A Simultaneous Iteration Algorithm for Solving Extended Split Equality

Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 9737062, 9 pages https://doi.org/10.1155/2017/9737062

Research Article A Simultaneous Iteration Algorithm for Solving Extended Split Equality Fixed Point Problem Meixia Li,1 Xiping Kao,2 and Haitao Che1 1

School of Mathematics and Information Science, Weifang University, Weifang, Shandong 261061, China College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

2

Correspondence should be addressed to Meixia Li; [email protected] Received 16 January 2017; Accepted 20 March 2017; Published 6 April 2017 Academic Editor: Roberto Fedele Copyright © 2017 Meixia Li et al. 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. We study a kind of split equality fixed point problem which is an extension of split equality problem. We propose a kind of simultaneous iterative algorithm with a way of selecting the step length which does not need any a priori information about the operator norms and prove that the sequences generated by the iterative method converge weakly to the solution of this problem. Some numerical results are shown to confirm the feasibility and efficiency of the proposed methods.

1. Introduction Let 𝐶 and 𝑄 be nonempty closed and convex subsets of the real Hilbert spaces 𝐻1 and 𝐻2 , respectively. The split feasibility problem is to find 𝑥∈𝐶

such that 𝐴𝑥 ∈ 𝑄.

(1)

It can be used in various disciplines such as image restoration and radiation therapy treatment planning [1, 2]. These applications are in finite-dimensional Hilbert spaces [3, 4]. It also can be found in an infinite-dimensional real Hilbert space [5, 6]. Recently, Moudafi [7] introduced a new split equality feasibility problem. Let 𝐻1 , 𝐻2 , and 𝐻3 be real Hilbert spaces. Let 𝐴 : 𝐻1 → 𝐻3 and 𝐵 : 𝐻2 → 𝐻3 be two bounded linear operators. The split equality feasibility problem is to find 𝑥 ∈ 𝐶, 𝑦∈𝑄

(2) such that 𝐴𝑥 = 𝐵𝑦,

which allows asymmetric and partial relations between the variables 𝑥 and 𝑦. The interest is to cover many situations, for instance, applications in decomposition methods for PDEs, in game theory and in intensity-modulated radiation therapy (for short, IMRT).

Moudafi [8] introduced the simultaneous iterative method to solve the split equality feasibility problem. Furthermore, Moudafi studied the fixed point formulation to avoid using the projection. Assume Fix(𝑇) and Fix(𝑆) are the sets of fixed points of 𝑇 and 𝑆, respectively, where 𝑇 : 𝐻1 → 𝐻1 and 𝑆 : 𝐻2 → 𝐻2 are nonlinear operators such that Fix(𝑇) ≠ 0 and Fix(𝑆) ≠ 0. So the split equality fixed point problem is to find 𝑥 ∈ Fix (𝑇) , 𝑦 ∈ Fix (𝑆)

(3) such that 𝐴𝑥 = 𝐵𝑦,

where 𝑇 and 𝑆 are firmly quasi-nonexpansive mappings. In order to find the solution of the split equality problem, Che and Li [9] proposed the following iterative algorithm: 𝑢𝑛 = 𝑥𝑛 − 𝛾𝑛 𝐴∗ (𝐴𝑥𝑛 − 𝐵𝑦𝑛 ) , 𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑢𝑛 , V𝑛 = 𝑦𝑛 + 𝛾𝑛 𝐵∗ (𝐴𝑥𝑛 − 𝐵𝑦𝑛 ) ,

(4)

𝑦𝑛+1 = 𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑆V𝑛 , and the weak convergence of the scheme (4) can also be established. Furthermore, Chang et al. [10] modified the

2

Mathematical Problems in Engineering

iterative scheme (4) and provided a unified framework for solving this problem without using the projection. The framework is as follows. 𝑢𝑛 = 𝑥𝑛 − 𝛾𝑛 𝐴∗ (𝐴𝑥𝑛 − 𝐵𝑦𝑛 ) ,

V𝑛 = 𝑦𝑛 + 𝛾𝑛 𝐵∗ (𝐴𝑥𝑛 − 𝐵𝑦𝑛 ) ,

(5)

+ (1 − 𝛼𝑛 ) [(1 − 𝜉) 𝐼 + 𝜉𝑆 ((1 − 𝜂) 𝐼 + 𝜂𝑆)] V𝑛 . They got the following conclusion that the sequence (𝑥𝑛 , 𝑦𝑛 ) generated by the above modification converges weakly to a solution of problem (3). Furthermore, some authors [11–14] studied the problems (1)–(3) in Banach space. They proposed effective algorithms and proved their convergence under some conditions. Recently, He and Sun [15] studied the problem of split convex feasibility and established a strongly convergent alternating algorithm. They proposed the following iterative algorithm to find 𝑥 ∈ 𝐶, (6)

2. Preliminaries In this paper, we recall some concepts, definitions, and conclusions, which are prepared for proving our main results. We write 𝑥𝑛 ⇀ 𝑥 to indicate that the sequence {𝑥𝑛 } converges weakly to 𝑥. 𝑥𝑛 → 𝑥 implies that {𝑥𝑛 } converges strongly to 𝑥. We denote by 𝐻1 a real Hilbert space with inner product ⟨⋅, ⋅⟩ and induced norm ‖ ⋅ ‖. A mapping 𝑇 : 𝐶 → 𝐶 is called

󵄩󵄩 󵄩 ∗󵄩 ∗󵄩 󵄩󵄩𝑇𝑥 − 𝑥 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑥 󵄩󵄩󵄩 ,

via the formula 𝐴𝑥𝑛 + 𝐵𝑦𝑛 ), 2

𝐾𝑛 = {𝑧 ∈ 𝐻3 : ℎ ( + ⟨𝜍𝑛 , 𝑧 − (

When 𝐾 = 𝐻3 , problem (8) is problem (3). Therefore, problem (8) is the extension of the split equality fixed point problem. We propose the simultaneous iterative algorithm for solving this problem, which avoids using the projection and the step length sequences do not depend on the operator norms ‖𝐴‖ and ‖𝐵‖. Furthermore, we prove the sequences generated by the algorithm weakly converge to a solution of the extended split equality fixed point problem. Numerical examples show the feasibility and efficiency of this algorithm.

(i) quasi-nonexpansive, if Fix(𝑇) ≠ 0,

such that 𝐴𝑥 = 𝐵𝑦 ∈ 𝐾,

𝜔𝑛 = 𝑃𝐾𝑛 (

(8) such that 𝐴𝑥 = 𝐵𝑦 ∈ 𝐾.

𝑦𝑛+1 = 𝛼𝑛 𝑦𝑛

𝑦∈𝑄

𝑥 ∈ Fix (𝐺) , 𝑦 ∈ Fix (𝑀)

𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) [(1 − 𝜉) 𝐼 + 𝜉𝑇 ((1 − 𝜂) 𝐼 + 𝜂𝑇)] 𝑢𝑛 ,

In this article, we study the following problem of extended split equality fixed point, which is to find

(9)

(ii) quasi-pseudo-contractive, if Fix(𝑇) ≠ 0, 󵄩 󵄩󵄩 ∗󵄩 ∗󵄩 󵄩󵄩𝑇𝑥 − 𝑥 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑥 󵄩󵄩󵄩 + ‖𝑇𝑥 − 𝑥‖ ,

𝐴𝑥𝑛 + 𝐵𝑦𝑛 ) 2

∀𝑥 ∈ 𝐶, 𝑥∗ ∈ Fix (𝑇) .

𝐴𝑥𝑛 + 𝐵𝑦𝑛 )⟩ ≤ 0} , 2

𝑢𝑛 = 𝑃𝐶 (𝑥𝑛 − 𝜉𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )) , V𝑛 = 𝑃𝑄 (𝑦𝑛 − 𝜉𝐵∗ (𝐵𝑥𝑛 − 𝜔𝑛 )) ,

∀𝑥 ∈ 𝐶, 𝑥∗ ∈ Fix (𝑇) ,

(7)

󵄩 󵄩2 𝐶𝑛+1 × 𝑄𝑛+1 = {(𝑥, 𝑦) ∈ 𝐶𝑛 × 𝑄𝑛 : 󵄩󵄩󵄩𝑢𝑛 − 𝑥󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 + 󵄩󵄩󵄩V𝑛 − 𝑦󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑦󵄩󵄩󵄩 , 𝑓 (𝑥𝑛 ) + ⟨𝜇𝑛 , 𝑥 − 𝑥𝑛 ⟩ ≤ 0, 𝑔 (𝑦𝑛 ) + ⟨𝜂𝑛 , 𝑦 − 𝑦𝑛 ⟩ ≤ 0} ,

A mapping 𝑃𝐶 is said to be metric projection of 𝐻1 onto 𝐶 if, for every point 𝑥 ∈ 𝐻1 , there exists a unique nearest point in 𝐶 denoted by 𝑃𝐶𝑥 such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑥 − 𝑃𝐶𝑥󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , ∀𝑦 ∈ 𝐶.

(11)

It is well known that 𝑃𝐶 is a nonexpansive mapping and is characterized by the following properties: 󵄩󵄩 󵄩2 󵄩󵄩𝑃𝐶𝑥 − 𝑃𝐶𝑦󵄩󵄩󵄩 ≤ ⟨𝑥 − 𝑦, 𝑃𝐶𝑥 − 𝑃𝐶𝑦⟩ , ∀𝑥, 𝑦 ∈ 𝐻1 ,

𝑥𝑛+1 = 𝑃𝐶𝑛+1 (𝑥1 ) ,

⟨𝑥 − 𝑃𝐶𝑥, 𝑦 − 𝑃𝐶𝑥⟩ ≤ 0,

𝑦𝑛+1 = 𝑃𝑄𝑛+1 (𝑦1 ) , ∀𝑛 ∈ 𝑁,

󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩2 󵄩 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩𝑥 − 𝑃𝐶𝑥󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦 − 𝑃𝐶𝑥󵄩󵄩󵄩 ,

where 𝜉 ∈ (0, min(1/‖𝐴‖2 , 1/‖𝐵‖2 )), 𝜇𝑛 ∈ 𝜕𝑓(𝑥𝑛 ), 𝜂𝑛 ∈ 𝜕𝑔(𝑦𝑛 ), and 𝜍𝑛 ∈ 𝜕ℎ((𝐴𝑥𝑛 + 𝐵𝑦𝑛 )/2). In some conditions such as Ω = {(𝑥, 𝑦) ∈ 𝐶 × 𝑄 : 𝐴𝑥 = 𝐵𝑦 ∈ 𝐾} ≠ 0, they proved that the sequences (𝑥𝑛 , 𝑦𝑛 ) → (𝑠, 𝑡), (𝑢𝑛 , V𝑛 ) → (𝑠, 𝑡), and 𝜔𝑛 → 𝑧∗ fl 𝐴𝑠 = 𝐵𝑡 ∈ 𝐾, where (𝑠, 𝑡) ∈ Ω.

(10)

∀𝑥 ∈ 𝐻1 , 𝑦 ∈ 𝐶,

∀𝑥 ∈ 𝐻1 , 𝑦 ∈ 𝐶, 󵄩󵄩 󵄩2 󵄩󵄩(𝑥 − 𝑦) − (𝑃𝐶𝑥 − 𝑃𝐶𝑦)󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 ≥ 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑃𝐶𝑥 − 𝑃𝐶𝑦󵄩󵄩󵄩 , ∀𝑥, 𝑦 ∈ 𝐻1 .

(12)


3

In the proof of our results, we need the following lemmas.

be sequences generated by

Lemma 1 (see [16]). Let 𝐻 be a real Hilbert space; then the following conclusions hold.


󵄩󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 = ‖𝑥‖ + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 − 2 ⟨𝑥, 𝑦⟩ ,

𝐴𝑥𝑛 + 𝐵𝑦𝑛 ), 2

𝐾𝑛 = {𝑧 ∈ 𝐻3 : ℎ (

∀𝑥, 𝑦 ∈ 𝐻, 󵄩󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩𝛼𝑥 + (1 − 𝛼) 𝑦󵄩󵄩󵄩 = 𝛼 ‖𝑥‖ + (1 − 𝛼) 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 󵄩 󵄩 − 𝛼 (1 − 𝛼) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 ,

+ ⟨𝜍𝑛 , 𝑧 −

(13)

(14)

If 0 < 𝜉 < 𝜂 < 1/(1 + √1 + 𝐿2 ), then the following conclusions hold. (i) Fix(𝑇) = Fix(𝑇((1 − 𝜂)𝐼 + 𝜂𝑇)) = Fix(𝐺). (ii) If 𝑇 is demiclosed at 0, then 𝐺 is also demiclosed at 0. (iii) In addition, if 𝑇 : 𝐻 → H is quasi-pseudo-contractive, then the mapping 𝐺 is quasi-nonexpansive; that is,

∀𝑥 ∈ 𝐻, 𝑢∗ ∈ Fix (𝑇) = Fix (𝐺) .

(15)

3. Main Results In this section, we assume that (i) 𝐺 : 𝐻1 → 𝐻1 and 𝑀 : 𝐻2 → 𝐻2 are two 𝐿Lipschitzian and quasi-pseudo-contractive mappings with 𝐿 ≥ 1, Fix(𝐺) ≠ 0, and Fix(𝑀) ≠ 0. (ii) 𝐻1 , 𝐻2 , and 𝐻3 be real Hilbert spaces. 𝐴 : 𝐻1 → 𝐻3 and 𝐵 : 𝐻2 → 𝐻3 are two bounded linear operators; 𝐴∗ and 𝐵∗ are their adjoint operators, respectively. Let 𝐾 ⊂ 𝐻3 be nonempty closed convex set. We consider the extended split equality fixed point problem (8). Theorem 3. Let 𝐻1 , 𝐻2 , and 𝐻3 be real Hilbert spaces and 𝐾 be a closed convex level set 𝐾 = {𝑥 ∈ 𝐻3 : ℎ (𝑥) ≤ 0} ,

(17)

𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑢𝑛 , V𝑛 = 𝑦𝑛 − 𝜉𝑛 𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 ) ,

Lemma 2 (see [16]). Let 𝐻 be a real Hilbert space and 𝑇 : 𝐻 → 𝐻 be a 𝐿-Lipschitzian mapping with 𝐿 ≥ 1. Denote

󵄩 󵄩󵄩 ∗󵄩 ∗󵄩 󵄩󵄩𝐺𝑥 − 𝑢 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥 − 𝑢 󵄩󵄩󵄩 ,

𝐴𝑥𝑛 + 𝐵𝑦𝑛 ⟩ ≤ 0} , 2

𝑢𝑛 = 𝑥𝑛 − 𝜉𝑛 𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 ) ,

∀𝑥, 𝑦 ∈ 𝐻, 𝛼 ∈ [0, 1] .

𝐺 fl (1 − 𝜉) 𝐼 + 𝜉𝑇 ((1 − 𝜂) 𝐼 + 𝜂𝑇) .


(16)

where ℎ : 𝐻3 → 𝑅 is convex function which is subdifferentiable on 𝐾 and its subdifferentials are bounded on bounded sets. 𝐴 : 𝐻1 → 𝐻3 and 𝐵 : 𝐻2 → 𝐻3 are two bounded linear operators with their adjoint operators 𝐴∗ and 𝐵∗ , respectively, 𝜎 ∈ (0, 1) is a parameter controlling step length, 0 < 𝛾 < 𝜂 < 1/(1 + √1 + 𝐿2 ), and {𝛼𝑛 } ⊂ (0, 1). Let {𝑥𝑛 }, {𝑦𝑛 }, {𝜔𝑛 }, {𝑢𝑛 }, and {V𝑛 }

𝑦𝑛+1 = 𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑆V𝑛 , where 𝜍𝑛 ∈ 𝜕ℎ (


𝜉𝑛 ∈ (0, (1 − 𝜎) 󵄩2 󵄩 󵄩2 󵄩󵄩 󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 ⋅󵄩 ), 󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩

(18)

𝑇 fl (1 − 𝛾) 𝐼 + 𝛾𝐺 ((1 − 𝜂) 𝐼 + 𝜂𝐺) , 𝑆 fl (1 − 𝛾) 𝐼 + 𝛾𝑀 ((1 − 𝜂) 𝐼 + 𝜂𝑀) , where 𝐺 and 𝑀 are demiclosed at 0. Assume Ω = {(𝑥, 𝑦) ∈ Fix(𝐺) × Fix(𝑀) : 𝐴𝑥 = 𝐵𝑦 ∈ 𝐾} ≠ 0. Then (𝑥𝑛 , 𝑦𝑛 ) ⇀ (𝑡, 𝑠), (𝑢𝑛 , V𝑛 ) ⇀ (𝑡, 𝑠), and 𝜔𝑛 ⇀ 𝑧∗ fl 𝐴𝑡 = 𝐵𝑠, where (𝑡, 𝑠) ∈ Ω. Proof. It is obvious that 𝐾 ⊂ 𝐾𝑛 for any 𝑛 ∈ 𝑁. Let (𝑥∗ , 𝑦∗ ) ∈ Ω; namely, 𝑥∗ ∈ Fix(𝐺), 𝑦∗ ∈ Fix(𝑀), and 𝜔∗ = 𝐴𝑥∗ = 𝐵𝑦∗ ∈ 𝐾. By Lemma 1, we have 󵄩󵄩 󵄩󵄩2 𝐴𝑥 + 𝐵𝑦𝑛 󵄩 󵄩 󵄩󵄩 ∗󵄩 ) − 𝑃𝐾𝑛 (𝜔∗ )󵄩󵄩󵄩 󵄩󵄩𝜔𝑛 − 𝜔 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑃𝐾𝑛 ( 𝑛 󵄩󵄩 2 󵄩󵄩 󵄩󵄩2 1 󵄩󵄩 1 ≤ 󵄩󵄩󵄩󵄩 (𝐴𝑥𝑛 − 𝜔∗ ) + (𝐵𝑦𝑛 − 𝜔∗ )󵄩󵄩󵄩󵄩 2 󵄩2 󵄩 1󵄩 󵄩2 1 󵄩 󵄩2 = 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔∗ 󵄩󵄩󵄩 2 2 −

(19)

1 󵄩󵄩 󵄩2 󵄩𝐴𝑥 − 𝐵𝑦𝑛 󵄩󵄩󵄩 . 4󵄩 𝑛

󵄩󵄩 󵄩 󵄩2 ∗ 󵄩2 ∗ ∗ 󵄩󵄩𝑢𝑛 − 𝑥 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑥𝑛 − 𝑥 − 𝜉𝑛 𝐴 (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 󵄩2 󵄩 󵄩 󵄩2 = 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 𝜉𝑛2 󵄩󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 − 2 ⟨𝑥𝑛 − 𝑥∗ , 𝜉𝑛 𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )⟩ .

(20)

4


By (20), we have 󵄩󵄩 󵄩 󵄩2 ∗ 󵄩2 ∗ 󵄩2 2󵄩 ∗ 󵄩󵄩𝑢𝑛 − 𝑥 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 + 𝜉𝑛 󵄩󵄩󵄩𝐴 (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 − 𝜉𝑛 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔∗ 󵄩󵄩󵄩 − 𝜉𝑛 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 󵄩 󵄩2 + 𝜉𝑛 󵄩󵄩󵄩𝜔𝑛 − 𝜔∗ 󵄩󵄩󵄩 .

(21)

󵄩 󵄩2 󵄩 󵄩2 ⋅ 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 𝜉𝑛2 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 󵄩2 󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔∗ 󵄩󵄩󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 )

Similarly, 󵄩 󵄩2 󵄩󵄩 ∗ 󵄩2 ∗ 󵄩2 2󵄩 ∗ 󵄩󵄩V𝑛 − 𝑦 󵄩󵄩󵄩 = 󵄩󵄩󵄩𝑦𝑛 − 𝑦 󵄩󵄩󵄩 + 𝜉𝑛 󵄩󵄩󵄩𝐵 (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 − 𝜉𝑛 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔∗ 󵄩󵄩󵄩 − 𝜉𝑛 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 󵄩 󵄩2 + 𝜉𝑛 󵄩󵄩󵄩𝜔𝑛 − 𝜔∗ 󵄩󵄩󵄩 .

󵄩2 󵄩2 1 󵄩 󵄩 ⋅ 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔∗ 󵄩󵄩󵄩 + 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔∗ 󵄩󵄩󵄩 − 2 󵄩2 󵄩 ⋅ 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝐵𝑦𝑛 󵄩󵄩󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) 󵄩2 󵄩 󵄩2 󵄩 ⋅ 󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑢𝑛 󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 )

󵄩2 󵄩2 󵄩 󵄩 ⋅ 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑆V𝑛 󵄩󵄩󵄩 (22)

By (21), (22), (17), and (19), we have 󵄩 󵄩󵄩 ∗ 󵄩2 ∗ 󵄩2 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛+1 − 𝑦 󵄩󵄩󵄩 󵄩2 󵄩 = 󵄩󵄩󵄩𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑢𝑛 − 𝑥∗ 󵄩󵄩󵄩 󵄩2 󵄩 + 󵄩󵄩󵄩𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑆V𝑛 − 𝑦∗ 󵄩󵄩󵄩 󵄩2 󵄩 = 󵄩󵄩󵄩𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑢𝑛 − (1 − 𝛼𝑛 ) 𝑥∗ − 𝛼𝑛 𝑥∗ 󵄩󵄩󵄩 󵄩2 󵄩 + 󵄩󵄩󵄩𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑆V𝑛 − (1 − 𝛼𝑛 ) 𝑦∗ − 𝛼𝑛 𝑦∗ 󵄩󵄩󵄩 󵄩 󵄩2 󵄩2 󵄩 = 𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑇𝑢𝑛 − 𝑥∗ 󵄩󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑢𝑛 󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑆V𝑛 − 𝑦∗ 󵄩󵄩󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑆V𝑛 󵄩󵄩󵄩 󵄩 󵄩2 󵄩2 󵄩 ≤ 𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑢𝑛 − 𝑥∗ 󵄩󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑢𝑛 󵄩󵄩󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩V𝑛 − 𝑦∗ 󵄩󵄩󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑆V𝑛 󵄩󵄩󵄩 󵄩 󵄩2 󵄩2 󵄩 = 𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩 󵄩2 󵄩2 󵄩 ⋅ 𝜉𝑛2 󵄩󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔∗ 󵄩󵄩󵄩 󵄩2 󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 2𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩2 󵄩2 󵄩 󵄩 ⋅ 󵄩󵄩󵄩𝜔𝑛 − 𝜔∗ 󵄩󵄩󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑢𝑛 󵄩󵄩󵄩 󵄩 󵄩2 󵄩2 󵄩 + 𝛼𝑛 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩 󵄩2 󵄩2 󵄩 ⋅ 𝜉𝑛2 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔∗ 󵄩󵄩󵄩 󵄩2 󵄩2 󵄩 󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑦𝑛 − 𝑆V𝑛 󵄩󵄩󵄩 󵄩 󵄩2 󵄩2 󵄩 ≤ 𝛼𝑛 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩 󵄩2 󵄩2 󵄩 ⋅ 𝜉𝑛2 󵄩󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔∗ 󵄩󵄩󵄩 󵄩2 󵄩 − 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 𝜉𝑛 (1 − 𝛼𝑛 )

󵄩2 󵄩 󵄩2 󵄩 = 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 + (1 − 𝛼𝑛 ) 󵄩 󵄩2 󵄩 󵄩2 ⋅ 𝜉𝑛2 (󵄩󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 ) 󵄩2 󵄩 󵄩2 󵄩 − (1 − 𝛼𝑛 ) 𝜉𝑛 (󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 ) 󵄩2 󵄩 󵄩2 󵄩 − 𝛼𝑛 (1 − 𝛼𝑛 ) (󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑢𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑆V𝑛 󵄩󵄩󵄩 ) −

1 󵄩 󵄩2 (1 − 𝛼𝑛 ) 𝜉𝑛 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝐵𝑦𝑛 󵄩󵄩󵄩 . 2 (23)

Notice that 󵄩󵄩 󵄩 󵄩 󵄩 ∗ 󵄩2 ∗ 󵄩2 ∗ 󵄩2 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛+1 − 𝑦 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 󵄩2 − 𝑦∗ 󵄩󵄩󵄩 + 𝜉𝑛 (1 − 𝛼𝑛 ) 󵄩 󵄩2 󵄩 󵄩2 ⋅ [𝜉𝑛 (󵄩󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 )

(24)

󵄩2 󵄩 󵄩2 󵄩 − (󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 )] . Thanks to 𝜉𝑛 ∈ (0, (1 − 𝜎) 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 ), ⋅󵄩 󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩

(25)

we obtain 󵄩󵄩 󵄩 ∗ 󵄩2 ∗ 󵄩2 󵄩󵄩𝑥𝑛+1 − 𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛+1 − 𝑦 󵄩󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 ≤ 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 .

(26)

󵄩2 󵄩 󵄩2 󵄩 𝑋𝑛 (𝑥∗ , 𝑦∗ ) = 󵄩󵄩󵄩𝑥𝑛 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑦∗ 󵄩󵄩󵄩 .

(27)

𝑋𝑛+1 (𝑥∗ , 𝑦∗ ) ≤ 𝑋𝑛 (𝑥∗ , 𝑦∗ ) .

(28)

Let

We obtain

This implies that 𝑋𝑛 (𝑥∗ , 𝑦∗ ) is a nonincreasing sequence; hence lim𝑛→∞ 𝑋𝑛 (𝑥∗ , 𝑦∗ ) exists. As a result, {𝑥𝑛 } and {𝑦𝑛 } are


5 that is,

bounded sequences. Rewrite (23) as 󵄩2 󵄩 󵄩2 󵄩 (1 − 𝛼𝑛 ) 𝜉𝑛 [(󵄩󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 )

𝐴𝑡 = 𝐵𝑠. ∗

󵄩 󵄩2 󵄩 󵄩2 − 𝜉𝑛 (󵄩󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩 )] 1 󵄩2 󵄩 󵄩2 󵄩 + 𝛼𝑛 (1 − 𝛼𝑛 ) (󵄩󵄩󵄩𝑥𝑛 − 𝑇𝑢𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑆V𝑛 󵄩󵄩󵄩 ) + (1 2 (29) 2 2 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩 󵄩 ∗󵄩 ∗ − 𝛼𝑛 ) 𝜉𝑛 󵄩󵄩𝐴𝑥𝑛 − 𝐵𝑦𝑛 󵄩󵄩 ≤ 󵄩󵄩𝑥𝑛 − 𝑥 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛 − 𝑦 󵄩󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩 − (󵄩󵄩󵄩𝑥𝑛+1 − 𝑥∗ 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦𝑛+1 − 𝑦∗ 󵄩󵄩󵄩 ) = 𝑋𝑛 (𝑥∗ , 𝑦∗ ) − 𝑋𝑛+1 (𝑥∗ , 𝑦∗ ) . 󵄩 󵄩󵄩 󵄩󵄩𝐴𝑥𝑛 − 𝐵𝑦𝑛 󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩 󵄩 󵄩󵄩𝑇𝑢𝑛 − 𝑥𝑛 󵄩󵄩󵄩 󳨀→ 0, 󵄩 󵄩󵄩 󵄩󵄩𝑆V𝑛 − 𝑦𝑛 󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩 󵄩󵄩 󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩 󳨀→ 0, 󵄩󵄩 󵄩󵄩 󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩 󳨀→ 0.

By (31), we have 𝜔𝑛 ⇀ 𝑧 fl 𝐴𝑡 = 𝐵𝑠. Now, we prove 𝑧 ∈ 𝐾. We know that {𝜍𝑛 } is bounded. There exists 𝑀 > 0, such that ‖𝜍𝑛 ‖ ≤ 𝑀, where 𝑀 is a constant. Note that 𝜔𝑛 = 𝑃𝐾𝑛 ((𝐴𝑥𝑛 + 𝐵𝑦𝑛 )/2) ∈ 𝐾𝑛 , and we have ℎ(

𝐴𝑥𝑛 + 𝐵𝑦𝑛 𝐴𝑥𝑛 + 𝐵𝑦𝑛 ) + ⟨𝜍𝑛 , 𝜔𝑛 − ⟩ ≤ 0. 2 2

(37)

𝐴𝑥𝑛 + 𝐵𝑦𝑛 𝐴𝑥𝑛 + 𝐵𝑦𝑛 ) ≤ − ⟨𝜍𝑛 , 𝜔𝑛 − ⟩ 2 2 󵄩󵄩 𝐴𝑥𝑛 + 𝐵𝑦𝑛 󵄩󵄩󵄩󵄩 󵄩 ≤ 𝑀 󵄩󵄩󵄩𝜔𝑛 − 󵄩󵄩 . 󵄩󵄩 󵄩󵄩 2

(38)

Hence, ℎ(

Letting 𝑛 → ∞ and taking the limit in (29), we have

(30)

(36) ∗

By the lower semicontinuity of ℎ, (38), and (31), we obtain 𝐴𝑥𝑛 + 𝐵𝑦𝑛 ) 2 󵄩󵄩 𝐴𝑥𝑛 + 𝐵𝑦𝑛 󵄩󵄩󵄩󵄩 󵄩 ≤ 𝑀 lim inf 󵄩󵄩󵄩𝜔𝑛 − 󵄩󵄩 = 0. 𝑛→∞ 󵄩 󵄩󵄩 2 󵄩

ℎ (𝑧∗ ) ≤ lim inf ℎ ( 𝑛→∞

(31)

(39)

Thus 𝑧∗ ∈ 𝐾. The proof is completed.

Then, 󵄩󵄩

󵄩 − 𝑥𝑛 󵄩󵄩󵄩 = 0,

4. Consequent Results

󵄩󵄩

󵄩 − 𝑦𝑛 󵄩󵄩󵄩 = 0,

In this section, we give some corollaries, which are easily obtained from Theorem 3. If 𝐺 = 𝑀, we have the following corollary.

lim 󵄩𝑢 𝑛→∞ 󵄩 𝑛 lim 󵄩V 𝑛→∞ 󵄩 𝑛

󵄩 󵄩 󵄩 󵄩 lim 󵄩󵄩𝑥 − 𝑥𝑛 󵄩󵄩󵄩 = lim 󵄩󵄩󵄩𝑇𝑢𝑛 − 𝑥𝑛 󵄩󵄩󵄩 = 0, 𝑛→∞ 󵄩 𝑛+1 𝑛→∞

(32)

󵄩 󵄩 󵄩 󵄩 lim 󵄩󵄩𝑦 − 𝑦𝑛 󵄩󵄩󵄩 = lim 󵄩󵄩󵄩𝑆V𝑛 − 𝑦𝑛 󵄩󵄩󵄩 = 0, 𝑛→∞ 󵄩 𝑛+1 𝑛→∞

Corollary 4. Let 𝐻1 , 𝐻2 , and 𝐻3 be real Hilbert spaces and 𝐾 be a closed convex level set

which imply that {𝑥𝑛 } and {𝑦𝑛 } are asymptotically regular. Furthermore, we get

𝐾 = {𝑥 ∈ 𝐻3 : ℎ (𝑥) ≤ 0} ,

󵄩 󵄩 lim 󵄩󵄩𝑇𝑢𝑛 − 𝑢𝑛 󵄩󵄩󵄩 = 0, 𝑛→∞ 󵄩 󵄩 󵄩 lim 󵄩󵄩𝑆V − V𝑛 󵄩󵄩󵄩 = 0. 𝑛→∞ 󵄩 𝑛

(33)

Since {𝑥𝑛 } and {𝑦𝑛 } are bounded sequences, there exist weakly convergent subsequences, say {𝑥𝑛𝑖 } ⊂ {𝑥𝑛 } such that 𝑥𝑛𝑖 ⇀ 𝑡; also {𝑦𝑛𝑖 } ⊂ {𝑦𝑛 } such that 𝑦𝑛𝑖 ⇀ 𝑠. The Opial property guarantees that the weakly subsequential limit of {(𝑥𝑛 , 𝑦𝑛 )} is unique. So we have 𝑥𝑛 ⇀ 𝑡, 𝑦𝑛 ⇀ 𝑠. Therefore 𝑢𝑛 ⇀ 𝑡, V𝑛 ⇀ 𝑠. Since 𝐺 and 𝑀 are demiclosed at 0, and from Lemma 2, by (33), we have 𝑇𝑡 = 𝑡, 𝑆𝑠 = 𝑠, which imply that 𝑡 ∈ Fix(𝐺), 𝑠 ∈ Fix(𝑀). Hence, (𝑡, 𝑠) ∈ Fix (𝐺) × Fix (𝑀) .

(34)

Furthermore, since 𝐴𝑥𝑛 −𝐵𝑦𝑛 ⇀ 𝐴𝑡−𝐵𝑠, by using the weakly lower semicontinuity of squared norm, we have 󵄩 󵄩 󵄩 󵄩 inf 󵄩󵄩𝐴𝑥𝑛 − 𝐵𝑦𝑛 󵄩󵄩󵄩 = lim 󵄩󵄩󵄩𝐴𝑥𝑛 − 𝐵𝑦𝑛 󵄩󵄩󵄩 ‖𝐴𝑡 − 𝐵𝑠‖ ≤ lim 𝑛→∞ 󵄩 𝑛→∞ = 0;

(35)

(40)

where ℎ : 𝐻3 → 𝑅 is convex function which is subdifferentiable on 𝐾 and its subdifferentials are bounded on bounded sets. 𝐴 : 𝐻1 → 𝐻3 and 𝐵 : 𝐻2 → 𝐻3 are two bounded linear operators with their adjoint operators 𝐴∗ and 𝐵∗ , respectively, 𝜎 ∈ (0, 1) is a parameter controlling step length, 0 < 𝛾 < 𝜂 < 1/(1 + √1 + 𝐿2 ), and {𝛼𝑛 } ⊂ (0, 1). Let {𝑥𝑛 }, {𝑦𝑛 }, {𝜔𝑛 }, {𝑢𝑛 }, and {V𝑛 } be sequences generated by 𝜔𝑛 = 𝑃𝐾𝑛 (


𝐾𝑛 = {𝑧 ∈ 𝐻3 : ℎ ( + ⟨𝜍𝑛 , 𝑧 −


𝐴𝑥𝑛 + 𝐵𝑦𝑛 ⟩ ≤ 0} , 2

𝑢𝑛 = 𝑥𝑛 − 𝜉𝑛 𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 ) , 𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑢𝑛 , V𝑛 = 𝑦𝑛 − 𝜉𝑛 𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 ) , 𝑦𝑛+1 = 𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑇V𝑛 ,

(41)

6

Mathematical Problems in Engineering where 𝐺 and 𝑀 are demiclosed at 0. Assume Ω = {(𝑥, 𝑦) ∈ Fix(𝐺) × Fix(𝑀) : 𝐴𝑥 = 𝐴𝑦 ∈ 𝐾} ≠ 0. Then (𝑥𝑛 , 𝑦𝑛 ) ⇀ (𝑠, 𝑡), (𝑢𝑛 , V𝑛 ) ⇀ (𝑠, 𝑡), and 𝜔𝑛 ⇀ 𝑧∗ fl 𝐴𝑠 = 𝐴𝑡, where (𝑠, 𝑡) ∈ Ω.

where 𝜍𝑛 ∈ 𝜕ℎ (


If 𝐺 = 𝑀 and 𝐴 = 𝐵, we have the following corollary.

𝜉𝑛 ∈ (0, (1 − 𝜎) 󵄩2 󵄩 󵄩2 󵄩󵄩 󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 ), ⋅󵄩 󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩

(42)

Corollary 6. Let 𝐻1 and 𝐻3 be real Hilbert spaces and 𝐾 be a closed convex level set 𝐾 = {𝑥 ∈ 𝐻3 : ℎ (𝑥) ≤ 0} ,

𝑇 fl (1 − 𝛾) 𝐼 + 𝛾𝐺 ((1 − 𝜂) 𝐼 + 𝜂𝐺) , where 𝐺 is demiclosed at 0. Assume Ω = {(𝑥, 𝑦) ∈ Fix(𝐺) × Fix(𝐺) : 𝐴𝑥 = 𝐵𝑦 ∈ 𝐾} ≠ 0. Then (𝑥𝑛 , 𝑦𝑛 ) ⇀ (𝑡, 𝑠), (𝑢𝑛 , V𝑛 ) ⇀ (𝑡, 𝑠) and 𝜔𝑛 ⇀ 𝑧∗ fl 𝐴𝑡 = 𝐵𝑠, where (𝑡, 𝑠) ∈ Ω. If 𝐴 = 𝐵, we have the following corollary.

where ℎ : 𝐻3 → 𝑅 is convex function which is subdifferentiable on 𝐾 and its subdifferentials are bounded on bounded sets. 𝐴 : 𝐻1 → 𝐻3 is a bounded linear operator with its adjoint operator 𝐴∗ , and 𝜎 ∈ (0, 1) is a parameter controlling step length, 0 < 𝛾 < 𝜂 < 1/(1 + √1 + 𝐿2 ), and {𝛼𝑛 } ⊂ (0, 1). Let {𝑥𝑛 }, {𝑦𝑛 }, {𝜔𝑛 }, {𝑢𝑛 }, and {V𝑛 } be sequences generated by



𝐴𝑥𝑛 + 𝐴𝑦𝑛 ⟩ ≤ 0} , 2

𝑢𝑛 = 𝑥𝑛 − 𝜉𝑛 𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 ) ,

∗

(44)

𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑢𝑛 , V𝑛 = 𝑦𝑛 − 𝜉𝑛 𝐴∗ (𝐴𝑦𝑛 − 𝜔𝑛 ) ,


𝜉𝑛 ∈ (0, (1 − 𝜎) 󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 ), ⋅󵄩 󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩

𝑦𝑛+1 = 𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑆V𝑛 ,

(48)

𝑇 fl (1 − 𝛾) 𝐼 + 𝛾𝐺 ((1 − 𝜂) 𝐼 + 𝜂𝐺) ,

where

where 𝐺 is demiclosed at 0. Assume Ω = {(𝑥, 𝑦) ∈ Fix(𝐺) × Fix(𝐺) : 𝐴𝑥 = 𝐴𝑦 ∈ 𝐾} ≠ 0. Then (𝑥𝑛 , 𝑦𝑛 ) ⇀ (𝑡, 𝑠), (𝑢𝑛 , V𝑛 ) ⇀ (𝑡, 𝑠), and 𝜔𝑛 ⇀ 𝑧∗ fl 𝐴𝑡 = 𝐴𝑠, where (𝑡, 𝑠) ∈ Ω.


If 𝐺 = 𝑀 = 𝐴 = 𝐵 is an identity operator, we have the following corollary.

𝜉𝑛 ∈ (0, (1 − 𝜎)

𝑆 fl (1 − 𝛾) 𝐼 + 𝛾𝑀 ((1 − 𝜂) 𝐼 + 𝜂𝑀) ,

(47)

𝑢𝑛 = 𝑥𝑛 − 𝜉𝑛 𝐴 (𝐴𝑥𝑛 − 𝜔𝑛 ) ,

𝜍𝑛 ∈ 𝜕ℎ (

V𝑛 = 𝑦𝑛 − 𝜉𝑛 𝐴∗ (𝐴𝑦𝑛 − 𝜔𝑛 ) ,

𝑇 fl (1 − 𝛾) 𝐼 + 𝛾𝐺 ((1 − 𝜂) 𝐼 + 𝜂𝐺) ,

𝐴𝑥𝑛 + 𝐴𝑦𝑛 ⟩ ≤ 0} , 2

where

𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑇𝑢𝑛 ,

󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩𝐴𝑥𝑛 − 𝜔𝑛 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝐵𝑦𝑛 − 𝜔𝑛 󵄩󵄩󵄩 ⋅󵄩 ), 󵄩󵄩𝐴∗ (𝐴𝑥𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 + 󵄩󵄩󵄩𝐵∗ (𝐵𝑦𝑛 − 𝜔𝑛 )󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩

𝐴𝑥𝑛 + 𝐴𝑦𝑛 ) 2

𝑦𝑛+1 = 𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) 𝑇V𝑛 ,

𝐴𝑥 + 𝐴𝑦𝑛 𝐾𝑛 = {𝑧 ∈ 𝐻3 : ℎ ( 𝑛 ) 2

𝜍𝑛 ∈ 𝜕ℎ (

+ ⟨𝜍𝑛 , 𝑧 −

𝐴𝑥𝑛 + 𝐴𝑦𝑛 ), 2

+ ⟨𝜍𝑛 , 𝑧 −

𝐴𝑥𝑛 + 𝐴𝑦𝑛 ), 2

𝐾𝑛 = {𝑧 ∈ 𝐻3 : ℎ (

(43)

where ℎ : 𝐻3 → 𝑅 is convex function which is subdifferentiable on 𝐾 and its subdifferentials are bounded on bounded sets. 𝐴 : 𝐻1 → 𝐻3 is a bounded linear operator with its adjoint operator 𝐴∗ , 𝜎 ∈ (0, 1) is a parameter controlling step length, 0 < 𝛾 < 𝜂 < 1/(1 + √1 + 𝐿2 ), and {𝛼𝑛 } ⊂ (0, 1). Let {𝑥𝑛 }, {𝑦𝑛 }, {𝜔𝑛 }, {𝑢𝑛 }, and {V𝑛 } be sequences generated by 𝜔𝑛 = 𝑃𝐾𝑛 (

(46)

(45)


(49)

where ℎ : 𝐻3 → 𝑅 is convex function which is subdifferentiable on 𝐾 and its subdifferentials are bounded on bounded sets.


7 ×104 16

25

14 20

12 10

15

8 10

6 4

5

2 0

0

5

10 15 20 Number of iterations

25

30

0

0

norm(Axn − Byn )

5

10

15 20 25 30 Number of iterations

35

40

45

35

40

45

norm(Axn − Byn )

×107 16000

3

14000

2.5

12000 2

10000 8000

1.5

6000

1

4000 0.5 0

2000 0

10

20 30 40 Number of iterations

50

60

norm(Axn − Byn )

0

0

5

10

15 20 25 30 Number of iterations

norm(Axn − Byn )

Figure 1: The behaviors of ‖𝐴𝑥𝑛 − 𝐵𝑦𝑛 ‖ for different initial points.

𝜎 ∈ (0, 1) is a parameter controlling step length, 0 < 𝛾 < 𝜂 < 1/(1 + √1 + 𝐿2 ), and {𝛼𝑛 } ⊂ (0, 1). Let {𝑥𝑛 }, {𝑦𝑛 }, {𝜔𝑛 }, {𝑢𝑛 }, and {V𝑛 } be sequences generated by 𝑥𝑛 + 𝑦𝑛 ), 2 𝑥 + 𝑦𝑛 𝑥 + 𝑦𝑛 ) + ⟨𝜍𝑛 , 𝑧 − 𝑛 ⟩ 𝐾𝑛 = {𝑧 ∈ 𝐻3 : ℎ ( 𝑛 2 2

5. Numerical Examples


≤ 0} , 𝑢𝑛 = (1 − 𝜉) 𝑥𝑛 + 𝜉𝜔𝑛 , 𝑥𝑛+1 = 𝛼𝑛 𝑥𝑛 + (1 − 𝛼𝑛 ) 𝑢𝑛 , V𝑛 = (1 − 𝜉) 𝑦𝑛 + 𝜉𝜔𝑛 , 𝑦𝑛+1 = 𝛼𝑛 𝑦𝑛 + (1 − 𝛼𝑛 ) V𝑛 ,

where 𝜍𝑛 ∈ 𝜕ℎ((𝑥𝑛 + 𝑦𝑛 )/2), 𝜉 ∈ (0, 1 − 𝜎). Assume Ω = {(𝑥, 𝑦) ∈ 𝐻1 × 𝐻1 : 𝑥 = 𝑦 ∈ 𝐾} ≠ 0. Then (𝑥𝑛 , 𝑦𝑛 ) ⇀ (𝑡, 𝑠), (𝑢𝑛 , V𝑛 ) ⇀ (𝑡, 𝑠), and 𝜔𝑛 ⇀ 𝑧∗ fl 𝑡 = 𝑠, where (𝑡, 𝑠) ∈ Ω.

(50)

In this section, we give an example to show some insight into the behavior of the algorithm presented in this paper. The whole codes are written in Matlab 7.0. All the numerical results are carried out on a personal Lenovo ThinkPad computer with Intel(R) Core(TM) i7-6500U CPU 2.50 GHz and RAM 8.00 GB. Let 𝐻1 = 𝑅2 , 𝐻2 = 𝑅3 . 𝐴 ∈ 𝑅3×2 and 𝐵 ∈ 𝑅3×3 are as follows. 0.8147 0.9134 𝐴 = (0.9058 0.6324) , 0.1270 0.0975

8

Mathematical Problems in Engineering Table 1: Number of iterations and the CPU time for different initial points. Initial point 𝑇

𝑥 = (1, 1) , 𝑥 = (5.2378, 2.6487)𝑇 , 𝑥 = (0.5221, 19.0936)𝑇 , 𝑥 = (0.0588, 5.4403)𝑇 ,

𝑦 = (1, 1, 1) 𝑦 = (6.8357, 43.6327, 17.3853)𝑇 𝑦 = (86.1193, 192.3197, 152.4829)𝑇 𝑦 = (14.1190, 12.9026, 11.0462)𝑇

36

38

34

37

32

36

30

35

28

34

26

33

24

32

22

31

20

30

18

0

100

200

300

400

500

29

60

83

59

82

58

81

57

80

56

79

55

78

54

77

53

76

52

75

51

0

100

Iter. 27 42 50 38

𝑇

200

300

400

500

74

Sec. 0.01 0.016 0.017 0.012

0

100

200

300

400

500

0

100

200

300

400

500

Figure 2: The behaviors of iterative numbers for different initial points.

0.2785 0.9649 0.9572 𝐵 = (0.5469 0.1576 0.4854) . 0.9575 0.9706 0.8003

𝑥 = (5.2378, 2.6487)𝑇 , 𝑦 = (6.8357, 43.6327, 17.3853)𝑇 , (51)

𝑛

𝑥13

𝑥22

Let 𝐾 = {𝑥 ∈ 𝑅 | ℎ(𝑥) = + + 𝑥3 − 2 ≤ 0}, where 𝑥 = (𝑥1 , 𝑥2 , 𝑥3 )𝑇 . For convenience, we take 𝜎 = 0.2, 𝛼𝑛 = 1/3 + 1/2𝑛 , and 𝐺 = 𝑀 = (1/2)𝐼. We choose ‖𝐴𝑥𝑛 − 𝐵𝑦𝑛 ‖ ≤ 10−6 as the stopping criterion. Figure 1 presents the behaviors of ‖𝐴𝑥𝑛 − 𝐵𝑦𝑛 ‖ in different initial points such as 𝑥 = (1, 1)𝑇 , 𝑦 = (1, 1, 1)𝑇 ,

𝑥 = (0.5221, 19.0936)𝑇 , 𝑦 = (86.1193, 192.3197, 152.4829)𝑇 , 𝑥 = (0.0588, 5.4403)𝑇 , 𝑦 = (14.1190, 12.9026, 11.0462)𝑇 . (52) It is easy to see that the presentation reveals that 𝐴𝑥 = 𝐵𝑦. Table 1 shows the number of iterations and the CPU time


9

for the above four initial points. We denote Iter. and Sec. as the number of iterations and the CPU time in seconds, respectively. Furthermore, for testing the stationary property of iterative numbers, we carry out 500 experiments for different initial points which are presented randomly, such as 𝑥 = rand (2, 1) , 𝑦 = rand (3, 1) , 𝑥 = 10 ∗ rand (2, 1) , 𝑦 = 10 ∗ rand (3, 1) , 𝑥 = 1000 ∗ rand (2, 1) ,

(53)

𝑦 = 600 ∗ rand (3, 1) , 𝑥 = 100000 ∗ rand (2, 1) , 𝑦 = 60000 ∗ rand (3, 1) , separately. In these cases, we take 𝜎 = 0.2, 𝛼𝑛 = 0.4 and choose ‖𝐴𝑥𝑛 − 𝑤𝑛 ‖ + ‖𝐵𝑦𝑛 − 𝑤𝑛 ‖ ≤ 10−6 as the stopping criterion. Figure 2 illustrates the behaviors of iterative numbers for different initial points, which reveals the stationary property of iterative numbers of the algorithm.

Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments This project is supported by the Natural Science Foundation of China (Grant nos. 11401438 and 11571120) and Shandong Provincial Natural Science Foundation (Grant no. ZR2013FL032) and the Project of Shandong Province Higher Educational Science and Technology Program (Grant no. J14LI52).

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