Abstract
Some new weakly singular versions of discrete nonlinear inequalities are established, which generalize some existing weakly singular inequalities and can be used in the analysis of nonlinear Volterra type difference equations with weakly singular kernels. A few applications to the upper bound and the uniqueness of solutions of nonlinear difference equations are also involved.
Highlights
Along with the development of the theory of integral inequalities and difference equations, many authors have researched some discrete versions of Gronwall-Bellman type inequalities [1,2,3,4,5]
Starting from the basic form, n−1 u (n) ≤ a (n) + ∑f (s) u (s) s=0 discussed originally by Pachpatte in [4], various such new inequalities have been established, which can be used as a powerful tool in the analysis of certain classes of finite difference equations
Discrete weakly singular integral inequalities play an important role in the study of the behavior and numerical solutions for singular integral equations [6, 7] and the theory for parabolic equations [8,9,10]
Summary
Along with the development of the theory of integral inequalities and difference equations, many authors have researched some discrete versions of Gronwall-Bellman type inequalities [1,2,3,4,5]. S=0 discussed originally by Pachpatte in [4], various such new inequalities have been established, which can be used as a powerful tool in the analysis of certain classes of finite difference equations. Among these results, discrete weakly singular integral inequalities play an important role in the study of the behavior and numerical solutions for singular integral equations [6, 7] and the theory for parabolic equations [8,9,10]. We investigate some new nonlinear discrete weakly singular inequalities n−1 xn ≤ an + ∑(tnα − tkα)β−1tkγ−1τkbkω (xk) ,.
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