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.

中文翻译：

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与时标上滞后项有关的一些非线性积分不等式

这项研究的目的是用时间标度理论来构造一类特定的Gronwall类积分不等式，涉及延迟项和非线性积分。我们的结果产生了一些新的不等式，这些不等式反映了连续和离散的形式，并为未知函数提供了上限估计。这种不平等的有效性源于以下信念：在没有有效利用各种可用不平等的独特情况下，它与广泛相关。此外，还代表了应用程序以显示组合假设的合法性。