Several New Types Of Wendroff-type Integral Inequalities With Integral Jump Conditions

With the development of the differential equation theory in various disciplines in recent decades, differential equations have been widely in social science and natural sci-ence. Gronwall inequalities, Gronwall-Bellman inequalities and their generalized forms would be a powerful tool for researching the existence, stability,uniqueness, bounded-ness, asymptotic property, oscillation and other character of solutions of the differential equations and the integral equations. A large amount of scholars conducted a series of promotion on it.It is well known that the impulsive differential equations play a dominant role in the theoretical study of differential and integral equations. Many authors devote themselves to investigating the characters of solutions with the problem of disturbance by using the impulsive differential equations and the impulsive integral inequalities.Borysenko researched the impulsive integral inequality with two dependence variables from [1]:Here ??t,x? is nonnegative with the exception of the points?ti,xi?,u?t_i + 0, x_i + 0? ? u?t_i - 0,x_i- 0?, i = 1,2, ….This paper on the basis of reference [1,3,5,9,15,19,20,21] spreads some new integral inequalities of Wendroff type for discontinuous functions with integral jump conditions.The thesis is divided into three chapters according to the contents.Chapter 1 Preference, this chapter mainly introduce the main contents and its background.Chapter 2 This chapter mainly study the inequality of Wendroff type, as follows:Here m > 0, t₀ ? 0, x₀ ? 0;?_i= const ?0,?_i = const ? 0, 0??_k??_k?t_k-t_k-1,0??_k??_k?x_k-x_k-1 for everyone of ?t,x? ??,a?t,x?>0. ??t,x? is nondecreasing.For ????p,g?? ?, ?P,Q???,we get a?p,q??a?P,Q? when ???p ? P,???q ? Q.And ??t,x? satisfies the condition as follow: If??,????ij,i?j,???i,j=1,2,….we have b?t,x??0 and b??,?? = 0.When t_k < t_k+1,x_k < x_k+1, ???k = 0,1,2,…, we have ?t_k,x_k? < (t_k+1,x_k+1), andThus we obtain some more extensive results which could be used as handy tools to study the boundedness of solutions of partial differential equations.Chapter 3 Based on chapter 2, we generalize the integral inequalities for discon-tinuous functions with integral jump conditions in chapter 2 to the equalities with retardation:Then we apply those inequalities to the research on the boundedness for solutions to the differential equations.