On Some Generalizations of Integral Inequalities in n Independent Variables and Their Applications

Abstract

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young’s method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities.

New Nonlinear Retarded Integral Inequalities and Their Applications to Nonlinear Retarded Integro-Differential Equations

“…On the other hand, integral inequalities occupy a very privileged position in all mathematical sciences, and they have many applications to questions of the existence, stability, boundedness, uniqueness, and asymptotic behavior of the solutions of nonlinear integro-differential equations. They can be used in various problems involving symmetry (see [1][2][3][4][5][6][7]). In 1919, Gronwall [8] was the first person to introduce the following inequality (which can be used to estimate the solution of a linear differential equation):…”