Volume 2011, Article ID 812813,11pages doi:10.1155/2011/812813
Research Article
Asymptotically Pseudocontractions,
Banach Operator Pairs and Best Simultaneous Approximations
N. Hussain
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to N. Hussain,[email protected] Received 3 December 2010; Accepted 12 January 2011
Academic Editor: Mohamed Amine Khamsi
Copyrightq2011 N. Hussain. 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 existence of common fixed points is established for the mappings whereT is asymptotically f-pseudo-contraction on a nonempty subset of a Banach space. As applications, the invariant best simultaneous approximation and strong convergence results are proved. Presented results are generalizations of very recent fixed point and approximation theorems of Khan and Akbar 2009, Chen and Li2007, Pathak and Hussain2008, and several others.
1. Introduction and Preliminaries
We first review needed definitions. LetMbe a subset of a normed spaceX, · . The set PMu {x∈M:x−udistu, M}is called the set of best approximants tou∈Xout of M, where distu, M inf{y−u:y∈M}. Suppose thatAandGare bounded subsets of X. Then, we write
rGA inf
g∈Gsup
a∈Aa−g,
centGA
g0∈G: sup
a∈Aa−g0rGA
.
1.1
The numberrGAis called the Chebyshev radius ofAw.r.t.G, and an elementy0 ∈centGA is called a best simultaneous approximation ofAw.r.t.G. IfA{u}, thenrGA distu, Gand centGAis the set of all best approximations,PGu, ofufromG. We also refer the reader to Milman1, and Vijayraju2for further details. We denote byÆ and clM wclM,
the set of positive integers and the closureweak closureof a setMinX, respectively. Let f, T:M → Mbe mappings. The set of fixed points ofTis denoted byFT. A pointx∈M is a coincidence pointcommon fixed pointoffandT iffx Tx xfx Tx. The pair {f, T}is called
1commuting3ifTfxfTxfor allx∈M,
2compatiblesee3,4if limnTfxn−fTxn0 whenever{xn}is a sequence such that limnTxnlimnfxntfor sometinM,
3weakly compatible if they commute at their coincidence points; that is, iffTx Tfx wheneverfxTx,
4Banach operator pair, if the set Ff is T-invariant, namely TFf ⊆ Ff.
Obviously, commuting pairT, fis a Banach operator pair but converse is not true in general, see 5,6. IfT, fis a Banach operator pair, thenf, Tneed not be a Banach operator pairsee, e.g.,5,7,8.
The setMis calledq-starshaped withq∈M, if the segmentq, x {1−kqkx: 0≤ k ≤1}joiningqtoxis contained inMfor allx∈M. The mapf defined on aq-starshaped setMis called affine if
f
1−kqkx
1−kfqkfx, ∀x∈M. 1.2
Suppose thatMisq-starshaped withq∈Ffand is bothT- andf-invariant. Then,Tandf are called,
5R-subweakly commuting onMsee9if for allx∈ M, there exists a real number R >0 such thatfTx−Tfx ≤Rdistfx,q, Tx,
6uniformlyR-subweakly commuting onM\ {q}see10if there exists a real number R >0 such thatfTnx−Tnfx ≤Rdistfx,q, Tnx, for allx∈M\ {q}andn∈Æ. The mapT :M → Xis said to be demiclosed at 0 if, for every sequence{xn}inM converging weakly toxand{Txn}converges to 0∈X, then 0Tx.
The classical Banach contraction principle has numerous generalizations, extensions and applications. While considering Lipschitzian mappings, a natural question arises whether it is possible to weaken contraction assumption a little bit in Banach contraction principle and still obtain the existence of a fixed point. In this direction the work of Edelstein 11, Jungck3, Park12–18and Suzuki19is worth to mention.
Schu20introduced the concept of asymptotically pseudocontraction and proved the existence and convergence of fixed points for this class of mapssee also21. Recently, Chen and Li5 introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Hussain 6, ´Ciri´c et al. 7, Khan and Akbar 22,23and Pathak and Hussain8. More recently, Zhou24established a demiclosedness principle for a uniformly L-Lipschitzian asymptotically pseudocontraction map and as an application obtained a fixed point result for asymptotically pseudocontraction in the setup of a Hilbert space. In this paper, we are able to join the concepts of uniformlyf-Lipschitzian, asymptoticallyf-pseudocontraction and Banach operator pair to get the result of Zhou24in the setting of a Banach space. As a consequence, the common fixed point and approximation results of Al-Thagafi25, Beg et al.10, Chidume et al.26, Chen and Li5, Cho et al.27, Khan and Akbar22,23, Pathak and Hussain8, Schu28and Vijayraju2are extended to the class of asymptoticallyf-pseudocontraction maps.
2. Main Results
LetX be a real Banach space andMbe a subset ofX. Letf, g T : M → Mbe mappings.
ThenTis called
aanf, g-contraction if there exists 0≤k < 1 such thatTx−Ty ≤kfx−gyfor anyx, y∈M; ifk1, thenTis calledf-nonexpansive,
basymptoticallyf, g-nonexpansive2if there exists a sequence{kn}of real numbers withkn≥1 and limn→ ∞kn1 such that
Tnx−Tny≤knfx−gy 2.1 for all x, y ∈ M and for eachn ∈ Æ; ifg id, then T is called f-asymptotically nonexpansive map,
cpseudocontraction if and only if for eachx, y ∈ M, there existsjx−y ∈Jx−y such that
Tx−Ty, j x−y
≤x−y2, 2.2
whereJ:X → 2X∗is the normalized duality mapping defined by Ju j∈X∗:
u, j
u2, ju
; 2.3
dstrongly pseudocontraction if and only if for eachx, y∈M, there existsk∈0,1and jx−y∈Jx−ysuch that
Tx−Ty, j x−y
≤kx−y2; 2.4 easymptoticallyf, g-pseudocontractive if and only if for eachn ∈ Æ andx, y ∈ M, there existsjx−y∈Jx−yand a constantkn≥1 with limn→ ∞kn1 such that
Tnx−Tny, j x−y
≤knfx−gy2. 2.5 Ifgid in2.5, thenTis called asymptoticallyf-pseudocontractive20,24,27, funiformlyf, g-Lipschitzian if there exists someL >0 such that
Tnx−Tny≤Lfx−gy, 2.6 for allx, y∈Mand for eachn∈Æ; ifg id, thenTis called uniformlyf-Lipschitzian 20,24,29.
The mapTis called uniformly asymptotically regular2,10onM, if for eachη >0, there existsNη Nsuch thatTnx−Tn1x< ηfor alln≥Nand allx∈M.
The class of asymptotically pseudocontraction contains properly the class of asymp- totically nonexpansive mappings and every asymptotically nonexpansive mapping is a uniformlyL-Lipschitzian2,24. For further details, we refer to21,24,27,29,30.
In 1974, Deimling30proved the following fixed point theorem.
Theorem D. LetT be self-map of a closed convex subsetKof a real Banach spaceX. Assume thatT is continuous strongly pseudocontractive mapping. Then,Thas a unique fixed point.
The following result extends and improves Theorem 3.4 of Beg et al.10, Theorem 2.10 in22, Theorems 2.2 of25and Theorem 4 in31.
Theorem 2.1. Let f, T be self-maps of a subset Mof a real Banach space X. Assume thatFf is closed (resp., weakly closed) and convex, T is uniformly f-Lipschitzian and asymptotically f- pseudocontractive which is also uniformly asymptotically regular on M. If clTM is compact (resp., wclTM is weakly compact and id−T is demiclosed at 0) and TFf ⊆ Ff, then FT∩Ff/∅.
Proof. For eachn≥1, define a self-mapTnonFfby Tnx
1−μn
qμnTnx, 2.7
where μn λn/kn and {λn} is a sequence of numbers in 0,1 such that limn→ ∞λn 1 andq ∈ Ff. SinceTnFf ⊂ FfandFfis convex withq ∈ Ff, it follows thatTn mapsFfintoFf. AsFfis convex and clTFf ⊆ Ff resp.wclTFf ⊆ Ff, so clTnFf ⊆ Ff resp. wclTnFf ⊆ Fffor each n ≥ 1. Since Tn is a strongly pseudocontractive onFf, by Theorem D, for eachn ≥ 1, there existsxn ∈ Ffsuch that xnfxnTnxn. AsTFfis bounded, soxn−Tnxn 1−μnTnxn−q → 0 asn → ∞.
Now,
xn−Txnxn−TnxnTnxn−Tn1xnTn1xn−Txn
≤ xn−TnxnTnxn−Tn1xnLfTnxn−fxn. 2.8 Since for eachn≥ 1,TnFf ⊆Ffandxn ∈Ff, thereforeTnxn ∈Ff. ThusfTnxn Tnxn. AlsoTis uniformly asymptotically regular, we have from2.8
xn−Txn ≤ xn−TnxnTnxn−Tn1xnLTnxn−xn −→0, 2.9 as n → ∞. Thus xn −Txn → 0 as n → ∞. As clTM is compact, so there exists a subsequence{Txm}of{Txn}such thatTxm → z ∈ clTMasm → ∞. Since{Txm}is a sequence inTFfand clTFf⊆Ff, thereforez∈Ff. Moreover,
Txm−Tz ≤Lfxm−fzLxm−z ≤Lxm−TxmLTxm−z. 2.10
Taking the limit asm → ∞, we getzTz. Thus,M∩FT∩Ff/∅proves the first case.
Since a weakly closed set is closed, by Theorem D, for eachn ≥ 1, there exists xn ∈ Ffsuch that xn fxn Tnxn. The weak compactness ofwclTM implies that there is a subsequence{Txm}of{Txn}converging weakly toy ∈ wclTMasm → ∞. Since {Txm}is a sequence inTFfandwclTFf ⊆ Ff, soy ∈ Ff. Moreover, we have, xm−Txm → 0 asm → ∞. If id−Tis demiclosed at 0, thenyTy. Thus,M∩FT∩Ff/∅.
Remark 2.2. By comparing Theorem 3.4 of Beg et al.10with the first case ofTheorem 2.1, their assumptions “M is closed and q-starshaped, fM M, TM\ {q} ⊂ fM\ {q}, f, T are continuous, f is linear, q ∈ Ff, clTM\ {q} is compact,T is asymptotically f-nonexpansive andT and f are uniformlyR-subweakly commuting on M” are replaced with “Mis nonempty set,Ffis closed, convex,TFf ⊆ Ff, clTMis compact,T is uniformlyf-Lipschitzian and asymptoticallyf-pseudocontractive”.
IfM is weakly closed andf is weakly continuous, thenFfis weakly closed and hence closed, thus we obtain the following.
Corollary 2.3. Letf, Tbe self-maps of a weakly closed subsetMof a Banach spaceX. Assume thatf is weakly continuous,Ffis nonempty and convex,Tis uniformlyf-Lipschitzian and asymptotically f-pseudocontractive which is also uniformly asymptotically regular onM. If clTMis compact (resp.wclTMis weakly compact and id−T is demiclosed at 0) andT, fis a Banach operator pair, thenFT∩Ff/∅.
A mappingfonMis called pointwise asymptotically nonexpansive32,33if there exists a sequence{αn}of functions such that
fnx−fny≤αnxx−y 2.11 for allx, y∈Mand for eachn∈Æ whereαn → 1 pointwise onM.
An asymptotically nonexpansive mapping is pointwise asymptotically nonexpansive.
A pointwise asymptotically nonexpansive mapfdefined on a closed bounded convex subset of a uniformly convex Banach space has a fixed point andFfis closed and convex32,33.
Thus we obtain the following.
Corollary 2.4. Letfbe a pointwise asymptotically nonexpansive self-map of a closed bounded convex subset M of a uniformly convex Banach space X. Assume that T is a self-map of M which is uniformlyf-Lipschitzian, asymptoticallyf-pseudocontractive and uniformly asymptotically regular.
If clTM is compact (resp. wclTM is weakly compact and id−T is demiclosed at 0) and TFf⊆Ff, thenFT∩Ff/∅.
Corollary 2.5see24, Theorem 3.3. LetTbe self-map of a closed bounded and convex subsetMof a real Hilbert spaceX. Assume thatTis uniformly Lipschitzian and asymptotically pseudocontractive which is also uniformly asymptotically regular onM. Then,FT/∅.
Corollary 2.6. Let X be a Banach space and T and f be self-maps ofX. If u ∈ X,D ⊆ PMu, D0:D∩Ffis closed (resp. weakly closed) and convex, clTDis compact (resp.wclTDis weakly compact and id−T is demiclosed at 0), T is uniformly f-Lipschitzian and asymptotically f-pseudocontractive which is also uniformly asymptotically regular on D, and TD0 ⊆ D0, thenPMu∩FT∩Ff/∅.
Remark 2.7. Corollary 2.6extends Theorems 4.1 and 4.2 of Chen and Li5to a more general class of asymptoticallyf-pseudocontractions.
Theorem 2.1 can be extended to uniformly f, g-Lipschitzian and asymptotically f, g-pseudocontractive map which extends Theorem 2.10 of 22to asymptoticallyf, g- pseudocontractions.
Theorem 2.8. Letf, g, Tbe self-maps of a subsetMof a Banach spaceX. Assume thatFf∩Fg is closed (resp. weakly closed) and convex, T is uniformly f, g-Lipschitzian and asymptotically f, g-pseudocontractive which is also uniformly asymptotically regular on M. If clTM is compact (resp.wclTMis weakly compact and id−T is demiclosed at 0) andTFf∩Fg⊆ Ff∩Fg, thenFT∩Ff∩Fg/∅.
Proof. For eachn≥1, define a self-mapTnonFf∩Fgby Tnx
1−μn
qμnTnx, 2.12
whereμn λn/knand{λn}is a sequence of numbers in0,1such that limn→ ∞λn 1 and q ∈ Ff∩Fg. Since TnFf∩Fg ⊂ Ff∩FgandFf∩Fgis convex withq ∈ Ff∩Fg, it follows thatTnmapsFf∩FgintoFf∩Fg. AsFf∩Fgis convex and clTFf∩Fg⊆Ff∩Fg resp.wclTFf∩Fg⊆Ff∩Fg, so clTnFf∩Fg⊆ Ff∩Fg resp.wclTnFf∩Fg ⊆ Ff∩Fgfor eachn ≥ 1. Further, sinceTn is a strongly pseudocontractive on Ff∩Fg, by Theorem D, for each n ≥ 1, there exists xn ∈ Ff∩Fgsuch thatxn fxn gxn Tnxn. Rest of the proof is similar to that of Theorem 2.1.
Corollary 2.9. Letf, g, Tbe self-maps of a subsetMof a Banach spaceX. Assume thatFf∩Fg is closed (resp. weakly closed) and convex, T is uniformly f, g-Lipschitzian and asymptotically f, g-pseudocontractive which is also uniformly asymptotically regular on M. If clTM is compact (resp.wclTMis weakly compact and id−Tis demiclosed at 0) andT, fandT, gare Banach operator pairs, thenFT∩Ff∩Fg/∅.
Corollary 2.10. LetX be a Banach space andT,f, andg be self-maps ofX. If y1, y2 ∈ X,D ⊆ centK{y1, y2}, where centKAis the set of best simultaneous approximations ofAw.r.tK. Assume thatD0 : D∩Ff∩Fgis closed (resp. weakly closed) and convex, clTDis compact (resp.
wclTDis weakly compact and id−Tis demiclosed at 0),T is uniformlyf, g-Lipschitzian and asymptoticallyf, g-pseudocontractive which is also uniformly asymptotically regular on D, and TD0⊆D0, then centK{y1, y2} ∩FT∩Ff∩Fg/∅.
Remark 2.11. 1 Theorem 2.2 and 2.7 of Khan and Akbar 23 are particular cases of Corollary 2.10.
2 By comparing Theorem 2.2 of Khan and Akbar 23 with the first case of Corollary 2.10, their assumptions “centK{y1, y2} is nonempty, compact, starshaped with respect to an elementq ∈Ff∩Fg, centK{y1, y2}is invariant underT,f andg,T, f andT, gare Banach operator pairs on centK{y1, y2},FfandFgareq-starshaped with q ∈ Ff∩Fg,f and g are continuous and T is asymptotically f, g-nonexpansive on D,” are replaced with “D ⊆ centK{y1, y2},D0 : D∩Ff∩Fg is closed and convex, TD0 ⊆ D0, clTDis compact andT is uniformlyf, g-Lipschitzian and asymptotically f, g-pseudocontractive onD.”
3 By comparing Theorem 2.7 of Khan and Akbar 23 with the second case of Corollary 2.10, their assumptions “centK{y1, y2}is nonempty, weakly compact, starshaped with respect to an elementq ∈ Ff∩Fg, centK{y1, y2}is invariant under T,f andg, T, fandT, gare Banach operator pairs on centK{y1, y2},FfandFgareq-starshaped with q ∈ Ff∩Fg, f and g are continuous under weak and strong topologies, f −T is demiclosed at 0 and T is asymptotically f, g-nonexpansive on D,” are replaced with
“D ⊆ centK{y1, y2}, D0 : D ∩Ff∩Fgis weakly closed and convex, TD0 ⊆ D0, wclTD is weakly compact and id − T is demiclosed at 0 and T is uniformly f, g- Lipschitzian and asymptoticallyf, g-pseudocontractive onD.”
We denote by0the class of closed convex subsets ofXcontaining 0. ForM∈0, we defineMu{x∈M:x ≤2u}. It is clear thatPMu⊂Mu∈0see9,25.
Theorem 2.12. Letf, g, T be self-maps of a Banach spaceX. Ifu ∈ X and M ∈ 0 such that TMu⊆M, clTMuis compact (resp.wclTMuis weakly compact) andTx−u ≤ x−u for allx∈Mu, thenPMuis nonempty, closed and convex withTPMu⊆PMu. If, in addition, D⊆PMu,D0:D∩Ff∩Fgis closed (resp. weakly closed) and convex, clTDis compact (resp.wclTDis weakly compact and id−Tis demiclosed at 0),Tis uniformlyf, g-Lipschitzian and asymptoticallyf, g-pseudocontractive which is also uniformly asymptotically regular onD, and TD0⊆D0, thenPMu∩FT∩Ff∩Fg/∅.
Proof. We may assume thatu /∈M. Ifx∈M\Mu, thenx>2u. Note that
x−u ≥ x − u>u ≥distu, M. 2.13 Thus, distu, Mu distu, M≤ u. If clTMuis compact, then by the continuity of norm, we getz−udistu,clTMufor somez∈clTMu.
If we assume thatwclTMuis weakly compact, using Lemma 5.5 in34, page 192, we can show the existence of az∈wclTMusuch that distu, wclTMu z−u.
Thus, in both cases, we have
distu, Mu≤distu,clTMu≤distu, TMu≤ Tx−u ≤ x−u, 2.14 for allx∈Mu. Hencez−udistu, Mand soPMuis nonempty, closed and convex with TPMu⊆PMu. The compactness of clTMu resp. weak compactness ofwclTMu implies that clTD is compact resp. wclTD is weakly compact. The result now follows fromTheorem 2.8.
Remark 2.13. Theorem 2.12 extends Theorems 4.1 and 4.2 in 25, Theorem 8 in 31, and Theorem 2.15 in22.
Definition 2.14. LetMbe a nonempty closed subset of a Banach spaceX,I, T :M → Mbe mappings andC{x∈M:hx minz∈Mhz}. ThenIandT are said to satisfy property S 10,27if the following holds: for any bounded sequence{xn}inM, limn→ ∞xn−Txn0 impliesC∩FI∩FT/∅.
The normal structure coefficient NX of a Banach space X is defined 10, 26 by NX inf{diamM/rCM : Mis nonempty bounded convex subset ofX with
diamM>0}, whererCM infx∈M{supy∈Mx−y}is the Chebyshev radius ofMrelative to itself and diamM supx,y∈Mx−yis diameter ofM. The spaceXis said to have the uniform normal structure ifNX > 1. A Banach limit LIM is a bounded linear functional onl∞such that lim infn→ ∞tn≤LIMtn ≤lim supn→ ∞tnand LIMtnLIMtn1for all bounded sequences{tn}inl∞. Let{xn}be bounded sequence inX. Then we can define the real-valued continuous convex functionfonXbyfz LIMxn−z2for allz∈X.
The following lemmas are well known.
Lemma 2.15see10,27. LetXbe a Banach space with uniformly Gateaux differentiable norm andu∈X. Let{xn}be bounded sequence inX. Thenfu infz∈Xfzif and only if LIMz, Jxn− u 0 for allz ∈X, whereJ :X → X∗is the normalized duality mapping and·,·denotes the generalized duality pairing.
Lemma 2.16see10,26. LetMbe a convex subset of a smooth Banach spaceX,Dbe a nonempty subset ofMandP be a retraction fromMontoD. ThenP is sunny and nonexpansive if and only if x−Px, Jz−Px ≤0 for allx∈Mandz∈D.
Now, we are ready to prove strong convergence to nearest common fixed points of asymptoticallyf-pseudocontraction mappings.
Theorem 2.17. Let M be a subset of a reflexive real Banach space X with uniformly Gateaux differentiable norm. Let f and T be self-maps on M such that Ff is closed and convex, T is continuous, uniformly asymptotically regular, uniformly f-Lipschitzian and asymptotically f- pseudocontractive with a sequence {kn}. Let {λn} be sequence of real numbers in0,1such that limn→ ∞λn1 and limn→ ∞kn−1/kn−λn 0. IfTFf⊂Ff, then we have the following.
AFor eachn≥1, there is exactly onexninMsuch that fxnxn
1−μn
qμnTnxn 2.15
BIf {xn} is bounded and f and T satisfy property S, then {xn} converges strongly to Pq∈FT∩Ff, wherePis the sunny nonexpansive retraction fromMontoFT.
Proof. PartAfollows from the proof ofTheorem 2.1.
BAs inTheorem 2.1, we get limn→ ∞xn−Txn0. Since{xn}is bounded, we can define a functionh:M → Rbyhz LIMxn−z2for allz∈M. Sincehis continuous and convex,hz → ∞asz → ∞andXis reflexive,hz0 minz∈Mhzfor somez0∈M.
Clearly, the setC{x∈M:hx minz∈Mhz}is nonempty. Since{xn}is bounded andf andTsatisfy propertyS, it follows thatC∩Ff∩FT/∅. Suppose thatv∈C∩Ff∩FT, then byLemma 2.15, we have
LIMx−v, Jxn−v ≤0 ∀x∈M. 2.16
In particular, we have
LIM
q−v, Jxn−v
≤0. 2.17
From2.8, we have
xn−Tnxn 1−μn
q−Tnxn 1−μn
μn
q−xn
. 2.18
Now, for anyv∈C∩Ff∩FT, we have
xn−Tnxn, Jxn−vxn−vTnv−Tnxn, Jxn−v
≥ −kn−1xn−v2
≥ −kn−1K2
2.19
for someK >0. It follows from2.18that xn−q, Jxn−v
≤ kn−1
kn−λnK2. 2.20
Hence we have
LIM
xn−q, Jxn−v
≤0. 2.21
This together with2.17implies that LIMxn−v, Jxn−vLIMxn−v20.
Thus there is a subsequence{xm}of{xn}which converges strongly tov. Suppose that there is another subsequence{xj} of{xn} which converges strongly to ysay. SinceT is continuous and limn→ ∞xn−Txn0,yis a fixed point ofT. It follows from2.21that
v−q, J v−y
≤0,
y−q, J y−v
≤0. 2.22
Adding these two inequalities, we get v−y, J
v−y
v−y2≤0 and thusvy. 2.23 Consequently, {xn} converges strongly to v ∈ Ff ∩FT. We can define now a mappingPfromMontoFTby limn→ ∞xn Pq. From2.21, we haveq−Pq, Jv−Pq ≤0 for allq∈Mandv∈FT. Thus byLemma 2.16,Pis the sunny nonexpansive retraction on M. Notice thatxn fxn and limn→ ∞xn Pq, so by the same argument as in the proof of Theorem 2.1we obtain,Pq∈Ff.
Remark 2.18. Theorem 2.17 extends Theorem 1 in 27. Notice that the conditions of the continuity and linearity off are not needed in Theorem 3.6 of Beg et al. 10; moreover, we have obtained the conclusion for more general class of uniformly f-Lipschitzian and asymptoticallyf-pseudocontractive mapTwithout any type of commutativity offandT.
Corollary 2.19see26, Theorem 3.1. LetMbe a closed convex bounded subset of a real Banach spaceX with uniformly Gˆateaux differentiable norm possessing uniform normal structure. Let T : M → Mbe an asymptotically nonexpansive mapping with a sequence{kn}. Letu∈Mbe fixed,{λn} be sequence of real numbers in0,1such that limn→ ∞λn1 and limn→ ∞kn−1/kn−λn 0.
Then,
Afor eachn≥1, there is uniquexninMsuch that xn
1−μn
uμnTnxn, 2.24
Bif limn→ ∞xn−Txn0, then{xn}converges strongly to a fixed point ofT.
Remark 2.20. 1Theorem 2.17improves and extends the results of Beg et al.10, Cho et al.
27, and Schu20,28to more general class of Banach operators.
2 It would be interesting to prove similar results in Modular Function Spacescf.
29.
3LetX Ê with the usual norm andM 0,1. A mappingT is defined byTx x, forx ∈ 0,1/2 andTx 0, forx ∈ 1/2,1and fx xon M. Clearly,T is not f- nonexpansive21 e.g.,T3/4−T1/21/2 andf3/4−f1/21/4. But,T is a f-pseudocontractive mapping.
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