Volume 2008, Article ID 902985,9pages doi:10.1155/2008/902985
Research Article
Convergence Theorems of Common Fixed Points
for Pseudocontractive Mappings
Yan Hao
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316004, China
Correspondence should be addressed to Yan Hao,[email protected] Received 10 June 2008; Accepted 24 September 2008
Recommended by Jerzy Jezierski
We consider an implicit iterative process with mixed errors for a finite family of pseudocontractive mappings in the framework of Banach spaces. Our results improve and extend the recent ones announced by many others.
Copyrightq2008 Yan Hao. 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 and preliminaries
LetEbe a real Banach space and letJ denote the normalized duality mapping fromEinto 2E∗given by
Jx f ∈E∗:x, fx2f2, x∈E, 1.1
whereE∗denotes the dual space ofEand·,·denotes the generalized duality pairing. In the sequel, we denote a single-valued normalized duality mapping byj. Throughout this paper, we useFTto denote the set of fixed points of the mappingT.and → denote weak and strong convergence, respectively. Let K be a nonempty subset ofE. For a given sequence {xn} ⊂K, letωωxndenote the weakω-limit set.
Recall thatT :K→Kis nonexpansive if the following inequality holds:
Tx−Ty ≤ x−y, ∀x, y∈K. 1.2
T is said to be strictly pseudocontractive in the terminology of Browder and Petryshyn1if for allx, y∈K,there existλ >0 andjx−y∈Jx−ysuch that
Tis said to be pseudocontractive if for allx, y∈K, there existsjx−y∈Jx−ysuch that
Tx−Ty, jx−y ≤ x−y2. 1.4
It is well known that2 1.4is equivalent to the following:
x−y ≤ x−y−sI−Tx−I−Ty, ∀s >0. 1.5
Recently, concerning the convergence problems of an implicitor nonimplicititerative process to a common fixed point,a finite family of nonexpansive mappings and its extensions in Hilbert spaces or Banach spaces have been considered by several authorssee1–18for more details.
In 2001, Xu and Ori17introduced the following implicit iteration process for a finite family of nonexpansive mappings{T1, T2, . . . , TN}with{αn}a real sequence in0,1and an initial pointx0∈K:
x1α1x0 1−α1T1x1,
x2α2x1 1−α2T2x2,
· · ·
xNαNxN−1 1−αNTNxN,
xN1αN1xN 1−αN1T1xN1,
· · ·
1.6
which can be written in the following compact form:
xnαnxn−1 1−αnTnxn, ∀n≥1, 1.7
whereTnTnmodNhere the modNtakes values in{1,2, . . . , N}.
Xu and Ori 17 proved weak convergence theorems of this iterative process to a common fixed point of the finite family of nonexpansive mappings in a Hilbert space. Chidume and Shahzad3improved Xu and Ori’s17results to some extent. They obtained a strong convergence theorem for a finite family of nonexpansive mappings if one of the mappings is semicompact. Osilike 8 improved the results of Xu and Ori 17 from nonexpansive mappings to strict pseudocontractions in the framework of Hilbert spaces. Recently, Chen et al.7obtained the following results in Banach spaces.
Theorem CSZ. LetE be a realq-uniformly smooth Banach space which is also uniformly convex and satisfies Opial’s condition. LetKbe a nonempty closed convex subset ofEandTi :K→K, i 1,2, . . . , Nbe strictly pseudocontractive mapping in the terminology of Browder-Petryshyn such that FNi1FTi/∅, and let{αn}be a real sequence satisfying the conditions:
Letx0 ∈Kand let{xn}be defined by1.7, whereTn TnmodN. Then{xn}weakly converges to a
common fixed point of the mappings{Ti}Ni1.
Very recently, Zhou18still considered the iterative Algorithm1.7in the framework of Banach spaces. Zhou 18 improved Theorem CSZ from strict pseudocontractions to Lipschitzian pseudocontractions. To be more precise, he proved the following theorem.
Theorem Z. LetEbe a real uniformly convex Banach space with a Fr´echet differentiable norm. Let K be a closed convex subset ofE, and{Ti}be a finite family of Lipschitzian pseudocontractive
self-mappings ofK such thatF ri1FTi/∅. Let{xn}be defined by1.7. If{αn}is chosen so that
αn ∈0,1withlim supαn < 1, then{xn}converges weakly to a common fixed point of the family
{Ti}ri1.
In this paper, motivated and inspired by Chidume and Shahzad3, Chen et al.7, Osilike8, Qin et al.10, Xu and Ori17, and Zhou18, we consider an implicit iteration process with mixed errors for a finite family of pseudocontractive mappings. To be more precise, we consider the following implicit iterative algorithm:
x0∈K, xnαnxn−1βnTnxnγnun, ∀n≥1, 1.9
where{αn}, {βn}, and{γn}are three sequences in0,1such thatαnβnγn1 and{un}is a bounded sequence inK.
We remark that, from the view of computation, the implicit iterative scheme1.7is often impractical since, for each step, we must solve a nonlinear operator equation. Therefore, one of the interesting and important problems in the theory of implicit iterative algorithm is to consider the iterative algorithm with errors. That is an efficient iterative algorithm to compute approximately fixed point of nonlinear mappings.
The purpose of this paper is to use a new analysis technique and establish weak and strong convergence theorems of the implicit iteration process 1.9 for a finite family of pseudocontractive mappings in Banach spaces. Our results improve and extend the corresponding ones announced by many others.
Next, we will recall some well-known concepts and results.
1A spaceEis said to satisfy Opial’s condition9if, for each sequence{xn}inE, the convergencexn→xweakly implies that
lim sup n→ ∞
xn−x<lim sup n→ ∞
xn−y, ∀y∈Ey /x. 1.10
2A mappingT : K→Kis said to be demiclosed at the origin if, for each sequence {xn}inK, the convergencesxn→x0weakly andTxn→0 strongly imply thatTx0
0.
3A mapping T : K→K is semicompact if any sequence {xn} in K satisfying limn→ ∞xn−Txn0 has a convergent subsequence.
Lemma 1.1 see 16. Let {rn}, {sn}, and {tn} be three nonnegative sequences satisfying the
following condition:
rn1≤1snrntn, ∀n≥1. 1.11
If∞n1sn<∞and
∞
n1tn<∞, thenlimn→ ∞rnexists.
Lemma 1.2 see 2. Let E be a real uniformly convex Banach space whose norm is Fr´echet differentiable. LetKbe a closed convex subset ofEand{Tn}be a family of Lipschitzian self-mappings
onKsuch that∞n1Ln−1<∞andF
r
i1FTi.For arbitraryx1∈K, definexn1Tnxn, for
alln≥1. Thenlimn→ ∞xn, jp−qexists for allp, q∈Fand, in particular, for allu, v∈ωωxn,
andp, q∈F, u−v, jp−q0.
Lemma 1.3see19. LetEbe a uniformly convex Banach space,Kbe a nonempty closed convex subset ofE, andT:K→Kbe a pseudocontractive mapping. ThenI−Tis demiclosed at zero.
Lemma 1.4see15. Suppose thatEis a uniformly convex Banach space and0< p≤tn≤q <1,
for alln∈N. Suppose further that{xn}and{yn}are sequences ofEsuch that
lim sup
n→ ∞ xn ≤r, lim supn→ ∞ yn ≤r,
lim
n→ ∞tnxn 1−tnynr
1.12
hold for somer ≥0.Thenlimn→ ∞xn−yn0.
2. Main results
Lemma 2.1. LetEbe a uniformly convex Banach space andKa nonempty closed convex subset of E. LetTibe anLi-Lipschitz pseudocontractive mappings fromKinto itself withFNi1FTi/∅.
Assume that the control sequences{αn}, {βn}, and{γn}satisfy the following conditions:
iαnβnγn1;
ii∞n1γn<∞;
iii0≤a≤αn≤b <1.
Let{xn}be defined by1.9. Then
1limn→ ∞xn−pexists, for allp∈F;
2limn→ ∞xn−Tmxn0, for all1≤m≤N.
Proof. SinceFNn1FTi/∅, for any givenp∈F, we have
xn−p2αnxn−1βnTnxnγnun−p, jxn−p
αnxn−1−p, jxn−pβnTnxn−p, jxn−pγnun−p, jxn−p
≤αnxn−1−pxn−pβnxn−p2γnun−pxn−p.
Simplifying the above inequality, we have
xn−p2≤ αn
αnγnxn−1−pxn−p
γn
αnγnun−pxn−p. 2.2
Ifxn−p0, then the result is apparent. Lettingxn−p>0, we obtain
xn−p ≤ αn
αnγnxn−1−p
γn
αnγnun−p
≤ xn−1−pγnM,
2.3
whereMis an appropriate constant such thatM≥supn≥1{un−p/a}. Noticing the condition
iiand applyingLemma 1.1to2.3, we have limn→ ∞xn−pexists. Next, we assume that
lim
n→ ∞xn−pd. 2.4
On the other hand, from1.5and1.9, we see
xn−p
xn−p 1−αn
2αn xn−Tnxn
xn−p1−αn
2αn αnxn−1−Tnxn γnun−Tnxn
xn−p 1−αn
2 xn−1−Tnxn
γn1−αn
2αn un−Tnxn
xn−1
2 xn− 1 2
αnxn−1 1−αnTnxnγnun−Tnxn
γn
2un−Tnxn
γn1−αn
2αn un−Tnxn
1
2xn−1−p 1
2xn−p
γn
2αnun−Tnxn
≤1
2xn−1−p 1
2xn−p
γn
2αnun−Tnxn.
2.5
Noting that the conditionsiiandiiiand2.4, we obtain
lim inf n→ ∞
12xn−1−p
1
2xn−p
≥d. 2.6
On the other hand, we have
lim sup n→ ∞
12xn−1−p 1
2xn−p
≤lim sup n→ ∞
1
2xn−1−p 1
2xn−p
Combining2.6with2.7, we arrive at
lim n→ ∞
12xn−1−p
1
2xn−p
d. 2.8
By usingLemma 1.4, we get
lim
n→ ∞xn−1−xn0. 2.9
That is,
lim
n→ ∞xni−xn0, ∀i∈ {1,2, . . . , N}. 2.10
It follows from1.9that
xn−1−Tnxn 1
1−αnxn−xn−1−γnun−Tnxn
≤ 1
1−αnxn−xn−1
γn
1−αnun−Tnxn.
2.11
From the conditionsiiandiii, we obtain
lim
n→ ∞xn−1−Tnxn0. 2.12
On the other hand, we have
xn−Tnxn ≤αnxn−1−Tnxnγnun−Tnxn. 2.13
From the conditioniiand2.12, we see
lim
n→ ∞xn−Tnxn0. 2.14
For each 1≤i≤N,we have
xn−Tnixn ≤1Lxn−xnixni−Tnixni, 2.15
whereLmax{Li : 1≤i≤N}. It follows from2.10and2.14that
lim
Therefore, for each 1≤m≤N,there exists somei∈ {1,2, . . . , N}such thatnimmodN. It follows that
xn−Tmxnxn−Tnixn, 2.17
which combines with2.16yields that
lim
n→ ∞xn−Tmxn0, ∀m∈ {1,2, . . . , N}. 2.18
This completes the proof.
Next, we give two weak convergence theorems.
Theorem 2.2. LetEbe a uniformly convex Banach space with a Fr´echet differentiable norm andKa nonempty closed convex subset ofE. LetTibe anLi-Lipschitz pseudocontractive mapping fromKinto
itself withF Ni1FTi/∅. If the control sequences{αn}, {βn}, and{γn}satisfy the followings
conditions:
iαnβnγn1;
ii∞n1γn<∞;
iii0≤a≤αn≤b <1,
then the sequence{xn}defined by1.9converges weakly to a common fixed point of{T1, T2, . . . , TN}.
Proof. FromLemma 1.3, we see thatωωxn ⊂ F.It follows fromLemma 1.2thatωωxnis singleton. Hence,{xn}converges weakly to a common fixed point of {T1, T2, . . . , TN}.This completes the proof.
Remark 2.3. Theorem 2.2includes Theorem 3.1 of Zhou18as a special case. If{γn} 0 for alln≥ 1, thenTheorem 2.2reduces to Theorem 3.1 of Zhou18. It derives to mention that the method in this paper is also different from Zhou’s18.
Theorem 2.4. Let E be a uniformly convex Banach space satisfying Opial’s condition and K a nonempty closed convex subset ofE. LetTibe anLi-Lipschitz pseudocontractive mapping fromKinto
itself withF Ni1FTi/∅. If the control sequences{αn}, {βn}, and{γn}satisfy the followings
conditions:
iαnβnγn1;
ii∞n1γn<∞;
iii0≤a≤αn≤b <1,
Proof. Since E is uniformly convex and {xn} is bounded, we see that there exists a subsequence {xni} ⊂ {xn} such that {xni} converges weakly to a point x∗ ∈ K.It follows fromLemma 1.3and arbitrariness ofm∈ {1,2, . . . , N}thatx∗∈F.
On the other hand, since the spaceEsatisfies Opial’s condition, we can prove that the sequence{xn}converges weakly to a common fixed point of{T1, T2, . . . , TN}by the standard proof. This completes the proof.
Remark 2.5. Theorem 2.4improves Theorem 2.6 of Chen et al.7in several respects.
aFrom q-uniformly smooth Banach spaces which both are uniformly convex and satisfy Opial’s condition extend to uniformly convex Banach spaces which satisfy the Opial’s condition.
bFrom strict pseudocontractions extend to Lipschitzian pseudocontractions.
cFrom view of computation, the iterative Algorithm1.9also can be viewed as an improvement of its analogue in7.
Now, we are in a position to state a strong convergence theorem.
Theorem 2.6. LetEbe a uniformly convex Banach space andKa nonempty closed convex subset of E. LetTibe anLi-Lipschitz pseudocontractive mappings fromKinto itself withF
N
i1FTi/∅.
Assume that the control sequences{αn}, {βn}, and{γn}satisfy the followings conditions:
iαnβnγn1;
ii∞n1γn<∞;
iii0≤a≤αn≤b <1.
Let the sequence{xn}be defined by1.9. If one of the mappings{T1, T2, . . . , TN}is semicompact, then
{xn}converges strongly to a common fixed point of{T1, T2, . . . , TN}.
Proof. Without loss of generality, we can assume thatT1is semicompact. It follows from2.18
that
lim
n→ ∞xn−T1xn0. 2.19
By the semicompactness ofT1, there exists a subsequence{xni}of{xn}such thatxni→x∗∈K strongly. From2.18, we have
lim
ni→ ∞xni−Tmxnix
∗−T
mx∗0, 2.20
for allm1,2, . . . , N.This implies thatx∗∈F.FromLemma 2.1, we know that limn→ ∞xn−
pexists for eachp∈F. That is, limn→ ∞xn−x∗exists. Fromxni→x∗, we have
lim
n→ ∞xn−x
∗0. 2.21
References
1 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 20, no. 2, pp. 197–228, 1967.
2 K.-K. Tan and H.-K. Xu, “The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces,”Proceedings of the American Mathematical Society, vol. 114, no. 2, pp. 399–404, 1992.
3 C. E. Chidume and N. Shahzad, “Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 6, pp. 1149–1156, 2005.
4 Y. J. Cho and X. Qin, “Viscosity approximation methods for a family ofm-accretive mappings in reflexive Banach spaces,”Positivity, vol. 12, no. 3, pp. 483–494, 2008.
5 Y. J. Cho, S. M. Kang, and X. Qin, “Convergence theorems of fixed points for a finite family of nonexpansive mappings in Banach spaces,”Fixed Point Theory and Applications, vol. 2008, Article ID 856145, 7 pages, 2008.
6 Y. J. Cho, X. Qin, M. Shang, and Y. Su, “Generalized nonlinear variational inclusions involvingA, η -monotone mappings in Hilbert spaces,”Fixed Point Theory and Applications, vol. 2007, Article ID 29653, 6 pages, 2007.
7 R. Chen, Y. Song, and H. Zhou, “Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings,”Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 701–709, 2006.
8 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.
9 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”Bulletin of the American Mathematical Society, vol. 73, no. 4, pp. 591–597, 1967.
10 X. Qin, Y. Su, and M. Shang, “On the convergence of strictly pseudo-contractive mappings in Banach spaces,”Journal of Prime Research in Mathematics, vol. 3, pp. 154–161, 2007.
11 X. Qin, Y. Su, and M. Shang, “On the convergence of implicit iteration process for a finite family of k-strictly asymptotically pseudocontractive mappings,”Kochi Journal of Mathematics, vol. 3, pp. 67–76, 2008.
12 X. Qin and Y. Su, “Approximation of a zero point of accretive operator in Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 415–424, 2007.
13 X. Qin and M. A. Noor, “General Wiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces,”Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 716–722, 2008.
14 Y. Su and X. Qin, “Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups,”Fixed Point Theory and Applications, vol. 2006, Article ID 96215, 11 pages, 2006.
15 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”
Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.
16 K.-K. Tan and H.-K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,”Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993.
17 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.
18 H. Zhou, “Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 10, pp. 2977–2983, 2008.
19 H. Zhou, “Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces,”
