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Generalized retarded nonlinear integral inequalities involving iterated integrals and an application
Journal of Inequalities and Applications volume 2013, Article number: 376 (2013)
Abstract
In this work, some new generalized retarded nonlinear integral inequalities, which include nonlinear composite functions of unknown functions between iterated integrals, are discussed. By adopting novel analysis techniques, the upper bounds of the embedded unknown functions are estimated explicitly. The derived results can be applied in the study of differential-integral equations and some practical problems in engineering.
MSC:26D15, 26D20, 34A12.
1 Introduction
Integral inequality that provides an explicit bound to the unknown function furnishes a handy tool to investigate qualitative properties of solutions of differential 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 u and f are nonnegative continuous functions on an interval satisfying
for some constant , then
It has become one of the very few classical and most influential results in the theory and applications of inequalities. Because of its fundamental importance, over the years, many generalizations and analogous results of (1.1) have been established, such as [3–23].
Among these references, Bainov et al. [[7], p.107] considered the following interesting Gronwall-type inequality
in which the unknown function only exists in the innermost layer of iterated integrals. In 2005 Kim [8] considered analogous Gronwall-type integral inequalities involving iterated integrals by replacing the unknown function u in the right-hand side of (1.2) with for some constant p. In 2007, Agarwal et al. [10] investigated some nonlinear retarded inequalities with iterated integrals to extend Kim’s results in [8],
which include the composite functions of unknown function only in the innermost layer of iterated integrals.
In 2011, Abdeldaim et al. [11] studied some new integral inequalities of Gronwall-Bellman-Pachpatte-type such as
and
which include the composite functions of unknown functions in every layer of iterated integrals, but the iterated integrals are double integrals.
In this paper, we extend certain results that were proved in [7–11] to obtain new generalizations of formerly famous Gronwall-Bellman-Pachpatte-type inequalities. There are not only composite functions of unknown functions in iterated integrals on the right hand side of our inequalities, but also the composite functions of unknown function exist in every layer of the iterated integrals. In this work, we give the upper bounds of the embedded unknown functions by adopting novel analysis techniques in three different scenarios and illustrate an application of our results, which verifies that our results are handy tools to study the qualitative properties of nonlinear differential equations and integral equations.
2 Main result
In this section, we state and prove some new integral inequalities of Gronwall-Bellman-Pachpatte-type, which can be used in the analysis of various problems in the theory of nonlinear ordinary differential and integral equations.
First, we give five assumptions for functions that will appear in our main results.
-
1.
and are nonnegative and continuous functions on . In addition, is nondecreasing;
-
2.
, are nonnegative and continuous functions for , and nondecreasing in t for fixed ;
-
3.
is a nondecreasing and continuous function on with for ;
-
4.
is an increasing continuous function with for and ;
-
5.
is a continuous, differentiable and nondecreasing function on with , .
In order to clearly present our main idea, we first consider a class of simple integral inequalities, namely, the composite function of unknown function is involved in the innermost layer of iterated integrals only.
Theorem 1 Assume that Assumptions 1-5 and the following inequality hold
for . Then we have
where
and , are the inverse functions of φ, W, respectively, and
Proof Choose arbitrarily. For , we obtain that
from (2.1), by Assumption 2 that are nondecreasing in t. Let be the right-hand side of (2.5), which is a positive and nondecreasing function on with . Then (2.5) can be written as
since the inverse function of φ exists by Assumption 4. From (2.5) and (2.6), we can obtain that
for all . Applying the monotonicity of w, φ and , (2.7) can be written as
for all . Integrating both sides of the above inequality from to t, we can obtain that
where W is defined as (2.4). In consequence, we get that
by (2.6) and (2.9). Let on both hand sides of (2.10), then we have that
Thus, we obtain that
from (2.11), where is defined as (2.3), since T is chosen arbitrarily. □
Next, consider a more general scenario: the composite function of unknown function exists not only in the innermost layer of iterated integrals, but also in the outermost layer of iterated integrals.
Theorem 2 Assume that Assumptions 1-5 and the following inequality hold
Then the integral inequality (2.12) implies that
where W is defined in (2.4) and
and , , are the inverse functions of φ, W, J, respectively, and
Proof Choose arbitrarily. For , from (2.12), we have that
by Assumption 2. Denote the right-hand side of (2.16) by , which can be proved that it is positive and nondecreasing on with . Then (2.16) can be written as
by Assumption 4. From (2.16) and (2.17), we obtain that
By the property of monotonicity of functions w, φ and , we can obtain that
from (2.18). Integrating both sides of the above inequality from to t, we have that
for all , where W is given in (2.4). Let denote the right-hand side of (2.19), which can be proved to be a positive and nondecreasing function on with . Then (2.19) is equivalent to
Differentiating , we get that
using (2.20), for all . By (2.21) and the monotonicity of w, , and , we further obtain that
for all . Integrating both sides of the above inequality from to t, we obtain
for all , where J is defined by (2.14). Hence, inequalities (2.17), (2.20) and (2.22) yield that
Let on both hand sides of (2.23), we have that
Due to the randomness of T, (2.13) is achieved immediately from (2.24). □
Obviously, the most general scenario is that the composite function of unknown function is involved in every layer of iterated integrals. For this kind of integral inequalities, we have the following result.
Theorem 3 Assume that Assumptions 1-5 and the following inequality hold
Then we have that
where
and , , are the inverse functions of φ, , , respectively, and
Proof Choose arbitrarily. For , we obtain that
from (2.25) and the monotonicity of , on t. Let be the right-hand side of (2.31), which is a positive and nondecreasing function on with . Then (2.31) is equivalent to
Differentiating , we can obtain that
from (2.32) and the monotonicity of , φ and , for all . Thus, we have
by (2.33) for all . Integrating both sides of the above inequality from to t, we obtain
where is defined by (2.27). Let denote the right-hand side of (2.34), which is a positive and nondecreasing function on with . Then (2.34) is equivalent to
Differentiating , we obtain
by (2.35). Applying (2.36) and the monotonicity of , , and , we can get that
for all . Integrating both sides of the above inequality from to t, we obtain
for all , where is defined by (2.28). Now, let be the right-hand side of (2.37), which is a positive and nondecreasing function on with
Then, (2.37) is equivalent to
Differentiating and applying (2.39), we can obtain that
for all . By (2.40) and the monotonicity of , , , and , we get
for all . Integrating both sides of the inequality above, from to t, we obtain
for all , where is defined by (2.29). By combining (2.32), (2.35), (2.39) and (2.41), we can obtain that
for all . Let on both hand sides of (2.42), we have
Since T is chosen arbitrarily in (2.43), thus (2.26) is proved. □
As a generalization of Theorem 3, we can obtain the following corollary, which can be proved similarly as Theorem 3.
Corollary 1 Assume that Assumptions 1-5 and the following inequality hold
Then we have that
where
and
and , , are the inverse functions of φ, , , respectively, and
3 Application
In this section, we apply our result in Theorem 3 to investigate the robust stability of a class of closed-loop control systems, which demonstrates that our results are handy tools to analyze the qualitative properties of solutions of some nonlinear ordinary differential and integral equations.
For a given control system
there is no doubt that controller design plays a pivotal role. Choosing the full state feedback controller with the appropriate gain F for (3.1), one can immediately obtain the following stable closed-loop system
where . However, in practice, some undesirable system factors, including nonlinear uncertainties and input disturbance, will be involved. As such, before applying the designed controller to real processes, the stability of a closed-loop system against external perturbations must be verified, which is the so-called robust stability analysis.
Consider a perturbed system of (3.2)
with
where is a nondecreasing function with , , is a continuous nonsingular matrix, and the function and satisfy the following conditions
where is a constant, and , , are nondecreasing in t for fixed , , , are positive and continuous functions defined on . In general, the perturbation term could result from modeling errors, aging, uncertainties, disturbances, or some other reasons. Suppose that the nominal system (3.2) has a uniformly asymptotically stable equilibrium at the origin, we next exploit the stability of the perturbed system (3.3). The result is presented in the following proposition.
Proposition 1 If there exists a constant such that the fundamental solution matrix of the linear system (3.2) satisfies
then we have that
where is a solution of the control system (3.3) with (3.4) and
and , are the inverse functions of , , respectively, and
Further, if there exists a positive constant b such that
any solution of the control system (3.3) with (3.4) is exponentially asymptotically stable.
Proof Firstly, we can obtain the solution of (3.3) with (3.4)
by using the variation of constants formula. Then we have that
by conditions (3.5) and (3.7) from (3.14). Further, by using conditions (3.6), we can obtain that
from (3.4) and (3.15). Then we have that
where we use the change . Let , (3.16) can be rewritten as
Letting , we have
from (3.10) and (3.17). Applying the result of Theorem 3, the inequality
is proved.
If there exists a positive constant b such that
then we have that
i.e., the nonlinear control system (3.3) with (3.4) is exponentially asymptotically stable. □
References
Gronwall TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 1919, 20: 292–296. 10.2307/1967124
Bellman R: The stability of solutions of linear differential equations. Duke Math. J. 1943, 10: 643–647. 10.1215/S0012-7094-43-01059-2
Bihari IA: A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation. Acta Math. Acad. Sci. Hung. 1956, 7: 81–94. 10.1007/BF02022967
Pachpatte BG: Inequalities for Differential and Integral Equations. Academic Press, London; 1998.
Lipovan O: A retarded Gronwall-like inequality and its applications. J. Math. Anal. Appl. 2000, 252: 389–401. 10.1006/jmaa.2000.7085
Dragomir SS, Kim YH: On certain new integral inequalities and their applications. J. Inequal. Pure Appl. Math. 2002., 3(4): Article ID 65
Bainov D, Simeonov P: Integral Inequalities and Applications. Kluwer Academic, Dordrecht; 1992.
Kim BI: On some Gronwall type inequalities for a system integral equation. Bull. Korean Math. Soc. 2005, 42(4):789–805.
Cho YJ, Dragomir SS, Kim YH: On some integral inequalities with iterated integrals. J. Korean Math. Soc. 2006, 43(3):563–578.
Agarwal RP, Ryoo CS, Kim YH: New integral inequalities for iterated integrals with applications. J. Inequal. Appl. 2007., 2007: Article ID 24385
Abdeldaim A, Yakout M: On some new integral inequalities of Gronwall-Bellman-Pachpatte type. Appl. Math. Comput. 2011, 217: 7887–7899. 10.1016/j.amc.2011.02.093
Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwall-like inequality and its applications. Appl. Math. Comput. 2005, 165: 599–612. 10.1016/j.amc.2004.04.067
Agarwal RP, Kim YH, Sen SK: New retarded integral inequalities with applications. J. Inequal. Appl. 2008., 2008: Article ID 908784
Cheung WS: Some new nonlinear inequalities and applications to boundary value problems. Nonlinear Anal. 2006, 64: 2112–2128. 10.1016/j.na.2005.08.009
Zhao X, Meng F: On some advanced integral inequalities and their applications. JIPAM. J. Inequal. Pure Appl. Math. 2005., 6(3): Article ID 60
Cheung WS, Ma QH: On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications. J. Inequal. Appl. 2005, 2005(4):347–361.
Kim YH: On some new integral inequalities for functions in one and two variables. Acta Math. Sin. 2005, 21: 423–434. 10.1007/s10114-004-0463-7
Pachpatte BG: ON certain nonlinear integral inequalities involving iterated integrals. Tamkang J. Math. 2006, 37(3):261–271.
Wang WS: A generalized retarded Gronwall-like inequality in two variables and applications to BVP. Appl. Math. Comput. 2007, 191: 144–154. 10.1016/j.amc.2007.02.099
Ma QH, Pečarić J: On certain new nonlinear retarded integral inequalities for functions in two variables and their applications. J. Korean Math. Soc. 2008, 45(1):121–136. 10.4134/JKMS.2008.45.1.121
Wang WS, Shen C: On a generalized retarded integral inequality with two variables. J. Inequal. Appl. 2008., 2008: Article ID 518646
Wang WS, Li Z, Li Y, Huang Y: Nonlinear retarded integral inequalities with two variables and applications. J. Inequal. Appl. 2010., 2010: Article ID 240790
Wang WS, Luo RC, Li Z: A new nonlinear retarded integral inequality and its application. J. Inequal. Appl. 2010., 2010: Article ID 462163
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Project No. 11161018), the SERC Research Grant (Project No. 092 101 00558), the Guangxi Natural Science Foundation (Project No. 0991265 and 2012GXNSFAA053009), the Scientific Research Foundation of the Education Department of Guangxi Province of China (Project No. 201106LX599), and the Key Discipline of Applied Mathematics of Hechi University of China (200725).
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Wang, WS., Huang, D. & Li, X. Generalized retarded nonlinear integral inequalities involving iterated integrals and an application. J Inequal Appl 2013, 376 (2013). https://doi.org/10.1186/1029-242X-2013-376
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DOI: https://doi.org/10.1186/1029-242X-2013-376