 Advances in Pure Mathematics, 2011, 1, 204-209 doi:10.4236/apm.2011.14036 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM A Modified Averaging Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of k–Strictly Asymptotically Pseudocontractive Mappings Donatus Igbokwe, Oku Ini Department of Mat hematics, University of Uyo, Uyo, Nigeria E-mail: {igbokwedi, inioku}@yahoo.com Received April 4, 2011; revised June 3, 2011; accepted June 10 , 20 1 1 Abstract The composite implicit iteration process introduced by Su and Li [J. Math. Anal. Appl. 320 (2006) 882-891] is modified. A strong convergence theorem for approximation of common fixed points of finite family of kstrictly asymptotically pseudo-contractive mappings is proved in Banach spaces using the modified itera-tion process. Keywords: Implicit Iteration Process, kStrictly Asymptotically Pseudo-Contractive Maps, Fixed Points 1. Introduction and Preliminaries Let E be an arbitrary real Banach space and let J denote normalized duality mapping from E into *2E given by 22 2*=:,=;=JxfExfxx f where *E denotes the dual space of E and , denotes the generalized duality pairing. If *E is strictly convex , then J is single-valued. In the sequel, we shall denote single-valued duality mappings by j. A ma- pping :TK K is called k-strictly asymptotically pseudocontractive with sequence [1, )na, 1limnna  (see, for example ) if for all ,xyK, there exists jx yJxy  and a constant [0,1)k such that 22,111122nnnnnTx Tyjxyaxy kxTxyTy  (1) for all nN. If I denotes the identity operator, then (1) can be written in the form 22,111122nnnnnIT xIT yjxykITxITya xy   (2) The class of kstrictly asymptotically pseudocon- tractive maps was first introduced in Hilbert spaces by Qihou . In Hilbert spaces, j is the identity and it is shown in Osilike  that (1) (and hence (2)) is equiva- lent to the the inequality 222nnn nnTTyaxykITxITy (3) which is the inequality considered by Qihou . A mapping T with domain DT and range RT in E is called strictly pseudo-contractive in the terminology of Browder and Petryshyn  if there exist 0 such that  22,TxTyjxyx yx yTxTy (4) for all ,xyDT and for all jx yJxy . Without loss of generality we may assume 0, 1. If I denotes the identity operator, then (1) can be written in the form 2,ITx ITyjxyITx ITy (5) In the Hilbert space ,H (4) (and hence (5)) is equivalent to the inequality 222=1 <1TxTyxyk I TxI Tyk (6) and we can assume also that 0k, so that [0,1]k. D. IGBOKWE ET AL. Copyright © 2011 SciRes. APM 205It is shown in  that a strictly ps eudocontractive map is L-Lipschitzian (Tx Tyxy, for all ,xyDT and for some 0L). It is also shown in  that a kstrictly asymptotically pseudocontractive mapping is uniformly LLipschitzian (i.e. for some 0L, nnTx TyLxy, for all ,xyK and nN). The class of kstrictly asymptotically pseudocon- tractive mappings and the class of strictly pseudo-contra- ctive mappings are independent (see ). The class of kstrictly asymptotically pseudocontractive mappings is a natural extension of the class of asymptotically nonexpansive mappings (i.e. mappings :TK K such that 1, ,nn nTx Tya xynxyK (7) and for some sequence [1, )na such that 1limnna .) If =0k, we have from (3) (and hence (1) ) that T is asymptotically nonexpansive. In fact, an asymptotically nonexpansive map is 0strictly asymp- totically pseudocontractive (see Remark 1 ). T is called asymptotically quasi-nonexpansive if there exists a sequence [1, )na such that 1limnna , and ,1nnTx pax pn  (8) for all xK and  :pFT xKTxx In , Xu and Ori introduced an implicit iteration process and proved weak convergence theorem for approximation of common fixed points of a finite family of nonexpansive mappings (i.e. a subclass of asympto- tically nonexpansive mappings for which Tx Tyxy, ,xyK). In , Osilike extended the results of  from nonexpansive mappings to strictly pseudocontractive mappings. In , Su and Li introduc- ed a new implicit iteration process and called it composite implicit iteration process. Using the new implicit iteration process, they proved the results estab- lished by Osilike in . In compact form, the composite iteration process introduced in  is the sequence nx generated from arbitrary 0xK by 11=1=1nnnnnnnnnnnnxxTyyx Tx   (9) where {},{}[0,1].nn In  Sun modified the implicit iteration process of Xu and Ori and applied the modified iteration process for the approximation of fixed points of a finite family of asymptotically quasi-nonex- pansive maps. In compact form, the modified implicit iteration process of Sun is the sequence {}nx generated from arbitrary 0xK by 1=1 ,1knnnninxx Txn (10) where 1, 1,2,,nk NiiIN . In this paper, we modify (9) as follows. Let K be a nonempty closed convex subset of E, =1NiiT a finite family of kstrictly asymptotically pseudocontractive self-maps of K, then for 0xK and , [0,1]nn.    1101111101112212222212221 12 21111111111221222222=1, =1=1, =1=1,=1=1,=1=1,=1NNNNNNNNNNNNNNNNNNNNNNNNNNNNNnxxTyyxTxxxTyyxTxxxTyyxTxxxTyyxTxxxTyyx              2222 222 212222 21223 3212122112121212 211213 322222122 22221222122 222==1,=1=1 ,=1=1 ,=1NNNNNNNNN NNNNNNNNNNNNNNNNNNNNNNNNNNTxxxTyyxTxxxTyyxTxxx Tyyx Tx          Our iteration process can be expressed in a compact form as 11=(1) 1=1knnn ninknnnninxx Tynyx Tx   (11) where 1,1,2,,nk NiiN . Observe that if :TK K is kstrictly asymptotically pseudocon- tractive mapping with sequence [1, )na such that 1limnna, then for every fixed uK and ,1,1ts LL, the operator ,, :tsnSKK defined for all xK by ,, =1 1nntsnSxtu tTsusTx satisfies  2,, ,, 11tsn tsnSxSyt sLxy , ,xyK. Since 211 0,1tsL, it follows that D. IGBOKWE ET AL. Copyright © 2011 SciRes. APM 206 ,,tsnS is a contraction map and hence has a unique fixed point ,,tsnx in K. This implies that there exists a unique ,,tsnxK such that  ,, ,,11nntsn tsnxtu tTsusTx . Thus our mod- ified composite implicit iteration process (11) is defined in K for the family =1NiiT of N kstrictly asymptotically pseudocontractive self maps of a nonempty convex subset K of a Banach space pro- vided ,,1nn where 1LL and 1max NiLL. The purpose of this paper is to study the convergence of the new modified averaging implicit iteration scheme (11) to a common fixed point of a finite family of kstrictly asymptotically pseudocontractive maps in arbitrary Banach spaces. The results presented in this paper, generalize the result of Su and Li  and several others in the literature (see for example , , , ). In the sequel, we shall need the following: Lemma 1.1 OAA (, p. 80): Let {}na, {}nb and {}n be three sequences of nonnegative real numbers satisfying the inequality 11,1nnnnaban  (12) If n and nb then limnna exists. If in addition {}na has a subsequence which converges strongly to zero, then 0limnna . Definition 1.1  A bounded convex subset K of a real Banach space E is said to have normal structure if every nontrivial convex subset C of K contains at least one nondimetrial point. That is, there exists 0xE such that 0:< :,=supx xxCsupxyxyCdC  where dC is the diameter of C Every uniformly convex Banach space and every compact convex subset K of a Banach space E has normal structure. For the definition of modulus of convexity of E and the characteristic of convexity 0 of E, see . Theorem 1.1 ( Corollary 3.6) Let E be a real Banch space with normal structure 0>1,NE max, 0>0, K a nonempty bounded closed convex subset of E and :TKK a uni- formly LLipschitzian mapping with 1. Then T has a fixed point. 2. Main Results Lemma 2.1 Let E be a real Banach space with normal structure 0>1,NE maxand let K be a nonempty closed convex subset of E. Let =1NiiT be N ik strictly asymptotically pseudo-contractive self-map s of K with sequences {}[1,)ina such that =1 11,NE max and let K be a nonempty closed convex subset of E. Let =1NiiT, n, n and nx be as in Lemma2.1. Then nx exists in K and converges strongly to a common fixed point of the mappings =1NiiT if and only if ,=0liminf nndxF where ,=inf pFnndxFx p. PROOF The existence of fixed point follows from Theorem 1.1. If nx converges strongly to a common fixed point of of the mappings =1NiiT, then =0liminf nnxp . Since 0,nndxFx p, we have ,=0liminf nnxF . Conversely, suppose ,=0liminf nnxF then our Lemma implies that ,=0limnndxF . Thus for arbitrary 0, there exists a positive integer 3N such that ,<4ndxF, 3nN . Furthermore =1nn implies that there exists a positive integer 4N such that =4<4jjnM, 4nN . Choose 34=max ,NNN, then ,4ndxF and =44jjNM. For all ,nm N and for all pF we have 44=1 =14=22nm nmnmnjNjjN jNNjjNxxxpxpxpMx pMxpM Taking infimum over all pF, we obtain 4=222,2 44nm NjjNMxxdxFMM  Thus nx is Cauchy. Suppose limnnxu . Then uK since K is closed. 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