- Open Access
Existence results for nonlinear fractional differential equations with closed boundary conditions and impulses
© Wang et al.; licensee Springer 2012
Received: 24 April 2012
Accepted: 20 August 2012
Published: 25 September 2012
This paper is concerned with the existence and uniqueness of solutions for impulsive nonlinear differential equations of fractional order with closed boundary conditions. By applying some standard fixed point theorems, we obtain the sufficient conditions for the existence and uniqueness of solutions of the problem at hand. An illustrative example is presented.
MSC:26A33, 34B15, 34B37.
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 [1, 2], sampled data or digital control system [3, 4], and impulsive switched system . 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  and the references therein.
Fractional calculus (differentiation and integration of arbitrary order) has proved to be an important tool in the modeling of dynamical systems associated with phenomena such as fractals and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, fitting of experimental data, etc. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. With this advantage, the fractional-order models become more realistic and practical than the classical integer-order models in which such effects are not taken into account. For some recent details and examples, see [7–22] and the references therein.
Impulsive differential equations are found to be important mathematical tools for better understanding of several real world problems in biology, physics, engineering, etc. In fact, the theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations and has emerged as an important area of investigation; for instance, see [23–25] and references therein. The recent surge in developing the theory of differential equations of fractional order has led several researchers to study the fractional differential equations with impulse effects. For some recent work on impulsive differential equations of fractional order, see [26–31] and the references therein.
where is the Caputo fractional derivative, , , (), , , , where and denote the right and the left limits of at , respectively. have a similar meaning for .
Here we remark that the boundary conditions in (1.1) include quasi-periodic boundary conditions () and interpolate between periodic (, ) and antiperiodic (, ) boundary conditions. For more details and applications of closed boundary conditions, see .
Let , and we introduce the spaces: with the norm , and with the norm . Obviously, and are Banach spaces.
In passing, we remark that indeed stands for for t in the subinterval .
Definition 2.1 A function with its Caputo derivative of order q existing on J is a solution of (1.1) if it satisfies (1.1).
Substituting the value of , in (2.3) and (2.4), we obtain (2.2). Conversely, assume that u is a solution of the impulsive fractional integral equation (2.2), then by a direct computation, it follows that the solution given by (2.2) satisfies (2.1). □
3 Main results
Observe that the problem (1.1) has a solution if and only if the operator T has a fixed point.
Lemma 3.1 The operator defined by (3.1) is completely continuous.
Proof It is obvious that is continuous in view of continuity of f, and .
which implies that .
This implies that is equicontinuous on all , . Thus, by the Arzela-Ascoli theorem, the operator is completely continuous. □
Theorem 3.1 Assume that
for , and .
Then the problem (1.1) has at least one solution.
where τ and ν are given by (3.3) and (3.4). So, . Thus is completely continuous. Therefore, by the Schauder fixed point theorem, the operator has at least one fixed point. Consequently, the problem (1.1) has at least one solution in . □
Theorem 3.2 Assume that
for , and .
Then the problem (1.1) has at least one solution.
Proof The proof is similar to that of Theorem 3.1, so we omit it. □
Theorem 3.3 ()
Let E be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then has a fixed point in E.
Theorem 3.4 If . In addition, assume that
for , and .
Then the problem (1.1) has at least one solution.
which implies that is bounded for any . So, the set V is bounded. Thus, by the conclusion of Theorem 3.3, the operator has at least one fixed point, which implies that (1.1) has at least one solution. □
Corollary 3.1 Assume that functions f, , () are bounded. Then the nonlinear problem (1.1) has at least one solution.
Theorem 3.5 Assume that
for , and .
Consequently, we have , where is given by (3.6). As , the conclusion of the theorem follows by the contraction mapping principle. This completes the proof. □
where , and .
In this case, , , , , and the conditions of Theorem 3.1 can readily be verified. Thus, by the conclusion of Theorem 3.1, the problem (4.1) has at least one solution.
We would like to express our gratitude to the anonymous reviewers and editors for their valuable comments and suggestions that led to the improvement of the original manuscript. The research of B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The research of Guotao Wang and Lihong Zhang was supported by the Natural Science Foundation for Young Scientists of Shanxi Province (2012021002-3), China.
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