Hybrid inertial accelerated algorithms for split fixed point problems of demicontractive mappings and equilibrium proble
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Hybrid inertial accelerated algorithms for split fixed point problems of demicontractive mappings and equilibrium problems Adisak Hanjing1 · Suthep Suantai2 Received: 14 May 2018 / Accepted: 14 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Our contribution in this paper, we introduce and analyze two new hybrid algorithms by combining Mann iteration and inertial method for solving split fixed point problems of demicontractive mappings and equilibrium problems in a real Hilbert space. By using a new technique of choosing step size, our algorithms do not need any prior information on the operator norm. In fact, an inertial type algorithm was proposed in order to accelerate its convergence rate. We then prove weak and strong convergence of proposed methods under some control conditions. Moreover, some numerical experiments for image restoration problems and oligopolistic market equilibrium problems are also provided for supporting our main results. Keywords Demicontractive mappings · Inertial method · Split fixed point problem · Equilibrium problem · Hilbert spaces
1 Introduction It is well-known that the split fixed point problems (SFPP) include many important problems in nonlinear analysis and optimization problems such as image restoration, radiation therapy treatment planning, antenna design, and material science and
Suthep Suantai
[email protected] Adisak Hanjing adisak [email protected] 1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
2
Data Science Research Center, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand
Numerical Algorithms
computerized tomography (see [7, 10, 12, 14]). Various algorithms were invented to solve above problems above (see [8, 11, 13, 15, 25, 27, 28, 33, 34]). Let X be a real Hilbert space with inner product ·, · and norm · . Let I denote the identity mapping on X. The split fixed point problem (SFPP) for mappings T and S which was first introduced by Censor and Segal [15] is to find v ∗ ∈ F ix(S) such that Av ∗ ∈ F ix(T ),
(1)
where A : X1 → X2 is a bounded linear operator, S : X1 → X1 and T : X2 → X2 are two mappings with F ix(S) and F ix(T ) are nonempty, where F ix(S) = {x ∈ X1 : Sx = x} and F ix(T ) = {x ∈ X2 : T x = x}. Let C be a nonempty subset of X and g : C × C → R be a bifunction. The equilibrium problem is to find a point v ∗ ∈ C such that g(v ∗ , y) ≥ 0,
(2)
for all y ∈ C and the set of all solutions of (2) is denoted by EP (g). Equilibrium problems have been extensively studied by many authors because they include many important problems in nonlinear analysis and optimization such as the Nase equilibrium problems, variational inequalities problems, saddle point problems, and game theory, see [1, 6, 9, 16, 17, 20, 22, 30–32, 35] for examples. In 2010, Moudafi [27] introduced the following algorithm for solving (1) for two demicontractive mappings: ⎧ ⎨ x1 ∈ H1 choose arbitrarily, un = xn + γ A∗ (T − I )Axn , (3) ⎩ n
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