Volume 2010, Article ID 754320,11pages doi:10.1155/2010/754320
Research Article
Strong Convergence Theorems of Common Fixed Points for a Family of Quasi- φ -Nonexpansive Mappings
Xiaolong Qin,
1Yeol Je Cho,
2Sun Young Cho,
3and Shin Min Kang
41Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, South Korea
3Department of Mathematics, Gyeongsang National University, Jinju 660-701, South Korea
4Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, South Korea
Correspondence should be addressed to Shin Min Kang,[email protected] Received 31 August 2009; Accepted 19 November 2009
Academic Editor: Tomonari Suzuki
Copyrightq2010 Xiaolong Qin 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.
We consider a modified Halpern type iterative algorithm for a family of quasi-φ-nonexpansive mappings in the framework of Banach spaces. Strong convergence theorems of the purposed iterative algorithms are established.
1. Introduction
LetEbe a Banach space,Ca nonempty closed and convex subset ofE, andT : C → Ca nonlinear mapping. Recall thatTis nonexpansive if
Tx−Ty≤x−y, ∀x, y∈C. 1.1 A pointx∈Cis a fixed point ofT providedTxx. Denote byFTthe set of fixed points of T, that is,FT {x∈C:Txx}.
One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping; see1,2. More precisely, taket∈0,1and define a contractionTt:C → Cby
Ttxtu 1−tTx, ∀x∈C, 1.2
whereu∈Cis a fixed element. Banach Contraction Mapping Principle guarantees thatTthas a unique fixed pointxtinC. It is unclear, in general, what the behavior ofxtis ast → 0 even ifThas a fixed point. However, in the case ofThaving a fixed point, Browder1proved the following well-known strong convergence theorem.
Theorem B. LetCbe a bounded closed convex subset of a Hilbert spaceH andT a nonexpansive mapping onC. Fixu ∈ Cand define zt ∈ Caszt tu 1−tTztfor anyt ∈ 0,1. Then{zt} converges strongly to an element ofFTnearest tou.
Motivated by Theorem B, Halpern3considered the following explicit iteration:
x0∈C, xn1αnu 1−αnTxn, ∀n≥0, 1.3
and obtained the following theorem.
Theorem H. LetCbe a bounded closed convex subset of a Hilbert spaceHand T a nonexpansive mapping onC. Define a real sequence{αn}in0,1byαn n−θ, 0< θ <1. Then the sequence{xn} defined by1.3converges strongly to the element ofFTnearest tou.
In4, Lions improved the result of Halpern 3, still in Hilbert spaces, by proving the strong convergence of{xn}to a fixed point ofT provided that the control sequence{αn} satisfies the following conditions:
C1limn→ ∞αn0;
C2∞
n1αn∞;
C3limn→ ∞αn1−αn/α2n1 0.
It was observed that both the Halpern’s and Lion’s conditions on the real sequence {αn}excluded the canonical choice{αn} 1/n1. This was overcome by Wittmann5, who proved, still in Hilbert spaces, the strong convergence of{xn}to a fixed point ofTif{αn} satisfies the following conditions:
C1limn→ ∞αn0;
C2∞
n1αn∞;
C4∞
n1|αn1−αn|<∞.
In 6, Shioji and Takahashi extended Wittmann’s results to the setting of Banach spaces under the assumptionsC1,C2, andC4imposed on the control sequences{αn}. In 7, Xu remarked that the conditionsC1andC2are necessary for the strong convergence of the iterative sequence defined in 1.3 for all nonexpansive self-mappings. It is well known that the iterative algorithm1.3is widely believed to have slow convergence because
the restriction of condition C2. Thus, to improve the rate of convergence of the iterative process1.3, one cannot rely only on the process itself.
Recently, hybrid projection algorithms have been studied for the fixed point problems of nonlinear mappings by many authors; see, for example,8–24. In 2006, Martinez-Yanes and Xu 10 proposed the following modification of the Halpern iteration for a single nonexpansive mappingT in a Hilbert space. To be more precise, they proved the following theorem.
Theorem MYX. LetH be a real Hilbert space,Ca closed convex subset ofH, andT : C → Ca nonexpansive mapping such thatFT/∅. Assume that{αn} ⊂0,1is such that limn→ ∞αn 0.
Then the sequence{xn}defined by
x0∈C chosen arbitrarily, yn αnx0 1−αnTxn, Cn
z∈C:yn−z2≤ xn−z2αn
x022xn−x0, z , Qn{z∈C:x0−xn, xn−z ≥0},
xn1PCn∩Qnx0, ∀n≥0,
1.4
converges strongly toPFTx0.
Very recently, Qin and Su17improved the result of Martinez-Yanes and Xu10from Hilbert spaces to Banach spaces. To be more precise, they proved the following theorem.
Theorem QS. Let E be a uniformly convex and uniformly smooth Banach space, C a nonempty closed convex subset ofE, andT :C → Ca relatively nonexpansive mapping. Assume that{αn}is a sequence in0,1such that limn→ ∞αn0. Define a sequence{xn}inCby the following algorithm:
x0∈C chosen arbitrarily, ynJ−1αnJx0 1−αnJTxn, Cn
v∈C:φ v, yn
≤αnφv, x0 1−αnφv, xn , Qn{v∈C:Jx0−Jxn, xn−v ≥0},
xn1 ΠCn∩Qnx0, ∀n≥0,
1.5
where J is the single-valued duality mapping on E. IfFTis nonempty, then {xn}converges to ΠFTx0.
In this paper, motivated by Kimura and Takahashi8, Martinez-Yanes and Xu10, Qin and Su17, and Qin et al.19, we consider a hybrid projection algorithm to modify the iterative process1.3to have strong convergence under conditionC1only for a family of closed quasi-φ-nonexpansive mappings.
2. Preliminaries
Let E be a Banach space with the dual space E∗. We denote by J the normalized duality mapping fromEto 2E∗defined by
Jx
f∗∈E∗: x, f∗
x2f∗2
, ∀x∈E, 2.1
where ·,· denotes the generalized duality pairing. It is well known that, ifE∗ is strictly convex, thenJis single-valued and, ifE∗is uniformly convex, thenJis uniformly continuous on bounded subsets ofE.
We know that, if C is a nonempty closed convex subset of a Hilbert space H and PC:H → Cis the metric projection ofHontoC, thenPCis nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber25recently introduced a generalized projection operator ΠC in a Banach spaceE, which is an analogue of the metric projection in Hilbert spaces.
A Banach spaceEis said to be strictly convex ifxy/2 <1 for allx, y ∈Ewith xy1 andx /y. The spaceEis said to be uniformly convex if limn→ ∞xn−yn0 for any two sequences{xn}and{yn}inEsuch thatxnyn1 and limn→ ∞xnyn/21.
LetU{x∈E:x1}be the unit sphere ofE. Then the spaceEis said to be smooth if
limt→0
xty− x
t 2.2
exists for eachx, y∈U.It is also said to be uniformly smooth if the limit is attained uniformly forx, y∈E. It is well known that, ifEis uniformly smooth, thenJis uniformly norm-to-norm continuous on each bounded subset ofE.
In a smooth Banach spaceE, we consider the functional defined by φ x, y
x2−2 x, Jy
y2, ∀x, y∈E. 2.3 Observe that, in a Hilbert spaceH,2.3reduces toφx, y x−y2 for allx, y ∈ H.The generalized projectionΠC : E → Cis a mapping that assigns to an arbitrary pointx ∈ E the minimum point of the functionalφx, y,that is,ΠCxx,wherexis the solution to the minimization problem:
φx, x min
y∈Cφ y, x
. 2.4
The existence and uniqueness of the operatorΠCfollows from some properties of the functionalφx, yand the strict monotonicity of the mappingJsee, e.g.,25–28. In Hilbert spaces,ΠCPC.It is obvious from the definition of the functionφthat
y− x2
≤φ y, x
≤ yx2
, ∀x, y∈E. 2.5
Remark 2.1. IfEis a reflexive, strictly convex, and smooth Banach space, then, for anyx, y∈E, φx, y 0 if and only ifxy. In fact, it is sufficient to show that, ifφx, y 0, thenxy.
From2.5, we havexy. This impliesx, Jyx2 Jy2.From the definition ofJ, one hasJxJy. Therefore, we havexysee27,29for more details.
LetCbe a nonempty closed and convex subset ofEandTa mapping fromCinto itself.
A pointp∈Cis said to be an asymptotic fixed point ofT30ifCcontains a sequence{xn} which converges weakly topsuch that limn→ ∞xn−Txn 0. The set of asymptotic fixed points ofT will be denoted byFT. A mappingT fromCinto itself is said to be relatively nonexpansive27,31,32ifFT FTandφp, Tx≤φp, xfor allx∈Candp∈FT. The asymptotic behavior of a relatively nonexpansive mapping was studied by some authors 27,31,32.
A mappingT :C → Cis said to beφ-nonexpansive18,19,24ifφTx, Ty≤φx, y for allx, y ∈ C. The mappingT is said to be quasi-φ-nonexpansive18,19,24ifFT/∅ andφp, Tx≤φp, xfor allx∈Candp∈FT.
Remark 2.2. The class of quasi-φ-nonexpansive mappings is more general than the class of relatively nonexpansive mappings, which requires the strong restriction:FT FT.
In order to prove our main results, we need the following lemmas.
Lemma 2.3see28. LetEbe a uniformly convex and smooth Banach space and{xn},{yn}two sequences ofE. Ifφxn, yn → 0 and either{xn}or{yn}is bounded, thenxn−yn → 0.
Lemma 2.4see25,28. LetCbe a nonempty closed convex subset of a smooth Banach spaceE andx∈E. Thenx0 ΠCx∈Cif and only if
x0−y, Jx−Jx0
≥0, ∀y∈C. 2.6
Lemma 2.5 see 25, 28. LetE be a reflexive, strictly convex, and smooth Banach space, C a nonempty closed convex subset ofEandx∈E.Then
φ y,ΠCx
φΠCx, x≤φ y, x
, ∀y∈C. 2.7
Lemma 2.6 see7,18. LetEbe a uniformly convex and smooth Banach space,Ca nonempty, closed, and convex subset ofEandTa closed quasi-φ-nonexpansive mapping fromCinto itself. Then FTis a closed and convex subset ofC.
3. Main Results
From now on, we useI to denote an index set. Now, we are in a position to prove our main results.
Theorem 3.1. LetCbe a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach spaceE and {Ti}i∈I : C → C a family of closed quasi-φ-nonexpansive mappings
such thatF
i∈IFTi/∅. Let{αn}be a real sequence in0,1such that limn→ ∞αn 0. Define a sequence{xn}inCin the following manner:
x0∈C chosen arbitrarily, yn,iJ−1αnJx0 1−αnJTixn, Cn,i
z∈C:φ z, yn,i
≤αnφz, x0 1−αnφz, xn , Cn
i∈ICn,i, Q0C,
Qn {z∈Qn−1:xn−z, Jx0−Jxn ≥0}, xn1 ΠCn∩Qnx0, ∀n≥0,
3.1
then the sequence{xn}defined by3.1converges strongly toΠFx0.
Proof. We first show thatCnandQnare closed and convex for eachn≥0. From the definitions ofCnandQn, it is obvious thatCnis closed andQnis closed and convex for eachn≥0. We, therefore, only show thatCnis convex for eachn≥0. Indeed, note that
φ z, yn,i
≤αnφz, x0 1−αnφz, xn 3.2
is equivalent to
2αnz, Jx021−αnz, Jxn −2
z, Jyn,i
≤αnx02 1−αnxn2−yn,i2. 3.3 This shows thatCn,iis closed and convex for eachn≥0 andi∈I.Therefore, we obtain that Cn
i∈ICn,iis convex for eachn≥0.
Next, we show thatF⊂Cnfor alln≥0. For eachw∈Fandi∈I, we have
φ w, yn,i φ
w, J−1αnJx0 1−αnJTixn
w2−2w, αnJx0 1−αnJTixnαnJx0 1−αnJTixn2
≤ w2−2αnw, Jx021−αnw, JTixnαnx02 1−αnTixn2
≤αnφw, x0 1−αnφw, Tixn
≤αnφw, x0 1−αnφw, xn,
3.4
which yields thatw ∈Cn,i for alln ≥ 0 andi∈ I.It follows thatw ∈ Cn
i∈ICn,i. This proves thatF⊂Cnfor alln≥0.
Next, we prove thatF ⊂ Qnfor alln≥ 0.We prove this by induction. Forn 0,we haveF ⊂CQ0.Assume thatF ⊂ Qn−1for somen≥1. Next, we show thatF ⊂Qnfor the samen. Sincexnis the projection ofx0ontoCn−1∩Qn−1,we obtain that
xn−z, Jx0−Jxn ≥0, ∀z∈Cn−1∩Qn−1. 3.5
SinceF ⊂Cn−1∩Qn−1by the induction assumption,3.5holds, in particular, for allw ∈F.
This together with the definition ofQnimplies thatF ⊂Qnfor alln≥0.Noticing thatxn1 ΠCn∩Qnx0 ∈Qnandxn ΠQnx0, one has
φxn, x0≤φxn1, x0, ∀n≥0. 3.6
We, therefore, obtain that{φxn, x0}is nondecreasing. FromLemma 2.5, we see that φxn, x0 φΠCnx0, x0
≤φw, x0−φw, xn
≤φw, x0, ∀w∈F⊂Cn, ∀n≥0.
3.7
This shows that{φxn, x0}is bounded. It follows that the limit of{φxn, x0}exists. By the construction ofQn, we see thatQm⊂Qnandxm ΠQmx0 ∈Qnfor any positive integerm≥n.
Notice that
φxm, xn φxm,ΠCnx0
≤φxm, x0−φΠCnx0, x0 φxm, x0−φxn, x0.
3.8
Taking the limit asm, n → ∞in3.8, we get thatφxm, xn → 0.FromLemma 2.3, one has xm−xn → 0 asm, n → ∞.It follows that{xn}is a Cauchy sequence inC. SinceEis a Banach space andCis closed and convex, we can assume thatxn → q∈Casn → ∞.
Finally, we show thatq ΠFx0.To end this, we first showq∈F. By takingmn1 in3.8, we have
φxn1, xn−→0 n−→ ∞. 3.9
FromLemma 2.3, we arrive at
xn1−xn−→0 n−→ ∞. 3.10
Noticing thatxn1 ∈Cn, we obtain φ xn1, yn,i
≤αnφxn1, x0 1−αnφxn1, xn. 3.11
It follows from the assumption on{αn}and3.9that limn→ ∞φxn1, yn,i 0 for eachi∈I.
FromLemma 2.3, we obtain
nlim→ ∞xn1−yn,i0, ∀i∈I. 3.12
On the other hand, we haveJyn,i−JTixn αnJx0 −JTixn.By the assumption on{αn}, we see that limn→ ∞Jyn,i−JTixn 0 for eachi∈ I.SinceJ−1 is also uniformly norm-to-norm continuous on bounded sets, we obtain that
nlim→ ∞yn,i−Tixn0. 3.13
On the other hand, we have
xn−Tixn ≤ xn−xn1xn1−yn,iyn,i−Tixn. 3.14
From3.10–3.13, we obtain limn→ ∞Tixn−xn0.From the closedness ofTi, we getq∈F.
Finally, we show thatq ΠFx0.Fromxn ΠCnx0, we see that
xn−w, Jx0−Jxn ≥0, ∀w∈F⊂Cn. 3.15
Taking the limit asn → ∞in3.15, we obtain that
q−w, Jx0−Jq
≥0, ∀w∈F, 3.16
and henceq ΠFx0byLemma 2.4. This completes the proof.
Remark 3.2. Comparing the hybrid projection algorithm3.1inTheorem 3.1with algorithm 1.5in Theorem QS, we remark that the setQnis constructed based on the setQn−1instead ofCfor eachn≥1.We obtain that the sequence generated by the algorithm3.1is a Cauchy sequence. The proof is, therefore, different from the one presented in Qin and Su17.
As a corollary ofTheorem 3.1, for a single quasi-φ-nonexpansive mapping, we have the following result immediately.
Corollary 3.3. LetCbe a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach spaceEandT :C → Ca closed quasi-φ-nonexpansive mappings with a fixed point.
Let{αn}be a real sequence in0,1such that limn→ ∞αn 0. Define a sequence{xn}in Cin the following manner:
x0∈C chosen arbitrarily, ynJ−1αnJx0 1−αnJTxn, Cn
z∈C:φ z, yn
≤αnφz, x0 1−αnφz, xn , Q0C,
Qn {z∈Qn−1:xn−z, Jx0−Jxn ≥0}, xn1 ΠCn∩Qnx0, ∀n≥0,
3.17
then the sequence{xn}converges strongly toΠFx0.
Remark 3.4. Corollary 3.3 mainly improves Theorem 2.2 of Qin and Su17 from the class of relatively nonexpansive mappings to the class of quasi-φ-nonexpansive mappings, which relaxes the strong restriction:FT FT.
In the framework of Hilbert spaces,Theorem 3.1is reduced to the following result.
Corollary 3.5. LetCbe a nonempty closed and convex subset of a Hilbert spaceHand{Ti}i∈I :C → Ca family of closed quasi-nonexpansive mappings such thatF
i∈IFTi/∅. Let{αn}be a real sequence in0,1such that limn→ ∞αn0. Define a sequence{xn}inCin the following manner:
x0∈C chosen arbitrarily, yn,iαnx0 1−αnTixn, Cn,i
z∈C:z−yn,i2≤αnz−x02 1−αnz−xn2 , Cn
i∈ICn,i, Q0C,
Qn{z∈Qn−1:xn−z, x0−xn ≥0}, xn1PCn∩Qnx0, ∀n≥0,
3.18
then the sequence{xn}converges strongly toPFx0.
Remark 3.6. Corollary 3.5includes the corresponding result of Martinez-Yanes and Xu10as a special case. To be more precise,Corollary 3.5improvesTheorem 3.1of Martinez-Yanes and Xu10from a single mapping to a family of mappings and from nonexpansive mappings to quasi-nonexpansive mappings, respectively.
Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean GovernmentKRF-2008-313-C00050.
References
1 F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.
2 S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,”
Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980.
3 B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol.
73, pp. 957–961, 1967.
4 P.-L. Lions, “Approximation de points fixes de contractions,” Comptes Rendus de l’Acad´emie des Sciences de Paris, S´erie. A-B, vol. 284, no. 21, pp. A1357–A1359, 1977.
5 R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol.
58, no. 5, pp. 486–491, 1992.
6 N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp.
3641–3645, 1997.
7 H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109–113, 2002.
8 Y. Kimura and W. Takahashi, “On a hybrid method for a family of relatively nonexpansive mappings in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 357, no. 2, pp. 356–363, 2009.
9 G. Lewicki and G. Marino, “On some algorithms in Banach spaces finding fixed points of nonlinear mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 3964–3972, 2009.
10 C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006.
11 S.-Y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.
12 S.-Y. Matsushita and W. Takahashi, “Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2004, no. 1, pp.
37–47, 2004.
13 K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence by the hybrid method for families of mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp.
112–119, 2009.
14 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–
379, 2003.
15 K. Nakajo, K. Shimoji, and W. Takahashi, “Strong convergence theorems by the hybrid method for families of mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 812–818, 2009.
16 S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2, pp. 103–115, 2007.
17 X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1958–1965, 2007.
18 X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.
19 X. Qin, Y. J. Cho, S. M. Kang, and H. Y. Zhou, “Convergence of a modified Halpern-type iteration algorithm for quasi-φ-nonexpansive mappings,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1051–
1055, 2009.
20 W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.
21 W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 528476, 11 pages, 2008.
22 W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 70, no. 1, pp. 45–57, 2009.
23 L. Wei, Y. J. Cho, and H. Y. Zhou, “A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 95–103, 2009.
24 H. Y. Zhou, G. Gao, and B. Tan, “Convergence theorems of a modified hybrid algorithm for a family of quasi-φ-asymptotically nonexpansive mappings,” Journal of Computational and Applied Mathematics, in press.
25 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G.
Kartsatos, Ed., vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.
26 Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” PanAmerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994.
27 I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1990.
28 S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
29 W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, Japan, 2000.
30 S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol.
178 of Lecture Notes in Pure and Applied Mathematics, pp. 313–318, Dekker, New York, NY, USA, 1996.
31 D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001.
32 D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508, 2003.
