Integrals Inequalities With Maxima In Two Variables And Their Applications

With the development of the theory of differential equation,there can be founded a lot of integral inequalities and their generalizations in various case.Thereinto,Gronwall-Bellman,Gamidov type and Volterra type integral inequality are widely used in the research on boundedness,uniqueness,and other qualitative analysis of differential(or integral)equations.In recent decades,along with the development of theory of automatic control and its applications to computational mathematics and modeling,attention were also put to differential equations with the maxima of the unknown function.Therefore,the integral inequality with maxima become a hot topic of research,the study for Gamidov type,Volterra-Fredholm type integral equalities with maxima have made some achievements.In the paper,based on the works of [6,16,26,27,33,35,42,45],we consider the Gronwall-Bellman-Gamidov type integral inequalities involving maxima of the unknown functions in two independent variables,their Bihari type and their weakly singular analogue.We also establish some new iterated integral inequalities of retarded nonlinear Volterra-Fredholm type with maxima in two independent variables.By analysis techniques,such as change of variable,amplification method,differential and integration,inverse function,and the dialectical relationship between constants and variables,the bounds of the unknown functions are estimated.The thesis is divided into four chapters according to the contents.Chapter 1 Preference,we introduce the main contents and its background of this paper.Chapter 2 Based on the works of [26,27,42],we consider the Gronwall-BellmanGamidov type integral inequalities involving maxima of the unknown function in two independent variables:(?)and their weakly singular analogue as follows:(?)Then we apply the obtained results to study the boundedness and uniqueness for weakly singular integral equations with maxima.Chapter 3 Motivated mainly by the work of [6,27,45],we give a new GamidovBihari type integral inequalities as follows:(?)and their weakly singular analogue:(?)Then,we give some examples to show the boundedness and uniqueness of solutions of weakly singular integral equations with maxima.Chapter 4 Motivated mainly by the integral inequalities of [16,33,35],we establish some new iterated integral inequalities of retarded Volterra-Fredholm type as follows:(?)Then we apply the results to research the boundness,uniqueness for solutions of retarded Volterra-Fredholm type integral equations with maxima in two variables.