A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi‐Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

Abstract

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings

“…In the framework of Hilbert spaces, (asymptotically) quasi-j-nonexpansive mappings is reduced to (asymptotically) quasi-nonexpansive mappings (cf. [29][30][31][32]). …”

“…common element in the fixed point set of a relatively nonexpansive mapping and in the solution set of the equilibrium problem (1.1) (cf. [32]). …”

Modified Proximal point algorithms for finding a zero point of maximal monotone operators, generalized mixed equilibrium problems and variational inequalities

“…In 2000, Kamimura and Takahashi [19] proved the following strong convergence theorem in Hilbert spaces, by the following algorithm 12) where J r = (I + rA) -1 J, then the sequence {x n } converges strongly to P A −1 0 (x), where P A −1 0 is the projection from H onto A -1 (0). These results were extended to more general Banach spaces see [20,21].…”

“…Let {x n } be a sequence generated by x 1 = x C and, 12) for all n N, where Π C is the generalized projection from E onto C, J is the duality mapping on E. The coefficient sequence {a n } ⊂ [0, 1], {b n } ⊂ (0, 1], {r n } ⊂ (0, ∞) satisfying ∞ n=1 α n < ∞ , lim sup n ∞ b n < 1, lim inf n ∞ r n > 0 and {l n } ⊂ [a, b] for some a, b…”

The shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings

“…Many authors developed the shrinking projection method for solving (mixed) equilibrium problems and fixed point problems in Hilbert and Banch spaces; see, [12,15,16,[47][48][49][50][51][52][53][54][55][56][57] and references therein.…”