A Generalization of Linear and Nonlinear Retarded Integral Inequalities in Two Independent Variables

Abstract

Integral inequalities, which provide explicit bounds on unknown functions, are used to serve as handy tools in the study of the qualitative properties of solutions to differential and integral equations. By utilizing some analysis techniques, such as amplification method, differential, and integration, several new types of linear and nonlinear retarded integral inequalities in two independent variables are provided. These results generalize and complement previous ones. An illustrative example is given to support the obtained results. The study of the numerical example shows that the new results presented in this paper work well in the analysis of retarded integral inequalities in two independent variables.

1. Introduction

With the development of science and technology, various inequalities have been paid more and more attention, and the generalization of inequalities has become one of the important research directions in modern mathematics. The integral inequality, which has integrals of unknown functions, is an important type of inequality. For nonlinear differential equations derived from the natural science and engineering technology, especially from various branches of mathematics, it is difficult or impossible to obtain explicit solutions in most cases. Therefore, it is of great significance to get the bounds of the solutions to those nonlinear differential equations. Integral inequalities just can provide the bounds of the solutions to the nonlinear differential equations and integral equations. Hence, integral inequalities are used to serve as handy tools in the study of the qualitative properties of solutions to differential and integral equations, such as existence, uniqueness, boundedness, oscillation, stability, and invariant manifold. For example, these inequalities have been widely employed to investigate the stability of switched systems which can be applied to modeling many engineering system problems in real world, such as traffic control, automobile engine control, switching power converters, and multiagent consensus [1–5]. For some related contributions on various classes of integral inequalities, we refer the reader to [1–20] and the references cited therein.

For convenience, throughout this paper, represents the set of real numbers, , and signifies the class of all continuous functions defined on set with range in the set .

In what follows, we provide some background details that motivated our study. One of the most famous and widespread integral inequalities in the study of differential and integral equations is Gronwall-Bellman-type inequality [6–8], which can be described as follows.

Theorem 1. Let and be nonnegative continuous functions on an interval satisfying for some constant . Then

In recent years, many scholars have done a lot of researches and generalization of the above integral inequality, which make the integral inequalities develop continually and the application fields expand gradually. Pachpatte [9, 10] investigated the inequality and the retarded inequality where is nondecreasing with on and and are constants. Abdeldaim and El-Deeb [11] generalized [9] and analyzed the following retarded linear and nonlinear inequalities:respectively. Tian et al. [16] introduced the retarded inequalities in two independent variables as follows.

Theorem 2 (see [16, Theorem  1]). Let , , and , , , , , and , be nondecreasing with and on . Moreover, let be an increasing function with and let on be a nondecreasing function with on . If then, for , where, , and are the inverses of , , and , respectively; is chosen so thatwith denoting the function domain.

Theorem 3 (see [16, Corollary  1]). Assume that , and are defined as in Theorem 2. Let and in Theorem 2, where are positive constants. If then, for all ,where

Motivated by the recent contributions of Abdeldaim and El-Deeb [11], Zhang and Meng [14], and Tian et al. [16], our principal goal is to extend the inequalities with one variable in [11] to those with two variables which include Theorems 2 and 3 as special cases.

The rest of the work is organized as follows. A useful lemma that plays a fundamental role in the proofs of the main theorems is presented in Section 2. In Section 3, we propose our main theorems and corollary on several new types of linear and nonlinear retarded integral inequalities in two independent variables. An illustrative example is given to indicate the usefulness of these inequalities in Section 4, which is followed by a short conclusion in Section 5.

2. Lemma

The subsequent lemma is helpful in proving our main theorems.

Lemma 4. Assume that , , and and is an increasing function with and is a nondecreasing function. Suppose that is a nonnegative constant and are nondecreasing with , , , and on . Ifthen, for ,where and are the inverses of and , respectively; is chosen so that

Proof. Define the nondecreasing positive function bywhere is an arbitrary small positive number. Utilizing inequality (13) and the monotonicity of , we getDifferentiating (17) with respect to and combining (18) and the monotonicities of , , and , we conclude that On account of , we deduce that Integrating the latter inequality on and letting , we haveowing to (15). By virtue of (16), (18), and the last inequality, we obtain inequality (14). The proof is complete.

Remark 5. Assume that . Then and (14) is valid on ; that is, one can select and .

3. Main Results

The following are the main results of this paper.

Theorem 6. Let , and and let be nondecreasing with , , , and on . If the inequality holds, for all , then

Proof. Letting then andOur assumptions on , and indicate that is a positive function which is nondecreasing with respect to each of the two variables. Differentiating with respect to and using (25), we arrive atBy virtue of the monotonicity of , we get Multiplying the latter inequality by yields Integrating this inequality on , we deduce thatCombining (25) with (29), we get inequality (23). This completes the proof.

Theorem 7. Let , , , , and , , , , and be nondecreasing with , , , and on . Moreover, let and be nondecreasing function with and on . If then, for ,where and are the inverses of and , respectively; is chosen so that for .

Proof. Define the nondecreasing function byThen Differentiating (36) and using (37) and the monotonicity of , we obtainLet and be arbitrary numbers. Utilizing (38) and the monotonicities of , and , we get that, for and , For and ,From another point of view, It follows from (40) and (41) that Integrating the above inequality on with respect to the second variable and taking into account, we haveFrom (33), the latter relation givesIntegrating the last inequality over , we get where is defined as in (32). Combining (37) and the monotonicity of and employing Lemma 4, we obtainwhere is defined as in (34). Taking and , we conclude thatAs and are arbitrary, we get the desired inequality (31). The proof is complete.

Theorem 8. Assume that , and are defined as in Theorem 7. Moreover, let be increasing function with and on . If then, for ,where , , and are the inverses of , , and , respectively; is chosen so that for .

Proof. Define function by (36). ThenThe rest of the proof is similar to that of Theorem 7 and hence is omitted.

Remark 9. Letting , , and in Theorem 8, Theorem 8 turns out to be Theorem 2. Therefore, the inequality established in Theorem 8 generalizes that of [16, Theorem  1].

If , , and in Theorem 8, where , and , and are positive constants, then we have the following corollary.

Corollary 10. Assume that , and are defined as in Theorem 8. Ifthen, for all ,where

Proof. Assume that and let , , and . Then we have , and sowhere and are defined in (55). Using Theorem 8, one can easily obtain When , where and are the same as in (55). By Theorem 8, similar discussions can giveThis completes the proof.

Remark 11. Letting , , , and , Corollary 10 reduces to Theorem 3. Hence, the inequality established in Corollary 10 includes the result of [16, Corollary ].

4. Example

Example 1. Consider the integral equationwhere and are continuous functions, is nondecreasing with , , , and on , and is a constant. Suppose thatwhere , and are defined as in Corollary 10. Combining (60)-(61) yieldsExploiting Corollary 10, we obtain an explicit bound to the solutions of (60): where and are defined as in Corollary 10.

5. Conclusions

This paper investigates some new types of linear and nonlinear retarded integral inequalities in two independent variables. Several theorems and a corollary of these inequalities are obtained based on some analysis techniques, such as amplification method, differential, and integration. An illustrative example is studied to demonstrate the effectiveness of the new results.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants 61304130 and 51277116).