New Integral Inequalities with Weakly Singular Kernel for Discontinuous Functions and Their Applications to Impulsive Fractional Differential Systems

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.


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][6][7][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][10][11][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][17][18][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,  ≥ 0,  ≥ 0,   ≥ 0, 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   () = ∑ ∞ =0 (  /Γ( + 1)) 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.

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 R, the fractional (arbitrary) order integral of the function ℎ ∈  1 ([, ], R) of order  ∈ R + is defined by where Γ(⋅) is the gamma function.When  = 0, we write , where   () =  −1 /Γ() for  > 0,   () = 0 for  ≤ 0, and   → () as  → 0, where  is the delta function.
Definition 2. For a given function ℎ on the interval [, ], the -order Caputo fractional order derivative of ℎ is defined by where  = [] + 1.
For calculation simplification, the symbols are defined as follows: where   (⋅) is the Mittag-Leffler function.
Remark 10.The results of Theorem 9 are valid when the function has only finite number of discontinuities points  0 ,  1 , . . .,   .