝛼>0, if 𝑓∈𝐶⁰([0,𝑇])∩𝐿¹_loc([0,𝑇]); moreover, 𝐼^𝛼₀𝑓(0)=0. Throughout the paper, we assume that 𝐼=[0,𝑇].

One can see that both of the fractional derivatives are actually nonlocal operator because integral is a nonlocal operator. Moreover, calculating time fractional derivative of a function at some time requires all the past history, and hence fractional derivatives can be used for modeling systems with memory. Fractional differential equations can be formulated using both Caputo and Riemann-Liouville fractional derivatives. A Riemann-Liouville initial value problem can be stated as follows: 𝐷^𝛼𝑥(𝑡)((𝑡[],𝐷=𝑓𝑡,𝑥)),𝑡∈𝐼=0,𝑇_𝛼−1𝑥(0)=𝑥₀, (3) or equivalently, 𝑥(𝑡)=𝑥₀𝑡^𝛼−1/Γ(𝛼)+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠,𝑥(𝑠))𝑑𝑠 in its integral representation [11, Theorem 3.24]. For a physical interpretation of the initial conditions in (3), see [26–28]. If derivatives of Caputo type are used instead of Riemann-Liouville type, then initial conditions for the corresponding Caputo fractional differential equations can be formulated as for classical ordinary differential equations, namely, 𝑥(0)=𝑥₀.

Our main objective is to discuss existence and uniqueness of solutions of the following impulsive fractional differential equation of Caputo type in a Banach space 𝑋 with norm ‖⋅‖_𝑋: ^𝐶𝐷^𝛼𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝐼=[0,𝑇],𝑡≠𝑡_𝑘,Δ𝑥(𝑡)∣_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡⁻_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑥₀, (4) where 𝑓∈𝐶(𝐼×𝑋,𝑋), 𝐼_𝑘∶𝑋→𝑋 and 𝑥₀∈𝑋. 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, Δ𝑥(𝑡)|_{𝑡=𝑡𝑘}=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝐾), 𝑥(𝑡⁺_𝑘)=lim_ℎ→0𝑥(𝑡+ℎ), and 𝑥(𝑡⁻_𝑘)=lim_ℎ→0𝑥(𝑡−ℎ). We break our function 𝑓 into two components which satisfy different conditions. We observed that this kind of functions occurs in ecological modeling. We have given the example of logistic equation in the last section. Our main tool is Sadovskii’s fixed point theorem.

Now, we define some important spaces and norm which will encounter frequently: PC(𝐼,𝑋)=𝑥∶[0,𝑇]→𝑋∣𝑥∈𝐶𝑡₀,𝑡₁𝑡,𝑋∪𝐶_𝑘,𝑡_𝑘+1𝑥𝑡,𝑋𝑘=1,2,…,𝑚,_+𝑘and𝑥𝑡⁻_𝑘exist,𝑥𝑡_𝑘=𝑥𝑡⁻_𝑘, (5) with sup-norm ‖⋅‖, defined by ‖𝑥‖=sup{‖𝑥(𝑡)‖_𝑋∶𝑡∈𝐼}.

Definition 1. A solution of fractional differential equation (4) is a piecewise continuous function 𝑥∈PC([0,𝑇],𝑋) which satisfied (4).

Definition 2 ([29], Definition 11.1). Kuratowskii noncompactness measure: let 𝑀 be a bounded set in metric space (𝑋,𝑑), then Kuratowskii noncompactness measure, 𝜇(𝑀) is defined asinf{𝜖:𝑀 covered by a finite many sets such that the diameter of each set ≤𝜖}.

Definition 3 ([29], Definition 11.6). Condensing map: Let Φ∶𝑋→𝑋 be a bounded and continuous operator on Banach space 𝑋 such that 𝜇(Φ(𝐵))<𝜇(𝐵) for all bounded set 𝐵⊂𝐷(Φ), where 𝜇 is the Kuratowskii noncompactness measure, then Φ is called condensing map.

Definition 4. Compact map: a map 𝑓∶𝑋→𝑋 is said to be compact if the image of every bounded subset of 𝑋 under 𝑓 is precompact (closure is compact).

Theorem 5 (see [30]). Let B be a convex, bounded, and closed subset of a Banach space 𝑋 and let Φ∶𝐵→𝐵 be a condensing map. Then, Φ has a fixed point in 𝐵.

Lemma 6 ([29], Example 11.7). A map Φ=Φ₁+Φ₂∶𝑋→𝑋 is 𝑘-contraction with 0≤𝑘<1 if  (a) Φ₁ is 𝑘-contraction, that is, ‖Φ₁(𝑥)−Φ₁(𝑦)‖_𝑋≤𝑘‖𝑥−𝑦‖_𝑋; (b) Φ₂ is compact,
and hence Φ is a condensing map.

The structure of the paper is as follows. In Section 2, we prove the existence (Theorem 8) and uniqueness of solutions to (4). We show in Section 3 the existence and uniqueness of solutions for a general class of impulsive functional differential equations of fractional order 𝛼∈(0,1). In Section 4, we give some examples in favor of our sufficient conditions.

2. Impulsive Fractional Differential Equation

Consider the initial value problem (4) on the cylinder 𝑅={(𝑡,𝑥)∈ℝ×𝑋∶𝑡∈[0,𝑇],𝑥∈𝐵(0,𝑟)} for some fixed 𝑇>0, 𝑟>0, and assume that there exist 𝑝∈(0,𝛼), 𝛼∈(0,1), 𝑀₁,𝑀₂,𝐿₁∈𝐿_1/𝑝([0,𝑇],ℝ⁺) and functions 𝑓₁,𝑓₂∈𝐶(𝑅×𝑋,𝑋) such that 𝑓=𝑓₁+𝑓₂, and the following assumptions are satisfied:(A.1)𝑓₁ is bounded and Lipschitz, in particular, ‖𝑓₁(𝑡,𝑥)‖_𝑋≤𝑀₁(𝑡) and ‖𝑓₁(𝑡,𝑥)−𝑓₁(𝑡,𝑦)‖_𝑋≤𝐿₁(𝑡)‖𝑥−𝑦‖_𝑋 for all (𝑡,𝑥),(𝑡,𝑦)∈𝑅,(A.2)𝑓₂ is compact and bounded, in particular, ‖𝑓₂(𝑡,𝑥)‖_𝑋≤𝑀₂(𝑡) for all (𝑡,𝑥)∈𝑅,(A.3)𝐼_𝑘∈𝐶(𝑋,𝑋) such that ‖𝐼_𝑘(𝑥)‖_𝑋≤𝑙₁ and ‖𝐼_𝑘(𝑥)−𝐼_𝑘(𝑦)‖_𝑋≤𝑙₂‖𝑥−𝑦‖_𝑋, where 𝐶(𝑅×𝑋,𝑋) denotes set of continuous functions from 𝑅×𝑋 to 𝑋 and 𝐿_1/𝑝([0,𝑇],ℝ⁺) denotes space 1/𝑝-Lesbegue measurable functions from [0,𝑇] to ℝ⁺ with norm ‖⋅‖_1/𝑝.

Lemma 7 (Fečken et al. [24], Lemma 2). The initial value problem (4) is equivalent to the nonlinear integral equation 𝑥(𝑡)=𝑥₀+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁((𝑠+1𝑠,𝑥))𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈0,𝑡₁=𝑥₀+𝐼₁𝑥𝑡⁻₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁((𝑠+1𝑠,𝑥))𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝑡₁,𝑡₂=𝑥₀+²_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁((𝑠+1𝑠,𝑥))𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝑡₂,𝑡₃=𝑥₀+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁((𝑠+1𝑠,𝑥))𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝑡_𝑚,𝑇. (6) In other words, every solution of the integral equation (6) is also solution of our original initial value problem (4) and conversely.

Theorem 8 (existence of solutions). Under the assumptions (A.1)–(A.3), the problem (4) has at least one solution in [0,𝑇], provided that 𝛾₁=𝑚𝑙₂+𝑐𝐿_11/𝑝𝑇^𝛼−𝑝Γ(𝛼+1)<1,where𝑐=1−𝑝𝛼−𝑝^1−𝑝. (7)

Proof. Let 𝐵_𝜆 be the closed bounded and convex subset of PC([0,𝑇],𝑋), where 𝐵_𝜆 is defined as 𝐵_𝜆={𝑥∶‖𝑥‖≤𝜆},𝜆=max{𝜆₀,𝜆₁,…,𝜆_𝑚}, and 𝜆_𝑘=𝑥_0𝑋+𝑘𝑙₁+𝑐𝑀_11/𝑝+𝑀_21/𝑝Γ(𝛼)𝑇^𝛼−𝑝,𝑘=0,1,2,…,𝑚. (8) Define a map 𝐹∶𝐵_𝜆→𝑋 such that 𝐹𝑥(𝑡)=𝑥₀+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁((𝑠+1𝑠,𝑥))𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠. (9) Let us consider that 𝐹₁𝑥(𝑡)=𝑥₀+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁((𝑠𝐹𝑠,𝑥))𝑑𝑠,₂𝑥(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠. (10)Step  1 (𝐹 is self mapping). Now, we show that 𝐹(𝐵_𝑟)⊂𝐵_𝑟. For 𝑡∈[0,𝑡₁]‖𝐹𝑥(𝑡)‖_𝑋≤𝑥_0𝑋+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠⋅𝑥(𝑠))_𝑋≤𝑥𝑑𝑠_0𝑋+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁(𝑠,𝑥(𝑠))_𝑋+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))_𝑋≤𝑥𝑑𝑠_0𝑋+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₁(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₂(𝑠)≤𝑥𝑑𝑠_0𝑋+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝑡₀𝑀₁^1/𝑝(𝑠)𝑑𝑠^𝑝+1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝𝑡₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝑥_0𝑋+𝑐𝑀_11/𝑝+𝑀_21/𝑝Γ(𝛼)𝑇^𝛼−𝑝=𝜆₀. (11) For 𝑡∈(𝑡₁,𝑡₂], ‖𝐹𝑥(𝑡)‖_𝑋≤𝑥_0𝑋+𝐼₁𝑥𝑡⁻₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠⋅𝑥(𝑠))_𝑋≤𝑥𝑑𝑠_0𝑋+𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁(𝑠,𝑥(𝑠))_𝑋+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂(𝑠,𝑥(𝑠))_𝑋≤𝑥𝑑𝑠_0𝑋+𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₁(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₂(𝑠)≤𝑥𝑑𝑠_0𝑋+𝑙₁+1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝𝑡₀𝑀₁^1/𝑝(𝑠)𝑑𝑠^𝑝+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝑡₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝑥_0𝑋+𝑙₁+𝑐𝑀_11/𝑝+𝑀_21/𝑝Γ(𝛼)𝑇^𝛼−𝑝=𝜆₁. (12) For 𝑡∈(𝑡₂,𝑡₃], ‖𝐹𝑥(𝑡)‖≤𝑥_0𝑋+²_𝑘=0𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠⋅𝑥(𝑠))_𝑋≤𝑥𝑑𝑠_0𝑋+2𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁(𝑠,𝑥(𝑠))_𝑋+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂((𝑠≤𝑥𝑠,𝑥))𝑑𝑠_0𝑋+2𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₁(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₂(𝑠)≤𝑥𝑑𝑠_0𝑋+2𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝑡₀𝑀₁^1/𝑝(𝑠)𝑑𝑠^𝑝+1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝𝑡₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝑥_0𝑋+2𝑙₁+𝑐𝑀_11/𝑝+𝑀_21/𝑝Γ(𝛼)𝑇^𝛼−𝑝=𝜆₂. (13) For 𝑡∈(𝑡_𝑚,𝑇], ‖𝐹𝑥(𝑡)‖_𝑋≤𝑥_0𝑋+^𝑚_𝑘=0𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠⋅𝑥(𝑠))_𝑋≤𝑥𝑑𝑠_0𝑋+𝑚𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₁(𝑠,𝑥(𝑠))_𝑋+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓₂((𝑠≤𝑥𝑠,𝑥))𝑑𝑠_0𝑋+𝑚𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₁(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₂(𝑠)≤𝑥𝑑𝑠_0𝑋+𝑚𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝑡₀𝑀₁^1/𝑝(𝑠)𝑑𝑠^𝑝+1Γ(𝛼)×^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝𝑡₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝑥_0𝑋+𝑚𝑙₁+𝑐𝑀_11/𝑝+𝑀_21/𝑝Γ(𝛼)𝑇^𝛼−𝑝=𝜆_𝑚, (14) and thus 𝐹(𝐵_𝜆)⊂𝐵_𝜆.
Step  2 (𝐹₁ is continuous and 𝛾-contraction). To prove the continuity of 𝐹₁ for 𝑡∈[0,𝑇], let us consider a sequence 𝑥_𝑛 converging to 𝑥. Taking the norm of 𝐹₁𝑥_𝑛(𝑡)−𝐹₁𝑥(𝑡), we have 𝐹₁𝑥_𝑛(𝑡)−𝐹₁𝑥(𝑡)_𝑋≤_{0<𝑡𝑘<𝑡}𝐼₁𝑥_𝑛𝑡⁻_𝑘−𝑥𝑡⁻_𝑘𝑋+1Γ𝛼^𝑡₀(𝑡−𝑠)^𝛼−1𝐿(𝑠)𝑥_𝑛(𝑠)−𝑥(𝑠)_𝑋𝑑𝑠≤_{0<𝑡𝑘<𝑡}𝑙₂(𝑥_𝑛𝑡⁻_𝑘−𝑥𝑡⁻_𝑘𝑋+1Γ𝛼^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝑡₀𝐿₁^1/𝑝(𝑠)𝑑𝑠^𝑝𝑥_𝑛≤−𝑥𝑚𝑙₂+𝑐𝐿_11/𝑝𝑇^𝛼−𝑝𝑥Γ𝛼^𝑛−𝑥. (15) From the pervoius analysis, we obtain 𝐹₁𝑥_𝑛−𝐹₁𝑥≤𝑚𝑙₂+𝑐𝐿_11/𝑝𝑇^𝛼−𝑝Γ(𝛼)𝑥_𝑛−𝑥. (16) To prove that 𝐹₁ is 𝛾₁-contraction, let us consider that, for 𝑥,𝑦∈𝐵_𝑟, 𝐹₁𝑥(𝑡)−𝐹₁𝑦(𝑡)_𝑋≤_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡⁻_𝑘−𝑦𝑡⁻_𝑘𝑋+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐿₁(𝑠)𝑥(𝑠)−𝑦(𝑠)_𝑋≤𝑑𝑠_{0<𝑡𝑘<𝑡}𝑙₂(𝑥𝑡⁻_𝑘−𝑦𝑡⁻_𝑘𝑋+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝑡₀𝐿₁^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝑥−𝑦𝑚𝑙₂+𝑐𝐿_11/𝑝𝑇^𝛼−𝑝Γ(𝛼)𝑥−𝑦. (17) Thus, for 𝛾₁=𝑚𝑙₂+𝑐𝐿_11/𝑝𝑇^𝛼−𝑝Γ(𝛼)<1, (18)𝐹₁ is 𝛾₁-contraction.
Step  3 (𝐹₂ is compact). For 0≤𝜏₁≤𝜏₂≤𝑇, we have 𝐹₂𝑥𝜏₂−𝐹₂𝑥𝜏_1𝑋≤1Γ(𝛼)^𝜏2₀𝜏₂−𝑠^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠<?rightdot?><?leftdot?>−^𝜏1₀𝜏₁−𝑠^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠_𝑋≤1Γ(𝛼)^𝜏1₀𝜏₂−𝑠^𝛼−1𝑓₂((𝑠+𝑠,𝑥))𝑑𝑠<?rightdot?>_𝜏2𝜏1𝜏₂−𝑠^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠<?leftdot?>−^𝜏1₀𝜏₁−𝑠^𝛼−1𝑓₂(𝑠,𝑥(𝑠))𝑑𝑠_𝑋≤1Γ(𝛼)^𝜏1₀𝜏₁−𝑠^𝛼−1−𝜏₂−𝑠^𝛼−1×𝑓₂(𝑠,𝑥(𝑠))_𝑋+1𝑑𝑠Γ(𝛼)^𝜏2_𝜏1𝜏₂−𝑠^𝛼−1𝑓₂(𝑠,𝑥(𝑠))_𝑋∵()𝑑𝑠𝜏−𝑠^𝛼−1≤1isadecreasingfunctionof𝜏−𝑠.Γ(𝛼)^𝜏1₀𝜏₁−𝑠^𝛼−1−𝜏₂−𝑠^𝛼−1𝑀₂(𝑠)+1𝑑𝑠Γ(𝛼)^𝜏2_𝜏1𝜏₂−𝑠^𝛼−1𝑀₂(𝑠)≤1𝑑𝑠Γ(𝛼)^𝜏1₀𝜏₁−𝑠^𝛼−1𝜏<?rightdot?><?rightdot?><?rightdot?><?leftdot?><?leftdot?><?leftdot?>−₂−𝑠^{𝛼−11/(1−𝑝)}𝑑𝑠^1−𝑝×^𝜏1₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝+1Γ(𝛼)×^𝜏2_𝜏1𝜏₂−𝑠^{(𝛼−1)/(1−𝑝)}𝑑𝑠^{1−𝑝𝜏1}₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝≤1Γ(𝛼)^𝜏1₀𝜏₁−𝑠^{(𝛼−1)/(1−𝑝)}𝜏<?rightdot?><?rightdot?><?rightdot?><?leftdot?><?leftdot?><?leftdot?>−₂−𝑠^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝜏1₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝∵𝑥^𝑧−𝑦^𝑧≥𝑥−𝑦^𝑧+1,∀𝑥≥𝑦≥0,𝑧>1.Γ(𝛼)^𝜏2_𝜏1𝜏₂−𝑠^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝×^𝜏1₀𝑀₂^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝑐<?displaybreak?>Γ(𝛼)𝜏₁^{(𝛼−𝑝)/(1−𝑝)}−𝜏₂^{(𝛼−𝑝)/(1−𝑝)}𝜏<?rightdot?><?leftdot?>+₂−𝜏₁^{(𝛼−𝑝)/(1−𝑝)1−𝑝}×𝑀_21/𝑝+𝑐Γ(𝛼)𝜏₂−𝜏₁^𝛼−𝑝𝑀_21/𝑝≤𝑐Γ(𝛼)𝜏₁−𝜏₂^{(𝛼−𝑝)/(1−𝑝)1−𝑝}𝑀_21/𝑝+𝑐Γ(𝛼)𝜏₂−𝜏₁^𝛼−𝑝𝑀_21/𝑝∵𝜏₂>𝜏₁.≤2𝑐𝑀_21/𝑝Γ(𝛼)𝜏₂−𝜏₁^𝛼−𝑝. (19) The right-hand side of the pervoiusly expression does not depend on 𝑥. Thus using Arzela-Ascoli theorem for equicontinuous functions (Diethelm, Theorem D.10 [10]), we conclude that 𝐹₂(𝐵_𝑟) is relatively compact, and hence 𝐹₂ is completely continuous on 𝐼−{𝑡₁,𝑡₂,…,𝑡_𝑚}. In a similar way, it can be prove the equi-continuity of 𝐹 on 𝑡=𝑡⁻_𝑘 and 𝑡=𝑡⁺_𝑘, 𝑘=1,2,…,𝑚. And thus 𝐹₂ is compact on [0,𝑇].
Step  4 (𝐹 is condensing). As 𝐹=𝐹₁+𝐹₂, 𝐹₁ is continuous contraction, 𝐹₂ is compact, so by using Lemma 6, 𝐹 is condensing map on 𝐵_𝑟.

And hence by using the Theorem 5, we conclude that (4) has a solution in 𝐵_𝑟.

Theorem 9. If 𝑓 is bounded and Lipschitz, in particular, ‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖_𝑋≤𝐿^∗₁(𝑡)‖𝑥−𝑦‖_𝑋 for all (𝑡,𝑥),(𝑡,𝑦)∈𝑅 and 𝐿^∗₁∈𝐿_1/𝑝([0,𝑇],ℝ⁺), then the problem (4) has a unique solution in 𝐵_𝜆, provided that 𝛾^∗₁=𝑚𝑙₂+𝑐𝐿^∗₁1/𝑝𝑇^𝛼−𝑝Γ(𝛼+1)<1, (20) where 𝑐=((1−𝑝)/(𝛼−𝑝))^1−𝑝.

3. Impulsive Fractional Differential Equations with Finite Delay

In this section, we discuss existence and uniqueness of solutions of the following impulsive fractional differential equations of Caputo type with finite delay in a Banach space 𝑋 with norm |⋅|^𝐶𝐷^𝛼𝑥(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡∈𝐼=[0,𝑇],𝑡≠𝑡_𝑘,Δ𝑥(𝑡)∣_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡⁻_𝑘𝑥(𝑡)(𝑡)[],,𝑘=1,2,…,𝑚,=𝜙,𝑡∈−𝑟,0 (21) where 𝑓∶𝐼×𝒞→𝑋, 𝒞=𝐶([−𝑟,0],𝑋) with norm ‖𝑥‖_𝑟=sup{‖𝑥(𝑡)‖_𝑋∶𝑡∈[−𝑟,0]}. 𝐼_𝑘∈𝐶(𝑋,𝑋), (𝑘=1,2,…,𝑚), and 𝑋 is a Banach space with a norm |⋅|_𝑋. For any 𝑥∶[−𝑟,𝑇]→𝑋 and 𝑡∈𝐼, 𝑥_𝑡∈𝒞 and is defined by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠), 𝑠∈[−𝑟,0]. Here, our tools are Banach and Schaefer fixed point theorems.

Define a new Banach space PC([−𝑟,𝑇],𝑋)PC([−𝑟,𝑇],𝑋)=𝑥∶[−𝑟,𝑇]→𝑋∣𝑥∈𝐶𝑡_𝑘,𝑡_𝑘+1])𝑥𝑡,𝑋∪𝐶([−𝑟,0,𝑋,𝑘=0,1,2,…,𝑚,_+𝑘and𝑥𝑡⁻_𝑘exist,𝑥𝑡_𝑘=𝑥𝑡⁻_𝑘, (22)

with sup-norm ‖⋅‖, defined by ‖𝑥‖=sup{‖𝑥(𝑡)‖_𝑋∶𝑡∈[−𝑟,𝑇]}.

Definition 10. A solution of fractional differential equation (21) is a piecewise continuous function 𝑥∈PC([−𝑟,𝑇],𝑋) which satisfies (21).

Consider the initial value problem (21) on 𝐼×𝒞 for some fixed 𝑇>0 and assume that there exist 𝑝∈(0,𝛼), 𝛼∈(0,1),𝑀₃,𝑀₄,𝐿₂∈𝐿_1/𝑝([0,𝑇],ℝ⁺) such that the following assumptions are satisfied:(A.4)𝑓∈𝐶(𝐼×𝒞,𝑋),(A.5)𝑓 bounded, in particular, ‖𝑓(𝑡,𝜙)‖_𝑋≤𝑀₃(𝑡) for all (𝑡,𝜙)∈𝐼×𝒞,(A.6)𝑓 is Lipschitz, in particular, ‖𝑓(𝑡,𝜙)−𝑓(𝑡,𝜓)‖_𝑋≤𝐿₂(𝑡)‖𝜙−𝜓‖_𝑟 for all (𝑡,𝜙),(𝑡,𝜓)∈𝐼×𝒞.

Lemma 11 (Fečken et al. [24], Lemma 2). The initial value problem (21) is equivalent to the nonlinear integral equation 𝑥(𝑡)(0)+1=𝜙Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑠𝑑𝑠,𝑡∈0,𝑡₁=𝜙(0)+𝐼₁𝑥𝑡⁻₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑠𝑑𝑠,𝑡∈𝑡₁,𝑡₂=𝜙(0)+²_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑠𝑑𝑠,𝑡∈𝑡₂,𝑡₃=𝜙(0)+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑠𝑑𝑠,𝑡∈𝑡_𝑚(𝑡)[].,𝑇=𝜙,𝑡∈−𝑟,0 (23) In other words, every solution of the integral equation (23) is also solution of our original initial value problem (21) and conversely.

Remark 12. Since history part/initial condition 𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝑟,0], is known, so we will investigate the existence and uniqueness of solution in 𝐼=[0,𝑇].

Theorem 13 (existence and uniqueness of solution). Under the assumptions (A.3)–(A.6), the problem (21) has a unique solution in [0,𝑇], provided that 𝛾₂=𝑚𝑙₂+𝑐𝐿_21/𝑝𝑇^𝛼−𝑝Γ(𝛼)<1, (24) where 𝑐=((1−𝑝)/(𝛼−𝑝))^1−𝑝.

Proof. In this case, we define the operator 𝐹∶PC(𝐼,𝑋)→PC(𝐼,𝑋) by 𝐹𝑥(𝑡)=^𝑚_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑠𝑑𝑠. (25)Step  1. To prove that 𝐹 is self mapping, we need to prove that for each 𝑥∈PC(𝐼,𝑋), 𝐹_𝑥∈PC(𝐼,𝑋).
The proof is similar to the proof of continuity of 𝐹₁ in Step  2 of Theorem 8, and hence we omit it.
Step  2 (𝐹 is continuous and 𝛾₂-contraction). The proof of this step is also similar to the proof of continuous and 𝛾₁-contraction of 𝐹₁ in Step 2 of Theorem 8.
Now by applying Banach’s fixed point theorem, we get that the operator 𝐹 has an unique fixed point in PC(𝐼,𝑋), and hence the problem (21) has a unique solution in PC([−𝑟,𝑇],𝑋).

Our next result is based on Schaefer’s fixed point theorem. In this case, we replace assumption (A.3) with the following linear growth condition: (A.3^′)𝐼_𝑘 bounded, in particular, ‖𝐼_𝑘(𝑥)‖_𝑋≤𝑙^∗₁, (A.5^′)𝑓 bounded, in particular, ‖𝑓(𝑡,𝜙)‖_𝑋≤𝑀₄(𝑡)(1+‖𝜙‖_𝑟) for all (𝑡,𝜙)∈𝐼×𝒞.

Theorem 14. Under the assumptions (A.3^') and (A.5^'), problem (21) has at least one solution.

Proof. We transform the problem into a fixed point problem. For this purpose, consider the operator 𝐹∶PC(𝐼,𝑋)→PC(𝐼,𝑋) defined by 𝐹𝑥(𝑡)=^𝑚_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑠𝑑𝑠. (26)Step  1 (𝐹 is continuous). Let {𝑥^𝑛} be a sequence such that 𝑥^𝑛→𝑥 in PC(𝐼,𝑋). Then, for every 𝑡∈𝐼, we have 𝐹𝑥^𝑛(𝑡)−𝐹𝑥(𝑡)_𝑋≤_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥^𝑛𝑡⁻_𝑘−𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ𝛼^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥^𝑛_𝑠−𝑓𝑠,𝑥_𝑠𝑋𝑑𝑠≤_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥^𝑛𝑡⁻_𝑘−𝐼_𝑘𝑥𝑡⁻_𝑘𝑋+𝑓⋅,𝑥^𝑛_.−𝑓⋅,𝑥_.1Γ𝛼^𝑡₀(𝑡−𝑠)^{𝛼−11−𝑝}≤𝑑𝑠_𝑚𝑘=1𝐼_𝑘𝑥^𝑛(⋅)−𝐼_𝑘(𝑥(⋅+𝑇))_𝛼Γ()𝑓𝛼+1⋅,𝑥^𝑛_.−𝑓⋅,𝑥_.. (27) We can see that if 𝑛→∞,𝐹𝑥^𝑛→𝐹𝑥, as 𝑥^𝑛→𝑥 and 𝐼_𝑘 and both 𝑓 are continuous. Hence, 𝐹 is continuous.
Step  2 (𝐹 maps bounded sets into bounded sets in PC(𝐼,𝑋)). It is enough to show that for any 𝛿>0, there exists an 𝑙=‖𝜙(0)‖_𝑟+𝑚𝑙^∗₁+(𝑐(1+𝛿)(‖𝑀₄‖_1/𝑝)/Γ(𝛼))𝑇^𝛼−𝑝>0 such that 𝑥∈𝐵_𝛿={𝑥∈PC(𝐼,𝑋)∣‖𝑥‖≤𝛿}, and we have ‖𝐹𝑥‖≤𝑙.
For 𝑡∈[0,𝑇], we have ‖𝐹𝑥(𝑡)‖_𝑋≤𝜙(0)_𝑟+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑠𝑋≤𝜙(0𝑑𝑠)_𝑟+𝑚𝑙^∗₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₄(𝑠)1+𝑥_𝑠𝑟≤𝜙(0𝑑𝑠)_𝑟+(1+‖𝑥‖)Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₄(𝑠)≤𝜙(0𝑑𝑠)_𝑟+𝑚𝑙^∗₁+(1+‖𝑥‖)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝𝑡₀𝑀₄^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝜙(0)_𝑟+𝑚𝑙^∗₁+𝑐(1+𝛿)𝑀_41/𝑝Γ(𝛼)𝑇^𝛼−𝑝. (28)Step  3 (𝐹 maps bounded sets into equicontinuous sets in PC(𝐼,𝑋)). The proof of this step is similar to the proof of compactness of 𝐹₂ in Step 3 of Theorem 8.
As a consequence of Steps 1–3 together with PC-type Arzela-Ascoli theorem (Fečken et al. [24], Theorem 2.11), the map 𝐹∶PC(𝐼,𝑋)→PC(𝐼,𝑋) is completely continuous.
Step  4 (A priori bounds). Now, we prove that the set 𝐸(𝐹)={𝑥∈PC(𝐼,𝑋)∣𝑥=𝜆𝐹𝑥, for some 𝜆∈(0,1)} is bounded.
We observe that for 𝑡∈[0,𝑇] and 𝑥∈𝐸(𝐹),𝑥(𝑡)=𝜆𝐹𝑥(𝑡) and ‖𝑥(𝑡)‖_𝑋≤‖𝐹𝑥(𝑡)‖_𝑋≤𝜙(0)_𝑟+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡⁻_𝑘𝑋+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠,𝑥_𝑠)_𝑋≤𝜙(0𝑑𝑠)_𝑟+𝑚𝑙^∗₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀₄(𝑠)1+𝑥_𝑠≤𝜙(0𝑑𝑠)_𝑟+𝑚𝑙^∗₁+(1+‖𝑥‖)Γ(𝛼)×^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑝)}𝑑𝑠^1−𝑝𝑡₀𝑀₄^1/𝑝(𝑠)𝑑𝑠^𝑝≤𝜙(0)_𝑟+𝑚𝑙^∗₁+𝑐(1+‖𝑥‖)𝑀_41/𝑝Γ(𝛼)𝑇^𝛼−𝑝, (29) and hence ‖𝑥‖≤𝜙(0)_𝑟+𝑚𝑙^∗₁+𝑐𝑀_41/𝑝/Γ(𝛼)𝑇^𝛼−𝑝1−𝑐𝑀_41/𝑝/Γ(𝛼)𝑇^𝛼−𝑝. (30) This shows that 𝐸(𝐹) is bounded.
As a consequence of Schaefer’s fixed point theorem, the problem (21) has at least one solution in PC([−𝑟,𝑇],𝑋).

Theorem 15. If 𝑓 is bounded and Lipschitz, in particular, ‖𝑓(𝑡,𝜙)−𝑓(𝑡,𝜓)‖_𝑋≤𝐿^∗₂(𝑡)‖𝜙−𝜓‖_𝑟, for all (𝑡,𝜙),(𝑡,𝜓)∈𝐼×𝒞 and 𝐿^∗₂∈𝐿_1/𝑝([0,𝑇],ℝ⁺), then problem (21) has a unique solution in 𝑃𝐶([−𝑟,𝑇],𝑋), provided that 𝛾^∗₂=𝑚𝑙₂+𝑐𝐿^∗_21/𝑝𝑇^𝛼−𝑝Γ(𝛼+1)<1, (31) where 𝑐=((1−𝑝)/(𝛼−𝑝))^1−𝑝.

Further, we consider the following more general Caputo fractional differential equation ^𝐶𝐷^𝛼𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥_𝑡,𝑡∈[0,𝑇],𝑡≠𝑡_𝑘,Δ𝑥(𝑡)∣_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡⁻_𝑘𝑥(𝑡)(𝑡)[],,𝑘=1,2,…,𝑚,=𝜙,𝑡∈−𝑟,0 (32) where 𝑓∶𝐼×𝑋×𝒞→𝑋 and 𝒞=𝐶([−𝑟,0],𝑋) with norm ‖𝑥‖_𝑟=sup{‖𝑥(𝑡)‖_𝑋∶𝑡∈[−𝑟,0]}, 𝐼_𝑘∈𝐶(𝑋,𝑋), (𝑘=1,2,…,𝑚), and 𝑋 is a separable real Banach space with the norm ‖⋅‖_𝑋. Here, our tools will be Banach and Schaefer fixed point theorems.

Definition 16. A solution of fractional differential equation (32) is a piecewise continuous function 𝑥∈PC([−𝑟,𝑇],𝑋) which satisfies (32).

Consider the initial value problem (32) on 𝐼×𝒞×𝑋 for some fixed 𝑇>0 and assume that there exist 𝑝∈(0,𝛼), 𝑀₅,𝑀₆,𝐿₃,𝐿₄∈𝐿_1/𝑝([0,𝑇],ℝ⁺) such that the following assumptions are satisfied: (A.7)𝑓∈𝐶(𝐼×𝑋×𝒞,𝑋), (A.8)𝑓 bounded, in particular, ‖𝑓(𝑡,𝑥,𝜙)‖_𝑋≤𝑀₅(𝑡) for all (𝑡,𝑥,𝜙)∈𝐼×𝑋×𝒞,(A.9)𝑓 is Lipschitz, in particular, ‖𝑓(𝑡,𝑥,𝜙)−𝑓(𝑡,𝑦,𝜓)‖_𝑋≤𝐿₃(𝑡)‖𝑥−𝑦‖_𝑋+𝐿₄(𝑡)‖𝜙−𝜓‖_𝑟 for all (𝑡,𝑥,𝜙),(𝑡,𝑦,𝜓)∈𝐼×𝑋×𝒞,(A.8^′)‖𝑓(𝑡,𝑥,𝜙)‖_𝑋≤𝑀₆(𝑡)(1+‖𝑥‖_𝑋+‖𝜙‖_𝑟).

Lemma 17 (Fečken et al. [24], Lemma 2). The initial value problem (32) is equivalent to the following nonlinear integral equation 𝑥(𝑡)(0)+1=𝜙Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥(𝑠),𝑥_𝑠𝑑𝑠,𝑡∈0,𝑡₁=𝜙(0)+𝐼₁𝑥𝑡⁻₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥(𝑠),𝑥_𝑠𝑑𝑠,𝑡∈𝑡₁,𝑡₂=𝜙(0)+²_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥(𝑠),𝑥_𝑠𝑑𝑠,𝑡∈𝑡₂,𝑡₃=𝜙(0)+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥(𝑠),𝑥_𝑠𝑑𝑠,𝑡∈𝑡_𝑚(𝑡)[].,𝑇=𝜙,𝑡∈−𝑟,0 (33) In other words, every solution of the integral equation (33) is also solution of our original initial value problem (32) and conversely.

Theorem 18. Under the assumptions (A.3), (A.7), (A.8), and (A.9), the problem (32) has a unique solution, provided that 𝛾₃=𝑚𝑙₂+𝑐𝐿_31/𝑝+𝐿_41/𝑝𝑇^𝛼−𝑝Γ(𝛼)<1. (34)

Proof. The proof is similar to the proof of Theorem 13.

Theorem 19. Under the assumptions (A.3^'), and (A.8^'), the problem (32) has at least one solution in 𝑃𝐶([−𝑟,𝑇],𝑋).

Proof. The proof is similar to the proof of Theorem 14.

Theorem 20. If 𝑓 is bounded and Lipschitz, in particular, ‖𝑓(𝑡,𝑥,𝜙)−𝑓(𝑡,𝑦,𝜓)‖_𝑋≤𝐿^∗₃(𝑡)‖𝑥−𝑦‖_𝑋+𝐿^∗₄(𝑡)‖𝜙−𝜓‖_𝑟, for all(𝑡,𝑥,𝜙),(𝑡,𝑦,𝜓)∈𝐼×𝑋×𝒞 and 𝐿^∗₃,𝐿^∗₄∈𝐿_1/𝑝([0,𝑇],ℝ⁺), then the problem (32) has a unique solution in 𝑃𝐶([−𝑟,𝑇],𝑋), provided that 𝛾^∗₃=𝑚𝑙₂+𝑐𝐿^∗_31/𝑝+𝐿^∗_21/𝑝𝑇^𝛼−𝑝Γ(𝛼+1)<1, (35) where 𝑐=((1−𝑝)/(𝛼−𝑝))^1−𝑝.

4. Examples

Example 21 (fractional impulsive logistic equation). Consider the following class of fractional logistic equations in banach space 𝑋 with norm ‖⋅‖_𝑋: ^𝐶𝐷^𝛼𝑥(𝑡)=𝑥(𝑡)(𝑎(𝑡)−𝑏(𝑡)𝑥(𝑡)),𝑡∈[0,𝑇],𝑡≠𝑡_𝑘,Δ𝑥(𝑡)∣_{𝑡=𝑡𝑘}=𝐼_𝑘𝑥𝑡⁻_𝑘𝑥(0),𝑘=1,2,…,𝑚,=𝑥₀, (36) where 𝑎(𝑡)∈[𝑎_∗,𝑎^∗] and 𝑏(𝑡)∈[𝑏_∗,𝑏^∗] with 𝑎_∗,𝑏_∗>0.

Lemma 22 (Fečken et al. [24], Lemma 2). The initial value problem (36) is equivalent to the nonlinear integral equation 𝑥(𝑡)=𝑥₀+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1(𝑥(𝑠)(𝑎(𝑠)−𝑏(𝑠)𝑥(𝑠)))𝑑𝑠,𝑡∈0,𝑡₁=𝑥₀+𝐼₁𝑥𝑡⁻₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑎(𝑠)−1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑏(𝑠)𝑥²(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂=𝑥₀+²_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑎(𝑠)−1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑏(𝑠)𝑥²(𝑠)𝑡𝑑𝑠,𝑡∈₂,𝑡₃=𝑥₀+^𝑚_𝑘=1𝐼_𝑘𝑥𝑡⁻_𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑥(𝑠)𝑎(𝑠)−1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑏(𝑠)𝑥²(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑇. (37) In other words, every solution of the integral equation (37) is also solution of our original initial value problem (36) and conversely.

We can easily see that for the problem (36), our functions are 𝑓₁(𝑡,𝑥)=𝑎(𝑡)𝑥 and 𝑓₂(𝑡,𝑥)=−𝑏(𝑡)𝑥². It is not difficult to deduce that 𝑓₁(𝑡,𝑥)_𝑋≤𝑎^∗‖𝑥‖_𝑋+𝑚𝑙₁,𝑓₁(𝑡,𝑥)−𝑓₁𝑡,𝑦_𝑋≤𝑎^∗𝑥−𝑦_𝑋. (38)

Also ‖𝑓₂(𝑡,𝑥)‖_𝑋≤𝑏^∗‖𝑥‖². From the integral representation of problem (36), we get ‖𝑥(𝑡)‖_𝑋≤𝑥_0𝑋+𝑚𝑙₁+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎^∗‖𝑥(𝑠)‖_𝑋𝑑𝑠. (39)

Using Gronwall’s inequality (Diethelm, Lemma 6.19 [10, 31]), we get ‖𝑥(𝑡)‖_𝑋≤𝑥_0𝑋+𝑚𝑙₁1expΓ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑎^∗≤𝑥𝑑𝑠_0𝑋+𝑚𝑙₁𝑎exp^∗Γ().𝛼+1 (40) Thus 𝑥 is bounded which implies that all the assumptions of Theorem 8 are satisfied, and hence there exists a solution of the problem (36).

We give some more examples which are inspired by [24].

Example 23. Consider the following Caputo impulsive delay fractional differential equations ^𝐶𝐷^𝛼𝑥(𝑡)=𝑒^−𝜈𝑡𝑥_𝑡(1+𝑒^𝑡)1+𝑥_𝑡,𝑡∈[0,1],𝑡≠𝑡₁(𝑡)∣,𝜈>0,Δ𝑥_𝑡=𝑡1=12,𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝑟,0]. (41)

Set 𝐶₁=𝐶([−𝑟,0],ℝ⁺),𝑓(𝑡,𝜙)=𝑒^−𝜈𝑡𝜙/((1+𝑒^𝑡)(1+𝜙)),(𝑡,𝜙)∈[0,1]×𝐶₁.

Let 𝜙₁,𝜙₂∈𝐶₁ and let 𝑡∈[0,1]. Then, we have 𝑓𝑡,𝜙₁−𝑓𝑡,𝜙₂=𝑒^−𝜈𝑡1+𝑒^𝑡𝜙₁1+𝜙₁−𝜙₂1+𝜙₂=𝑒^−𝜈𝑡‖𝜙₁−𝜙₂‖_𝑟(1+𝑒^𝑡)1+𝜙₁1+𝜙₂≤𝑒^−𝜈𝑡‖𝜙₁−𝜙₂‖_𝑟(1+𝑒^𝑡)≤𝐿^∗(𝑡)‖𝜙₁−𝜙₂‖_𝑟, (42) where 𝐿^∗(𝑡)=𝑒^−𝜈𝑡/2.

Again, for all 𝜙∈𝐶₁ and each 𝑡∈[0,𝑇], 𝑓=𝑒𝑡,𝜙^−𝜈𝑡1+𝑒^𝑡𝜙≤𝑒1+𝜙(1+𝑒^{−𝜈𝑡𝑡})<𝑚₁(𝑡), (43) where 𝑚₁(𝑡)=𝑒^−𝜈𝑡/2.

For 𝑡∈[0,1] and some 𝑝∈(0,𝛼),𝐿^∗(𝑡)=𝑚₁(𝑡)=𝑒^−𝜈𝑡/2∈𝐿_1/𝑝([0,1],ℝ⁺) with 𝑀^∗₁=‖𝑒^−𝜈𝑡/10‖_1/𝑝, we can arrive at the inequality 1/4+𝑐𝑀^∗₁/Γ(𝛼)<1. We can see that all the assumptions of Theorem 13 are satisfied, and hence the problem (41) has a unique solution in [0,1].

Example 24. Consider the following Caputo impulsive delay fractional differential equation ^𝐶𝐷^𝛼𝑥(𝑡)=𝑥_𝑡(1+𝑒^𝑡)1+𝑥_𝑡,𝑡∈[0,1],𝑡≠𝑡₁(𝑡)∣,𝜈>0,Δ𝑥_𝑡=𝑡1=12,𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝑟,0]. (44)

Set 𝐶₂=𝐶([0,1],ℝ⁺),𝑓(𝑡,𝜙)=𝜙/((1+𝑒^𝑡)(1+𝜙)),(𝑡,𝜙)∈[−𝑟,0]×𝐶₂.

Again, for all 𝜙∈𝐶₂ and each 𝑡∈[0,𝑇], 𝑓=𝑒𝑡,𝜙^−𝑡1+𝑒^𝑡𝜙≤𝑒1+𝜙_−𝑡(1+𝑒^𝑡)<𝑚₂(𝑡)1+𝜙, (45) where 𝑚₂(𝑡)=𝑒^−𝑡/4.

For 𝑡∈[0,1] and some 𝑝∈(0,𝛼),𝑚₂(𝑡)=𝑒^−𝑡/4∈𝐿_1/𝑝([0,1],ℝ⁺) with 𝑀^∗₂=‖𝑒^−𝑡/4‖_1/𝑝, we can arrive at the inequality 1/4+𝑐𝑀/Γ(𝛼)<1. We can see that all the assumptions of Theorem 14 are satisfied, and hence the problem (44) has a solution in [0,1].

Acknowledgments

The authors are thankful to the anonymous reviewer for his/her valuable comments and suggestions. S. Abbas acknowledge “Erasmus Mundus Lot13, India 4EU” for providing him fellowship to visit University of Bologna, Italy. He also acknowledges Professor Stefan Siegmund, Center for Dynamics of TU Dresden, Germany, for his fruitful discussion on this topic and for hosting his short visit.