Research On Inertial Projection Algorithm For A Class Of Variational Inequalities And Fixed Point Problems Of Demicontractive Mappings

The research on the algorithm of variational inequalities and fixed point problems provides a powerful tool for revealing the internal mechanism of such problems,and the exploration of a common element has always been a hot research topic in nonlinear functional analysis.It plays an important role in pure and applied mathematics research,and is widely used in economics,medicine,control,engineering fields and other fields.Therefore,it is of great significance to study the algorithm of variational inequalities and fixed point problems.This dissertation focuses on the research of a common element of variational inequalities and fixed point problems,especially in the variational inequalities and fixed point problems of demicontractive mappings.Systematic use of the Mann-type iterative method,viscous method and projection algorithm,especially the inertial method.That three modified projection algorithms are proposed,which proves the strong convergence results of the common element of the given algorithm under Lipschitz continuous and nonempty conditions.Firstly,a modified inertial Mann-type contraction projection algorithm is proposed in real Hilbert space.A Mann-type iterative algorithm and inertial method are introduced.The Armijo method is used to search for the step size.Strong convergence theorems of a monotone variational inequality problem and a fixed point problem with a demicontractive mapping is established.I implement some computational tests to show the efficiency of the proposed algorithm.Secondly,a modified viscous inertial contraction projection algorithm is proposed in real Hilbert space.By combining the inertial projection algorithm with a viscosity method,weakening the monotonic mapping,and selecting the adaptive step size to find the common element of a pseudo-monotonic variational inequality problem and a fixed point problem with a demicontractive mapping.I implement some computational tests to show the efficiency of the proposed algorithm.Finally,A modified inertial two-subgradient extragradient algorithm is studied in the constructed half-space.Only two-step projections are used to design the proposed algorithm in the same half-space each iteration.Based on the subgradient extragradient algorithm and the inertia method,the common element of a monotone variational inequality problem and a fixed point problem with a demicontractive mapping is found under the condition of adaptive step size.Strong convergence theorems of a monotone variational inequality problem and a fixed point problem with a demicontractive mapping is established.