Abstract
Some new integral inequalities with weakly singular kernel for discontinuous functions are established using the method of successive iteration and properties of Mittag-Leffler function, which can be used in the qualitative analysis of the solutions to certain impulsive fractional differential systems.
1. Introduction
Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. There has been a significant development in the study of fractional differential equations in recent years; see the monographs of Kilbas et al. [1], Lakshmikantham et al. [2], and Podlubny [3] and the survey by Diethelm [4]. Integral inequalities with weakly singular kernels play an important role in the qualitative analysis of the solutions to fractional differential equations. With the development of fractional differential equations, integral inequalities with weakly singular kernels have drawn more and more researchers’ attention and lead to inspiring results; see, for example, [5–8].
Impulsive differential equations, that is, differential equations involving impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. Many processes studied in applied sciences are represented by impulsive differential equations. However, the situation is quite different in many physical phenomena that have a sudden change in their states such as mechanical systems with impact, biological systems such as heart beats, blood flow, population dynamics theoretical physics, pharmacokinetics, mathematical economy, chemical technology, electric technology, metallurgy, ecology, industrial robotics, and biotechnology processes (see the monographs [9–12] for details).
The theory of impulsive differential equations is an important branch of differential equations. In spite of its importance, the development of the theory has been quite slow due to special features possessed by impulsive differential equations in general, such as pulse phenomena, confluence, and loss of autonomy. Among these results, integrosum inequalities for discontinuous functions play increasingly important roles in the study of quantitative properties of solutions of impulsive differential systems. In 2005, Borysenko et al. [13] considered some integrosum inequalities and devoted them to investigate the properties of motion represented by essential nonlinear system of differential equations with impulsive effect. In 2007 and 2009, Gallo and Piccirillo [14, 15] presented some new nonlinear integral inequalities like Gronwall-Bellman-Bihari type with delay for discontinuous functions and applied them to investigate the properties of solutions of impulsive differential systems.
The theory of impulsive fractional differential equations is a new topic of research which involve both the fractional order integral (or differentiation) and the impulsive effect; most of the results related to this topic are the existence of solutions (see [16–19] and the references therein). To our best knowledge, there is no result on other qualitative properties (such as boundedness and stability), and impulsive fractional differential equations involving the Caputo fractional derivative have not been studied very perfectly, so we set up a new kind of integral inequalities with weakly singular kernel for discontinuous functions and use the new inequalities to study the qualitative properties of the solutions to certain impulsive fractional differential systems.
On the basis of previous studies, in this paper, we consider the following integral inequalities with weakly singular kernel for discontinuous functions: where , , and are constants, , , , and is a nonnegative piecewise-continuous function with the 1st kind of discontinuous points: , , . In general, due to the existence of weak singular integral kernel, the methods of these inequalities for discontinuous functions are quite different to that of classical Gronwall-Bellman-Bihari inequalities. We use the properties of the Mittag-Leffler function defined by and the successive iterative technique to establish the new type of integral inequalities for discontinuous functions. These inequalities are applied to investigate the qualitative analysis of the solutions to certain impulsive fractional differential systems.
2. Preliminary Knowledge
In this section, we give some definitions, symbols, and known inequalities, which will be used in the remainder of this paper.
Definition 1. Given an interval of , the fractional (arbitrary) order integral of the function of order is defined by where is the gamma function. When , we write , where for , for , and as , where is the delta function.
Definition 2. For a given function on the interval , the -order Caputo fractional order derivative of is defined by where .
For calculation simplification, the symbols are defined as follows: where is the Mittag-Leffler function.
Lemma 3 (see [6]). Suppose that , is a nonnegative and nondecreasing function which is locally integrable on (for some ), and is a nonnegative, nondecreasing continuous function defined on , (constant), and suppose that is a nonnegative and locally integrable function on which satisfies and then
Using Lemma 3, we can easily get the following corollary.
Corollary 4. Let be constants, and . And suppose is a nonnegative and locally integrable function on with and then
Lemma 5. Let and . Then where is the Beta function.
Proof. Let ; we obtain for which is the desired result.
Remark 6. If we replace the integration interval by for , we can get the following equality:
Corollary 7. Let . Then
Proof. By the definition of , we have Since the series of function is the Mittag-Leffler function , which is convergent uniformly on , we permute the sum and the integral to obtain Using Lemma 5, we get
Remark 8. If we replace the integration interval by , we can get the similar conclusion
3. Main Results
Theorem 9. Let be constants: , , and is a nonnegative piecewise-continuous function with the 1st kind of discontinuous points: and . If where , then the following assertions hold:
Proof. If , the inequality (17) is reduced to the following form: Using Lemma 3, we get If , then Using Corollary 7, we get Applying Lemma 3, we obtain that Similarly, for , using Remark 8, we have Again, Lemma 3 implies that Using the inductive method, suppose that, for , and then for each and from (26), we have Then applying Lemma 3, we obtain the inequality (17). This completes the proof of Theorem 9.
Remark 10. The results of Theorem 9 are valid when the function has only finite number of discontinuities points .
4. Application to Impulsive Fractional Differential Systems
Let us consider the following system of Caputo fractional differential equations: where is the Caputo fractional derivatives, , is Lebesgue measurable with respect to on , and is continuous with respect to on . are continuous for . and represent the right and left limits of at , and is a constant-valued vector.
Using Lemma 6.2 in [3], is a solution of Cauchy problem for system (28) if only if satisfies
Theorem 11. Suppose that and , where , are constants, and . If is any solution of the initial value problem (28), then the following estimations hold: where , , and is a suitable complete norm in .
Proof. From (29), it is easy to obtain that Using Theorem 9, we get the desired conclusion.
Remark 12. In [13, 15], during considering the qualitative properties of the impulsive differential equations, the functions and are defined in the domain for some . However, we do not add such a restriction since our conclusion in Theorem 11 does not involve .
Corollary 13. Let the right-hand side of the initial value problem (28) satisfy the following conditions:(i), is a constant and ;(ii), are constants and ;(iii)there exists a constant such that , , . Then all solutions of the problem (28) are bounded, and the trivial solution of the problem (28) is stable in the sense of Lyapunov stability.
Corollary 14. Suppose that(i), , for all ;(ii), where , are constants, , , ;(iii)there exists a constant such that , , . Then all solutions of the problem (28) are bounded.
Conflict of Interests
The author declares that there is no conflict of interests.
Acknowledgments
The author sincerely thanks the referees for their constructive suggestions and corrections. This research was partially supported by the NSF of China (Grants 11171178 and 11271225), the NSF of Shandong Province (Grant ZR2013AL005), the Science and Technology Project of High Schools of Shandong Province (Grant J12LI52), and the Key Projects of Jining University Servicing the Local Economics (Grant 2013FWDF14).