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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 414831, 35 pages
doi:10.1155/2012/414831
Research Article

A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

Poom Kumam1,2 and Phayap Katchang2,3

1Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

Received 17 November 2011; Accepted 23 January 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Poom Kumam and Phayap Katchang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨ ⋅ , ⋅ ⟩ and norm ‖ ⋅ ‖ . Let 𝐶 be a nonempty closed convex subset of 𝐻 . Recall that a mapping 𝑇 ∶ 𝐶 → 𝐶 is nonexpansive if ‖ 𝑇 𝑥 − 𝑇 𝑦 ‖ ≤ ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 . ( 1 . 1 ) We denote the set of fixed points of 𝑇 by 𝐹 ( 𝑇 ) , that is 𝐹 ( 𝑇 ) = { 𝑥 ∈ 𝐶 ∶ 𝑥 = 𝑇 𝑥 } . A mapping 𝑓 ∶ 𝐶 → 𝐶 is said to be an 𝛼 -contraction if there exists a coefficient 𝛼 ∈ ( 0 , 1 ) such that ‖ 𝑓 ( 𝑥 ) − 𝑓 ( 𝑦 ) ‖ ≤ 𝛼 ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 . ( 1 . 2 ) Let 𝐵 ∶ 𝐶 → 𝐻 be a mapping. Then 𝐵 is called:(1)monotone if ⟨ 𝐵 𝑥 − 𝐵 𝑦 , 𝑥 − 𝑦 ⟩ ≥ 0 , ∀ 𝑥 , 𝑦 ∈ 𝐶 ; ( 1 . 3 ) (2) 𝑑 -strongly monotone if there exists a positive real number 𝑑 such that ⟨ 𝐵 𝑥 − 𝐵 𝑦 , 𝑥 − 𝑦 ⟩ ≥ 𝑑 ‖ 𝑥 − 𝑦 ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 1 . 4 ) for constant 𝑑 > 0 , this implies that ‖ 𝐵 𝑥 − 𝐵 𝑦 ‖ ≥ 𝑑 ‖ 𝑥 − 𝑦 ‖ , ( 1 . 5 ) that is, 𝐵 is 𝑑 -expansive and when 𝑑 = 1 , it is expansive;(3) 𝐿 -Lipschitz continuous if there exists a positive real number 𝐿 such that ‖ 𝐵 𝑥 − 𝐵 𝑦 ‖ ≤ 𝐿 ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 ; ( 1 . 6 ) (4) 𝑐 -cocoercive [1, 2] if there exists a positive real number 𝑐 such that ⟨ 𝐵 𝑥 − 𝐵 𝑦 , 𝑥 − 𝑦 ⟩ ≥ 𝑐 ‖ 𝐵 𝑥 − 𝐵 𝑦 ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 1 . 7 ) Clearly, every 𝑐 -cocoercive map 𝐵 is ( 1 / 𝑐 ) -Lipschitz continuous;(5)relaxed 𝑐 -cocoercive, if there exists a positive real number 𝑐 such that ⟨ 𝐵 𝑥 − 𝐵 𝑦 , 𝑥 − 𝑦 ⟩ ≥ ( − 𝑐 ) ‖ 𝐵 𝑥 − 𝐵 𝑦 ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐶 ; ( 1 . 8 ) (6)relaxed ( 𝑐 , 𝑑 ) -cocoercive, if there exists a positive real number 𝑐 , 𝑑 such that ⟨ 𝐵 𝑥 − 𝐵 𝑦 , 𝑥 − 𝑦 ⟩ ≥ ( − 𝑐 ) ‖ 𝐵 𝑥 − 𝐵 𝑦 ‖ 2 + 𝑑 ‖ 𝑥 − 𝑦 ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 1 . 9 ) for 𝑐 = 0 , 𝐵 is 𝑑 -strongly monotone. This class of mapping is more general than the class of strongly monotone mapping. It is easy to see that we have the following implication: 𝑑 -strongly monotonicity implying relaxed ( 𝑐 , 𝑑 ) -cocoercivity,(7) 𝑘 -strictly pseudocontractive, if there exists a constant 𝑘 ∈ [ 0 , 1 ) such that ‖ 𝐵 𝑥 − 𝐵 𝑦 ‖ 2 ≤ ‖ 𝑥 − 𝑦 ‖ 2 ‖ + 𝑘 ‖ ( 𝐼 − 𝐵 ) 𝑥 − ( 𝐼 − 𝐵 ) 𝑦 2 , ∀ 𝑥 , 𝑦 ∈ 𝐶 . ( 1 . 1 0 )

Remark 1.1 (see [3, Remark  1.1 pages 135-136]). If 𝐵 ∶ 𝐶 → 𝐻 is a 𝐿 𝐵 -Lipschitz continuous and relaxed ( 𝑐 , 𝑑 ) -cocoercive mapping with 𝑑 > 𝑐 𝐿 2 𝐵 and 0 < 𝜏 < 2 ( 𝑑 − 𝑐 𝐿 2 𝐵 ) / 𝐿 2 𝐵 , then 𝐼 − 𝜏 𝐵 satisfies the following: ‖ ( 𝐼 − 𝜏 𝐵 ) 𝑥 − ( 𝐼 − 𝜏 𝐵 ) 𝑦 ‖ ≤ ( 1 − 𝜏 𝜉 ) ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 1 . 1 1 ) where 𝜉 = ( 𝐿 2 𝐵 / 2 ) [ 2 ( 𝑑 − 𝑐 𝐿 2 𝐵 ) / 𝐿 2 𝐵 − 𝜏 ] .
Similarly, if 𝐷 ∶ 𝐶 → 𝐻 i s 𝐿 𝐷 -Lipschitz continuous and relaxed ( 𝑐 ′ , 𝑑 ′ ) -cocoercive mapping with 𝑑 ′ > 𝑐 ′ 𝐿 2 𝐷 and 0 < 𝛿 < 2 ( 𝑑 ′ − 𝑐 ′ 𝐿 2 𝐷 ) / 𝐿 2 𝐷 , then the mapping 𝐼 − 𝛿 𝐷 satisfies the following: ‖ ( 𝐼 − 𝛿 𝐷 ) 𝑥 − ( 𝐼 − 𝛿 𝐷 ) 𝑦 ‖ ≤ 1 − 𝛿 𝜉 ‖ 𝑥 − 𝑦 ‖ , ( 1 . 1 2 ) where 𝜉 ′ = ( 𝐿 2 𝐷 / 2 ) [ 2 ( 𝑑 ′ − 𝑐 ′ 𝐿 2 𝐷 ) / 𝐿 2 𝐷 − 𝛿 ] .

Let 𝐴 be a strongly positive linear bounded operator on 𝐻 if there is a constant 𝛾 > 0 with the property ⟨ 𝐴 𝑥 , 𝑥 ⟩ ≥ 𝛾 ‖ 𝑥 ‖ 2 , ∀ 𝑥 ∈ 𝐻 . ( 1 . 1 3 ) We recall optimization problem (for short, OP) as the following m i n 𝑥 ∈ 𝐹 𝜇 2 1 ⟨ 𝐴 𝑥 , 𝑥 ⟩ + 2 ‖ 𝑥 − 𝑢 ‖ 2 − ℎ ( 𝑥 ) , ( 1 . 1 4 ) where 𝐹 = ∩ ∞ 𝑛 = 1 𝐶 𝑛 , 𝐶 1 , 𝐶 2 , … are infinitely closed convex subsets of 𝐻 such that ∩ ∞ 𝑛 = 1 𝐶 𝑛 ≠ ∅ , 𝑢 ∈ 𝐻 , 𝜇 ≥ 0 is a real number, 𝐴 is a strongly positive linear bounded operator on 𝐻 , and ℎ is a potential function for 𝛾 𝑓 (i.e., ℎ ′ ( 𝑥 ) = 𝛾 𝑓 ( 𝑥 ) for 𝑥 ∈ 𝐻 ). This kind of optimization problem has been studied extensively by many authors, see, for example, [4–7] when 𝐹 = ∩ ∞ 𝑛 = 1 𝐶 𝑛 and ℎ ( 𝑥 ) = ⟨ 𝑥 , 𝑏 ⟩ , where 𝑏 is a given point in 𝐻 .

On the other hand, a family 𝒮 = { 𝑆 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } of mappings of 𝐶 into itself is called a nonexpansive semigroup on 𝐶 if it satisfies the following conditions:(i) 𝑆 ( 0 ) 𝑥 = 𝑥 for all 𝑥 ∈ 𝐶 ;(ii) 𝑆 ( 𝑠 + 𝑡 ) = 𝑆 ( 𝑠 ) 𝑆 ( 𝑡 ) for all 𝑠 , 𝑡 ≥ 0 ;(iii) ‖ 𝑆 ( 𝑠 ) 𝑥 − 𝑆 ( 𝑠 ) 𝑦 ‖ ≤ ‖ 𝑥 − 𝑦 ‖ for all 𝑥 , 𝑦 ∈ 𝐶 and 𝑠 ≥ 0 ;(iv)for all 𝑥 ∈ 𝐶 , 𝑠 ↦ 𝑆 ( 𝑠 ) 𝑥 is continuous.

We denote by 𝐹 ( 𝒮 ) the set of all common fixed points of 𝒮 = { 𝑆 ( 𝑠 ) ∶ 𝑠 ≥ 0 } , that is, 𝐹 ( 𝒮 ) = ∩ 𝑠 ≥ 0 𝐹 ( 𝑆 ( 𝑠 ) ) . It is known that 𝐹 ( 𝒮 ) is closed and convex.

Let 𝜙 ∶ 𝐶 → ℝ be a real-valued function and let { Θ 𝑘 ∶ 𝐶 × 𝐶 → ℝ , 𝑘 = 1 , 2 , … , 𝑁 } be a finite family of equilibrium functions, that is, Θ 𝑘 ( 𝑢 , 𝑢 ) = 0 for each 𝑢 ∈ 𝐶 . The system of mixed equilibrium problems (for short, SMEP) for function ( Θ 1 , Θ 2 , … , Θ 𝑁 , 𝜙 ) is to find 𝑧 ∈ 𝐶 such that Θ 1 Θ ( 𝑧 , 𝑦 ) + 𝜙 ( 𝑦 ) − 𝜙 ( 𝑧 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 , 2 ⋮ Θ ( 𝑧 , 𝑦 ) + 𝜙 ( 𝑦 ) − 𝜙 ( 𝑧 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 , 𝑁 ( 𝑧 , 𝑦 ) + 𝜙 ( 𝑦 ) − 𝜙 ( 𝑧 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 . ( 1 . 1 5 ) The set of solutions of (1.15) is denoted by ∩ 𝑁 𝑘 = 1 M E P ( Θ 𝑘 , 𝜙 ) , where M E P ( Θ 𝑘 , 𝜙 ) is the set of solutions of the mixed equilibrium problem (for short, MEP), which is to find 𝑧 ∈ 𝐶 such that Θ 𝑘 ( 𝑧 , 𝑦 ) + 𝜙 ( 𝑦 ) − 𝜙 ( 𝑧 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 . ( 1 . 1 6 ) In particular, if 𝜙 ≡ 0 , and 𝑁 = 1 , then the problem (1.15) reduces to the equilibrium problem (for short, EP), which is to find 𝑧 ∈ 𝐶 such that Θ ( 𝑧 , 𝑦 ) ≥ 0 , ∀ 𝑦 ∈ 𝐶 . ( 1 . 1 7 ) It is well known that the SMEP includes fixed point problem, optimization problem, variational inequality problem, and Nash equilibrium problem as its special cases (see [8–13] for more details).

For solving the solutions of a nonexpansive semigroup and the solutions of the system of mixed equilibrium problems were studied by many authors see [14–23] and reference therein. In 2010, Chang et al. [24] studied the following approximation method: Θ 1 𝑢 𝑛 ( 1 ) 𝑢 , 𝑥 + 𝜙 ( 𝑥 ) − 𝜙 𝑛 ( 1 ) + 1 𝑟 1 𝐾 𝑢 𝑛 ( 1 ) − 𝐾 𝑥 𝑛 , 𝜂 𝑥 , 𝑢 𝑛 ( 1 ) Θ ≥ 0 , ∀ 𝑥 ∈ 𝐶 , 2 𝑢 𝑛 ( 2 ) 𝑢 , 𝑥 + 𝜙 ( 𝑥 ) − 𝜙 𝑛 ( 2 ) + 1 𝑟 2 𝐾 𝑢 𝑛 ( 2 ) − 𝐾 𝑥 𝑛 , 𝜂 𝑥 , 𝑢 𝑛 ( 2 ) ⋮ Θ ≥ 0 , ∀ 𝑥 ∈ 𝐶 , 𝑁 𝑢 𝑛 ( 𝑁 ) 𝑢 , 𝑥 + 𝜙 ( 𝑥 ) − 𝜙 𝑛 ( 𝑁 ) + 1 𝑟 𝑁 𝐾 𝑢 𝑛 ( 𝑁 ) − 𝐾 𝑥 𝑛 , 𝜂 𝑥 , 𝑢 𝑛 ( 𝑁 ) 𝑥 ≥ 0 , ∀ 𝑥 ∈ 𝐶 , 𝑛 + 1 = 𝛼 𝑛 𝑓 𝑊 𝑛 𝑥 𝑛 + 𝛽 𝑛 𝑥 𝑛 + 𝛾 𝑛 1 𝑡 𝑛 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑢 𝑛 ( 𝑁 ) 𝑑 𝑠 , ( 1 . 1 8 ) where 𝑢 𝑛 ( 1 ) = 𝐽 Θ 1 𝑟 1 𝑥 𝑛 , 𝑢 𝑛 ( 𝑘 ) = 𝐽 Θ 𝑘 𝑟 𝑘 𝑢 𝑛 ( 𝑘 − 1 ) = 𝐽 Θ 𝑘 𝑟 𝑘 𝐽 Θ 𝑘 − 1 𝑟 𝑘 − 1 𝑢 𝑛 ( 𝑘 − 2 ) = 𝐽 Θ 𝑘 𝑟 𝑘 ⋯ 𝐽 Θ 2 𝑟 2 𝑢 𝑛 ( 1 ) , = 𝐽 Θ 𝑘 𝑟 𝑘 ⋯ 𝐽 Θ 2 𝑟 2 𝐽 Θ 1 𝑟 1 𝑥 𝑛 , 𝑘 = 2 , 3 , … , 𝑁 , ( 1 . 1 9 ) 𝐽 Θ 𝑘 𝑟 𝑘 ∶ 𝐶 → 𝐶 , 𝑘 = 1 , 2 , … , 𝑁 is the mapping defined by (2.22) below, 𝑊 𝑛 is the mapping defined by (2.12), and 𝒮 = { 𝑆 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } is a nonexpansive semigroup. They proved that { 𝑥 𝑛 } converges strongly to a fixed point of 𝐹 ( 𝒮 ) ∩ 𝐹 ( 𝑊 ) ∩ ( ∩ 𝑁 𝑘 = 1 M E P ( Θ 𝑘 , 𝜙 ) ) under control conditions on the parameters.

Let 𝐵 , 𝐷 ∶ 𝐶 → 𝐻 be two mappings. The general system of variational inequalities problem (see [25]) is to find ( 𝑥 ∗ , 𝑦 ∗ ) ∈ 𝐶 × 𝐶 such that ⟨ 𝜏 𝐵 𝑦 ∗ + 𝑥 ∗ − 𝑦 ∗ , 𝑥 − 𝑥 ∗ ⟩ ≥ 0 , ∀ 𝑥 ∈ 𝐶 , ⟨ 𝛿 𝐷 𝑥 ∗ + 𝑦 ∗ − 𝑥 ∗ , 𝑥 − 𝑦 ∗ ⟩ ≥ 0 , ∀ 𝑥 ∈ 𝐶 , ( 1 . 2 0 ) where 𝜏 and 𝛿 are two positive real numbers. The set of solutions of the general system of variational inequalities problem is denoted by S V I ( 𝐶 , 𝐵 , 𝐷 ) . In particular, if 𝐵 = 𝐷 , then the problem (1.20) reduces to the following equation: ⟨ 𝜏 𝐵 𝑦 ∗ + 𝑥 ∗ − 𝑦 ∗ , 𝑥 − 𝑥 ∗ ⟩ ≥ 0 , ∀ 𝑥 ∈ 𝐶 , ⟨ 𝛿 𝐵 𝑥 ∗ + 𝑦 ∗ − 𝑥 ∗ , 𝑥 − 𝑦 ∗ ⟩ ≥ 0 , ∀ 𝑥 ∈ 𝐶 , ( 1 . 2 1 ) which is defined by Verma [26] (see also Verma [27]), and is called the new system of variational inequalities. Further, if we set 𝐷 = 0 , then problem (1.20) reduces to the classical variational inequality is to find 𝑥 ∗ ∈ 𝐶 such that ⟨ 𝐵 𝑥 ∗ , 𝑥 − 𝑥 ∗ ⟩ ≥ 0 , ∀ 𝑥 ∈ 𝐶 . ( 1 . 2 2 ) We denoted by V I ( 𝐶 , 𝐵 ) the set of solutions of the variational inequality problem. The variational inequality problem has been extensively studied in literature, see, for example, [28–31] and references therein. In order to find the solutions of the general system of variational inequality problem (1.20), Wangkeeree and Kamraksa [32] considered the following iterative algorithm: Θ 𝑢 𝑛 𝑢 , 𝑥 + 𝜙 ( 𝑥 ) − 𝜙 𝑛 + 1 𝑟 𝐾 𝑢 𝑛 − 𝐾 𝑥 𝑛 , 𝜂 𝑥 , 𝑢 𝑛 𝑧 ≥ 0 , ∀ 𝑥 ∈ 𝐶 , 𝑛 = 𝑃 𝐶 𝑢 𝑛 − 𝛿 𝐷 𝑢 𝑛 , 𝑥 𝑛 + 1 = 𝛼 𝑛 𝑥 𝛾 𝑓 𝑛 + 𝛽 𝑛 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 𝐴 𝑊 𝑛 𝑃 𝐶 𝑧 𝑛 − 𝜏 𝐵 𝑧 𝑛 , ( 1 . 2 3 ) where 𝐵 , 𝐷 ∶ 𝐶 → 𝐻 is a 𝐿 𝐵 -Lipschitz continuous and relaxed ( 𝑐 , 𝑑 ) -cocoercive mapping and 𝐿 𝐷 -Lipschitz continuous and relaxed ( 𝑐 ′ , 𝑑 ′ ) -cocoercive mapping, respectively. They proved that { 𝑥 𝑛 } converges strongly to a fixed point of 𝐹 ( 𝑊 𝑛 ) ∩ M E P ( Θ , 𝜙 ) ∩ S V I ( 𝐶 , 𝐵 , 𝐷 ) which is a solution of general system of variational inequality (1.20). Very recently, Jaiboon and Kumam [33] studied a new general iterative method for finding a common element of the set of solution of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces, which solves some optimization problems.

Inspired and motivated by Chang et al. [24], Jaiboon and Kumam [33], Kumam and Jaiboon [34] and Wangkeeree and Kamraksa [32], the purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroup, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings. Consequently, we prove the strong convergence theorem in Hilbert spaces under control conditions on the parameters. Furthermore, we can apply our results for solving some optimization problems. Our results extend and improve the corresponding results in Chang et al. [24], Kumam and Jaiboon [34], Wangkeeree and Kamraksa [32], and many others.

2. Preliminaries

Let 𝐻 a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻 . We denote strong convergence (weak convergence) by notation → ( ⇀ ) . In a real Hilbert space 𝐻 , it is well known that ‖ 𝑥 − 𝑦 ‖ 2 = ‖ 𝑥 ‖ 2 − ‖ 𝑦 ‖ 2 − 2 ⟨ 𝑥 − 𝑦 , 𝑦 ⟩ , ( 2 . 1 ) ‖ 𝑥 + 𝑦 ‖ 2 ≤ ‖ 𝑥 ‖ 2 + 2 ⟨ 𝑦 , 𝑥 + 𝑦 ⟩ , ( 2 . 2 ) ‖ 𝑥 + 𝑦 ‖ 2 ≥ ‖ 𝑥 ‖ 2 + 2 ⟨ 𝑦 , 𝑥 ⟩ , ( 2 . 3 ) ‖ 𝜆 𝑥 + ( 1 − 𝜆 ) 𝑦 ‖ 2 = 𝜆 ‖ 𝑥 ‖ 2 + ( 1 − 𝜆 ) ‖ 𝑦 ‖ 2 − 𝜆 ( 1 − 𝜆 ) ‖ 𝑥 − 𝑦 ‖ 2 ( 2 . 4 ) for all 𝑥 , 𝑦 ∈ 𝐻 and 𝜆 ∈ ℝ .

Recall that for every point 𝑥 ∈ 𝐻 , there exists a unique nearest point in 𝐶 , denoted by 𝑃 𝐶 𝑥 , such that ‖ ‖ 𝑥 − 𝑃 𝐶 𝑥 ‖ ‖ ≤ ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑦 ∈ 𝐶 . ( 2 . 5 ) 𝑃 𝐶 is called the metric projection of 𝐻 onto 𝐶 . It is well known that 𝑃 𝐶 is a nonexpansive mapping of 𝐻 onto 𝐶 and satisfies ⟨ 𝑥 − 𝑦 , 𝑃 𝐶 𝑥 − 𝑃 𝐶 ‖ ‖ 𝑃 𝑦 ⟩ ≥ 𝐶 𝑥 − 𝑃 𝐶 𝑦 ‖ ‖ 2 ( 2 . 6 ) for every 𝑥 , 𝑦 ∈ 𝐻 . Obviously, this immediately implies that ‖ ‖ 𝑃 ( 𝑥 − 𝑦 ) − 𝐶 𝑥 − 𝑃 𝐶 𝑦 ‖ ‖ 2 ≤ ‖ 𝑥 − 𝑦 ‖ 2 − ‖ ‖ 𝑃 𝐶 𝑥 − 𝑃 𝐶 𝑦 ‖ ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐻 . ( 2 . 7 ) Moreover, 𝑃 𝐶 𝑥 is characterized by the following properties: 𝑃 𝐶 𝑥 ∈ 𝐶 and ⟨ 𝑥 − 𝑃 𝐶 𝑥 , 𝑦 − 𝑃 𝐶 𝑥 ⟩ ≤ 0 , ‖ 𝑥 − 𝑦 ‖ 2 ≥ ‖ ‖ 𝑥 − 𝑃 𝐶 𝑥 ‖ ‖ 2 + ‖ ‖ 𝑦 − 𝑃 𝐶 𝑥 ‖ ‖ 2 ( 2 . 8 ) for all 𝑥 ∈ 𝐻 , 𝑦 ∈ 𝐶 .

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [35]). Let 𝑉 ∶ 𝐶 → 𝐻 be a k -strict pseudo-contraction, then(1)the fixed point set 𝐹 ( 𝑉 ) of 𝑉 is closed convex so that the projection 𝑃 𝐹 ( 𝑉 ) is well defined;(2)define a mapping 𝑇 ∶ 𝐶 → 𝐻 by 𝑇 𝑥 = 𝑡 𝑥 + ( 1 − 𝑡 ) 𝑉 𝑥 , ∀ 𝑥 ∈ 𝐶 . ( 2 . 9 ) If 𝑡 ∈ [ 𝑘 , 1 ) , then 𝑇 is a nonexpansive mapping such that 𝐹 ( 𝑉 ) = 𝐹 ( 𝑇 ) .

A family of mappings { 𝑉 𝑖 ∶ 𝐶 → 𝐻 } ∞ 𝑖 = 1 is called a family of uniformly 𝑘 -strict pseudo-contractions, if there exists a constant 𝑘 ∈ [ 0 , 1 ) such that ‖ ‖ 𝑉 𝑖 𝑥 − 𝑉 𝑖 𝑦 ‖ ‖ 2 ≤ ‖ 𝑥 − 𝑦 ‖ 2 ‖ ‖ + 𝑘 𝐼 − 𝑉 𝑖 𝑥 − 𝐼 − 𝑉 𝑖 𝑦 ‖ ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ∀ 𝑖 ≥ 1 . ( 2 . 1 0 ) Let { 𝑉 𝑖 ∶ 𝐶 → 𝐶 } ∞ 𝑖 = 1 be a countable family of uniformly 𝑘 -strict pseudo-contractions. Let { 𝑇 𝑖 ∶ 𝐶 → 𝐶 } ∞ 𝑖 = 1 be the sequence of nonexpansive mappings defined by (2.9), that is, 𝑇 𝑖 𝑥 = 𝑡 𝑥 + ( 1 − 𝑡 ) 𝑉 𝑖 [ 𝑥 , ∀ 𝑥 ∈ 𝐶 , ∀ 𝑖 ≥ 1 , 𝑡 ∈ 𝑘 , 1 ) . ( 2 . 1 1 )

Let { 𝑇 𝑖 } be a sequence of nonexpansive mappings of 𝐶 into itself defined by (2.11) and let { 𝜇 𝑖 } be a sequence of nonnegative numbers in [ 0 , 1 ] . For each 𝑛 ≥ 1 , define a mapping 𝑊 𝑛 of 𝐶 into itself as follows: 𝑈 𝑛 , 𝑛 + 1 𝑈 = 𝐼 , 𝑛 , 𝑛 = 𝜇 𝑛 𝑇 𝑛 𝑈 𝑛 , 𝑛 + 1 + 1 − 𝜇 𝑛 𝑈 𝐼 , 𝑛 , 𝑛 − 1 = 𝜇 𝑛 − 1 𝑇 𝑛 − 1 𝑈 𝑛 , 𝑛 + 1 − 𝜇 𝑛 − 1 ⋮ 𝑈 𝐼 , 𝑛 , 𝑘 = 𝜇 𝑘 𝑇 𝑘 𝑈 𝑛 , 𝑘 + 1 + 1 − 𝜇 𝑘 𝑈 𝐼 , 𝑛 , 𝑘 − 1 = 𝜇 𝑘 − 1 𝑇 𝑘 − 1 𝑈 𝑛 , 𝑘 + 1 − 𝜇 𝑘 − 1 ⋮ 𝑈 𝐼 , 𝑛 , 2 = 𝜇 2 𝑇 2 𝑈 𝑛 , 3 + 1 − 𝜇 2 𝑊 𝐼 , 𝑛 = 𝑈 𝑛 , 1 = 𝜇 1 𝑇 1 𝑈 𝑛 , 2 + 1 − 𝜇 1 𝐼 . ( 2 . 1 2 ) Such a mapping 𝑊 𝑛 is nonexpansive from 𝐶 to 𝐶 and it is called the 𝑊 -mapping generated by 𝑇 1 , 𝑇 2 , … , 𝑇 𝑛 and 𝜇 1 , 𝜇 2 , … , 𝜇 𝑛 .

For each 𝑛 , 𝑘 ∈ ℕ , let the mapping 𝑈 𝑛 , 𝑘 be defined by (2.12). Then we can have the following crucial conclusions concerning 𝑊 𝑛 . You can find them in [36]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.2 (see [36]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 . Let 𝑇 1 , 𝑇 2 , … be nonexpansive mappings of 𝐶 into itself such that ∩ ∞ 𝑛 = 1 𝐹 ( 𝑇 𝑛 ) is nonempty, let 𝜇 1 , 𝜇 2 , … be real numbers such that 0 ≤ 𝜇 𝑛 ≤ 𝑏 < 1 for every 𝑛 ≥ 1 . Then,(1) 𝑊 𝑛 is nonexpansive and 𝐹 ( 𝑊 𝑛 ) = ∩ 𝑛 𝑖 = 1 𝐹 ( 𝑇 𝑖 ) , for all 𝑛 ≥ 1 ;(2)for every 𝑥 ∈ 𝐶 and 𝑘 ∈ ℕ , the limit l i m 𝑛 → ∞ 𝑈 𝑛 , 𝑘 𝑥 exists;(3)a mapping 𝑊 ∶ 𝐶 → 𝐶 defined by 𝑊 𝑥 ∶ = l i m 𝑛 → ∞ 𝑊 𝑛 𝑥 = l i m 𝑛 → ∞ 𝑈 𝑛 , 1 𝑥 , ∀ 𝑥 ∈ 𝐶 ( 2 . 1 3 ) is a nonexpansive mapping satisfying 𝐹 ( 𝑊 ) = ∩ ∞ 𝑖 = 1 𝐹 ( 𝑇 𝑖 ) and it is called the 𝑊 -mapping generated by 𝑇 1 , 𝑇 2 , … and 𝜇 1 , 𝜇 2 , … .

Lemma 2.3 (see [37]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 , { 𝑇 𝑖 ∶ 𝐶 → 𝐶 } a countable family of nonexpansive mappings with ∩ ∞ 𝑖 = 1 𝐹 ( 𝑇 𝑖 ) ≠ ∅ , { 𝜇 𝑖 } a real sequence such that 0 < 𝜇 𝑖 ≤ 𝑏 < 1 , f o r a l l 𝑖 ≥ 1 . If 𝐷 is any bounded subset of 𝐶 , then l i m 𝑛 → ∞ s u p 𝑥 ∈ 𝐷 ‖ ‖ 𝑊 𝑥 − 𝑊 𝑛 𝑥 ‖ ‖ = 0 . ( 2 . 1 4 )

Lemma 2.4 (see [38]). Each Hilbert space 𝐻 satisfies Opial’s condition, that is, for any sequence { 𝑥 𝑛 } ⊂ 𝐻 with 𝑥 𝑛 ⇀ 𝑥 , the inequality l i m i n f 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑥 < l i m i n f 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑦 ( 2 . 1 5 ) holds for each 𝑦 ∈ 𝐻 with 𝑦 ≠ 𝑥 .

Lemma 2.5 (see [39]). Assume 𝐴 is a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾 > 0 and 0 < 𝜌 ≤ ‖ 𝐴 ‖ − 1 . Then, ‖ 𝐼 − 𝜌 𝐴 ‖ ≤ 1 − 𝜌 𝛾 .

For solving the system of mixed equilibrium problems (1.15), let us assume that function Θ 𝑘 ∶ 𝐻 × 𝐻 → ℝ , 𝑘 = 1 , 2 , … , 𝑁 satisfies the following conditions:(H1) Θ 𝑘 is monotone, that is, Θ 𝑘 ( 𝑥 , 𝑦 ) + Θ 𝑘 ( 𝑦 , 𝑥 ) ≤ 0 , for all 𝑥 , 𝑦 ∈ 𝐻 ;(H2) for each fixed 𝑦 ∈ 𝐻 , 𝑥 ↦ Θ 𝑘 ( 𝑥 , 𝑦 ) is convex and upper semicontinuous;(H3) for each 𝑥 ∈ 𝐻 , 𝑦 ↦ Θ 𝑘 ( 𝑥 , 𝑦 ) is convex.

Let 𝜂 ∶ 𝐻 × 𝐻 → 𝐻 and 𝐵 ∶ 𝐻 → 𝐻 be two mappings. 𝐵 is said to be(1)monotone if ⟨ 𝐵 𝑥 − 𝐵 𝑦 , 𝜂 ( 𝑥 , 𝑦 ) ⟩ ≥ 0 , ∀ 𝑥 , 𝑦 ∈ 𝐻 ; ( 2 . 1 6 ) (2) 𝑑 -strongly monotone if there exists a positive real number 𝑑 such that ⟨ 𝐵 𝑥 − 𝐵 𝑦 , 𝜂 ( 𝑥 , 𝑦 ) ⟩ ≥ 𝑑 ‖ 𝑥 − 𝑦 ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐻 ; ( 2 . 1 7 ) (3) 𝐿 -Lipschitz continuous if there exists a constant 𝐿 > 0 such that ‖ 𝜂 ( 𝑥 , 𝑦 ) ‖ ≤ 𝐿 ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐻 . ( 2 . 1 8 )

Let 𝐾 ∶ 𝐻 → ℝ be a differentiable functional on 𝐻 , which is called:(1) 𝜂 -convex [40] if 𝐾 𝐾 ( 𝑦 ) − 𝐾 ( 𝑥 ) ≥ ( 𝑥 ) , 𝜂 ( 𝑦 , 𝑥 ) , ∀ 𝑥 , 𝑦 ∈ 𝐻 , ( 2 . 1 9 ) where 𝐾 ( 𝑥 ) is the Fréchet derivative of 𝐾 at 𝑥 ;(2) 𝜂 -strongly convex [41] if there exists a constant 𝜎 > 0 such that 𝐾 𝐾 ( 𝑦 ) − 𝐾 ( 𝑥 ) − ≥ 𝜎 ( 𝑥 ) , 𝜂 ( 𝑦 , 𝑥 ) 2 ‖ 𝑥 − 𝑦 ‖ 2 , ∀ 𝑥 , 𝑦 ∈ 𝐻 . ( 2 . 2 0 )

In particular, if 𝜂 ( 𝑥 , 𝑦 ) = 𝑥 − 𝑦 for all 𝑥 , 𝑦 ∈ 𝐻 , then 𝐾 is said to be strongly convex.

Lemma 2.6 (see [42]). Let 𝐻 be a real Hilbert space and let 𝜙 be a lower semicontinuous and convex functional from 𝐻 to ℝ . Let Θ be a bifunction from 𝐻 × 𝐻 to ℝ satisfying (H1)–(H3). Assume that(i) 𝜂 ∶ 𝐻 × 𝐻 → 𝐻 is 𝜆 -Lipschitz continuous with constant 𝜆 > 0 such that(a) 𝜂 ( 𝑥 , 𝑦 ) + 𝜂 ( 𝑦 , 𝑥 ) = 0 , f o r a l l 𝑥 , 𝑦 ∈ 𝐻 , (b) 𝜂 ( ⋅ , ⋅ ) is affine in the first variable,(c)for each fixed 𝑥 ∈ 𝐻 , 𝑦 ↦ 𝜂 ( 𝑥 , 𝑦 ) is sequentially continuous from the weak topology to the weak topology;(ii) 𝐾 ∶ 𝐻 → ℝ is 𝜂 -strongly convex with constant 𝜎 > 0 and its derivative 𝐾 is sequentially continuous from the weak topology to the strong topology;(iii)for each 𝑥 ∈ 𝐻 , there exist bounded subsets 𝐸 𝑥 ⊂ 𝐻 and 𝑧 𝑥 ∈ 𝐻 such that for any 𝑦 ∈ 𝐻 ⧵ 𝐸 𝑥 , Θ 𝑦 , 𝑧 𝑥 𝑧 + 𝜙 𝑥 1 − 𝜙 ( 𝑦 ) + 𝑟 𝐾 ( 𝑦 ) − 𝐾 𝑧 ( 𝑥 ) , 𝜂 𝑥 , 𝑦 < 0 . ( 2 . 2 1 ) For given 𝑟 > 0 , let 𝐽 Θ 𝑟 ∶ 𝐻 → 𝐻 be the mapping defined by 𝐽 Θ 𝑟 1 ( 𝑥 ) = 𝑦 ∈ 𝐻 ∶ Θ ( 𝑦 , 𝑧 ) + 𝜙 ( 𝑧 ) − 𝜙 ( 𝑦 ) + 𝑟 𝐾 ( 𝑦 ) − 𝐾 ( 𝑥 ) , 𝜂 ( 𝑧 , 𝑦 ) ≥ 0 , ∀ 𝑧 ∈ 𝐻 ( 2 . 2 2 ) for all 𝑥 ∈ 𝐻 . Then(1) 𝐽 Θ 𝑟 is single-valued.(2) 𝐹 ( 𝐽 Θ 𝑟 ) = M E P ( Θ , 𝜙 ) , where M E P ( Θ , 𝜙 ) is the set of solution of the mixed equilibrium problem, Θ ( 𝑥 , 𝑦 ) + 𝜙 ( 𝑦 ) − 𝜙 ( 𝑥 ) ≥ 0 , ∀ 𝑦 ∈ 𝐻 . ( 2 . 2 3 ) (3) M E P ( Θ , 𝜙 ) is closed and convex.

Lemma 2.7 (see [43]). Let { 𝑥 𝑛 } and { 𝑣 𝑛 } be bounded sequences in a Banach space 𝑋 and let { 𝛽 𝑛 } be a sequence in [ 0 , 1 ] with 0 < l i m i n f 𝑛 → ∞ 𝛽 𝑛 ≤ l i m s u p 𝑛 → ∞ 𝛽 𝑛 < 1 . Suppose 𝑥 𝑛 + 1 = ( 1 − 𝛽 𝑛 ) 𝑣 𝑛 + 𝛽 𝑛 𝑥 𝑛 for all integers 𝑛 ≥ 0 and l i m s u p 𝑛 → ∞ ( ‖ 𝑣 𝑛 + 1 − 𝑣 𝑛 ‖ − ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ) ≤ 0 . Then, l i m 𝑛 → ∞ ‖ 𝑣 𝑛 − 𝑥 𝑛 ‖ = 0 .

Lemma 2.8 (see [44]). Assume { 𝑥 𝑛 } is a sequence of nonnegative real numbers such that 𝑥 𝑛 + 1 ≤ 1 − 𝑎 𝑛 𝑥 𝑛 + 𝑏 𝑛 , ∀ 𝑛 ≥ 0 , ( 2 . 2 4 ) where { 𝑎 𝑛 } is a sequence in ( 0 , 1 ) and { 𝑏 𝑛 } is a sequence in ℝ such that(1) ∑ ∞ 𝑛 = 1 𝑎 𝑛 = ∞ , (2) l i m s u p 𝑛 → ∞ ( 𝑏 𝑛 / 𝑎 𝑛 ) ≤ 0 or ∑ ∞ 𝑛 = 1 | 𝑏 𝑛 | < ∞ .Then, l i m 𝑛 → ∞ 𝑥 𝑛 = 0 .

Lemma 2.9 (see [45]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑔 ∶ 𝐶 → ℝ ∪ { ∞ } a proper lower-semicontinuous differentiable convex function. If 𝑧 is a solution to the minimization problem 𝑔 ( 𝑧 ) = i n f 𝑥 ∈ 𝐶 𝑔 ( 𝑥 ) , ( 2 . 2 5 ) then 𝑔 ( 𝑥 ) , 𝑥 − 𝑧 ≥ 0 , 𝑥 ∈ 𝐶 . ( 2 . 2 6 ) In particular, if 𝑧 solves problem 𝑂 𝑃 , then [ ] ⟨ 𝑢 + 𝛾 𝑓 − ( 𝐼 + 𝜇 𝐴 ) 𝑧 , 𝑥 − 𝑧 ⟩ ≤ 0 . ( 2 . 2 7 )

Lemma 2.10 (see [46]). Let 𝐶 be a nonempty bounded closed convex subset of a Hilbert space 𝐻 and let 𝒮 = { 𝑆 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } be a nonexpansive semigroup on 𝐶 , then for any ℎ ≥ 0 , l i m 𝑡 → ∞ s u p 𝑥 ∈ 𝐶 ‖ ‖ ‖ 1 𝑡 𝑡 0 1 𝑇 ( 𝑠 ) 𝑥 𝑑 𝑠 − 𝑇 ( ℎ ) 𝑡 𝑡 0 ‖ ‖ ‖ 𝑇 ( 𝑠 ) 𝑥 𝑑 𝑠 = 0 . ( 2 . 2 8 )

Lemma 2.11 (see [47]). Let C be a nonempty bounded closed convex subset of 𝐻 , { 𝑥 𝑛 } a sequence in C, and 𝒮 = { 𝑆 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } a nonexpansive semigroup on 𝐶 . If the following conditions are satisfied:(i) 𝑥 𝑛 ⇀ 𝑧 ;(ii) l i m s u p 𝑠 → ∞ l i m s u p 𝑛 → ∞ ‖ 𝑆 ( 𝑠 ) 𝑥 𝑛 − 𝑥 𝑛 ‖ = 0 , then 𝑧 ∈ 𝒮 .

Lemma 2.12 (see [25]). For given 𝑥 ∗ , 𝑦 ∗ ∈ 𝐶 and ( 𝑥 ∗ , 𝑦 ∗ ) is a solution of the problem (1.20) if and only if 𝑥 ∗ is a fixed point of the mapping 𝐺 ∶ 𝐶 → 𝐶 is defined by 𝐺 ( 𝑥 ) = 𝑃 𝐶 𝑃 𝐶 ( 𝑥 − 𝛿 𝐷 𝑥 ) − 𝜏 𝐵 𝑃 𝐶 ( 𝑥 − 𝛿 𝐷 𝑥 ) , ∀ 𝑥 ∈ 𝐻 , ( 2 . 2 9 ) where 𝑦 ∗ = 𝑃 𝐶 ( 𝑥 − 𝛿 𝐷 𝑥 ) , 𝛿 and 𝜏 are positive constants and 𝐵 , 𝐷 ∶ 𝐻 → 𝐻 are two mappings.

Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by S V I ( 𝐶 , 𝐵 , 𝐷 ) .

Lemma 2.13 (see [32]). Let 𝐺 ∶ 𝐶 → 𝐶 be defined in Lemma 2.12. If 𝐵 ∶ 𝐻 → 𝐻 is a 𝐿 𝐵 -Lipschitzian and relaxed ( 𝑐 , 𝑑 ) -cocoercive mapping and 𝐷 ∶ 𝐻 → 𝐻 is a 𝐿 𝐷 -Lipschitz and relaxed ( 𝑐 ′ , 𝑑 ′ ) -cocoercive mapping where 𝜏 ≤ 2 ( 𝑑 − 𝑐 𝐿 2 𝐵 ) / 𝐿 2 𝐵 and 𝛿 ≤ 2 ( 𝑑 ′ − 𝑐 ′ 𝐿 2 𝐷 ) / 𝐿 2 𝐷 , then G is nonexpansive.

Lemma 2.14 (demiclosedness principle [48]). Assume that 𝑆 is a nonexpansive self-mapping of a nonempty closed convex subset 𝐶 of a real Hilbert space 𝐻 . If 𝑆 has a fixed point, then 𝐼 − 𝑆 is demiclosed; that is, whenever { 𝑥 𝑛 } is a sequence in 𝐶 converging weakly to some 𝑥 ∈ 𝐶 (for short, 𝑥 𝑛 ⇀ 𝑥 ∈ 𝐶 ), and the sequence { ( 𝐼 − 𝑆 ) 𝑥 𝑛 } converges strongly to some 𝑦 (for short, ( 𝐼 − 𝑆 ) 𝑥 𝑛 → 𝑦 ), it follows that ( 𝐼 − 𝑆 ) 𝑥 = 𝑦 . Here 𝐼 is the identity operator of 𝐻 .

3. Main Results

In this section, we prove a strong convergence theorem of an iterative algorithm (3.1) for finding the solutions of a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 which 𝐶 + 𝐶 ⊂ 𝐶 and let 𝑓 be a contraction of 𝐶 into itself with 𝛼 ∈ ( 0 , 1 ) . Let 𝜙 be a lower semicontinuous and convex functional from 𝐻 to ℝ and let { Θ 𝑘 ∶ 𝐻 × 𝐻 → ℝ , 𝑘 = 1 , 2 , … , 𝑁 } be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮 = { 𝑆 ( 𝑠 ) ∶ 0 ≤ 𝑠 < ∞ } be a nonexpansive semigroup on 𝐶 and let { 𝑡 𝑛 } be a positive real divergent sequence. Let { 𝑉 𝑖 ∶ 𝐶 → 𝐶 } ∞ 𝑖 = 1 be a countable family of uniformly 𝑘 -strict pseudo-contractions, let { 𝑇 𝑖 ∶ 𝐶 → 𝐶 } ∞ 𝑖 = 1 be the countable family of nonexpansive mappings defined by 𝑇 𝑖 𝑥 = 𝑡 𝑥 + ( 1 − 𝑡 ) 𝑉 𝑖 𝑥 , f o r a l l 𝑥 ∈ 𝐶 , f o r a l l 𝑖 ≥ 1 , 𝑡 ∈ [ 𝑘 , 1 ) , let 𝑊 𝑛 be the 𝑊 -mapping defined by (2.12), and let 𝑊 be a mapping defined by (2.13) with 𝐹 ( 𝑊 ) ≠ ∅ . Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾 > 0 and let 0 < 𝛾 < ( 1 + 𝜇 𝛾 ) / 𝛼 , 𝐵 ∶ 𝐻 → 𝐻 be a 𝐿 𝐵 -Lipschitz continuous and relaxed ( 𝑐 , 𝑑 ) -cocoercive mapping with 𝑑 > 𝑐 𝐿 2 𝐵 , and let 𝐷 ∶ 𝐻 → 𝐻 be a 𝐿 𝐷 -Lipschitz continuous and relaxed ( 𝑐 ′ , 𝑑 ′ ) -cocoercive mapping with 𝑑 ′ > 𝑐 ′ 𝐿 2 𝐷 . Suppose that Ω ∶ = 𝐹 ( 𝒮 ) ∩ 𝐹 ( 𝑊 ) ∩ 𝔉 ∩ S V I ( 𝐶 , 𝐵 , 𝐷 ) ≠ ∅ , where 𝔉 = ( ∩ 𝑁 𝑘 = 1 M E P ( Θ 𝑘 , 𝜙 ) ) . Let 𝜇 > 0 , 𝛾 > 0 and 𝑟 𝑘 > 0 , 𝑘 = 1 , 2 , … , 𝑁 , which are constants. For given 𝑥 1 ∈ 𝐻 arbitrarily and fixed 𝑢 ∈ 𝐻 , suppose { 𝑥 𝑛 } , { 𝑦 𝑛 } , { 𝑧 𝑛 } and { 𝑢 𝑛 ( 𝑘 ) } , 𝑘 = 1 , 2 , … , 𝑁 are the sequences generated iteratively by Θ 1 𝑢 𝑛 ( 1 ) 𝑢 , 𝑥 + 𝜙 ( 𝑥 ) − 𝜙 𝑛 ( 1 ) + 1 𝑟 1 𝐾 𝑢 𝑛 ( 1 ) − 𝐾 𝑥 𝑛 , 𝜂 𝑥 , 𝑢 𝑛 ( 1 ) Θ ≥ 0 , ∀ 𝑥 ∈ 𝐻 , 2 𝑢 𝑛 ( 2 ) 𝑢 , 𝑥 + 𝜙 ( 𝑥 ) − 𝜙 𝑛 ( 2 ) + 1 𝑟 2 𝐾 𝑢 𝑛 ( 2 ) − 𝐾 𝑥 𝑛 , 𝜂 𝑥 , 𝑢 𝑛 ( 2 ) ⋮ Θ ≥ 0 , ∀ 𝑥 ∈ 𝐻 , 𝑁 𝑢 𝑛 ( 𝑁 ) 𝑢 , 𝑥 + 𝜙 ( 𝑥 ) − 𝜙 𝑛 ( 𝑁 ) + 1 𝑟 𝑁 𝐾 𝑢 𝑛 ( 𝑁 ) − 𝐾 𝑥 𝑛 , 𝜂 𝑥 , 𝑢 𝑛 ( 𝑁 ) 𝑧 ≥ 0 , ∀ 𝑥 ∈ 𝐻 , 𝑛 = 𝑃 𝐶 𝑢 𝑛 ( 𝑁 ) − 𝛿 𝐷 𝑢 𝑛 ( 𝑁 ) , 𝑦 𝑛 = 𝑃 𝐶 𝑧 𝑛 − 𝜏 𝐵 𝑧 𝑛 , 𝑥 𝑛 + 1 = 𝛼 𝑛 𝑊 𝑢 + 𝛾 𝑓 𝑛 𝑥 𝑛 + 𝛽 𝑛 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 1 ( 𝐼 + 𝜇 𝐴 ) 𝑡 𝑛 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 𝑑 𝑠 , ( 3 . 1 ) where 𝑢 𝑛 ( 1 ) = 𝐽 Θ 1 𝑟 1 𝑥 𝑛 , 𝑢 𝑛 ( 𝑘 ) = 𝐽 Θ 𝑘 𝑟 𝑘 𝑢 𝑛 ( 𝑘 − 1 ) = 𝐽 Θ 𝑘 𝑟 𝑘 𝐽 Θ 𝑘 − 1 𝑟 𝑘 − 1 𝑢 𝑛 ( 𝑘 − 2 ) = 𝐽 Θ 𝑘 𝑟 𝑘 ⋯ 𝐽 Θ 2 𝑟 2 𝑢 𝑛 ( 1 ) , = 𝐽 Θ 𝑘 𝑟 𝑘 ⋯ 𝐽 Θ 2 𝑟 2 𝐽 Θ 1 𝑟 1 𝑥 𝑛 , 𝑘 = 2 , 3 , … , 𝑁 , ( 3 . 2 ) 𝐽 Θ 𝑘 𝑟 𝑘 ∶ 𝐻 → 𝐻 , 𝑘 = 1 , 2 , … , 𝑁 is the mapping defined by (2.22) and { 𝛼 𝑛 } and { 𝛽 𝑛 } are two sequences in ( 0 , 1 ) for all 𝑛 ∈ ℕ . Assume the following conditions are satisfied:(C1) 𝜂 ∶ 𝐻 × 𝐻 → 𝐻 is 𝜆 -Lipschitz continuous with constant 𝜆 > 0 such that(a) 𝜂 ( 𝑥 , 𝑦 ) + 𝜂 ( 𝑦 , 𝑥 ) = 0 , f o r a l l 𝑥 , 𝑦 ∈ 𝐻 , (b) 𝑥 ↦ 𝜂 ( 𝑥 , 𝑦 ) is affine,(c)for each fixed 𝑦 ∈ 𝐻 , 𝑦 ↦ 𝜂 ( 𝑥 , 𝑦 ) is sequentially continuous from the weak topology to the weak topology;(C2) 𝐾 ∶ 𝐻 → ℝ is 𝜂 -strongly convex with constant 𝜎 > 0 and its derivative 𝐾 is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant 𝜈 > 0 such that 𝜎 > 𝜆 𝜈 ;(C3) for each 𝑘 ∈ { 1 , 2 , … , 𝑁 } and for all 𝑥 ∈ 𝐻 , there exist bounded subsets 𝐸 𝑥 ⊂ 𝐻 and 𝑧 𝑥 ∈ 𝐻 such that for any 𝑦 ∈ 𝐻 ⧵ 𝐸 𝑥 , Θ 𝑘 𝑦 , 𝑧 𝑥 𝑧 + 𝜙 𝑥 1 − 𝜙 ( 𝑦 ) + 𝑟 𝑘 𝐾 ( 𝑦 ) − 𝐾 𝑧 ( 𝑥 ) , 𝜂 𝑥 , 𝑦 < 0 ; ( 3 . 3 ) (C4) l i m 𝑛 → ∞ 𝛼 𝑛 = 0 and ∑ ∞ 𝑛 = 1 𝛼 𝑛 = ∞ ;(C5) 0 < l i m i n f 𝑛 → ∞ 𝛽 𝑛 ≤ l i m s u p 𝑛 → ∞ 𝛽 𝑛 < 1 ; (C6) 0 < 𝜏 < 2 ( 𝑑 − 𝑐 𝐿 2 𝐵 ) / 𝐿 2 𝐵 and 0 < 𝛿 < 2 ( 𝑑 ′ − 𝑐 ′ 𝐿 2 𝐷 ) / 𝐿 2 𝐷 .Then, { 𝑥 𝑛 } converges strongly to 𝑥 ∗ ∈ Ω , which solves the following optimization problem (OP): m i n 𝑥 ∗ ∈ Ω 𝜇 2 ⟨ 𝐴 𝑥 ∗ , 𝑥 ∗ 1 ⟩ + 2 ‖ 𝑥 ∗ ‖ − 𝑢 2 𝑥 − ℎ ∗ , ( 3 . 4 ) and ( 𝑥 ∗ , 𝑦 ∗ ) is a solution of the general system of variational inequality problem (1.20) such that 𝑦 ∗ = 𝑃 𝐶 ( 𝑥 ∗ − 𝛿 𝐷 𝑥 ∗ ) .

Proof. By the condition (C4) and (C5), we may assume, without loss of generality, that 𝛼 𝑛 ≤ ( 1 − 𝛽 𝑛 ) ( 1 + 𝜇 ‖ 𝐴 ‖ ) − 1 for all 𝑛 ∈ ℕ . Indeed, 𝐴 is a strongly positive bounded linear operator on 𝐻 , we have | | | | ‖ 𝐴 ‖ = s u p ⟨ 𝐴 𝑥 , 𝑥 ⟩ ∶ 𝑥 ∈ 𝐻 , ‖ 𝑥 ‖ = 1 . ( 3 . 5 ) Observe that 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 ( 𝐼 + 𝜇 𝐴 ) 𝑥 , 𝑥 = 1 − 𝛽 𝑛 − 𝛼 𝑛 − 𝛼 𝑛 𝜇 ⟨ 𝐴 𝑥 , 𝑥 ⟩ ≥ 1 − 𝛽 𝑛 − 𝛼 𝑛 − 𝛼 𝑛 𝜇 ‖ 𝐴 ‖ ≥ 0 , ( 3 . 6 ) so this shows that ( 1 − 𝛽 𝑛 ) 𝐼 − 𝛼 𝑛 ( 𝐼 + 𝜇 𝐴 ) is positive. It follows that ‖ ‖ 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 ‖ ‖ | | ( 𝐼 + 𝜇 𝐴 ) = s u p 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 | | ( 𝐼 + 𝜇 𝐴 ) 𝑥 , 𝑥 ∶ 𝑥 ∈ 𝐻 , ‖ 𝑥 ‖ = 1 = s u p 1 − 𝛽 𝑛 − 𝛼 𝑛 − 𝛼 𝑛 𝜇 ⟨ 𝐴 𝑥 , 𝑥 ⟩ ∶ 𝑥 ∈ 𝐻 , ‖ 𝑥 ‖ = 1 ≤ 1 − 𝛽 𝑛 − 𝛼 𝑛 − 𝛼 𝑛 𝜇 𝛾 . ( 3 . 7 ) We shall divide the proofs into several steps.Step 1. We show that { 𝑥 𝑛 } is bounded.
Let 𝑥 ∗ ∈ Ω ∶ = 𝐹 ( 𝒮 ) ∩ 𝐹 ( 𝑊 ) ∩ ( ∩ 𝑁 𝑘 = 1 M E P ( Θ 𝑘 , 𝜙 ) ) ∩ S V I ( 𝐶 , 𝐵 , 𝐷 ) . In fact, by the assumption that for each 𝑘 ∈ { 1 , 2 , … , 𝑁 } , 𝐽 Θ 𝑘 𝑟 𝑘 is nonexpansive. Let 𝒜 𝑁 ∶ = 𝐽 Θ 𝑁 𝑟 𝑁 ⋯ 𝐽 Θ 2 𝑟 2 𝐽 Θ 1 𝑟 1 and 𝒜 0 = 𝐼 . Then, we have 𝑥 ∗ = 𝒜 𝑁 𝑥 ∗ and 𝑢 𝑛 ( 𝑁 ) = 𝒜 𝑁 𝑥 𝑛 . Since 𝑥 ∗ ∈ S V I ( 𝐶 , 𝐵 , 𝐷 ) , then 𝑥 ∗ = 𝑃 𝐶 𝑃 𝐶 𝑥 ∗ − 𝛿 𝐷 𝑥 ∗ − 𝜏 𝐵 𝑃 𝐶 𝑥 ∗ − 𝛿 𝐷 𝑥 ∗ = 𝑃 𝐶 𝑃 𝐶 ( 𝐼 − 𝛿 𝐷 ) 𝒜 𝑁 𝑥 ∗ − 𝜏 𝐵 𝑃 𝐶 ( 𝐼 − 𝛿 𝐷 ) 𝒜 𝑁 𝑥 ∗ . ( 3 . 8 ) Putting 𝑦 ∗ = 𝑃 𝐶 ( 𝑥 ∗ − 𝛿 𝐷 𝑥 ∗ ) = 𝑃 𝐶 ( 𝐼 − 𝛿 𝐷 ) 𝒜 𝑁 𝑥 ∗ , we have 𝑥 ∗ = 𝑃 𝐶 ( 𝑦 ∗ − 𝜏 𝐵 𝑦 ∗ ) . Since 𝑥 ∗ = 𝑆 ( 𝑠 ) 𝑥 ∗ , f o r a l l 𝑠 ≥ 0 and 𝑥 ∗ = 𝑊 𝑛 𝑥 ∗ , f o r a l l 𝑛 ≥ 1 , therefore, we have 𝑥 ∗ = 𝒜 𝑁 𝑥 ∗ = 𝑃 𝐶 𝑦 ∗ − 𝜏 𝐵 𝑦 ∗ = 𝑊 𝑛 𝑃 𝐶 𝑦 ∗ − 𝜏 𝐵 𝑦 ∗ = 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑃 𝐶 𝑦 ∗ − 𝜏 𝐵 𝑦 ∗ . ( 3 . 9 ) Because 𝑃 𝐶 and 𝒜 𝑁 are nonexpansive mappings and from Remark 1.1, we have ‖ ‖ 𝑦 𝑛 − 𝑥 ∗ ‖ ‖ = ‖ ‖ 𝑃 𝐶 𝑧 𝑛 − 𝜏 𝐵 𝑧 𝑛 − 𝑃 𝐶 𝑦 ∗ − 𝜏 𝐵 𝑦 ∗ ‖ ‖ ≤ ‖ ‖ ( 𝐼 − 𝜏 𝐵 ) 𝑧 𝑛 − ( 𝐼 − 𝜏 𝐵 ) 𝑦 ∗ ‖ ‖ ≤ ‖ ‖ 𝑧 𝑛 − 𝑦 ∗ ‖ ‖ = ‖ ‖ 𝑃 𝐶 𝑢 𝑛 ( 𝑁 ) − 𝛿 𝐷 𝑢 𝑛 ( 𝑁 ) − 𝑃 𝐶 𝑥 ∗ − 𝛿 𝐷 𝑥 ∗ ‖ ‖ ≤ ‖ ‖ ( 𝐼 − 𝛿 𝐷 ) 𝑢 𝑛 ( 𝑁 ) − ( 𝐼 − 𝛿 𝐷 ) 𝑥 ∗ ‖ ‖ ≤ ‖ ‖ 𝑢 𝑛 ( 𝑁 ) − 𝑥 ∗ ‖ ‖ = ‖ ‖ 𝒜 𝑁 𝑥 𝑛 − 𝒜 𝑁 𝑥 ∗ ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ ( 3 . 1 0 ) which yields that ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 ∗ ‖ ‖ = ‖ ‖ ‖ 𝛼 𝑛 𝑢 + 𝛼 𝑛 𝑊 𝛾 𝑓 𝑛 𝑥 𝑛 − ( 𝐼 + 𝜇 𝐴 ) 𝑥 ∗ + 𝛽 𝑛 𝑥 𝑛 − 𝑥 ∗ + 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 ( 1 𝐼 + 𝜇 𝐴 ) 𝑡 𝑛 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 𝑑 𝑠 − 𝑥 ∗ ‖ ‖ ‖ ≤ 𝛼 𝑛 ‖ 𝑢 ‖ + 𝛼 𝑛 ‖ ‖ 𝑊 𝛾 𝑓 𝑛 𝑥 𝑛 − ( 𝐼 + 𝜇 𝐴 ) 𝑥 ∗ ‖ ‖ + 𝛽 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ + 1 − 𝛽 𝑛 − 𝛼 𝑛 1 + 𝜇 𝛾 ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ ≤ 𝛼 𝑛 ‖ 𝑢 ‖ + 𝛼 𝑛 ‖ ‖ 𝑊 𝛾 𝑓 𝑛 𝑥 𝑛 𝑥 − 𝛾 𝑓 ∗ ‖ ‖ + 𝛼 𝑛 ‖ ‖ 𝑥 𝛾 𝑓 ∗ − ( 𝐼 + 𝜇 𝐴 ) 𝑥 ∗ ‖ ‖ + 𝛽 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ + 1 − 𝛽 𝑛 − 𝛼 𝑛 1 + 𝜇 𝛾 ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ ≤ 𝛼 𝑛 ‖ 𝑢 ‖ + 𝛼 𝑛 ‖ ‖ 𝑥 𝛾 𝛼 𝑛 − 𝑥 ∗ ‖ ‖ + 𝛼 𝑛 ‖ ‖ 𝑥 𝛾 𝑓 ∗ − ( 𝐼 + 𝜇 𝐴 ) 𝑥 ∗ ‖ ‖ + 𝛽 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ + 1 − 𝛽 𝑛 − 𝛼 𝑛 1 + 𝜇 𝛾 ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ = 𝛼 𝑛 ‖ ‖ 𝑥 ‖ 𝑢 ‖ + 𝛾 𝑓 ∗ − ( 𝐼 + 𝜇 𝐴 ) 𝑥 ∗ ‖ ‖ + 1 − 𝛼 𝑛 1 + 𝜇 𝛾 + 𝛼 𝑛 ‖ ‖ 𝑥 𝛾 𝛼 𝑛 − 𝑥 ∗ ‖ ‖ = 1 − 𝛼 𝑛 1 + 𝜇 𝛾 ‖ ‖ 𝑥 − 𝛾 𝛼 𝑛 − 𝑥 ∗ ‖ ‖ + 𝛼 𝑛 1 + 𝜇 𝛾 ‖ ‖ 𝑥 − 𝛾 𝛼 ‖ 𝑢 ‖ + 𝛾 𝑓 ∗ − ( 𝐼 + 𝜇 𝐴 ) 𝑥 ∗ ‖ ‖ 1 + 𝜇 𝛾 . − 𝛾 𝛼 ( 3 . 1 1 ) It follows from (3.11) and induction that ‖ ‖ 𝑥 𝑛 − 𝑥 ∗ ‖ ‖ ‖ ‖ 𝑥 ≤ m a x 1 ‖ ‖ , ‖ ‖ 𝑥 − 𝑝 ‖ 𝑢 ‖ + 𝛾 𝑓 ∗ − ( 𝐼 + 𝜇 𝐴 ) 𝑥 ∗ ‖ ‖ 1 + 𝜇 𝛾 − 𝛾 𝛼 , 𝑛 ≥ 1 . ( 3 . 1 2 ) Hence, { 𝑥 𝑛 } is bounded, so are { 𝑦 𝑛 } , { 𝑧 𝑛 } , { 𝑊 𝑛 𝑥 𝑛 } , { 𝑓 ( 𝑊 𝑛 𝑥 𝑛 ) } , { 𝑢 𝑛 ( 𝑘 ) } for all 𝑘 = 1 , 2 , … , 𝑁 and { 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 } , where 𝐾 𝑛 = ( 1 / 𝑡 𝑛 ) ∫ 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑑 𝑠 .Step 2. We prove that l i m 𝑛 → ∞ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ = 0 and l i m 𝑛 → ∞ ‖ 𝑢 ( 𝑁 ) 𝑛 + 1 − 𝑢 𝑛 ( 𝑁 ) ‖ = 0 .
Again, from Remark 1.1, we have the following estimates: ‖ ‖ 𝑦 𝑛 + 1 − 𝑦 𝑛 ‖ ‖ = ‖ ‖ 𝑃 𝐶 𝑧 𝑛 + 1 − 𝜏 𝐵 𝑧 𝑛 + 1 − 𝑃 𝐶 𝑧 𝑛 − 𝜏 𝐵 𝑧 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑧 𝑛 + 1 − 𝜏 𝐵 𝑧 𝑛 + 1 − 𝑧 𝑛 − 𝜏 𝐵 𝑧 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑧 𝑛 + 1 − 𝑧 𝑛 ‖ ‖ = ‖ ‖ 𝑃 𝐶 𝑢 ( 𝑁 ) 𝑛 + 1 − 𝛿 𝐷 𝑢 ( 𝑁 ) 𝑛 + 1 − 𝑃 𝐶 𝑢 𝑛 ( 𝑁 ) − 𝛿 𝐷 𝑢 𝑛 ( 𝑁 ) ‖ ‖ ≤ ‖ ‖ 𝑢 ( 𝑁 ) 𝑛 + 1 − 𝛿 𝐷 𝑢 ( 𝑁 ) 𝑛 + 1 − 𝑢 𝑛 ( 𝑁 ) − 𝛿 𝐷 𝑢 𝑛 ( 𝑁 ) ‖ ‖ ≤ ‖ ‖ 𝑢 ( 𝑁 ) 𝑛 + 1 − 𝑢 𝑛 ( 𝑁 ) ‖ ‖ = ‖ ‖ 𝒜 𝑁 𝑥 𝑛 + 1 − 𝒜 𝑁 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ . ( 3 . 1 3 ) On the other hand, since 𝑇 𝑖 and 𝑈 𝑛 , 𝑖 are nonexpansive, we have ‖ ‖ 𝑊 𝑛 + 1 𝑦 𝑛 − 𝑊 𝑛 𝑦 𝑛 ‖ ‖ = ‖ ‖ 𝜇 1 𝑇 1 𝑈 𝑛 + 1 , 2 𝑦 𝑛 − 𝜇 1 𝑇 1 𝑈 𝑛 , 2 𝑦 𝑛 ‖ ‖ ≤ 𝜇 1 ‖ ‖ 𝑈 𝑛 + 1 , 2 𝑦 𝑛 − 𝑈 𝑛 , 2 𝑦 𝑛 ‖ ‖ = 𝜇 1 ‖ ‖ 𝜇 2 𝑇 2 𝑈 𝑛 + 1 , 3 𝑦 𝑛 − 𝜇 2 𝑇 2 𝑈 𝑛 , 3 𝑦 𝑛 ‖ ‖ ≤ 𝜇 1 𝜇 2 ‖ ‖ 𝑈 𝑛 + 1 , 3 𝑦 𝑛 − 𝑈 𝑛 , 3 𝑦 𝑛 ‖ ‖ ⋮ ≤ 𝜇 1 𝜇 2 ⋯ 𝜇 𝑛 ‖ ‖ 𝑈 𝑛 + 1 , 𝑛 + 1 𝑦 𝑛 − 𝑈 𝑛 , 𝑛 + 1 𝑦 𝑛 ‖ ‖ ≤ 𝑀 1 𝑛 𝑖 = 1 𝜇 𝑖 , ( 3 . 1 4 ) where 𝑀 1 ≥ 0 is a constant such that ‖ 𝑈 𝑛 + 1 , 𝑛 + 1 𝑦 𝑛 − 𝑈 𝑛 , 𝑛 + 1 𝑦 𝑛 ‖ ≤ 𝑀 1 for all 𝑛 ≥ 0 . It follows from (3.13) and (3.14) that we have ‖ ‖ 𝑊 𝑛 + 1 𝑦 𝑛 + 1 − 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑊 𝑛 + 1 𝑦 𝑛 + 1 − 𝑊 𝑛 + 1 𝑦 𝑛 ‖ ‖ + ‖ ‖ 𝑊 𝑛 + 1 𝑦 𝑛 − 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑦 𝑛 + 1 − 𝑦 𝑛 ‖ ‖ + 𝑀 1 𝑛 𝑖 = 1 𝜇 𝑖 ≤ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ + 𝑀 1 𝑛 𝑖 = 1 𝜇 𝑖 . ( 3 . 1 5 ) It follows that ‖ ‖ 𝐾 𝑛 + 1 𝑊 𝑛 + 1 𝑦 𝑛 + 1 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ = ‖ ‖ ‖ 1 𝑡 𝑛 + 1 𝑡 𝑛 + 1 0 𝑆 ( 𝑠 ) 𝑊 𝑛 + 1 𝑦 𝑛 + 1 1 𝑑 𝑠 − 𝑡 𝑛 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ‖ ≤ 1 𝑑 𝑠 𝑡 𝑛 + 1 𝑡 𝑛 + 1 0 ‖ ‖ 𝑆 ( 𝑠 ) 𝑊 𝑛 + 1 𝑦 𝑛 + 1 − 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 ‖ ‖ + ‖ ‖ ‖ 1 𝑑 𝑠 𝑡 𝑛 + 1 𝑡 𝑛 + 1 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 1 𝑑 𝑠 − 𝑡 𝑛 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ‖ ≤ ‖ ‖ 𝑊 𝑑 𝑠 𝑛 + 1 𝑦 𝑛 + 1 − 𝑊 𝑛 𝑦 𝑛 ‖ ‖ + ‖ ‖ ‖ 1 𝑡 𝑛 + 1 𝑡 𝑛 + 1 𝑡 𝑛 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 1 𝑑 𝑠 + 𝑡 𝑛 + 1 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 1 𝑑 𝑠 − 𝑡 𝑛 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ‖ ≤ ‖ ‖ 𝑊 𝑑 𝑠 𝑛 + 1 𝑦 𝑛 + 1 − 𝑊 𝑛 𝑦 𝑛 ‖ ‖ + 1 𝑡 𝑛 + 1 𝑡 𝑛 + 1 𝑡 𝑛 ‖ ‖ 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 ‖ ‖ + | | | | 1 𝑑 𝑠 𝑡 𝑛 + 1 − 1 𝑡 𝑛 | | | | 𝑡 𝑛 0 ‖ ‖ 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑊 𝑑 𝑠 𝑛 + 1 𝑦 𝑛 + 1 − 𝑊 𝑛 𝑦 𝑛 ‖ ‖ 𝑡 + 2 1 − 𝑛 𝑡 𝑛 + 1 𝑀 2 ≤ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ + 𝑀 1 𝑛 𝑖 = 1 𝜇 𝑖 𝑡 + 2 1 − 𝑛 𝑡 𝑛 + 1 𝑀 2 , ( 3 . 1 6 ) where 𝑀 2 = m a x { ‖ 𝑆 ( 𝑠 ) 𝑊 𝑛 𝑦 𝑛 ‖ } .
Setting 𝑥 𝑛 + 1 = ( 1 − 𝛽 𝑛 ) 𝑣 𝑛 + 𝛽 𝑛 𝑥 𝑛 , for all 𝑛 ≥ 1 , we have 𝑣 𝑛 = 𝑥 𝑛 + 1 − 𝛽 𝑛 𝑥 𝑛 1 − 𝛽 𝑛 = 𝛼 𝑛 𝑊 𝑢 + 𝛾 𝑓 𝑛 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 𝐾 ( 𝐼 + 𝜇 𝐴 ) 𝑛 𝑊 𝑛 𝑦 𝑛 1 − 𝛽 𝑛 . ( 3 . 1 7 ) Then, we obtain 𝑣 𝑛 + 1 − 𝑣 𝑛 = 𝛼 𝑛 + 1 𝑊 𝑢 + 𝛾 𝑓 𝑛 + 1 𝑥 𝑛 + 1 + 1 − 𝛽 𝑛 + 1 𝐼 − 𝛼 𝑛 + 1 𝐾 ( 𝐼 + 𝜇 𝐴 ) 𝑛 + 1 𝑊 𝑛 + 1 𝑦 𝑛 + 1 1 − 𝛽 𝑛 + 1 − 𝛼 𝑛 𝑊 𝑢 + 𝛾 𝑓 𝑛 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 ( 𝐾 𝐼 + 𝜇 𝐴 ) 𝑛 𝑊 𝑛 𝑦 𝑛 1 − 𝛽 𝑛 = 𝛼 𝑛 + 1 1 − 𝛽 𝑛 + 1 𝑊 𝑢 + 𝛾 𝑓 𝑛 + 1 𝑥 𝑛 + 1 − 𝛼 𝑛 1 − 𝛽 𝑛 𝑊 𝑢 + 𝛾 𝑓 𝑛 𝑥 𝑛 + 𝐾 𝑛 + 1 𝑊 𝑛 + 1 𝑦 𝑛 + 1 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 + 𝛼 𝑛 1 − 𝛽 𝑛 ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 − 𝛼 𝑛 + 1 1 − 𝛽 𝑛 + 1 ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 + 1 𝑊 𝑛 + 1 𝑦 𝑛 + 1 = 𝛼 𝑛 + 1 1 − 𝛽 𝑛 + 1 𝑊 𝑢 + 𝛾 𝑓 𝑛 + 1 𝑥 𝑛 + 1 − ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 + 1 𝑊 𝑛 + 1 𝑦 𝑛 + 1 + 𝛼 𝑛 1 − 𝛽 𝑛 ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 𝑊 − 𝑢 − 𝛾 𝑓 𝑛 𝑥 𝑛 + 𝐾 𝑛 + 1 𝑊 𝑛 + 1 𝑦 𝑛 + 1 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 . ( 3 . 1 8 ) It follows from (3.16) and (3.18) that ‖ ‖ 𝑣 𝑛 + 1 − 𝑣 𝑛 ‖ ‖ − ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ ≤ 𝛼 𝑛 + 1 1 − 𝛽 𝑛 + 1 ‖ ‖ 𝑊 ‖ 𝑢 ‖ + 𝛾 𝑓 𝑛 + 1 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 + 1 𝑊 𝑛 + 1 𝑦 𝑛 + 1 ‖ ‖ + 𝛼 𝑛 1 − 𝛽 𝑛 ‖ ‖ ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ‖ ‖ 𝑊 + ‖ 𝑢 ‖ + 𝛾 𝑓 𝑛 𝑥 𝑛 ‖ ‖ + 𝑀 1 𝑛 𝑖 = 1 𝜇 𝑖 𝑡 + 2 1 − 𝑛 𝑡 𝑛 + 1 𝑀 2 . ( 3 . 1 9 ) By the conditions (C4), (C5) and from 𝑡 𝑛 ∈ ( 0 , ∞ ) , 𝑡 𝑛 → ∞ and 0 < 𝜇 𝑖 ≤ 𝑏 < 1 , f o r a l l 𝑖 ≥ 1 , we have l i m s u p 𝑛 → ∞ ‖ ‖ 𝑣 𝑛 + 1 − 𝑣 𝑛 ‖ ‖ − ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ ≤ 0 . ( 3 . 2 0 ) Hence, by Lemma 2.7, we obtain l i m 𝑛 → ∞ ‖ ‖ 𝑣 𝑛 − 𝑥 𝑛 ‖ ‖ = 0 . ( 3 . 2 1 ) It follows that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ = l i m 𝑛 → ∞ 1 − 𝛽 𝑛 ‖ ‖ 𝑣 𝑛 − 𝑥 𝑛 ‖ ‖ = 0 . ( 3 . 2 2 ) Applying (3.22) into (3.13), we obtain that l i m 𝑛 → ∞ ‖ ‖ 𝑦 𝑛 + 1 − 𝑦 𝑛 ‖ ‖ = l i m 𝑛 → ∞ ‖ ‖ 𝑧 𝑛 + 1 − 𝑧 𝑛 ‖ ‖ = l i m 𝑛 → ∞ ‖ ‖ 𝑢 ( 𝑁 ) 𝑛 + 1 − 𝑢 𝑛 ( 𝑁 ) ‖ ‖ = 0 . ( 3 . 2 3 ) Step 3. We show that l i m 𝑛 → ∞ ‖ 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 − 𝑦 𝑛 ‖ = 0 , l i m 𝑛 → ∞ ‖ 𝑦 𝑛 − 𝑆 ( 𝑠 ) 𝑦 𝑛 ‖ = 0 , and l i m 𝑛 → ∞ ‖ 𝑢 𝑛 ( 𝑘 + 1 ) − 𝑢 𝑛 ( 𝑘 ) ‖ = 0 , where 𝐾 𝑛 = ( 1 / 𝑡 𝑛 ) ∫ 𝑡 𝑛 0 𝑆 ( 𝑠 ) 𝑑 𝑠 .
Since 𝑥 𝑛 + 1 = 𝛼 𝑛 ( 𝑢 + 𝛾 𝑓 ( 𝑊 𝑛 𝑥 𝑛 ) ) + 𝛽 𝑛 𝑥 𝑛 + ( ( 1 − 𝛽 𝑛 ) 𝐼 − 𝛼 𝑛 ( 𝐼 + 𝜇 𝐴 ) ) 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 , we have ‖ ‖ 𝑥 𝑛 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ = ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝛼 𝑛 𝑊 𝑢 + 𝛾 𝑓 𝑛 𝑥 𝑛 + 𝛽 𝑛 𝑥 𝑛 + 1 − 𝛽 𝑛 𝐼 − 𝛼 𝑛 𝐾 ( 𝐼 + 𝜇 𝐴 ) 𝑛 𝑊 𝑛 𝑦 𝑛 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ = ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝛼 𝑛 𝑊 𝑢 + 𝛾 𝑓 𝑛 𝑥 𝑛 − ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 + 𝛽 𝑛 𝑥 𝑛 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + 𝛼 𝑛 ‖ ‖ 𝑊 ‖ 𝑢 ‖ + 𝛾 𝑓 𝑛 𝑥 𝑛 ‖ ‖ + ‖ ‖ ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ + 𝛽 𝑛 ‖ ‖ 𝑥 𝑛 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ , ( 3 . 2 4 ) that is ‖ ‖ 𝑥 𝑛 − 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ ≤ 1 1 − 𝛽 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + 𝛼 𝑛 1 − 𝛽 𝑛 ‖ ‖ 𝑊 ‖ 𝑢 ‖ + 𝛾 𝑓 𝑛 𝑥 𝑛 ‖ ‖ + ‖ ‖ ( 𝐼 + 𝜇 𝐴 ) 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 ‖ ‖ . ( 3 . 2 5 ) By (C4), (C5), and (3.22) it follows that l i m 𝑛 → ∞ ‖ ‖ 𝐾 𝑛 𝑊 𝑛 𝑦 𝑛 − 𝑥 𝑛 ‖ ‖ = 0 . ( 3 . 2 6 )
Since 𝐽 Θ 𝑁 𝑟 𝑁 ∶ 𝐶 → 𝐶 is firmly nonexpansive, 𝑢 𝑛 ( 𝑁 ) = 𝒜 𝑁 𝑥 𝑛 , where 𝒜 𝑁 ∶ = 𝐽 Θ 𝑁 𝑟 𝑁 ⋯ 𝐽 Θ 2 𝑟 2 𝐽 Θ