Nonlinear Retarded Integral Inequalities Research Articles

Further Nonlinear Retarded Integral Inequalities for Gronwall–Bellman Type and Their Applications

In this article, we will derive several new nonlinear retarded integral inequalities for Gronwall–Bellman type and then apply them as handy tools to examine explicit bounds for solutions of initial value problems, and will also use to evaluate the qualitative as well as the quantitative behavior of solutions of nonlinear retarded differential and integral equations. These inequalities extend some present inequalities to the current literature. An application is also given to demonstrate the strength of our results.

On some generalizations of certain nonlinear retarded integral inequalities for Volterra–Fredholm integral equations and their applications in delay differential equations

Abstract By some new analysis techniques, we generalize the results presented by Pachpatte in [Integral and Finite Difference Inequalities and Applications, volume 205, Elsevier, 2006] and and by S. D. Kendre in [Some nonlinear integral inequalities for Volterra–Fredholm integral equations, Adv. Inequal. App, 2014:Article 21, 2014.] to nonlinear retarded inequalities, and investigate also some new forms. Some applications are also presented in order to illustrate the usefulness of some of our results.

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

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.

On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation

Abstract In this paper, we investigate some new nonlinear retarded integral inequalities of Gronwall–Bellman–Pachpatte type. These inequalities generalize some former famous inequalities and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some nonlinear retarded differential and integral equations. An application is also presented to illustrate the usefulness of some of our results in estimation of solution of certain retarded nonlinear differential equations with the initial conditions.

On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations

In this paper, some new nonlinear retarded integral inequalities of Gronwall–Bellman–Pachpatte type are investigated. Some applications are also presented to illustrate the usefulness of some of our results in estimation of solution of certain retarded integro-differential equations with the initial conditions.

Some Retarded Integral Inequalities and their Applications in Retarded Integral Equations

The aim of the present paper is to establish some new nonlinear retarded integral inequalities which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain integral equations

Some generalized nonlinear retarded integral inequalities with applications

In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions. Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40.

Nonlinear Retarded Integral Inequalities with Two Variables and Applications

We consider some new nonlinear retarded integral inequalities with two variables, which extend the results in the work of W.-S. Wang (2007), and the one in the work of Y.-H. kim (2009). These inequalities include not only a nonconstant term outside the integrals but also more than one distinct nonlinear integrals without assumption of monotonicity. Finally, we give some applications to the boundary value problem of a partial differential equation for boundedness and uniqueness.

Gronwall, Bellman and Pachpatte type integral inequalities with applications

In this article, a number of two-dimensional nonlinear retarded integral inequalities of Gronwall–Bellman type which can be used as ready and powerful tools in the analysis of various classes of partial differential, integral and integro-differential equations are presented. Applications of nonlinear retarded integral inequalities are also presented.

EXPLICIT BOUNDS ON SOME NEW NONLINEAR RETARDED INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

In this paper, some new explicit bounds on solutions to a class of new nonlinear retarded integral inequalities are established, which can be used as effective tools in the study of certain nonlinear retarded integral equations. Applications examples are also indicated.

EXPLICIT BOUNDS ON SOME NEW NONLINEAR RETARDED INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

In this paper, some new explicit bounds on solutions to a class of new nonlinear retarded integral inequalities are established, which can be used as effective tools in the study of certain nonlinear retarded integral equations. Applications examples are also indicated.

ON SOME NEW NONLINEAR RETARDED INTEGRAL INEQUALITIES WITH ITERATED INTEGRALS AND THEIR APPLICATIONS

Some new nonlinear retarded integral inequalities of Gronwall-like type are established, which mainly generalized some results given by Cho, Dragomir and Kim (J. Korean Math. Soc. 43 (2006), No.3, pp. 563-578) and can be used in the analysis of various problems in the theory of certain classes of differential equations and integral equations. Applications examples are also indicated.

New Integral Inequalities for Iterated Integrals with Applications

Some new nonlinear retarded integral inequalities of Gronwall type are established. These inequalities can be used as basic tools in the study of certain classes of integrodifferential equations.

Some new nonlinear retarded integral inequalities and their applications

In the present paper, priori bound on certain new retarded integral inequalities in two independent variables are established. Applications are given to illustrate the usefulness of these inequalities. Mathematics subject classification (2000): 26D15, 26D20.

Integral inequalities for retarded Volterra equations

Integral inequalities are very useful in the qualitative analysis of differential and integral equations. Starting with [O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl. 252 (2000) 389–401], several recent investigations, see [O. Lipovan, A retarded integral inequality and its applications, J. Math. Anal. Appl. 285 (2003) 436–443; B.G. Pachpatte, Explicit bounds on certain integral inequalities, J. Math. Anal. Appl. 267 (2002) 48–61; B.G. Pachpatte, On some retarded integral inequalities and applications, J. Inequal. Pure Appl. Math. 3 (2002), Article 18; B.G. Pachpatte, On a certain retarded integral inequality and its applications, J. Inequal. Pure Appl. Math. 5 (2004), Article 19; B.G. Pachpatte, On some new nonlinear retarded integral inequalities, J. Inequal. Pure Appl. Math. 5 (2004), Article 80], were devoted to retarded integral inequalities. In this paper we consider the case of retarded Volterra integral equations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the asymptotic behaviour of the solutions.