Existence of Solutions for Fractional-Order Neutral Differential Inclusions with Impulsive and Nonlocal Conditions

Abstract

This paper investigates the existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions. Utilizing the fractional calculus and fixed point theorem for multivalued maps, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. Finally, an illustrative example is given to show the effectiveness of theoretical results.

1. Introduction

This paper deals with the existence of solutions for the following fractional-order differential inclusions with impulsive nonlocal conditions: ^𝑐𝐷^𝑞𝑥(𝑡)−𝑔(𝑡,𝑥(𝑡))∈𝐹(𝑡,𝑥(𝑡)),𝑡∈𝐼=[0,𝑇],𝑡≠𝑡_𝑘,Δ𝑥|_{𝑡=𝑡𝑘}∈𝐼_𝑘𝑥𝑡⁻_𝑘,𝑘=1,…,𝑚,0<𝑡₁<⋯<𝑡_𝑚𝑥(0)<𝑇,=𝑥₀+ℎ(𝑥), (1.1) where 0<𝑞<1 and ^𝑐𝐷^𝑞 is the Caputo fractional derivative. 𝐼_𝑘∶𝑅→𝒫(𝑅)(𝑘=1,…,𝑚,) and 𝐹∶𝐼×𝑅→𝒫(𝑅) are multivalued maps (𝒫(𝑅) is the family of all nonempty subsets of 𝑅). Δ𝑥|_{𝑡=𝑡𝑘}=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), 𝑥(𝑡⁺_𝑘),𝑥(𝑡⁻_𝑘) represents the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively. 𝑔,ℎ are given functions to be specified later.

The nonlocal problem was more general and has better effect than the classical Cauchy problems. So it has been studied extensively under various conditions in the literature [1–7] and the references contained therein. In [2], Byszewski and Lakshmikantham considered the existence and uniqueness of mild solutions when 𝑓 and 𝑔 satisfied Lipschitz conditions. In [3], Ntouyas and Tsamatos studied the case of compactness conditions of 𝑔 and 𝑇(𝑡). Liang et al. in [5] discussed the nonlinear nonlocal Cauchy problems when 𝑇(𝑡) was compact and 𝑔 was Lipschitz and 𝑔 was not Lipschitz and not compact, respectively. Xue in [6] established existence results of mild solutions for semilinear differential equations under various conditions on 𝑓,𝑔, and 𝑇(𝑡). Subsequently in [7], the author examined the semilinear differential equations when 𝑇(𝑡) was compact and 𝑔 failed to be compact.

The theory of impulsive differential equations and differential inclusions has received much attention for the past decades because of its wide applicability in control, electrical engineering, mechanics, biology, and so on. For more details on this theory and applications, we refer to the monograph of Lakshmikantham et al. [8], Samoilenko and Perestyuk [9] and the references therein. In [10], Abada et al. discussed the impulsive differential inclusions with delays when 𝐴 was the infinitesimal generator of a strongly continuous semigroup 𝑇(𝑡). In the following work [11], they obtained some existence and controllability results when 𝐴 was a nondensely defined closed linear operator.

Since fractional-order differential equations have proved to be valuable tools in the modeling of many phenomena in physics and technical sciences, differential equations involving Riemann-Liouville as well as Caputo derivatives have been investigated extensively in the last decades (see [12–28] and the references therein). In [13], Mophou obtained the existence and uniqueness of mild solutions for the semilinear impulsive fractional differential equations by means of the Schauder fixed point theorem in Banach spaces. In [18], Benchohra and Slimani established sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential equations. In [21], Chang and Nieto studied the existence of solutions for a class of fractional differential inclusions with boundary conditions by means of Bohnenblust-Karlin fixed point theorem. Balachandran et al. in [25] investigated fractional-order impulsive integrodifferential equations in Banach spaces. In [27], Agarwal and Ahmad studied the existence of solutions for nonlinear fractional differential equations and inclusions of order 𝑞∈(3,4] with antiperiodic boundary conditions. The existence results were established for convex as well as nonconvex multivalued maps by some fixed point theorems, such as Leray-Schauder degree theory, and nonlinear alternative of Leray-Schauder type.

However, there is little information in the literature on neutral fractional-order impulsive differential inclusions with nonlocal conditions. Motivated by works mentioned above, we consider the existence results of (1.1) by using fixed point theorem for multivalued maps.

This paper is organized as follows. In Section 2, we will recall briefly some basic definitions and preliminary results which will be used throughout the paper. In Section 3, based on the fractional calculus and fixed point theorem for multivalued maps, we will prove the existence of solution to the problem (1.1). In Section 4, An example is given to show the effectiveness of our abstract results.

2. Preliminaries

In this section, we introduce some definitions, notes, and preliminary facts which will be used in this paper.

Let 𝐶(𝐼,𝑅) be a Banach space, with the norm ‖𝑥‖_𝐶=sup{|𝑥(𝑡)|,𝑡∈𝐼}. Let 𝐿^𝑝(𝐼,𝑅) denote the space of Bochner integrable functions on 𝐼 with the norm ‖𝑥‖_𝐿𝑝=(∫^𝑇₀|𝑥(𝑡)|^𝑝𝑑𝑡)^1/𝑝.

Let (𝑋,‖⋅‖) be a Banach space. 𝑃(𝑋) is the family of all nonempty subsets of 𝑋. 𝑃_bd(𝑋),𝑃_cl(𝑋),𝑃_bd,cl(𝑋),𝑃_cp,cv(𝑋), and 𝑃_bd,cl,cv(𝑋) denote, respectively, the family of all nonempty bounded, closed, bounded-closed, compact-convex, and bounded-closed-convex subsets of 𝑋. A multivalued map 𝐺∶𝑋→𝑃(𝑋) is convex (closed) valued if 𝐺(𝑥) is convex (closed) for all 𝑥∈𝑋. We say that 𝐺 is bounded on bounded sets if 𝐺(𝐵)=∪_𝑥∈𝐵𝐺(𝑥) is bounded in 𝑋 for all 𝐵∈𝑃_bd(𝑋).

A mapping 𝐺 is called upper semicontinuous (u.s.c.) on 𝑋 if, for each 𝑥₀∈𝑋, the set 𝐺(𝑥₀) is a nonempty closed subset of 𝑋 and if, for each open set 𝑁 of 𝑋 containing 𝐺(𝑥₀), there exists an open neighborhood 𝑁₀ of 𝑥₀ such that 𝐺(𝑁₀)⊆𝑁.  𝐺 is completely continuous if 𝐺(𝑉) is relatively compact for every 𝑉∈𝑃_bd(𝑋). If the multivalued map 𝐺 is completely continuous with nonempty compact values, then 𝐺 is u.s.c. if and only if 𝐺 has a closed graph (i.e., 𝑥_𝑛→𝑥_∗,𝑦_𝑛→𝑦_∗,𝑦_𝑛∈𝐺(𝑥_𝑛) imply 𝑦_∗∈𝐺(𝑥_∗)).

A mapping 𝐺 has a fixed point if there exists 𝑥∈𝑋 such that 𝑥∈𝐺(𝑥). A multivalued map 𝐺∶𝐼→𝑃_cl(𝑋) is said to be measurable if, for each 𝑥∈𝑋, the function 𝑡⟼𝑑(𝑥,𝐺(𝑡))=inf{|𝑥−𝑧|∶𝑧∈𝐺(𝑡)} (2.1) is measurable.

Definition 2.1. A multivalued map 𝐹∶𝐼×𝑋→𝒫(𝑋) is 𝐿¹-Carathéodory if (i)𝑡↦𝐹(𝑡,𝑢) is measurable for each 𝑢∈𝑋; (ii)𝑢↦𝐹(𝑡,𝑢) is upper semicontinuous for almost all 𝑡∈𝐼; (iii)for each 𝑞>0, there exists 𝜙_𝑞∈𝐿¹(𝐼,𝑅⁺) such that ‖𝐹(𝑡,𝑢)‖_𝒫=sup{‖𝑣‖∶𝑣∈𝐹(𝑡,𝑢)}≤𝜙_𝑞(𝑡), (2.2) for all ‖𝑢‖≤𝑞 and a.e.𝑡∈𝐼.

Definition 2.2. A multivalued operator 𝑁∶𝑋→𝑃_cl(𝑋) is called (a)𝛾-Lipschitz if there exists 𝛾>0 such that 𝐻_𝑑𝑁(𝑥),𝑁𝑦≤𝛾𝑑𝑥,𝑦,𝑥,𝑦∈𝑋; (2.3)(b)a contraction if it is 𝛾-Lipschitz with 𝛾<1.
The key tool in our approach is the following fixed point theorem.

Lemma 2.3 2.3(see [29]). Let 𝐵(0,𝑟) and 𝐵[0,𝑟] denote, respectively, the open and closed balls in a Banach space 𝑋 centered at origin and of radius 𝑟, and let Φ₁∶𝑋→𝑃_bd,cl,cv(𝑋) and Φ₂∶𝐵[0,𝑟]→𝑃_cp,cv(𝑋) be two multivalued operators satisfying the following:(a)Φ₁ is a contraction,(b)Φ₂ is completely continuous. Then either(i)the operator inclusion 𝑥∈Φ₁𝑥+Φ₂𝑥 has a solution in 𝐵[0,𝑟] or(ii)there exists an 𝑢∈𝑋 with ‖𝑢‖=𝑟 such that 𝜆𝑢∈Φ₁𝑢+Φ₂𝑢, for some 𝜆>1.

Lemma 2.4 (see [30]). Let 𝐼 be a compact real interval and 𝑋 a Banach space. Let 𝐹 be an 𝐿¹-Caratheodory multivalued mapping with compact convex values with 𝑆_𝐹,𝑦≠∅, where 𝑆_𝐹,𝑦={𝑓∈𝐿¹(𝐼,𝑋)∶𝑓(𝑡)∈𝐹(𝑡,𝑦(𝑡)),𝑎.𝑒.𝑡∈𝐼} and let Γ∶𝐿¹(𝐼,𝑋)→𝐶(𝐼,𝑋) be a linear continuous mapping. Then the operator Γ∘𝑆_𝐹∶𝐶(𝐼,𝑋)⟶𝑃_cp,cv(𝐶(𝐼,𝑋)),𝑥⟼Γ∘𝑆_𝐹(𝑥)=Γ𝑆_𝐹,𝑥 (2.4) is a closed graph operator in 𝐶(𝐼,𝑋)×𝐶(𝐼,𝑋).

For more details on multivalued map, see the books of Aubin and Cellina [31], Deimling [32].

Now, we recall some definitions and facts about fractional derivatives and fractional integrals of arbitrary order, see [33].

Definition 2.5. The Riemann-Liouville fractional integral operator of order 𝑞>0 of a function ℎ is defined by 𝐼^𝑞_𝑎ℎ(𝑡)=1Γ𝑞^𝑡_𝑎(𝑡−𝑠)^𝑞−1ℎ(𝑠)𝑑𝑠, (2.5) where Γ is the gamma function.

Definition 2.6. The Caputo fractional-order derivative of order 𝑞>0 of a function ℎ(𝑡) is defined by ^𝑐𝐷^𝑞_𝑎+ℎ(𝑡)=1Γ𝑛−𝑞^𝑡_𝑎(𝑡−𝑠)^{𝑛−𝑞−1}ℎ^(𝑛)(𝑠)𝑑𝑠,𝑛−1<𝑞<𝑛,𝑛∈ℕ⁺. (2.6)

Lemma 2.7 2.7(see [34]). Let 𝑞>0; then the differential equation ^𝑐𝐷^𝑞𝑦(𝑡)=ℎ(𝑡) (2.7) has solutions 𝑦(𝑡)=𝐼^𝑞ℎ(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1, where 𝑐_𝑖∈𝑅,𝑛=[𝑞]+1.

Lemma 2.8 2.8( see [34]). Let 𝛼>0; then 𝐼^𝛼𝑐𝐷^𝛼ℎ(𝑡)=ℎ(𝑡)+𝑎₀+𝑎₁𝑡+𝑎₂𝑡²+⋯+𝑎_𝑛−1𝑡^𝑛−1 (2.8) for some 𝑎_𝑖∈𝑅,𝑖=0,1,2,…,𝑛−1,𝑛=[𝛼]+1.

3. Main results

In this section, main results are presented.

Let 𝑃𝐶(𝐼,𝑅)={𝑥∶𝐼→𝑅∶𝑥∈𝐶((𝑡_𝑘−1,𝑡_𝑘],𝑅),𝑘=0,1,…,𝑚, and there exist 𝑥(𝑡⁻_𝑘),𝑥(𝑡⁺_𝑘),𝑘=1,…,𝑚 with 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘)}. This set is a Banach space with the norm ‖𝑥‖_𝑃𝐶=sup_𝑡∈𝐼|𝑥(𝑡)|.

As a consequence of Lemmas 2.7 and 2.8, we can define the solution of problem (1.1).

Definition 3.1. The function 𝑥∈𝑃𝐶(𝐼,𝑅) is said to be a solution of the problem (1.1) if 𝑥(0)=𝑥₀+ℎ(𝑥), and there exist 𝑓(𝑡)∈𝑆_𝐹,𝑥,𝐽_𝑘∈𝐼_𝑘(𝑥(𝑡⁻_𝑘)) such that integral equation 𝑥(𝑡)=𝑥₀(𝑥)((0((𝑡+1+ℎ−𝑔0,𝑥))+𝑔𝑡,𝑥))Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠)+1𝑑𝑠Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠+_{0<𝑡𝑖<𝑡}𝐽_𝑘,𝑡∈𝐼, (3.1) is satisfied.

For the study of the system (1.1), the following hypotheses are given:(H1)𝐹 is a Carathéodory map with compact convex values, and for each fixed 𝑥∈𝑅,𝑆_𝐹,𝑥 is nonempty;(H2) there exist a function 𝑎(⋅)∈𝐿^𝑝(𝐼,𝑅⁺), where 𝑝>1/𝑞,𝑝∈ℕ⁺ and a nondecreasing continuous function Ω∶𝑅⁺→𝑅⁺ such that ‖𝐹(𝑡,𝑥)‖_𝒫=sup𝑓(𝑡)∶𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡))≤𝑎(𝑡)Ω(‖𝑥‖), (3.2)  for 𝑡∈𝐼 and each 𝑥∈𝑅;(H3)𝑔∶𝐼×𝑅→𝑅 is Lipschtiz continuous with Lipschtiz constant 𝑙_𝑔, and there exists 𝑀>0 such that |𝑔(𝑡,𝑥(𝑡))|≤𝑀;(H4)ℎ∶𝑃𝐶(𝐼,𝑅)→𝑅 is Lipschtiz continuous with Lipschtiz constant 𝑙_ℎ;(H5)𝐼_𝑘∶𝑅→𝒫(𝑅)(𝑘=1,…,𝑚) are multivalued maps with bounded, closed, and convex values. And there exist 𝑐_𝑘>0,𝑘=1,…,𝑚, such that 𝐻_𝑑𝐼_𝑘𝑦,𝐼_𝑘(𝑥)≤𝑐_𝑘𝑦−𝑥, (3.3)  for all 𝑥,𝑦∈𝑅;(H6)𝐶=((𝑝−1)/(𝑝𝑞−1))^{(𝑝−1)/𝑝}𝑇^{𝑞−1/𝑝},𝐶₀=|𝑥₀|+|ℎ(0)|+2𝑀+∑^𝑚_𝑘=1|𝐼_𝑘(0)|, 𝑙_ℎ+2𝑙_𝑔+^𝑚_𝑘=1𝑐_𝑘<1. (3.4)

The existence of solutions is now presented.

Theorem 3.2. Assume that (H1)–(H6) are satisfied; then the problem (1.1) has at least one solution on 𝐼 provided that there exists a real number 𝑟>0 such that 𝐶₀Γ𝑞+𝐶(𝑚+1)Ω(𝑟)‖𝑎‖_𝐿𝑝1−𝑙_ℎ−∑^𝑚_𝑘=1𝑐_𝑘Γ𝑞<𝑟. (3.5)

Proof. We transform the problem (1.1) into a fixed point problem by considering the multivalued operator 𝑁(𝑥)∶𝑃𝐶(𝐼,𝑅)→𝒫(𝑃𝐶(𝐼,𝑅)): 𝑁(𝑥)=𝜑∈𝑃𝐶(𝐼,𝑅)∶𝜑(𝑡)=𝑥₀+ℎ(𝑥)−𝑔(0,𝑥(0))+𝑔(𝑡,𝑥(𝑡))+_{0<𝑡𝑖<𝑡}𝐽_𝑘+1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠, (3.6) where 𝑓∈𝑆_𝐹,𝑥,𝐽_𝑘∈𝐼_𝑘(𝑥(𝑡⁻_𝑘)),𝑘=1,…,𝑚.
Consider the multivalued operators 𝛼,𝛽: 𝛼(𝑥)=𝜑∈𝑃𝐶(𝐼,𝑅)∶𝜑(𝑡)=𝑥₀+ℎ(𝑥)−𝑔(0,𝑥(0))+𝑔(𝑡,𝑥(𝑡))+_{0<𝑡𝑖<𝑡}𝐽_𝑘,𝛽(𝑥)=()(𝑡)=1𝜑∈𝑃𝐶𝐼,𝑅∶𝜑Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠. (3.7) It is clear that 𝑁=𝛼+𝛽. This problem of finding a solution of (1.1) is reduced to find a solution of the operator inclusions 𝑥∈𝛼(𝑥)+𝛽(𝑥). In what follows, we aim to show that the operator 𝛼+𝛽 has a fixed point, which is a solution of the system (1.1). For this purpose, We will show that 𝛼+𝛽 satisfies all the conditions of Lemma 2.3. For better readability, the proof will be given in several steps.
Step 1. A priori bounded on solution: we show that the second assertion of Lemma 2.3 is not true.
Define an open ball 𝐵(0,𝑟)∈𝑃𝐶(𝐼,𝑅), where the real number 𝑟 satisfies the inequality given in condition (3.5). Let 𝑥 be a possible solution of 𝜆𝑥∈𝛼(𝑥)+𝛽(𝑥), for some real number 𝜆>1 with ‖𝑥‖=𝑟. Then we have 𝑥(𝑡)∈𝜆⁻¹𝑥₀+ℎ(𝑥)−𝑔(0,𝑥(0))+𝑔(𝑡,𝑥(𝑡))+_{0<𝑡𝑖<𝑡}𝐽_𝑘+𝜆⁻¹Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠)𝜆𝑑𝑠+⁻¹Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠. (3.8) Hence by (H2)–(H5), we have |𝑥(𝑡)|≤𝑥₀+ℎ(𝑥)−𝑔(0,𝑥(0))+𝑔(𝑡,𝑥(𝑡))+_{0<𝑡𝑖<𝑡}𝐽_𝑘+1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠1)𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠≤𝑥)𝑑𝑠₀+|ℎ(0)|+𝑙_ℎ‖𝑥‖+2𝑀+^𝑚_𝑘=1𝑐_𝑘𝑥𝑡⁻_𝑘+^𝑚_𝑘=1𝐼_𝑘(0+1)Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑎(𝑠)Ω𝑥1(‖‖)𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑎(𝑠)Ω(‖𝑥‖)𝑑𝑠≤𝐶₀+𝑙_ℎ+^𝑚_𝑘=1𝑐_𝑘‖𝑥‖+𝐶(𝑚+1)Ω(‖𝑥‖)‖𝑎‖_𝐿𝑝Γ𝑞. (3.9) Taking the supremum over 𝑡, we get ‖𝑥‖≤𝐶₀+𝑙_ℎ+^𝑚_𝑘=1𝑐_𝑘‖𝑥‖+𝐶(𝑚+1)Ω(‖𝑥‖)‖𝑎‖_𝐿𝑝Γ𝑞. (3.10) Substituting ‖𝑥‖=𝑟 in the above inequality yields 𝐶𝑟≤⁰Γ𝑞+𝐶(𝑚+1)Ω(‖𝑟‖)‖𝑎‖_𝐿𝑝1−𝑙_ℎ−∑^𝑚_𝑘=1𝑐_𝑘Γ𝑞, (3.11) which is a contradiction to (3.5).
Step 2. 𝛼∶𝑃𝐶(𝐼,𝑅)→𝒫(𝑃𝐶(𝐼,𝑅)) is a contraction mapping.
Indeed, let 𝑥₁,𝑥₂∈𝑃𝐶(𝐼,𝑅), and from (H5), we have 𝐻_𝑑𝛼𝑥₁,𝛼𝑥₂=𝐻_𝑑𝑥₀+ℎ𝑥₁−𝑔0,𝑥₁(0)+𝑔𝑡,𝑥₁(𝑡)+_{0<𝑡𝑖<𝑡}𝐼_𝑘𝑥₁𝑡⁻_𝑘,𝑥₀+ℎ𝑥₂−𝑔0,𝑥₂(0)+𝑔𝑡,𝑥₂(𝑡)+_{0<𝑡𝑖<𝑡}𝐼_𝑘𝑥₂𝑡⁻_𝑘≤𝐻_𝑑ℎ𝑥₁,ℎ𝑥₂+𝐻_𝑑−𝑔0,𝑥₁(0),−𝑔0,𝑥₂(0)+𝐻_𝑑𝑔𝑡,𝑥₁(𝑡),𝑔𝑡,𝑥₂(𝑡)+𝐻_{𝑑0<𝑡𝑖<𝑡}𝐼_𝑘𝑥₁𝑡⁻_𝑘,_{0<𝑡𝑖<𝑡}𝐼_𝑘𝑥₂𝑡⁻_𝑘≤𝑙_ℎ+2𝑙_𝑔+^𝑚_𝑘=1𝑐_𝑘𝑥₁−𝑥₂. (3.12) Hence by (3.4), 𝛼 is a contraction mapping.
Step 3. 𝛽 has compact, convex values, and it is completely continuous.

Claim 1. 𝛽 is convex for each 𝑥∈𝑃𝐶(𝐼,𝑅).
In fact, if 𝜑₁ and 𝜑₂ belong to 𝛽(𝑥), then there exist 𝑓₁, 𝑓₂∈𝑆_𝐹,𝑥 such that 𝜑_𝑖(𝑡)=1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓_𝑖(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓_𝑖(𝑠)𝑑𝑠,𝑖=1,2. (3.13) Let 0≤𝜆≤1. Then, for each 𝑡∈𝐼, we have 𝜆𝜑₁+(1−𝜆)𝜑₂(𝑡)=1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝜆𝑓₁(𝑠)+(1−𝜆)𝑓₂(𝑠)+1𝑑𝑠Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝜆𝑓₁(𝑠)+(1−𝜆)𝑓₂(𝑠)𝑑𝑠. (3.14) Since 𝑆_𝐹,𝑥 is convex (because 𝐹 has convex values), we have 𝜆𝜑₁+(1−𝜆)𝜑₂∈𝛽(𝑥). (3.15)

Claim 2. 𝛽 maps bounded sets into bounded sets in 𝑃𝐶(𝐼,𝑅). Let 𝑟>0,𝐵_𝑟={𝑥∈𝑃𝐶(𝐼,𝑅)∶‖𝑥‖≤𝑟} be a bounded set in 𝑃𝐶(𝐼,𝑅). For 𝑥∈𝐵_𝑟,𝜑∈𝛽(𝑥), then there exists 𝑓∈𝑆_𝐹,𝑥 such that 𝜑(𝑡)=1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠. (3.16) Therefore, 𝜑(𝑡≤1)Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠1)𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠≤1)𝑑𝑠Γ𝑞^𝑚_𝑘=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑎(𝑠)Ω𝑥1(‖‖)𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑎(𝑠)Ω𝑥≤𝐶()Ω(𝑟𝑎‖(‖‖)𝑑𝑠𝑚+1)‖_𝐿𝑝Γ𝑞. (3.17) Thus 𝜑≤𝐶(𝑚+1)Ω(𝑟)‖𝑎‖_𝐿𝑝Γ𝑞∶=𝑚^∗. (3.18)

Claim 3. 𝛽 maps bounded sets into equicontinuous sets. Let 𝜏₁,𝜏₂∈𝐼, 𝜏₁<𝜏₂, 𝐵_𝑟 be a bounded set of 𝑃𝐶(𝐼,𝑅) as in Claim 2 and 𝑥∈𝐵_𝑟. Then there exists 𝑓∈𝑆_𝐹,𝑥 such that 𝜑(𝑡)=1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠. (3.19)
By elementary computation, we have 𝜑𝜏₂−𝜑𝜏₁≤1Γ𝑞^𝜏1_𝑡𝑖|𝜏₂−𝑠^𝑞−1−𝜏₁−𝑠^𝑞−1𝑓(𝑠1)𝑑𝑠+Γ𝑞^𝜏2_𝜏1𝜏₂−𝑠^𝑞−1𝑓(𝑠≤‖𝑎‖)𝑑𝑠_𝐿𝑝Ω(𝑟)Γ𝑞^𝜏1_𝑡𝑖𝜏₂−𝑠^𝑞−1−𝜏₁−𝑠^{𝑞−1𝑝/(𝑝−1)}𝑑𝑠^{(𝑝−1)/𝑝}+‖𝑎‖_𝐿𝑝Ω(𝑟)Γ𝑞^𝜏2_𝜏1𝜏₂−𝑠^{(𝑝𝑞−𝑝)/(𝑝−1)}𝑑𝑠^{(𝑝−1)/𝑝}≤‖𝑎‖_𝐿𝑝Ω(𝑟)Γ𝑞𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}×𝜏₂−𝜏₁^{(𝑝𝑞−1)/(𝑝−1)}−𝜏₂−𝑡_𝑖^{(𝑝𝑞−1)/(𝑝−1)}+𝜏₁−𝑡_𝑖^{(𝑝𝑞−1)/(𝑝−1)(𝑝−1)/𝑝}+‖𝑎‖_𝐿𝑝Ω(𝑟)Γ𝑞𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}𝜏₂−𝜏₁^{(𝑝𝑞−1)/𝑝}. (3.20) As 𝜏₁→𝜏₂, the right-hand side of the above inequality tends to zero. As a consequence of Claims 1 to 3 together with the Arzelá-Ascoli theorem, we conclude that 𝛽 is completely continuous.

Claim 4. 𝛽 has a closed graph. Let 𝑥_𝑛→𝑥_∗,𝜑_𝑛∈𝛽(𝑥_𝑛), and 𝜑_𝑛→𝜑_∗; we prove that 𝜑_∗∈𝛽(𝑥_∗). Now 𝜑_𝑛∈𝛽(𝑥_𝑛) means that there exists 𝑓_𝑛∈𝑆_{𝐹,𝑥𝑛} such that 𝜑_𝑛(𝑡)=1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓_𝑛(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓_𝑛(𝑠)𝑑𝑠,𝑡∈𝐼. (3.21)
We will prove that there exists 𝑓_∗∈𝑆_𝐹,𝑥∗ such that

𝜑_∗(𝑡)=1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓_∗(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓_∗(𝑠)𝑑𝑠. (3.22)

Consider the continuous operator 𝜅∶𝐿¹(𝐼,𝑋)→𝐶(𝐼,𝑋)(𝑡)=1𝜅𝑓Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠. (3.23)

From Lemma 2.4, it follows that 𝜅∘𝑆_𝐹 is a closed graph operator. Since 𝜑_𝑛∈𝜅∘𝑆_{𝐹,𝑥𝑛}, we have 𝜑_∗∈𝜅∘𝑆_𝐹,𝑥∗. That is, there exists 𝑓_∗∈𝑆_𝐹,𝑥∗ such that 𝜑_∗(𝑡)=1Γ𝑞_{0<𝑡𝑖<𝑡}^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝑞−1𝑓_∗(𝑠)1𝑑𝑠+Γ𝑞^𝑡_𝑡𝑖(𝑡−𝑠)^𝑞−1𝑓_∗(𝑠)𝑑𝑠. (3.24) Therefore, 𝛽 has a closed graph.

As a consequence of Lemma 2.3, the operator inclusion 𝑥∈𝛼(𝑥)+𝛽(𝑥) has a solution which implies that the problem (1.1) has a solution on 𝐼.

Remark 3.3. If we take 𝐹(𝑡,𝑥)={𝑓(𝑡,𝑥)}, where 𝑓 is a continuous operator, then the result corresponds to a single-value problem. In this case, our result is also new in the present configuration.

Corollary 3.4. Assume that (H1)–(H6) are satisfied; then the problem (1.1) has at least one solution on 𝐼 provided that ‖𝑎‖_𝐿𝑝<liminf_𝑅→+∞𝑅1−𝑙_ℎ−∑^𝑚_𝑘=1𝑐_𝑘−𝐶₀Γ𝑞𝐶()Ω(𝑅).𝑚+1 (3.25)

Proof. From (3.25), we know that there exists a constant 𝑟>0, such that ‖𝑎‖_𝐿𝑝<𝑟1−𝑙_ℎ−∑^𝑚_𝑘=1𝑐_𝑘−𝐶₀Γ𝑞𝐶()Ω(𝑟).𝑚+1 (3.26)
Since the proof of Corollary 3.4 follows the very same lines as the proof of Theorem 3.2, we do not give details.

Corollary 3.5. Assume that (H1), (H3)–(H6), and the following condition are satisfied,
(H2^') There exists a function 𝜌(⋅)∈𝐿^𝑝(𝐼,𝑅), such that ‖𝐹(𝑡,𝑥)‖_𝒫=sup𝑓(𝑡)∶𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡))≤𝜌(𝑡)‖𝑥‖, (3.27) for 𝑡∈𝐼 and each 𝑥∈𝑅.
Then the problem (1.1) has at least one solution on 𝐼 provided that 𝜌_𝐿𝑝<1−𝑙_ℎ−∑^𝑚_𝑘=1𝑐_𝑘𝐶().𝑚+1 (3.28)

Proof. In this case, we take 𝑎(𝑡)=𝜌(𝑡),Ω(𝑥)=𝑥 in Corollary 3.4.
At last we would like to discuss the impulsive condition.(H5^′)𝐼_𝑘∶𝑅→𝑅(𝑘=1,…,𝑚) are single continuous operator, and there exists 𝑐_𝑘>0,𝑘=1,…,𝑚, such that |𝐼_𝑘(𝑦)−𝐼_𝑘(𝑥)|≤𝑐_𝑘|𝑦−𝑥|, for all 𝑥,𝑦∈𝑅.

Corollary 3.6. Assume that (H1)–(H4), (𝐻5^′), (H 6) are satisfied; then the problem (1.1) has at least one solution on 𝐼 provided that ‖𝑎‖_𝐿𝑝<liminf_𝑅→+∞𝑅1−𝑙_ℎ−∑^𝑚_𝑘=1𝑐_𝑘−𝐶₀Γ𝑞𝐶()Ω(𝑅).𝑚+1 (3.29)

4. An Illustrative Example

In this section, an example is given to show the effectiveness of our theoretical result. Consider the following fractional-order differential inclusions with impulsive and nonlocal conditions: ^𝑐𝐷^𝑞𝑥(𝑡)−𝑒^−𝑡|𝑥(𝑡)|(9+𝑒^𝑡|𝑥(𝑡((𝑡[]1)(1+)|)∈𝐹𝑡,𝑥)),𝑡∈𝐼=0,1,𝑡≠2,Δ𝑥|_𝑡=1/2=|𝑥(1/2⁻)|3+|𝑥(1/2⁻,𝑥(0)(𝑥),)|=ℎ (4.1) where ℎ(𝑥)=(1/9)𝑥(0)+(1/10)𝑥(1),𝑔(𝑡,𝑥)=(𝑒^−𝑡𝑥)/(9+𝑒^𝑡)(1+𝑥),𝐼_𝑘(𝑥)=𝑥/(3+𝑥),𝑥∈[0,∞), and 𝐹∶𝐼×𝑅→𝑃(𝑅) is a multivalued map given by()=𝑒𝑥⟶𝐹𝑡,𝑥^−2𝑡|𝑥(𝑡)|(9+𝑒^𝑡|𝑥(𝑡,𝑒)(1+)|)^−𝑡|𝑥(𝑡)|(9+𝑒^𝑡|𝑥(𝑡.)(1+)|) (4.2)

Thus, ‖𝐹(𝑡,𝑥)‖_𝒫=sup𝑓(𝑡)∶𝑓(𝑡)∈𝐹(𝑡,𝑥(𝑡))≤𝑎(𝑡)Ω(‖𝑥‖), (4.3) where 𝑎(𝑡)=1/(9+𝑒^𝑡),Ω(‖𝑥‖)=|𝑥(𝑡)|/(1+|𝑥(𝑡)|).

By computation, we have ℎ(𝑥)−ℎ𝑦≤19,𝑔()≤190𝑥−𝑦𝑡,𝑥−𝑔𝑡,𝑦,𝐼10𝑥−𝑦_𝑘(𝑥)−𝐼_𝑘𝑦≤13,𝑔(≤1𝑥−𝑦𝑡,𝑥).10 (4.4)

Suppose that 𝑥₀=ℎ(0)=𝐼_𝑘(0)=0; we have 𝑙_𝑔=1/10, 𝑙_ℎ=2/9, 𝑀=1/10, 𝑐_𝑘=1/3, 𝐶₀=2𝑀=1/5, 𝐶=((𝑝−1)/(𝑝𝑞−1))^{(𝑝−1)/𝑝}.

Clearly, for suitable 𝑞∈(0,1), all the conditions of Corollary 3.6 are satisfied. So there exists at least one solution of problem (4.1) on 𝐼.

Acknowledgments

The authors would like to thank the Associate Editor and all the anonymous reviewers for their valuable comments and constructive suggestions, which lead to the improvement of the presentation of this paper. This work was funded by Scientific Research Foundation of Nanjing Institute of Technology under Grant Nos. QKJB2010028, QKJA2011009 and also jointly supported by the National Natural Science Foundation of China under Grant Nos. 60874088 and 11072059 and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20110092110017, the JSPS Innovation Program under Grant CXZZ12_0080.