On Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaces

Abstract

We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points F(T) of a multivalued (or single-valued) k-strictly pseudocontractive-type mapping T and the set of solutions EP(F) of an equilibrium problem for a bifunction F in a real Hilbert space H. This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence {Kn}n=1∞ of closed convex subsets of H from an arbitrary x0∈H and a sequence {xn}n=1∞ of the metric projections of x0 into Kn. The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.

New Iteration Scheme for Approximating a Common Fixed Point of a Finite Family of Mappings

“…., N. Furthermore, lim n⟶∞ ‖x n − u n ‖ � 0. Consequently, lim n⟶∞ ‖x n+1 − x n ‖ 2 � lim n⟶∞ ‖1/2(x n − u n )‖ 2 � 0 which implies that x n is a Cauchy sequence in K. Also, since K is convex and closed, x n converges strongly to some q ∈ K. From the Opial condition of H and the demiclosedness property of T i , we have that q ∈ T i q, for all i − 1, 2,... , N. e remaining part of the proof is similar to the method of [34], eorem 20. erefore, it is omitted.…”

“…Motivated by Algorithm 19 of Isiogugu et al [34], we obtain the following result using a selection of Algorithm 4.2 above in the sense of [34].…”

Strong convergence of a selection of ishikawa-reich-sabach-type algorithm

“…that lim n→∞ p * − x n − p * − u n = 0. (17)Now from(16) p * − y n ≤ p * − x n . (18)Also, using u n = T r n y n , Lemma 2.3 and (18) we have u n p * − u n 2 .…”