Some New Pachpatte Type Inequalities on Time Scales and Their Applications

On Some Nonlinear Integral Inequalities on Time Scales

In this paper we study some nonlinear Pachpatte type integral inequalities on time scales by using a Bihari type inequality. Our results unify some continuous inequalities and their corresponding discrete analogues, and extend these inequalities to dynamic inequalities on time scales. Furthermore, we give some examples concerning our results.

Pachpatte Inequalities on Time Scales

In the study of dynamic equations on time scales we deal with certain dynamic inequalities which provide explicit bounds on the unknown functions and their derivatives. Most of the inequalities presented are of comparison or Gronwall type and, more specifically, of Pachpatte type.

Some Pachpatte type inequalities on time scales

Some new variants of interval-valued Gronwall type inequalities on time scales

By using an efficient partial order and concept of gH-differentiability oninterval-valued functions, we investigate some new variants of Gronwall typeinequalities on time scales, which provide explicit bounds on unknownfunctions. Our results not only unify and extend some continuousinequalities, but also in discrete case, all are new.

Explicit Bounds to Some New Gronwall-Bellman-Type Delay Integral Inequalities in Two Independent Variables on Time Scales

Some new Gronwall-Bellman-type delay integral inequalities in two independent variables on time scales are established, which provide a handy tool in the research of qualitative and quantitative properties of solutions of delay dynamic equations on time scales. The established inequalities generalize some of the results in the work of Zhang and Meng 2008, Pachpatte 2002, and Ma 2010.