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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 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: RudongΒ Chen

Copyright Β© 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 𝐴∢𝐻1→𝐻2 be a bounded linear operator. Given intergers 𝑝,π‘Ÿβ‰₯1, let us recall that the multiple-set split feasibility problem (MSSFP) was recently introduced [1] and is to find a point:π‘₯βˆ—βˆˆπ‘ξ™π‘–=1𝐢𝑖,𝐴π‘₯βˆ—βˆˆπ‘Ÿξ™π‘—=1𝑄𝑗,(1.1) where {𝐢𝑖}𝑝𝑖=1 and {𝑄𝑗}π‘Ÿπ‘—=1 are nonempty closed convex subsets of 𝐻1 and 𝐻2, respectively. If 𝑝=π‘Ÿ=1, the MSSFP (1.1) becomes the so-called split feasibility problem (SFP) [2] which is to find a point:π‘₯βˆ—βˆˆπΆ,𝐴π‘₯βˆ—βˆˆπ‘„,(1.2) where 𝐢 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, [3–10].

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:π‘₯βˆ—βˆˆπ‘ξ™π‘–=1ξ€·π‘ˆFix𝑖,𝐴π‘₯βˆ—βˆˆπ‘Ÿξ™π‘—=1𝑇Fix𝑗,(1.3) where π‘ˆπ‘–βˆΆπ»1→𝐻1(𝑖=1,2,…,𝑝) and π‘‡π‘—βˆΆπ»2→𝐻2(𝑗=1,2,…,π‘Ÿ) are nonlinear operators. If 𝑝=π‘Ÿ=1, the problem (1.3) reduces to the so-called two-set SCFPP, which is to find a point:π‘₯βˆ—βˆˆFix(π‘ˆ),𝐴π‘₯βˆ—βˆˆFix(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-β„‘ operators:π‘₯𝑛+1ξ€·π‘₯=π‘ˆπ‘›βˆ’π›Ύπ΄βˆ—(πΌβˆ’π‘‡)𝐴π‘₯𝑛,𝑛β‰₯0,(1.5) where π‘₯0∈𝐻1, 0<𝛾<2/‖𝐴‖2 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-β„‘ operators and introduced the following iterative algorithm for solving the SCFPP (1.3):π‘₯𝑛+1=π‘ˆ[𝑛]ξ€·π‘₯π‘›βˆ’π›Ύπ΄βˆ—ξ€·πΌβˆ’π‘‡[𝑛]𝐴π‘₯𝑛,𝑛β‰₯0.(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-β„‘ 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 [13–15].

Motivated and inspired by the aforementioned research works, we introduce a generalized split common fixed point problem (GSCFPP) which is to find a point:π‘₯βˆ—βˆˆβˆžξ™π‘–=1ξ€·π‘ˆFix𝑖,𝐴π‘₯βˆ—βˆˆβˆžξ™π‘—=1𝑇Fix𝑗.(1.7) Then, we show that the GSCFPP (1.7) for nonexpansive operators is equivalent to the following common fixed point problem:π‘₯βˆ—βˆˆβˆžξ™π‘–=1ξ€·π‘ˆFix𝑖,π‘₯βˆ—βˆˆβˆžξ™π‘—=1𝑉Fix𝑗,(1.8) where 𝑉𝑗=πΌβˆ’π›Ύπ΄βˆ—(πΌβˆ’π‘‡π‘—)𝐴(0<𝛾≀1/‖𝐴‖2) for every π‘—βˆˆβ„•. 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 π‘₯𝑛⇀π‘₯ and π‘₯𝑛→π‘₯ to indicate that {π‘₯𝑛} converges weakly to π‘₯ and converges strongly to π‘₯, respectively.

An operator π‘‡βˆΆπ»β†’π» is said to be nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπ». The set of fixed points of 𝑇 is denoted by 𝐹(𝑇). It is known that 𝐹(𝑇) is closed and convex. An operator π‘“βˆΆπ»β†’π» is called contraction if there exists a constant 𝜌∈[0,1) such that ‖𝑓(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœŒβ€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπ». Let 𝐢 be a nonempty closed convex subset of 𝐻. For each π‘₯∈𝐻, there exists a unique nearest point in 𝐢, denoted by 𝑃𝐢π‘₯, such that β€–π‘₯βˆ’π‘ƒπΆπ‘₯‖≀‖π‘₯βˆ’π‘¦β€– for every π‘¦βˆˆπΆ. 𝑃𝐢 is called a metric projection of 𝐻 onto 𝐢. It is known that for each π‘₯∈𝐻,⟨π‘₯βˆ’π‘ƒπΆπ‘₯,π‘¦βˆ’π‘ƒπΆπ‘₯βŸ©β‰€0(2.1) for all π‘¦βˆˆπΆ.

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⋂})=βˆžπ‘›=1𝐹(𝑇𝑛). A sequence {𝑇𝑛} is said to be strongly nonexpansive if each {𝑇𝑛} is nonexpansive andπ‘₯π‘›βˆ’π‘¦π‘›βˆ’ξ€·π‘‡π‘›π‘₯π‘›βˆ’π‘‡π‘›π‘¦π‘›ξ€ΈβŸΆ0(2.2) whenever {π‘₯𝑛} and {𝑦𝑛} are sequences in 𝐢 such that {π‘₯π‘›βˆ’π‘¦π‘›} is bounded and β€–π‘₯π‘›βˆ’π‘¦π‘›β€–βˆ’β€–π‘‡π‘›π‘₯π‘›βˆ’π‘‡π‘›π‘¦π‘›β€–β†’0; see [16, 17]. A sequence {𝑧𝑛} in 𝐻 is said to be an approximate fixed point sequence of {𝑇𝑛} if π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β†’0. 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 {π‘₯𝑛} is a common fixed point whenever {π‘₯𝑛}∈𝐹({𝑇𝑛}). A sequence {𝑇𝑛} of nonexpansive mappings of 𝐻 into itself is said to satisfy the condition (𝑅) iflimπ‘›β†’βˆžsupπ‘¦βˆˆπ·β€–β€–π‘‡π‘›+1π‘¦βˆ’π‘‡π‘›π‘¦β€–β€–=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 𝐢 be a nonempty subset of a Hilbert space 𝐻. Let {𝑇𝑛} be a sequence of nonexpansive mappings of 𝐢 into 𝐻. Let {πœ†π‘›} be a sequence in [0,1] such that liminfπ‘›β†’βˆžπœ†π‘›>0. Let {π‘ˆπ‘›} be a sequence of mappings of 𝐢 into 𝐻 defined by π‘ˆπ‘›=πœ†π‘›πΌ+(1βˆ’πœ†π‘›)𝑇𝑛 for π‘›βˆˆβ„•, where 𝐼 is the identity mapping on 𝐢. Then {π‘ˆπ‘›} is a strongly nonexpansive sequence.

Lemma 2.2 (see [16]). Let 𝐻 be a Hilbert space, 𝐢 a nonempty subset of 𝐻, and {𝑆𝑛} and {𝑇𝑛} sequences of nonexpansive self-mappings of 𝐢. 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 𝐢 a nonempty subset of 𝐻. Both {𝑆𝑛} and {𝑇𝑛} satisfy the condition (𝑅) and {π‘‡π‘›π‘¦βˆΆπ‘›βˆˆβ„•,π‘¦βˆˆπ·} is bounded for any bounded subset 𝐷 of 𝐢. Then {𝑆𝑛𝑇𝑛} satisfies the condition (𝑅).

Lemma 2.4 (see [19]). Let {π‘₯𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1. Suppose π‘₯𝑛+1=(1βˆ’π›½π‘›)𝑦𝑛+𝛽𝑛π‘₯𝑛for all integers 𝑛β‰₯0 and limsupπ‘›β†’βˆžξ€·β€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(2.4) Then limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.5 (see [20]). Assume that {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+𝛿𝑛,𝑛β‰₯0,(2.5) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that(i)βˆ‘βˆžπ‘›=1𝛾𝑛=∞,(ii)limsupπ‘›β†’βˆžπ›Ώπ‘›/𝛾𝑛≀0 or βˆ‘βˆžπ‘›=1|𝛿𝑛|<∞. Then limπ‘›β†’βˆžπ‘Žπ‘›=0.

3. Main Results

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

Lemma 3.1. Let 𝐴∢𝐻1→𝐻2 be a given bounded linear operator and let π‘‡π‘›βˆΆπ»2→𝐻2 be a sequence of nonexpansive operators. Assume π΄βˆ’1𝑇Fix𝑛=ξ€½ξ€Ύξ€Έξ€Έπ‘₯∈𝐻1π‘‡βˆΆπ΄π‘₯∈Fixξ€·ξ€½π‘›ξ€Ύξ€Έξ€Ύβ‰ βˆ….(3.1) For each constant 𝛾>0, 𝑉𝑛 is defined by the following: 𝑉𝑛=πΌβˆ’π›Ύπ΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄.(3.2) Then Fix({𝑉𝑛})=π΄βˆ’1(Fix({𝑇𝑛})). Moreover, for 0<𝛾≀1/‖𝐴‖2, 𝑉𝑛 is nonexpansive on 𝐻1 for π‘›βˆˆβ„•.

Proof. Since the inclusion π΄βˆ’1(Fix({𝑇𝑛}))βŠ†Fix({𝑉𝑛}) is evident, now we only need to show the converse inclusion. If π‘§βˆˆFix({𝑉𝑛}), then we have π΄βˆ—(πΌβˆ’π‘‡π‘›)𝐴𝑧=0. Since π΄βˆ’1(Fix({𝑇𝑛}))β‰ βˆ…, we take an arbitrary π‘βˆˆπ΄βˆ’1(Fix({𝑇𝑛})). Hence β€–π΄π‘§βˆ’π‘‡π‘›π΄π‘§β€–2=βŸ¨π΄π‘§βˆ’π‘‡π‘›π΄π‘§,π΄π‘§βˆ’π‘‡π‘›π΄π‘§βŸ©=βŸ¨π΄π‘§βˆ’π‘‡π‘›π΄π‘§,π΄π‘§βˆ’π΄π‘+π΄π‘βˆ’π‘‡π‘›=ξ«π΄π΄π‘§βŸ©βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έξ¬π΄π‘§,π‘§βˆ’π‘+βŸ¨π΄π‘§βˆ’π‘‡π‘›π΄π‘§,π΄π‘βˆ’π‘‡π‘›1π΄π‘§βŸ©=βˆ’2β€–π΄π‘§βˆ’π΄π‘β€–2+12β€–β€–π΄π‘§βˆ’π‘‡π‘›β€–β€–π΄π‘§2+12β€–β€–π΄π‘βˆ’π‘‡π‘›β€–β€–π΄π‘§2≀12β€–β€–π΄π‘§βˆ’π‘‡π‘›β€–β€–π΄π‘§2.(3.3) It follows that (1/2)β€–π΄π‘§βˆ’π‘‡π‘›π΄π‘§β€–2≀0, then 𝐴𝑧=𝑇𝑛𝐴𝑧 for every π‘›βˆˆβ„•, hence π‘§βˆˆπ΄βˆ’1(Fix({𝑇𝑛})). Next we turn to show that 𝑉𝑛 is a nonexpansive operator for π‘›βˆˆβ„•. Since 𝑇𝑛 is nonexpansive, we have β€–β€–(πΌβˆ’π‘‡π‘›)𝐴π‘₯βˆ’(πΌβˆ’π‘‡π‘›β€–β€–)𝐴𝑦2=‖𝐴π‘₯βˆ’π΄π‘¦β€–2+‖‖𝑇𝑛𝐴π‘₯βˆ’π‘‡π‘›β€–β€–π΄π‘¦2βˆ’2⟨𝐴π‘₯βˆ’π΄π‘¦,𝑇𝑛𝐴π‘₯βˆ’π‘‡π‘›π΄π‘¦βŸ©β‰€2‖𝐴π‘₯βˆ’π΄π‘¦β€–2βˆ’2⟨𝐴π‘₯βˆ’π΄π‘¦,𝑇𝑛𝐴π‘₯βˆ’π‘‡π‘›ξ«ξ€·π‘‡π΄π‘¦βŸ©β‰€2𝐴π‘₯βˆ’π΄π‘¦,𝐴π‘₯βˆ’π΄π‘¦βˆ’π‘›π΄π‘₯βˆ’π‘‡π‘›.𝐴𝑦(3.4) Hence ‖‖𝑉𝑛π‘₯βˆ’π‘‰π‘›π‘¦β€–β€–2=β€–β€–ξ€·πΌβˆ’π›Ύπ΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄ξ€Έπ‘₯βˆ’(πΌβˆ’π›Ύπ΄βˆ—(πΌβˆ’π‘‡π‘›β€–β€–)𝐴)𝑦2=β€–π‘₯βˆ’π‘¦β€–2+𝛾2‖𝐴‖2β€–β€–ξ€·πΌβˆ’π‘‡π‘›ξ€Έξ€·π΄π‘₯βˆ’πΌβˆ’π‘‡π‘›ξ€Έβ€–β€–π΄π‘¦2ξ«ξ€·βˆ’2𝛾𝐴π‘₯βˆ’π΄π‘¦,πΌβˆ’π‘‡π‘›ξ€Έξ€·π΄π‘₯βˆ’πΌβˆ’π‘‡π‘›ξ€Έξ¬π΄π‘¦β‰€β€–π‘₯βˆ’π‘¦β€–2ξ€·+𝛾𝛾‖𝐴‖2ξ€Έβ€–β€–ξ€·βˆ’1πΌβˆ’π‘‡π‘›ξ€Έξ€·π΄π‘₯βˆ’πΌβˆ’π‘‡π‘›ξ€Έβ€–β€–π΄π‘¦2.(3.5) For 0<𝛾≀1/‖𝐴‖2, we can immediately obtain that 𝑉𝑛 is a nonexpansive operator for every π‘›βˆˆβ„•.

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 {π‘ˆπ‘›} and {𝑉𝑛} be sequences of nonexpansive operators on Hilbert space 𝐻1. Both {π‘ˆπ‘›} and {𝑉𝑛} satisfy the conditions (𝑅) and (𝑍). Let π‘“βˆΆπ»1→𝐻1 be a contraction with coefficient 𝜌∈[0,1). Suppose Ξ©=Fix(π‘ˆπ‘›)β‹‚Fix(𝑉𝑛)β‰ βˆ…. Take an initial guess π‘₯1∈𝐻1 and define a sequence {π‘₯𝑛} by the following algorithm: 𝑦𝑛=πœ†π‘›π‘₯𝑛+ξ€·1βˆ’πœ†π‘›ξ€Έπ‘‰π‘›π‘₯𝑛,π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘ˆπ‘›π‘¦π‘›,(3.6) where {𝛼𝑛}, {𝛽𝑛}, {𝛾𝑛}, and {πœ†π‘›} are sequences in [0,1]. If the following conditions are satisfied:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛β‰₯1;(ii)limπ‘›β†’βˆžπ›Όπ‘›=0  and   Ξ£βˆžπ‘›=1𝛼𝑛=∞;(iii)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1;(iv)0<liminfπ‘›β†’βˆžπœ†π‘›β‰€limsupπ‘›β†’βˆžπœ†π‘›<1;(v)limπ‘›β†’βˆž|πœ†π‘›+1βˆ’πœ†π‘›|=0, then {π‘₯𝑛} converges strongly to π‘€βˆˆΞ© where 𝑀=𝑃Ω𝑓(𝑀).

Proof. We proceed with the following steps.
Step 1. First show that there exists π‘€βˆˆΞ© such that 𝑀=𝑃Ω𝑓(𝑀).
In fact, since 𝑓 is a contraction with coefficient 𝜌, we have ‖‖𝑃Ω𝑓(π‘₯)βˆ’π‘ƒΞ©β€–β€–β‰€π‘“(𝑦)‖𝑓(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœŒβ€–π‘₯βˆ’π‘¦β€–(3.7) for every π‘₯, 𝑦. Hence 𝑃Ω𝑓 is also a contraction. Therefore, there exists a unique π‘€βˆˆΞ© such that 𝑀=𝑃Ω𝑓(𝑀).
Step 2. Now we show that {π‘₯𝑛} is bounded.
Let π‘βˆˆΞ©, then π‘βˆˆFix({π‘ˆπ‘›}) and π‘βˆˆFix({𝑉𝑛}). Hence β€–π‘ˆπ‘›π‘¦π‘›βˆ’π‘β€–β‰€β€–π‘¦π‘›βˆ’π‘β€–β‰€πœ†π‘›β€–π‘₯π‘›ξ€·βˆ’π‘β€–+1βˆ’πœ†π‘›ξ€Έβ€–π‘‰π‘›π‘₯π‘›βˆ’π‘β€–β‰€β€–π‘₯π‘›βˆ’π‘β€–.(3.8) Then β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘+𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘+π›Ύπ‘›β€–β€–π‘ˆπ‘›π‘¦π‘›β€–β€–βˆ’π‘β‰€π›Όπ‘›πœŒβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘+𝛼𝑛‖𝑓(𝑝)βˆ’π‘β€–+𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘+𝛾𝑛‖‖π‘₯π‘›β€–β€–β‰€ξ€·βˆ’π‘1βˆ’π›Όπ‘›(ξ€Έβ€–β€–π‘₯1βˆ’πœŒ)π‘›β€–β€–βˆ’π‘+𝛼𝑛(11βˆ’πœŒ)(ξ‚»β€–β€–π‘₯1βˆ’πœŒβ€–π‘“π‘)βˆ’π‘β€–β‰€max𝑛‖‖,1βˆ’π‘ξ‚Ό.1βˆ’πœŒβ€–π‘“(𝑝)βˆ’π‘β€–(3.9) By induction on 𝑛, ‖‖𝑉𝑛π‘₯𝑛‖‖≀‖‖π‘₯βˆ’π‘π‘›β€–β€–ξ‚»β€–β€–π‘₯βˆ’π‘β‰€max1β€–β€–,1βˆ’π‘β€–ξ‚Ό1βˆ’πœŒβ€–π‘“(𝑝)βˆ’π‘(3.10) for every π‘›βˆˆβ„•. This shows that {π‘₯𝑛} and {𝑉𝑛π‘₯𝑛} are bounded, and hence, {π‘ˆπ‘›π‘¦π‘›}, {𝑦𝑛}, and {𝑓(π‘₯𝑛)} are also bounded.
Step 3. We claim that 𝐹({𝐴𝑛})=𝐹({𝑉𝑛}) and 𝐹({π‘ˆπ‘›π΄π‘›ξ‚})=𝐹({π‘ˆπ‘›ξ‚})∩𝐹({𝑉𝑛}), where 𝐴𝑛=πœ†π‘›πΌ+(1βˆ’πœ†π‘›)𝑉𝑛.
We first show the former equality. Let {𝑧𝑛} be a bounded sequence in 𝐻1. If {𝑧𝑛}∈𝐹({𝑉𝑛}), then β€–β€–π΄π‘›π‘§π‘›βˆ’π‘§π‘›β€–β€–=β€–β€–πœ†π‘›π‘§π‘›+ξ€·1βˆ’πœ†π‘›ξ€Έπ‘‰π‘›π‘§π‘›βˆ’π‘§π‘›β€–β€–=ξ€·1βˆ’πœ†π‘›ξ€Έβ€–β€–π‘‰π‘›π‘§π‘›βˆ’π‘§π‘›β€–β€–βŸΆ0.(3.11) Hence {𝑧𝑛}∈𝐹({𝐴𝑛}). On the other hand, if {𝑧𝑛}∈𝐹({𝐴𝑛}), combining (3.11) and limsupπ‘›β†’βˆžπœ†π‘›<1, we obtain that β€–π‘‰π‘›π‘§π‘›βˆ’π‘§π‘›β€–β†’0. 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({π‘ˆπ‘›ξ‚})∩𝐹({𝐴𝑛})=𝐹({π‘ˆπ‘›ξ‚})∩𝐹({𝑉𝑛})β‰ βˆ…, from Lemma 2.2 we have ξ‚πΉπ‘ˆξ€·ξ€½π‘›π΄π‘›=ξ‚πΉπ‘ˆξ€Ύξ€Έξ€·ξ€½π‘›βˆ©ξ‚πΉπ΄ξ€Ύξ€Έξ€·ξ€½π‘›=ξ‚πΉπ‘ˆξ€Ύξ€Έξ€·ξ€½π‘›βˆ©ξ‚πΉπ‘‰ξ€Ύξ€Έξ€·ξ€½π‘›ξ€Ύξ€Έ.(3.12)
Step 4. {𝑆𝑛} satisfies the condition (𝑅), where 𝑆𝑛=π‘ˆπ‘›π΄π‘›.
Let 𝐷 be a nonempty bounded subset of 𝐻1. From the definition of {𝐴𝑛}, we have, for all π‘¦βˆˆπ·, ‖‖𝐴𝑛+1π‘¦βˆ’π΄π‘›π‘¦β€–β€–=β€–β€–πœ†π‘›+1𝑦+1βˆ’πœ†π‘›+1𝑉𝑛+1π‘¦βˆ’πœ†π‘›ξ€·π‘¦βˆ’1βˆ’πœ†π‘›ξ€Έπ‘‰π‘›π‘¦β€–β€–β‰€||πœ†π‘›+1βˆ’πœ†π‘›||‖‖𝑉‖𝑦‖+𝑛+1π‘¦βˆ’π‘‰π‘›π‘¦β€–β€–+β€–β€–πœ†π‘›+1𝑉𝑛+1π‘¦βˆ’πœ†π‘›π‘‰π‘›π‘¦β€–β€–β‰€||πœ†π‘›+1βˆ’πœ†π‘›||‖‖‖𝑉𝑦‖+𝑛+1π‘¦βˆ’π‘‰π‘›π‘¦β€–β€–+β€–β€–πœ†π‘›+1𝑉𝑛+1π‘¦βˆ’πœ†π‘›π‘‰π‘›+1𝑦‖‖+β€–β€–πœ†π‘›π‘‰π‘›+1π‘¦βˆ’πœ†π‘›π‘‰π‘›π‘¦β€–β€–=||πœ†π‘›+1βˆ’πœ†π‘›||‖‖𝑉‖𝑦‖+𝑛+1π‘¦βˆ’π‘‰π‘›π‘¦β€–β€–+||πœ†π‘›+1βˆ’πœ†π‘›||‖‖𝑉𝑛+1𝑦‖‖+πœ†π‘›β€–β€–π‘‰π‘›+1π‘¦βˆ’π‘‰π‘›π‘¦β€–β€–=||πœ†π‘›+1βˆ’πœ†π‘›||‖‖‖𝑉𝑦‖+𝑛+1𝑦‖‖+ξ€·1+πœ†π‘›ξ€Έβ€–β€–π‘‰π‘›+1π‘¦βˆ’π‘‰π‘›π‘¦β€–β€–.(3.13) It follows that supπ‘¦βˆˆπ·β€–β€–π΄π‘›+1π‘¦βˆ’π΄π‘›π‘¦β€–β€–β‰€||πœ†π‘›+1βˆ’πœ†π‘›||supπ‘¦βˆˆπ·ξ€·β€–β€–π‘‰β€–π‘¦β€–+𝑛+1𝑦‖‖+ξ€·1+πœ†π‘›ξ€Έsupπ‘¦βˆˆπ·β€–β€–π‘‰π‘›+1π‘¦βˆ’π‘‰π‘›π‘¦β€–β€–.(3.14) Since {𝑉𝑛} satisfies the condition (𝑅) and limπ‘›β†’βˆž|πœ†π‘›+1βˆ’πœ†π‘›|=0, we have limπ‘›β†’βˆžsupπ‘¦βˆˆπ·β€–β€–π΄π‘›+1π‘¦βˆ’π΄π‘›π‘¦β€–β€–=0,(3.15) that is, {𝐴𝑛} satisfies the condition (𝑅). Since {π΄π‘›π‘¦βˆΆπ‘›βˆˆβ„•,π‘¦βˆˆπ·} 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 β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖→0.
We can write (3.6) as π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+(1βˆ’π›½π‘›)𝑧𝑛 where 𝑧𝑛=(𝛼𝑛𝑓(π‘₯𝑛)+𝛾𝑛𝑆𝑛π‘₯𝑛)/1βˆ’π›½π‘›. It follows that 𝑧𝑛+1βˆ’π‘§π‘›=𝛼𝑛+1𝑓π‘₯𝑛+1ξ€Έ+𝛾𝑛+1𝑆𝑛+1π‘₯𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›π‘“ξ€·π‘₯𝑛+𝛾𝑛𝑆𝑛π‘₯𝑛1βˆ’π›½π‘›=𝛼𝑛+11βˆ’π›½π‘›+1𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›+𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+𝛾𝑛+11βˆ’π›½π‘›+1𝑆𝑛+1π‘₯𝑛+1βˆ’π‘†π‘›π‘₯𝑛+𝛾𝑛+11βˆ’π›½π‘›+1βˆ’π›Ύπ‘›1βˆ’π›½π‘›ξ‚Άπ‘†π‘›π‘₯𝑛.(3.16) From Step 2, we may assume that {π‘₯𝑛}βŠ‚π·ξ…ž, where 𝐷′ is a bounded set of 𝐻1. Then from (3.16), we obtain ‖𝑧𝑛+1βˆ’π‘§π‘›||||𝛼‖≀𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛‖‖+‖‖𝑆𝑛π‘₯𝑛‖‖+𝛼𝑛+11βˆ’π›½π‘›+1πœŒβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝛾𝑛+11βˆ’π›½π‘›+1‖‖𝑆𝑛+1π‘₯𝑛+1βˆ’π‘†π‘›π‘₯𝑛+1β€–β€–+𝛾𝑛+11βˆ’π›½π‘›+1‖‖𝑆𝑛π‘₯𝑛+1βˆ’π‘†π‘›π‘₯𝑛‖‖≀||||𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛‖‖+‖‖𝑆𝑛π‘₯𝑛‖‖+𝛼1βˆ’π‘›+11βˆ’π›½π‘›+1ξ‚Ήβ€–β€–π‘₯(1βˆ’πœŒ)𝑛+1βˆ’π‘₯𝑛‖‖+𝛾𝑛+11βˆ’π›½π‘›+1supπ‘¦βˆˆπ·β€²β€–β€–π‘†π‘›+1π‘¦βˆ’π‘†π‘›π‘¦β€–β€–.(3.17) It follows that ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀||||𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛‖‖+‖‖𝑆𝑛π‘₯𝑛‖‖+𝛾𝑛+11βˆ’π›½π‘›+1supπ‘¦βˆˆπ·ξ…žβ€–β€–π‘†π‘›+1π‘¦βˆ’π‘†π‘›π‘¦β€–β€–βˆ’π›Όπ‘›+11βˆ’π›½π‘›+1β€–β€–π‘₯(1βˆ’πœŒ)𝑛+1βˆ’π‘₯𝑛‖‖.(3.18) Since {𝑆𝑛} satisfies the condition (𝑅), combining 𝛼𝑛→0 as π‘›β†’βˆž, we have limsupπ‘›β†’βˆžξ€·β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(3.19) Hence by Lemma 2.4, we get β€–π‘§π‘›βˆ’π‘₯𝑛‖→0 as π‘›β†’βˆž. Consequently, limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=limπ‘›β†’βˆžξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘§π‘›βˆ’π‘₯𝑛‖‖=0.(3.20)
Step 6. We claim that {π‘₯𝑛}∈𝐹({π‘ˆπ‘›ξ‚})∩𝐹({𝑉𝑛}).
From (3.6), we have ‖‖𝑆𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖≀‖‖𝑆𝑛π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=‖‖𝑆𝑛π‘₯π‘›βˆ’π›Όπ‘›π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π›½π‘›π‘₯π‘›βˆ’π›Ύπ‘›π‘†π‘›π‘₯𝑛‖‖+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀𝛼𝑛‖‖𝑆𝑛π‘₯𝑛π‘₯βˆ’π‘“π‘›ξ€Έβ€–β€–+𝛽𝑛‖‖𝑆𝑛π‘₯π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖,(3.21) and hence ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘†π‘›π‘₯π‘›βˆ’π‘₯𝑛‖‖≀𝛼𝑛‖‖𝑆𝑛π‘₯𝑛π‘₯βˆ’π‘“π‘›ξ€Έβ€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.22) Since β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖→0, 𝛼𝑛→0 and limsupπ‘›β†’βˆžπ›½π‘›<1, we derive ‖‖𝑆𝑛π‘₯π‘›βˆ’π‘₯π‘›β€–β€–βŸΆ0.(3.23) Thus (3.23) and Steps 2 and 3 imply that ξ€½π‘₯π‘›ξ€Ύβˆˆξ‚πΉπ‘†ξ€·ξ€½π‘›=ξ‚πΉπ‘ˆξ€Ύξ€Έξ€·ξ€½π‘›βˆ©ξ‚πΉπ‘‰ξ€Ύξ€Έξ€·ξ€½π‘›ξ€Ύξ€Έ.(3.24)
Step 7. Show limsupπ‘›β†’βˆžβŸ¨π‘“(𝑀)βˆ’π‘€,π‘₯π‘›βˆ’π‘€βŸ©β‰€0, where 𝑀=𝑃Ω𝑓(𝑀).
Since {π‘₯𝑛} is bounded, there exist a point π‘£βˆˆπ»1 and a subsequence {π‘₯𝑛𝑖} of {π‘₯𝑛} such that limsupπ‘›β†’βˆžβŸ¨π‘“(𝑀)βˆ’π‘€,π‘₯π‘›βˆ’π‘€βŸ©=limπ‘–β†’βˆžξ«π‘“(𝑀)βˆ’π‘€,π‘₯π‘›π‘–ξ¬βˆ’π‘€(3.25) and π‘₯𝑛𝑖⇀𝑣. Since {π‘ˆπ‘›} and {𝑉𝑛} satisfy the condition (𝑍), from Step 6, we have π‘£βˆˆπΉ({π‘ˆπ‘›})∩𝐹({𝑉𝑛}). Using (2.1), we get limsupπ‘›β†’βˆžβŸ¨π‘“(𝑀)βˆ’π‘€,π‘₯π‘›βˆ’π‘€βŸ©=limπ‘–β†’βˆžξ«π‘“(𝑀)βˆ’π‘€,π‘₯π‘›π‘–ξ¬βˆ’π‘€=βŸ¨π‘“(𝑀)βˆ’π‘€,π‘£βˆ’π‘€βŸ©β‰€0.(3.26)
Step 8. Show π‘₯𝑛→𝑀=𝑃Ω𝑓(𝑀).
Since π‘€βˆˆΞ©, using (3.8), we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘€2=𝛼𝑛𝑓π‘₯π‘›ξ€Έξ€Έβˆ’π‘€+𝛽𝑛π‘₯π‘›ξ€Έβˆ’π‘€+π›Ύπ‘›ξ€·π‘ˆπ‘›π‘¦π‘›ξ€Έβˆ’π‘€,π‘₯𝑛+1ξ¬βˆ’π‘€β‰€π›Όπ‘›ξ«π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘“(𝑀),π‘₯𝑛+1ξ¬βˆ’π‘€+𝛼𝑛𝑓(𝑀)βˆ’π‘€,π‘₯𝑛+1ξ¬βˆ’π‘€+𝛽𝑛‖‖π‘₯𝑛‖‖⋅‖‖π‘₯βˆ’π‘€π‘›+1β€–β€–βˆ’π‘€+𝛾𝑛‖‖𝑦𝑛‖‖⋅‖‖π‘₯βˆ’π‘€π‘›+1‖‖≀1βˆ’π‘€2π›Όπ‘›πœŒξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘€2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘€2+𝛼𝑛𝑓(𝑀)βˆ’π‘€,π‘₯𝑛+1+1βˆ’π‘€2𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘€2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘€2+12𝛾𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘€2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘€2≀12ξ€Ί1βˆ’π›Όπ‘›ξ€»β€–β€–π‘₯(1βˆ’πœŒ)π‘›β€–β€–βˆ’π‘€2+12β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘€2+𝛼𝑛𝑓(𝑀)βˆ’π‘€,π‘₯𝑛+1,βˆ’π‘€(3.27) which implies that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘€2≀1βˆ’π›Όπ‘›ξ€»β€–β€–π‘₯(1βˆ’πœŒ)π‘›β€–β€–βˆ’π‘€2+2𝛼𝑛1(1βˆ’πœŒ)1βˆ’πœŒπ‘“(𝑀)βˆ’π‘€,π‘₯𝑛+1ξ¬βˆ’π‘€,(3.28) for every π‘›βˆˆβ„•. Consequently, according to Step 7, 𝜌∈[0,1), and Lemma 2.5, we deduce that {π‘₯𝑛} converges strongly to 𝑀=𝑃Ω(𝑀). 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 {π‘ˆπ‘›} and {𝑇𝑛} be sequences of nonexpansive operators on Hilbert space 𝐻1 and 𝐻2, respectively. Both {π‘ˆπ‘›} and {𝑇𝑛} satisfy the conditions (𝑅) and (𝑍). Let π‘“βˆΆπ»1→𝐻1 be a contraction with coefficient 𝜌∈[0,1). Suppose that the solution set Ξ© of GSCFPP (1.7) is nonempty. Take an initial guess π‘₯1∈𝐻1 and define a sequence {π‘₯𝑛} by the following algorithm: 𝑦𝑛=π‘₯π‘›ξ€·βˆ’π›Ύ1βˆ’πœ†π‘›ξ€Έπ΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄π‘₯𝑛,π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘ˆπ‘›π‘¦π‘›,(3.29) where π›Ύβˆˆ(0,1/‖𝐴‖2), and {𝛼𝑛}, {𝛽𝑛}, {𝛾𝑛}, {πœ†π‘›} are sequences in [0,1]. If the following conditions are satisfied:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛β‰₯1;(ii)limπ‘›β†’βˆžπ›Όπ‘›=0  andβ€‰β€‰Ξ£βˆžπ‘›=1𝛼𝑛=∞;(iii)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1;(iv)0<liminfπ‘›β†’βˆžπœ†π‘›β‰€limsupπ‘›β†’βˆžπœ†π‘›<1;(v)limπ‘›β†’βˆž|πœ†π‘›+1βˆ’πœ†π‘›|=0, then {π‘₯𝑛} converges strongly to π‘€βˆˆΞ© where 𝑀=𝑃Ω𝑓(𝑀).

Proof. Set 𝑉𝑛=πΌβˆ’π›Ύπ΄βˆ—(πΌβˆ’π‘‡π‘›)𝐴. By Lemma 3.1, 𝑉𝑛 is a nonexpansive operator for every π‘›βˆˆβ„•. We can rewrite (3.29) as 𝑦𝑛=πœ†π‘›π‘₯𝑛+ξ€·1βˆ’πœ†π‘›ξ€Έπ‘‰π‘›π‘₯𝑛,π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘ˆπ‘›π‘¦π‘›.(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 π‘¦βˆˆπ·, we have β€–β€–ξ€·πΌβˆ’π›Ύπ΄βˆ—ξ€·πΌβˆ’π‘‡π‘›+1ξ€Έπ΄ξ€Έξ€·π‘¦βˆ’πΌβˆ’π›Ύπ΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄ξ€Έπ‘¦β€–β€–β€–β€–π΄β‰€π›Ύβˆ—ξ€·πΌβˆ’π‘‡π‘›+1ξ€Έπ΄π‘¦βˆ’π΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έβ€–β€–β€–β€–π‘‡π΄π‘¦β‰€π›Ύβ€–π΄β€–π‘›+1(𝐴𝑦)βˆ’π‘‡π‘›β€–β€–.(𝐴𝑦)(3.31) Since {𝑇𝑛} satisfies the condition (𝑅), and π·ξ…ž={π΄π‘¦βˆΆπ‘¦βˆˆπ·} is bounded, it follows from (3.31) that supπ‘¦βˆˆπ·β€–β€–ξ€·πΌβˆ’π›Ύπ΄βˆ—ξ€·πΌβˆ’π‘‡π‘›+1ξ€Έπ΄ξ€Έξ€·π‘¦βˆ’πΌβˆ’π›Ύπ΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄ξ€Έπ‘¦β€–β€–β‰€π›Ύβ€–π΄β€–supπ‘¦βˆˆπ·β€–β€–π‘‡π‘›+1(𝐴𝑦)βˆ’π‘‡π‘›β€–β€–(𝐴𝑦)=𝛾‖𝐴‖supπ‘§βˆˆπ·β€²β€–β€–π‘‡π‘›+1π‘§βˆ’π‘‡π‘›π‘§β€–β€–βŸΆ0.(3.32) Therefore, {𝑉𝑛} satisfies the condition (𝑅).
Assume that π‘₯𝑛⇀𝑧 and π‘₯π‘›βˆ’π‘‰π‘›π‘₯𝑛→0; we next show that 𝑉𝑛𝑧=𝑧. By using π‘₯π‘›βˆ’π‘‰π‘›π‘₯𝑛→0, we have π΄βˆ—(πΌβˆ’π‘‡π‘›)𝐴π‘₯𝑛→0. Since π΄βˆ’1(Fix({𝑇𝑛}))β‰ βˆ…, we choose an arbitrary point π‘βˆˆπ΄βˆ’1(Fix({𝑇𝑛})); then for every π‘›βˆˆβ„•, ‖𝐴π‘₯π‘›βˆ’π‘‡π‘›π΄π‘₯𝑛‖2=⟨𝐴π‘₯π‘›βˆ’π‘‡π‘›π΄π‘₯𝑛,𝐴π‘₯π‘›βˆ’π΄π‘+π΄π‘βˆ’π‘‡π‘›π΄π‘₯π‘›βŸ©=ξ«π΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄π‘₯𝑛,π‘₯π‘›ξ¬βˆ’π‘+⟨𝐴π‘₯π‘›βˆ’π‘‡π‘›π΄π‘₯𝑛,π΄π‘βˆ’π‘‡π‘›π΄π‘₯π‘›βŸ©=ξ«π΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄π‘₯𝑛,π‘₯π‘›ξ¬βˆ’1βˆ’π‘2‖‖𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘2+12‖‖𝐴π‘₯π‘›βˆ’π‘‡π‘›π΄π‘₯𝑛‖‖2+12β€–β€–π΄π‘βˆ’π‘‡π‘›π΄π‘₯𝑛‖‖2β‰€ξ«π΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄π‘₯𝑛,π‘₯𝑛+1βˆ’π‘2‖‖𝐴π‘₯π‘›βˆ’π‘‡π‘›π΄π‘₯𝑛‖‖2.(3.33) Hence 12‖‖𝐴π‘₯π‘›βˆ’π‘‡π‘›π΄π‘₯𝑛‖‖2β‰€ξ«π΄βˆ—ξ€·πΌβˆ’π‘‡π‘›ξ€Έπ΄π‘₯𝑛,π‘₯π‘›ξ¬βˆ’π‘βŸΆ0.(3.34) Then we get 𝐴π‘₯π‘›βˆˆξ‚πΉ({𝑇𝑛}). Since {𝑇𝑛} satisfies the condition (𝑍) and 𝐴π‘₯𝑛⇀𝐴𝑧, we have π΄π‘§βˆˆπΉ({𝑇𝑛}). From Lemma 3.1, we have π‘§βˆˆFix({𝑉𝑛}).

Let π‘‡βˆΆπ»β†’π» be a nonexpansive mapping with a fixed point, and define 𝑇𝑛=𝑇 for all π‘›βˆˆβ„•. 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 π‘ˆ and 𝑇 be nonexpansive operators on Hilbert space 𝐻1 and 𝐻2, respectively. Let π‘“βˆΆπ»1→𝐻1 be a contraction with coefficient 𝜌∈[0,1). Suppose that the solution set Ξ© of SCFPP (1.4) is nonempty. Take an initial guess π‘₯1∈𝐻1 and define a sequence {π‘₯𝑛} by the following algorithm in (3.29), where π›Ύβˆˆ(0,1/‖𝐴‖2), and {𝛼𝑛}, {𝛽𝑛}, {𝛾𝑛}, {πœ†π‘›} are sequences in [0,1]. If the following conditions are satisfied:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛β‰₯1;(ii)limπ‘›β†’βˆžπ›Όπ‘›=0 and Ξ£βˆžπ‘›=1𝛼𝑛=∞;(iii)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1;(iv)0<liminfπ‘›β†’βˆžπœ†π‘›β‰€limsupπ‘›β†’βˆžπœ†π‘›<1;(v)limπ‘›β†’βˆž|πœ†π‘›+1βˆ’πœ†π‘›|=0. Then {π‘₯𝑛} converges strongly to π‘€βˆˆΞ© where 𝑀=𝑃Ω𝑓(𝑀).

Remark 3.5. By adding more operators to the families {π‘ˆπ‘›} and {𝑇𝑛} by setting π‘ˆπ‘–=𝐼 for 𝑖β‰₯𝑝+1 and 𝑇𝑗=𝐼 for 𝑗β‰₯π‘Ÿ+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).

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