On Impulsive Boundary Value Problems of Fractional Differential Equations with Irregular Boundary Conditions

Abstract

We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.

1. Introduction

Boundary value problems of nonlinear fractional differential equations have recently been studied by several researchers. Fractional differential equations appear naturally in various fields of science and engineering and constitute an important field of research. As a matter of fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes [1–4]. Some recent work on boundary value problems of fractional order can be found in [5–23] and the references therein. In [24], some existence and uniqueness results were obtained for an irregular boundary value problem of fractional differential equations.

Dynamical systems with impulse effect are regarded as a class of general hybrid systems. Impulsive hybrid systems are composed of some continuous variable dynamic systems along with certain reset maps that define impulsive switching among them. It is the switching that resets the modes and changes the continuous state of the system. There are three classes of impulsive hybrid systems, namely, impulsive differential systems [25, 26], sampled data or digital control system [27, 28], and impulsive switched system [29, 30]. Applications of such systems include air traffic management [31], automotive control [32, 33], real-time software verification [34], transportation systems [35, 36], manufacturing [37], mobile robotics [38], and process industry [39]. In fact, hybrid systems have a central role in embedded control systems that interact with the physical world. Using hybrid models, one may represent time and event-based behaviors more accurately so as to meet challenging design requirements in the design of control systems for problems such as cut-off control and idle speed control of the engine. For more details, see [40] and the references therein.

The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modelling of a wide variety of practical situations and has emerged as an important area of investigation. The impulsive differential equations of fractional order have also attracted a considerable attention and a variety of results can be found in the papers [41–50].

In this paper, motivated by [24], we study a nonlinear impulsive hybrid system of fractional differential equations with irregular boundary conditions given by ^𝐶𝐷^𝛼𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),1<𝛼≤2,𝑡∈𝐽^′,Δ𝑢𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘,Δ𝑢^′𝑡_𝑘=𝐼^∗_𝑘𝑢𝑡_𝑘𝑢,𝑘=1,2,…,𝑝,_′(0)+(−1)^𝜃𝑢^′(𝑇)+𝑏𝑢(𝑇)=0,𝑢(0)+(−1)^𝜃+1𝑢(𝑇)=0,𝜃=1,2, (1.1) where ^𝐶𝐷^𝛼 is the Caputo fractional derivative, 𝑓∈𝐶(𝐽×ℝ,ℝ),𝐼_𝑘,𝐼^∗_𝑘∈𝐶(ℝ,ℝ),𝑏∈ℝ,𝑏≠0,𝐽=[0,𝑇](𝑇>0),0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯<𝑡_𝑝<𝑡_𝑝+1=𝑇,𝐽^′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑝},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) have a similar meaning for 𝑢^′(𝑡).

Here, we remark that irregular boundary value problems for ordinary and partial differential equations occur in scientific and engineering disciplines and have been addressed by many authors, for instance, see [24] and the references.

The paper is organized as follows. Section 2 deals with some definitions and preliminary results, while the main results are presented in Section 3.

2. Preliminaries

Let us fix 𝐽₀=[0,𝑡₁],𝐽_𝑘−1=(𝑡_𝑘−1,𝑡_𝑘],𝑘=2,…,𝑝+1 with 𝑡_𝑝+1=𝑇 and introduce the spaces: 𝑃𝐶(𝐽,ℝ)=𝑢∶𝐽→ℝ∣𝑢∈𝐶𝐽_𝑘,𝑘=0,1,…,𝑝,and𝑢𝑡⁺_𝑘exist,𝑘=1,2,…,𝑝, (2.1) with the norm ‖𝑢‖=sup_𝑡∈𝐽|𝑢(𝑡)|, and 𝑃𝐶¹(𝐽,ℝ)=𝑢∶𝐽→ℝ∣𝑢∈𝐶¹𝐽_𝑘,𝑘=0,1,…,𝑝,and𝑢𝑡⁺_𝑘,𝑢^′𝑡⁺_𝑘exist,,𝑘=1,2,…,𝑝 (2.2) with the norm ‖𝑢‖_𝑃𝐶¹=max{‖𝑢‖,‖𝑢^′‖}. Obviously, 𝑃𝐶(𝐽,ℝ) and 𝑃𝐶¹(𝐽,ℝ) are Banach spaces.

Definition 2.1. A function 𝑢∈𝑃𝐶¹(𝐽,ℝ) with its Caputo derivative of order 𝛼 existing on 𝐽 is a solution of (1.1) if it satisfies (1.1).

To prove the existence of solutions of problem (1.1), we need the following fixed-point theorems.

Theorem 2.2 (see [51]). Let 𝐸 be a Banach space. Assume that Ω is an open bounded subset of 𝐸 with 𝜃∈Ω and let 𝑇∶Ω→𝐸 be a completely continuous operator such that ‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝜕Ω. (2.3) Then 𝑇 has a fixed point in Ω.

Lemma 2.3 (see [1]). For 𝛼>0, the general solution of fractional differential equation^𝐶𝐷^𝛼𝑢(𝑡)=0 is 𝑢(𝑡)=𝐶₀+𝐶₁𝑡+𝐶₂𝑡²+⋯+𝐶_𝑛−1𝑡^𝑛−1, (2.4) where 𝐶_𝑖∈ℝ,𝑖=0,1,2,…,𝑛−1,𝑛=[𝛼]+1 ([𝛼] denotes integer part of 𝛼).

Lemma 2.4 (see [1]). Let 𝛼>0. Then 𝐼^𝛼𝐶𝐷^𝛼𝑢(𝑡)=𝑢(𝑡)+𝐶₀+𝐶₁𝑡+𝐶₂𝑡²+⋯+𝐶_𝑛−1𝑡^𝑛−1 (2.5) for some 𝐶_𝑖∈ℝ,𝑖=1,2,…,𝑛−1,𝑛=[𝛼]+1.

Lemma 2.5. For a given 𝑦∈𝐶[0,𝑇], a function 𝑢 is a solution of the following impulsive irregular boundary value problem ^𝐶𝐷^𝛼𝑢(𝑡)=𝑦(𝑡),1<𝛼≤2,𝑡∈𝐽^′,Δ𝑢𝑡_𝑘=𝐼_𝑘𝑢𝑡_𝑘,Δ𝑢^′𝑡_𝑘=𝐼^∗_𝑘𝑢𝑡_𝑘𝑢,𝑘=1,2,…,𝑝,_′(0)+(−1)^𝜃𝑢^′(𝑇)+𝑏𝑢(𝑇)=0,𝑢(0)+(−1)^𝜃+1𝑢(𝑇)=0,𝜃=1,2,𝑏≠0, (2.6) if and only if 𝑢 is a solution of the impulsive fractional integral equation 𝑢(𝑡)=∫^𝑡₀(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)()𝑑𝑠+1−−1^𝜃+1∫𝑏𝑇^𝑇𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)+()𝑑𝑠1+−1^𝜃+1𝑡∫𝑏𝑇^𝑇𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)−1𝛼−1𝑑𝑠𝑏∫^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)𝑡𝛼−1𝑑𝑠−𝑇∫^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝒜,𝑡∈𝐽₀;∫^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)()𝑑𝑠+1−−1^𝜃+1∫𝑏𝑇^𝑇𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)+()𝑑𝑠1+−1^𝜃+1𝑡∫𝑏𝑇^𝑇𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)−1𝛼−1𝑑𝑠𝑏∫^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)𝑡𝛼−1𝑑𝑠−𝑇∫^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)+𝑑𝑠_𝑘𝑖∑₌₁∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+∑^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+∑^𝑘_𝑖=1𝑡−𝑡_𝑘∫^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+𝒜,𝑡∈𝐽_𝑘,𝑘=1,2,…,𝑝, (2.7) where ()𝒜=1+−1^𝜃+1𝑡−𝑇𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+1−(−1)^𝜃+1−𝑏𝑡𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖. (2.8)

Proof. Let 𝑢 be a solution of (2.6). Then, by Lemma 2.4, we have 𝑢(𝑡)=𝐼^𝛼𝑦(𝑡)−𝑐₁−𝑐₂1𝑡=Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠−𝑐₁−𝑐₂𝑡,𝑡∈𝐽₀, (2.9) for some 𝑐₁,𝑐₂∈ℝ. Differentiating (2.9), we get 𝑢^′(𝑡)=1Γ(𝛼−1)^𝑡₀(𝑡−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−𝑐₂,𝑡∈𝐽₀. (2.10)
If 𝑡∈𝐽₁, then 𝑢(𝑡)=1Γ(𝛼)^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝑦(𝑠)𝑑𝑠−𝑑₁−𝑑₂𝑡−𝑡₁,𝑢^′(𝑡)=1Γ(𝛼−1)^𝑡_𝑡1(𝑡−𝑠)^𝛼−2𝑦(𝑠)𝑑𝑠−𝑑₂, (2.11) for some 𝑑₁,𝑑₂∈ℝ. Thus, 𝑢𝑡⁻₁=1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1𝑦(𝑠)𝑑𝑠−𝑐₁−𝑐₂𝑡₁,𝑢𝑡⁺₁=−𝑑₁,𝑢^′𝑡⁻₁=1Γ(𝛼−1)^𝑡1₀𝑡₁−𝑠^𝛼−2𝑦(𝑠)𝑑𝑠−𝑐₂,𝑢^′𝑡⁺₁=−𝑑₂. (2.12) Using the impulse conditions Δ𝑢𝑡₁=𝑢𝑡⁺₁−𝑢𝑡⁻₁=𝐼₁𝑢𝑡₁,Δ𝑢^′𝑡₁=𝑢^′𝑡⁺₁−𝑢^′𝑡⁻₁=𝐼^∗₁𝑢𝑡₁, (2.13) we find that −𝑑₁=1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1𝑦(𝑠)𝑑𝑠−𝑐₁−𝑐₂𝑡₁+𝐼₁𝑢𝑡₁,−𝑑₂=1Γ(𝛼−1)^𝑡1₀𝑡₁−𝑠^𝛼−2𝑦(𝑠)𝑑𝑠−𝑐₂+𝐼^∗₁𝑢𝑡₁. (2.14) Consequently, we obtain 𝑢(𝑡)=1Γ(𝛼)^𝑡_𝑡1(𝑡−𝑠)^𝛼−1𝑦(𝑠)1𝑑𝑠+Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1𝑦(𝑠)+𝑑𝑠𝑡−𝑡₁Γ(𝛼−1)^𝑡1₀𝑡₁−𝑠^𝛼−2𝑦(𝑠)𝑑𝑠+𝐼₁𝑢𝑡₁+𝑡−𝑡₁𝐼^∗₁𝑢𝑡₁−𝑐₁−𝑐₂𝑡,𝑡∈𝐽₁. (2.15) By a similar process, we get 𝑢(𝑡)=^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖−𝑐₁−𝑐₂𝑡,𝑡∈𝐽_𝑘,𝑘=1,2,…,𝑝. (2.16)
Applying the boundary conditions 𝑢^′(0)+(−1)^𝜃𝑢^′(𝑇)+𝑏𝑢(𝑇)=0 and 𝑢(0)+(−1)^𝜃+1𝑢(𝑇)=0, we find that 𝑐₁()=−1−−1^𝜃+1𝑏𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)1𝑑𝑠+𝑏^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)+𝛼−1𝑑𝑠_𝑝𝑖=11𝑏^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖−1−(−1)^𝜃+1𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖,𝑐₂()=−1+−1^𝜃+1𝑏𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑦(𝑠)1𝛼−1𝑑𝑠+𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑦(𝑠)−()𝑑𝑠1+−1^𝜃+1𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+1𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑦(𝑠)𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑦(𝑠)𝛼−1𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖. (2.17) Substituting the value of 𝑐_𝑖(𝑖=1,2) in (2.9) and (2.16), we obtain (2.7). Conversely, assume that 𝑢 is a solution of the impulsive fractional integral equation (2.7), then by a direct computation, it follows that the solution given by (2.7) satisfies (2.6). This completes the proof.

Remark 2.6. With 𝑇=𝜋, the first five terms of the solution (2.7) correspond to the solution for the problem without impulses [24].

3. Main Results

Define an operator 𝒢∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) by 𝒢𝑢(𝑡)=^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑢))𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑢))𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+1−(−1)^𝜃+1𝑏𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠()𝑠,𝑢))𝑑𝑠+1+−1^𝜃+1𝑡𝑏𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠−1𝛼−1𝑠,𝑢))𝑑𝑠𝑏^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠𝑡𝛼−1𝑠,𝑢))𝑑𝑠−𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠+()𝑠,𝑢))𝑑𝑠1+−1^𝜃+1𝑡−𝑇𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑢))𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+1−(−1)^𝜃+1−𝑏𝑡𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑢))𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑢))𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖. (3.1) Notice that problem (1.1) has a solution if and only if the operator 𝒢 has a fixed point.

For the sake of convenience, we set the following notations: 1𝜇=21+𝑝1+|𝑏|𝑇𝐼^𝛼𝑎(𝑇)+22𝑝−1𝑇+5𝑝+1|𝑏|𝐼^𝛼−11+21+|𝑏|𝑇𝑝𝐿₂+22𝑝−1𝑇+5𝑝−2|𝑏|𝐿₃,1𝜈=21+𝑝1+|𝑏|𝑇𝑇^𝛼Γ()+2𝛼+12𝑝−1𝑇+5𝑝+1|𝑏|𝑇^𝛼−1Γ(𝛼). (3.2)

Theorem 3.1. Assume that (𝐇_𝟏) there exists a nonnegative function 𝑎(𝑡)∈𝐿(0,𝑇) such that 𝑓(𝑡,𝑢)≤𝑎(𝑡)+𝜉|𝑢|^𝜌,0<𝜌<1, (3.3) where 𝜉 is a nonnegative constant;(𝐇_𝟐) there exist positive constants 𝐿₂ and 𝐿₃ such that 𝐼_𝑘(𝑢)≤𝐿₂,𝐼^∗_𝑘(𝑢)≤𝐿₃,for𝑡∈𝐽,𝑢∈ℝ,𝑘=1,2,…,𝑝. (3.4)Then problem (1.1) has at least one solution.

Proof. As a first step, we show that the operator 𝒢∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) is completely continuous. Observe that continuity of 𝒢 follows from the continuity of 𝑓,𝐼_𝑘 and 𝐼^∗_𝑘.
Let Ω⊂𝑃𝐶(𝐽,ℝ) 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,forall𝑢∈Ω. Thus, forall𝑢∈Ω, we have |𝒢𝑢(𝑡)|≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖+1−(−1)^𝜃+1|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠+()𝑠,𝑢))𝑑𝑠1+−1^𝜃+1𝑡|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠+1𝛼−1𝑠,𝑢))𝑑𝑠|𝑏|^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠𝑡𝛼−1𝑠,𝑢))𝑑𝑠+𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠+()𝑠,𝑢))𝑑𝑠1+−1^𝜃+1𝑡−𝑇𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖+1−(−1)^𝜃+1−𝑏𝑡𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖≤𝐿₁^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑑𝑠+^𝑝_𝑖=1𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑑𝑠+𝐿₂+^𝑝−1_𝑖=1𝑇𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃+^𝑝_𝑖=1𝑇𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃+2𝐿₁|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑑𝑠+2𝐿₁|𝑏|^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()+𝐿𝛼−1𝑑𝑠₁|𝑏|^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₁^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)1𝑑𝑠+|𝑏|^𝑝_𝑖=1𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃+2+|𝑏|𝑇|𝑏|𝑇^𝑝_𝑖=1𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑑𝑠+𝐿₂+^𝑝−1_𝑖=1𝑇𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃+^𝑝_𝑖=1𝑇𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃≤1|𝑏|21+𝑝(|𝑏|𝑇+1)𝑇^𝛼−1𝐿₁Γ()+|𝑏|𝑇𝑇𝛼+15𝑝+1+22𝑝−1^𝛼−1𝐿₁Γ(𝛼)+2(1+|𝑏|𝑇)𝑝𝐿₂𝑇+5𝑝−2+2|𝑏|𝑇2𝑝−1𝐿₃, (3.5) which implies ‖‖≤1𝒢𝑢|𝑏|21+𝑝(|𝑏|𝑇+1)𝑇^𝛼−1𝐿₁Γ()+|𝑏|𝑇𝑇𝛼+15𝑝+1+22𝑝−1^𝛼−1𝐿₁Γ(𝛼)+2(1+|𝑏|𝑇)𝑝𝐿₂𝑇+5𝑝−2+2|𝑏|𝑇2𝑝−1𝐿₃∶=𝐿. (3.6)
On the other hand, for any 𝑡∈𝐽_𝑘,0≤𝑘≤𝑝, we get (𝒢𝑢)^′(𝑡)≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−2Γ()𝑓((𝑠𝛼−1𝑠,𝑢))𝑑𝑠+^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖+1+(−1)^𝜃+1|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠1𝛼−1𝑠,𝑢))𝑑𝑠+𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠+()𝑠,𝑢))𝑑𝑠1+−1^𝜃+1|𝑏|𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖+1𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠𝐼𝛼−1𝑠,𝑢))𝑑𝑠+^∗_𝑖𝑢𝑡_𝑖≤𝐿₁^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−2Γ(𝛼−1)𝑑𝑠+^𝑝_𝑖=1𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃+2𝐿₁|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()+𝐿𝛼−1𝑑𝑠₁𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)2𝑑𝑠+|𝑏|𝑇^𝑝_𝑖=1𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃+1𝑇^𝑝_𝑖=1𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑑𝑠+𝐿₂+^𝑝−1_𝑖=1𝑇𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃+^𝑝_𝑖=1𝑇𝐿₁^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ(𝛼−1)𝑑𝑠+𝐿₃≤1+𝑝𝑇^𝛼−1𝐿₁Γ()+𝛼+13𝑝𝑇^𝛼−1𝐿₁Γ(𝛼)+21+2𝑝𝑇^𝛼−2𝐿₁|𝑏|Γ(𝛼)+𝑝𝐿₂𝑇+2𝑝𝐿₃|𝑏|𝑇+3𝑝−1𝐿₃∶=𝐿. (3.7)
Hence, for 𝑡₁,𝑡₂∈𝐽_𝑘 with 𝑡₁<𝑡₂,0≤𝑘≤𝑝, we have (𝒢𝑢)𝑡₂−(𝒢𝑢)𝑡₁≤^𝑡2_𝑡1(𝒢𝑢)^′(𝑠)𝑑𝑠≤𝐿𝑡₂−𝑡₁. (3.8) This implies that 𝐺 is equicontinuous on all 𝐽_𝑘,𝑘=0,1,2,…,𝑝 and hence, by the Arzela-Ascoli theorem, the operator 𝒢∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) is completely continuous.
Next, we prove that 𝒢∶ℬ→ℬ. For that, let us choose 𝑅≥max{2𝜇,(2𝜈𝜉)^1/(1−𝜌)}and define a ball ℬ={𝑢∈𝑃𝐶(𝐽,ℝ)∶‖𝑢‖≤𝑅}. For any 𝑢∈ℬ, by the assumptions (𝐇_𝟏) and (𝐇_𝟐), we have |𝒢𝑢(𝑡)|≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑎(𝑠)+𝜉|𝑢(𝑠)|^𝜌𝑑𝑠+^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑎(𝑠)+𝜉|𝑢(𝑠)|^𝜌𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑎(𝑠)|𝑢(𝑠𝛼−1+𝜉)|^𝜌𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑎(𝑠)|𝑢(𝑠𝛼−1+𝜉)|^𝜌𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+1−(−1)^𝜃+1|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑎(𝑠)+𝜉|𝑢(𝑠)|^𝜌+()𝑑𝑠1+−1^𝜃+1𝑡|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑎(𝑠)||𝛼−1+𝜉𝑢(𝑠)^𝜌+1𝑑𝑠|𝑏|^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑎(𝑠)||𝛼−1+𝜉𝑢(𝑠)^𝜌𝑡𝑑𝑠+𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑎(𝑠)+𝜉|𝑢(𝑠)|^𝜌+()𝑑𝑠1+−1^𝜃+1𝑡−𝑇𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑎(𝑠)|𝑢(𝑠𝛼−1+𝜉)|^𝜌𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+1−(−1)^𝜃+1−𝑏𝑡𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑎(𝑠)+𝜉|𝑢(𝑠)|^𝜌𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑎(𝑠)|𝑢(𝑠𝛼−1+𝜉)|^𝜌𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑎(𝑠)|𝑢(𝑠𝛼−1+𝜉)|^𝜌𝑑𝑠+𝐼^∗_𝑖𝑢𝑡_𝑖1≤21+𝑝1+|𝑏|𝑇𝐼^𝛼𝑎(𝑇)+22𝑝−1𝑇+5𝑝+1|𝑏|𝐼^𝛼−1𝑎(𝑇)1+21+|𝑏|𝑇𝑝𝐿₂+22𝑝−1𝑇+5𝑝−2|𝑏|𝐿₃1+21+𝑝1+|𝑏|𝑇𝑇^𝛼𝜉𝑅^𝜌Γ()+2𝛼+12𝑝−1𝑇+5𝑝+1|𝑏|𝑇^𝛼−1𝜉𝑅^𝜌Γ(𝛼). (3.9) Thus, ‖𝒢𝑢‖≤𝜇+𝜈𝜉𝑅^𝜌≤𝑅2+𝑅2=𝑅, (3.10) where 𝜇 and 𝜈 are given by (3.2). This implies 𝒢∶ℬ→ℬ. Hence, 𝒢∶ℬ→ℬ is completely continuous. Therefore, by the Schauder fixed-point theorem, the operator 𝒢 has at least one fixed point. Consequently, problem (1.1) has at least one solution in ℬ.

Remark 3.2. For 𝜌=1 in (𝐇_𝟏), if 𝜈𝜉<1, we can take 𝑅≥𝜇/(1−𝜈𝜉), then the conclusion of Theorem 3.1 holds.

Theorem 3.3. Suppose that there exist a nonnegative functions 𝑎₁∈𝐿(0,1) and a nonnegative number 𝜉₁ such that |𝑓(𝑡,𝑢)|≤𝑎₁(𝑡)+𝜉₁|𝑢|^𝜌 for 𝜌>1. Furthermore, the assumption (𝐇_𝟐) holds. Then problem (1.1) has at least one solution.

Proof. The proof is similar to that of Theorem 3.1, so we omit it.

Theorem 3.4. Suppose that lim_𝑢→0𝑓(𝑡,𝑢)𝑢=0,lim_𝑢→0𝐼_𝑘(𝑢)𝑢=0,lim_𝑢→0𝐼^∗_𝑘(𝑢)𝑢=0. (3.11) Then problem (1.1) has at least one solution.

Proof. By Theorem 3.1, we know that the operator 𝒢∶𝑃𝐶(𝐽,ℝ)→𝑃𝐶(𝐽,ℝ) is completely continuous. In view of (3.11), we can find 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 21+𝑝(|𝑏|𝑇+1)𝑇^𝛼−1𝛿₁Γ()+|𝑏|𝑇𝑇𝛼+15𝑝+1+22𝑝−1^𝛼−1𝛿₁Γ(𝛼)+2(1+|𝑏|𝑇)𝑝𝛿₂𝑇+5𝑝−2+2|𝑏|𝑇2𝑝−1𝛿₃≤|𝑏|. (3.12)
Let Ω={𝑢∈𝑃𝐶(𝐽,ℝ)∣‖𝑢‖<𝑟}. Take 𝑢∈𝑃𝐶(𝐽,ℝ) such that ‖𝑢‖=𝑟, which means 𝑢∈𝜕Ω. Then, as in the proof of Theorem 3.1, we have |(𝑡≤1𝒢𝑢)||𝑏|21+𝑝(|𝑏|𝑇+1)𝑇^𝛼−1𝛿₁Γ()+|𝑏|𝑇𝑇𝛼+15𝑝+1+22𝑝−1^𝛼−1𝛿₁Γ(𝛼)+2(1+|𝑏|𝑇)𝑝𝛿₂𝑇+5𝑝−2+2|𝑏|𝑇2𝑝−1𝛿₃‖𝑢‖, (3.13) which, in view of (3.12), implies that ‖𝒢𝑢‖≤‖𝑢‖, 𝑢∈𝜕Ω. Therefore, by Theorem 2.2, the operator 𝒢 has at least one fixed point. Thus we conclude that problem (1.1) has at least one solution 𝑢∈Ω.

Theorem 3.5. Assume that(𝐇_𝟑) 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|, (3.14) for 𝑡∈𝐽,𝑢,𝑣∈ℝand𝑘=1,2,…,𝑝.
Then problem (1.1) has a unique solution if 2𝑏|)𝑇Λ=1+𝑝(|𝑇+1^𝛼−1𝐾₁Γ()+|𝑏|𝑇𝑇𝛼+15𝑝+1+22𝑝−1^𝛼−1𝐾₁Γ(𝛼)+2(1+|𝑏|𝑇)𝑝𝐾₂𝑇+5𝑝−2+2|𝑏|𝑇2𝑝−1𝐾₃<|𝑏|. (3.15)

Proof. For 𝑢,𝑣∈𝑃𝐶(𝐽,ℝ), we have |(𝒢𝑢)(𝑡)−(𝒢𝑣)(𝑡)|≤^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠((𝑠+𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_{𝑘𝑖=1𝑡𝑖𝑡𝑖−1}𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))−𝑓(𝑠,𝑣(𝑠))𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑘−1_𝑖=1𝑡_𝑘−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠((𝑠+𝐼𝛼−1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_∗𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑘_𝑖=1𝑡−𝑡_𝑘^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠((𝑠+𝐼𝛼−1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_∗𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+1−(−1)^𝜃+1|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠((𝑠+()𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠1+−1^𝜃+1𝑡|𝑏|𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠((𝑠+1𝛼−1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠|𝑏|^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−2Γ()𝑓((𝑠((𝑠+𝑡𝛼−1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠𝑇^𝑇_𝑡𝑝(𝑇−𝑠)^𝛼−1Γ(𝛼)𝑓((𝑠((𝑠+()𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠1+−1^𝜃+1𝑡−𝑇𝑏𝑇^𝑝_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠((𝑠+𝐼𝛼−1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_∗𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+1−(−1)^𝜃+1−𝑏𝑡×𝑏𝑇_{𝑝𝑖=1𝑡𝑖𝑡𝑖−1}𝑡_𝑖−𝑠^𝛼−1Γ(𝛼)𝑓(𝑠,𝑢(𝑠))−𝑓(𝑠,𝑣(𝑠))𝑑𝑠+𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖𝑣𝑡_𝑖+^𝑝−1_𝑖=1𝑡_𝑝−𝑡_𝑖^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠((𝑠+𝐼𝛼−1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_∗𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖+^𝑝_𝑖=1𝑇−𝑡_𝑝^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−2Γ()𝑓((𝑠((𝑠+𝐼𝛼−1𝑠,𝑢))−𝑓𝑠,𝑣))𝑑𝑠_∗𝑖𝑢𝑡_𝑖−𝐼^∗_𝑖𝑣𝑡_𝑖≤1|𝑏|21+𝑝(|𝑏|𝑇+1)𝑇^𝛼−1𝐾₁Γ()+|𝑏|𝑇𝑇𝛼+15𝑝+1+22𝑝−1^𝛼−1𝐾₁Γ(𝛼)+2(1+|𝑏|𝑇)𝑝𝐾₂𝑇+5𝑝−2+2|𝑏|𝑇2𝑝−1𝐾₃‖=Λ‖𝑢−𝑣|𝑏|‖𝑢−𝑣‖, (3.16) which, by (3.15), yields ‖𝑇𝑢−𝑇𝑣‖<‖𝑢−𝑣‖. So, 𝒢 is a contraction. Therefore, by the Banach contraction mapping principle, problem (1.1) has a unique solution.

Example 3.6. Consider the following fractional impulsive irregular boundary value problem ^𝐶𝐷^𝛼𝑢(𝑡)=𝑒^3𝑡cos⁵𝑢(𝑡)+𝑒^𝑢(𝑡)1+𝑢⁴(𝑡)+sin(𝑡+1)5+𝑢²(𝑡)|𝑢|^𝜌1,0<𝑡<1,𝑡≠4,1Δ𝑢4=2+3sin²ln1+2𝑢²14,Δ𝑢^′14=7+2𝑢²(1/4)2+𝑢²(),𝑢1/4_′(0)+(−1)^𝜃𝑢^′(1)+𝑏𝑢(1)=0,𝑢(0)+(−1)^𝜃+1𝑢(1)=0,𝜃=1,2,𝑏≠0, (3.17) where 1<𝛼≤2 and 𝑝=1.

Observe that 𝑓(=𝑒𝑡,𝑢)^3𝑡cos⁵𝑢(𝑡)+𝑒^𝑢(𝑡)1+𝑢⁴(𝑡)+sin(𝑡+1)5+𝑢²(𝑡)|𝑢|^𝜌≤𝑒^3𝑡+|𝑢|^𝜌. (3.18) Clearly, 𝑎(𝑡)=𝑒^3𝑡,𝜉=1,𝐿₂=5,𝐿₃=7/2, and the conditions of Theorem 3.1 hold for 0<𝜌<1. Thus, by Theorem 3.1, problem (3.17) has at least one solution. In a similar way, for 𝜌>1, the impulsive irregular fractional boundary value problem (3.17) has at least one solution by means of Theorem 3.3.

Example 3.7. Consider the impulsive fractional irregular boundary value problem given by ^𝐶𝐷^𝛼𝑢(𝑡)=𝑡²(1−cos𝑢(𝑡))+𝑒^(3+𝑡)𝑢⁴(𝑡)1,0<𝑡<1,𝑡≠5,1Δ𝑢5=arctan²𝑢(1/5)5,Δ𝑢^′15=𝑒^𝑢3(1/5)𝑢−1,_′(0)+(−1)^𝜃𝑢^′(1)+𝑏𝑢(1)=0,𝑢(0)+(−1)^𝜃+1𝑢(1)=0,𝜃=1,2,𝑏≠0, (3.19) where 1<𝛼≤2 and 𝑝=1.

It can easily be verified that all the assumptions of Theorem 3.4 are satisfied. Thus, by the conclusion of Theorem 3.4, we deduce that the problem (3.19) has at least one solution.

Example 3.8. Consider ^𝐶𝐷^7/4𝑢(𝑡)=100𝑡⁵+𝑡²𝑒200^{−cos2𝑢(𝑡)}3,0<𝑡<1,𝑡≠4,3Δ𝑢4=193cos𝑢4,Δ𝑢^′34=|𝑢(3/4)|5(|𝑢(,𝑢1+3/4)|)_′(0)+(−1)^𝜃𝑢^′(1)+8𝑢(1)=0,𝑢(0)+(−1)^𝜃+1𝑢(1)=0,𝜃=1,2. (3.20)

Here 𝑞=7/4,𝑏=8,𝑇=1, and 𝑝=1. With 𝐾₁=1200,𝐾₂=19,𝐾₃=15, (3.21) we find that 2𝑏|)𝑇Λ=1+𝑝(|𝑇+1^𝛼−1𝐾₁Γ()+|𝑏|𝑇𝑇𝛼+15𝑝+1+22𝑝−1^𝛼−1𝐾₁Γ(𝛼)+2(1+|𝑏|𝑇)𝑝𝐾₂𝑇+5𝑝−2+2|𝑏|𝑇2𝑝−1𝐾₃<6.031587<|𝑏|=8. (3.22) Thus, all the conditions of Theorem 3.5 are satisfied. Consequently, the conclusion of Theorem 3.5 applies and the fractional order impulsive irregular boundary value problem (3.20) has a unique solution on [0,1].

Acknowledgment

The research of G. Wang and L. Zhang was supported by the Natural Science Foundation for Young Scientists of Shanxi Province (2012021002-3), China.