This paper deals with a split equality hierarchical fixed point problem in real Hilbert spaces which is an important and natural extension of hierarchical fixed point problem and split equality fixed point problem. An iterative algorithm where the stepsizes do not depend on the operator norms, so called simultaneous Krasnoselski-Mann algorithm is suggested for solving the split equality hierarchical fixed point problem. Further we prove a weak convergence theorem for the sequence generated by this algorithm. This special aspect of the algorithm together with the convergence result makes it an interesting scheme. Furthermore, we give some examples to justify the main result. Finally, we show that our purposed iterative algorithm is more efficient than some other known iterative algorithms. On the other hand, the framework is general and allows us to treat in a unified way several iterative algorithms, recovering, developing and improving some recently known related convergence results in the literature.
Abstract. In this paper, we introduce and study an explicit hybrid relaxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.
Abstract. In this paper, we introduce an iterative scheme for approximating a common solution to a new system of unrelated split mixed vector variational inequality problems, multivalued variational inequality problems, and common fixed point problem for a family of nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the sequence generated by the proposed iterative scheme. The results presented in this paper generalize and unify previously known results in this area. Key Words and Phrases: system of unrelated split mixed vector variational inequality problems, system of unrelated multivalued variational inequality problems, common fixed-point problem, iterative scheme.
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