Abstract
Several new retarded integral inequalities of Gronwall-Bellman-Pachpatte type are presented which generalize the inequalities to the more general nonlinear case in the literature and provide explicit bounds on unknown functions. These results include many existing ones as special cases and can be used as tools in the qualitative analysis of certain classes of integrodifferential equations.
1. Introduction
It is well known that integral inequalities are very useful to investigate the existence, uniqueness, boundedness, oscillation, and other qualitative properties of solutions to differential equations and integrodifferential equations [1–19]. One of the fundamental inequalities is the Gronwall inequality, which was established in 1919 by Gronwall [1]. As a generalization of Gronwall inequality, Gronwall-Bellman inequality plays a key role in studying stability and asymptotic behavior of solutions to differential equations and integrodifferential equations. An important nonlinear generalization of Gronwall-Bellman inequality is Bihari’s inequality [2]. In recent years, there has been much research activity concerning linear and nonlinear generalizations of Gronwall-Bellman-Bihari type inequalities. To mention a few, Pachpatte [3] presented a new integral inequality and studied the boundedness properties of some linear integrodifferential equations, one of which we give below for the convenience of the reader.
Theorem 1 (see Pachpatte [3]). Let , , and be a nonnegative constant. If then for .
Very recently, Abdeldaim and El-Deeb [4, 5] investigated the following integral inequalities which generalized the main results of [3].
Theorem 2 (see [4, Theorem 2.1]). Let and suppose that are increasing functions with and , for , and is a nonnegative constant. If then
Theorem 3 (see [4, Theorem ] and [5, Theorem ]). Let and assume that are increasing functions with , , , and , for , and and are positive constants. If then where and and stand for the inverses of the functions and , respectively.
Note that the function of inequalities established in [4, 5] satisfies ; that is, is global Lipchitz. However, in the real system there may exist the non-Lipschitz , such as or , and then the obtained results can not be applied to this kind of systems. The natural question now is as follows: is it possible to relax this condition? The aim of this paper is to give an affirmative answer to this question.
In this paper, we study certain classes of integral inequalities which generalize the inequalities established in [3–5] to the more general nonlinear case. The obtained results can be used as tools in the study of qualitative theory of certain classes of integrodifferential equations with more general nonlinearities. At the end, two examples are provided to illustrate the main results.
2. Main Results
In what follows, denotes the set of real numbers, and denote the sets of all continuous functions and all continuously differentiable functions defined on set with range in the set , respectively, and stands for the inverse to .
The following lemmas are very useful in the proof of our main results.
Lemma 4 (see [6, Lemma ]). Assume that and . Then for any .
Lemma 5 (see [7, Corollary ]). Assume that , , and is a positive constant. If thenwhere is the largest number such that
Theorem 6. Let , , , and be nonnegative constants satisfying , , , , , , and . If then, for any , where
Proof. Define a function by Then, , is nondecreasing, and Differentiating and using (16), we conclude that By virtue of Lemma 4, for any , Combining (17) and (18), we deduce that Define another function by Then is nondecreasing, , and It follows from (19) thatDifferentiating and using (22), we obtain Integrating the latter inequality from to , we conclude that Therefore, which yields By virtue of , (13) holds. This completes the proof.
Remark 7. If , then Theorem 6 reduces to Theorem 2.
Remark 8. If and , then Theorem 6 reduces to Theorem 1.
Theorem 9. Let . Assume further that , , , , , , , , , , , and is a positive constant. If, for , then where is the largest number such that
Proof. Define a function by Then, , is nondecreasing, and It is not difficult to obtain where is nondecreasing, , and . Differentiating , we have Noting that is increasing and is nondecreasing, we get , and so Integrating the latter inequality from to , we have By virtue of Lemma 5, This and (33) imply that and thus Integrating (40) from to , which yields (28) due to (32). The proof is complete.
Remark 10. The condition in [4, 5] is relaxed to , where and are nonnegative constants. On the basis of this new assumption, the classes of are enlarged. For example, satisfies when choosing and .
If we choose , , and in Theorem 9, then . Letting , , we have , and hence . Therefore, satisfies with and , and then we can obtain the following result.
Corollary 11. Assume that , , , , , is a positive constant, and, for , Then where , and is the largest number such that
Proof. By virtue of and , we obtain and An application of Theorem 9 implies that This completes the proof.
3. Applications
In this section, we will use the inequalities established in Theorems 6 and 9 to study the boundedness of certain integrodifferential equations.
Example 1. Consider the following integral equation:where is a continuous function defined on , , is a nonnegative constant, and and satisfy where , , and are nonnegative constants satisfying , , and . Then every solution of (48) satisfies Now, using Theorem 6, we obtain that, for any , where which illustrates that the solution of (48) is global boundedness.
Example 2. Consider the integrodifferential equation where is a constant, , and .
Assume that where . Then the solution of (53) satisfies and hence A suitable application of Corollary 11 to (56) yields where and is the largest number such that
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61174217, 61374074, 61473133, and 11671227) and the Doctoral Scientific Research Foundation of University of Jinan (1008398).