Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space
Lu-Chuan Ceng, Sangho Kum, Jen‐Chih Yao · Fixed Point Theory and Applications · 2010
Abstract Let "Equation missing" be a real Banach space with the dual space "Equation missing". Let "Equation missing" be a proper functional and let "Equation missing" be a bifunction. In this paper, a new concept of "Equation missing"-proximal mapping of "Equation missing" with respect to "Equation missing" is introduced. The existence and Lipschitz continuity of the "Equation missing"-proximal mapping of "Equation missing" with respect to "Equation missing" are proved. By using properties of the "Equation missing"-proximal mapping of "Equation missing" with respect to "Equation missing", a generalized mixed equilibrium problem with perturbation (for short, GMEPP) is introduced and studied in Banach space "Equation missing". An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space "Equation missing", and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in "Equation missing" a Hilbert space.