Henry-Gronwall Integral Inequalities with ‘ ‘ Maxima ’ ’ and Their Applications to Fractional Differential Equations

and Applied Analysis 3 (H 11 ) the function φ ∈ C([βt 0 , t 0 ],R + ) with max s∈[βt0 ,t0] φ(s) > 0, where 0 < β < 1; (H 12 ) the function u ∈ C([βt 0 , T),R + ) with


Introduction
It is well known that Gronwall-Bellman type integral inequalities play a dominant role in the study of quantitative properties of solutions of differential and integral equations [1][2][3][4][5].Usually, the integrals concerning these type inequalities have regular or continuous kernels, but some problems of theory and practicality require us to solve integral inequalities with singular kernels.For example, Henry [6] proposed a method to find solutions and proved some results concerning linear integral inequalities with weakly singular kernel.Moreover, Medved' [7,8] presented a new approach to solve integral inequalities of Henry-Gronwall type and their Bihari version and obtained global solutions of semilinear evolution equations.Ye and Gao [9] considered the integral inequalities of Henry-Gronwall type and their applications to fractional differential equations with delay.Ma and Pečarić [10] established some weakly singular integral inequalities of Gronwall-Bellman type and used them in the analysis of various problems in the theory of certain classes of differential equations, integral equations, and evolution equations.Shao and Meng [11] studied a certain class of nonlinear inequalities of Gronwall-Bellman type, which is used to a qualitative analysis to certain fractional differential equations.For other results on the subject we refer to [12][13][14][15][16][17][18] and references cited therein.Differential equations with "maxima" are a special type of differential equations that contain the maximum of the unknown function over a previous interval.Several integral inequalities have been established in the case when maxima of the unknown scalar function are involved in the integral; see [19,20] and references cited therein.
Recently in [21] some new types of integral inequalities on time scales with "maxima" are established, which can be used as a handy tool in the investigation of making estimates for bounds of solutions of dynamic equations on time scales with "maxima." In this paper we establish some Henry-Gronwall type integral inequalities with "maxima." The significance of our work lies in the fact that "maxima" are taken on intervals [, ] which have nonconstant length, where 0 <  < 1.Most of the papers take the "maxima" on [ − ℎ, ], where ℎ > 0 is a given constant.We apply our results to demonstrate the bound of solutions and the dependence of solutions on the orders with initial conditions for Caputo fractional differential equations with "maxima" The paper is organized as follows.In Section 2 we recall some results from [21] in the special case T = R which
Lemma 1 (see [21]).Let the following conditions be satisfied: Then holds, where By splitting the initial function  to be two functions, we deduce the following corollary.

Main Results
Theorem 5. Suppose that the following conditions are satisfied: where  > 0.
Then the following assertions hold.
Then the following assertions hold.

Applications to Fractional Differential Equations with ''Maxima''
In this section, we apply our results to demonstrate the bound of solutions and the dependence of solutions on the orders with initial conditions for Caputo fractional differential equations with "maxima." We consider the following fractional differential equations (FDEs) with "maxima" and initial condition where   represents the Caputo fractional derivative of order ), and 0 <  < 1.We denote  () ( 0 ) =   ,  = 0, 1, 2, . . .,  − 1.