Some Gronwall{bellman Type Inequalities on Time Scales for Volterra-Fredholm Dynamic Integral Equations

Abstract

In this paper, we prove several new explicit estimations for the solutions of some classes of nonlinear dynamic inequalities of Gronwall-Bellman-Pachpatte type on time scales. Our results formulate some integral and discrete inequalities discussed in the literature as special cases and extend some known dynamic inequalities on time scales. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of dynamic equations on time scales. Some examples are presented to demonstrate the applications of our results.

On some explicit bounds of integral inequalities related to time scales

“…and B(u, v) is as defined in (7). Fixing any two time-scale points u 1 ∈ [u , ∞) T and v 1 ∈ [v , ∞) T , then from (22) and (23) it is easy to observe that ) to the right-hand side of the last relation, therefore…”

On Some Generalizations of Reverse Dynamic Hardy Type Inequalities on Time Scales

“…For more details on Hardy-type inequalities and other types on time scales, we suggest [17][18][19][20][21][22][23][24][25][26][27][28][29] for the reader. ).…”

Dynamic Hardy–Copson-Type Inequalities via (γ,a)-Nabla-Conformable Derivatives on Time Scales

“…In books [13,14], Bohner and Peterson introduce most basic concepts and definitions related with the theory of time scales. In [1,3,13,20,21,24,32,50], several mathematicians investigate new forms of dynamic inequalities. Řehák seems to be the first mathematician to have introduced a time-scale version of Hardy's inequality, by obtaining in 2005 a dynamic inequality that unifies inequalities (1) and (2).…”