Abstract
The objective of this research is to formulate a specific class of integral inequalities of Gronwall kind concerning retarded term and nonlinear integrals with time scales theory. Our results generate several new inequalities that reflect continuous and discrete form, as well as giving the unknown function an upper bound estimate. The effectiveness of such inequalities arises from the belief that it is widely relevant in unique circumstances where there is no valid utilization of various available inequalities. Applications are additionally represented to display the legitimacy of built-up hypotheses.
Highlights
A variety of basic and critical inequalities have been inspected with improvement in the methods of differential and integral equations, which are anticipating a great deal of study in the analysis of boundedness, global existence, and stability of solutions of differential and integral equations as well as difference equations [1, 2]
Pachpatte [4] replaced the constant m from the prior integral inequality by a nondecreasing function m ðħÞ and contemplated ðħ LðħÞ ≤ mðħÞ + H ðħÞ GðP ÞLðP ÞdP, ħ ∈ 1⁄20,∞Þ: ð2Þ
Unlike some proven and defined inequalities in the literature, Theorem 10, Theorem 12, and Theorem 14 have examined some dynamic integral inequalities of the GronwallBellman form in a single independent variable with a retarded and nonlinear term that can be used to overcome the qualitative properties of integral equations
Summary
A variety of basic and critical inequalities have been inspected with improvement in the methods of differential and integral equations, which are anticipating a great deal of study in the analysis of boundedness, global existence, and stability of solutions of differential and integral equations as well as difference equations [1, 2]. Bellman [3] has proven the integral inequality ðħ LðħÞ ≤ m + GðP ÞLðP ÞdP , ħ ∈ 1⁄2ħ1, ħ, ð1Þ ħ1 for some m ≥ 0, which is significantly dedicated to evaluate the equilibrium and asymptotic behavior with a view to find solutions for integral equations. The ultimate point is to study an equation or an inequality which can be dynamic such that a time scale T be a domain of an unknown function. Our primary concern of this work is to analyze nonlinear integral inequalities with retarded term and to explore the well-known existing results which determine the explicit bounds of the solutions of the unknown functions of the particular dynamic equations on time scales. A few concluding feedback and suggestions for future research are provided in Section 5 and completes this work
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