The Study Of Several Gronwall-type Inequalities On Time Scales

Integral inequalities play an extremely important role in the study of the existence and stability of solutions and other qualitative and quantitative properties of differential equations. Among the promotion and research of many inequalities, since the Gronwall-Bellman type inequalities have been widely applied in the research of boundedness and estimation of the solutions for differential equations, difference equations, dynamic e-quations on time scales, many mathematicians have been committed to promote these inequalities.On the basis of integral inequalities which have been founded, this paper will fur-ther promote some Gronwall-Bellman type retarded inequalies with one variable or two independent variables on time scales and establish several new results.The paper is divided into five chapters.In Chapter 1, we mainly introduce the research of the situations and some related definitions and lemmas which have been used in this paper.In Chapter 2, we consider the Gronwall type inequalities with delay on time scales, such as and Gronwall-Bellman-Ou-Iang type retarded integral inequalities. The left side is square term and the delay of right side is more general, such as We use the related lemmas and the knowledge of time scales, through analysis and cal-culation to get some new integral inequalities and generalize the previous results.In Chapter 3, we consider the following Gronwall-like retarded inequalities in two independent variables on time scales. By introducing a new function, we differentiate the new function through using algebraic inequalities and get an estimation of unknown function about u. we also get some other generalized inequalities.In Chapter 4, we study the following Gronwall-Bellman type integral inequalities’ system with two variables in two dimension, Using the known inequalities, we get some new results. Applying the new result to a two-dimensional integral equation system, we obtain a sufficient condition for boundedness of the solution.In Chapter 5, we mainly discuss the Gronwall-Bellman-Bihari type integral inequal-ities with both delay and impulsive points on time scales, such as By considering the segmental definition domain, we use the monotonicity of functions and mathematical induction method to obtain the upper bound of u function.