Viscosity iterative process for demicontractive mappings and multivalued mappings and equilibrium problems
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Viscosity iterative process for demicontractive mappings and multivalued mappings and equilibrium problems M. Eslamian1 · R. Saadati2 · J. Vahidi2
Received: 7 November 2013 / Revised: 14 October 2015 / Accepted: 19 October 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract In this paper, we introduce a general iterative scheme based on the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of all fixed points of a demicontractive mapping and a generalized nonexpansive multivalued mapping. Then, we prove the strong convergence of the iterative scheme to find a unique solution of the variational inequality which is the optimality condition for the minimization problem. The main results presented in this paper extend various results existing in the current literature. Keywords Demicontractive mapping · Equilibrium problem · Fixed point · Generalized nonexpansive multivalued mapping Mathematics Subject Classification
47H10 · 47H09
1 Introduction Let H be a real Hilbert space. For a mapping T : H → H, the fixed point set of T is denoted by F(T ), that is, F(T ) = {x ∈ H : x = T x}. A mapping T : H → H is said to be demicontractive mapping Hicks and Kubicek (1977), Naimpally and Singh (1983) if there exists k ∈ [0, 1) such that T x − p2 ≤ x − p2 + kx − T x2 ,
∀x ∈ H, ∀ p ∈ F(T ).
Communicated by Carlos Conca.
B
M. Eslamian [email protected]
1
Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran
2
Department of Applied Mathematics, Iran University of Science and Technology, Tehran, Iran
123
M. Eslamian et al.
A mapping T : H → H is called nonexpansive if T x − T y ≤ x − y,
∀x, y ∈ H.
A mapping T : H → H is called quasi-nonexpansive if F(T ) = ∅ and T x − p ≤ x − p,
∀x ∈ H, ∀ p ∈ F(T ).
It is also clear from the definitions that every quasi-nonexpansive mapping is demicontractive. In 2010, Kohsaka and Takahashi (2008a, b) introduced the following nonlinear mappping. Let E be a smooth, strictly convex and reflexive Banach space, let J be the duality mapping of E and let C be a nonempty closed convex subset of E. Then, a mapping T : C → C is said to be nonspreading if φ(T x, T y) + φ(T y, T x) ≤ φ(T x, y) + φ(T y, x) for all x, y ∈ C, where φ(x, y) = x2 − 2 x, J (y) + y2 , x, y ∈ E. They considered the class of nonspreading mappings to study the resolvents of a maximal monotone operators in the Banach space. In the case when E is a Hilbert space, we know that φ(x, y) = x − y2 for all x, y ∈ E. So, a nonspreading mapping T : C → C in a Hilbert space H is defined as follows: 2 T x − T y2 ≤ T x − y2 + T y − x2
∀x, y ∈ C.
Iemoto and Takahashi (2009) proved that T : C → C is nonspreading if and only if T x − T y2 ≤ x − y2 + 2 x − T x, y − T y
∀x, y ∈ C.
Observe that if T is nonspreading and F(T ) = ∅, then T is quasi-nonexpansive and hence F(T ) is closed and convex. Recently, Osilike and Isiogugu (2011) introduced a n
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