Abstract
The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature.
Highlights
Throughout, let H be a real Hilbert space endowed with inner product h·, ·i and induced norm p k · k(k x k = h x, x i, ∀ x ∈ H )
We propose an iterative algorithm for seeking a common solution of the pseudomonotone equilibrium problem and fixed point of pseudocontractive operators
We first present our algorithm to solve the pseudomonotone equilibrium problem and fixed point problem and, we prove the convergence of the suggested algorithm, see
Summary
Let ∅ 6= C ⊂ H be a closed and convex set. Recall that f is said to be monotone if f (u† , v† ) + f (v† , u† ) ≤ 0, ∀u† , v† ∈ C. F is said to be pseudomonotone if f (u† , v† ) ≥ 0 implies f (v† , u† ) ≤ 0, ∀u† , v† ∈ C. Our research is associated with the equilibrium problem [1] of seeking an element ũ ∈ C such that f (ũ, u) ≥ 0, ∀u ∈ C. The solution set of the equilibrium problem in Equation (3) is denoted by EP( f , C ). Equilibrium problems have been studied extensively in the literature (see, e.g., [2,3,4,5])
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