Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications

Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two... In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inequalities and Applications Springer Journals

Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications

Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications

Department of Mathematics, Qufu In this paper, we establish some new retarded nonlinear Volterra-Fredholm type Normal University, Qufu, Shandong 273165, People’s Republic of China integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations. MSC: 42B20; 26D07; 26D15 Keywords: Volterra-Fredholm type; integral inequality; integral equations; maxima; iterated integral inequality 1 Introduction Gronwall-Bellman inequality [, ] and Bihari inequality [] provided important devices in the study of existence, uniqueness, boundedness, oscillation, stability, invariant mani- folds and other qualitative properties of solutions to differential equations, integral equa- tions and integro-differential equations. In the past few decades, a number of studies have focused on generalizations of the Gronwall-Bellman inequality. For example, in [–], the Gronwall-Bellman-Gamidov type integral inequalities and their generalizations were studied; in [–], the Gronwall-like inequalities and their deformations were investi- gated; in [–], the Volterra type iterated inequalities were discussed; in [–], the Volterra-Fredholm type inequalities were examined. The Gronwall-Bellman inequality can be stated as follows. If u and f are nonnegative continuous functions on an interval [a, b], and u satisfies the following inequality: u(t) ≤ c + f (s)u(s) ds, t ∈ [a, b], (.) where c ≥ is a constant. Then u(t) ≤ c exp f (s) ds.(.) © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and...
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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by The Author(s)
Subject
Mathematics; Analysis; Applications of Mathematics; Mathematics, general
eISSN
1029-242X
D.O.I.
10.1186/s13660-017-1460-6
Publisher site
See Article on Publisher Site

Abstract

In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations.

Journal

Journal of Inequalities and ApplicationsSpringer Journals

Published: Aug 15, 2017

References

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