Some Random Contractive Type Mappings And Related Fixed Point Theorems

In this thesis,we introduce the concept of k-α-ψ-contractive type random map-pings and establish fixed point theorems for such mappings in complete metric s-paces.In the first chapter,we have listed some fundamental knowledge.In the second chapter,we introduce the k-α-ψ-contractive type random map-pings T,the mapping satisfy condition α(x,y)d(T(ω)x,T(ω)y)≤k(ω)φ(d(x,y)), and establish some fixed point theorems. Moreover,we consider the existence of random solutions of the problem where f:Ω×I×X×Xâ†'X,for all u∈C(I,X),for each ω∈Ω,f(ω,t,u,S(ω)u(t)) is measurable,for each t∈I,f(ω,t,u,S(ω)u(t))is continuous and x0(ω)is a mea-surable random variable.Where S(ω)u(t)=f0ak(ω,t,s)u(s)ds,for all ω∈Ω,t∈I,k:Ω×I×Iâ†'R+.The function ωâ†'k(ω,t,s)is measurable for all(t,s)∈I×I.In the third chapter,we introduce the generalized k-α-ψ-contractive type ran-dom mappings T,the mappings satisfy conditions α(x,y)d(T(ω)x,T(ω)y)≤k(ω)φ(D(ω,y)), and establish some fixed point for such mappings in complete metric spaces.More-over,as applications,we consider the following two-points boundary value problems of second order random differential equations where x:Ωâ†'C(I,R),f:Ω×I×Râ†'and I[0,2Ï€].