JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation34195310.1155/2012/341953341953Research ArticleAn Iterative Algorithm on Approximating Fixed Points of Pseudocontractive MappingsYuYouli1YaoYonghong1School of Mathematics and Information Engineering, Taizhou University, Linhai 317000Chinatzc.edu.cn20123112011201204092011160920112012Copyright © 2012 Youli Yu.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.

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let f:KK a contractive mapping and T:KK be a uniformly continuous pseudocontractive mapping with F(T). Let {λn}(0,1/2) be a sequence satisfying the following conditions: (i) limnλn=0; (ii) n=0λn=. Define the sequence {xn} in K by x0K, xn+1=λnf(xn)+(12λn)xn+λnTxn, for all n0. Under some appropriate assumptions, we prove that the sequence {xn} converges strongly to a fixed point pF(T) which is the unique solution of the following variational inequality: f(p)p,j(zp)0, for all zF(T).

1. Introduction

Let E be a real Banach space with dual E*. We denote by J the normalized duality mapping from E to 2E* defined by J(x)={f*E*:x,f*=x2=f*2},  xE, where ·,· denotes the generalized duality pairing.

It is well known that, if E is smooth, then J is single-valued. In the sequel, we will denote the single-valued normalized duality mapping by j. We use D(T),R(T) to denote the domain and range of T, respectively.

An operator T:D(T)R(T) is called pseudocontractive if there exists j(x-y)J(x-y) such that Tx-Ty,j(x-y)x-y2,  x,yD(T).

A point xK is a fixed point of T provided Tx=x. Denote by F(T) the set of fixed points of T, that is, F(T)={xK:Tx=x}.

Within the past 40 years or so, many authors have been devoted to the iterative construction of fixed points of pseudocontractive mappings (see ).

In 1974, Ishikawa  introduced an iterative scheme to approximate the fixed points of Lipschitzian pseudocontractive mappings and proved the following result.

Theorem 1.1 (see [<xref ref-type="bibr" rid="B7">11</xref>]).

If K is a compact convex subset of a Hilbert space H,T:KK is a Lipschitzian pseudocontractive mapping. Define the sequence {xn} in K by x0K,xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n0, where {αn},{βn} are sequences of positive numbers satisfying the conditions

0αnβn<1,

limnβn=0,

n=1αnβn=.

Then, the sequence {xn} converges strongly to a fixed point of T.

In connection with the iterative approximation of fixed points of pseudo-contractions, in 2001, Chidume and Mutangadura  provided an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iterative algorithm failed to converge. Chidume and Zegeye  introduced a new iterative scheme for approximating the fixed points of pseudocontractive mappings.

Theorem 1.2 (see [<xref ref-type="bibr" rid="B6">13</xref>]).

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty closed convex subset of E. Let T:KK be a L-Lipschitzian pseudocontractive mapping such that F(T). Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings. Let {λn} and {θn} be two sequences in (0,1] satisfying the following conditions:

limnθn=0,

λn(1+θn)1,  n=0λnθn=,  limn(λn/θn)=0,

limn((θn-1/θn-1)/λnθn)=0.

For given x1K arbitrarily, let the sequence {xn} be defined iteratively by xn+1=(1-λn)xn+λnTxn-λnθn(xn-x1),  n1. Then, the sequence {xn} defined by (1.4) converges strongly to a fixed point of T.

Prototypes for the iteration parameters are, for example, λn=1/(n+1)a and θn=1/(n+1)b for 0<b<a and a+b<1. But we observe that the canonical choices of λn=1/n and θn=1/n are impossible. This bring us a question.

Question 1.

Under what conditions, limnλn=0 and n=0λn= are sufficient to guarantee the strong convergence of the iterative scheme (1.4) to a fixed point of T?

In this paper, we explore an iterative scheme to approximate the fixed points of pseudocontractive mappings and prove that, under some appropriate assumptions, the proposed iterative scheme converges strongly to a fixed point of T, which solves some variational inequality. Our results improve and extend many results given in the literature.

2. Preliminaries

Let K be a nonempty closed convex subset of a real Banach space E. Recall that a mapping f:KK is called contractive if there exists a constant α(0,1) such that f(x)-f(y)αx-y,  x,yK.

Let μ be a continuous linear functional on l and s=(a0,a1,)l. We write μn(an) instead of μ(s). We call μ a Banach limit if μ satisfies μ=μ(1)=1 and μn(an+1)=μn(an) for all (a0,a1,)l.

If μ is a Banach limit, then we have the following.

For all n1, ancn implies μn(an)μn(cn).

μn(an+r)=μn(an) for any fixed positive integer r.

liminfnanμn(an)limsupnan for all (a0,a1,)l.

If s=(a0,a1,)l with ana, then μ(s)=μn(an)=a for any Banach limit μ.

For more details on Banach limits, we refer readers to . We need the following lemmas for proving our main results.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B11">15</xref>]).

Let E be a Banach space. Suppose that K is a nonempty closed convex subset of E and T:KE is a continuous pseudocontractive mapping satisfying the weakly inward condition: T(x)IK(x)¯(IK(x)¯ is the closure of IK(x)) for each xK, where IK(x)={x+c(u-x):uE  and  c1}. Then, for each zK, there exists a unique continuous path tztK for all t[0,1), satisfying the following equation zt=tTzt+(1-t)z. Furthermore, if E is a reflexive Banach space with a uniformly Gâteaux differentiable norm and every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings, then, as t1, zt converges strongly to a fixed point of T.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B3">16</xref>]).

(1) If E is smooth Banach space, then the duality mapping J is single valued and strong-weak* continuous.

(2) If E is a Banach space with a uniformly Gâteaux differentiable norm, then the duality mapping J:EE* is single valued and norm to weak star uniformly continuous on bounded sets of E.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B8">17</xref>]).

Let {an} be a sequence of nonnegative real numbers satisfying an+1(1-αn)an+αnβn for all n0, where {αn}(0,1), and {βn} two sequences of real numbers such that n=0αn= and limsupnβn0. Then {an} converges to zero as n.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B2">18</xref>]).

Let E be a real Banach space, and let J be the normalized duality mapping. Then, for any given x,yE, x+y2x2+2y,j(x+y),  j(x+y)J(x+y).

Lemma 2.5 (see [<xref ref-type="bibr" rid="B14">14</xref>]).

Let a be a real number, and let (x0,x1,,)l such that μnxna for all Banach limits. If   limsupn(xn+1-xn)0, then limsupnxna.

3. Main Results

Now, we are ready to give our main results in this paper.

Theorem 3.1.

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings. Let f:KK a contractive mapping and T:KK be a uniformly continuous pseudocontractive mapping with F(T). Let {λn}(0,1/2] be a sequence satisfying the conditions:

limnλn=0,

n=0λn=.

Define the sequence {xn} in K by x0K,xn+1=λnf(xn)+(1-2λn)xn+λnTxn,  n0. If limnxn-Txn=0, then the sequence {xn} converges strongly to a fixed point pF(T), which is the unique solution of the following variational inequality: f(p)-p,j(z-p)0,  zF(T).

Proof.

Take pF(T), and let S=I-T. Then, we have Sx-Sy,j(x-y)0. From (3.1), we obtain xn=xn+1+λnxn-λnTxn+λnxn-λnf(xn)=xn+1+λnxn+λnSxn-λnf(xn)=xn+1+λn[xn+1+λnxn+λnSxn-λnf(xn)]+λnSxn-λnf(xn)=(1+λn)xn+1+λn2(xn+Sxn)-λn2f(xn)+λnSxn-λnf(xn)=(1+λn)xn+1+λnSxn+1+λn2(xn+Sxn)-λn2f(xn)+λn(Sxn-Sxn+1)-λnf(xn). By (3.4), we have xn-p=(1+λn)(xn+1-p)+λn(Sxn+1-Sp)+λn2(xn+Sxn)-λn2f(xn)+λn(Sxn-Sxn+1)+λn(p-f(xn)). Combining (3.3) and (3.5), we have xn-p-λn2(xn+Sxn)+λn2f(xn)-λn(Sxn-Sxn+1)+λn(f(xn)-p),j(xn+1-p)=(1+λn)xn+1-p2+λnSxn+1-Sp,j(xn+1-p)(1+λn)xn+1-p2.

Next, we prove that limsupnf(p)-p,j(xn-p)0. Indeed, taking z=f(p) in Lemma 2.1, we have zt-xn=(1-t)(Tzt-xn)+t(f(p)-xn), and, hence, zt-xn2=(1-t)Tzt-xn,j(zt-xn)+tf(p)-xn,j(zt-xn)=(1-t)Tzt-Txn,j(zt-xn)+(1-t)Txn-xn,j(zt-xn)+tf(p)-zt,j(zt-xn)+tzt-xn2zt-xn2+(1-t)Txn-xnzt-xn+tf(p)-zt,j(zt-xn). Therefore, we have zt-f(p),j(zt-xn)1-ttTxn-xnzt-xnM11-ttTxn-xn, where M1>0 is some constant such that zt-xnM1 for all t(0,1] and n1. Letting n, we have limsupnzt-f(p),j(zt-xn)0. From Lemma 2.1, we know ztp as t0. Since the duality mapping J:EE* is norm to weak star uniformly continuous from Lemma 2.2, we have limsupnf(p)-p,j(xn-p)0. From (3.6), we have (1+λn)xn+1-p2xn-p-λn2(xn+Sxn)+λn2f(xn)-λn(Sxn-Sxn+1),j(xn+1-p)xn-pxn+1-p+M2λn2+M2λnSxn+1-Sxn+λnf(xn)-f(p)xn+1-p+λnf(p)-p,j(xn+1-p)xn-pxn+1-p+M2λn2+M2λnSxn+1-Sxn+λnαxn-pxn+1-p+λnf(p)-p,j(xn+1-p)1+λnα2(xn-p2+xn+1-p2)+M2λn2+M2λnSxn+1-Sxn+λnf(p)-p,j(xn+1-p), where M2 is a constant such that sup{xn+Sxnxn+1-p+f(xn)xn+1-p+xn+1-p,  n0}M2. It follows that xn+1-p21+λnα1+(2-α)λnxn-p2+M2λn2+M2λnSxn+1-Sxn+λn1+(2-α)λnf(p)-p,j(xn+1-p)=[1-2(1-α)1+(2-α)λnλn]xn-p2+2(1-α)λn1+(2-α)λn×{1+(2-α)λn2(1-α)M2λn+1+(2-α)λn2(1-α)M2Sxn+1-Sxn+12(1-α)f(p)-p,j(xn+1-p)}=(1-αn)xn-p2+αnβn, where αn=2(1-α)1+(2-α)λnλn,βn=1+(2-α)λn2(1-α)M2λn+1+(2-α)λn2(1-α)M2Sxn+1-Sxn+12(1-α)f(p)-p,j(xn+1-p). Note that xn+1-xnλnxn+λnTxn+λnxn-f(xn)0(n). By the uniformly continuity of T, we have Sxn+1-Sxn0(n). Hence, it is clear that n=0αn= and limsupnβn0.

Finally, applying Lemma 2.3 to (3.14), we can conclude that xnp. This completes the proof.

From Theorem 3.1, we can prove the following corollary.

Corollary 3.2.

Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive self-mappings. Let T:KK be a uniformly continuous pseudocontractive mapping with F(T). Let {λn}(0,1/2] be a sequence satisfying the conditions:

limnλn=0,

n=0λn=.

Define the sequence {xn} in K by u,x0K,xn+1=λnu+(1-2λn)xn+λnTxn,  n0. Then, the sequence {xn} converges strongly to a fixed point of T if and only if limnxn-Txn=0.

Theorem 3.3.

Let E be a uniformly smooth Banach space and K a nonempty bounded closed convex subset of E. Let f:KK be a contractive mapping and T:KK a uniformly continuous pseudocontractive mapping with F(T). Let {λn}(0,1/2] be a sequence satisfying the conditions:

limnλn=0,

n=0λn=.

If limnxn-Txn=0, then the sequence {xn} defined by (3.1) converges strongly to a fixed point pF(T), which is the unique solution of the following variational inequality: f(p)-p,j(z-p)0,  zF(T).

Proof.

Since every uniformly smooth Banach space E is reflexive and whose norm is uniformly Gâteaux differentiable, at the same time, every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Hence, from Theorem 3.1, we can obtain the result. This completes the proof.

From Theorem 3.3, we can prove the following corollary.

Corollary 3.4.

Let E be a uniformly smooth Banach space and K a nonempty bounded closed convex subset of E. Let T:KK be a uniformly continuous pseudocontractive mapping with F(T). Let {λn}(0,1/2] be a sequence satisfying the conditions:

limnλn=0,

n=0λn=.

Define the sequence {xn} in K by u,x0K,xn+1=λnu+(1-2λn)xn+λnTxn,  n0. Then, the sequence {xn} converges strongly to a fixed point of T if and only if limnxn-Txn=0.

Theorem 3.5.

Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gâteaux differentiable norm. Let f:KK a contractive mapping and T:KK be a uniformly continuous pseudocontractive mapping. Let {λn}(0,1/2] be a sequence satisfying the conditions:

limnλn=0,

n=0λn=.

If DF(T), where D is defined as (3.22) below, then the sequence {xn} defined by (3.1) converges strongly to a fixed point pF(T), which is the unique solution of the following variational inequality: f(p)-p,j(z-p)0,  zF(T).

Proof.

First, we note that the sequence {xn} is bounded. Now, if we define g(x)=μnxn-x2, then g(x) is convex and continuous. Also, we can easily prove that g(x) as x. Since E is reflexive, there exists yK such that g(y)=infxKg(x). So the set D={yK:g(y)=infxKg(x)}. Clearly, D is closed convex subset of K.

Now, we can take pDF(T) and t(0,1). By the convexity of K, we have that (1-t)p+tf(p)K. It follows that g(p)g((1-t)p+tf(p)). By Lemma 2.4, we have xn-p-t(f(p)-p)2xn-p2-2tf(p)-p,j(xn-p-t(f(p)-p)). Taking the Banach limit in (3.24), we have μnxn-p-t(f(p)-p)2μnxn-p2-2tμnf(p)-p,j(xn-p-t(f(p)-p)). This implies 2tμnf(p)-p,j(xn-p-t(f(p)-p))g(p)-g((1-t)p+tf(p)). Therefore, it follows from (3.23) and (3.26) that μnf(p)-p,j(xn-p-t(f(p)-p))0. Since the normalized duality mapping j is single valued and norm-weak* uniformly continuous on bounded subset of E, we have f(p)-p,j(xn-p)-f(p)-p,j(xn-p-t(f(p)-p))0(t0). This implies that, for any ϵ>0, there exists δ>0 such that, for all t(0,δ) and n1, f(p)-p,j(xn-p)-f(p)-p,j(xn-p-t(f(p)-p))<ϵ. Taking the Banach limit and noting that (3.27), we have μnf(p)-p,j(xn-p)μnf(p)-p,j(xn-p-t(f(p)-p))+ϵϵ. By the arbitrariness of ϵ, we obtain μnf(p)-p,j(xn-p)0. At the same time, we note that xn+1-xnλn(f(xn)+2xn+Txn)0(n). Since {xn-p},  {f(p)-p} are bounded and the duality mapping j is single valued and norm topology to weak star topology uniformly continuous on bounded sets in Banach space E with a uniformly Gâteaux differentiable norm, it follows that limn{f(p)-p,j(xn+1-p)-f(p)-p,j(xn-p)}=0. From (3.31), (3.33), and Lemma 2.5, we conclude that limsupnf(p)-p,j(xn+1-p)0.

Finally, by the similar arguments as that the proof in Theorem 3.1, it is easy prove that the sequence {xn} converges to a fixed point of T. This completes the proof.

From Theorem 3.5, we can easily to prove the following result.

Corollary 3.6.

Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gâteaux differentiable norm. Let f:KK be a contractive mapping and T:KK a uniformly continuous pseudocontractive mapping. Let {λn}(0,1/2] be a sequence satisfying the conditions:

limnλn=0,

n=0λn=.

Define the sequence {xn} in K by u,x0K,xn+1=λnu+(1-2λn)xn+λnTxn,  n0. If DF(T), where D is defined as (3.22), then the sequence {xn} defined by (3.35) converges strongly to a fixed point pF(T).

Acknowledgment

This research was partially supported by Youth Foundation of Taizhou University (2011QN11).

ChangS. S.On the convergence problems of Ishikawa and Mann iterative processes with error for Φ-pseudo contractive type mappingsApplied Mathematics and Mechanics200021111210.1007/BF024585331765333ChidumeC. E.MooreC.The solution by iteration of nonlinear equations in uniformly smooth Banach spacesJournal of Mathematical Analysis and Applications19972151132146147885510.1006/jmaa.1997.5628ZBL0906.47050LiuQ. H.The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappingsJournal of Mathematical Analysis and Applications199014815562105204410.1016/0022-247X(90)90027-DZBL0729.47052MannW. R.Mean value methods in iterationProceedings of the American Mathematical Society19534506510005484610.1090/S0002-9939-1953-0054846-3ZBL0050.11603OsilikeM. O.Iterative solution of nonlinear equations of the Φ-strongly accretive typeJournal of Mathematical Analysis and Applications19962002259271139114810.1006/jmaa.1996.0203ReichS.Iterative methods for accretive setsNonlinear Equations in Abstract Spaces1978New York, NY, USAAcademic Press317326502549ZBL0495.47034YaoY.MarinoG.LiouY. C.A hybrid method for monotone variational inequalities involving pseudocontractionsFixed Point Theory and Applications20112011818053410.1155/2011/1805342774718ZBL1215.49021YaoY.LiouY. C.KangS. M.Iterative methods for k-strict pseudo-contractive mappings in Hilbert spacesAnalele Stiintifice ale Universitatii Ovidius Constanta20111913133302785706YaoY.LiouY. C.YaoJ.-C.New relaxed hybrid-extragradient method for fixed point problems, a general system of variational inequality problems and generalized mixed equilibrium problemsOptimization201160339541210.1080/023319309031969412780926YaoY.ChoY. J.LiouY. C.Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problemsEuropean Journal of Operational Research20112122242250278420210.1016/j.ejor.2011.01.042IshikawaS.Fixed points and iteration of a nonexpansive mapping in a Banach spaceProceedings of the American Mathematical Society19765916571041290910.1090/S0002-9939-1976-0412909-XZBL0352.47024ChidumeC. E.MutangaduraS. A.An example of the Mann iteration method for Lipschitz pseudocontractionsProceedings of the American Mathematical Society200112982359236310.1090/S0002-9939-01-06009-91823919ChidumeC. E.ZegeyeH.Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive mapsProceedings of the American Mathematical Society2004132383184010.1090/S0002-9939-03-07101-62019962ZBL1051.47041ShiojiN.TakahashiW.Strong convergence of approximated sequences for nonexpansive mappings in Banach spacesProceedings of the American Mathematical Society19971251236413645141537010.1090/S0002-9939-97-04033-1ZBL0888.47034MoralesC. H.JungJ. S.Convergence of paths for pseudocontractive mappings in Banach spacesProceedings of the American Mathematical Society20001281134113419170752810.1090/S0002-9939-00-05573-8ChangS. S.ChoY. J.ZhouH.Iterative Methods for Nonlinear Operator Equations in Banach Spaces2002Huntington, NY, USANova Sciencexiv+4592016857KimT. H.XuH. K.Strong convergence of modified Mann iterationsNonlinear Analysis, Theory, Methods & Applications2005611-2516010.1016/j.na.2004.11.0112122242ZBL1091.47055ChangS. S.ChoY. J.LeeB. S.JungJ. S.KangS. M.Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spacesJournal of Mathematical Analysis and Applications19982241149165163297010.1006/jmaa.1998.5993ZBL0933.47040