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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 575014, 13 pages
http://dx.doi.org/10.1155/2012/575014
Research Article

Strong Convergence Theorems for the Generalized Split Common Fixed Point Problem

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 9 January 2012; Accepted 17 February 2012

Academic Editor: Rudong Chen

Copyright © 2012 Cuijie Zhang. 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 the generalized split common fixed point problem (GSCFPP) and show that the GSCFPP for nonexpansive operators is equivalent to the common fixed point problem. Moreover, we introduce a new iterative algorithm for finding a solution of the GSCFPP and obtain some strong convergence theorems under suitable assumptions.

1. Introduction

Let 𝐻1 and 𝐻2 be real Hilbert spaces and let 𝐴𝐻1𝐻2 be a bounded linear operator. Given intergers 𝑝,𝑟1, let us recall that the multiple-set split feasibility problem (MSSFP) was recently introduced [1] and is to find a point:𝑥𝑝𝑖=1𝐶𝑖,𝐴𝑥𝑟𝑗=1𝑄𝑗,(1.1) where {𝐶𝑖}𝑝𝑖=1 and {𝑄𝑗}𝑟𝑗=1 are nonempty closed convex subsets of 𝐻1 and 𝐻2, respectively. If 𝑝=𝑟=1, the MSSFP (1.1) becomes the so-called split feasibility problem (SFP) [2] which is to find a point:𝑥𝐶,𝐴𝑥𝑄,(1.2) where 𝐶 and 𝑄 are nonempty closed convex subsets of 𝐻1 and 𝐻2, respectively. Recently, the SFP (1.2) and MSSFP (1.1) have been investigated by many researchers; see, [310].

Since every closed convex subset in a Hilbert space is looked as the fixed point set of its associating projection, the MSSFP (1.1) becomes a special case of the split common fixed point problem (SCFPP), which is to find a point:𝑥𝑝𝑖=1𝑈Fix𝑖,𝐴𝑥𝑟𝑗=1𝑇Fix𝑗,(1.3) where 𝑈𝑖𝐻1𝐻1(𝑖=1,2,,𝑝) and 𝑇𝑗𝐻2𝐻2(𝑗=1,2,,𝑟) are nonlinear operators. If 𝑝=𝑟=1, the problem (1.3) reduces to the so-called two-set SCFPP, which is to find a point:𝑥Fix(𝑈),𝐴𝑥Fix(𝑇).(1.4)

Censor and Segal in [11] firstly introduced the concept of SCFPP in finite-dimensional Hilbert spaces and considered the following iterative algorithm for the two-set SCFPP (1.4) for Class- operators:𝑥𝑛+1𝑥=𝑈𝑛𝛾𝐴(𝐼𝑇)𝐴𝑥𝑛,𝑛0,(1.5) where 𝑥0𝐻1, 0<𝛾<2/𝐴2 and 𝐼 is the identity operator. They proved the convergence of the algorithm (1.5) to a solution of problem (1.4). Moreover, they introduced a parallel iterative algorithm, which converges to a solution of the SCFPP (1.3). However, the parallel iterative algorithm does not include the algorithm (1.5) as a special case.

Very recently, Wang and Xu in [12] considered the SCFPP (1.3) for Class- operators and introduced the following iterative algorithm for solving the SCFPP (1.3):𝑥𝑛+1=𝑈[𝑛]𝑥𝑛𝛾𝐴𝐼𝑇[𝑛]𝐴𝑥𝑛,𝑛0.(1.6) Under some mild conditions, they proved some weak and strong convergence theorems. Their iterative algorithm (1.6) includes Censor and Segal’s algorithm (1.5) as a special case for the two-set SCFPP (1.4). Moreover, they prove that the SCFPP (1.3) for the Class- operators is equivalent to a common fixed point problem. This is also a classical method. Many problems eventually converted to a common fixed point problem; see [1315].

Motivated and inspired by the aforementioned research works, we introduce a generalized split common fixed point problem (GSCFPP) which is to find a point:𝑥𝑖=1𝑈Fix𝑖,𝐴𝑥𝑗=1𝑇Fix𝑗.(1.7) Then, we show that the GSCFPP (1.7) for nonexpansive operators is equivalent to the following common fixed point problem:𝑥𝑖=1𝑈Fix𝑖,𝑥𝑗=1𝑉Fix𝑗,(1.8) where 𝑉𝑗=𝐼𝛾𝐴(𝐼𝑇𝑗)𝐴(0<𝛾1/𝐴2) for every 𝑗. Moreover, we give a new iterative algorithm for solving the GSCFPP (1.7) for nonexpansive operators and obtain some strong convergence theorems.

2. Preliminaries

Throughout this paper, we write 𝑥𝑛𝑥 and 𝑥𝑛𝑥 to indicate that {𝑥𝑛} converges weakly to 𝑥 and converges strongly to 𝑥, respectively.

An operator 𝑇𝐻𝐻 is said to be nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦 for all 𝑥,𝑦𝐻. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇). It is known that 𝐹(𝑇) is closed and convex. An operator 𝑓𝐻𝐻 is called contraction if there exists a constant 𝜌[0,1) such that 𝑓(𝑥)𝑓(𝑦)𝜌𝑥𝑦 for all 𝑥,𝑦𝐻. Let 𝐶 be a nonempty closed convex subset of 𝐻. For each 𝑥𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃𝐶𝑥, such that 𝑥𝑃𝐶𝑥𝑥𝑦 for every 𝑦𝐶. 𝑃𝐶 is called a metric projection of 𝐻 onto 𝐶. It is known that for each 𝑥𝐻,𝑥𝑃𝐶𝑥,𝑦𝑃𝐶𝑥0(2.1) for all 𝑦𝐶.

Let {𝑇𝑛} be a sequence of operators of 𝐻 into itself. The set of common fixed points of {𝑇𝑛} is denoted by 𝐹({𝑇𝑛}), that is, 𝐹({𝑇𝑛})=𝑛=1𝐹(𝑇𝑛). A sequence {𝑇𝑛} is said to be strongly nonexpansive if each {𝑇𝑛} is nonexpansive and𝑥𝑛𝑦𝑛𝑇𝑛𝑥𝑛𝑇𝑛𝑦𝑛0(2.2) whenever {𝑥𝑛} and {𝑦𝑛} are sequences in 𝐶 such that {𝑥𝑛𝑦𝑛} is bounded and 𝑥𝑛𝑦𝑛𝑇𝑛𝑥𝑛𝑇𝑛𝑦𝑛0; see [16, 17]. A sequence {𝑧𝑛} in 𝐻 is said to be an approximate fixed point sequence of {𝑇𝑛} if 𝑧𝑛𝑇𝑛𝑧𝑛0. The set of all bounded approximate fixed point sequences of {𝑇𝑛} is denoted by 𝐹({𝑇𝑛}); see [16, 17]. We know that if {𝑇𝑛} has a common fixed point, then 𝐹({𝑇𝑛}) is nonempty; that is, every bounded sequence in the common fixed point set is an approximate fixed point sequence. A sequence {𝑇𝑛} with a common fixed point is said to satisfy the condition (𝑍) if every weak cluster point of {𝑥𝑛} is a common fixed point whenever {𝑥𝑛}𝐹({𝑇𝑛}). A sequence {𝑇𝑛} of nonexpansive mappings of 𝐻 into itself is said to satisfy the condition (𝑅) iflim𝑛sup𝑦𝐷𝑇𝑛+1𝑦𝑇𝑛𝑦=0(2.3) for every nonempty bounded subset 𝐷 of 𝐻; see [18].

In order to prove our main results, we collect the following lemmas in this section.

Lemma 2.1 (see [16]). Let 𝐶 be a nonempty subset of a Hilbert space 𝐻. Let {𝑇𝑛} be a sequence of nonexpansive mappings of 𝐶 into 𝐻. Let {𝜆𝑛} be a sequence in [0,1] such that liminf𝑛𝜆𝑛>0. Let {𝑈𝑛} be a sequence of mappings of 𝐶 into 𝐻 defined by 𝑈𝑛=𝜆𝑛𝐼+(1𝜆𝑛)𝑇𝑛 for 𝑛, where 𝐼 is the identity mapping on 𝐶. Then {𝑈𝑛} is a strongly nonexpansive sequence.

Lemma 2.2 (see [16]). Let 𝐻 be a Hilbert space, 𝐶 a nonempty subset of 𝐻, and {𝑆𝑛} and {𝑇𝑛} sequences of nonexpansive self-mappings of 𝐶. Suppose that {𝑆𝑛} or {𝑇𝑛} is a strongly nonexpansive sequence and 𝐹({𝑆𝑛})𝐹({𝑇𝑛}) is nonempty. Then 𝐹({𝑆𝑛})𝐹({𝑇𝑛})=𝐹({𝑆𝑛𝑇𝑛}).

Lemma 2.3 (see [17]). Let 𝐻 be a Hilbert space, and 𝐶 a nonempty subset of 𝐻. Both {𝑆𝑛} and {𝑇𝑛} satisfy the condition (𝑅) and {𝑇𝑛𝑦𝑛,𝑦𝐷} is bounded for any bounded subset 𝐷 of 𝐶. Then {𝑆𝑛𝑇𝑛} satisfies the condition (𝑅).

Lemma 2.4 (see [19]). Let {𝑥𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1. Suppose 𝑥𝑛+1=(1𝛽𝑛)𝑦𝑛+𝛽𝑛𝑥𝑛for all integers 𝑛0 and limsup𝑛𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛0.(2.4) Then lim𝑛𝑦𝑛𝑥𝑛=0.

Lemma 2.5 (see [20]). Assume that {𝑎𝑛} is a sequence of nonnegative real numbers such that 𝑎𝑛+11𝛾𝑛𝑎𝑛+𝛿𝑛,𝑛0,(2.5) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that(i)𝑛=1𝛾𝑛=,(ii)limsup𝑛𝛿𝑛/𝛾𝑛0 or 𝑛=1|𝛿𝑛|<. Then lim𝑛𝑎𝑛=0.

3. Main Results

Now we state and prove our main results of this paper.

Lemma 3.1. Let 𝐴𝐻1𝐻2 be a given bounded linear operator and let 𝑇𝑛𝐻2𝐻2 be a sequence of nonexpansive operators. Assume 𝐴1𝑇Fix𝑛=𝑥𝐻1𝑇𝐴𝑥Fix𝑛.(3.1) For each constant 𝛾>0, 𝑉𝑛 is defined by the following: 𝑉𝑛=𝐼𝛾𝐴𝐼𝑇𝑛𝐴.(3.2) Then Fix({𝑉𝑛})=𝐴1(Fix({𝑇𝑛})). Moreover, for 0<𝛾1/𝐴2, 𝑉𝑛 is nonexpansive on 𝐻1 for 𝑛.

Proof. Since the inclusion 𝐴1(Fix({𝑇𝑛}))Fix({𝑉𝑛}) is evident, now we only need to show the converse inclusion. If 𝑧Fix({𝑉𝑛}), then we have 𝐴(𝐼𝑇𝑛)𝐴𝑧=0. Since 𝐴1(Fix({𝑇𝑛})), we take an arbitrary 𝑝𝐴1(Fix({𝑇𝑛})). Hence 𝐴𝑧𝑇𝑛𝐴𝑧2=𝐴𝑧𝑇𝑛𝐴𝑧,𝐴𝑧𝑇𝑛𝐴𝑧=𝐴𝑧𝑇𝑛𝐴𝑧,𝐴𝑧𝐴𝑝+𝐴𝑝𝑇𝑛=𝐴𝐴𝑧𝐼𝑇𝑛𝐴𝑧,𝑧𝑝+𝐴𝑧𝑇𝑛𝐴𝑧,𝐴𝑝𝑇𝑛1𝐴𝑧=2𝐴𝑧𝐴𝑝2+12𝐴𝑧𝑇𝑛𝐴𝑧2+12𝐴𝑝𝑇𝑛𝐴𝑧212𝐴𝑧𝑇𝑛𝐴𝑧2.(3.3) It follows that (1/2)𝐴𝑧𝑇𝑛𝐴𝑧20, then 𝐴𝑧=𝑇𝑛𝐴𝑧 for every 𝑛, hence 𝑧𝐴1(Fix({𝑇𝑛})). Next we turn to show that 𝑉𝑛 is a nonexpansive operator for 𝑛. Since 𝑇𝑛 is nonexpansive, we have (𝐼𝑇𝑛)𝐴𝑥(𝐼𝑇𝑛)𝐴𝑦2=𝐴𝑥𝐴𝑦2+𝑇𝑛𝐴𝑥𝑇𝑛𝐴𝑦22𝐴𝑥𝐴𝑦,𝑇𝑛𝐴𝑥𝑇𝑛𝐴𝑦2𝐴𝑥𝐴𝑦22𝐴𝑥𝐴𝑦,𝑇𝑛𝐴𝑥𝑇𝑛𝑇𝐴𝑦2𝐴𝑥𝐴𝑦,𝐴𝑥𝐴𝑦𝑛𝐴𝑥𝑇𝑛.𝐴𝑦(3.4) Hence 𝑉𝑛𝑥𝑉𝑛𝑦2=𝐼𝛾𝐴𝐼𝑇𝑛𝐴𝑥(𝐼𝛾𝐴(𝐼𝑇𝑛)𝐴)𝑦2=𝑥𝑦2+𝛾2𝐴2𝐼𝑇𝑛𝐴𝑥𝐼𝑇𝑛𝐴𝑦22𝛾𝐴𝑥𝐴𝑦,𝐼𝑇𝑛𝐴𝑥𝐼𝑇𝑛𝐴𝑦𝑥𝑦2+𝛾𝛾𝐴21𝐼𝑇𝑛𝐴𝑥𝐼𝑇𝑛𝐴𝑦2.(3.5) For 0<𝛾1/𝐴2, we can immediately obtain that 𝑉𝑛 is a nonexpansive operator for every 𝑛.

From Lemma 3.1, we can obtain that the solution set of GSCFPP (1.7) is identical to the solution set of problem (1.8).

Theorem 3.2. Let {𝑈𝑛} and {𝑉𝑛} be sequences of nonexpansive operators on Hilbert space 𝐻1. Both {𝑈𝑛} and {𝑉𝑛} satisfy the conditions (𝑅) and (𝑍). Let 𝑓𝐻1𝐻1 be a contraction with coefficient 𝜌[0,1). Suppose Ω=Fix(𝑈𝑛)Fix(𝑉𝑛). Take an initial guess 𝑥1𝐻1 and define a sequence {𝑥𝑛} by the following algorithm: 𝑦𝑛=𝜆𝑛𝑥𝑛+1𝜆𝑛𝑉𝑛𝑥𝑛,𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝛽𝑛𝑥𝑛+𝛾𝑛𝑈𝑛𝑦𝑛,(3.6) where {𝛼𝑛}, {𝛽𝑛}, {𝛾𝑛}, and {𝜆𝑛} are sequences in [0,1]. If the following conditions are satisfied:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛1;(ii)lim𝑛𝛼𝑛=0  and   Σ𝑛=1𝛼𝑛=;(iii)0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1;(iv)0<liminf𝑛𝜆𝑛limsup𝑛𝜆𝑛<1;(v)lim𝑛|𝜆𝑛+1𝜆𝑛|=0, then {𝑥𝑛} converges strongly to 𝑤Ω where 𝑤=𝑃Ω𝑓(𝑤).

Proof. We proceed with the following steps.
Step 1. First show that there exists 𝑤Ω such that 𝑤=𝑃Ω𝑓(𝑤).
In fact, since 𝑓 is a contraction with coefficient 𝜌, we have 𝑃Ω𝑓(𝑥)𝑃Ω𝑓(𝑦)𝑓(𝑥)𝑓(𝑦)𝜌𝑥𝑦(3.7) for every 𝑥, 𝑦. Hence 𝑃Ω𝑓 is also a contraction. Therefore, there exists a unique 𝑤Ω such that 𝑤=𝑃Ω𝑓(𝑤).
Step 2. Now we show that {𝑥𝑛} is bounded.
Let 𝑝Ω, then 𝑝Fix({𝑈𝑛}) and 𝑝Fix({𝑉𝑛}). Hence 𝑈𝑛𝑦𝑛𝑝𝑦𝑛𝑝𝜆𝑛𝑥𝑛𝑝+1𝜆𝑛𝑉𝑛𝑥𝑛𝑝𝑥𝑛𝑝.(3.8) Then 𝑥𝑛+1𝑝𝛼𝑛𝑓𝑥𝑛𝑝+𝛽𝑛𝑥𝑛𝑝+𝛾𝑛𝑈𝑛𝑦𝑛𝑝𝛼𝑛𝜌𝑥𝑛𝑝+𝛼𝑛𝑓(𝑝)𝑝+𝛽𝑛𝑥𝑛𝑝+𝛾𝑛𝑥𝑛𝑝1𝛼𝑛(𝑥1𝜌)𝑛𝑝+𝛼𝑛(11𝜌)(𝑥1𝜌𝑓𝑝)𝑝max𝑛,1𝑝.1𝜌𝑓(𝑝)𝑝(3.9) By induction on 𝑛, 𝑉𝑛𝑥𝑛𝑥𝑝𝑛𝑥𝑝max1,1𝑝1𝜌𝑓(𝑝)𝑝(3.10) for every 𝑛. This shows that {𝑥𝑛} and {𝑉𝑛𝑥𝑛} are bounded, and hence, {𝑈𝑛𝑦𝑛}, {𝑦𝑛}, and {𝑓(𝑥𝑛)} are also bounded.
Step 3. We claim that 𝐹({𝐴𝑛})=𝐹({𝑉𝑛}) and 𝐹({𝑈𝑛𝐴𝑛})=𝐹({𝑈𝑛})𝐹({𝑉𝑛}), where 𝐴𝑛=𝜆𝑛𝐼+(1𝜆𝑛)𝑉𝑛.
We first show the former equality. Let {𝑧𝑛} be a bounded sequence in 𝐻1. If {𝑧𝑛}𝐹({𝑉𝑛}), then 𝐴𝑛𝑧𝑛𝑧𝑛=𝜆𝑛𝑧𝑛+1𝜆𝑛𝑉𝑛𝑧𝑛𝑧𝑛=1𝜆𝑛𝑉𝑛𝑧𝑛𝑧𝑛0.(3.11) Hence {𝑧𝑛}𝐹({𝐴𝑛}). On the other hand, if {𝑧𝑛}𝐹({𝐴𝑛}), combining (3.11) and limsup𝑛𝜆𝑛<1, we obtain that 𝑉𝑛𝑧𝑛𝑧𝑛0. Hence {𝑧𝑛}𝐹({𝑉𝑛}). Therefore, 𝐹({𝐴𝑛})=𝐹({𝑉𝑛}).
Next, we show the latter equality. Using Lemma 2.1, we know that {𝐴𝑛} is a strongly nonexpansive sequence. Thus, since 𝐹({𝑈𝑛})𝐹({𝐴𝑛})=𝐹({𝑈𝑛})𝐹({𝑉𝑛}), from Lemma 2.2 we have 𝐹𝑈𝑛𝐴𝑛=𝐹𝑈𝑛𝐹𝐴𝑛=𝐹𝑈𝑛𝐹𝑉𝑛.(3.12)
Step 4. {𝑆𝑛} satisfies the condition (𝑅), where 𝑆𝑛=𝑈𝑛𝐴𝑛.
Let 𝐷 be a nonempty bounded subset of 𝐻1. From the definition of {𝐴𝑛}, we have, for all 𝑦𝐷, 𝐴𝑛+1𝑦𝐴𝑛𝑦=𝜆𝑛+1𝑦+1𝜆𝑛+1𝑉𝑛+1𝑦𝜆𝑛𝑦1𝜆𝑛𝑉𝑛𝑦||𝜆𝑛+1𝜆𝑛||𝑉𝑦+𝑛+1𝑦𝑉𝑛𝑦+𝜆𝑛+1𝑉𝑛+1𝑦𝜆𝑛𝑉𝑛𝑦||𝜆𝑛+1𝜆𝑛||𝑉𝑦+𝑛+1𝑦𝑉𝑛𝑦+𝜆𝑛+1𝑉𝑛+1𝑦𝜆𝑛𝑉𝑛+1𝑦+𝜆𝑛𝑉𝑛+1𝑦𝜆𝑛𝑉𝑛𝑦=||𝜆𝑛+1𝜆𝑛||𝑉𝑦+𝑛+1𝑦𝑉𝑛𝑦+||𝜆𝑛+1𝜆𝑛||𝑉𝑛+1𝑦+𝜆𝑛𝑉𝑛+1𝑦𝑉𝑛𝑦=||𝜆𝑛+1𝜆𝑛||𝑉𝑦+𝑛+1𝑦+1+𝜆𝑛𝑉𝑛+1𝑦𝑉𝑛𝑦.(3.13) It follows that sup𝑦𝐷𝐴𝑛+1𝑦𝐴𝑛𝑦||𝜆𝑛+1𝜆𝑛||sup𝑦𝐷𝑉𝑦+𝑛+1𝑦+1+𝜆𝑛sup𝑦𝐷𝑉𝑛+1𝑦𝑉𝑛𝑦.(3.14) Since {𝑉𝑛} satisfies the condition (𝑅) and lim𝑛|𝜆𝑛+1𝜆𝑛|=0, we have lim𝑛sup𝑦𝐷𝐴𝑛+1𝑦𝐴𝑛𝑦=0,(3.15) that is, {𝐴𝑛} satisfies the condition (𝑅). Since {𝐴𝑛𝑦𝑛,𝑦𝐷} is bounded for any bounded subset 𝐷 of 𝐻1, by using Lemma 2.3, we have that {𝑉𝑛𝐴𝑛} satisfies the condition (𝑅), that is, {𝑆𝑛} satisfies the condition (𝑅).
Step 5. We show 𝑥𝑛+1𝑥𝑛0.
We can write (3.6) as 𝑥𝑛+1=𝛽𝑛𝑥𝑛+(1𝛽𝑛)𝑧𝑛 where 𝑧𝑛=(𝛼𝑛𝑓(𝑥𝑛)+𝛾𝑛𝑆𝑛𝑥𝑛)/1𝛽𝑛. It follows that 𝑧𝑛+1𝑧𝑛=𝛼𝑛+1𝑓𝑥𝑛+1+𝛾𝑛+1𝑆𝑛+1𝑥𝑛+11𝛽𝑛+1𝛼𝑛𝑓𝑥𝑛+𝛾𝑛𝑆𝑛𝑥𝑛1𝛽𝑛=𝛼𝑛+11𝛽𝑛+1𝑓𝑥𝑛+1𝑥𝑓𝑛+𝛼𝑛+11𝛽𝑛+1𝛼𝑛1𝛽𝑛𝑓𝑥𝑛+𝛾𝑛+11𝛽𝑛+1𝑆𝑛+1𝑥𝑛+1𝑆𝑛𝑥𝑛+𝛾𝑛+11𝛽𝑛+1𝛾𝑛1𝛽𝑛𝑆𝑛𝑥𝑛.(3.16) From Step 2, we may assume that {𝑥𝑛}𝐷, where 𝐷 is a bounded set of 𝐻1. Then from (3.16), we obtain 𝑧𝑛+1𝑧𝑛||||𝛼𝑛+11𝛽𝑛+1𝛼𝑛1𝛽𝑛||||𝑓𝑥𝑛+𝑆𝑛𝑥𝑛+𝛼𝑛+11𝛽𝑛+1𝜌𝑥𝑛+1𝑥𝑛+𝛾𝑛+11𝛽𝑛+1𝑆𝑛+1𝑥𝑛+1𝑆𝑛𝑥𝑛+1+𝛾𝑛+11𝛽𝑛+1𝑆𝑛𝑥𝑛+1𝑆𝑛𝑥𝑛||||𝛼𝑛+11𝛽𝑛+1𝛼𝑛1𝛽𝑛||||𝑓𝑥𝑛+𝑆𝑛𝑥𝑛+𝛼1𝑛+11𝛽𝑛+1𝑥(1𝜌)𝑛+1𝑥𝑛+𝛾𝑛+11𝛽𝑛+1sup𝑦𝐷𝑆𝑛+1𝑦𝑆𝑛𝑦.(3.17) It follows that 𝑧𝑛+1𝑧𝑛𝑥𝑛+1𝑥𝑛||||𝛼𝑛+11𝛽𝑛+1𝛼𝑛1𝛽𝑛||||𝑓𝑥𝑛+𝑆𝑛𝑥𝑛+𝛾𝑛+11𝛽𝑛+1sup𝑦𝐷𝑆𝑛+1𝑦𝑆𝑛𝑦𝛼𝑛+11𝛽𝑛+1𝑥(1𝜌)𝑛+1𝑥𝑛.(3.18) Since {𝑆𝑛} satisfies the condition (𝑅), combining 𝛼𝑛0 as 𝑛, we have limsup𝑛𝑧𝑛+1𝑧𝑛𝑥𝑛+1𝑥𝑛0.(3.19) Hence by Lemma 2.4, we get 𝑧𝑛𝑥𝑛0 as 𝑛. Consequently, lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛1𝛽𝑛𝑧𝑛𝑥𝑛=0.(3.20)
Step 6. We claim that {𝑥𝑛}𝐹({𝑈𝑛})𝐹({𝑉𝑛}).
From (3.6), we have 𝑆𝑛𝑥𝑛𝑥𝑛𝑆𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑥𝑛=𝑆𝑛𝑥𝑛𝛼𝑛𝑓𝑥𝑛𝛽𝑛𝑥𝑛𝛾𝑛𝑆𝑛𝑥𝑛+𝑥𝑛+1𝑥𝑛𝛼𝑛𝑆𝑛𝑥𝑛𝑥𝑓𝑛+𝛽𝑛𝑆𝑛𝑥𝑛𝑥𝑛+𝑥𝑛+1𝑥𝑛,(3.21) and hence 1𝛽𝑛𝑆𝑛𝑥𝑛𝑥𝑛𝛼𝑛𝑆𝑛𝑥𝑛𝑥𝑓𝑛+𝑥𝑛+1𝑥𝑛.(3.22) Since 𝑥𝑛+1𝑥𝑛0, 𝛼𝑛0 and limsup𝑛𝛽𝑛<1, we derive 𝑆𝑛𝑥𝑛𝑥𝑛0.(3.23) Thus (3.23) and Steps 2 and 3 imply that 𝑥𝑛𝐹𝑆𝑛=𝐹𝑈𝑛𝐹𝑉𝑛.(3.24)
Step 7. Show limsup𝑛𝑓(𝑤)𝑤,𝑥𝑛𝑤0, where 𝑤=𝑃Ω𝑓(𝑤).
Since {𝑥𝑛} is bounded, there exist a point 𝑣𝐻1 and a subsequence {𝑥𝑛𝑖} of {𝑥𝑛} such that limsup𝑛𝑓(𝑤)𝑤,𝑥𝑛𝑤=lim𝑖𝑓(𝑤)𝑤,𝑥𝑛𝑖𝑤(3.25) and 𝑥𝑛𝑖𝑣. Since {𝑈𝑛} and {𝑉𝑛} satisfy the condition (𝑍), from Step 6, we have 𝑣𝐹({𝑈𝑛})𝐹({𝑉𝑛}). Using (2.1), we get limsup𝑛𝑓(𝑤)𝑤,𝑥𝑛𝑤=lim𝑖𝑓(𝑤)𝑤,𝑥𝑛𝑖𝑤=𝑓(𝑤)𝑤,𝑣𝑤0.(3.26)
Step 8. Show 𝑥𝑛𝑤=𝑃Ω𝑓(𝑤).
Since 𝑤Ω, using (3.8), we have 𝑥𝑛+1𝑤2=𝛼𝑛𝑓𝑥𝑛𝑤+𝛽𝑛𝑥𝑛𝑤+𝛾𝑛𝑈𝑛𝑦𝑛𝑤,𝑥𝑛+1𝑤𝛼𝑛𝑓𝑥𝑛𝑓(𝑤),𝑥𝑛+1𝑤+𝛼𝑛𝑓(𝑤)𝑤,𝑥𝑛+1𝑤+𝛽𝑛𝑥𝑛𝑥𝑤𝑛+1𝑤+𝛾𝑛𝑦𝑛𝑥𝑤𝑛+11𝑤2𝛼𝑛𝜌𝑥𝑛𝑤2+𝑥𝑛+1𝑤2+𝛼𝑛𝑓(𝑤)𝑤,𝑥𝑛+1+1𝑤2𝛽𝑛𝑥𝑛𝑤2+𝑥𝑛+1𝑤2+12𝛾𝑛𝑥𝑛𝑤2+𝑥𝑛+1𝑤2121𝛼𝑛𝑥(1𝜌)𝑛𝑤2+12𝑥𝑛+1𝑤2+𝛼𝑛𝑓(𝑤)𝑤,𝑥𝑛+1,𝑤(3.27) which implies that 𝑥𝑛+1𝑤21𝛼𝑛𝑥(1𝜌)𝑛𝑤2+2𝛼𝑛1(1𝜌)1𝜌𝑓(𝑤)𝑤,𝑥𝑛+1𝑤,(3.28) for every 𝑛. Consequently, according to Step 7, 𝜌[0,1), and Lemma 2.5, we deduce that {𝑥𝑛} converges strongly to 𝑤=𝑃Ω(𝑤). This completes the proof.

Combining Lemma 3.1 and Theorem 3.2, we can obtain the following strong convergence theorem for solving the GSCFPP (1.7).

Theorem 3.3. Let {𝑈𝑛} and {𝑇𝑛} be sequences of nonexpansive operators on Hilbert space 𝐻1 and 𝐻2, respectively. Both {𝑈𝑛} and {𝑇𝑛} satisfy the conditions (𝑅) and (𝑍). Let 𝑓𝐻1𝐻1 be a contraction with coefficient 𝜌[0,1). Suppose that the solution set Ω of GSCFPP (1.7) is nonempty. Take an initial guess 𝑥1𝐻1 and define a sequence {𝑥𝑛} by the following algorithm: 𝑦𝑛=𝑥𝑛𝛾1𝜆𝑛𝐴𝐼𝑇𝑛𝐴𝑥𝑛,𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝛽𝑛𝑥𝑛+𝛾𝑛𝑈𝑛𝑦𝑛,(3.29) where 𝛾(0,1/𝐴2), and {𝛼𝑛}, {𝛽𝑛}, {𝛾𝑛}, {𝜆𝑛} are sequences in [0,1]. If the following conditions are satisfied:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛1;(ii)lim𝑛𝛼𝑛=0  and  Σ𝑛=1𝛼𝑛=;(iii)0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1;(iv)0<liminf𝑛𝜆𝑛limsup𝑛𝜆𝑛<1;(v)lim𝑛|𝜆𝑛+1𝜆𝑛|=0, then {𝑥𝑛} converges strongly to 𝑤Ω where 𝑤=𝑃Ω𝑓(𝑤).

Proof. Set 𝑉𝑛=𝐼𝛾𝐴(𝐼𝑇𝑛)𝐴. By Lemma 3.1, 𝑉𝑛 is a nonexpansive operator for every 𝑛. We can rewrite (3.29) as 𝑦𝑛=𝜆𝑛𝑥𝑛+1𝜆𝑛𝑉𝑛𝑥𝑛,𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝛽𝑛𝑥𝑛+𝛾𝑛𝑈𝑛𝑦𝑛.(3.30)
We only need to prove that {𝑉𝑛} satisfies the conditions (𝑅) and (𝑍). Assume that 𝐷 is a nonempty bounded subset of 𝐻1. For every 𝑦𝐷, we have 𝐼𝛾𝐴𝐼𝑇𝑛+1𝐴𝑦𝐼𝛾𝐴𝐼𝑇𝑛𝐴𝑦𝐴𝛾𝐼𝑇𝑛+1𝐴𝑦𝐴𝐼𝑇𝑛𝑇𝐴𝑦𝛾𝐴𝑛+1(𝐴𝑦)𝑇𝑛.(𝐴𝑦)(3.31) Since {𝑇𝑛} satisfies the condition (𝑅), and 𝐷={𝐴𝑦𝑦𝐷} is bounded, it follows from (3.31) that sup𝑦𝐷𝐼𝛾𝐴𝐼𝑇𝑛+1𝐴𝑦𝐼𝛾𝐴𝐼𝑇𝑛𝐴𝑦𝛾𝐴sup𝑦𝐷𝑇𝑛+1(𝐴𝑦)𝑇𝑛(𝐴𝑦)=𝛾𝐴sup𝑧𝐷𝑇𝑛+1𝑧𝑇𝑛𝑧0.(3.32) Therefore, {𝑉𝑛} satisfies the condition (𝑅).
Assume that 𝑥𝑛𝑧 and 𝑥𝑛𝑉𝑛𝑥𝑛0; we next show that 𝑉𝑛𝑧=𝑧. By using 𝑥𝑛𝑉𝑛𝑥𝑛0, we have 𝐴(𝐼𝑇𝑛)𝐴𝑥𝑛0. Since 𝐴1(Fix({𝑇𝑛})), we choose an arbitrary point 𝑝𝐴1(Fix({𝑇𝑛})); then for every 𝑛, 𝐴𝑥𝑛𝑇𝑛𝐴𝑥𝑛2=𝐴𝑥𝑛𝑇𝑛𝐴𝑥𝑛,𝐴𝑥𝑛𝐴𝑝+𝐴𝑝𝑇𝑛𝐴𝑥𝑛=𝐴𝐼𝑇𝑛𝐴𝑥𝑛,𝑥𝑛𝑝+𝐴𝑥𝑛𝑇𝑛𝐴𝑥𝑛,𝐴𝑝𝑇𝑛𝐴𝑥𝑛=𝐴𝐼𝑇𝑛𝐴𝑥𝑛,𝑥𝑛1𝑝2𝐴𝑥𝑛𝐴𝑝2+12𝐴𝑥𝑛𝑇𝑛𝐴𝑥𝑛2+12𝐴𝑝𝑇𝑛𝐴𝑥𝑛2𝐴𝐼𝑇𝑛𝐴𝑥𝑛,𝑥𝑛+1𝑝2𝐴𝑥𝑛𝑇𝑛𝐴𝑥𝑛2.(3.33) Hence 12𝐴𝑥𝑛𝑇𝑛𝐴𝑥𝑛2𝐴𝐼𝑇𝑛𝐴𝑥𝑛,𝑥𝑛𝑝0.(3.34) Then we get 𝐴𝑥𝑛𝐹({𝑇𝑛}). Since {𝑇𝑛} satisfies the condition (𝑍) and 𝐴𝑥𝑛𝐴𝑧, we have 𝐴𝑧𝐹({𝑇𝑛}). From Lemma 3.1, we have 𝑧Fix({𝑉𝑛}).

Let 𝑇𝐻𝐻 be a nonexpansive mapping with a fixed point, and define 𝑇𝑛=𝑇 for all 𝑛. Then {𝑇𝑛} satisfies the conditions (𝑅) and (𝑍). Thus, one obtains the algorithm for solving the two-set SCFPP (1.4).

Corollary 3.4. Let 𝑈 and 𝑇 be nonexpansive operators on Hilbert space 𝐻1 and 𝐻2, respectively. Let 𝑓𝐻1𝐻1 be a contraction with coefficient 𝜌[0,1). Suppose that the solution set Ω of SCFPP (1.4) is nonempty. Take an initial guess 𝑥1𝐻1 and define a sequence {𝑥𝑛} by the following algorithm in (3.29), where 𝛾(0,1/𝐴2), and {𝛼𝑛}, {𝛽𝑛}, {𝛾𝑛}, {𝜆𝑛} are sequences in [0,1]. If the following conditions are satisfied:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛1;(ii)lim𝑛𝛼𝑛=0 and Σ𝑛=1𝛼𝑛=;(iii)0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1;(iv)0<liminf𝑛𝜆𝑛limsup𝑛𝜆𝑛<1;(v)lim𝑛|𝜆𝑛+1𝜆𝑛|=0. Then {𝑥𝑛} converges strongly to 𝑤Ω where 𝑤=𝑃Ω𝑓(𝑤).

Remark 3.5. By adding more operators to the families {𝑈𝑛} and {𝑇𝑛} by setting 𝑈𝑖=𝐼 for 𝑖𝑝+1 and 𝑇𝑗=𝐼 for 𝑗𝑟+1, the SCFPP (1.3) can be viewed as a special case of the GSCFPP (1.7).

Acknowledgment

This research is supported by the science research foundation program in Civil Aviation University of China (07kys09), the Fundamental Research Funds for the Central Universities (Program No. ZXH2011D005), and the NSFC Tianyuan Youth Foundation of Mathematics of China (No. 11126136).

References

  1. Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2-4, pp. 221–239, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441–453, 2002. View at Publisher · View at Google Scholar
  4. C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. F. Wang and H. K. Xu, “Approximating curve and strong convergence of the CQ algorithm for the split feasibility problem,” Journal of Inequalities and Applications, Article ID 102085, 13 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. F. Wang and H. K. Xu, “Strongly convergent iterative algorithms for solving a class of variational inequalities,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 407–421, 2010. View at Zentralblatt MATH
  7. H. K. Xu, “A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006. View at Publisher · View at Google Scholar
  8. H. K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, p. 17, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. Y. Yao, W. Jigang, and Y.-C. Liou, “Regularized methods for the split feasibility problem,” Abstract and Applied Analysis, vol. 2012, Article ID 140679, 13 pages, 2012. View at Publisher · View at Google Scholar
  10. A. Moudafi, “A note on the split common fixed-point problem for quasi-nonexpansive operators,” Nonlinear Analysis, vol. 74, no. 12, pp. 4083–4087, 2011. View at Publisher · View at Google Scholar
  11. Y. Censor and A. Segal, “The split common fixed point problem for directed operators,” Journal of Convex Analysis, vol. 16, no. 2, pp. 587–600, 2009. View at Zentralblatt MATH
  12. F. Wang and H. K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,” Nonlinear Analysis, vol. 74, no. 12, pp. 4105–4111, 2011. View at Publisher · View at Google Scholar
  13. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters, vol. 6, no. 4, pp. 621–628, 2012. View at Publisher · View at Google Scholar
  14. Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1506–1515, 2012. View at Publisher · View at Google Scholar
  15. Y. Yao, Y. Je Cho, and Y.-C. Liou, “Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequalities,” Fixed Point Theory and Applications, vol. 2011, article 101, 2011. View at Publisher · View at Google Scholar
  16. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “On a strongly nonexpansive sequence in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 3, pp. 471–489, 2007. View at Zentralblatt MATH
  17. K. Aoyama and Y. Kimura, “Strong convergence theorems for strongly nonexpansive sequences,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7537–7545, 2011. View at Publisher · View at Google Scholar
  18. K. Aoyama, “An iterative method for fixed point problems for sequences of nonexpansive mappings,” in Fixed Point Theory and Applications, pp. 1–7, Yokohama Publication, Yokohama, Japan, 2010.
  19. T. Suzuki, “Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH