Hierarchical Optimization Problem And Modified Mann-External Gradient Metho

In this dissertation,we study the variational inequality problems with pseudomonotone mapping,common fixed point problems,and generalized mixed equilibrium problems in an infinite dimensional real Hilbert space.In order to find solutions of the above problems,we constructs several iterative methods in Chapters 3,4,and 5 with the help of inertial type method,Mann iteration method,subgradient extragradient method with linear-search process,and hybrid deepest-descent technique.Under suitable assumptions,we demonstrate the convergence of the iterative methods.In Chapter 1,firstly,we introduced the concepts of variational inequality problems and fixed point problems,as well as the development of convergence criteria for iterative methods for solving these two types of problems.Then we introduced the structural arrangement,and innovative points of this dissertation.In Chapter 2,we recall some basic concepts,important lemmas and propositions,etc.that need to be used in the process of proving convergence criteria.In Chapter 3,the purpose of this chapter is to study the iterative method for finding common solutions of the variational inequality problem with monotone mapping and common fixed point problem of countable nonexpansive operators and asymptotically nonexpansive operator.In the current literatures of solving common solution problems involving variational inequalities,there is not much studies on introducing countable nonexpansive operators for common fixed point problems.In order to seek a common solution to the above problem,we constructed a modified Mann-type extragradient method with linear-search process.The proposed algorithms are based on inertial type method,Mann iteration method,and subgradient extragradient method with linear-search process.Under appropriate conditions,we proved that the iterative method weakly converges to the common solution of the above problems.In Chapter 4,the purpose of this chapter is to study the iterative method for finding common solutions of the variational inequality problem with pseudomonotone mapping and common fixed point problem of countable nonexpansive operators and asymptotically nonexpansive operator.In order to seek a common solution to the above problem,we further weaken the conditions of the operator and improve the weak convergence criterion to a strong convergence criterion based on the problem studied in the previous chapter.We constructed several modified extragradient methods with linear-search process,the proposed algorithms are based on the inertial type method,Mann iteration method,subgradient extragradient method with linear-search process,and hybrid deepest-descent technique.Under suitable assumptions,we have demonstrated the convergence of the iterative methods,and the convergence point is also the unique solution of a certain hierarchical fixed point problem.In Chapter 5,the purpose of this chapter is to study the iterative method for finding common solutions of the generalized mixed equilibrium problem,variational inequality problem with pseudomonotone mapping,and common fixed point problem for countable and asymptotic nonexpansive operators.In order to seek a common solution to the above problem,we introduced the generalized mixed equilibrium problem in the problem of common solutions in Chapter 4.We constructed a modified Mann-type extragradient method with adaptive step size technique,which is based on the modified Mann-type iteration method and the hybrid deepest-descent method.Under suitable conditions,we proved that the iterative method strongly converges to the above common solution,and the convergence point is also the unique solution of a certain hierarchical fixed point problem.In Chapter 6,We provide several examples to verify the feasibility and practicality of the constructed convergence criteria.