On Some Gronwall-Bellman-Bihari Type Retarded Integral Inequalities And Their Applications

Inequalities play a fundamental role in the development of the theoryof differential equations and difference equations, which can be used as toolsin the theory of partial differential and integral equations with time delays.Gronwall-Bellman-Bihari inequalities were established some more general re-tarded integral inequalities from linear to nonlinear, and from one nonlinearto more nonlinear terms. In recently years,many scholars take on the researchof this field, they have achieved many good results. Especially R.P.Agarwal,Pachpatte,O.Lipovan,E.H Yang,F.W.Meng established some similartypes with discrete and continuous types of Gronwall-Bellman-Bihari inequal-ities. With the increasing.The purposes of this paper is to generalize some Gronwall-Bellman-Bihariinequalities and establish some more general retarded integral inequalities,which generalized and improved some known inequalities .The thesis is divided into three sections according to contents.In chapter 1, Preface.we introduce the main contents of this paper.In chapter 2, The chapter is divided into three sections to investigateGronwall-Bellman-Bihari nonlinear inequalities. In 2006 O.Lipovan [5] estab-lished main inequalites as follows: In section 1, we mainly generalized above inequalities to two independent vari-ables,and obtain some new integral inequalities, which can be used as tools inthe theory of partial differential and integral equations with time delays. Insection 2, we will generalize (2.1.1),(2.1.2),(2.1.3) to more general tpyesand make them more easy in study the theory of differential and integral equa-tions. In chapter 3, we will investigate the asymptotic behavior of solutions ofintegro-differential equations ofWe obtain some interesting results which improve the main results of[25,26,27,28,31].