Bounding Unknown Functions in Nonlinear Integral Inequalities: A Comprehensive Study

This article presents a comprehensive examination of linear and non-linear systems of integral inequalities involving two real-valued unknown functions in "n" independent variables. The primary objective of this investigation is to establish upper bounds for these unknown functions and to analyze their practical implications within broader mathematical frameworks. The results obtained not only extend the classical Grönwall-Bellman integral inequalities but also introduce novel and explicit bounds within the contexts of Young and Pachpatte integral inequalities. These contributions significantly enhance the theoretical understanding of integral inequalities and their utility in addressing complex analytical problems. Moreover, the results yield important insights into the qualitative analysis of nonlinear hyperbolic partial integro-differential equations, particularly with regard to the existence, uniqueness, and boundedness of solutions. To derive the main theoretical results, Young's method based on the Riemann approach is employed. Additionally, the analysis highlights the essential role of symmetry in the selection of appropriate methods for treating dynamic inequalities.