Henry-Gronwall Integral Inequalities with “Maxima” and Their Applications to Fractional Differential Equations

Abstract

Some new weakly singular Henry-Gronwall type integral inequalities with “maxima” are established in this paper. Applications to Caputo fractional differential equations with “maxima” are also presented.

1. 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 [1–5]. 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 [12–18] 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 . Most of the papers take the “maxima” on , where 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 which are used to prove our main results which are presented in Section 3. In the last Section 4 we give applications of our results for an initial value problem for a Caputo fractional differential equation with “maxima.”

2. Preliminaries

For convenience we let throughout . The following results in Lemmas 1 and 3 are obtained by reducing the time scales, , and for all in Theorems 3.3 and 3.2 ([21], page 8 and page 6), respectively.

Lemma 1 (see [21]). Let the following conditions be satisfied: the functions and ;the function with , where ;the function and satisfies the inequalities Then holds, where

By splitting the initial function to be two functions, we deduce the following corollary.

Corollary 2. Let the following conditions be satisfied:the functions , , and ;the function with and , where ;the function and satisfies the inequalities Then holds, where with

Lemma 3 (see [21]). Let the condition of Lemma 1 be satisfied. In addition, assume thatthe function is nondecreasing;the function , where ;the function and satisfies the inequalities Then holds, where

The following lemma is a consequence of Jensen’s inequality which can be found in [22].

Lemma 4 (see [22]). Let , and let be nonnegative real numbers. Then, for ,

3. Main Results

Theorem 5. Suppose that the following conditions are satisfied:the functions and ;the function with , where ;the function with where .
Then the following assertions hold.Suppose ; then  where  with  Moreover, if is a nondecreasing function, then  where Suppose ; then  where  Moreover, if is a nondecreasing function, then  where

Proof. Consider . Using the Cauchy-Schwarz inequality with (13), we get for The first integral of (30) implies the estimate Therefore, from (30) and (31), we obtain Applying Lemma 4 with , , we get Now, taking , we have and, for , where and are defined by (16) and (17), respectively.
Applying Corollary 2 for (34) and (35), we obtain where is defined by (18). Therefore, we get the required inequality in (15).
Moreover, if is a nondecreasing function, then, by applying Lemma 3 for (34) and (35), we obtain the estimate where is defined by (21). Thus, we get the desired inequality in (20). This completes the proof of the first part.
Consider . Let , be defined by (23) and (24), respectively. It is obvious that   +  . Using the Hölder inequality in (13), for , we have Repeating the process to get (31), the first integral of (38) implies the estimate Obviously, and . From (38) and (39), it follows that where is defined by (26). Applying Lemma 4 with , , we have By setting , we get and, for , where is defined by (25). Consequently, applying Corollary 2 with (42) and (43), we have where is defined by (27). Therefore, the desired inequality (22) is established.
Furthermore, if is a nondecreasing function, then by applying Lemma 3 for (42) and (43) we deduce that Thus, inequality (28) is proved. This completes the proof.

Theorem 6. Assume thatthe conditions , of Theorem 5 are satisfied;the function ;the function with where .
Then the following assertions hold.Suppose ; then  where  with being defined by (19). Furthermore, if is a nondecreasing function, then  where is defined by (21).Suppose ; then  where , , and are defined by (23), (24), and (26), respectively,  Furthermore, if is a nondecreasing function, then  where is defined by (29).

Proof. Consider . By using the Cauchy-Schwarz inequality in (46), for , we have Applying Lemma 4 with , , we get Taking , we have where and are defined by (49) and (50), respectively. Using Corollary 2 for (59) and (60), it follows that where is defined by (51). Thus, we get the result in (48).
If is a nondecreasing function, then Lemma 3 with (59) and (60) implies the estimate where is defined by (21). Thus, the required inequality (52) is established. This completes the proof of the first part.
Consider . Let , be defined by (23) and (24), respectively. Applying the Hölder inequality in (46), we have that for where is defined by (26). By using Lemma 4 with , , we obtain the estimate Substituting , we get and, for , where is defined by (54). An application of Corollary 2 to (65) and (66) gives where is defined by (55). Therefore, we deduce inequality (53).
As a special case, if is a nondecreasing function, then, by Lemma 3 with (65) and (66), we get Therefore, the desired inequality (56) is established. This completes the proof of Theorem 6.

4. 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 , , is a given continuously differentiable function on up to order , and . We denote , . For more details on fractional differential equations, see [23, 24].

Theorem 7. Assume that there exist functions such that, for , , If is solution of the initial value problem (69)-(70), then the following estimates hold. Suppose . Then Suppose . Then Suppose . Then where and , , , , and are defined as in Theorems 5 and 6.

Proof. The solution of the initial value problem (69)-(70) satisfies the following equations (see [23]): For , by using the assumption , it follows that Hence, Theorem 6 yields the estimate inequalities (72) and (73).
For , by using the assumption in (76), we have Since is a nondecreasing function, Theorem 6 yields the estimate inequality (74). This completes the proof.

Theorem 8. Let and such that . Also let be a continuous function satisfying the following assumption:there exist constants such that , for each and .
If and are the solutions of the initial value problem (69)-(70) and with initial condition respectively, where is a given continuous function on such that for all up to order . we denote , . Then the following estimates hold for .Suppose . Then for Suppose . Then for where with

Proof. The solutions and of the initial value problems (69)-(70) and (80)-(81) satisfy the following equations: respectively. So, using the assumption , it follows that where is defined by (84) and Applying Theorem 6 yields the desired inequalities (82) and (83). This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.