JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation38563810.1155/2012/385638385638Research ArticleThe Split Common Fixed Point Problem for Total Asymptotically Strictly Pseudocontractive MappingsChangS. S.1WangL.1TangY. K.1YangL.2YaoYonghong1Department of MathematicsCollege of Statistics and MathematicsYunnan University of Finance and EconomicsKunmingYunnan 650221Chinaynufe.edu.cn2Department of MathematicsSouth West University of Science and TechnologyMianyang, Sichuan 621010Chinaswust.edu.cn201215122011201204092011211020112012Copyright © 2012 S. S. Chang 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.

The purpose of this paper is to propose an algorithm for solving the split common fixed point problems for total asymptotically strictly pseudocontractive mappings in infinite-dimensional Hilbert spaces. The results presented in the paper improve and extend some recent results of Moudafi (2011 and 2010), Xu (2010 and 2006), Censor and Segal (2009), Censor et al. (2005), Masad and Reich (2007), Censor et al. (2007), Yang (2004), and others.

1. Introduction

Throughout this paper, we always assume that H1,  H2 are real Hilbert spaces, “,” denote by strong and weak convergence, respectively, and F(T) is the fixed point set of a mapping T.

The split common fixed point problem (SCFP) is a generalization of the split feasibility problem (SEP) and the convex feasibility problem (CFP). It is worth mentioning that SFP in finite-dimensional spaces was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and in medical image reconstruction . Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning .

SEP in an infinite-dimensional Hilbert space can be found in [2, 4, 68]. Moreover the convex feasibility formalism is at the core of the modeling of many inverse problems and has been used to model significant real-world problems.

The split common fixed point problems for a class of quasi-nonexpansive mappings and demicontractive mappings in the setting of Hilbert space were first introduced and studied by Moudafi [9, 10].

The purpose of this paper is to introduce and study the following split common fixed point problem for a more general class of total asymptotically strict pseudocontraction (SCFP) in the framework of an infinite-dimensional Hilbert spaces which contains the quasi-nonexpansive mappings and the demicontractive mappings as its special cases:find  x*C  such  that  Ax*Q, where A:H1H2 is a bounded linear operator, S:H1H1 and T:H2H2 are mappings C:=F(S), and Q:=F(T). In the sequel we use Γ to denote the set of solutions of (SCFP), that is,Γ={xC,AxQ}.

2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed in proving our main results.

Let E be a Banach space. A mapping T:EE is said to be demiclosed at origin, if for any sequence {xn}E with xnx* and (I-T)xn0, x*=Tx*.

A Banach space E is said to have the Opial property, if for any sequence {xn} with xnx*,liminfnxn-x*<liminfnxn-y,  yE  with  yx*.

Remark 2.1.

It is well known that each Hilbert space possesses the Opial property.

Definition 2.2.

Let H be a real Hilbert space, and let K be nonempty and closed convex subset of H.

A mapping G:KK is said to be (γ,{μn},{ξn},ϕ)-totally asymptotically strictly pseudocontractive, if there exist a constant γ[0,1) and sequences {μn}[0,) and {ξn}[0,) with μn0 and ξn0 such that for all x,yKGnx-Gny2x-y2+γ(I-Gn)x-(I-Gn)y2+μnϕ(x-y)+ξn,n1, where ϕ:[0,)[0,) is a continuous and strictly increasing function with ϕ(0)=0.

A mapping G:KK is said to be (γ,{kn})-asymptotically strictly pseudocontractive, if there exist a constant γ[0,1) and a sequence {kn}[1,) with kn1 such that

Gnx-Gny2knx-y2+γ(I-Gn)x-(I-Gn)y2,x,yK.

Especially, if there exists γ[0,1) such that Gx-Gy2x-y2+γ(I-G)x-(I-G)y2,x,yK, then G:KK is called a γ-strictly pseudocontractive mapping.

A mapping G:KK is said to be uniformly L-Lipschitzian, if there exists a constant L>0, such that Gnx-GnyLx-y,  x,yK,  n1.

A mapping G:KK is said to be semicompact, if for any bounded sequence {xn}K with limnxn-Gxn=0, there exists a subsequence {xni}{xn} such that {xni} converges strongly to some point x*K.

Remark 2.3.

If ϕ(λ)=λ2,  λ0, and ξn=0, then a (γ,{μn},{ξn},ϕ)-total asymptotically strictly pseudocontractive mapping is an (γ,{kn})-asymptotically strict pseudocontractive mapping, where {kn=1+μn}.

Proposition 2.4.

Let G:KK be a (γ,{μn},{ξn},ϕ)-total asymptotically strictly pseudocontractive mapping. If F(G), then for each qF(G) and for each xK, the following inequalities hold and they are equivalent: Gnx-q2x-q2+γx-Gnx2+μnϕ(x-q)+ξn;x-Gnx,x-q1-γ2x-Gnx2-μn2ϕ(x-q)-ξn2;x-Gnx,q-Gnxγ+12Gnx-x2+μn2ϕ(x-q)+ξn2.

Proof.

(I) Inequality (2.6) can be obtained from (2.2) immediately.

(II) (2.6) (2.7) In fact, since Gnx-q2=Gnx-x+x-q2=Gnx-x2+x-q2+2Gnx-x,x-q,xK,qF(G), from (2.6) we have that Gnx-x2+x-q2+2Gnx-x,x-qx-q2+γx-Gnx2+μnϕ(x-q)+ξn. Simplifying it, the inequality (2.7) is obtained.

Conversely, from (2.7) the inequality (2.6) can be obtained immediately.

(III) (2.7) (2.8) In fact, sincex-Gnx,x-q=x-Gnx,x-Gnx+Gnx-q=x-Gnx2+x-Gnx,Gnx-q, it follows from (2.7) that x-Gnx2+x-Gnx,Gnx-q1-γ2x-Gnx2-μn2ϕ(x-q)-ξn2. Simplifying it, the inequality (2.8) is obtained.

Conversely, the inequality (2.7) can be obtained from (2.8) immediately.

This completes the proof of Proposition 2.4.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let {an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying an+1(1+δn)an+bn,n1. If i=1δn< and i=1bn<, then the limit limnan exists.

Lemma 2.6 (see [<xref ref-type="bibr" rid="B12">12</xref>]).

Let H be a real Hilbert space. If {xn} is a sequence in H weakly convergent to z, then limsupnxn-y2=limsupnxn-z2+z-y2,yH.

Proposition 2.7.

Let H be a real Hilbert space and let T:HH be a uniformly L-Lipschitzian and γ,{μn},{ξn},ϕ-total asymptotically strictly pseudocontractive mapping. Then the demiclosedness principle holds for T in the sense that if {xn} is a sequence in H such that xnx*, and limsupmlimsupnxn-Tmxn=0, then (I-T)x*=0. In particular, if xnx*, and (I-T)xn0, then (I-T)x*=0, that is, T is demiclosed at origin.

Proof.

Since {xn} is bounded, we can define a function f on H by f(x)=limsupnxn-x2,xH. Since xnx*, it follows from Lemma 2.6 that f(x)=f(x*)+x-x*2,xH. In particular, for each m1, f(Tmx*)=f(x*)+Tmx*-x*2. On the other hand, since T is a (γ,{μn},{ξn},ϕ)-total asymptotically strictly pseudocontraction mapping, we get f(Tmx*)=limsupnxn-Tmx*2=limsupnxn-Tmxn+Tmxn-Tmx*2=limsupn(xn-Tmxn2+2xn-Tmxn,Tmxn-Tmx*+Tmxn-Tmx*2)limsupnxn-Tmxn(xn-Tmxn+2Lxn-x*)+limsupn(xn-x*2+γxn-Tmxn-(x*-Tmx*)2+μmϕ(xn-x*)+ξm). Taking limsupm on both sides and observing the facts that limmμm=0, limmξm=0, and limsupmlimsupnxn-Tmxn=0, we derive that limsupmf(Tmx*)limsupmxn-x*2+γlimsupmx*-Tmx*2=f(x*)+γlimsupmx*-Tmx*2. On the other hand, it follows from (2.17) that limsupmf(Tmx*)=f(x*)+limsupmx*-Tmx*2. Since κ<1, this together with (2.19) shows that limsupmx*-Tmx*2=0. That is, limmTmx*=x*; hence Tx*=x*.

3. Split Common Fixed Point Problem

For solving the split common fixed point problem (1.1), let us assume that the following conditions are satisfied.

(1) H1 and H2 are two real Hilbert spaces, and A:H1H2 is a bounded linear operator.

(2) S:H1H1 is a uniformly L-Lipschitzian and (β,{μn(1)},{ξn(1)},ϕ1)-total asymptotically strictly pseudocontractive mapping and T:H2H2 is a uniformly L̃-Lipschitzian and (κ,{μn(2)},{ξn(2)},ϕ2)-total asymptotically strictly pseudocontractive mapping satisfying the following conditions:

C:=F(S), Q:=F(T);

μn=max{μn(1),μn(2)},  ξn=max{ξn(1),ξn(2)},  n1, and n=1μn<, n=1ξn<;

ϕ=max{ϕ1,ϕ2} and there exist two positive constants M and M* such that ϕ(λ)M*λ2 for all λM.

We are now in a position to give the following result.

Theorem 3.1.

Let H1,  H2,  A,  S,  T,  C,  Q,  β,  κ,  L,  L̃,  {μn},  {ξn}, and ϕ be the same as mentiond before. Let {xn} be the sequence generated by x1H1  chosen  arbitrarily,xn+1=(1-αn)un+αnSn(un),un=xn+γA*(Tn-I)Axn,n1, where {αn} is a sequence in [0,1] and γ>0 is a constant satisfying the following conditions:

αn(δ,1-β),for all n1 and γ(0,(1-κ)/A2), where δ(0,1-β) is a positive constant.

If Γ (where Γ is the set of solutions to (SCFP)-(1.1)), then {xn} converges weakly to a point x*Γ.

In addition, if S is also semicompact, then {xn} and {un} both converge strongly to x*Γ.

Proof.

The following is the proof of Theorem 3.1.

The Proof of Conclusion (I)

(1) First we prove that for each pΓ, the following limits exist and limnxn-p=limnun-p.

In fact, since ϕ is a continuous and increasing function, it results that ϕ(λ)ϕ(M), if λM, and ϕ(λ)M*λ2, if λM. In either case, we can obtain that ϕ(λ)ϕ(M)+M*λ2,λ0.

For any given pΓ, hence pC:=F(S), and ApQ:=F(T), from (3.1) and (2.7) we have xn+1-p2=un-p-αn(un-Snun)2=un-p2-2αnun-p,un-Snun+αn2un-Snun2un-p2-αn(1-β)un-Snun2+αnμnϕ(un-p)+αnξn+αn2un-Snun2  (by  (2.7))un-p2-αn(1-β-αn)un-Snun2+αnμn(ϕ(M)+M*(un-p)2)+αnξn=(1+αnμnM*)un-p2-αn(1-β-αn)un-Snun2+αnμnϕ(M)+αnξn. On the other hand, since un-p2=xn-p+γA*(Tn-I)Axn2=xn-p2+γ2A*(Tn-I)Axn2+2γxn-p,A*(Tn-I)Axn,γ2A*(Tn-I)Axn2=γ2A*(Tn-I)Axn,A*(Tn-I)Axn=γ2AA*(Tn-I)Axn,(Tn-I)Axnγ2A2TnAxn-Axn2,2γxn-p,A*(Tn-I)Axn=2γAxn-Ap,(Tn-I)Axn=2γAxn-Ap)+(Tn-I)Axn-(Tn-I)Axn,(Tn-I)Axn=2γ{TnAxn-Ap,TnAxn-Axn-(Tn-I)Axn2}. In (2.8) taking x=Axn,  Gn=Tn,   and   q=Ap and noting ApF(T), from (2.8) we have TnAxn-Ap,TnAxn-Axn-(Tn-I)Axn21+κ2(Tn-I)Axn2+μn2ϕ(Axn-Ap)+ξn2-(Tnn-I)Axn2κ-12(Tn-I)Axn2+μn2(ϕ(M)+M*Axn-Ap2)+ξn2κ-12(Tn-I)Axn2+μn2M*A2xn-p2+μn2ϕ(M)+ξn2. Substituting (3.8) into (3.7), after simplifying it and then substituting the resultant result into (3.5), we have un-p2xn-p2+γ2A2TnAxn-Axn2+γ(κ-1)(Tn-I)Axn2+γμnM*A2xn-p2+γμnϕ(M)+γξn=xn-p2-γ(1-κ-γA2)TnAxn-Axn2+γμnM*A2xn-p2+γμnϕ(M)+γξn(1+γμnM*A2)xn-p2-γ(1-κ-γA2)TnAxn-Axn2+γμnϕ(M)+γξn. Substituting (3.9) into (3.4) and simplifying it we have xn+1-p2(1+αnμnM*){(1+γμnM*A2)xn-p2-γ(1-κ-γA2)TnAxn-Axn2+γμnϕ(M)+γξn}-αn(1-β-αn)un-Snun2+αnμnϕ(M)+αnξn=(1+δn)xn-p2-γ(1-κ-γA2)TnAxn-Axn2-αn(1-β-αn)un-Snun2+bn, where δn=αnμnM*+γμnM*A2+γA2αnμn2(M*)2,bn=((1+αnμnM*)γ+αn)μnϕ(M)+((1+αnμnM*)γ+αn)ξn. By condition (iii) we have xn+1-p2(1+δn)xn-p2+bn. By condition (ii), n=1δn< and n=1bn<. Hence it follows from Lemma 2.5 that the following limit exists: limnxn-p. Consequently, from (3.10) and (3.13) we have that γ(1-κ-γA2)(Tn-I)Axn2+αn(1-β-αn)un-Snun2xn-p2-xn+1-p2+δnxn-p20(as  n). This together with the condition (iii) implies that limnun-Snun=0;limn(Tn-I)Axn=0. It follows from (3.5), (3.13), and (3.16) that the limit limnun-p exists and limnun-p=limnxn-p. The conclusion (1) is proved.

(2) Next we prove that limnxn+1-xn=0,limnun+1-un=0.

In fact, it follows from (3.1) that xn+1-xn=(1-αn)un+αnSn(un)-xn=(1-αn)(xn+γA*(Tn-I)Axn)+αnSn(un)-xn=(1-αn)γA*(Tn-I)Axn+αn(Sn(un)-xn)=(1-αn)γA*(Tn-I)Axn+αn(Sn(un)-un)+αn(un-xn)=(1-αn)γA*(Tn-I)Axn+αn(Sn(un)-un)+αnγA*(Tn-I)Axn=γA*(Tn-I)Axn+αn(Sn(un)-un). In view of (3.15) and (3.16) we have that limnxn+1-xn=0. Similarly, it follows from (3.1), (3.16), and (3.20) that un+1-un=xn+1+γA*(Tn+1-I)Axn+1-(xn+γA*(Tn-I)Axn)xn+1-xn+γA*(Tn+1-I)Axn+1+γA*(Tn-I)Axn0(as  n). The conclusion (3.18) is proved.

(3) Next we prove that un-Sun0,  Axn-TAxn0(as  n).

In fact, from (3.15) we have ζn:=un-Snun0(as  n). Since S is uniformly L-Lipschitzian continuous, it follows from (3.18) and (3.23) that un-Sunun-Snun+Snun-Sunζn+LSn-1un-unζn+L{Sn-1un-Sn-1un-1+Sn-1un-1-un}ζn+L2un-un-1+LSn-1un-1-un-1+un-1-unζn+L(1+L)un-un-1+Lζn-10(as  n).

Similarly, from (3.16) we have Axn-TnAxn0(as  n). Since T is uniformly L̃-Lipschitzian continuous, by the same way as above, from (3.18) and (3.25), we can also prove that Axn-TAxn0(as  n).

(4) Finally we prove that xnx* and unx* which is a solution of (SCFP)-(1.1).

Since {un} is bounded, there exists a subsequence {uni}{un} such that unix* (some point in H1). From (3.22) we have uni-Suni0(as  ni). By Proposition 2.7, S is demiclosed at zero; hence we know that x*F(S).

Moreover, from (3.1) and (3.16) we have xni=uni-γA*(Tni-I)Axnix*. Since A is a linear bounded operator, it gets AxniAx*. In view of (3.22) we have Axni-TAxni0(as  ni). Again by Proposition 2.7, T is demiclosed at zero, and we have Ax*F(T). Summing up the above argument, it shows that x*Γ; that is, x* is a solution to the (SCFP)-(1.1).

Now we prove that xnx* and unx*.

Suppose to the contrary, if there exists another subsequence {unj}{un} such that unjy*Γ with y*x*, then by virtue of (3.2) and the Opial property of Hilbert space, we have liminfniuni-x*<liminfniuni-y*=limnun-y*=limnjunj-y*<liminfnjunj-x*=limnun-x*=liminfniuni-x*. This is a contradiction. Therefore, unx*. By using (3.1) and (3.16), we have xn=un-γA*(Tnn-I)Axnx*.

The Proof of Conclusion (II)

By the assumption that S is semicompact, it follows from (3.27) that there exists a subsequence of {uni} (without loss of generality, we still denote it by {uni}) such that uniu*H (some point in H). Since unix*, this implies that x*=u*. And so unix*Γ. By virtue of (3.2) we know that limnun-x*=0 and limnxn-x*=0; that is, {un} and {xn} both converge strongly to x*Γ.

This completes the proof of Theorem 3.1.

The following theorem can be obtained from Theorem 3.1 immediately.

Theorem 3.2.

Let H1 and H2 be two real Hilbert spaces, let A:H1H2 be a bounded linear operator, let S:H1H1 be a uniformly L-Lipschitzian and (β,{kn(1)})-asymptotically strictly pseudocontractive mapping, and let T:H2H2 be a uniformly L̃-Lipschitzian and (κ,{kn(2)})- asymptotically strictly pseudocontractive mapping satisfying the following conditions:

C:=F(S), Q:=F(T);

kn=max{kn(1),kn(2)}, and n=1(kn-1)<.

Let {xn} be the sequence defined by (3.1), where {αn} is a sequence in [0,1] and γ>0 is a constant satisfying the following condition:

αn(δ,1-β), for all n1 and γ(0,(1-κ)/A2), where δ(0,1-β) is a constant. If Γ, then the conclusions of Theorem 3.1 still hold.

From Theorems 3.1 and 3.2 we can obtain the following.

Theorem 3.3.

Let H1 and H2 be two real Hilbert spaces, let A:H1H2 be a bounded linear operator, S:H1H1 be a uniformly L-Lipschitzian and β-strictly pseudocontractive mapping, and let T:H2H2 be a uniformly L̃-Lipschitzian and κ-strictly pseudocontractive mapping satisfying the following conditions:

C:=F(S), Q:=F(T);

T and S both are demiclosed at origin.

Let {xn} be the sequence generated by x1H1  chosen  arbitrarily,xn+1=(1-αn)un+αnSun,un=xn+γA*(T-I)Axn,n1, where {αn} is a sequence in [0,1] and γ>0 is a constant satisfying the following condition:

αn(δ,1-β), for all n1 and γ(0,(1-κ)/A2), where δ(0,1-β) is a constant. If Γ, then the conclusions of Theorem 3.1 still hold.

Proof.

By the same way as given in the proof of Theorems 3.1 and 3.2 and noting that in the case of strictly pseudocontractive mapping the sequence {kn=1} in Theorem 3.2. Therefore we can prove that for each pΓ, the limits limnxn-p and limnun-p exist and limnxn-p  =  limnun-p;un-Sun0;Axn-TAxn0;un-un+10;xn-xn+10;xnx*;unx*Γ. In addition, if S is also semicompact, we can also prove that {xn} and {un} both converge strongly to x*.

Remark 3.4.

Theorems 3.1 and 3.2 improve and extend the corresponding results of Censor et al. [4, 5], Yang , Moudafi [9, 10], Xu , Censor and Segal , Masad and Reich , and others.

Acknowledgments

The authors would like to express their thanks to the editors and the referees for their helpful comments and suggestions. This work was supported by the Natural Science Foundation of Yunnan Province and Yunnan University of Finance and Economics.

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