Volume 2008, Article ID 280908,10pages doi:10.1155/2008/280908
Research Article
Strong Convergence of an Implicit Iteration
Algorithm for a Finite Family of Pseudocontractive Mappings
Yonghong Yao1and Yeong-Cheng Liou2
1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
Correspondence should be addressed to Yonghong Yao,[email protected] Received 2 December 2007; Accepted 2 January 2008
Recommended by Ram Verma
Strong convergence theorems for approximation of common fixed points of a finite family of pseudocontractive mappings are proven in Banach spaces using an implicit iteration scheme. The results presented in this paper improve and extend the corresponding results of Osilike, Xu and Ori, Chidume and Shahzad, and others.
Copyrightq2008 Y. Yao and Y.-C. Liou. 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.
1. Introduction
LetEbe a real Banach space and letJdenote the normalized duality mapping fromEinto 2E∗ given by
Jx
x∗∈E∗: x, x∗
x2x∗2
, 1.1
whereE∗denotes the dual space ofEand·,·denotes the generalized duality pairing. IfE∗is strictly convex, thenJis single valued. In the sequel, we will denote the single-value duality mapping byj.
LetCbe a nonempty closed convex subset ofE. Recall that a self-mappingf :C→Cis said to be a contraction if there exists a constantδ∈0,1such that
fx−fy≤δx−y, ∀x, y∈C. 1.2 We use ΠC to denote the collection of all contractions on C. That is, ΠC {f : f : C → C a contraction}.
A mappingT with domain DTandRTinE is called pseudocontractive if, for all x, y∈DT, there existsjx−y∈Jx−ysuch that
Tx−Ty, jx−y
≤ x−y2. 1.3
We use FixTto denote the fixed point set ofT, that is, FixT {x∈C:Txx}.
Recently, Xu and Ori1have introduced an implicit iteration process below for a finite family of nonexpansive mappings. LetT1, T2, . . . , TN be N self-mappings of Eand suppose thatN
i1FixTi/∅, the set of common fixed points ofTi, i1,2, . . . , N.An implicit iteration process for a finite family of nonexpansive mappings is defined as follows with{tn} a real sequence in0,1,x0∈E:
x1t1x0
1−t1 T1x1, x2t2x1
1−t2 T2x2, ...
xNtNxN−1
1−tN TNxN, xN1tN1xN
1−tN1 T1xN1, ...
1.4
which can be written in the following compact form:
xntnxn−1
1−tn Tnxn, n≥1, 1.5
whereTnTnmodN.
Xu and Ori proved the weak convergence of the above iterative process 1.5 to a common fixed point of a finite family of nonexpansive mappings{Tn}Nn1 in a Hilbert space.
They further remarked that it is yet unclear what assumptions on the mapping and/or the parameters{tn}are sufficient to guarantee the strong convergence of the sequence{xn}.
Very recently, Osilike2first extended Xu and Ori1from the class of nonexpansive mappings to the more general class of strictly pseudocontractive mappings in a Hilbert space.
He proved the following two convergence theorems.
Theorem O1. LetH be a real Hilbert space and letCbe a nonempty closed convex subset ofH. Let {Ti}Ni1beNstrictly pseudocontractive self-mappings ofCsuch thatN
i1FixTi/∅. Letx0 ∈Cand let{αn}∞n1be a sequence in0,1such that limn→∞αn0. Then the sequence{xn}∞n1defined by
xnαnxn−1
1−αn Tnxn, n≥1, 1.6
whereTnTnmodN, converges weakly to a common fixed point of the mappings{Ti}Ni1.
Theorem O2. LetEbe a real Banach space and letCbe a nonempty closed convex subset of E. Let {Ti}Ni1beNstrictly pseudocontractive self-mappings ofCsuch thatN
i1FixTi/∅, and let{αn}∞n1 be a real sequence satisfying the conditions 0< αn <1,∞
n11−αn ∞and∞
n11−αn2 <∞.
Letx0∈Cand let{xn}∞n1be defined by xnαnxn−1
1−αn Tnxn, n≥1, 1.7
whereTnTnmodN. Then{xn}converges strongly to a common fixed point of the mappings{Ti}Ni1if and only if lim infn→∞dxn, F 0.
Remark 1.1. We note that Theorem O1 has only weak convergence even in a Hilbert space and Theorem O2 has strong convergence, but imposed condition lim infn→∞dxn, F 0.
In 2005, Chidume and Shahzad3also proved the strong convergence of the implicit iteration process1.5to a common fixed point for a finite family of nonexpansive mappings.
They gave the following theorem.
Theorem CS. LetEbe a uniformly convex Banach space, let Cbe a nonempty closed convex subset ofE. Let{Ti}Ni1 beN nonexpansive self-mappings ofCwith N
i1FixTi/∅. Suppose that one of the mappings in{Ti}Ni1 is semicompact. Let{tn} ⊂ δ,1−δfor some δ ∈ 0,1. From arbitrary x0∈C, define the sequence{xn}by1.5. Then{xn}converges strongly to a common fixed point of the mappings{Ti}Ni1.
Remark 1.2. Chidume and Shahzad gave an affirmative response to the question raised by Xu and Ori1, but they imposed compactness condition on some mapping of{Ti}Ni1.
In this paper, we will consider a process for a finite family of pseudocontractive mappings which include the nonexpansive mappings as special cases. Let f : C → Cbe a contraction. Let{αn},{βn}, and{γn}be three real sequences in0,1and an initial pointx0∈C.
Let the sequence{xn}be defined by x1α1f
x0 β1x0γ1T1x1, x2α2f
x1 β2x1γ2T2x2, ...
xNαNf
xN−1 βNxN−1γNTNxN, xN1αN1f
xN βN1xNγN1T1xN1, ...
1.8
which can be written in the following compact form:
xnαnf
xn−1 βnxn−1γnTnxn, n≥1, 1.9 whereTnTnmodN.
Motivated by the works in 1–6, our purpose in this paper is to study the implicit iteration process1.9in the general setting of a uniformly smooth Banach space and prove the strong convergence of the iterative process1.9to a common fixed point of a finite family of pseudocontractive mappings{Ti}Ni1. The results presented in this paper generalize and extend the corresponding results of Chidume and Shahzad3, Osilike2, Xu and Ori1, and others.
2. Preliminaries
LetEbe a Banach space. Recall the norm ofEis said to be Gateaux differentiableandEis said to be smoothif
limt→0
xty − x
t 2.1
exists for eachx, yin its unit sphereU {x∈E:x1}. It is said to be uniformly Frechet differentiableandEis said to be uniformly smoothif the limit in2.1is attained uniformly forx, y∈U×U.It is well known that a Banach spaceEis uniformly smooth if and only if the duality mapJis single valued and norm-to-norm uniformly continuous on bounded sets ofE.
Recall that ifCandDare nonempty subsets of a Banach spaceEsuch thatCis nonempty closed convex andD⊂C, then a mapQ:C→Dis called a retraction fromContoDprovided Qx xfor allx∈D. A retractionQ:C→Dis sunny providedQxtx−Qx Qx for allx∈ Candt≥ 0 wheneverxtx−Qx ∈C. A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive.
We need the following lemmas for proof of our main results.
Lemma 2.1 see 7. Let Ebe a uniformly smooth Banach space, C a closed convex subset ofE, T :C→Ca nonexpansive with FixT/∅. For eachf∈ΠCand everyt∈0,1, then{xt}defined by
xt tf
xt 1−tTxt 2.2
converges strongly ast→0 to a fixed point ofT.
In particular, if f u ∈ C is a constant, then2.2is reduced to the sunny nonexpansive retraction of Reich fromConto FixT,
Qu−u, J
Qu−p ≤0, p∈FixT. 2.3
Lemma 2.2see8. LetEbe a real uniformly smooth Banach space, then there exists a nondecreasing continuous functionb:0,∞→0,∞satisfying
ibct≤cbtfor allc≥1;
iilimt→0bt 0;
iiixy2≤ x22y, jxmax{x,1}yby, for allx, y∈E.
The inequalityiiiis called Reich’s inequality.
Lemma 2.3see9. Let{an}∞n0 be a sequences of nonegative real numbers satisfying the property an1≤1−γnanγnσn, n≥0,where{γn}∞n0⊂0,1and{σn}∞n0are such that
i∞
n0γn∞;
iieither lim supn→∞σn≤0 or∞
n0|γnσn|<∞.
Then{an}∞n0converges to 0.
3. Main results
Theorem 3.1. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset ofE. Let{Ti}Ni1beNpseudocontractive self-mappings ofCsuch thatN
i1FixTi/∅. Let{αn},{βn}, and{γn}be three real sequences in0,1satisfying the following conditions:
iαnβnγn1;
iilimn→∞βn0 and limn→∞αn/βn 0;
iii∞
n0αn/αnβn ∞.
Forf ∈ΠC and givenx0 ∈Carbitrarily, let the sequence{xn}be defined by1.9. Then{xn} converges strongly to a common fixed pointpof the mappings{Ti}Ni1, wherep Qfis the unique solution of the following variational inequality:
f−IQf, j
z−Qf ≤0 ∀z∈N
i1
Fix
Ti . 3.1
Proof. First, we observe that{xn}is bounded. Indeed, if we take a fixed pointpofT, noting that
xn−p
1−γn αn
1−γnf
xn−1 βn
1−γnxn−1
γnTnxn−p
1−γn αn
1−γn
f
xn−1 −fp αn
fp−p 1−γn βn
xn−1−p 1−γn
γn
Tnxn−p .
3.2 It follows that
xn−p2
1−γn αn
1−γn
f
xn−1 −fp αn
fp−p 1−γn βn
xn−1−p 1−γn , j
xn−p γn
Tnxn−p, j xn−p
≤
1−γn αn
1−γn
f
xn−1 −fp αn
fp−p 1−γn βn
xn−1−p 1−γn
xn−pγnxn−p2, 3.3 which implies that
xn−p≤ αn
f
xn−1 −fp 1−γn αn
fp−p 1−γn βn
xn−1−p 1−γn
≤ αn
1−γn
fp−pδαnβn
1−γn
xn−1−p 1−δαn
1−γn ×fp−p 1−δ
1−1−δαn
1−γn
xn−1−p
≤maxfp−p
1−δ ,xn−1−p .
3.4
Now, an induction yields
xn−p≤maxfp−p
1−δ ,x0−p
. 3.5
Hence{xn}is bounded, so are{fxn}and{Tixn}for alli1,2, . . . , N.
Observe that
xn−Tnxn≤αnf
xn−1 −Tnxnβnxn−1−Tnxn−→0. 3.6 SetAn 2I−Tn−1for alln1,2, . . . , N, it is well known that{An}Nn1are all nonexpansive mappings and FixAn FixTnas a consequence of10, Theorem 6. Then we have
xn−AnxnAnA−1n xn−Anxn≤xn−Tnxn. 3.7 It also follows from3.6that limn→∞xn−Anxn0.
Next, we claim that lim sup
n→∞
fp−p, j
xn−p ≤0 p∈N
i1
Fix
Ti , 3.8
where p Qf limt→0zt with zt being the fixed point of z → tfz 1−tAnz see Lemma 2.1.
Indeed,ztsolves the fixed point equation zt tf
zt 1−tAnzt. 3.9
Then we have
zt−xn 1−t
Anzt−xn t f
zt −xn . 3.10
Thus we obtain
zt−xn2≤1−t2Anzt−Anxnxn−Anxn2 2t
f
zt −zt, j
zt−xn 2tzt−xn2.
3.11
Noting that f
zt −zt, j
zt−xn f
zt −fp, j
zt−xn
fp−zt, j zt−xn
≤δzt−pzt−xn
fp−zt, j
zt−xn . 3.12 Thus3.11gives
zt−xn2≤1−t2zt−xnxn−Anxn22δtzt−pzt−xn 2t
fp−zt, j
zt−xn 2tzt−xn2
≤1−t2zt−xn2ant 2δtzt−pzt−xn 2t
fp−zt, j
zt−xn 2tzt−xn2,
3.13
where
ant
2zt−xnxn−Anxn xn−Anxn−→0 asn−→ ∞. 3.14
It follows that
zt−fp, j
zt−xn ≤ t
2zt−xn2 1
2tant δzt−pzt−xn. 3.15 Lettingn→ ∞in3.15and noting3.14yields
lim sup
n→∞
zt−fp, j
zt−xn ≤ t
2MδMzt−p, 3.16
whereM >0 is a constant.
For 3.9, since zt strongly converges to p, then {zt} is bounded. Hence we obtain immediately that the set{zt −xn}is bounded. At the same time, we note that the duality mapjis norm-to-norm uniformly continuous on bounded sets ofE. By lettingt→0 in3.16, it is not hard to find that the two limits can be interchanged and3.8is thus proven.
Finally, we show thatxn→pstrongly.
Indeed, usingLemma 2.2and noting that3.4, we obtain xn−p2
≤ αn
1−γn
f
xn−1 −p βn
1−γn
xn−1−p
2
≤ βn
1−γn
2
xn−1−p22 αnβn
1−γn 2
f
xn−1 −p, j xn−1−p max
βn
1−γn
xn−1−p ,1 αn
1−γn
f
xn−1 −pb αn
1−γn
f
xn−1 −p
βn
1−γn
2
xn−1−p22 αnβn
1−γn 2
fp−p, j xn−1−p 2 αnβn
1−γn 2
f
xn−1 −fp, j
xn−1−p max
βn
1−γn
xn−1−p ,1f
xn−1 −pαn
1−γn bf
xn−1 −pαn
1−γn
≤ 1−αn
αn2βn
αnβn 2
xn−1−p2 2αnβn
αnβn 2δxn−1−p2 2αnβn
αnβn 2
fp−p, j
xn−1−p max
βn
1−γn
xn−1−p ,1f
xn−1 −pαn
αnβn bf
xn−1 −pαn
αnβn
1−αn
αn21−δβn
αnβn 2
xn−1−p2αn
αn21−δβn
αnβn 2
×
2βn
αn21−δβn
fp−p, j xn−1−p max
βn
1−γn
xn−1−p ,1f
xn−1 −pαnβn
αn21−δβn bf
xn−1 −pαn
αnβn
1−λn xn−1−p2λnσn,
3.17 whereλnαnαn21−δβn/αnβn2and
σn 2βn
αn21−δβn
fp−p, j
xn−1−p max βn
xn−1−p 1−γn
,1
×f
xn−1 −pαnβn
αn21−δβn ×bf
xn−1 −pαn
αnβn
.
3.18
We observe that limn→∞αn 21− δβn/αn βn 21− δ, then ∞
n0λn ∞ and max{βn/1−γnxn−1−p,1}fxn−1−pαn βn/αn21−δβnis bounded. At the same time, from limn→∞αn/αnβn 0, we have thatbfxn−1−pαn/αnβn→0.
This implies that lim supn→∞σn≤0.
Now, we applyLemma 2.3 and use3.8to see thatxn−p → 0. This completes the proof.
Remark 3.2. Theorem 3.1 proves the strong convergence in the framework of real uniformly smooth Banach spaces. Our theorem extends Theorem O1 to the more general real Banach spaces. Our result improves Theorem O2 without condition lim infn→∞dxn, F 0 and at the same time extends the mappings from nonexpansive mappings to pseudocontractive mappings.
Corollary 3.3. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset ofE. Let{Ti}Ni1beNpseudocontractive self-mappings ofCsuch thatN
i1 FixTi/∅. Let{αn},{βn}, and{γn}be three real sequences in0,1satisfying the following conditions:
iαnβnγn1;
iilimn→∞βn0 and limn→∞αn/βn 0;
iii∞
n0αn/αnβn ∞.
For fixedu∈Cand givenx0∈Carbitrarily, let the sequence{xn}be defined by
xnαnuβnxn−1γnTnxn, n≥1. 3.19 Then{xn}converges strongly to a common fixed pointpof the mappings{Ti}Ni1, wherep Quis the unique solution of the following inequality:
u−Qu, j
z−Qu ≤0 ∀z∈N
i1
Fix
Ti , 3.20
whereQis a sunny nonexpansive retraction fromContoN
i1 FixTi.
Corollary 3.4. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset ofE. Let{Ti}Ni1beNnonexpansive self-mappings ofCsuch thatN
i1FixTi/∅. Let{αn},{βn}, and {γn}be three real sequences in0,1satisfying the following conditions:
iαnβnγn1;
iilimn→∞βn0 and limn→∞αn/βn 0;
iii∞
n0αn/αnβn ∞.
Forf ∈ΠC and givenx0 ∈Carbitrarily, let the sequence{xn}be defined by1.9. Then{xn} converges strongly to a common fixed pointpof the mappings{Ti}Ni1, wherep Qfis the unique solution of the following variational inequality:
f−IQf, j
z−Qf ≤0 ∀z∈N
i1
Fix
Ti . 3.21
Corollary 3.5. LetEbe a uniformly smooth Banach space and letCbe a nonempty closed convex subset ofE. Let{Ti}Ni1beNnonexpansive self-mappings ofCsuch thatN
i1FixTi/∅. Let{αn},{βn}, and {γn}be three real sequences in0,1satisfying the following conditions:
iαnβnγn1;
iilimn→∞βn0 and limn→∞αn/βn 0;
iii∞
n0αn/αnβn ∞.
For fixedu∈Cand givenx0∈Carbitrarily, let the sequence{xn}be defined by
xnαnuβnxn−1γnTnxn, n≥1. 3.22 Then{xn}converges strongly to a common fixed pointpof the mappings{Ti}Ni1, wherep Quis the unique solution of the following inequality:
u−Qup, j
z−Qu ≤0 ∀z∈N
i1
Fix
Ti , 3.23
whereQis a sunny nonexpansive retraction fromContoN
i1FixTi.
Remark 3.6. Corollary 3.5 improves Theorem CS without compactness assumption of map- pings.
Acknowledgments
The authors are extremely grateful to the referee for his/her careful reading. The first author was partially supposed by National Natural Science Foundation of China, Grant no. 10771050.
The second author was partially supposed by the Grant no. NSC 96-2221-E-230-003.
References
1 H. K. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001.
2 M. O. Osilike, “Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 73–81, 2004.
3 C. E. Chidume and N. Shahzad, “Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings,” Nonlinear Analysis, vol. 62, no. 6, pp. 1149–1156, 2005.
4 Y. C. Liou, Y. Yao, and R. Chen, “Iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive Mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 29091, 10 pages, 2007.
5 Y. C. Liou, Y. Yao, and K. Kimura, “Strong convergence to common fixed points of a finite family of nonexpansive mappings,” Journal of Inequalities and Applications, vol. 2007, Article ID 37513, 10 pages, 2007.
6 L. C. Ceng, N. C. Wong, and J. C. Yao, “Implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive mappings,” Journal of Inequalities and Applications, vol. 2006, Article ID 65983, 11 pages, 2006.
7 H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
8 S. Reich, “An iterative procedure for constructing zeros of accretive sets in Banach spaces,” Nonlinear Analysis, vol. 2, no. 1, pp. 85–92, 1978.
9 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
10 R. H. Martin Jr., “Differential equations on closed subsets of a Banach space,” Transactions of the American Mathematical Society, vol. 179, pp. 399–414, 1973.
