A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families o
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A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems L. O. Jolaoso1 · T. O. Alakoya1 · A. Taiwo1 · O. T. Mewomo1 Received: 16 December 2018 / Accepted: 19 June 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, we introduce a new parallel combination extragradient method for solving a finite family of pseudo-monotone equilibrium problems and finding a common fixed point of a finite family of demicontractive mappings in Hilbert space. The algorithm is designed such that at each iteration a single strongly convex program is solved and the stepsize is determined via an Armijo line searching technique. Also, the algorithm make a single projection onto a sub-level set which is constructed by the convex combination of finite convex functions. Under certain mild-conditions, we state and prove a strong convergence theorem for approximating a common solution of a finite family of equilibrium problems with pseudo-monotone bifunctions and a finite family of demicontractive mappings. Finally, we present numerical examples to illustrate the applicability of the algorithm proposed. This method improves many of the existing methods in the literature. Keywords Pseudo-monotone · Equilibrium problem · Extragradient method · Fixed point problem · Projection method · Iterative method Mathematics Subject Classification 65K15 · 47J25 · 65J15 · 90C33
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O. T. Mewomo [email protected] L. O. Jolaoso [email protected] T. O. Alakoya [email protected]; [email protected] A. Taiwo [email protected]
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School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
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L. O. Jolaoso et al.
1 Introduction Let C be a nonempty, closed and convex subset of a real Hilbert space H and g : C × C → R be a bifunction satisfying g(x, x) = 0 for every x ∈ C. The Equilibrium Problem (shortly, EP) is stated as follows: Find
x ∗ ∈ C such that g(x ∗ , y) ≥ 0 ∀ y ∈ C.
(1.1)
The set of solutions of (1.1) is denoted by E P(g). The bifunction g : C × C → R is said to be • monotone on C if g(x, y) + g(y, x) ≤ 0 ∀ x, y ∈ C, • pseudo-monotone on C if g(x, y) ≥ 0 ⇒ g(y, x) ≤ 0 ∀ x, y ∈ C. It is obvious that every monotone bifunction is pseudo-monotone but not vice-versa. The theory of EP has served as an important tool in studying a wide class of important nonlinear problems arising in several branches of pure and applied sciences in a unified and general framework (see, for e.g [3,4,8,16,25,30,31,34]). Several iterative methods have also been developed for solving EP and related optimization problems, see [1,6,24,26,27,29,40,42– 45,47,48,52]. Pseudo-monotone operators in the sense of Karamardian were introduced back in 1976 as a generalization of monotone operators. This has been studied for the last 40 years and has found many applications in variational inequality and economics. In case of gradient maps, this generalized monotonicity characterized
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