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

and Applied Analysis 3 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 Du t 0 is u t C0 C1t C2t · · · Cn−1tn−1, 2.4 where Ci ∈ 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 Du t u t C0 C1t C2t · · · Cn−1tn−1 2.5 for some Ci ∈ 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 Du t y t , 1 < α ≤ 2, t ∈ J ′, Δu tk Ik u tk , Δu′ tk I∗ k u tk , k 1, 2, . . . , p, u′ 0 −1 θu′ T bu T 0, u 0 −1 θ u T 0, θ 1, 2, b / 0, 2.6 4 Abstract and Applied Analysis if and only if u is a solution of the impulsive fractional integral equation u t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∫ t 0 t − s α−1 Γ α y s ds 1 − −1 θ 1 bT ∫T tp T − s α−1 Γ α y s ds [ 1 −1 θ 1 ] t bT ∫T tp T − s α−2 Γ α − 1 y s ds − b ∫T tp T − s α−2 Γ α − 1 y s ds − t T ∫T tp T − s α−1 Γ α y s ds A, t ∈ J0; ∫ t tk t − s α−1 Γ α y s ds 1 − −1 θ 1 bT ∫T tp T − s α−1 Γ α y s ds [ 1 −1 θ 1 ] t bT ∫T tp T − s α−2 Γ α − 1 y s ds − b ∫T tp T − s α−2 Γ α − 1 y s ds − t T ∫T tp T − s α−1 Γ α y s ds


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 .

Abstract and Applied Analysis
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 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.

Preliminaries
Let us fix J 0 0, t 1 , J k−1 t k−1 , t k , k 2, . . . , p 1 with t p 1 T and introduce the spaces: Then T has a fixed point in Ω.

Lemma 2.5. For a given y ∈ C 0, T , a function u is a solution of the following impulsive irregular boundary value problem
Δu t k I k u t k , Δu t k I * k u t k , k 1, 2, . . . , p,

2.8
Proof. Let u be a solution of 2.6 . Then, by Lemma 2.4, we have Abstract and Applied Analysis 5 for some c 1 , c 2 ∈ R. Differentiating 2.9 , we get

2.12
Using the impulse conditions

2.14
Consequently, we obtain Abstract and Applied Analysis By a similar process, we get

2.16
Applying the boundary conditions u 0 −1 θ u T bu T 0 and u 0 −1 θ 1 u T 0, we find that

2.17
Abstract and Applied Analysis 7 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 .

3.1
Notice that problem 1.1 has a solution if and only if the operator G has a fixed point.

Abstract and Applied Analysis
For the sake of convenience, we set the following notations:

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 | ≤ L 1 , |I k u | ≤ L 2 , and |I * k u | ≤ L 3 , for all u ∈ Ω. Thus, for all u ∈ Ω, we have Abstract and Applied Analysis which implies Gu ≤ 1 |b|

3.6
On the other hand, for any t ∈ J k , 0 ≤ k ≤ p, we get

3.7
Abstract and Applied Analysis 11 Hence, for t 1 , t 2 ∈ J k with t 1 < t 2 , 0 ≤ k ≤ p, we have 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 3.9 Thus, 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. Observe that f t, u e 3t cos 5 u t e u t 1 u 4 t sin t 1 5 u 2 t |u| ρ ≤ e 3t |u| ρ .

3.18
Clearly, a t e 3t , ξ 1, L 2 5, L 3 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. where 1 < α ≤ 2 and p 1.