Abstract
A general nonlinear framework for an Ishikawa-hybrid proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding to -accretive mapping due to Lan-Cho-Verma in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem with the convergence rate is proved.
Highlights
The set-valued inclusions problem, which was introduced and studied by Di Bella 1, Huang et al 2, and Jeong 3, is a useful extension of the mathematics analysis
Ding and Luo 4, Verma 5, Huang 6, Fang and Huang 7, Lan et al 8, Fang et al 9, and Zhang et al 10 introduced the concepts of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, A-monotone operators, H, η monotone operators, A, η -accretive mappings, G, η -monotone operators, and defined
Verma has developed a hybrid version of the Eckstein and Bertsekas proximal point algorithm, introduced the algorithm based on the A, η -maximal monotonicity framework, and studied convergence of the algorithm
Summary
The set-valued inclusions problem, which was introduced and studied by Di Bella 1 , Huang et al 2 , and Jeong 3 , is a useful extension of the mathematics analysis. The variational inclusion inequality is an important context in the set-valued inclusions problem It provides us with a unified, natural, novel, innovative, and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences. In 2008, Li 13 studied the existence of solutions and the stability of perturbed Ishikawa iterative algorithm for nonlinear mixed quasivariational inclusions involving A, η -accretive mappings in Banach spaces by using the resolvent operator technique in 14. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear setvalued inclusions problem are explored along with some results on the resolvent operator corresponding to A, η -accretive mapping due to Lan et al in Banach space. The result that sequence {xn} generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem as the convergence rate θ is proved
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