Some New Integral Inequality On The Time Scale Space

The time scales theory was introduced by Stefan Hilger[1] in his PhD in1988in order to contain both diference and diferential calculus in a consistentway, which has received a lot of attention recently. Since then many authorshave expounded on various aspects of the theory of dynamic equations on timescales. It is applied widely in various parts of applied mathematics and physics,especially in the aspects of computer and biochemistry. Therefore, by studyingthe theory on time scales, we obtain practical meaning. Since then many authorshave expounded on various aspects of the theory of dynamiy equations on timescales, they also get many better results.The delay integral inequalities on time scales are applied very widely oncontrol theory. It is a hot question about applied mathematies, biochemistry,physics and control theory. Since then many authors have expounded on variousaspects of the theory of dynamic equations on time scales. However, very littlepaper has studied the delay integral inequalities on time scales. In this paper, werefer the reader to the paper [2,3] and investigate some delay integral inequalitieson time scales. The main objective of this paper is to establish and extend somenew Pachpatte type inequalities and Gronwall type inequalities on time scales byusing some inequalities, which provide explicit bounds on unknown functions, alsounify and extend some continuous inequalities and their corresponding discreteanalogues. Otherwise, it is well known that Gronwall integral inequalities play animportant role in the study quantitative analysis of the solutions to diferentialand integral equations.The thesis is divided into four sections according to contents.Chapter1Preference, we introduce the main contents of this paper and theintroduction of time scales.Chapter2Based on some known dynamic inequalities, this chapter givesthe existence certain new Bellman type inequalities on time scales of [4],[5],[7]by using Lemma5as a tool, and then have the following inequalities: Chapter3Combine the feature of Pachpatte type inequalities with time scalestheory by using Lemma9as a tool, we extend them of [8],[9] on time scales, andthen have the following inequalities:Chapter4Using Gronwall’s inequality, we unify and extend some delay integralinequalities of [15],[16] on time scales and then have the following inequalities:...