Abstract
We consider the split generalized equilibrium problem and the fixed point problem for a countable family of nonexpansive multivalued mappings in real Hilbert spaces. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a common solution of the considered problems. A numerical example is presented to illustrate the convergence result. Our results improve and extend the corresponding results in the literature.
Highlights
Let H be a real Hilbert space with inner product ·, · and induced norm ·
Motivated and inspired by the results mentioned and related literature, we propose an iterative algorithm based on the shrinking projection method for finding a common element of the set of solutions of split generalized equilibrium problems and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces
3 Main results we prove strong convergence theorems for finding a common element of the set of solutions of split generalized equilibrium problems and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces and give a numerical example to support our main result
Summary
Let H be a real Hilbert space with inner product ·, · and induced norm ·. Motivated and inspired by the results mentioned and related literature, we propose an iterative algorithm based on the shrinking projection method for finding a common element of the set of solutions of split generalized equilibrium problems and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. Lemma 2.5 ([22]) Let C be a nonempty closed convex subset of a real Hilbert space H. We have: (6) For each v ∈ H2, Ts(F2,φ2) = ∅, (7) Ts(F2,φ2) is single-valued, (8) Ts(F2,φ2) is firmly nonexpansive, (9) F(Ts(F2,φ2)) = GEP(F2, φ2), (10) GEP(F2, φ2) is closed and convex, where GEP(F2, φ2) is the solution set of the following generalized equilibrium problem: Find y∗ ∈ Q such that F2(y∗, y) + φ2(y∗, y) ≥ 0 for all y ∈ Q. By Assumption 2.7, (A1)–(A7), we have 0 ≤ F1(yt, yt) + φ1(yt, yt) ≤ t F1(yt, y) + φ1(yt, y) + (1 – t) F1(yt, q) + φ1(yt, q) ≤ t F1(yt, y) + φ1(yt, y) + (1 – t) F1(q, yt) + φ1(q, yt) ≤ F1(yt, y) + φ1(yt, y)
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