Hybrid iterative scheme for solving split equilibrium and hierarchical fixed point problems
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Hybrid iterative scheme for solving split equilibrium and hierarchical fixed point problems Monairah Alansari1 · K. R. Kazmi2,3 · Rehan Ali4 Received: 14 February 2019 / Accepted: 18 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We suggest and analyze a hybrid projected subgradient–proximal iterative scheme to approximate a common solution of a split equilibrium problem for pseudomonotone and monotone bifunctions and a hierarchical fixed point problem for nonexpansive and quasi-nonexpansive mappings. We prove that sequences generated by the proposed scheme converge weakly to a common solution of these problems. Further, we discuss some consequences of the main result and a numerical example for the proposed scheme. Keywords Split equilibrium problem · Hierarchical fixed point problem · Hybrid projected subgradient–proximal iterative scheme · Quasi-nonexpansive mapping · Pseudomonotone bifunction
1 Introduction Let H1 and H2 be two real Hilbert spaces and their inner products and induced norms are respectively, denoted by the notations ·, · and · . Let C and Q be nonempty, closed and convex subsets of H1 and H2 , respectively. Recall that a mapping T : C → C is nonexpansive if T x − T y ≤ x − y, for all x, y ∈ C. It is known that if
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K. R. Kazmi [email protected]; [email protected] Monairah Alansari [email protected] Rehan Ali [email protected]
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Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia
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Department of Mathematics, Faculty of Science and Arts - Rabigh, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Kingdom of Saudi Arabia
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Department of Mathematics, Aligarh Muslim University, Aligarh, India
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Department of Mathematics, Jamia Millia Islamia, New Delhi, India
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M. Alansari et al.
Fix(T ) := {x ∈ C : T x = x} = ∅ then Fix(T ) is closed and convex. The split equilibrium problem (in short, SpEP): Find x ∗ ∈ C such that f (x ∗ , x) ≥ 0, ∀x ∈ C,
(1.1)
y ∗ = Ax ∗ ∈ Q solves g(y ∗ , y) ≥ 0, ∀y ∈ Q,
(1.2)
and such that
where f : C × C → R and g : Q × Q → R are two bifunctions and A : H1 → H2 is a bounded linear operator. Sp EP(1.1)–(1.2) is initially given by Moudafi [18] and studied by Kazmi and Rizvi [11] for monotone bifunctions. Sp EP(1.1)–(1.2) include as special case, the split variational inequalities [6] and split feasibility problem [5] which have wide range of applications, see [2,3,6]. Sp EP(1.1)–(1.2) is a special class of split inverse problems. The inverse problems arise in signal processing, specially in phase retrieval and other image restoration problems, see for example [19]. In particular, if g = 0 and Q = H2 , then Sp EP(1.1)–(1.2) reduces to the equilibrium problem (in short, EP): Find x ∗ ∈ C such that f (x ∗ , x) ≥ 0, ∀x ∈ C.
(1.3)
EP(1.3) was introduced and studied by Blum and Oettli [1]. The solution set of EP(1.3) is denoted by Sol(EP(1.3)). The solution set of Sp EP(1.1)–(1.2) is denoted by = {x ∗ ∈ C : x ∗ ∈ Sol(EP(1.1)) and Ax ∗ ∈ Sol(EP(1.2)) }. Recently, Hieu [9
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