Estimation on Certain Nonlinear Discrete Inequality and Applications to Boundary Value Problem

Various generalizations of the Gronwall inequality 1, 2 are fundamental tools in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them such as 3–8 . Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are paid to some discrete versions of Gronwall-Bellman-type inequalities such as 9–11 . Some recent works can be found, for example, in 12–17 and some references therein. We first introduce two lemmas which are useful in our main result.