Contraction of the proximal mapping and applications to the equilibrium problem

Contraction-Mapping Algorithm for the Equilibrium Problem Over the Fixed Point Set of a Nonexpansive Semigroup

“…We note that there exist several definitions of the Lipschitz-type continuity of bifunctions [1,4,8,15]. The Lipschitz-type condition used in this paper is a relaxation of the one introduced in [8].…”

“…Before proving the convergence theorem, we need the following lemma: Lemma 4. [8] Suppose that the bifunction f satisfies assumption (A2) and lim n→∞ λ n = 0, then for all x ∈ H, we have…”

“…Very recently, in [8], Hai introduced a contraction-mapping algorithm for solving (1.1). Under the assumptions that f is strongly monotone and Lipschitz type continuous, the author proved that the mapping U λ defined by…”

“…Motivated by the results in [8], we introduce a contraction algorithm for solving problem (1.3). In contrast to [8], the main contribution of this paper is twofold: the considered problem is more general and the convergence conditions are weakened.…”

“…Motivated by the results in [8], we introduce a contraction algorithm for solving problem (1.3). In contrast to [8], the main contribution of this paper is twofold: the considered problem is more general and the convergence conditions are weakened. In this paper, we assume that the bifunction f is Lipschitz-type continuous on bounded sets instead of the whole space H. Moreover, to prove the contraction of the proximal mapping U λ , we introduce a new type of the Lipschitz continuity, which is a relaxation of the one used in [8].…”