 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 kstrictly asymptotically pseudo-contractive mappings is proved in Banach spaces using the modified itera- tion process. Keywords: Implicit Iteration Process, kStrictly Asymptotically Pseudo-Contractive Maps, Fixed Points 1. Introduction and Preliminaries Let E be an arbitrary real Banach space and let denote normalized duality mapping from E into * 2 given by 22 2 * =:,=;= xfExfxx f where * E denotes the dual space of E and , denotes the generalized duality pairing. If * E is strictly convex , then 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, ) n a, 1 limnn a (see, for example [1]) if for all , yK , there exists jx yJxy and a constant [0,1)k such that 2 2 , 11 11 22 nn nn n Tx Tyjxy axy kxTxyTy (1) for all nN. If denotes the identity operator, then (1) can be written in the form 22 , 11 11 22 nn nnn IT xIT yjxy kITxITya xy (2) The class of k strictly asymptotically pseudocon- tractive maps was first introduced in Hilbert spaces by Qihou [2]. In Hilbert spaces, j is the identity and it is shown in Osilike [3] that (1) (and hence (2)) is equiva- lent to the the inequality 2 22 nnn n n TTyaxykITxITy (3) which is the inequality considered by Qihou [2]. A mapping T with domain DT and range RT in E is called strictly pseudo-contractive in the terminology of Browder and Petryshyn [4] if there exist 0 such that 2 2 ,TxTyjxyx yx yTxTy (4) for all , yDT and for all jx yJxy . Without loss of generality we may assume 0, 1 . If denotes the identity operator, then (1) can be written in the form 2 , Tx ITyjxyITx ITy (5) In the Hilbert space , (4) (and hence (5)) is equivalent to the inequality 2 22 =1 <1 TxTyxyk I TxI Ty k (6) and we can assume also that 0k, so that [0,1]k .  D. IGBOKWE ET AL. Copyright © 2011 SciRes. APM 205 It is shown in [5] that a strictly ps eudocontractive map is L-Lipschitzian (Tx Tyxy, for all , yDT and for some 0L). It is also shown in [3] that a kstrictly asymptotically pseudocontractive mapping is uniformly LLipschitzian (i.e. for some 0L, nn Tx TyLxy, for all , yK and nN ). The class of kstrictly asymptotically pseudocon- tractive mappings and the class of strictly pseudo-contra- ctive mappings are independent (see [1]). The class of kstrictly asymptotically pseudocontractive mappings is a natural extension of the class of asymptotically nonexpansive mappings (i.e. mappings :TK K such that 1, , nn n Tx Tya xynxyK (7) and for some sequence [1, ) n a such that 1 limnn a .) If =0k, we have from (3) (and hence (1) ) that T is asymptotically nonexpansive. In fact, an asymptotically nonexpansive map is 0strictly asymp- totically pseudocontractive (see Remark 1 [6]). T is called asymptotically quasi-nonexpansive if there exists a sequence [1, ) n a such that 1 limnn a , and ,1 nn Tx pax pn (8) for all K and :pFT xKTxx In [7], 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 , , yK ). In [8], Osilike extended the results of [7] from nonexpansive mappings to strictly pseudocontractive mappings. In [9], 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 [8]. In compact form, the composite iteration process introduced in [9] is the sequence n generated from arbitrary 0 K by 1 1 =1 =1 nnnnnn nnnnnn xTy yx Tx (9) where {},{}[0,1]. nn In [10] 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 {} n generated from arbitrary 0 K by 1 =1 ,1 k nnnnin xx Txn (10) where 1, 1,2,,nk NiiIN . In this paper, we modify (9) as follows. Let be a nonempty closed convex subset of E, =1 ii T a finite family of k strictly asymptotically pseudocontractive self-maps of K, then for 0 K and , [0,1] nn . 110111110111 221222221222 1 1 2 2 1111111111 2 2122222 2 =1, =1 =1, =1 =1,=1 =1,=1 =1,=1 NNNNNNNNNNNN NNNNNNNNNN NNNNNNN n xxTyyxTx xxTyyxTx xxTyyxTx xxTyyxTx xxTyyx 2 22 2 2 22 212222 2122 3 3 212122112121212 21121 3 3 22222122 22221222122 222 = =1,=1 =1 ,=1 =1 ,=1 NNN NNNNNN NNNNNN NNNNNNNNNN NNNNNNNNNN Tx xxTyyxTx xxTyyxTx xx Tyyx Tx Our iteration process can be expressed in a compact form as 1 1 =(1) 1 =1 k nnn nin k nnnnin xx Ty n yx Tx (11) where 1,1,2,,nk NiiN . Observe that if :TK K is kstrictly asymptotically pseudocon- tractive mapping with sequence [1, ) n a such that 1 limnn a , then for every fixed uK and ,1,1ts LL, the operator ,, : tsn SKK defined for all K by ,, =1 1 nn tsn Sxtu tTsusTx satisfies 2 ,, ,, 11 tsn tsn SxSyt sLxy , , yK . Since 2 11 0,1tsL, it follows that  D. IGBOKWE ET AL. Copyright © 2011 SciRes. APM 206 ,,tsn S is a contraction map and hence has a unique fixed point ,,tsn in K. This implies that there exists a unique ,,tsn K such that ,, ,, 11 nn tsn tsn xtu tTsusTx . Thus our mod- ified composite implicit iteration process (11) is defined in for the family =1 ii T of N k strictly asymptotically pseudocontractive self maps of a nonempty convex subset K of a Banach space pro- vided ,,1 nn where 1LL and 1 max Ni LL . 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 kstrictly asymptotically pseudocontractive maps in arbitrary Banach spaces. The results presented in this paper, generalize the result of Su and Li [9] and several others in the literature (see for example [8], [11], [10], [7]). In the sequel, we shall need the following: Lemma 1.1 OAA ([3], p. 80): Let {} n a, {} n b and {} n be three sequences of nonnegative real numbers satisfying the inequality 11,1 nnnn aban (12) If n and n b then limnn a exists. If in addition {} n a has a subsequence which converges strongly to zero, then 0 limnn a . Definition 1.1 [12] A bounded convex subset of a real Banach space E is said to have normal structure if every nontrivial convex subset C of contains at least one nondimetrial point. That is, there exists 0 E such that 0:< :,= upx xxCsupxyxyCdC where dC is the diameter of C Every uniformly convex Banach space and every compact convex subset 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 [13]. Theorem 1.1 ([13] 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 LLipschitzian mapping with <L , >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 be a nonempty closed convex subset of E. Let =1 ii T be N i k strictly asymptotically pseudo-contractive self-map s of K with sequences {}[1,) in a such that =1 1< in na , and let =i FFT. Let {} n , {} ,1 n be two real sequences satisfying the condi- tions: ()i =1 1n na , ()ii 2 =1 1< n n , ()iii =1 1n n , ()iv 2 11 <1 nn L , where 1LL and 1 =max iN i LL , i L the Lipschitzian constants of =1 ii T. Let n be the implicit iteration sequence generated by (11). Then ()i limnn p exists for all pF. ()ii , n dxF exists, where ,inf pF nn dxFx p ()iii 0 liminf nnnn xTx . Proof The existence of fixed point follows from Theorem 1.1. We shall use the well known inequality (see for example [7,14]) 22 , yxyjxy (13) which holds for all , yE and for all jx yJx y . Let pF, then using (11) and (13) we obtain 2 2 1 2 21 2 21 2 21 2 =1 21 , =21 ,, 21 21 , k nnn nin k nnn inn nn n kk k inin nin n nn n nnn n k nninn xpx pTyp pTypjxp xp TyTx jxpTx jxp xp Lyx xpxp xTxjxp (14) Since each : i TK K, iI, is i k strictly asymptotically pseudocontractive, then 22 , 11 11 22 nn ii nn ii iim IT xIT yjxy kxTx yTyaxy  D. IGBOKWE ET AL. Copyright © 2011 SciRes. APM 207 [0, 1) i k. Let 1 min iN i kk . Then 22 , 11 11 22 nn ii nn ii im IT xITyjxy kxTx yTyaxy Thus it follows from (14) that 22 21 2 2 1 12 1 11 nnn nnnn nnn k nnin pxpLyxxp xp kxTx (15) Observe that 1 11 kk nn nninnnnin yxTyx xTx (16) 2 11 11 k innnnn n TyxLxp Lxp (17) and 1 k nin n TxLx p (18) Substituting (16)-(18) into (15), we obtain 23 2 2 2 21 2 1 121(1 )2111 12 1 21 1 11 nnnn n nik n nnnnn k nn nnin LLL axp xp LL xpxp kxTx (19) Observe that 11 ik in aa , kn , since 1nkNi , 1, 2,,iI N . Setting 22 2 21 1 2111 11 nnnn nn nn bL LL a then it follows from (19) that 2 22 1 2 1 2 1 1121 21 1 121 (1)(1 ) 12(1) nn nn nn nnn nn nn k nnin nn b xpx p b LL px p b kxTx b (20) Since 3 121 =1 121 211 211 nn nin nn nn b a LLL and ,1 nn , then we obtain that 3 3 2121 1211 21221 inn nnn m aL LL aLLL Setting 3 12221MLLL , then there must exist a natural number 1 N, such that if 1 nN then 12 121 nn b , (since 2 =1 1n n and =1 1 in na ). Therefore it follows from (20) that 2 22 1 2 1 2 2 121 221 1 11 11 nnnn nnn nn k nnin k nnin xpb xp LLx pxp kx Tx kx Tx (21) Observe that, 2 1 1 1 2 =, =, 1 , =, 1 ,1 , 1 1 nnn nn nn k in n nn nn kk in innn k in n nn nn nnnn n xp xpjxp xpjxp Ty pjx p xpjxp TyTx jxp Tx pjx p xpxpL yxxp Lxp (22) Substituting (16)-(18) into (21) and simplifying this inequalities, we have 2 3 2 2 1 111 1 11 1 11 nnnn nn n nnnnnn LL LLxp LLxpxp  D. IGBOKWE ET AL. Copyright © 2011 SciRes. APM 208 2 1 2 3 2 2 3 1 32 2 3 (1 )1 11111(1)1 1(1)1(1)1 111 =1 11(1)1111 111111 1 nnnn n n nnnnnn nnnnnnn nnn nnnnnn nn nnn LL xpx p LLL L LLLLL p LLL L LLLL 2 1 2 3 1 11111 11 nn nn nnnnnn Lxp LLL L (23) Now, we consider the second term on the right-hand side of (23) . Since ,1 nn , then 3 3 11111 11 nnnnn n LLL L LL LL Set 3 2=1MLLLL . Since 10 limnn , then there exists a natural number 2 N, such that if 2 nN then 2 3 111 1 1 11 1 2 nnnn nn LL LL Again it follows from the condition ,1 nn , that 2 3 2 22 3 11111 11 11111 nn nnn nn n nn n LLL LL LLLLL Theref o r e i t follow s f rom (23) that 2 3 2 1 1 12 111 11 =1 nnn nn nn xpLL L LLxp xp (24) where 22 3 21 111 1 nnn n LLLLL From conditions ()ii, ()iii it is easy to see that 22 3 =1 2111 11 < nn n n LLLLL Thus using Lemma ,OOA we have limnn p exists, completing the proof of ().i Also it follows from (24) that 1 ,1, nnn dxFdxF , and it again follows from Lemma OOA that limn exists, this completes the proof of ()ii. Now, we consider the second term on the right-hand side of (21). Since {} n is bounded, ,1 n , then there exists a constant 30M such that 2 1 2 3 221 1 41 nnn nn n LLx pxp M Thus, it follows from (21) that 2 22 1 2 3 121 211 1 nnnn k nnnin xpb xp kx Tx (25) Since {} n is bounded, then there exists a constant 40M such that 2 4 n pM. It follows from (25) that 2 22 43 22 1 11 21 41 k nnin nn n nn kxTx MbM xpxp Hence, 2 =1 22 43 =1 =1 2 11 21 41 < nk jjij jN nn jn j jN jN N kxTx MbM xp (26) Using condition ()ii and ()iii, it follows from (26) that 2 =1 1< k nnin nxTx , and using condition ()i, =0 liminf k nnin xTx . Thus =0 liminf k nnnn xTx . For all nN we have nnN TT so that 1 kk n nnn nnnn nn kk nnn nnn TxxTxTxTx xTxLTxx  D. IGBOKWE ET AL. Copyright © 2011 SciRes. APM 209 Thus, =0 liminf nnnn xTx , completing the proof of ().iii Theorem 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 =1 ii T, n , n and n be as in Lemma2.1. Then n exists in K and converges strongly to a common fixed point of the mappings =1 ii T if and only if ,=0 liminf nn dxF where ,= inf pF nn dxFx p . PROOF The existence of fixed point follows from Theorem 1.1. If n converges strongly to a common fixed point of of the mappings =1 ii T, then =0 liminf nn xp . Since 0, nn dxFx p, we have ,=0 liminf nn xF . Conversely, suppose ,=0 liminf nn xF then our Lemma implies that ,=0 limnn dxF . Thus for arbitrary 0 , there exists a positive integer 3 N such that ,<4 n dxF , 3 nN . Furthermore =1n n implies that there exists a positive integer 4 N such that =4 <4 j jn , 4 nN . Choose 34 =max ,NNN, then ,4 n dxF and =4 4 j jN . For all ,nm N and for all pF we have 44 =1 =1 4= 22 nm nm nm njNj jN jN Nj jN xxxpxp xpMx pM xpM Taking infimum over all pF , we obtain 4= 22 2,2 44 nm Nj jN M xxdxFM Thus n is Cauchy. Suppose limnn u . Then uK since K is closed. Furthermore, since i T is closed for all iI, we have that is closed. Since ,0 limnn dxF , we must have that uF. 3. References [1] M. O. Osilike, A. Udomene, D. I. Igbokwe and B. G. Akuchu, “Demiclosedness Principle and Convergence Theorems for K-Strictly Asymptotically Pseudocontrac- tive Maps,” Journal of Mathematical Analysis and Ap- plications, Vol. 326, No. 2, 2007, pp. 1334-1345. HHHHHHH0UUUUUdoi:10.1016/j.jmaa.2005.UU12UU.052UUUU [2] L. Qihou, “Convergence Theorems of the Sequence of Iterates for Asymptotially Demicontractive and Hemi- contractive Mappings,” Nonlinear Analysis, Vol. 26, No. 11, 1996, 1835-1842. HHHHHHH1UUUUUdoi:10.1016/0362-546X(94)00351-HUUUU [3] M. O. Osilike, S. C. Aniagboso and G. B. Akachu, “Fixed Points of Asymptotically Demicontractive Mapping in Arbitrary Banach Space,” Pan American Mathematical Journal, Vol. 12, No. 2, 2002, pp. 77-88. [4] F. E. Browder and W. V. Petryshyn, “Construction of Fixed Points of Nonlinear Mappings in Hilbert Spaces,” Journal of Mathematical Analysis and Applications, Vol. 20, No. 2, 1967, pp. 197-228. HHHHHHH2UUUUUdoi:10.1016/0022-247X(67)90085-6UUUU [5] M. O. Osilike and A. Udomene, “Demiclosedness Princi- ple and Convergence Results for Strictly Pseudocontrac- tive Mappings of Browder-Petryshyn Type,” Journal of Mathematical Analysis and Applications, Vol. 256, No. 2, 2001, pp. 431-445. HHHHHHH3UUUUUdoi:10.1006/jmaa.2000.7257UUUU [6] H. Zhou, “Demiclosedness Principle with Applictions for Asymptotically Pseudocontractions in Hilbert Spaces,” Nonlinear Analysis, Vol. 70, No. 9, 2009, pp. 3140-3145. HHHHHHH4UUUUUdoi:10.1016/j.na.2008.04.017UUUU [7] H. K Xu and R. G. Ori, “An Implicit Iteration Process for Nonexpansive Mappings,” Numerical Functional Analy- sis and Optimization, Vol. 22, No. 5-6, 2001, pp. 767-733. HHHHHHH5UUUUUdoi:10.1081/NFA-100105317UUUU [8] M. O. Osilike, “Implicit Iteration Process for Common Fixed Point of a Finite Family of Strictly Pseudo-contrac- tive Maps,” Journal of Mathematical Analysis and Ap- plications, Vol. 294, No. 1, 2004, pp. 73-81. HHHHHHH6UUUUUdoi:10.1016/j.jmaa.2004.01.038UUUU [9] Y. Su and S. Li, “Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Maps,” Journal of Mathematical Analy- sis and Applications, Vol. 320, No. 2, 2006, pp. 882-891. HHHHHHH7UUUUUdoi:10.1016/j.jmaa.2005.07.038UUUU [10] Z. H. Sun, “Strong Convergence of an Implicit Iteration Process for a Finite Family of Asymptotically Quasi- nonexpansive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 286, No. 1, 2003, pp. 351-358. HHHHHHH8UUUUdoi:10.1016/S0022-247X(03)00537-7UUUU [11] M. O. Osilike and B. G. Akuchu, “Common Fixed Points of a Finite Family of Asymptotically Pseudocontractive Maps,” Journal Fixed Points and Applications, Vol. 2004, No. 2, 2004, pp. 81-88. HHHHHHH9UUUUUdoi:10.1155/S16871820043UU12UU027UUUU [12] C. E. Chidume, “Functional Analysis: Fundamental Theorems with Application,” International Centre for Theoretical Physics, Trieste Italy, 1995. [13] L. C. Ceng, H. K. Xu and J. C. Yao, “Uniformly Normal Structure and Uniformly Lipschitzian Semigroups,” Nonlinear Analysis, Vol. 73, No. 12, 2010, pp. 3742-3750. UUUUdoi:10.1016/j.na 2010.o7.044 . [14] W. V. Petryshyn, “A Characterization of Strict Convexity of Banach Spaces and Other Uses of Duality Mappings,” Journal of Functional Analysis, Vol. 6, No. 2, 1970, pp. 282-291. HHHHHHH1UUUUUdoi:10.1016/0022-UU12UU36(70)90061-3UUUU
|
