The Split Common Fixed Point Problem for 𝜚-Strictly Pseudononspreading Mappings

Abstract

We introduce and analyze the viscosity approximation algorithm for solving the split common fixed point problem for the strictly pseudononspreading mappings in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related results.

1. Introduction

Throughout this paper, we always assume that 𝐻 is a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐼 denote the identity operator on 𝐻. Let 𝐻₁ and 𝐻₂ be two real Hilbert spaces and let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator. Given closed convex subsets 𝐶 and 𝑄 of 𝐻₁ and 𝐻₂, respectively.

The split feasibility problem (SFP) (Censor and Elfving 1994 [1]), modeling phase retrieval problems, is to find a point 𝑥^∗ with the property 𝑥^∗∈𝐶,𝐴𝑥^∗∈𝐷. (1) Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy [2–8]. A special case of the SFP (1) is the convexly constrained linear problem: 𝐴𝑥=𝑏,𝑥∈𝐶. (2) This problem, due to its applications in many applied disciplines, has extensively been investigated in the literature ever since Landweber [9] introduced his iterative method in 1951.

Note that the split feasibility problem (1) can be formulated as fixed point equation by using the fact 𝑥^∗=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗, (3) where 𝑃_𝐶 and 𝑃_𝑄 are the projections onto 𝐶 and 𝑄, respectively, 𝛾>0 is any positive constant, and 𝐴^∗ denotes the adjoint of 𝐴; that is, 𝑥^∗ solves the SFP (1) if and only if 𝑥^∗ solves the fixed point equation (3) (see [10] for more details). This implies that we can use fixed point algorithms to solve SFP.

In 2002, Byrne [2] proposed his CQ algorithm to solve (1). The sequence {𝑥_𝑛} is generated by the following iteration scheme: 𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛∈𝑁, (4) where 𝛾∈(0,2/𝜆), with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴.

The CQ algorithm (4) is a special case of the K-M algorithm. Due to the fixed point formulation (2) of the SFP, Moudafi [11] applied the K-M algorithm to the operator 𝑃_𝐶(𝐼−𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴) to obtain a sequence given by 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛−𝛼_𝑛𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛∈𝑁, (5) where 𝛾∈(0,2/𝜆), with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴, and the sequence {𝛼_𝑛} satisfies the condition ∑^∞_𝑛=1𝛼_𝑛(1−𝛼_𝑛)=+∞; he proved weak convergence result of the algorithm (5) in Hilbert spaces.

In 2009, Censor and Segal [12] considered the following algorithm to be solved (1).

Algorithm 1. Initialization: let 𝑥^∗∈𝐻₁=𝑅^𝑛 be arbitrary.
Iterative step: for 𝑛∈𝑁 let 𝑥_𝑛+1=𝑈𝑥_𝑛+𝛾𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛,𝑛∈𝑁, (6) where 𝛾∈(0,2/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴 and 𝑈,𝑉 be a single pair of directed operators.

In 2010, Moudafi [13] extended the Algorithm 1 and introduced the following algorithm with weak convergence for the split common fixed point problem.

Algorithm 2. Initialization: let 𝑥^∗∈𝐻₁=𝑅^𝑛 be arbitrary.
Iterative step: for 𝑛∈𝑁 let 𝑢_𝑛=𝑥_𝑛+𝛾𝛽𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑈𝑢_𝑛,𝑛∈𝑁, (7) where 𝛽∈(0,1), 𝛼_𝑛∈(0,1), and 𝛾∈(0,1/𝜆𝛽) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴 and 𝑈,𝑉 be a pair of quasi-nonexpansive operators.

In 2012, Zhao and He [14] continue to consider the split common fixed point problem with quasi-nonexpansive operators and to use the following algorithm to obtain the strong convergence of the viscosity method for solving the split common fixed point problem.

Algorithm 3. Initialization: let 𝑥^∗∈𝐻₁=𝑅^𝑛 be arbitrary.
Iterative step: for 𝑛∈𝑁 let 𝑇=𝑈𝐼+𝛾𝐴^∗()𝐴𝑥𝑉−𝐼_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛×1−𝜔_𝑛𝑥_𝑛+𝜔_𝑛𝑇𝑥_𝑛,𝑛∈𝑁, (8) where 𝑓∶𝐻→𝐻 is a contractive mapping with constant 𝛽∈(0,1), 𝛼_𝑛∈(0,1), and 𝛾∈(0,1/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴 and 𝑈,𝑉 be a pair of quasi-nonexpansive operators.

Motivated and inspired by Censor and Segal [12], Moudafi [11], and Zhao and He [14], we introduce the following relaxed algorithm.

Algorithm 4. Initialization: let 𝑥^∗∈𝐻₁=𝑅^𝑛 be arbitrary.
Iterative step: for 𝑛∈𝑁 let 𝑇=𝑈𝐼+𝛾𝐴^∗()𝐴𝑥𝑉−𝐼_𝑛+1=𝛼_𝑛𝜎𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵×1−𝜔_𝑛𝑥_𝑛+𝜔_𝑛𝑇𝑥_𝑛,𝑛∈𝑁, (9) where 𝑓∶𝐻→𝐻 is a contractive mapping with constant 𝛽∈(0,1), 𝐵∶𝐻→𝐻 is 𝜂-strongly monotone and boundedly Lipschitzian, 𝛼_𝑛∈(0,1), and 𝛾∈(0,1/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴 and 𝑈,𝑉 be a pair of 𝜚_𝑖-strictly pseudononspreading mappings 𝑖=1,2.

This paper establishes the strong convergence of the sequence given by (9) to the unique solution of solving the split common fixed point problem and the following variational inequality problem VIP(𝜇𝐵−𝜎𝑓,𝑇): find𝑥^∗∈Γsuchthat𝜇𝐵−𝜎𝑓𝑥^∗,𝑣−𝑥^∗≥0,𝑣∈Γ. (10)

2. Preliminaries

In this section, we introduce the concepts of contraction mappings, nonexpansive mappings, quasi-nonexpansive mappings, and 𝜚-strictly pseudononspreading mappings and some Lemmas.

Assume that 𝐶 is a nonempty closed and convex subset of Hilbert space 𝐻. Recall that the (nearest point or metric) projection from 𝐻 onto 𝐾, that denoted 𝑃_𝐶, assigns, to each 𝑥∈𝐻, the unique point 𝑃_𝐶𝑥∈𝐶 with the property 𝑥−𝑃_𝐶𝑥=inf𝑥−𝑦∶𝑦∈𝐶. (11)

Definition 5. A mapping 𝑇∶𝐻→𝐻 is said to be (1)contraction, if ‖𝑇𝑥−𝑇𝑦‖≤𝛽‖𝑥−𝑦‖, ∀𝑥,𝑦∈𝐻 and 𝛽∈(0,1);(2)nonexpansive, if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, ∀𝑥,𝑦∈𝐻;(3)quasi-nonexpansive, ‖𝑇𝑥−𝑞‖≤‖𝑥−𝑞‖, ∀(𝑥,𝑞)∈𝐻×𝐹_𝑖𝑥(𝑇).

Remark 6. From the Definition 5, It is easy to see that(i)iterative methods for quasi-nonexpansive mappings have been extensively investigated; see [13–17];(ii)a nonexpansive mapping is a quasi-nonexpansive mapping.

Following the terminology of Browder and Petryshyn [18], we obtain the following definitions.

Definition 7. A mapping 𝑇∶𝐷(𝑇)⊆𝐻→𝐻 is 𝜚-strictly pseudononspreading if there exists 𝜚∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝜚𝑥−𝑇𝑥−𝑦−𝑇𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦, (12) for all 𝑥,𝑦∈𝐷(𝑇).

Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see [19–23].

Lemma 8 (see [24]). Let 𝐻 be a Hilbert spaces, and 𝑓∶𝐻→𝐻 is a contractive mapping with constant 𝛽∈(0,1). 𝐵∶𝐻→𝐻 is 𝑘-Lipschitzian and 𝜂-strongly monotone operator with 𝑘>0, 𝜂>0. Then for 0<𝜎<𝜇𝜂/𝛽, 𝑦≥𝑥−𝑦,𝜇𝐵−𝜎𝑓𝑥−𝜇𝐵−𝜎𝑓𝜇𝜂−𝜎𝛽𝑥−𝑦²,𝑥,𝑦∈𝐻. (13) That is, 𝜇𝐵−𝜎𝑓 is strongly monotone with coefficient 𝜇𝜂−𝜎𝛽.

Lemma 9. Let 𝐻 be a real Hilbert space. Then the following well-known results hold: for all 𝑥,𝑦∈𝐻 and 𝑡∈[0,1](i)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩; (ii)‖𝑡𝑥+(1−𝑡)𝑦‖²=𝑡‖𝑥‖²+(1−𝑡)‖𝑦‖²−𝑡(1−𝑡)‖𝑥−𝑦‖²; (iii)⟨𝑥,𝑦⟩=−(1/2)‖𝑥−𝑦‖²+(1/2)‖𝑥‖²+(1/2)‖𝑦‖².

Lemma 10. Let 𝑇 be a 𝜚-strictly pseudononspreading mapping with 𝜚∈(0,1), and set 𝑇_𝛼=(1−𝛼)𝐼+𝛼𝑇, 𝛼∈(𝜚,1). The following properties are reached for each (𝑥,𝑝)∈𝐻×𝐹_𝑖𝑥(𝑇): (1)⟨𝑥−𝑇𝑥,𝑥−𝑝⟩≥((1−𝜚)/2)‖𝑥−𝑇𝑥‖² and ⟨𝑥−𝑇𝑥,𝑝−𝑇𝑥⟩≤((1+𝜚)/2)‖𝑥−𝑇𝑥‖²; (2)‖𝑇_𝛼𝑥−𝑝‖²≤‖𝑥−𝑝‖²−𝛼(1−𝜚−𝛼)‖𝑥−𝑇𝑥‖²; (3)⟨𝑥−𝑇_𝛼𝑥,𝑥−𝑝⟩≥(𝛼(1−𝜚)/2)‖𝑥−𝑇𝑥‖².

Proof. Note that property (1) is easily deduced from the Lemma 8(iii) and the fact that 𝑇 is 𝜚-strictly pseudononspreading mapping, we obtain 1𝑥−𝑇𝑥,𝑥−𝑝=−2𝑇𝑥−𝑝²+12‖𝑥−𝑇𝑥‖²+12𝑥−𝑝²1=−2𝑇𝑥−𝑝²+1−𝜚2‖𝑥−𝑇𝑥‖²+12𝑥−𝑝²+𝜚‖𝑥−𝑇𝑥‖²1≥−2𝑇𝑥−𝑝²+12𝑇𝑥−𝑝²+1−𝜚2‖𝑥−𝑇𝑥‖²≥1−𝜚2‖𝑥−𝑇𝑥‖²,1𝑥−𝑇𝑥,𝑝−𝑇𝑥=−2𝑥−𝑝²+12‖𝑥−𝑇𝑥‖²+12𝑇𝑥−𝑝²1≤−2𝑇𝑥−𝑝²+𝑇𝑥−𝑝²+12+𝜚𝑇𝑥−𝑝²≤1+𝜚2‖𝑥−𝑇𝑥‖². (14)
Property (2) is obtained from property (1) and by 𝑇_𝛼𝑥−𝑝²=𝑥−𝑝²−2𝛼𝑥−𝑝,𝑥−𝑇𝑥+𝛼²‖𝑇𝑥−𝑥‖². (15)
Property (3) is given by 𝐼−𝑇_𝛼=𝛼(𝐼−𝑇) and property (1).

Lemma 11 (see [25]). Let {𝒯_𝑛} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝒯_𝑛𝑗}_𝑗≥0 of {𝒯_𝑛} which satisfies 𝒯_𝑛𝑗<𝒯_𝑛𝑗+1 for all 𝑗≥0. Also consider the sequence of integers {𝛿(𝑛)}_𝑛≥𝑛0 defined by 𝛿(𝑛)=max𝑘≤𝑛∣𝒯_𝑘<𝒯_𝑘+1. (16) Then {𝛿(𝑛)}_𝑛≥𝑛0 is a nondecreasing sequence verifying lim_𝑛→∞𝛿(𝑛)=∞, for all 𝑛≥𝑛₀; it holds that 𝒯_𝛿(𝑛)<𝒯_𝛿(𝑛)+1 and one has 𝒯_𝑛<𝒯_𝛿(𝑛)+1. (17)

Lemma 12. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐻, given 𝑥∈𝐻 and 𝑦∈𝐾. Then 𝑦=𝑃_𝐾𝑥 if and only if there holds the inequality 𝑥−𝑦,𝑦−𝑧≥0,∀𝑧∈𝐾. (18)

3. Main Results

In what follows, we will focus our attention on the following general two operator split common fixed point problem in real Hilbert space 𝐻: find𝑥^∗∈𝐶suchthat𝐴𝑥^∗∈𝐷, (19) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, 𝑈∶𝐻₁→𝐻₁ and 𝑉∶𝐻₂→𝐻₂ are two 𝜚_𝑖-strictly pseudononspreading mappings 𝑖=1,2 with nonempty fixed point sets 𝐹_𝑖𝑥(𝑈)=𝐶 and 𝐹_𝑖𝑥(𝑉)=𝑄, and denote the solution set of the two-operator SCFP by Γ=𝑦∈𝐶;𝐴𝑦∈𝑄. (20) On the other hand, 𝑥^∗∈Γ is also unique solution of solving the variational inequality problem VIP(𝜇𝐵−𝜎𝑓,𝑇): 𝑥^∗∈Γ,𝜇𝐵−𝜎𝑓𝑥^∗,𝑦−𝑥^∗≥0,𝑦∈Γ, (21) where 𝐵∶𝐻→𝐻 is 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0. Let 0<𝜇<2𝜂/𝑘², 0<𝜎<𝜇(𝜂−(𝜇𝑘²/2))/𝛽=𝜏/𝛽.

Before stating our main convergence result, we establish the boundedness of the iterates given by (9).

Lemma 13. The sequence {𝑥_𝑛} is generated by (9), and let 𝑈∶𝐻₁→𝐻₁ and 𝑉∶𝐻₂→𝐻₂ be two 𝜚_𝑖-strictly pseudononspreading mappings on 𝐻, 𝑖=1,2, and 𝑓∶𝐻→𝐻 is a contractive mapping with constant 𝛽∈(0,1), {𝛼_𝑛}⊂(0,1) and 0<𝜚_𝑖<𝜔_𝑛<1/2, 𝑖=1,2. Then {𝑥_𝑛} is bounded.

Proof. Set 𝑇_𝜔𝑛=(1−𝜔_𝑛)𝐼+𝜔_𝑛𝑇. Then 𝑥_𝑛+1=𝛼_𝑛𝜎𝑓(𝑥_𝑛)+(𝐼−𝜇𝛼_𝑛𝐵)𝑇_𝜔𝑛𝑥_𝑛.
Taking 𝑦∈Γ, that is, 𝑦∈𝐹_𝑖𝑥(𝑈) and 𝐴𝑦∈𝐹_𝑖𝑥(𝑉). We obtain 𝑥_𝑛+1−𝑦=𝛼_𝑛𝜎𝑓𝑥_𝑛−𝑓𝑦+𝛼_𝑛𝑦+𝜎𝑓−𝜇𝐵𝑦𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔𝑛𝑥_𝑛−𝑦≤𝛼_𝑛𝜎𝑓𝑥_𝑛−𝑓𝑦+𝛼_𝑛𝑦+𝜎𝑓−𝜇𝐵𝑦1−𝛼_𝑛𝜏𝑇_𝜔𝑛𝑥_𝑛−𝑦≤𝛼_𝑛𝜎𝛽𝑥_𝑛−𝑦+𝛼_𝑛𝑦+𝜎𝑓−𝜇𝐵𝑦1−𝛼_𝑛𝜏𝑇_𝜔𝑛𝑥_𝑛−𝑦. (22) From the definition of 𝑇_𝜔𝑛, we have 𝑇_𝜔𝑛𝑥_𝑛−𝑦²=1−𝜔_𝑛𝑥_𝑛+𝜔_𝑛𝑇𝑥_𝑛−𝑦²=𝑥_𝑛−𝑦+𝜔_𝑛𝑇𝑥_𝑛−𝑥_𝑛²=𝑥_𝑛−𝑦²−2𝜔_𝑛𝑥_𝑛−𝑦,𝑥_𝑛−𝑇𝑥_𝑛+𝜔²_𝑛𝑇𝑥_𝑛−𝑥_𝑛². (23) On the other hand, we obtain 𝑇𝑥_𝑛−𝑦²=𝑈𝐼+𝛾𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛−𝑦²≤(𝐼+𝛾𝐴^∗(𝑉−𝐼)𝐴)𝑥_𝑛−𝑦²=𝑥_𝑛−𝑦²+𝛾²𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛²+2𝛾𝑥_𝑛−𝑦,𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛=𝑥−𝑦_𝑛−𝑦²+𝛾²(𝑉−𝐼)𝐴𝑥_𝑛,𝐴𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛+2𝛾𝑥_𝑛−𝑦,𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛. (24) According to the definition of 𝜆, we have 𝛾²(𝑉−𝐼)𝐴𝑥_𝑛,𝐴𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛≤𝜆𝛾²(𝑉−𝐼)𝐴𝑥_𝑛,(𝑉−𝐼)𝐴𝑥_𝑛=𝜆𝛾²(𝑉−𝐼)𝐴𝑥_𝑛². (25) Now, by using property (1) of Lemma 9, we obtain 2𝛾𝑥_𝑛−𝑦,𝐴^∗(𝑉−𝐼)𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑦,(𝑉−𝐼)𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑦+(𝑉−𝐼)𝐴𝑥_𝑛−(𝑉−𝐼)𝐴𝑥_𝑛,(𝑉−𝐼)𝐴𝑥_𝑛=2𝛾𝑉𝐴𝑥_𝑛−𝐴𝑦,(𝑉−𝐼)𝐴𝑥_𝑛−(𝑉−𝐼)𝐴𝑥_𝑛²≤2𝛾1−𝜚₂2(𝑉−𝐼)𝐴𝑥_𝑛²−(𝑉−𝐼)𝐴𝑥_𝑛²≤−𝛾1−𝜚₂(𝑉−𝐼)𝐴𝑥_𝑛². (26) Combining (24)–(26), we obtain 𝑇𝑥_𝑛−𝑦²=𝑥_𝑛−𝑦²+𝜆𝛾²(𝑉−𝐼)𝐴𝑥_𝑛²−𝛾1−𝜚₂(𝑉−𝐼)𝐴𝑥_𝑛²=𝑥_𝑛−𝑦²−𝛾1−𝜚₂×)−𝜆𝛾(𝑉−𝐼𝐴𝑥_𝑛²≤𝑥_𝑛−𝑦². (27) From property (i) of Lemma 9 and (23), we get 𝑇_𝜔𝑛𝑥_𝑛−𝑦²≤𝑥_𝑛−𝑦²−1−𝜚₂𝜔_𝑛𝑥_𝑛−𝑇𝑥_𝑛²+𝜔²_𝑛𝑇𝑥_𝑛−𝑥_𝑛²=𝑥_𝑛−𝑦²−𝜔_𝑛1−𝜚₂−𝜔_𝑛×𝑥_𝑛−𝑇𝑥_𝑛²≤𝑥_𝑛−𝑦². (28) Combining (22), (23), and (28), we have 𝑥_𝑛+1−𝑦≤𝛼_𝑛𝜎𝛽𝑥_𝑛−𝑦+𝛼_𝑛𝑓𝑦+−𝜇𝐵𝑦1−𝛼_𝑛𝜏𝑇_𝜔𝑛𝑥_𝑛=−𝑦1−𝛼_𝑛𝜏−𝜎𝛽𝑥_𝑛−𝑦+𝛼_𝑛𝑓𝑦𝑥−𝜇𝐵𝑦≤max_𝑛,1−𝑦𝑓𝑦.𝜏−𝜎𝛽−𝜇𝐵𝑦 (29) It follows from (29) and induction that 𝑥_𝑛+1−𝑦≤max𝑥₀,1−𝑦𝑓𝑦,𝜏−𝜎𝛽−𝜇𝐵𝑦 (30) and hence {𝑥_𝑛} is bounded.

Now we are in position to claim the main convergence result.

Theorem 14. Given a bounded linear operator 𝐴∶𝐻₁→𝐻₂, let 𝑈∶𝐻₁→𝐻₁ and 𝑉∶𝐻₂→𝐻₂ be two 𝜚_𝑖-strictly pseudononspreading mappings, and 𝑖=1,2 with fixed point 𝐹_𝑖𝑥(𝑈)=𝐶 and 𝐹_𝑖𝑥(𝑉)=𝑄. Assume that 𝑈−𝐼 and 𝑉−𝐼 are demiclosed at origin. Let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0, and 𝑓∶𝐻→𝐻 is a contractive mapping with constant 𝛽∈(0,1). Assume that {𝑥_𝑛} is the sequence given by Algorithm 4 with 𝛾∈(0,1/𝜆), 0<𝜚_𝑖<𝜔_𝑛<1/2, and 𝑖=1,2 such that 0<liminf_𝑛→∞𝜔_𝑛≤limsup_𝑛→∞𝜔_𝑛<1/2 and 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. If Γ≠∅, then the sequence {𝑥_𝑛} strongly converges to a split common fixed point 𝑦∈Γ, verifying 𝑦=𝑃_Γ(𝐼−𝜇𝐵+𝜎𝑓)(𝑦) which equivalently solves the following variational inequality problem: 𝑦∈Γ,𝜇𝐵−𝜎𝑓𝑦,𝑥^∗−𝑦≥0,𝑥^∗∈Γ. (31)

Proof. Let 𝑦 be the solution of (31). From (9) we obtain that 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝜇𝐵𝑥_𝑛−𝜎𝑓𝑥_𝑛=𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛, (32) hence 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝜇𝐵𝑥_𝑛−𝜎𝑓𝑥_𝑛,𝑥_𝑛=−𝑦𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛=−𝑦1−𝛼_𝑛𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑦+𝛼_𝑛𝐼−𝜇𝐵𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛≤−𝑦1−𝛼_𝑛𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑦+𝛼_𝑛𝐼−𝜇𝐵𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛𝑥_𝑛≤−𝑦1−𝛼_𝑛𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑦+𝛼_𝑛(1−𝜏)𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛𝑥_𝑛=−𝑦1−𝛼_𝑛𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛,𝑥_𝑛−𝑦+𝜔_𝑛𝛼_𝑛(1−𝜏)𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑦. (33) By (28), we obtain that 𝑥_𝑛−𝑇_𝜔𝑛𝑥_𝑛,𝑥_𝑛=𝜔−𝑦^2𝑛2𝑥_𝑛−𝑇_𝜔𝑛𝑥_𝑛²+12𝑥_𝑛−𝑦²−12𝑇_𝜔𝑛𝑥_𝑛−𝑦²≥𝜔²_𝑛2𝑥_𝑛−𝑇𝑥_𝑛²+12𝑥_𝑛−𝑦²−12𝑥_𝑛−𝑦²+𝜔_𝑛21−𝜚₂−𝜔_𝑛𝑥_𝑛−𝑇𝑥_𝑛²=𝜔_𝑛1−𝜚₂2𝑥_𝑛−𝑇𝑥_𝑛². (34) It follows from (33) that 𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝜇𝐵𝑥_𝑛−𝜎𝑓𝑥_𝑛,𝑥_𝑛𝜔−𝑦≤−^𝑛21−𝜚₂1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛²+𝜔_𝑛𝛼_𝑛(1−𝜏)𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑦 (35) or equivalently −𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑦≤−𝛼_𝑛𝜇𝐵−𝜎𝑓𝑥_𝑛,𝑥_𝑛−𝜔−𝑦_𝑛21−𝜚₂1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛²+𝜔_𝑛𝛼_𝑛(1−𝜏)𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑦. (36) Furthermore, using the classical equality (iii) in Lemma 10 and setting 𝒯_𝑛=(1/2)‖𝑥_𝑛−𝑦‖², we have 𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑦=𝒯_𝑛−𝒯_𝑛+1+12𝑥_𝑛−𝑥_𝑛+1². (37) So that (36) can be equivalently rewritten as 𝒯_𝑛+1−𝒯_𝑛−12𝑥_𝑛−𝑥_𝑛+1²≤−𝛼_𝑛𝜇𝐵−𝜎𝑓𝑥_𝑛,𝑥_𝑛−𝜔−𝑦_𝑛21−𝜚₂1−𝛼_𝑛𝑥_𝑛−𝑇𝑥_𝑛²+𝜔_𝑛𝛼_𝑛(1−𝜏)𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑦. (38) Now using (32) again, we have 𝑥_𝑛+1−𝑥_𝑛²=𝛼_𝑛𝜎𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐵𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛². (39) Since 𝐵∶𝐻→𝐻 is 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻, hence it is a classical matter to see that 𝑥_𝑛+1−𝑥_𝑛²≤2𝛼²_𝑛𝜎𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛²+21−𝛼_𝑛𝜏²𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛², (40) which by ‖𝑇_𝜔𝑛𝑥_𝑛−𝑥_𝑛‖=𝜔_𝑛‖𝑥_𝑛−𝑇𝑥_𝑛‖ yields 12𝑥_𝑛+1−𝑥_𝑛²≤𝛼²_𝑛𝜎𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛²+1−𝛼_𝑛𝜏²𝜔²_𝑛𝑥_𝑛−𝑇𝑥_𝑛². (41) Then from (38) and (41), we have 𝒯_𝑛+1−𝒯_𝑛+𝜔_𝑛21−𝜚₂1−𝛼_𝑛−𝜔²_𝑛1−𝛼_𝑛𝜏²×𝑥_𝑛−𝑇𝑥_𝑛²≤𝛼_𝑛𝛼_𝑛𝜎𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛²−𝜇𝐵−𝜎𝑓𝑥_𝑛,𝑥_𝑛−𝑦+𝜔_𝑛(1−𝜏)𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑦. (42) The rest of the proof will be divided into two parts.
Case  1.  Suppose that there exists 𝑛₀ such that {𝒯_𝑛}_𝑛≥𝑛0 is nonincreasing. In this situation, {𝒯_𝑛} is then convergent because it is also nonnegative (hence it is bounded from below), so that lim_𝑛→∞(𝒯_𝑛+1−𝒯_𝑛)=0; hence, in light of (42) together with lim_𝑛→∞𝛼_𝑛=0, the boundedness of {𝑥_𝑛} and 0<liminf_𝑛→∞𝜔_𝑛≤limsup_𝑛→∞𝜔_𝑛<1/2, we obtain lim_𝑛→∞𝑥_𝑛−𝑇𝑥_𝑛=0. (43) It also follows from (42) that 𝒯_𝑛−𝒯_𝑛+1≥𝛼_𝑛−𝛼_𝑛𝜎𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛²+𝜇𝐵−𝜎𝑓𝑥_𝑛,𝑥_𝑛−𝑦−𝜔_𝑛(1−𝜏)𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑦. (44) Then, by ∑^∞_𝑛=0𝛼_𝑛=∞, we obviously deduce that liminf_𝑛→∞𝛼_𝑛−𝛼_𝑛𝜎𝑓𝑥_𝑛−𝜇𝐵𝑥_𝑛²+𝜇𝐵−𝜎𝑓𝑥_𝑛,𝑥_𝑛−𝑦−𝜔_𝑛(1−𝜏)𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑦≤0 (45) or equivalently (as 𝛼_𝑛‖𝜎𝑓(𝑥_𝑛)−𝜇𝐵𝑥_𝑛‖²→0 and lim_𝑛→∞(𝒯_𝑛+1−𝒯_𝑛)=0). From (45), we get liminf_𝑛→∞𝜇𝐵−𝜎𝑓𝑥_𝑛,𝑥_𝑛−𝑦≤0. (46) Moreover, by Lemma 8, we have 2𝜇𝜂−𝜎𝛽𝒯_𝑛+𝜇𝐵−𝜎𝑓𝑦,𝑥_𝑛≤𝑥−𝑦𝜇𝐵−𝜎𝑓_𝑛,𝑥_𝑛−𝑦, (47) which by (46) entails liminf_𝑛→∞2𝜇𝜂−𝜎𝛽𝒯_𝑛+𝜇𝐵−𝜎𝑓𝑦,𝑥_𝑛−𝑦≤0. (48) Hence, recalling that lim_𝑛→∞𝒯_𝑛 exists, we equivalently obtain 2𝜇𝜂−𝜎𝛽lim_𝑛→∞𝒯_𝑛+liminf_𝑛→∞𝜇𝐵−𝜎𝑓𝑦,𝑥_𝑛−𝑦≤0, (49) Namely, 2𝜇𝜂−𝜎𝛽lim_𝑛→∞𝒯_𝑛≤−liminf_𝑛→∞𝜇𝐵−𝜎𝑓𝑦,𝑥_𝑛−𝑦. (50) Now we prove that liminf_𝑛→∞𝜇𝐵−𝜎𝑓𝑦,𝑥_𝑛−𝑦≤0. (51) It follows from (27) and (43) that 𝛾1−𝜚₂−𝜆𝛾(𝑉−𝐼)𝐴𝑥_𝑛²≤𝑥_𝑛−𝑦²−𝑇𝑥_𝑛−𝑦²=𝑥_𝑛−𝑦−𝑇𝑥_𝑛×𝑥−𝑦_𝑛−𝑦+𝑇𝑥_𝑛=𝑥−𝑦_𝑛−𝑇𝑥_𝑛𝑥_𝑛−𝑦+𝑇𝑥_𝑛(),−𝑦⟶0,𝑛⟶∞ (52) and hence lim_𝑛→∞(𝑉−𝐼)𝐴𝑥_𝑛=0. (53) Taking 𝑥^∗∈𝜔_𝑤(𝑥_𝑛), from the demiclosedness of 𝑉−𝐼 at 0, we have 𝑉𝐴𝑥^∗=𝐴𝑥^∗. (54) Now, by setting 𝑢_𝑛=𝑥_𝑛+𝛾𝐴^∗𝑉−𝐼𝐴𝑥_𝑛, it follows that 𝑥^∗∈𝜔_𝑤(𝑢_𝑛). On the other hand, 𝑈𝑢_𝑛−𝑢_𝑛=𝑇𝑥_𝑛−𝑥_𝑛−𝛾𝐴^∗𝑉−𝐼𝐴𝑥_𝑛≤𝑇𝑥_𝑛−𝑥_𝑛+𝛾𝐴^∗𝑉−𝐼𝐴𝑥_𝑛⟶0, (55) which, combined with the demiclosedness of 𝑈−𝐼 at 0, yields 𝑈𝑥^∗=𝑥^∗. (56) Hence, 𝑥^∗∈𝐶 and 𝑥^∗∈Γ. We can take subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that lim_𝑛→∞𝑥_𝑛𝑗=𝑥^∗ and liminf_𝑛→∞𝜎𝑓−𝜇𝐵𝑦,𝑥_𝑛−𝑦=lim_𝑗→∞𝜎𝑓−𝜇𝐵𝑦,𝑥_𝑛𝑗−𝑦, (57) which leads to liminf_𝑛→∞𝜎𝑓−𝜇𝐵𝑦,𝑥_𝑛−𝑦=𝜎𝑓−𝜇𝐵𝑦,𝑥^∗−𝑦≤0. (58) By (50), we have lim_𝑛→∞Γ_𝑛=0, and hence {𝑥_𝑛} converges strongly to 𝑦.
Case  2.  Suppose that there exists a subsequence {𝒯_𝑛𝑘}_𝑘≥0 of {𝒯_𝑛}_𝑛≥0 such that 𝒯_𝑛𝑘≤𝒯_𝑛𝑘+1 for all 𝑘≥0. In this situation, we consider the sequence of indices {𝛿(𝑛)} as defined in Lemma 11. It follows that 𝒯_𝛿(𝑛+1)−𝒯_𝛿(𝑛)>0, which by (42) amounts to 𝜔_𝑛21−𝜚₂1−𝛼_𝛿(𝑛)−𝜔²_𝑛1−𝛼_𝛿(𝑛)𝜏²𝑥_𝛿(𝑛)−𝑇𝑥_𝛿(𝑛)²≤𝛼_𝛿(𝑛)𝛼_𝛿(𝑛)𝜎𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝑛²−𝜇𝐵−𝜎𝑓𝑥_𝑛,𝑥_𝑛−𝑦+𝜔_𝑛(1−𝜏)𝑇𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)𝑥_𝛿(𝑛)−𝑦. (59) By the boundedness of {𝑥_𝑛} and lim_𝑛→∞𝛼_𝑛=0, we immediately obtain lim_𝑛→∞𝑥_𝛿(𝑛)−𝑇𝑥_𝛿(𝑛)=0. (60) Using (9), we have 𝑥_𝛿(𝑛)+1−𝑥_𝛿(𝑛)≤𝛼_𝛿(𝑛)𝜎𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)+1−𝛼_𝛿(𝑛)𝜏𝑇_𝜔𝑛𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)≤𝛼_𝛿(𝑛)𝜎𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)+1−𝛼_𝛿(𝑛)𝜔_𝑛𝜏𝑇𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛), (61) which together with (60) and lim_𝑛→∞𝛼_𝑛=0 yields lim_𝑛→∞𝑥_𝛿(𝑛)+1−𝑥_𝛿(𝑛)=0. (62) Similar to Case 1, we have liminf_𝑛→∞𝜇𝐵−𝜎𝑓𝑦,𝑥_𝛿(𝑛)−𝑦≥0. (63) Now by (59) we clearly have 𝛼_𝛿(𝑛)𝜎𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)²()×+𝛼1−𝜏𝑇𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)𝑥_𝛿(𝑛)≥𝑥−𝑦𝜇𝐵−𝜎𝑓_𝛿(𝑛),𝑥_𝛿(𝑛)−𝑦, (64) which in the light of (47) yields 2𝜇𝜂−𝜎𝛽𝒯_𝛿(𝑛)+𝜇𝐵−𝜎𝑓𝑦,𝑥_𝛿(𝑛)−𝑦≤𝛼_𝛿(𝑛)𝜎𝑓𝑥_𝛿(𝑛)−𝜇𝐵𝑥_𝛿(𝑛)²+𝛼(1−𝜏)𝑇𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)𝑥_𝛿(𝑛)−𝑦. (65) Hence (as lim_𝑛→∞𝛼_𝛿(𝑛)‖𝜎𝑓(𝑥_𝛿(𝑛))−𝜇𝐵𝑥_𝛿(𝑛)‖²=0 and lim_𝑛→∞‖𝑇𝑥_𝛿(𝑛)−𝑥_𝛿(𝑛)‖=0) it follows that 2𝜇𝜂−𝜎𝛽limsup_𝑛→∞𝒯_𝛿(𝑛)≤−liminf_𝑛→∞𝜇𝐵−𝜎𝑓𝑦,𝑥_𝛿(𝑛)−𝑦. (66) From (59) and (63), we obtain lim_𝑛→∞𝜇𝐵−𝜎𝑓𝑦,𝑥_𝛿(𝑛)−𝑦≥0, (67) which by (60) yields limsup_𝑛→∞𝒯_𝛿(𝑛)=0, so that lim_𝑛→∞𝒯_𝛿(𝑛)=0. Combining (62), we have lim_𝑛→∞𝒯_𝛿(𝑛)+1=0. Then, recalling that 𝒯_𝑛<𝒯_𝛿(𝑛)+1 (by Lemma 11), we get lim_𝑛→∞𝒯_𝑛=0, so that 𝑥_𝑛→𝑦 strongly.
In addition, the variational inequality (50) and (67) can be written as 𝐼−𝜇𝐵+𝜎𝑓𝑦−𝑦,𝑥^∗−𝑦≤0,𝑥^∗∈Γ. (68) So, by the Lemma 12, it is equivalent to the fixed point equation 𝑃_Γ𝐼−𝜇𝐵+𝜎𝑓𝑦=𝑦. (69)

4. Application in Other Nonlinear Operators

In order to define our motivations, we recall some definitions of classed of operators as follows

Definition 15. 𝑇∶𝐷(𝑇)⊆𝐻→𝐻 is said to be (1)nonspreading in [26, 27], if 𝑇𝑥−𝑇𝑦²≤𝑇𝑥−𝑦²+𝑇𝑦−𝑥²,∀𝑥,𝑦∈𝐶, (70)(2)demicontractive in [28], if there exists a constant 𝛼<1 such that 𝑇𝑥−𝑞²≤𝑥−𝑞²+𝛼‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻×𝐹_𝑖𝑥(𝑇). (71)

Remark 16. Iemoto and Takahashi [29] proved that (70) is equivalent to 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+2𝑥−𝑇𝑥,𝑦−𝑇𝑦,∀𝑥,𝑦∈𝐶. (72) Iterative methods for nonspreading mapping have been extensively investigated; see [30–34].

Remark 17. From the Definition 5 (3), Definition 7, and Definition 15, we have the following facts. (i)Observe that every nonspreading mapping is 0-strictly pseudononspreading.(ii)If 𝑇 is nonspreading mapping and the set of fixed point is nonempty, then 𝑇 is quasi-nonexpansive mapping.(iii)Every pseudononspreading mapping with a nonempty fixed point set 𝐹_𝑖𝑥(𝑇) is demicontractive (see [28]).

Corollary 18. Given a bounded linear operator 𝐴∶𝐻₁→𝐻₂, let 𝑈∶𝐻₁→𝐻₁ and 𝑉∶𝐻₂→𝐻₂ be two nonspreading mappings with fixed point 𝐹_𝑖𝑥(𝑈)=𝐶 and 𝐹_𝑖𝑥(𝑉)=𝑄. Assume that 𝑈−𝐼 and 𝑉−𝐼 are demiclosed at origin. Let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0, and let 𝑓∶𝐻→𝐻 be a contractive mapping with constant 𝛽∈(0,1). Let {𝑥_𝑛} be the sequence given by (9) with 𝛾∈(0,1/𝜆), 0<𝜚_𝑖<𝜔_𝑛<1/2, and 𝑖=1,2 such that 0<liminf_𝑛→∞𝜔_𝑛≤limsup_𝑛→∞𝜔_𝑛<1/2 and 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. If Γ≠∅, then the sequence {𝑥_𝑛} strongly converges to a split common fixed point 𝑥^∗∈Γ, verifying 𝑥^∗=𝑃_Γ(𝐼−𝜇𝐵+𝜎𝑓)(𝑥^∗) which equivalently solves the following variational inequality problem: 𝑥^∗∈Γ,𝜇𝐵−𝜎𝑓𝑥^∗,𝑦−𝑥^∗≥0,𝑦∈Γ. (73)

Proof. Form the proof of the Theorem 14, we can easily certify this theorem by nonspreading mapping (i.e., nonspreading is 0-strictly pseudononspreading).

From the Remark 17(ii) and the Corollary 18, we have the following corollary.

Corollary 19. Given a bounded linear operator 𝐴∶𝐻₁→𝐻₂, let 𝑈∶𝐻₁→𝐻₁ and 𝑉∶𝐻₂→𝐻₂ be two quasi-nonexpansives with fixed point 𝐹_𝑖𝑥(𝑈)=𝐶 and 𝐹_𝑖𝑥(𝑉)=𝑄. Assume that 𝑈−𝐼 and 𝑉−𝐼 are demiclosed at origin. Let 𝐵∶𝐻→𝐻 be 𝜂-strongly monotone and 𝑘-Lipschitzian on 𝐻 with 𝑘>0, 𝜂>0, and let 𝑓∶𝐻→𝐻 be a contractive mapping with constant 𝛽∈(0,1). Let {𝑥_𝑛} be the sequence given by (9) with 𝛾∈(0,1/𝜆), 𝜔_𝑛∈(0,1/2) such that 0<liminf_𝑛→∞𝜔_𝑛≤limsup_𝑛→∞𝜔_𝑛<1/2 and 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. If Γ≠∅, then the sequence {𝑥_𝑛} strongly converges to a split common fixed point 𝑥^∗∈Γ, verifying 𝑥^∗=𝑃_Γ𝑓(𝑥^∗) which equivalently solves the following variational inequality problem: 𝑥^∗∈Γ,𝜇𝐵−𝜎𝑓𝑥^∗,𝑦−𝑥^∗≥0,𝑦∈Γ. (74)

If 𝜎=𝜇=1 and 𝐵=𝐼 in (9), thus 𝑘=𝜂=1 and 𝛽∈(0,1/2), and then we obtain (8) and the following corollary. On the other hand, this corollary was proven by Zhao and He [14].

Corollary 20. Given a bounded linear operator 𝐴∶𝐻₁→𝐻₂, let 𝑈∶𝐻₁→𝐻₁ and 𝑉∶𝐻₂→𝐻₂ be two quasi-nonexpansives with fixed point 𝐹_𝑖𝑥(𝑈)=𝐶 and 𝐹_𝑖𝑥(𝑉)=𝑄. Assume that 𝑈−𝐼 and 𝑉−𝐼 are demiclosed at origin. Let 𝑓∶𝐻→𝐻 be a contractive mapping with constant 𝛽∈(0,1/2). Let {𝑥_𝑛} be the sequence given by (8) with 𝛾∈(0,1/𝜆), 𝜔_𝑛∈(0,1/2) such that 0<liminf_𝑛→∞𝜔_𝑛≤limsup_𝑛→∞𝜔_𝑛<1/2, and 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. IfΓ≠∅, then the sequence {𝑥_𝑛} strongly converges to a split common fixed point 𝑥^∗∈Γ, verifying 𝑥^∗=𝑃_Γ(𝐼−𝜇𝐵+𝜎𝑓)(𝑥^∗) which equivalently solves the following variational inequality problem: 𝑥^∗∈Γ,𝐼−𝑓𝑥^∗,𝑦−𝑥^∗≥0,𝑦∈Γ. (75)