A Nonlinear Weakly Singular Retarded Henry-Gronwall Type Integral Inequality and Its Application

Abstract

We establish a class of new nonlinear retarded weakly singular integral inequality. Under several practical assumptions, the inequality is solved by adopting novel analysis techniques, and explicit bounds for the unknown functions are given clearly. An application of our result to the fractional differential equations with delay is shown at the end of the paper.

1. Introduction

Integral inequalities play increasingly important roles in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds, and other qualitative properties of solutions of ordinary differential equations and integral equations. One of the best known and widely used inequalities in the study of nonlinear differential equations is Gronwall-Bellman inequality [1, 2], which can be stated as follows. If and are nonnegative continuous functions on an interval satisfying , , then , . Many papers are devoted to different generalizations of Bellman-Gronwall inequality. Very well-known generalization of Bellman-Gronwall inequality to the nonlinear case is the Bihari inequality [3]. In 1956, Bihari [3] discussed the integral inequality where is a constant. In recent years, many researchers have devoted much effort to investigating weakly singular integral inequalities. For example, Henry [4] proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu [5] gave a modified version of Henry type inequality. However, such results are expressed by a complicated power series which are sometimes inconvenient for their applications. To avoid the shortcomings of these results, Medved’ [6] presented a new method to discuss nonlinear singular integral inequalities of Henry type and their Bihari version is as follows: and the estimates of solutions are given. From then on, more attention has been paid to such inequalities with singular kernel; see [7–24] and the references cited therein. Ye and Gao [20] considered the integral inequality of Henry-Gronwall type with delay and Henry-Gronwall type retarded integral inequality with singular kernel In this paper, motivated by [6, 20], we discuss the nonlinear integral inequality of Henry-Gronwall type with delay and Henry-Gronwall type nonlinear retarded integral inequality with singular kernel

2. Main Results

Throughout this paper, denotes the set of real numbers, . For convenience, before giving our main results, we cite some useful lemmas and definitions in the discussion of our proof as follows.

Definition 1 (see [6]). Let be a real number and . We say that a function satisfies a condition (), if where is a continuous, nonnegative function.

Lemma 2 (discrete Jensen inequality [25]). Let be nonnegative real numbers, is real numbers, and is a natural number. Then

Lemma 3 (see [6]). (1) Let ; then where is the gamma function.
(2) Let ; then

Proof. (1) Using a change of variables and successively, we have the estimate Since , and .
(2) Using a change of variables and successively, we have the estimate Since , , , and .

Theorem 4. Suppose that are nonnegative continuous functions on , is a nonnegative continuous function on , , and , , are constants. Suppose that the function satisfies the following conditions:(1)) condition, that is, satisfies inequality (7);(2)subadditivity, that is, for all , .
If satisfies (5), then where

Proof. Define a function by the right side of (5), that is, Then , , and is a nonnegative, nondecreasing, and continuous function with , .
For , by the subadditivity satisfied by , we conclude Letting in (18) and integrating both sides of inequality (18) from to , we obtain where is chosen arbitrarily.
Define a function by the right side of (19), that is, Then, the function is a nonnegative, nondecreasing, and continuous function with Differentiating , we have From (22), we obtain Using (21), from (23) we obtain where are defined by (14) and (15), respectively. From (24), we observe Let in (25); we have Since is chosen arbitrarily, from (26), we have the estimation
For , using the subadditivity of and monotony of , from (17) we have Letting in (28) and integrating both sides of inequality (28) from to and using (27) we obtain where , is seen as a constant, and is defined by (16).
Define a function by the right side of (29), that is, Obviously, is a nonnegative, nondecreasing, and continuous function with Differentiating , we have From (33), we have Using (31), from (34), we have It follows that In (36), let , and then we have Since is chosen arbitrarily, from (32) and (37), we obtain the estimation Noting that , from (27) and (38), we obtain our required estimations (13).

Remark 5. When . The estimations (13) in Theorem 4 are reduced to the corresponding estimations in [20].

Theorem 6. Suppose that satisfy the corresponding conditions in Theorem 4; is a constant. If satisfies (6), then the following assertions hold.
(1) Suppose . Then where is defined by (14) in Theorem 4, and is defined in (7) in Definition 1.
(2) Suppose that . Then where and .