Nonlinear Volterra-stieltjes Integral Equations In Banach Space Random Product Of Convergence

The paper is divided into two parts .The first part is to discuss the solvability of Volterra-Stieltjes integral equations.The theory of integral operators and integral equations finds numerous applications and creats an important and significant subject of nonlinear functional analysis. Let us mention that both the theory of integral operators and integral equations find nurmerous applications in mathematical physics,Engineering biology ,economics,and among others.Many scholars had done many works in this area,especially, they obtained that many integral equations have a continuous solution x(t)∈C[a,b],because this result is the best one that we want.In this paper,the main tool used in our study is Schauder fixed point theorem. I use a new method and some results to show the solvability of a nonlinear Volterra-Stieltjes integral equationFirst,I prove that the integral equation(E) must have a continuous solution x(t)∈C[0,t₁], (t₁≤1).Second,by introducing the concept of locally saturated solution,I obtain that the integral equation (E) has a locally saturated solution,and under some conditions,the integral equation (E) has a continuous solution in [0,1]. By a example, we can find this theorem is better than the major theorems of [7,8].Another theorem of my paper is that the integral equation(E) has a continuous solution x(t)∈C[0,1]for special u and g ,this result omits the key conditions of[9,10].The last major theorem of my paper is that the integral equation (E) has a saturated solution,and under some conditions,the integral equation (E) has a continuous solution in [0,∞).The second part is to discuss the convergence of unrestricted iterations in Banach spaces.Let{T₁,T₂,...T_N} be a sequence of nonexpansive mappings that satisfy condition (W),and let r be a mapping from the set of natural numbers N onto {1,2,..N},which assumes each value infinitely often.An unrestricted(or random)product of these operators is the sequence{S_n : n=1,2.....}defined by S_n = T_r(n)T_r(n－1)....T_r(1).Our purpose...