Some New Inequalities On Time Scales

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger [1] in his PhD in 1988 in order to contain both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scales. It is applied widely in various parts of applied mathematics and physics, especially in the aspects of computer and biochemistry. Therefore, by studying the theory on time scales, we obtain practical meaning. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scales, they also get many results.The main objective of this paper is to establish and extend some new Pachpatte type inequalities and Bellman-Bihari's type inequalities on time scales by using an inequality. Since then many authors have expounded on various aspects of the the-ory of dynamic equations on time scales. However, very little paper has studied the delay integral inequalities on time scales. Otherwise, the delay integral inequalities on time scales are applied very widely on control theory. It is a hot question about applied mathematics, biochemistry, physics and control theory. In this paper, we refer the reader to the paper [2,3] and investigate some delay integral inequalities on time scales, which provide explicit bounds on unknown functions, also unify and extend some continuous inequalities and their corresponding discrete analogues. Otherwise, it is well known that Gronwall integral inequalities play an important role in the study quantitative analysis of the solutions to differential and integral equations. The lit-erature on such inequalities and their applications is vast. Basic on this, sometimes we also need some different form, so the purpose of the following chapter is to pop-ularize the Gronwall inequality in aid to study the solution about the order and the initial condition for fractional differential equations with Riemann-Liouville fractional derivatives.The thesis is divided into five sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 In this chapter, we combine the feature of Pachpatte type inequalities with time scales theory by using Lemma [2.2.1] as a tool and extend them into Higher-order Scalar. We also obtain some new result and make it as a general rule. Chapter 3 Based on some known dynamic inequalities, this chapter gives the ex-istence certain new Bellman-Bihari's type inequalities on time scales by using Lemma [3.2.1] as a tool, which unify and extend some continuous inequalities and their corre-sponding discrete analogues.Chapter 4 The main objective of this chapter is to establish some new delay integral inequalities on time scales by using Gronwall's inequality, which provide ex-plicit bounds on unknown function. Our result unify and extend some delay integral inequalities and their corresponding discrete analogues. The inequalities given can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations on time scales.Chapter 5 We give a particular Gronwall integral inequality and apply it into a fractional differential equation.