On the Strong Convergence of an Algorithm about Firmly Pseudo-Demicontractive Mappings for the Split Common Fixed-Point Problem

Abstract

Based on the recent work by Censor and Segal (2009 J. Convex Anal.16), and inspired by Moudafi (2010 Inverse Problems 26), we modify the algorithm of demicontractive operators proposed by Moudafi and study the modified algorithm for the class of firmly pseudodemicontractive operators to solve the split common fixed-point problem in a Hilbert space. We also give the strong convergence theorem under some appropriate conditions. Our work improves and/or develops the work of Moudafi, Censor and Segal, and other results.

1. Introduction

Throughout, let 𝐻₁ and 𝐻₂ be real Hilbert spaces, and let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator. The split feasibility problem (SFP) [1–4] is to find a point𝑥∈𝐶suchthat𝐴𝑥∈𝑄, (1.1) where 𝐶 is a closed convex subset of a Hilbert space 𝐻₁ and 𝑄 is a closed convex subset of a Hilbert space 𝐻₂. If the two closed convex subsets 𝐶 and 𝑄 are fixed point sets of 𝑈 and 𝑇, respectively, where 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ are nonlinear operators, we obtain the two-set split common fixed-point problem (SCFP).

The split common fixed-point problem [5–8] requires to find a common fixed point of a family of operators in one space such that its image under a linear transformation is a common fixed point of another family of operators in the image space. This generalizes the split feasibility problem (SFP) and the convex feasibility problem (CFP).

In 2008, Censor and Segal proposed the split common fixed-point problem (SCFP) in [5] for directed operators in finite-dimensional Hilbert spaces. They invented an algorithm for the two-set SCFP which generated a sequence {𝑥_𝑛} according to the iterative procedure𝑥_𝑘+1=𝑈𝑥_𝑘+𝛾𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘,𝑘∈𝑁, (1.2) where 𝛾∈(0,2/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴. Let 𝑥₀∈𝑅^𝑛 be arbitrary.

They proved the convergence of the algorithm in finite-dimension spaces. Inspired by the work of Censor and Segal, Moudafi [6] introduced the following algorithm for 𝜇-demicontractive operators in Hilbert spaces:𝑢_𝑘=𝑥_𝑘+𝛾𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘, (1.3) and let 𝑥_𝑘+1=1−𝑡_𝑘𝑢_𝑘+𝑡_𝑘𝑈𝑢_𝑘,𝑘∈𝑁, (1.4) where 𝛾∈(0,(1−𝜇)/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴, 𝑡_𝑘∈(0,1). Let 𝑥₀∈𝐻₁ be arbitrary. Using Féjer-monotone and the demiclosed properties of 𝑈−𝐼 and 𝑇−𝐼 at the origin, Moudafi proved the convergence theorem. Based on the work of Censor, Segal, and Moudafi, Sheng and Chen gave their results of pseudo-demicontractive operators for the split common fixed-point problem recently.

Furthermore, we modify the algorithm (1.4) proposed by Moudafi and extend the operators to the class of firmly pseudo-demicontractive operators [9] in this paper. The firmly pseudo-demicontractive operators are more general class, which properly includes the class of demicontractive operators, pseudo-demicontractive operators, and quasi-nonexpansive mappings and is more desirable, for example, in fixed-point methods in image recovery where, in many cases, it is possible to map the set of images possessing a certain property to the fixed-point set of a nonlinear quasi-nonexpansive operator. Also for the hybrid steepest descent method, see [10], which is an algorithmic solution to the variational inequality problem over the fixed-point set of certain quasi-nonexpansive mappings and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Our work is related to significant real-world applications, see, for instance, [2–4, 11], where such methods were applied to the inverse problem of intensity-modulated radiation therapy (IMRT) and to the dynamic emission tomographic image reconstruction. Based on the very recent work in this field, we give an extension of their unified framework to firmly pseudo-demicontractive operators and obtain convergence results of a modified algorithm in the context of general Hilbert spaces.

Our paper is organized as follows. Section 2 reviews some preliminaries. Section 3 gives a modified algorithm and shows its strong convergence under some appropriate conditions. Section 4 gives some conclusions briefly.

2. Preliminaries

To begin with, let us recall that the split common fixed point problem [5] proposed by Censor and Segal in finite spaces.

Given operators 𝑈_𝑖∶𝑅^𝑛→𝑅^𝑛, 𝑖=1,2,…,𝑝, and 𝑇_𝑗∶𝑅^𝑚→𝑅^𝑚, 𝑗=1,2,…,𝑟, with nonempty fixed points sets 𝐶_𝑖, 𝑖=1,2,…,𝑝 and 𝑄_𝑗, 𝑗=1,2,…,𝑟, respectively. The split common fixed point problem (SCFP) is to find a vector𝑥^∗∈𝐶∶=^𝑝_𝑖=1𝐶_𝑖suchthat𝐴𝑥^∗∈𝑄∶=^𝑟_𝑗=1𝑄_𝑗. (2.1)

In the sequel, we concentrate on the study of the two-set split common fixed-point problem, which is to find that𝑥^∗∈𝐶suchthat𝐴𝑥^∗∈𝑄, (2.2) where 𝐻₁ and 𝐻₂ are two Hilbert spaces, 𝐶 and 𝑄 are two nonempty closed convex subsets, 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, 𝑈∶𝐻₁→𝐻₁,𝑇∶𝐻₂→𝐻₂ are two firmly pseudo-demicontractive operators with Fix(𝑈)=𝐶 and Fix(𝑇)=𝑄. 𝛼,𝛽, and 𝜇,𝜃 are coefficients of 𝑈,𝑇, respectively.

Definition 2.1. We say that 𝑇 is demicontractive [6] means that there exists constant 𝛽<1 such that 𝑇𝑥−𝑞²≤𝑥−𝑞²+𝛽‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻×Fix(𝑇), (2.3)𝑇 is pseudo-demicontractive [9] means that there exists constant 𝛼>1 such that 𝑇𝑥−𝑞²≤𝛼𝑥−𝑞²+‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻×Fix(𝑇). (2.4)

Definition 2.2. We say that 𝑇 is firmly pseudo-demicontractive means that there exist constants 𝛼>1,𝛽>1 such that 𝑇𝑥−𝑞²≤𝛼𝑥−𝑞²+𝛽‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻×Fix(𝑇). (2.5) The inequality (2.5) is equivalent to 𝑥−𝑇𝑥,𝑥−𝑞≥1−𝛼2𝑥−𝑞²+1−𝛽2‖𝑇𝑥−𝑥‖²,∀𝑥,𝑞∈𝐻×Fix(𝑇), (2.6)
An operator satisfying (2.5) will be referred to as a 𝛼,𝛽 firmly pseudo-demicontractive mapping. It is worth noting that the class of firmly pseudo-demicontractive maps contains important operators such as the demicontractive maps, quasi-nonexpansive maps, and the strictly pseudo-contractive maps with fixed points.
Next, let us recall several concepts:(1)a mapping 𝑇∶𝐻→𝐻 is said to be nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻×𝐻; (2.7)(2)a mapping 𝑇∶𝐻→𝐻 is said to be quasi-nonexpansive if 𝑇𝑥−𝑞≤𝑥−𝑞,∀𝑥,𝑞∈𝐻×Fix(𝑇); (2.8)(3)a mapping 𝑇∶𝐻→𝐻 is said to be strictly pseudo-contractive if 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝛽𝑥−𝑦−𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐻×𝐻𝛽∈[0,1). (2.9)Obviously, the nonexpansive operators are both quasi-nonexpansive and strictly pseudo-contractive maps and are well known for being demiclosed.

Lemma 2.3 (see [5]). An operator T is said to be closed at a point 𝑦∈𝑅^𝑛 if for every 𝑥∈𝑅^𝑛 and every sequence 𝑥_𝑘 in 𝑅^𝑛, such that, 𝑥_𝑘→𝑥(𝑘→∞) and 𝑇(𝑥_𝑘)→𝑦, we have 𝑇𝑥=𝑦.
In what follows, only the particular case of closed at zero will be used, which is the particular case when 𝑦=0.

Lemma 2.4 (see [12]). Let {𝑎_𝑛},{𝑏_𝑛} and {𝛿_𝑛} be sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,𝑛≥1, (2.10) where {𝛿_𝑛}⊂𝑙¹, {𝑏_𝑛}⊂𝑙¹, then lim_𝑛→∞𝑎_𝑛 exists.

Motivated by the former works in [5–9], we modify the algorithm proposed by Moudafi in [6] for solving SCFP in the more general case when the operators are firmly pseudo-demicontractive, defined on a general Hilbert space and also change several conditions. Then, we prove a strong convergence theorem of the modified algorithm about firmly pseudo-demicontractive operators, which improves and/or develops several corresponding results in this field. We present in this paper only theoretical results of algorithmic developments and convergence theorems. Experimental computational work in other literatures [4, 10] shows the practical viability of this class of algorithms.

3. Main Results

Let us consider now the two operators split common fixed-point problem (SCFP) [5, 6]find𝑥^∗∈𝐶suchthat𝐴𝑥^∗∈𝑄, (3.1) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, 𝑈∶𝐻₁→𝐻₁,𝑇∶𝐻₂→𝐻₂ are two firmly pseudo-demicontractive operators with Fix(𝑈)=𝐶 and Fix(𝑇)=𝑄,𝑇𝑥−𝑞²≤𝛼𝑥−𝑞²+𝛽‖𝑥−𝑇𝑥‖²,∀𝑥,𝑞∈𝐻₂×Fix(𝑇),𝑈𝑥−𝑞²≤𝜇𝑥−𝑞²+𝜃‖𝑥−𝑈𝑥‖²,∀𝑥,𝑞∈𝐻₁×Fix(𝑈), (3.2) where 𝛼,𝛽 and 𝜇,𝜃 are two firmly pseudo-demicontractive coefficients of 𝑈,𝑇, respectively. 𝛼>1,𝛽>1,𝜇>1,𝜃>1, 𝐻₁,𝐻₂ are two Hilbert spaces.

In what follows we always assume that the solution set of the two-operator SCFP is not empty, which denotes byΓ=𝑦∈𝐶∣𝐴𝑦∈𝑄. (3.3)

Based on the algorithm of [5, 6], we develop the following modified algorithm to solve (3.1).

Algorithm 3.1. Initialization: let 𝑥₀∈𝐻₁ be arbitrary.
Iterative step: for 𝑘∈𝑁 set 𝑢_𝑘=𝑥_𝑘+𝛾𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘 and let 𝑥_𝑘+1=1−𝑡_𝑘𝑢_𝑘+𝑡_𝑘𝑈𝑢_𝑘,𝑘∈𝑁, (3.4) where (1−𝛽)/𝜆<𝛾<0 with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴, 𝑡_𝑘>0.

Theorem 3.2. Let 𝐻₁, 𝐻₂ are two Hilbert spaces, 𝐴 is a bounded linear operator, 𝑈∶𝐻₁→𝐻₁,𝑇∶𝐻₂→𝐻₂ are two firmly pseudo-demicontractive operators with Fix(𝑈)=𝐶 and Fix(𝑇)=𝑄,𝛼,𝛽, and 𝜇,𝜃 are two firmly pseudo-demicontractive coefficients of 𝑈,𝑇, respectively. Let 𝑏_𝑘=𝑡_𝑘(𝜃+𝑡_𝑘−1)‖𝑈(𝑢_𝑘)−𝑢_𝑘‖², 𝛿_𝑘=𝑡_𝑘(𝜇−1) and {𝑏_𝑘}⊂𝑙¹,{𝛿_𝑘}⊂𝑙¹. 𝑇−𝐼 is closed at the origin, 𝜆 the spectral radius of the operator 𝐴^∗𝐴, then the sequence generated by the modified algorithm (3.4) converges strongly to the solution of (3.1).

Proof. Taking 𝑦∈Γ, that is, 𝑦∈Fix(𝑈),𝐴𝑦∈Fix(𝑇), and using (2.6), we obtain that 𝑥_𝑘+1−𝑦²=1−𝑡_𝑘𝑢_𝑘+𝑡_𝑘𝑈𝑢_𝑘−𝑦²=𝑢_𝑘−𝑦²−2𝑡_𝑘𝑢_𝑘−𝑦,𝑢_𝑘−𝑈𝑢_𝑘+𝑡²_𝑘𝑈𝑢_𝑘−𝑢_𝑘²≤𝑢_𝑘−𝑦²−2𝑡_𝑘1−𝜇2𝑢_𝑘−𝑦²+1−𝜃2𝑈𝑢_𝑘−𝑢_𝑘²+𝑡²_𝑘𝑈𝑢_𝑘−𝑢_𝑘²≤1−𝑡_𝑘1−𝜇𝑢_𝑘−𝑦²−𝑡_𝑘1−𝜃−𝑡_𝑘𝑈𝑢_𝑘−𝑢_𝑘². (3.5) Using the expression of 𝑢_𝑘 in Algorithm 3.1, we also have 𝑢_𝑘−𝑦²=𝑥_𝑘+𝛾𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘−𝑦²=𝑥_𝑘−𝑦²+𝛾²𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘²+2𝛾𝑥_𝑘−𝑦,𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘=𝑥_𝑘−𝑦²+𝛾²(𝑇−𝐼)𝐴𝑥_𝑘,𝐴𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘+2𝛾𝐴𝑥_𝑘−𝐴𝑦,(𝑇−𝐼)𝐴𝑥_𝑘. (3.6) From the definition of 𝜆, it follows that 𝛾²(𝑇−𝐼)𝐴𝑥_𝑘,𝐴𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘≤𝜆𝛾²(𝑇−𝐼)𝐴𝑥_𝑘². (3.7)
Now, by setting 𝜃=2𝛾⟨𝑥_𝑘−𝑦,𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑘⟩ and using the fact (2.5) and its equivalent form (2.6), we infer that 𝜃=2𝛾𝐴𝑥_𝑘−A𝑦,𝑇𝐴𝑥_𝑘−𝐴𝑥_𝑘≤2𝛾𝛼−12𝐴𝑥_𝑘−𝐴𝑦²+𝛽−12𝑇𝐴𝑥_𝑘−𝐴𝑥_𝑘²=𝛾(𝛼−1)𝐴𝑥_𝑘−𝐴𝑦²+𝛾𝛽−1𝑇𝐴𝑥_𝑘−𝐴𝑥_𝑘². (3.8) Substituting (3.8), (3.7), and (3.6) into (3.5), we get the following inequality: 𝑥_𝑘+1−𝑦²≤1−𝑡_𝑘1−𝜇𝑢_𝑘−𝑦²−𝑡_𝑘1−𝜃−𝑡_𝑘𝑈𝑢_𝑘−𝑢_𝑘²≤1−𝑡_𝑘1−𝜇𝑥_𝑘−𝑦²+𝛾(𝛼−1)𝐴𝑥_𝑘−𝐴𝑦²+𝛾𝛽−1+𝛾𝜆(𝑇−𝐼)𝐴𝑥_𝑘²−𝑡_𝑘1−𝜃−𝑡_𝑘𝑈𝑢_𝑘−𝑢_𝑘²=1+𝑡_𝑘𝜇−1𝑥_𝑘−𝑦²+1−𝑡_𝑘1−𝜇𝛾(𝛼−1)𝐴𝑥_𝑘−𝐴𝑦²+1−𝑡_𝑘1−𝜇𝛾𝛽−1+𝛾𝜆(𝑇−𝐼)𝐴𝑥_𝑘²+𝑡_𝑘𝜃+𝑡_𝑘−1𝑈𝑢_𝑘−𝑢_𝑘². (3.9) Since ((1−𝛽)/𝜆)<𝛾<0, 𝑡_𝑘>0 and 𝛼>1,𝛽>1,𝜇>1,𝜃>1, we obtain that −𝑡_𝑘(1−𝜇)>0, (1−𝑡_𝑘(1−𝜇))𝛾(𝛼−1)<0, (1−𝑡_𝑘(1−𝜇))𝛾(𝛽−1+𝛾𝜆)<0, and 𝑡_𝑘(𝜃+𝑡_𝑘−1)>0, thus, 𝑥_𝑘+1−𝑦²≤1+𝑡_𝑘𝜇−1𝑥_𝑘−𝑦²+𝑡_𝑘𝜃+𝑡_𝑘−1𝑈𝑢_𝑘−𝑢_𝑘². (3.10) Setting 𝑏_𝑘=𝑡_𝑘(𝜃+𝑡_𝑘−1)‖𝑈(𝑢_𝑘)−𝑢_𝑘‖² and 𝛿_𝑘=𝑡_𝑘(𝜇−1), (3.10) can be formulated as 𝑥_𝑘+1−𝑦²≤1+𝛿_𝑘𝑥_𝑘−𝑦²+𝑏_𝑘. (3.11) We also can denote that 𝑎_𝑘+1=‖𝑥_𝑘+1−𝑦‖², 𝑎_𝑘=‖𝑥_𝑘−𝑦‖², thus (3.11) can be rewritten as 𝑎_𝑘+1≤1+𝛿_𝑘𝑎_𝑘+𝑏_𝑘. (3.12)
Obviously, {𝑎_𝑘}, {𝑏_𝑘}, and {𝛿_𝑘} are sequences of nonnegative real numbers. Since {𝑏_𝑘}⊂𝑙¹, {𝛿_𝑘}⊂𝑙¹, that is, ∑^∞_𝑘=1𝛿_𝑘<∞, ∑^∞_𝑘=1𝑏_𝑘<∞ from Lemma 2.4, lim_𝑘→∞‖𝑥_𝑘−𝑦‖² exists.
Since 𝜆 being the spectral radius of the operator 𝐴^∗𝐴, (3.9) also can be reformulated as the following: 𝑥_𝑘+1−𝑦²≤1+𝑡_𝑘𝜇−1𝑥_𝑘−𝑦²+1−𝑡_𝑘1−𝜇𝛾(𝛼−1)𝐴𝑥_𝑘−𝐴𝑦²+1−𝑡_𝑘1−𝜇𝛾𝛽−1+𝛾𝜆(𝑇−𝐼)𝐴𝑥_𝑘²+𝑡_𝑘𝜃+𝑡_𝑘−1𝑈𝑢_𝑘−𝑢_𝑘²=1+𝑡_𝑘𝜇−1𝑥_𝑘−𝑦²+1−𝑡_𝑘1−𝜇𝛾(𝛼−1)𝑥_𝑘−𝑦,𝐴^∗𝐴𝑥_𝑘+−𝑦1−𝑡_𝑘1−𝜇𝛾𝛽−1+𝛾𝜆(𝑇−𝐼)𝐴𝑥_𝑘²+𝑡_𝑘𝜃+𝑡_𝑘−1𝑈𝑢_𝑘−𝑢_𝑘²≤1+𝑡_𝑘𝜇−1𝑥_𝑘−𝑦²+1−𝑡_𝑘1−𝜇𝛾(𝛼−1)𝜆𝑥_𝑘−𝑦²+1−𝑡_𝑘1−𝜇𝛾𝛽−1+𝛾𝜆(𝑇−𝐼)𝐴𝑥_𝑘²+𝑡_𝑘𝜃+𝑡_𝑘−1𝑈𝑢_𝑘−𝑢_𝑘². (3.13) Taking limits from both side of (3.13), (1−𝑡_𝑘(1−𝜇))𝛾(𝛼−1)𝜆<0, 𝑡_𝑘(𝜇−1)→0, (1−𝑡_𝑘(1−𝜇))𝛾(𝛽−1+𝛾𝜆)<0, 𝑡_𝑘(𝜃+𝑡_𝑘−1)‖𝑈(𝑢_𝑘)−𝑢_𝑘‖²→0, we obtain that lim_𝑘→∞𝑥_𝑘−𝑦²⟶0,thatis,𝑥_𝑘⟶𝑦(𝑘⟶∞). (3.14) If we take limits from both sides of (3.9), we can get the following: lim_𝑘→∞𝐴𝑥_𝑘−𝐴𝑦²⟶0,thatis,𝐴𝑥_𝑘⟶𝐴𝑦(𝑘⟶∞),lim_𝑘→∞(𝑇−𝐼)𝐴𝑥_𝑘²⟶0,thatis,(𝑇−𝐼)𝐴𝑥_𝑘⟶0(𝑘⟶∞). (3.15)(3.16) Because 𝑇−𝐼 is closed at the origin, from (3.15) and (3.16), using Lemma 2.3, we have (𝑇−𝐼)𝐴𝑦=0,thatis,𝑇𝐴𝑦=𝐴𝑦. (3.17)
The sequence generated by modified algorithm (3.1) converges strongly to the solution of SCFP. The proof is completed.

Under the same conditions as in Thereom 3.2, if we take 𝛽=𝜃=1 (i.e., 𝑈,𝑇 are pseudo-demicontractive operators), the strong convergence also holds, so we get the following corollary.

Corollary 3.3. Let 𝐻₁,𝐻₂ are two Hilbert spaces, 𝐴 is a bounded linear operator, 𝑈∶𝐻₁→𝐻₁,𝑇∶𝐻₂→𝐻₂ are two pseudo-demicontractive operators with Fix(𝑈)=𝐶 and Fix(𝑇)=𝑄, 𝛼 and 𝜇 are two pseudo-demicontractive coefficients of 𝑈,𝑇, respectively. Let 𝑏_𝑘=(1−𝑡_𝑘(1−𝜇))𝛾²𝜆‖(𝑇−𝐼)(𝐴𝑥_𝑘)‖²+𝑡²_𝑘‖𝑈(𝑢_𝑘)−𝑢_𝑘‖², 𝛿_𝑘=𝑡_𝑘(𝜇−1), and {𝑏_𝑘}⊂𝑙¹,{𝛿_𝑘}⊂𝑙¹. 𝑇−𝐼 is closed at the origin, and 𝜆 is the spectral radius of the operator 𝐴^∗𝐴, then the sequence generated by the modified algorithm (3.4) converges strongly to the solution of (3.1).

Proof. Taking 𝛽=𝜃=1, here (3.9) as the following: 𝑥_𝑘+1−𝑦²≤1+𝑡_𝑘𝜇−1𝑥_𝑘−𝑦²+1−𝑡_𝑘1−𝜇𝛾(𝛼−1)𝐴𝑥_𝑘−𝐴𝑦²+1−𝑡_𝑘1−𝜇𝛾²𝜆(𝑇−𝐼)(𝐴𝑥_𝑘)²+𝑡²_𝑘𝑈(𝑢_𝑘)−𝑢_𝑘². (3.18) Because 𝑏_𝑘=1−𝑡_𝑘1−𝜇𝛾²𝜆(𝑇−𝐼)𝐴𝑥_𝑘²+𝑡²_𝑘𝑈(𝑢_𝑘)−𝑢_𝑘²,𝛿_𝑘=𝑡_𝑘𝜇−1, (3.19) from (3.18), we get 𝑎_𝑘+1≤1+𝛿_𝑘𝑎_𝑘+𝑏_𝑘. (3.20) The same lines as the proof of Thereom 3.2, we obtain the desired result.

At last, we want to say that the condition {𝑏_𝑘}⊂𝑙¹,{𝛿_𝑘}⊂𝑙¹ actually controls the distance between independent variable and dependent variable of the related operators. Provided that the distance may not fluctuate too severely, the condition can always be satisfied with some appropriate coefficients, for example, taking 𝑡_𝑘=1/𝑘².

4. Conclusion

In this paper, we generalize the algorithm proposed by Moudafi for demicontractive operators to firmly pseudo-demicontractive operators for SCFP and use some beautiful lemmas to prove the strong convergence of the modified algorithm. Our results improve and/or develop Moudafi, Censor, and some other people’s work.

Acknowledgment

This research was supported partly by NSFC Grant no: 11071279.