Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations of Order 𝜶∈(2,3] with Nonlocal Boundary Conditions

Abstract

We investigate the existence and uniqueness of solutions to the nonlocal boundary value problem for nonlinear impulsive fractional differential equations of order 𝛼∈(2,3]. By using some well-known fixed point theorems, sufficient conditions for the existence of solutions are established. Some examples are presented to illustrate the main results.

1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines such as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, control theory, signal and image processing, biophysics, electrodynamics of complex medium, polymer rheology, and fitting of experimental data [1–6]. For example, one could mention the problem of anomalous diffusion [7–9], the nonlinear oscillation of earthquake can be modeled with fractional derivative [10], and fluid-dynamic traffic model with fractional derivatives [11] can eliminate the deficiency arising from the assumption to continuum traffic flow and many other [12, 13] recent developments in the description of anomalous transport by fractional dynamics. For some recent development on nonlinear fractional differential equations, see [14–29] and the references therein.

In this paper, we investigate a three-point boundary value problems for nonlinear impulsive fractional differential equations of order 𝛼∈(2,3]: ^𝐶𝐷^𝛼𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈𝐽^′,Δ𝑢^′𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘,𝑘=1,2,…,𝑝,Δ𝑢^′′𝑡_𝑘=𝐼^∗_𝑘𝑢𝑡_𝑘𝑢(0),𝑘=1,2,…,𝑝,=𝑢^′(0)=0,𝛽𝑢𝜂=𝑢(1), (1.1) where ^𝐶𝐷^𝛼 is the Caputo fractional derivative, 2<𝛼≤3,  𝑓∈𝐶(𝐽×𝑅,𝑅),𝐼_𝑘,𝐼^∗_𝑘∈𝐶(𝑅,𝑅),  𝐽=[0,1],0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯<𝑡_𝑝<𝑡_𝑝+1=1,  𝐽′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑝},  0<𝜂<1,  𝜂≠𝑡_𝑘(𝑘=1,2,…,𝑝),  0<𝛽<1/𝜂²,  Δ𝑢′(𝑡_𝑘)=𝑢′(𝑡⁺_𝑘)−𝑢′(𝑡⁻_𝑘), where 𝑢′(𝑡⁺_𝑘) and 𝑢′(𝑡⁻_𝑘) denote the right and the left limits of 𝑢′(𝑡) at 𝑡=𝑡_𝑘(𝑘=1,2,…,𝑝), respectively. Δ𝑢^′′(𝑡_𝑘) has a similar meaning for 𝑢^′′(𝑡).

Impulsive differential equations arise in many engineering and scientific disciplines as the important mathematical modeling of systems and processes in the fields of biology, physics, engineering, and so forth. Due to their significance, it is important to study the solvability of impulsive differential equations. The theory of impulsive differential equations of integer order has emerged as an important area of investigation. Recently, the impulsive differential equations of fractional order have also attracted a considerable attention, and a variety of results can be found in the papers [30–42].

The study of multipoint boundary-value problems was initiated by Bicadze and Samarskiĭ in [43]. Many authors since then considered nonlinear multipoint boundary-value problems, see [44–51] and the references therein. The multipoint boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the sensors located at intermediate points. For example, the vibrations of a guy wire of uniform cross-section and composed of 𝑁 parts of different densities can be set up as a multipoint boundary-value problem.

To our knowledge, no paper has considered nonlinear impulsive fractional differential equations of order 𝛼∈(2,3] with nonlocal boundary conditions, that is, problem (1.1). This paper fills this gap in the literature. Our purpose here is to give the existence and uniqueness of solutions for nonlinear impulsive fractional differential equations (1.1). Our results are based on some well-known fixed point theorems.

2. Preliminaries

Let 𝐽₀=[0,𝑡₁],𝐽₁=(𝑡₁,𝑡₂],…,𝐽_𝑝−1=(𝑡_𝑝−1,𝑡_𝑝],𝐽_𝑝=(𝑡_𝑝,1].

We introduce the space: 𝑃𝐶²(𝐽,𝑅)=𝑢∶𝐽⟶𝑅∣𝑢∈𝐶²𝐽_𝑘,𝑘=0,1,…,𝑝,𝑢^′𝑡⁺_𝑘,𝑢^′′𝑡⁺_𝑘exist,𝑘=1,2,…,𝑝,, (2.1) with the norm: ‖𝑢‖=sup_𝑡∈𝐽|𝑢(𝑡)|,‖𝑢‖_𝑃𝐶2=max‖𝑢‖,𝑢^′,𝑢^′′. (2.2) Obviously, 𝑃𝐶²(𝐽,𝑅) is a Banach space.

Definition 2.1. A function 𝑢∈𝑃𝐶²(𝐽,𝑅) with its Caputo derivative of order 𝛼 existing on 𝐽 is a solution of (1.1) if it satisfies (1.1).

Theorem 2.2 (see [52]). Let 𝐸 be a Banach space. Assume that 𝐴∶𝐸→𝐸 is a completely continuous operator, and the set 𝑉={𝑢∈𝐸∣𝑢=𝜇𝐴𝑢,0<𝜇<1} is bounded. Then 𝐴 has a fixed point in 𝐸.

Theorem 2.3 (see [52]). Let 𝐸 be a Banach space. Assume that Ω is an open bounded subset of 𝐸 with 𝜃∈Ω, and let 𝐴∶Ω→𝐸 be a completely continuous operator such that ‖𝐴𝑢‖≤‖𝑢‖,∀𝑢∈𝜕Ω. (2.3) Then, 𝐴 has a fixed point in Ω.

Lemma 2.4. Let 2<𝛼≤3,1≠𝛽𝜂²,𝜂∈(𝑡_𝑚,𝑡_𝑚+1),  𝑚 is a nonnegative integer, 0≤𝑚≤𝑝. For a given 𝑦∈𝐶[0,1], a function 𝑢 is a solution of the following impulsive boundary value problem: ^𝐶𝐷^𝛼𝑢(𝑡)=𝑦(𝑡),𝑡∈𝐽^′,Δ𝑢^′𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘,𝑘=1,2,…,𝑝,Δ𝑢^′′𝑡_𝑘=𝐼^∗_𝑘𝑢𝑡_𝑘𝑢(0),𝑘=1,2,…,𝑝,=𝑢^′(0)=0,𝛽𝑢𝜂=𝑢(1), (2.4) if and only if 𝑢 is a solution of the impulsive fractional integral equation: 𝑢(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠+𝐶𝑡²,𝑡∈𝐽₀;1Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑦(𝑠)1𝑑𝑠+Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1𝑦(𝑠)+𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑦(𝑠)+𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑦(𝑠)+𝑑𝑠_{𝑘−1𝑖=1}𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑦(𝑠)𝑑𝑠+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘²2𝐼^∗_𝑖𝑢𝑡_𝑖+𝐶𝑡²,𝑡∈𝐽_𝑘,𝑘=1,2,…,𝑝, (2.5) where 1𝐶=1−𝛽𝜂²×𝛽Γ(𝛼)^𝜂_𝑡𝑚𝜂−𝑠^𝛼−1𝑦(𝑠)𝑑𝑠+^𝑚_𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑦(𝑠)+𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝛽(𝑡_𝑚−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑚𝑖=1𝛽(𝜂−𝑡_𝑚)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)1𝑑𝑠−Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑦(𝑠)−𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)−𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)−𝑑𝑠_𝑝𝑖=1(1−𝑡_𝑝)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽(𝑡_𝑚−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽(𝜂−𝑡_𝑚)²2𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝_𝑖=11−𝑡_𝑝²2𝐼^∗_𝑖𝑢𝑡_𝑖. (2.6)

Proof. Let 𝑢 is a solution of (2.4), it holds 𝑢(𝑡)=𝐼^𝛼𝑦(𝑡)−𝑐₁−𝑐₂𝑡−𝑐₃𝑡²=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠−𝑐₁−𝑐₂𝑡−𝑐₃𝑡²,𝑡∈𝐽₀, (2.7) for some 𝑐₁,𝑐₂,𝑐₃∈𝑅. Then, we have (𝑡)=1𝑢′Γ(𝛼−1)^𝑡₀(𝑡−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−𝑐₂−2𝑐₃𝑡,𝑡∈𝐽₀,𝑢^′′(𝑡)=1Γ(𝛼−2)^𝑡₀(𝑡−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠−2𝑐₃,𝑡∈𝐽₀. (2.8) In view of 𝑢(0)=𝑢′(0)=0, it follows 𝑐₁=𝑐₂=0.
If 𝑡∈𝐽₁, then 𝑢(𝑡)=1Γ(𝛼)^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠−𝑑₁−𝑑₂𝑡−𝑡₁−𝑑₃(𝑡−𝑡₁)²,(𝑡)=1𝑢′Γ(𝛼−1)^𝑡_𝑡1(𝑡−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−𝑑₂−2𝑑₃𝑡−𝑡₁,𝑢^′′(𝑡)=1Γ(𝛼−2)^𝑡_𝑡1(𝑡−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠−2𝑑₃, (2.9) for some 𝑑₁,𝑑₂,𝑑₃∈𝑅.
Thus, we have 𝑢′𝑡⁻₁=1Γ(𝛼−1)^𝑡1₀(𝑡₁−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−2𝑐₃𝑡₁,𝑢′𝑡⁺₁=−𝑑₂,𝑢^′′𝑡⁻₁=1Γ(𝛼−2)^𝑡1₀𝑡₁−𝑠^𝛼−3𝑦(𝑠)𝑑𝑠−2𝑐₃,𝑢^′′𝑡⁺₁=−2𝑑₃. (2.10)
In view of 𝑢(𝑡⁻₁)=𝑢(𝑡⁺₁),Δ𝑢′(𝑡₁)=𝑢′(𝑡⁺₁)−𝑢′(𝑡⁻₁)=𝐼₁(𝑢(𝑡₁)) and Δ𝑢^′′(𝑡₁)=𝑢^′′(𝑡⁺₁)−𝑢^′′(𝑡⁻₁)=𝐼^∗₁(𝑢(𝑡₁)), we have −𝑑₁=1Γ(𝛼)^𝑡1₀(𝑡₁−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠−𝑐₃𝑡²₁,−𝑑₂=1Γ(𝛼−1)^𝑡1₀(𝑡₁−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−2𝑐₃𝑡₁+𝐼₁𝑢𝑡₁,−2𝑑₃=1Γ(𝛼−2)^𝑡1₀(𝑡₁−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠−2𝑐₃+𝐼^∗₁𝑢𝑡₁. (2.11)
Consequently, 𝑢(𝑡)=1Γ(𝛼)^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝑦(𝑠)1𝑑𝑠+Γ(𝛼)^𝑡1₀(𝑡₁−𝑠)^𝛼−1𝑦(𝑠)+𝑑𝑠𝑡−𝑡₁Γ(𝛼−1)^𝑡1₀(𝑡₁−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠+(𝑡−𝑡₁)²2Γ(𝛼−2)^𝑡1₀(𝑡₁−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠𝑡−𝑡₁𝐼₁𝑢𝑡₁+12(𝑡−𝑡₁)²𝐼^∗₁𝑢𝑡₁−𝑐₃𝑡²,𝑡∈𝐽₁. (2.12)
Similarly, we can get 𝑢(𝑡)=1Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑦(𝑠)1𝑑𝑠+Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑦(𝑠)+𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠+^𝑘−1_𝑖=1(𝑡_𝑘−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑘𝑖=1(𝑡−𝑡_𝑘)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1(𝑡_𝑘−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝑐₃𝑡²,𝑡∈𝐽_𝑘,𝑘=1,2,…,𝑝. (2.13)
By (2.13), it follows 𝑢(1)=1Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1𝑦(𝑠)𝑑𝑠+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)+𝑑𝑠_{𝑝−1𝑖=1}(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠+^𝑝_𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)+𝑑𝑠_{𝑝−1𝑖=1}1−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠+^𝑝_𝑖=1(1−𝑡_𝑝)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=11−𝑡_𝑝²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝑐₃,𝑢𝜂=1Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑦(𝑠)1𝑑𝑠+Γ(𝛼)^𝑚_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑦(𝑠)+𝑑𝑠_{𝑚−1𝑖=1}𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝑡_𝑚−𝑡_𝑖²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑚𝑖=1𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑚𝑖=1(𝜂−𝑡_𝑚)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1(𝑡_𝑚−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1(𝜂−𝑡_𝑚)²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝑐₃𝜂². (2.14)
In view of the condition 𝛽𝑢(𝜂)=𝑢(1), we have 𝑐₃1=−1−𝛽𝜂²𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠+^𝑚_𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑦(𝑠)+𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝛽(𝑡_𝑚−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)+𝑑𝑠_{𝑚−1𝑖=1}𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)+𝑑𝑠_𝑚𝑖=1𝛽(𝜂−𝑡_𝑚)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)1𝑑𝑠−Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑦(𝑠)−𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)−𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑦(𝑠)−𝑑𝑠_𝑝𝑖=11−𝑡_𝑝²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑦(𝑠)𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽(𝜂−𝑡_𝑚)²2𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝_𝑖=1(1−𝑡_𝑝)²2𝐼^∗_𝑖𝑢𝑡_𝑖. (2.15)
Substituting the value of 𝑐_𝑖(𝑖=1,2,3) in (2.7) and (2.13) and letting 𝐶=−𝑐₃, we can get (2.5). Conversely, assume that 𝑢 is a solution of the impulsive fractional integral equation (2.5), then by a direct computation, it follows that the solution given by (2.5) satisfies (2.4).
This completes the proof.

3. Main Results

Let 2<𝛼≤3,1≠𝛽𝜂²,𝜂∈(𝑡_𝑚,𝑡_𝑚+1),  𝑚 is a nonnegative integer, and 0≤𝑚≤𝑝. Define the operator 𝐴∶𝐶(𝐽)→𝐶(𝐽) as follows: (𝑡)=1𝐴𝑢Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠1𝑠,𝑢))𝑑𝑠+Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘²2𝐼^∗_𝑖𝑢𝑡_𝑖+𝑡²1−𝛽𝜂²×𝛽Γ(𝛼)^𝜂_𝑡𝑚𝜂−𝑠^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠1𝑠,𝑢))𝑑𝑠−Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1𝑓((𝑠−𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠−^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠−𝑠,𝑢))𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠−𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}1−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠−𝑠,𝑢))𝑑𝑠_𝑝𝑖=11−𝑡_𝑝²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚²2𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖−^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖−^𝑝_𝑖=11−𝑡_𝑝²2𝐼^∗_𝑖𝑢𝑡_𝑖, (3.1)

then (1.1) has a solution if and only if the operator 𝐴 has a fixed point.

Lemma 3.1. The operator 𝐴∶𝐶(𝐽)→𝐶(𝐽) is completely continuous.

Proof. Obviously, 𝐴 is continuous in view of continuity of 𝑓, 𝐼_𝑘, and 𝐼^∗_𝑘.
Let Ω⊂𝐶(𝐽) be bounded. Then, there exist positive constants 𝐿_𝑖>0(𝑖=1,2,3) such that |𝑓(𝑡,𝑢)|≤𝐿₁,|𝐼_𝑘(𝑢)|≤𝐿₂ and |𝐼^∗_𝑘(𝑢)|≤𝐿₃, for all 𝑢∈Ω. Thus, for all 𝑢∈Ω, we have 𝑡≤1|(𝐴𝑢)()|Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠1𝑠,𝑢))𝑑𝑠+Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘−1_𝑖=1(𝑡_𝑘−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1(𝑡−𝑡_𝑘)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1(𝑡_𝑘−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1(𝑡−𝑡_𝑘)²2𝐼^∗_𝑖𝑢𝑡_𝑖+𝑡²1−𝛽𝜂²×𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽(𝑡_𝑚−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽(𝜂−𝑡_𝑚)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠1𝑠,𝑢))𝑑𝑠+Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}1−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=1(1−𝑡_𝑝)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽(𝑡_𝑚−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽(𝜂−𝑡_𝑚)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1(1−𝑡_𝑝)²2𝐼^∗_𝑖𝑢𝑡_𝑖≤𝐿₁Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝐿𝑑𝑠+¹Γ(𝛼)^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑑𝑠+^𝑝−1_𝑖=1𝐿₁Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2+𝑑𝑠_{𝑝−1𝑖=1}𝐿₁2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝_𝑖=1𝐿₁Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2+𝑑𝑠_{𝑝−1𝑖=1}𝐿₁Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝_𝑖=1𝐿₁2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝−1_𝑖=1𝐿₂+^𝑝−1_𝑖=112𝐿₃+^𝑝_𝑖=1𝐿₂+^𝑝−1_𝑖=1𝐿₃+^𝑝_𝑖=112𝐿₃+11−𝛽𝜂²×𝐿₁𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑑𝑠+^𝑝_𝑖=1𝐿₁𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1+𝑑𝑠_{𝑝−1𝑖=1}𝐿₁𝛽Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁𝛽2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁𝛽Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁𝛽Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁𝛽2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝+1_𝑖=1𝐿₁Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1+𝑑𝑠_{𝑝−1𝑖=1}𝐿₁Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝−1_𝑖=1𝛽𝐿₂+^𝑝−1_𝑖=1𝛽2𝐿₃+^𝑝_𝑖=1𝛽𝐿₂+^𝑝−1_𝑖=1𝛽𝐿₃+^𝑝_𝑖=1𝛽2𝐿₃+^𝑝−1_𝑖=1𝐿₂+^𝑝−1_𝑖=112𝐿₃+^𝑝_𝑖=1𝐿₂+^𝑝−1_𝑖=1𝐿₃+^𝑝_𝑖=112𝐿₃≤𝐿₁Γ()+𝛼+1𝑝𝐿₁Γ()+𝐿𝛼+1𝑝−1₁Γ(𝛼)+𝑝−1𝐿₁()+2Γ𝛼−1𝑝𝐿₁Γ(𝛼)+𝑝−1𝐿₁Γ()+𝛼−1𝑝𝐿₁()+𝐿2Γ𝛼−1𝑝−1₂+𝑝−12𝐿₃+𝑝𝐿₂+𝑝−1𝐿₃+𝑝2𝐿₃+11−𝛽𝜂²×𝛽𝐿₁Γ()+𝛼+1𝛽𝑝𝐿₁Γ()+𝛽𝐿𝛼+1𝑝−1₁Γ(𝛼)+𝛽𝑝−1𝐿₁()+2Γ𝛼−1𝛽𝑝𝐿₁Γ(𝛼)+𝛽𝑝−1𝐿₁Γ()+𝛼−1𝛽𝑝𝐿₁()+𝐿2Γ𝛼−1𝑝+1₁Γ()+𝐿𝛼+1𝑝−1₁Γ(𝛼)+𝑝−1𝐿₁()+2Γ𝛼−1𝑝𝐿₁Γ(𝛼)+𝑝−1𝐿₁Γ()+𝛼−1𝑝𝐿₁()𝐿2Γ𝛼−1+𝛽𝑝−1₂+𝛽𝑝−12𝐿₃+𝛽𝑝𝐿₂+𝛽𝑝−1𝐿₃+𝛽𝑝2𝐿₃+𝑝−1𝐿₂+𝑝−12𝐿₃+𝑝𝐿₂+𝑝−1𝐿₃+𝑝2𝐿₃=2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1𝐿₁Γ()+𝐿𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝐿₁()+𝐿2Γ𝛼−12𝑝−1₂+32𝑝−2𝐿₃, (3.2) which implies ‖𝐴𝑢‖≤2+𝛽1−𝜂²1−𝛽𝜂²×𝑝+1𝐿₁Γ()+𝐿𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝐿₁()+𝐿2Γ𝛼−12𝑝−1₂+32𝑝−2𝐿₃∶=𝐿. (3.3)
On the other hand, for any 𝑡∈𝐽_𝑘,0≤𝑘≤𝑝, we have (𝐴𝑢)^′(𝑡≤1)Γ(𝛼−1)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−2𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘_𝑖=11Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘𝐼^∗_𝑖𝑢𝑡_𝑖+2𝑡1−𝛽𝜂²×𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽(𝑡_𝑚−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+1𝑠,𝑢))𝑑𝑠Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}1−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=1(1−𝑡_𝑝)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1(1−𝑡_𝑝)²2𝐼^∗_𝑖𝑢𝑡_𝑖≤𝐿₁Γ(𝛼−1)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−2𝑑𝑠+^𝑝_𝑖=1𝐿₁Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2+𝑑𝑠_{𝑝−1𝑖=1}𝐿₁Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝_𝑖=1𝐿₁Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝_𝑖=1𝐿₂+^𝑝−1_𝑖=1𝐿₃+^𝑝_𝑖=1𝐿₃+21−𝛽𝜂²×𝐿₁𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑑𝑠+^𝑝_𝑖=1𝐿₁𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1+𝑑𝑠_{𝑝−1𝑖=1}𝐿₁𝛽Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁𝛽2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁𝛽Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁𝛽Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁𝛽2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝+1_𝑖=1𝐿₁Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1+𝑑𝑠_{𝑝−1𝑖=1}𝐿₁Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑑𝑠+^𝑝−1_𝑖=1𝐿₁Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3+𝑑𝑠_𝑝𝑖=1𝐿₁2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑑𝑠+^𝑝−1_𝑖=1𝛽𝐿₂+^𝑝−1_𝑖=1𝛽2𝐿₃+^𝑝_𝑖=1𝛽𝐿₂+^𝑝−1_𝑖=1𝛽𝐿₃+^𝑝_𝑖=1𝛽2𝐿₃+^𝑝−1_𝑖=1𝐿₂+^𝑝−1_𝑖=112𝐿₃+^𝑝_𝑖=1𝐿₂+^𝑝−1_𝑖=1𝐿₃+^𝑝_𝑖=112𝐿₃≤𝐿₁Γ(𝛼)+𝑝𝐿₁Γ(𝛼)+𝑝−1𝐿₁Γ()+𝛼−1𝑝𝐿₁Γ(𝛼−1)𝑝𝐿₂+𝑝−1𝐿₃+𝑝𝐿₃+21−𝛽𝜂²×𝛽𝐿₁Γ()+𝛼+1𝛽𝑝𝐿₁Γ()+𝛽𝐿𝛼+1𝑝−1₁Γ(𝛼)+𝛽𝑝−1𝐿₁()+2Γ𝛼−1𝛽𝑝𝐿₁Γ(𝛼)+𝛽𝑝−1𝐿₁Γ()+𝛼−1𝛽𝑝𝐿₁()+𝐿2Γ𝛼−1𝑝+1₁Γ()+𝐿𝛼+1𝑝−1₁Γ(𝛼)+𝑝−1𝐿₁()+2Γ𝛼−1𝑝𝐿₁Γ(𝛼)+𝑝−1𝐿₁Γ()+𝛼−1𝑝𝐿₁()𝐿2Γ𝛼−1+𝛽𝑝−1₂+𝛽𝑝−12𝐿₃+𝛽𝑝𝐿₂+𝛽𝑝−1𝐿₃+𝛽𝑝2𝐿₃+𝑝−1𝐿₂+𝑝−12𝐿₃+𝑝𝐿₂+𝑝−1𝐿₃+𝑝2𝐿₃=𝑝+1𝐿₁Γ(𝛼)+2𝑝−1𝐿₁Γ(𝛼−1)𝑝𝐿₂+2𝑝−1𝐿₃+2𝛽+11−𝛽𝜂²×𝑝+1𝐿₁Γ()+𝐿𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝐿₁()+𝐿2Γ𝛼−12𝑝−1₂+32𝑝−2𝐿₃∶=𝐿. (3.4)
Hence, let 𝑡₁,𝑡₂∈𝐽_𝑘,𝑡₁<𝑡₂,0≤𝑘≤𝑝, we have (𝐴𝑢)𝑡₂−(𝐴𝑢)𝑡₁≤^𝑡2_𝑡1(𝐴𝑢)^′(𝑠)𝑑𝑠≤𝐿𝑡₂−𝑡₁. (3.5) This implies that 𝐴 is equicontinuous on all 𝐽_𝑘,𝑘=0,1,2,…,𝑝. The Arzela-Ascoli Theorem implies that 𝐴∶𝐶(𝐽)→𝐶(𝐽) is completely continuous.

Theorem 3.2. Assume that the nonlinearity 𝑓 is bounded and that the impulse functions 𝐼_𝑘,𝐼^∗_𝑘,𝑘=1,2,…,𝑝 are bounded. Then, the nonlinear impulsive fractional three-point boundary value problems (1.1) have at least one solution.

Proof. Firstly, by Lemma 3.1, we know that the operator 𝐴∶𝐶(𝐽)→𝐶(𝐽) is completely continuous.
Let 𝐿_𝑖(𝑖=1,2,3) be nonnegative constants such that 𝑓(𝑡,𝑢)≤𝐿₁,𝐼_𝑘(𝑢)≤𝐿₂,𝐼^∗_𝑘(𝑢)≤𝐿₃,𝑡∈𝐽,𝑢∈𝑅. (3.6)
For 0<𝜇<1, consider the equation: 𝑢=𝜇𝐴𝑢. (3.7) If 𝑢 is a solution of (3.7), then for every 𝑡∈𝐽 we have that |𝑢(𝑡|(𝑡≤1)|=𝜇𝐴𝑢)|Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠1𝑠,𝑢))𝑑𝑠+Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}(𝑡_𝑘−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1(𝑡−𝑡_𝑘)²2𝐼^∗_𝑖𝑢𝑡_𝑖+𝑡²1−𝛽𝜂²×𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚_𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽(𝑡_𝑚−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+1𝑠,𝑢))𝑑𝑠Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}1−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=1(1−𝑡_𝑝)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1(1−𝑡_𝑝)²2𝐼^∗_𝑖𝑢𝑡_𝑖≤2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1𝐿₁Γ()+𝐿𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝐿₁()+𝐿2Γ𝛼−12𝑝−1₂+32𝑝−2𝐿₃, (3.8) which implies for any 𝑡∈𝐽, it holds that ‖𝑢‖≤2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1𝐿₁Γ()+𝐿𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝐿₁()+𝐿2Γ𝛼−12𝑝−1₂+32𝑝−2𝐿₃. (3.9)
This shows that all the solutions of (3.7) are bounded independently of 0<𝜇<1. Using Schaeffer's theorem (see Theorem 2.2), we get that 𝐴 has at least one fixed point, which implies that (1.1) has at least one solution.

Now, we present some existence results when the nonlinearity and the impulse functions have sublinear growth.

Theorem 3.3. Assume that  (H₁) there exist nonnegative constants 𝑎 and 𝑏 such that 𝑓(𝑡,𝑢)≤𝑎+𝑏|𝑢|^𝜌,0≤𝜌<1,𝑡∈𝐽,𝑢∈ℝ; (3.10) (H₂) there exist nonnegative constants 𝑎_𝑘,𝑏_𝑘,𝑎^∗_𝑘,𝑏^∗_𝑘 such that 𝐼_𝑘(𝑢)≤𝑎_𝑘+𝑏_𝑘|𝑢|^𝜌𝑘,𝐼^∗_𝑘(𝑢)≤𝑎^∗_𝑘+𝑏^∗_𝑘|𝑢|^𝜌∗𝑘,0≤𝜌_𝑘,𝜌^∗_𝑘<1,𝑢∈ℝ,𝑘=1,2,…,𝑝. (3.11)Then, problem (1.1) has at least one solution.

Proof. If 𝑢∈𝐶(𝐽), then we can write that 𝑓(𝑠,𝑢(𝑠))≤𝑎+𝑏|𝑢(𝑠)|^𝜌𝐼,𝑠∈𝐽,_𝑘𝑢𝑡_𝑘≤𝑎_𝑘+𝑏_𝑘𝑢𝑡_𝑘^𝜌𝑘,𝐼^∗_𝑘𝑢𝑡_𝑘≤𝑎^∗_𝑘+𝑏^∗_𝑘𝑢𝑡_𝑘^𝜌∗𝑘,𝑘=1,2,…,𝑝. (3.12) So, if 𝑢 is a solution of (3.7), then, by the similar process used to obtain (3.2), for every 𝑡∈𝐽 we have that |𝑢(𝑡≤𝜇)|Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠𝜇𝑠,𝑢))𝑑𝑠+Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝜇𝑡_𝑘−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝜇(𝑡_𝑘−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘_𝑖=1𝜇𝑡−𝑡_𝑘Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑘−1𝑖=1}𝜇𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑘𝑖=1𝜇𝑡−𝑡_𝑘²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘−1_𝑖=1𝜇𝑡_𝑘−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝜇𝑡_𝑘−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝜇𝑡−𝑡_𝑘𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝜇𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝜇(𝑡−𝑡_𝑘)²2𝐼^∗_𝑖𝑢𝑡_𝑖+𝜇𝑡²1−𝛽𝜂²×𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑚𝑖=1𝛽(𝜂−𝑡_𝑚)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠1𝑠,𝑢))𝑑𝑠+Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2𝑓((𝑠+𝑠,𝑢))𝑑𝑠_{𝑝−1𝑖=1}1−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠+𝑠,𝑢))𝑑𝑠_𝑝𝑖=1(1−𝑡_𝑝)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝑡_𝑚−𝑡_𝑖²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑚_𝑖=1𝛽(𝜂−𝑡_𝑚)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1(1−𝑡_𝑝)²2𝐼^∗_𝑖𝑢𝑡_𝑖≤𝑀₁𝑎+𝑀₂^𝑝_𝑘=1𝑎_𝑘+^𝑝−1_𝑘=1𝑎_𝑘+𝑀₂2^𝑝_𝑘=1𝑎^∗_𝑘+3^𝑝−1_𝑘=1𝑎^∗_𝑘+𝑀₁𝑏‖𝑢‖^𝜌+𝑀₂^𝑝_𝑘=1𝑏_𝑘‖𝑢‖^𝜌𝑘+𝑀₂^𝑝−1_𝑘=1𝑏_𝑘‖𝑢‖^𝜌𝑘+𝑀₂2^𝑝_𝑘=1𝑏^∗_𝑘‖𝑢‖^𝜌∗𝑘+3𝑀₂2^𝑝−1_𝑘=1𝑏^∗_𝑘‖𝑢‖^𝜌∗𝑘, (3.13) where 𝑀₁=2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1Γ()+𝛼+12𝑝−1Γ(𝛼)+4𝑝−32Γ(𝛼−1),𝑀₂=2+𝛽1−𝜂²1−𝛽𝜂². (3.14)
In consequence, ‖𝑢‖≤𝑀+𝑀₁𝑏‖𝑢‖^𝜌+𝑀₂^𝑝_𝑘=1𝑏_𝑘‖𝑢‖^𝜌𝑘+𝑀₂^𝑝−1_𝑘=1𝑏_𝑘‖𝑢‖^𝜌𝑘+𝑀₂2^𝑝_𝑘=1𝑏^∗_𝑘‖𝑢‖^𝜌∗𝑘+3𝑀₂2^𝑝−1_𝑘=1𝑏^∗_𝑘‖𝑢‖^𝜌∗𝑘, (3.15) where 𝑀=𝑀₁𝑎+𝑀₂(∑^𝑝_𝑘=1𝑎_𝑘+∑^𝑝−1_𝑘=1𝑎_𝑘)+(𝑀₂/2)(∑^𝑝_𝑘=1𝑎^∗_𝑘+3∑^𝑝−1_𝑘=1𝑎^∗_𝑘).
Now, taking into account that 0≤𝜌,𝜌_𝑘,𝜌^∗_𝑘<1,  𝑘=1,2,…,𝑝, we can conclude that there exists a constant 𝐶>0 such that ‖𝑢‖≤𝐶 for any solution of (3.7), 0<𝜇<1. Therefore, by Schaeffer's theorem (Theorem 2.2), we obtain the existence of a fixed point for 𝐴, which implies the problem (1.1) has at least one solution.

Theorem 3.4. Assume that  (H₃) there exist nonnegative constants 𝑎 and 𝑏 such that 𝑓(𝑡,𝑢)≤𝑎+𝑏|𝑢|,𝑡∈𝐽,𝑢∈ℝ; (3.16) (H₄) there exist nonnegative constants 𝑎_𝑘,𝑏_𝑘,𝑎^∗_𝑘,𝑏^∗_𝑘 such that 𝐼_𝑘(𝑢)≤𝑎_𝑘+𝑏_𝑘|𝑢|,𝐼^∗_𝑘(𝑢)≤𝑎^∗_𝑘+𝑏^∗_𝑘|𝑢|,𝑢∈ℝ,𝑘=1,2,…,𝑝. (3.17)
If 𝑀₁𝑏+𝑀₂^𝑝_𝑘=1𝑏_𝑘+12𝑏^∗_𝑘+^𝑝−1_𝑘=1𝑏_𝑘+32𝑏^∗_𝑘<1, (3.18) where 𝑀₁=2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1Γ()+𝛼+12𝑝−1Γ(𝛼)+4𝑝−32Γ(𝛼−1),𝑀₂=2+𝛽1−𝜂²1−𝛽𝜂². (3.19) Then problem (1.1) has at least one solution.

Proof. The proof is similar to that of Theorem 3.3, so we omit it.

Theorem 3.5. Let lim_𝑢→0𝑓(𝑡,𝑢)/𝑢=0,lim_𝑢→0𝐼_𝑘(𝑢)/𝑢=0, and lim_𝑢→0𝐼^∗_𝑘(𝑢)/𝑢=0, then problem (1.1) has at least one solution.

Proof. Now, in view of lim_𝑢→0𝑓(𝑡,𝑢)/𝑢=0,lim_𝑢→0𝐼_𝑘(𝑢)/𝑢=0, and lim_𝑢→0𝐼^∗_𝑘(𝑢)/𝑢=0, there exists a constant 𝑟>0 such that |𝑓(𝑡,𝑢)|≤𝛿₁|𝑢|,|𝐼_𝑘(𝑢)|≤𝛿₂|𝑢|, and |𝐼^∗_𝑘(𝑢)|≤𝛿₃|𝑢| for 0<|𝑢|<𝑟, where 𝛿_𝑖>0(𝑖=1,2,3) satisfy 2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1𝛿₁Γ()+𝛿𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝛿₁()+𝛿2Γ𝛼−12𝑝−1₂+32𝑝−2𝛿₃≤1. (3.20)
Let Ω={𝑢∈𝐶(𝐽)∣‖𝑢‖<𝑟}. Take 𝑢∈𝐶(𝐽) such that ‖𝑢‖=𝑟, that is, 𝑢∈𝜕Ω. By Lemma 3.1, we know that 𝐴 is completely continuous, and |(𝐴𝑢)(𝑡)|≤2+𝛽1−𝜂²1−𝛽𝜂²×𝑝+1𝛿₁Γ()+𝛿𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝛿₁()+𝛿2Γ𝛼−12𝑝−1₂+32𝑝−2𝛿₃‖𝑢‖. (3.21)
Thus, in view of (3.21), we obtain ‖𝐴𝑢‖≤‖𝑢‖, 𝑢∈𝜕Ω. Therefore, by Theorem 2.3, the operator 𝐴 has at least one fixed point, which implies that (1.1) has at least one solution 𝑢∈Ω.

For the forthcoming analysis, we need the following assumption: (H₅) there exist positive constants 𝐾_𝑖(𝑖=1,2,3) such that 𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)≤𝐾₁|𝑢−𝑣|,𝐼_𝑘(𝑢)−𝐼_𝑘(𝑣)≤𝐾₂|𝑢−𝑣|,𝐼^∗_𝑘(𝑢)−𝐼^∗_𝑘(𝑣)≤𝐾₃|𝑢−𝑣|, (3.22)for 𝑡∈𝐽,𝑢,𝑣∈𝑅  and  𝑘=1,2,…,𝑝.

Define the constants: Λ=2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1𝐾₁Γ()+𝐾𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝐾₁()+𝐾2Γ𝛼−12𝑝−1₂+32𝑝−2𝐾₃. (3.23)

Theorem 3.6. If condition (H₅) holds. Then problem (1.1) has a unique solution provided Λ<1, where Λ is given by (3.23).

Proof. Let 𝑢,𝑣∈𝐶(𝐽), by a simple computation, we can get 𝑡)−(𝑡≤1|(𝐴𝑢)(𝐴𝑣)()|Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠((𝑠+1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑘−1𝑖=1}(𝑡_𝑘−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑘𝑖=1𝑡−𝑡_𝑘Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑘−1𝑖=1}𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑘𝑖=1(𝑡−𝑡_𝑘)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑘−1𝑖=1}𝑡_𝑘−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑘−1_𝑖=1(𝑡_𝑘−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑘−1_𝑖=1𝑡−𝑡_𝑘𝑡_𝑘−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑘_𝑖=1(𝑡−𝑡_𝑘)²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+𝑡²1−𝛽𝜂²×𝛽Γ(𝛼)^𝜂_𝑡𝑚(𝜂−𝑠)^𝛼−1𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑚𝑖=1𝛽Γ(𝛼)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑚−1𝑖=1}𝛽(𝑡_𝑚−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑚𝑖=1𝛽𝜂−𝑡_𝑚Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑚𝑖=1𝛽(𝜂−𝑡_𝑚)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠Γ(𝛼)^𝑝+1_𝑖=1^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−1𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑝−1𝑖=1}𝑡_𝑝−𝑡_𝑖Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑝−1𝑖=1}(𝑡_𝑝−𝑡_𝑖)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑝𝑖=11−𝑡_𝑝Γ(𝛼−1)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−2𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑝−1𝑖=1}1−𝑡_𝑝𝑡_𝑝−𝑡_𝑖Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_𝑝𝑖=1(1−𝑡_𝑝)²2Γ(𝛼−2)^𝑡𝑖_𝑡𝑖−1(𝑡_𝑖−𝑠)^𝛼−3𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑚−1𝑖=1}𝛽𝑡_𝑚−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑚−1_𝑖=1𝛽(𝑡_𝑚−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑚_𝑖=1𝛽𝜂−𝑡_𝑚𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑚−1_𝑖=1𝛽𝜂−𝑡_𝑚𝑡_𝑚−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑚_𝑖=1𝛽(𝜂−𝑡_𝑚)²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑝−1_𝑖=1(𝑡_𝑝−𝑡_𝑖)²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑝_𝑖=11−𝑡_𝑝𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑝−1_𝑖=11−𝑡_𝑝𝑡_𝑝−𝑡_𝑖𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑝_𝑖=1(1−𝑡_𝑝)²2𝐼^∗_𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖≤2+𝛽1−𝜂²1−𝛽𝜂²𝑝+1𝐾₁Γ()+𝐾𝛼+12𝑝−1₁Γ(𝛼)+4𝑝−3𝐾₁()+𝐾2Γ𝛼−12𝑝−1₂+32𝑝−2𝐾₃×‖‖‖‖.𝑢−𝑣≤Λ𝑢−𝑣 (3.24)
Thus, ‖𝐴𝑢−𝐴𝑣‖≤Λ‖𝑢−𝑣‖.
As Λ<1, therefore, 𝐴 is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.

4. Examples

Example 4.1. Consider the following nonlinear impulsive fractional differential equations: ^𝐶𝐷^𝛼𝑢(𝑡)=𝑒^𝑡cos³𝑢(𝑡)1+𝑢²(𝑡)1,0<𝑡<1,𝑡≠2,Δ𝑢^′12=5+3𝑢⁴(1/2)1+𝑢⁴(1/2),Δ𝑢^′′121=3sin𝑢2,𝑢(0)=𝑢^′(0)=0,𝛽𝑢𝜂=𝑢(1), (4.1) where 2<𝛼≤3,0<𝜂<1,𝜂≠1/2,0<𝛽<1/𝜂²,  and  𝑝=1.

Obviously, for 𝐿₁=𝑒,𝐿₂=5, and 𝐿₃=3, it is easy to verify condition of Theorem 3.2 holds. Hence, by Theorem 3.2, we can get that the above equation (4.1) has at least one solution.

Example 4.2. Consider the following fractional impulsive the three-point boundary value problem: ^𝐶𝐷^𝛼𝑢(𝑡)=𝑒^5𝑡cos⁵3𝑢(𝑡)+𝑒^8𝑢(𝑡)1+𝑢⁴(𝑡)+2sin(2𝑡+5)3+𝑢²(𝑡)|𝑢(𝑡)|^𝜌1,0<𝑡<1,𝑡≠4,Δ𝑢^′14=3+2cos²ln1+8𝑢²141+3cos𝑢4𝑢14^𝜌1,Δ𝑢^′′14=12+5+2𝑢²(1/4)2+𝑢²(1/4)+3𝑒^{−𝑢2(1/4)}𝑢14^𝜌2,𝑢(0)=𝑢^′(0)=0,𝛽𝑢𝜂=𝑢(1), (4.2) where 2<𝛼≤3,0<𝜂<1,𝜂≠1/4,0<𝛽<1/𝜂² ,  and  𝑝=1.

Obviously 𝑎=𝑒⁵,𝑏=2,𝑎₁=5,𝑏₁=𝑎^∗₁=𝑏^∗₁=3. It is easy to verify that for 0<𝜌,𝜌_𝑘,𝜌^∗_𝑘<1, the conditions of Theorem 3.3 hold. Therefore, by Theorem 3.3, the impulsive the three-point fractional boundary value problem (4.2) has at least one solution.

Example 4.3. Consider the following nonlinear impulsive fractional differential equations: ^𝐶𝐷^𝛼𝑢(𝑡)=𝑡²𝑢²(𝑡)(𝑡)(𝑡)1+𝑢arctan𝑢,0<𝑡<1,𝑡≠3,Δ𝑢^′13=𝑒^𝑢(𝑡)𝑢³13,Δ𝑢^′′13=3𝑢²(1/3)1+2𝑢⁴(),𝑢(0)1/3=𝑢^′(0)=0,𝛽𝑢𝜂=𝑢(1), (4.3) where 2<𝛼≤3,0<𝜂<1,𝜂≠1/3,0<𝛽<1/𝜂², and 𝑝=1.

Clearly, all the assumptions of Theorem 3.5 hold. Thus, by the conclusion of Theorem 3.5 we can get that (4.3) has at least one solution.

5. Conclusion

The existence and uniqueness of solutions to a three-point boundary value problem for fractional nonlinear differential equation with impulses have been discussed. We apply the concepts of fractional calculus together with fixed point theorems to establish the existence results. First of all, we investigate a linear three-point boundary value problem involving fractional derivatives and impulses, which in fact provides the platform to prove the existence of solutions for the associated nonlinear fractional equation with impulses. It is worth mentioning that the conditions of our theorems are easily to verify and that our approach is simple, so they are applicable to a variety of real world problems.

Acknowledgments

The authors would like to express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which improve the quality of present paper. They also would like to thank Professor Bashir Ahmad for his significant suggestions on the improvement of this paper. This paper was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant no. 2012021002-3) and Science Foundation of Shanxi Normal University, China (Grant no. ZR1113).