Weak Generalized Mixed Variational Inequality Problems And A Gradual And Relatively Nonexpansive Mapping On The Fixed Point Iterative Approximation

In this paper, we study a class of weakly generalized mixed variational inequalities which are new, also study the convergences of iterative sequences to a comman fixed point of two asymptotically nonexpansive mappings and two relatively nonexpanseive mappings, respectively.In the first chaper, we simply introduce the background and development of variational inequalities and nonexpansive mappings.In the second chaper, we use the auxiliary principle technique to study a class of weakly generalized mixed variational inequalities which are new. We prove the existence of the solution of the auxiliary problem for the weakly generalized mixed variational inequalities and show the existence of the solution by using the auxiliary principle technique. We also prove the convergence of iterative sequences generated by the algorithm.In the third chaper, we introduce the scheme x_n+1=λ(1 - a_n)Sⁿx_n+λa_nTⁿx_n + (1 -λ)x_n for two asymptotically nonexpansive mappings S and T and study the strong convergence of {xn} to a common fixed point of S and T in Banach spaces.In the forth chaper, we consider the following iterative sequence: for a pair of relatively nonexpansive mappings S and T in Banach space and prove the sequence {xn} converges weakly to a common fixed point of S and T under some assumptions.