Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems

doi:10.1016/j.na.2009.02.106

Nonlinear Analysis: Theory, Methods & Applications

Volume 71, Issue 9, 1 November 2009, Pages 4203-4214

https://doi.org/10.1016/j.na.2009.02.106 Get rights and content

Abstract

The purpose of this work is to introduce a hybrid projection method for finding a common element of the set of a generalized equilibrium problem, the set of solutions to a variational inequality and the set of fixed points of a strict pseudo-contraction in a real Hilbert space.

Section snippets

Introduction and preliminaries

Let $H$ be a real Hilbert space, whose inner product and norm are denoted by $〈 \cdot, \cdot 〉$ and $‖ \cdot ‖$ , respectively. Let $C$ be a nonempty closed convex subset of $H$ and $B : C \to H$ a nonlinear mapping. Let $P_{C}$ be the projection of $H$ onto the closed convex subset $C$ .

Recall the following definitions:

(1) $B$ is said to be monotone if $〈 B x - B y, x - y 〉 \geq 0, \forall x, y \in C .$

(2) $B$ is said to be strongly monotone if there exists a constant $α > 0$ such that $〈 B x - B y, x - y 〉 \geq α {‖ x - y ‖}^{2}, \forall x, y \in C .$

For such a case, $B$ is said to be $α$ -strongly-monotone.

(3) $B$ is

Main results

Now, we are ready to give our main results in this paper.

Theorem 2.1

Let $C$ be a closed convex subset of a real Hilbert space $H$ and let $f : C \times C \to R$ be a bifunction satisfying (A1)–(A4) . Let $A$ be an $α$ -inverse-strongly monotone mapping of $C$ into $H$ and $B$ be a $β$ -inverse-strongly monotone mapping of $C$ into $H$ , respectively. Let $S : C \to C$ be a $k$ -strict pseudo-contraction with a fixed point. Define a mapping $S_{k} : C \to C$ by $S_{k} x = k x + (1 - k) S x$ for all $x \in C$ . Assume that $F ≔ F (S) \cap V I (C, B) \cap E P (f, A) \neq 0̸$ . Let ${x_{n}}$ be a sequence generated by

Applications

In this section, we prove some strong convergence theorems by using Theorem 2.1.

Theorem 3.1

Let $C$ be a closed convex subset of a real Hilbert space $H$ and $f : C \times C \to R$ be a bifunction satisfying (A1)–(A4) . Let $A$ be an $α$ -inverse-strongly monotone mapping of $C$ into $H$ and $S : C \to C$ be a $k$ -strict pseudo-contraction with a fixed point. Define a mapping $S_{k} : C \to C$ by $S_{k} x = k x + (1 - k) S x$ for all $x \in C$ . Assume that $F ≔ F (S) \cap E P (f, A) \neq 0̸$ . Let ${x_{n}}$ be a sequence generated by the following algorithm: ${\begin{cases} x_{1} \in C, \\ C_{1} = C, \\ f (u_{n}, y) + 〈 A x_{n}, y - u_{n} 〉 + \frac{1}{r_{n}} 〈 y - u_{n}, u \end{cases}$

Acknowledgement

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

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Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems

Abstract

Section snippets

Introduction and preliminaries

Main results

Applications

Acknowledgement

Nonlinear Anal.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

J. Comput. Appl. Math.

Appl. Math. Comput.

Nonlinear Anal.

J. Math. Anal. Appl.

Appl. Math. Comput.

J. Math. Anal. Appl.

Math. Comput. Modeling

J. Comput. Appl. Math.

Nonlinear Anal.

J. Math. Anal. Appl.

Nonlinear Anal.