Abstract
The object of this paper is to establish some nonlinear integrodifferential integral inequalities in independent variables. These new inequalities represent a generalization of the results obtained by Pachpatte in the case of a function with one and two variables. Our results can be used as tools in the qualitative theory of a certain class of partial integrodifferential equation.
Highlights
It is well known that the integral inequalities involving functions of one and more than one independent variables, which provide explicit bounds on unknown functions, play a fundamental role in the development of the theory of differential equations
In the past few years, a number of integral inequalities had been established by many scholars, which are motivated by certain applications
Our aim in this paper is to establish some integrodifferential inequalities in n independent variables, an application of our results is given
Summary
It is well known that the integral inequalities involving functions of one and more than one independent variables, which provide explicit bounds on unknown functions, play a fundamental role in the development of the theory of differential equations. The study of integrodifferential inequalities for functions of one or n independent variables is a very important tool in the study of stability, existence, bounds, and other qualitative properties of differential equation solutions, integrodifferential equations, and in the theory of hyperbolic partial differential equations see 6–9. One of the most useful inequalities is given in the following lemma see 1, 10. Let Φ x, y and c x, y be nonnegative continuous functions defined for x ≥ 0, y ≥ 0, for which the inequality xy. In 10 , Pachpatte considered some new integrodifferential inequalities of the Wendroff type for functions of two independent variables. Our aim in this paper is to establish some integrodifferential inequalities in n independent variables, an application of our results is given
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