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.
Highlights
Gronwall-Bellman inequality [, ] and Bihari inequality [ ] provided important devices in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds and other qualitative properties of solutions to differential equations, integral equations and integro-differential equations
4 Conclusion In this paper, we established several new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables in Theorem . and Theorem . , and gave their specific cases in Corollary . and Corollary . , respectively, which can be used in the analysis of the qualitative properties to solutions of integral equations with maxima
In Theorem . and Theorem . , we presented the applications to research the boundedness of solutions of retarded nonlinear Volterra-Fredholm type integral equations
Summary
Gronwall-Bellman inequality [ , ] and Bihari inequality [ ] provided important devices in the study of existence, uniqueness, boundedness, oscillation, stability, invariant manifolds and other qualitative properties of solutions to differential equations, integral equations and integro-differential equations. In , Pachpatte [ ] investigated the retarded linear Volterra-Fredholm type integral inequality in two independent variables: α(x) β(y) α(M) β(N) In , Wang [ ] investigated a retarded Volterra type integral inequality with two variables: x ψ u(x, y) ≤ a(x, y) + b(x, y) c(s, y)ψ u(s, y) ds x
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