Nonlinear Analysis: Theory, Methods & Applications
Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems
Section snippets
Introduction and preliminaries
Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of and a nonlinear mapping. Let be the projection of onto the closed convex subset .
Recall the following definitions:
(1) is said to be monotone if
(2) is said to be strongly monotone if there exists a constant such that
For such a case, is said to be -strongly-monotone.
(3) is
Main results
Now, we are ready to give our main results in this paper.
Theorem 2.1 Let be a closed convex subset of a real Hilbert space and let be a bifunction satisfying (A1)–(A4) . Let be an -inverse-strongly monotone mapping of into and be a -inverse-strongly monotone mapping of into , respectively. Let be a -strict pseudo-contraction with a fixed point. Define a mapping by for all . Assume that . Let be a sequence generated by
Applications
In this section, we prove some strong convergence theorems by using Theorem 2.1.
Theorem 3.1 Let be a closed convex subset of a real Hilbert space and be a bifunction satisfying (A1)–(A4) . Let be an -inverse-strongly monotone mapping of into and be a -strict pseudo-contraction with a fixed point. Define a mapping by for all . Assume that . Let be a sequence generated by the following algorithm:
Acknowledgement
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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