Inertial Tseng's Extragradient Methods For Variational Inequalities With Fixed-point Problems

In this paper,we mainly discussed the inertial Tseng's extragradient methods for variational inequalities with fixed-point problems in Hilbert space,and for finding the common solution of the splitting feasibility problem and fixed point problem in Hilbert space.For solving these problems,we designed several iterative algorithms,Some strong convergence theorems are given under certain assumptions.The main results of this paper are divided into five parts below.In Part 1,we introduced the research background and current situation of relevant problems in this paper.In Part 2,we introduced some basic theories and concepts that will be used in this paper.In Part 3,we designed two inertial-like Tseng's extragradient algorithms with adaptive stepsizes for finding a common solution of the variational inequality problem with Lipschitzian,pseudomonotone operator and fixed-point problem of quasi-nonexpansive operator with a demiclosedness property.Under appropriate assumptions,we prove a strong convergence theorem for the iterative algorithms.Numerical example illustrates the feasibility of theoretical result.In Part 4,we studied common solution of the variational inequality problem with Lipschitzian,pseudomonotone operator and fixed-point problem of quasi-nonerator with a demiclosedness property,and designed new algorithms.The algorithms does not depend on z constant of the cost operator,and the step size is updated in each iteration,which makes the calculation process relatively simple.At the same time,the strong convergence of the algorithm is proved under appropriate parameter conditions.In Part 5,The common solutions of the splitting feasibility problem and fixed point problem in Hilbert space are studied,we introduced a modified viscosity implicit rules of one asymptotically nonexpansive mapping for finding the common solutions of the splitting feasibility problem and fixed point problem in Hilbert space.Under appropriate assumptions,we prove a strong convergence theorem for the iterative algorithms.