Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space

Let [inline-graphic not available: see fulltext] be a real Banach space with the dual space [inline-graphic not available: see fulltext]. Let [inline-graphic not available: see fulltext] be a proper functional and let [inline-graphic not available: see fulltext] be a bifunction. In this paper, a new concept of [inline-graphic not available: see fulltext]-proximal mapping of [inline-graphic not available: see fulltext] with respect to [inline-graphic not available: see fulltext] is introduced. The existence and Lipschitz continuity of the [inline-graphic not available: see fulltext]-proximal mapping of [inline-graphic not available: see fulltext] with respect to [inline-graphic not available: see fulltext] are proved. By using properties of the [inline-graphic not available: see fulltext]-proximal mapping of [inline-graphic not available: see fulltext] with respect to [inline-graphic not available: see fulltext], a generalized mixed equilibrium problem with perturbation (for short, GMEPP) is introduced and studied in Banach space [inline-graphic not available: see fulltext]. 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 [inline-graphic not available: see fulltext], and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in [inline-graphic not available: see fulltext] a Hilbert space.