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ABSTRACT Inequalities play a very important role in almost all branches of mathematics such as functional analysis, real analysis, theory of differential and integral equations. Integral inequalities play a fundamental role in the modern theory of differential equations and are used to study uniqueness, existence, boundedness stability, asymptotic behavior of solutions of differential equations.We introduce some important mathematical theorems regarding the inequality under consideration "Gronwall inequality". We study linear and nonlinear integral inequalities in one and two variables of Gronwall type. Some useful generalizations are given here. Some applications are given to study certain qualitative properties of solutions of hyperbolic partial differential equations such as existence and uniqueness. |
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