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Advances in Pure Mathematics, 2011, 1, 33-41 doi:10.4236/apm.2011.13009 Published Online May 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM A Note on Convergence of a Sequence and its Applications to Geometry of Banach Spaces Hemant Kumar Pathak School of Studies in Mathematics, Pandit Ravishankar Shukla University, Raipur, India E-mail: hkpathak05@gmail.com Received January 9, 2011; revised April 22, 2011; accepted April 30, 2011 Abstract The purpose of this note is to point out several obscure places in the results of Ahmed and Zeyada [J. Math. Anal. Appl. 274 (2002) 458-465]. In order to rectify and improve the results of Ahmed and Zeyada, we in- troduce the concepts of locally quasi-nonexpansive, biased quasi-nonexpansive and conditionally biased qu- asi-nonexpansive of a mapping w.r.t. a sequence in metric spaces. In the sequel, we establish some theorems on convergence of a sequence in complete metric spaces. As consequences of our main result, we obtain some results of Ghosh and Debnath [J. Math. Anal. Appl. 207 (1997) 96-103], Kirk [Ann. Univ. Mariae Cu- rie-Sklodowska Sec. A LI.2, 15 (1997) 167-178] and Petryshyn and Williamson [J. Math. Anal. Appl. 43 (1973) 459-497]. Some applications of our main results to geometry of Banach spaces are also discussed. Keywords: Locally Quasi-Nonexpansive, Biased Quasi-Nonexpansive, Conditionally Biased Quasi-Nonexpansive, Drop, Super Drop 1. Introduction In the last four decades of the last century, there have been a multitude of results on fixed points of nonexpan- sive and quasi-nonexpansive mappings in Banach spaces (e.g., [5-7, 9-11]) . Our aim in this note is to point out several obscure places in the results of Ahmed and Zeyada [J. Math. Anal. Appl. 274 (2002) 458-465]. In order to rectify and improve the results of Ahmed and Zeyada, we introduce the concepts of locally qu asi-nonexpansive, biased quasi- nonexpansive and conditionally biased quasi-nonexpan- sive of a mapping w.r.t. a sequence in metric spaces. Let X be a metric space and D a nonempty subset of X . Let T be a mapping of D into X and let F T be the set of all fixed points of T. For a given 0 x D , the sequence of iterate n x is determined by 10 ,1,2,3 n nn xTx Txn (I) Let X be a normed space, 0, 1 and 0,1 , the sequence of iterates n x are defined by 10 , 1,1,2,3 n nn xTx Tx TI Tn (II) ,1,0 , , 1[1 ], 1,2, 3.... n nn xT xT x TITIT n (III) The iteration scheme (I) is called Teoplitz iteration and the iteration scheme (II) was introduced by Mann [12] while the iteration scheme (III) was introduced by Ishikawa [9]. The concept of quasi-nonexpansive mapping was ini- tiated by Tricomi in 1941 for real functions. It was fur- ther studied by Diaz and Metcalf [5] and Doston [6,7] for mappings in Banach spaces. Recently, this concept was given by Kirk [10] in metric spaces as follows: Definition 1.1. The mapping T is said to be quasi- nonexpansive if for each x D and for every pFT, ,,.dT xpd xp A mapping T is conditionally quasi-nonexpansive if it is quasi-nonexpansive whenever FT . We now introduce the following definition : Definition 1.2. The mapping T is said to be locally quasi-nonexpansive at pFT if for each x D , ,,dT xpdxp. Obviously, quasi-nonexpansive locally quasi-nonex- pansive at each pFT but the reverse implication H. K. PATHAK Copyright © 2011 SciRes. APM 34 may not be true. To this end, we observe the following example. Example 1.1. Let 0,1X and 3 0, 4 D be en- dowed with the Euclidean metric d. Define the mapping :TD X by 2 3 2 Tx x for each . x D Then we observe that 2 0, 3 FT , for all x D and 0p TF , we have that 2 3 ,00, 2 dT xpxxdxp, i.e., T is locally quasi-nonexpansive at 0pFT . However, one can easily see that T is not locally quasi- nonexpansive at 2 3 pFT . Indeed, for all 2 0, 3 x and 2 3 pFT we have 2 32 2 ,, 23 3 dT xpxxdxp . Hence we conclude that T is no t quasi-n onexpansive, although it is locally quasi-nonexpansive at 0 p FT . The concept of asymptotic regularity was formally in- troduced by Browder and Petryshyn [3] for mappings in Hilbert spaces. Recently, it was defined by Kirk [11] in metric spaces as follows: Definition 1.3. The mapping T is said to be asymp- totically regular if 1 lim ,0 nn ndTx Tx for each x D. 2. Main Results Let N denote the set of all positive integers and 0N Ahmed and Zeyada [1] introduce-ed the fol- lowing: Definition 2.1. The mapping T is said to be quasi- nonexpansive w.r.t. a sequence n x if for all n and for each pFT, 1,, nn dxpdx p . The following lemma was quoted by Ahmed and Zeyada [1] without proof. Lemma A. If T is quasi-nonexpansive, then T is quasi-nonexpansive w.r.t. a sequence 0 n Tx (respec- tively, 0,0 , nn TxT x ) for each 0 x D. Remark 2.1. We notice that the above lemma is valid if 0 n Tx D for each n and a given 0 x D (or D is T -invarient). So the correct version of Lemma A should be read as follows: Lemma 2.1. If T is quasi-nonexpansive and for a given 0 x D and each n , 0 n Tx D, then T is quasi-nonexpansive w.r.t. a sequence 0 n Tx (re- spectively, 0,0 , nn TxT x ) for each 0 x D. Further, they claimed that the reverse implication in Lemma A may not be true in their Example 2.1. We again notice that there are several obscure places in this example. We now quote Example 2.1 of Ahmed and Zeyada [1] in the following: Example A. Let 0,1X and 4 0, 5 D be en- dowed with the Euclidean metric d We define the mapping :TD X by 2 2Tx x for each x D . For a given 01 4 x D we have 1 21 21 10 0 11 ,00 22 , nn n n dTx p dT xp where 21 14 120 n n TDn N and FT 0, i.e., T is quasi-nonexpansive w.r.t. a sequence 14 n T Furthermore, the map T is quasi-nonexpan- sive w.r.t. a sequence 12 12 n T and 12,12 12 n T. They found that T is neither conditionally quasi-non- expansive nor quasi-nonexpansive, for 3 4 x D and 0, 34,034,0 p Fd dT and D is not closed. Remark 2.2. We notice that the following claims made in Example A were false: 1) :TDX is a mapping. In fact, 32 0, 0,1 25 TD X . 2) FT 0, In fact, 1 0, 2 FT . 3) T is quasi-nonexpansive w.r.t. a sequence n T 14 . 4) T is quasi-nonexpansive w.r.t. a sequence 12 12 n T and 12,12 12 n T. However, (i) can be rectified by taking X as 32 0, 25 or any superset of 32 0, 25 in 0, Even if this correction is made we find that the remaining state- ments 2) - 4) will remain false. Consequently, the claim of Ahmed and Zeyada [1] that the reverse implication in Lemma 2.1 may not be true seems false. We now introduce the following definition . Definition 2.2. The mapping T is said to be locally quasi-nonexpansive at pFT w.r.t. a sequence n x H. K. PATHAK Copyright © 2011 SciRes. APM 35 if for all n , 1,, nn dxpdx p . Obviously, locally quasi-nonexpansiveness at p F T locally quasi-nonexpansiveness at p F T w.r.t. a sequence n x . We now state the followin g le m ma without proof. Lemma 2.2. If T is quasi-nonexpansive w.r.t. a se- quence n x then T is locally quasi-nonexpansive at each pFT w.r.t. the sequence n x . The reverse implication in Lemma 2.2 may not be true as shown in the following example: Example 2.1. Let 0, 1X and 2 0, 3 D be en- dowed with the Euclidean metric d Define the map- ping :TD X by 2 2Tx x for each x D Then we observe that 1 0, 2 FT . For a given 01 4 x D and 0pFT we have that 1 21 21 10 0 11 ,00 22 , nn n n dTx p dT xp * where 21 1 2 1 4 n n TD i.e., T is locally quasi- nonexpansive at 0pFT w.r.t. a sequence 1 4 n T However, one can easily see that T is not locally quasi-nonexpansive at 1 2 pFT w.r.t. the sequence . 1 4 n T Indeed, we ha ve 11 2121 10 0 1111 ,2222 , nn n n dTx p dT xp ** for all n Consequently, T is neither quasi-nonex- pansive nor quasi-nonexpansive w.r.t. the sequence 1 4 n T . We now introduce the following: Definition 2.3. The mapping :TD X is said to be biased quasi-nonexpansive (b.q.n) w.r.t. a sequence n xX if for all n and at each cond p F T, 1,, nn dxpdx p where lico msup l nd: , ,iminf n n n n pF dFT T T x p dx F A mapping T is conditionally biased quasi-nonex- pansive (c.b.q.n) w.r.t. a sequence n x if cond FT . Remark 2.3. We observe that the following implica- tions are obvious: (a) Conditional biased quasi-nonexpansiveness w.r.t. a sequence n x biased quasi-nonexpansiveness w.r.t. a sequence n x but the reverse impl ication may not be true (Indeed, any mapping :TDX for which cond FT is a biased quasi-nonexpansive w.r.t. a sequence n x but not conditionally biased quasi- nonexpansive w.r.t. a sequence n x . However, under certain conditions a biased quasi-nonexpansive map w.r.t. a sequence n x may be a conditional biased quasi- nonexpansive w.r.t. a sequence n x (see Lemma 2.6 below). (b) If T is conditionally biased quasi-nonexpansive w.r.t. a sequence n x and cond TFTF then T is locally quasi-nonexpansive at each pTF w.r.t. a sequence n x . (c) If T is biased quasi-nonexpansive w.r.t. a se- quence n x and cond FTFT Ö then T is locally quasi-nonexpansive at each cond p TF w.r.t. a sequence n x . (d) Quasi-nonexpansivenes locally quasi-nonex- pansiveness at pFT locally quasi-nonexpan- siveness at pFT w.r.t. a sequence n x . In Example 2.1 above, we observe that 1) for 0pFT, we have 21 21 1 ,sup 0 2 10 2 lim suplim lim n n n n n ndxp 2) for 1 2 pFT , we have 21 21 l11 ,sup 22 111 imsup lim li 2 m22 n n nn n n dx p 21 21 1 liminfliminfl 1 ,0 22 im nn n nnn dx TF H. K. PATHAK Copyright © 2011 SciRes. APM 36 Here cond 0FT and in view of (*) and (**), it is evident that T is conditionally biased quasi-non- expansive (c.b.q.n.) w.r.t. a sequence 1 4 n T and hence it is biased quasi-nonexpansive (b.q.n.) w.r.t. a sequence 1 4 n T . We now show in the following example that cond FT need not be a singleton set. Example 2.2. Let 0,2X and 0,1 1,2D be endowed with the Euclidean metric d Define the mapping :TDX by Txx for 0,1x 1, 2 and 2Tx for 2x. Clearly, FT 0,2 Consider the sequence 1 n x in X then we observe that 1) for 0pFT, we have lim suplim supli,10m11 nn nn dx p ; 2) for 2pFT , we have lim suplim supli,12m11 nn nn dx p ; and ,lim inf1lim 1 n nn dx FT . Thus we have cond F0,2T and it is evident that T is conditionally biased quasi nonexpansive (c.b.q.n.) w.r.t. the sequence 1 n x in X , and hence it is biased quasi-nonexpansive (b.q.n.) w.r.t. the sequence 1 n x in X . However, interested reader can check that if we con- sider the sequence n x such that 1 n x then condFT2 Further, we observe that for2p Fcond T and for all n we have 1,, nn dxpdx p Thus, T is conditionally biased quasi-nonexpansive (c.b.q.n .) w.r.t. the sequ ence n x in X . On the other hand, if we consider the sequence n x such that 1 n x then cond F0T and T is conditionally biased quasi-nonexpansive (c.b.q.n.) w.r.t. the sequence n x in X . Remark 2.4. Example 2.2 above also shows that condFT is a closed set even though T is discon- tinuous at 2p. We need the following lemmas to prove our main theorem: Lemma 2.3. Let T be locally quasinonexpansive at pFT w.r.t. n x and lim ,0 n ndx TF . Then n x is a Cauchy sequence. Proof. Since lim ,0 n ndx TF then for any given 0 there exists 1 n such that for each 1,nn ,2 n dx FT So, there exists qFT such that for all 1,,2 n nndxq . Thus, for any 1 ,mn n we have ,,, 22 mn mn dxxdxqdx q , qFT, Hence n x is a Cauchy sequence. Lemma 2.4. Let T be conditionally biased quasi- nonexpansive w.r.t. n x ,and lim inf,0 n ndx TF Then: 1) n x converges to a point p in cond FT and T is locally quasi-nonexpansive at dFcon p T w.r.t. n x . 2) n x is a Cauchy sequence. Proof. 1) Since T is conditionally biased quasi- nonexpansive w.r.t. n x , it follows that cond FT . As lim inf,0 n ndxTF we have that lim sup,0 n ndx p for some dFconpT.So, we have lim, 0 n ndx p for some dFconpT; i.e., n x converges to a point p in condF T and T is locally quasi-nonexpansive at dFcon p T w.r.t. n x . 2) From lim, 0 n ndx p it follows that for any given 0 there exists 1 nN such that for each 1,,2 n nndxp . Thus, for any 1 ,mn n, we have ,,, 22 mn mn dxxdxqdx q qFT, Hence n x is a Cauchy sequence. The following lemma follows easily. Lemma 2.5. Let T be biased quasi-nonexpansive w.r.t. n x , and let n x converges to a point p in TF Then: 1) n x converges to a point p in cond FT and T is conditionally biased quasi-nonexpansive w.r.t. n x ; 2) n x is a Cauchy sequence. We now state our main theorem in the present paper. Theorem 2.1. Let FT be a nonempty closed set. Then 1) lim ,0 n ndx TF if n x converges to a po- int p in FT; 2) n x converges to a point in FT if H. K. PATHAK Copyright © 2011 SciRes. APM 37 lim ,0 n ndx TF , T is locally quasi-nonexpansive at w.r.t. n FpT x and X is complete. Proof. 1) Since FT is closed, FpT and the mapping ,xTdxF is continuo us (see [1, p. 13]), then lim,lim ,,0 nn nn dx Fdx FdpFTTT 2) From Lemma 2.3, n x is a Cauchy sequence. Since X is complete, then n x converges to a point, say q in X . Since FT is closed, then 0 lim,lim ,, nn nn dx Fdx FTTTdpF implies that FqT. As consequences of Theorem 2.1, we have the fol- lowing: Corollary 2.1. Let FT a nonempty closed set and for a given 0 x D and each 0 ,n nTxD Then 1) 0 lim ,0 n nTdTx F if 0 n Tx converges to a point p in FT; 2) 0 n Tx co n verges to a poi nt in FT if, 0 lim ,0, n ndTx FT T is locally qusi-nonexpansive at FpT w.r.t. 0 n Tx and X is complete. Corollary 2.2. Let X be a normed linear space, FT a nonempty closed set and for a given 0 x D and each n , 0 n Tx D . (1) If the sequence 0 n Tx converges to a point p in FT , then 0 lim ,0 n ndTx FT (2) If 0 lim ,0 n ndTx FT T is locally quasi- nonexpansive at FpT w.r.t. 0 n Tx and X is complete, then 0 n Tx converges to a point p in FT. Corollary 2.3. Let X be a normed linear space, FT a nonempty closed set and for a given 0 x D and each n , ,0 n Tx D Then (1) ,0 lim ,0 n ndTx FT if the sequence ,0 n Tx converges to a point p in FT ; (2) ,0 n Tx converges to a point p in FT if ,0 lim ,0 n ndTx FT , T is locally quasi-nonexpan- sive at FpT w.r.t. ,0 n Tx and X is complete. Note that the continuity of T implies that FT is closed but the converse need not be true. To effect this consider the following example. Example 2.3. Let 0,X and 0, 1D be endowed with the Euclidean metric d. Define the map- ing :TD X by Tx x if 1 0, 2 x and Tx 2 3 x if 1,1 2 x Obviously, 0,1 2FT is a nonempty closed but T is not continuous at 12x . Remark 2.5. (a) In order to support the above fact Ahmed and Zeyada [1] stated wrongly in their Example 2.2, where 0,1X, 0,1 41 2,5 6D , Tx x . If 0,1 4X and 2Tx x if 12,56x that T is not continuous. In fact, we observe that in this example T is continuous. (b) From Lemma 2.1, Examples 2.1 and 2.3, the con- tinuity of T implies that FT is closed but the con- verse may not be true; then we have that Corollaries 2.1, 2.2 and 2.3 are improvement of Theorem 1.1 in [13, p.462], Theorem 1.1 in [13, p. 469], and Theorem 3.1 in [8, p. 98], respectively. (c) Since every quasi-nonexpansive map w.r.t. a se- quence n x is locally quasi-nonexpansive at each p F T w.r.t. a sequence n x , but the converse may not be true; we have that Theorem 2.1, Corollaries 2.1, 2.2 and 2.3 are improvement of corresponding Theorem 2.1, Corollary 2.1, 2.2 and 2.3 of Ahmed and Zeyada [1]. (d) By considering the closedness of F T in lieu of the continuity of T and :TD X instead of :TX X we have that our Corollary 2.1 improves Proposition 1.1 of Kirk [10, p. 168]. (e) The closedness condition of D in Theorem 1.1 and 1.1 of Petryshyn and Williamson [12, p. 462, 469] and Theorem 3.1 in [8, p. 98] is superfluous. (f) The convexity condition of D in Theorem 1.1 of Petryshyn and Williamson [12, p. 469] is superfluous because the author assumed in their theorem that 0 n Tx D for each n and a given 0 x D in condition 1.3 . Theorem 2.2. Let cond FT be a nonempty closed set. Then n x converges to a point in cond FT if inf ,cond0lim n ndx TF , T is condionally biased quasi-nonexpansive w.r.t. n x and X is complete. Proof. Since cond FT TF we have that inf ,cond0lim n ndx TF implies ,iminfln ndx 0FT Now using the technique of the proof of Theorem 2.1 the conclusion follows from Lemma 2.3. The following results follows easily from Lemma 2.5. Theorem 2.3. Let FT be a nonempty closed set. Then n x converges to a point in cond FT if n x converges to a point p in FT , T is biased quasi-nonexpansive w.r.t. n x and X is complete. H. K. PATHAK Copyright © 2011 SciRes. APM 38 Theorem 2.4. Let X be a complete metric space and let cond FT be a nonempty closed set. Assume that 1) T is biased quasi-nonexpa ns ive w.r.t. n x ; 2) 1 lim ,0 nn ndx x or n x is a Cauchy se- quence; 3) if the sequence n y satisfies 1 lim ,0 nn ndy y then lim in,n0fcod nn dy FT or lim su,n0pcod nn dy FT . Then n x converges to a point in cond FT . Proof. Since cond FT it follows from (i) that T is condionally biased quasi-nonexpansive w.r.t. n x and the sequence ,cond n dxFT is mono- tonically decreasing and bounded from below by zero. Then inf ,cim ndlo n ndx TF exists. From 2) and 3), we have that lim in,n0fcod nn dx TF or lim su,n0pcod nn dx TF . Then ,lcim ond0 n ndx TF Therefore, by The- orem 2.2, the sequence n x converges to a point in cond FT . As consequences of Theorem 2.4, we obtain the following: Corollary 2.4. Let X be a complete metric space and let cond FT be a nonempty closed set. Assume that 1) T is biased quasi-nonexpan sive w.r .t. n x ; 2) T is asymptoticc regular at 0 x D( or 0 n Tx is a Cauchy sequence ); 3) if the sequence n y satisfies 1 lim ,0 nn ndy y then limin, n0fcod nn dy FT or lim su,n0pcod nn dy FT . Then 0 n Tx converges to a point in cond FT . Corollary 2.5. Let X be a Banach space and let cond FT be a nonempty closed set. Assume that 1) T is biased quasi-nonexpan sive w.r .t. 0 n Tx ; 2) T is asymptoticc regular at 0 x D (or 0 n Tx is a Cauchy sequence ); 3) if the sequence n y satisfies lim 0 nn nyTy , then lim in,n0fcod nn dy FT or lim su,n0pcod nn dy FT . Then 0 n Tx converges to a point in cond FT . Corollary 2.6. Let X be a Banach space and let cond FT be a nonempty closed set. Assume that 1) T is biased quasi-none xpansive w.r.t. ,0 n Tx ; 2) T is asymptoticc regular at 0 x D (or ,0 n Tx is a Cauchy sequence ); 3) if the sequence n y satisfies , lim 0 nn nyTy , then limin, n0fcod nn dy FT or lim su,n0pcod nn dy FT . Then ,0 n Tx converges to a point in cond FT. Remark 2.6. From Lemmas 2.1 and 2.2, Examples 2.1 and 2.3, Remark 2.3, the continuity of T implies that F T is closed but the converse may not be true; we obtain that Corollary 2.4 include Theorem 1.2 in [12, p. 464] and Theorem 3.2 in [7, p. 99] as special cases. As another consequence of Theorem 2.1, we establish the following theorem: Theorem 2.5. Let X be a complete metric space and let cond FT be a nonempty closed set. Assume that 1) T is biased quasi-no ne xpansive w.r.t. n x ; 2) for every condDTxF there exists x p cond FT such that 1,, nx nx dxpdx p ; 3) the sequence n x contains a subsequence j n x converging to x D . Then n x converges to a point in cond FT . Proof. Since cond FT it follows from (i) that T is condionally biased quasi-nonexpansive w.r.t. n x and the sequence ,cond n dxFT is mono- tonically decreasing and bounded from below by zero. Then l,conimdlim ,cond nn nn FT FTdxd x H. K. PATHAK Copyright © 2011 SciRes. APM 39 0r exists. We now apply Theorem 2.4. It suf fices to show that r=0. If lim cond n nxx F T then =0r. If cond F Tx then condDTxF Thus there exists cond x F Tp such that 11 ,lim ,lim, lim,lim ,, x x nn xx nn n xx nn dx pdxpdxp dx pdx pdx p This is a contradiction. So, cond F Tx. Corollary 2.7. Let X be a complete metric space, cond F T a nonempty closed set and for a given 0 x D and each n , 0 n Tx D. Assume that 1) T is biased quasi-no ne xpansive w.r.t. 0 n Tx ; 2) for every condDTxF there exists cond xTpF such that 100 ,, nn x x dTxpdTxp ; 3) the sequence 0 n Tx contains a subsequence 0 j n Tx converging to x D . Then 0 n Tx converges to a point in cond FT . Corollary 2.8. Let X be a Banach space, cond F T a nonempty closed set and for a given 0 x D and each n , ,0 nDTx Assume that 1) T is biased quasi-no ne xpansive w.r.t. ,0 n Tx ; 2) for every condDTxF there exists x p cond FT such that 100 nn x x Txp Txp ; 3) the sequence 0 n Tx contains a subsequence 0 j n Tx converging to x D . Then 0 j n Tx converges to a point in cond FT . Corollary 2.9. Let X be a Banach space, cond F T a nonempty closed set and for a given 0 x D and each n , ,0 nDTx Assume that 1) T is biased quasi-nonexpa nsive w.r .t. ,0 n Tx ; 2) for every condDTxF there exists cond xTpF such that 1 ,0 ,0 nn x x Txp Txp ; 3) the sequence 0 n Tx contains a subsequence ,0 j n Tx converging to x D . Then ,0 n Tx converges to a point in cond FT . Remark 2.7. From Lemmas 2.1 and 2.2, Examples 2.1 and 2.3, Remark 2.3, the continuity of T implies that FT is closed but the converse may not be true ; we obtain that Corollary 2.7 is an improvement of Theo- rem 1.3 in [13, p. 466]. 3. Applications to Geometry of Banach Spaces Throughout this section, let R denote the set of real numbers. Let , K Kzr be a closed ball in a Banach space X . For a sequence 0 nn x K Ú converging to X we define lim DSD, n n x K where 00 Dconv x K and 1 Dconv D nnn xn and SD , x K is called a superdrop . Clearly, for a constant sequence n x x con- verging to x we have 1 DD nn n so that D ,conv x kxK and is called a drop Thus the concept of a drop is a special case of superdrop It is also clear that if D, y xK then D, D, y KxK and if 0z then y x. Recall that a function :X R is called a lower semicontinuous whenever : x Xxa is closed for each a R. Caristi [4] proved the following: Theorem A. Let , X d be complete and :X R a lower semicontinuous function with a finite lower bound. Let :TXX be any function such that ,dxTxx Tx for each . x X Then T has a fixed point. We now state and prove some applications of our main results in section 2 to geometry of Banach Spaces. Theorem 3.1. Let C be a closed subset of a Banach space X let zXC and let , K Kzr be a closed ball of radius ,rdzCR Let x be an arbitrary element of C let n x be a sequence in C converging to X and let :TC X be any continuous function defined implicitly by SD ,Tx CxK for each x C in the sense that D nn Tx C for each n . Then H. K. PATHAK Copyright © 2011 SciRes. APM 40 1) lim ,0 n ndx TF if n x converges to a point p in FT ; 2) n x converges to a point in FT if lim ,0 n ndx TF , T is locally quasi-nonexpansive at p F T w.r.t. n x . Proof. Without loss of generality we may assume that 0z. Let x R and let SD , X AxK Then it is clear that T maps X into itself. For given y X and a sequence n y converging to y, we shall estimate yTy on X . For given y X and the corresponding sequence n y there is a sequence n b in X with 1,01 nn n Ty tbtyt Now nn Ty tb 1n ty, we have nn nn ty by Ty so because nn ybR , we find that nn yTy tR . Thus, nnnn nn nn yTytyb ty br ryTy Rr Define ,,dxyx yxyX and y ry Rr then X is complete as a metric space and :X R is a continuous function. So, is a lower-semicon tinuous function. Also, the above inequal- ity takes the form , nn nn dyTyy Ty . Proceeding to the limit as n we obtain ,dyTy yTy for each y X . There- fore, applying the theorem of Caristi we obtain that T has a fixed point ppx for each , x C i.e., FT . By continuity of T, FT is closed. Hence the conclusion follows from Theorem 2.1. Since drop is a special case of super drop, we have the following: Corolla ry 3 .1. Let C be a closed subset of a Banach space X let zXC and let , K Kzr be a closed ball of radius ,rdzCR Let x be an arbitrary element of C, and let :TC X be any (not necessarily continuous) function defined implicitly by D,Tx CxK for each x C. Then (1) lim ,0 n ndx TF if n x converges to a point p in FT ; (2) n x converges to a point in FT if lim ,0 n ndx TF , T is locally quasi-nonexpansive at p F T w.r.t. n x . We now prove the following result for biased quasi- nonexpansi ve m a pping w. r.t. a seque nce n x . Theorem 3.2. Let C be a closed subset of a Banach space X let zXC and let , K Kzr be a closed ball of radius ,rdzCR Let x be an arbitrary element of C, n x a sequence in C con- verging to X , and let :TCX be any con- tinuous function defined implicitly by Tx C SD , x K for each x C in the sense that n Tx Dn C for each n . If n x converges to a point in ,FTT is biased quasi-nonexpansive w.r.t. n x then n x converges to a point in cond FT . Proof. Using Theorem 2.3. instead of Theorem 2.1 the conclusion fo llows on the lines of the proof tech nique of Theorem 3.1. As a consequence of Theorem 3.2, we obtain the following: Corolla ry 3 .2. Let C be a closed subset of a Banach space X let zXC and let , K Kzr be a closed ball of radius ,.rdzC R Let x be an arbitrary element of C, and let :TC X be any (not necessarily continuous) function defined implicitly by D,Tx CxK for each x C. If n x conver- ges to a point in FT,T is biased quasi-nonexpansive w.r.t. n x then n x converges to a point in cond FT . Open Question. To what extent can the continuity hypothesis on T be muted in Theorems 3.1 and 3.2? 4. References [1] M. A. Ahmed and F. M. Zeyad, “On Convergence of a Sequence in Complete Metric Spaces and its Applications to Some Iterates of Quasi-Nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 274, No. 1, 2002, pp. 458-465. doi:10.1016/S0022-247X(02)00242-1 [2] J.-P. Aubin, “Applied Abstract Analysis,” Wiley-Inter- science, New York, 1977. [3] F. E. Browder and W. V. Petryshyn, “The Solution by Iteration of Nonlinear Functional Equations in Banach Spaces,” Bulletin of the American Mathematical Soci- ety, Vol. 272, 1966, pp. 571-575. doi:10.1090/S0002-9904-1966-11544-6 [4] J. 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