Selçuk J. Appl. Math. Selçuk Journal of Vol. 14. No. 1. pp. 21-30, 2013 Applied Mathematics
Approximating Common Fixed Point of a Finite Family with Errors for Generalized Asymptotically Quasi-Nonexpansive Mappings in Banach Spaces
Esra Yolacan1, Hukmi K¬z¬ltunc2
Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkiye.
e-mail:2hukmu@ atauni.edu.tr, 1yolacanesra@ gm ail.com
Received Date: March 10, 2011 Accepted Date: May 17, 2012
Abstract. In this paper, we de…ne and study su¢ cient and necessary conditions for modi…ed …nite-step iterative sequences with mean errors for a …nite family of generalized asymptotically quasi-nonexpansive mappings in real Banach spaces to converge to a common …xed point. The results of this paper can be viewed as an improvement and extension the corresponding results of [12], [13] and others. Key words: Generalized asymptotically quasi-nonexpansive mapping; Com-mon …xed point; Modi…ed iterative process; Strong convergence; Banach space. AMS Classi…cation: 47H10; 47H09.
1. Introduction and Preliminaries
Let K be a nonempty subset of Banach space E. A mapping T : K ! K is said to be
(a) asymptotically nonexpansive [1] if there exists a sequence frng in [0; 1) such that rn ! 0 and
(1) kTnx Tnyk (1 + rn) kx yk
for all x; y 2 K and n 1;
(b) asymptotically quasi nonexpansive [2] if F (T ) := fp 2 K : T p = pg 6= ? and there exists a sequence frng in [0; 1) such that rn! 0 and
(2) kTnx pk (1 + rn) kx pk
for all x 2 K; p 2 F (T ) and n 1;
(c) generalized asymptotically nonexpansive [4]if there exist sequences frng ; flng in [0; 1) such that rn; ln! 0 and
(3) kTnx Tnyk (1 + rn) kx yk + ln for all x; y 2 K and n 1;
(d) generalized asymptotically quasi nonexpansive [3] if F (T ) 6= ? and there exist sequences frng ; flng in [0; 1) such that rn; ln ! 0
(4) kTnx pk (1 + rn) kx pk + ln for all x 2 K; p 2 F (T ) and n 1:
It is well known that the concept of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] who proved that every asymptotically nonexpansive self-mapping of nonempty closed bounded and convex subset of a uniformly convex Banach space has …xed point. In 2003, Zhou et al. [4] in-troduced a new class of generalized asymptotically nonexpansive mapping and gave a necessary and su¢ cient condition for the modi…ed Ishikawa and Mann iterative sequences to converge to …xed points for the class of mappings. At-sushiba [5] studied the necessary and su¢ cient condition for the convergence of iterative sequences to a common …xed point of the …nite family of asymp-totically nonexpansive mappings in Banach spaces. Suzuki [6], Zeng and Yao [7] discussed a necessary and su¢ cient condition for common …xed points of two nonexpansive mappings and a …nite family of nonexpansive mappings, and proved some convergence theorems for approximating a common …xed point, respectively.
In 2006, Lan [8] introduced a new class of generalized asymptotically quasi-nonexpansive mappings and gave necessary and su¢ cient condition for the 2-step modi…ed Ishikawa iterative sequences to converge to …xed points for the class of mappings. In 2007, Yang [17] established convergence theorems for the modi…ed multistep iterative process for some common …xed point of a …nite family of nonself asymptotically nonexpansive mappings. In 2008, Nantadilok [9] extension and improvement the result of Lan [8] and gave a necessary and su¢ cient condition for convergence of common …xed point for three-step it-erative sequence with errors for generalized asymptotically quasi-nonexpansive mappings. Lan [8] and many authors ( e.g., [9], [10], [11]) have investigated con-vergence theorems for such mappings without awareness that Lan’s mappings are not new ones. In 2009, Suantai et al. [12] and Saejung [13] introduced a general iteration scheme for a …nite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces.
Inspired and motivated by this facts, we de…ned and study the convergence theo-rems of …nite steps iterative sequences with errors for generalized asymptotically quasi-nonexpansive mappings. The scheme (5) is de…ned as follows:
Let E be a normed space, K be a nonempty closed convex subset of E. Let Ti: K ! K (i = 1; 2; : : : ; k) be mappings and F := \ki=1F (Ti) 6= ?: Then for a given x1 2 K and n 1; compute the iterative sequences fxng ; fyng ; : : : ; fyn+k 2g de…ned by (5) yn = nkxn+ nkTknxn+ nkunk; yn+1= n(k 1)xn+ n(k 1)Tkn 1yn+ n(k 1)Tkn 1xn+ n(k 1)un(k 1); yn+2= n(k 2)xn+ n(k 2)Tkn 2yn+1+ n(k 2)Tkn 2yn+ n(k 2)un(k 2); .. . yn+k 2 = n2xn+ n2T2nyn+k 3+ n2T2nyn+k 4+ n2un2; xn+1= n1xn+ n1T1nyn+k 2+ n1T1nyn+k 3+ n1un1;
where fun1g ; fun2g ; : : : ; funkg are bounded sequences in K with f nig ; f nig ; f nig and f nig are appropriate real sequences in [0; 1] such that nk+ nk+
nk= 1 and ni+ ni+ ni+ ni= 1 for all i = 1; 2; : : : ; k 1 and all n: The purpose of this paper is to study the convergence theorems of …nite steps it-erative sequences with errors for generalized asymptotically quasi-nonexpansive mappings in Banach spaces.
In the sequel, the following lemmas are needed to prove our main results. A family fTi: i = 1; 2; : : : ; kg of self-mappings of K with F := \ki=1F (Ti) 6= ? is said to satisfy the following conditions.
(1) Condition A [14]. If there is a nondecreasing function f : [0;1) ! [0; 1) with f (0) = 0 and f (t) > 0 for all t 2 (0; 1) such that 1
k k X i=1
kx Tixk f (d (x; F)) for all x 2 K; where d (x; F) = inf fkx pk : p 2 Fg :
(2) Condition B [14]. If there is a nondecreasing function f : [0; 1) ! [0; 1) with f (0) = 0 and f (t) > 0 for all t 2 (0; 1) such that max1 i kfkx Tixkg f (d (x; F)) for all x 2 K:
(3) Condition C [14]. If there is a nondecreasing function f : [0; 1) ! [0; 1) with f (0) = 0 and f (t) > 0 for all t 2 (0; 1) such that kx Tixk f (d (x; F)) for all x 2 K and for at least one Ti; i = 1; 2; : : : ; k:
Note that B and C are equivalent, condition B reduces to condition (I) when all but one of Ti’s are identities, and in addition, it also condition A : It is well known that every continuous and demicompact mapping must satisfy condition (I) (see [15]). Since every completely continuous T : K ! K is continuous and demicompact so that it satis…es condition (I): Thus we will use condition C instead of the demicompactness and complete continuity of a family fTi: i = 1; 2; : : : ; kg :
Lemma 1. [16] Let fang, fbng and fcng be sequences of nonnegative real numbers satisfying the inequality
(6) an+1 (1 + cn) an+ bn; n 1; ifP1n=1bn < 1 andP1n=1cn < 1,
(i) then lim
n!1an exists; (ii) lim
n!1an= 0 whenever lim infn!1an= 0 . 2. Main Results
Our …rst result is the strong convergence theorems of the iterative scheme (5) for a …nite family of generalized asymptotically quasi-nonexpansive mappings in Banach space. In order that prove our main results, the following lemma is needed.
Lemma 2. Let E be a Banach space and K a nonempty closed and convex sub-set of E, and fTi: i = 1; 2; : : : ; kg a …nite family of generalized asymptotically quasi-nonexpansive self-mappings of K with the sequences frnig ; flnig [0; 1) such that P1 n=1 rni < 1 and 1 P n=1
lni < 1 for all i = 1; 2; : : : ; k: Assume that F 6= ? and P1
n=1 ni< 1 for each i = 1; 2; : : : ; k: For a given x
1 2 K; let the sequences fxng ; fyng ; : : : ; fyn+k 2g be de…ned by (5). Then
(a) there exist sequences fbng and fcnig in [0; 1) such that 1 P n=1 bn < 1; 1 P n=1
cni < 1; and kyn+j pk (1 + bn)j+1kxn pk + cn(j+1) for all j = 0; 1; : : : ; k 2 and all p 2 F;
(b) limn!1kxn pk exists for all p 2 F;
(c) there exist constant M > 0 and fsig in [0; 1) such that 1 P i=1 si< 1 and kxn+m pk M kxn pk + 1 P i=n
si for all p 2 F and n; m 2 N:
Proof. (a)Let p 2 F; bn= max1 i kfrnig and dn= max1 i kflnig for all n: Since P1 n=1 rni < 1 and 1 P n=1
lni < 1 for all i = 1; 2; : : : ; k; therefore 1 P n=1 bn< 1 and 1 P n=1
dn < 1: For each n 1; we note that (7) kyn pk k nkxn+ nkTknxn+ nkunk pk nkkxn pk + nkkTknxn pk + nkkunk pk nkkxn pk + nk(1 + rnk) kxn pk + nklnk+ nkkunk pk nk(1 + bn) kxn pk + nk(1 + bn) kxn pk + nkdn+ nkkunk pk nk(1 + bn) kxn pk + nk(1 + bn) kxn pk + nkdn+ nkkunk pk
where cn1 = nkdn+ nkkunk pk : Since funkg is bounded, 1 P n=1 nk< 1 and 1 P n=1 dn < 1; we obtain that 1 P n=1
cn1< 1: It follows from (7) that (8) kyn+1 pk n(k 1)xn+ n(k 1)Tkn 1yn+ n(k 1)Tkn 1xn + n(k 1)un(k 1) p n(k 1)kxn pk + n(k 1) Tkn 1yn p + n(k 1) Tkn 1xn p + n(k 1) un(k 1) p n(k 1)kxn pk + n(k 1)(1 + bn) kyn pk + n(k 1)dn + n(k 1)(1 + bn) kxn pk + n(k 1)dn+ n(k 1) un(k 1) p n(k 1)kxn pk + n(k 1)(1 + bn) [(1 + bn) kxn pk + cn1] + n(k 1)dn + n(k 1)(1 + bn)2kxn pk + n(k 1)dn+ n(k 1) un(k 1) p n(k 1)+ n(k 1)+ n(k 1) (1 + bn)2kxn pk + n(k 1)(1 + bn) cn1 + n(k 1)dn+ n(k 1)dn+ n(k 1) un(k 1) p (1 + bn)2kxn pk + cn2; where cn2= n(k 1)(1 + bn) cn1+ n(k 1)dn+ n(k 1)dn+ n(k 1) un(k 1) p : Since un(k 1) ; fbng are bounded,
1 P n=1 cn1< 1; 1 P n=1 dn< 1; and 1 P n=1 n(k 1) < 1; it follows that P1 n=1
cn2< 1: Moreover, we see that (9) kyn+2 pk n(k 2)xn+ n(k 2)Tkn 2yn+1+ n(k 2)Tkn 2yn + n(k 2)un(k 2) p n(k 2)kxn pk + n(k 2) Tkn 2yn+1 p + n(k 2) Tkn 2yn p + n(k 2) un(k 2) p n(k 2)kxn pk + n(k 2)(1 + bn) kyn+1 pk + n(k 2)dn + n(k 2)(1 + bn) kyn pk + n(k 2)dn+ n(k 2) un(k 2) p n(k 2)kxn pk + n(k 2)(1 + bn) h (1 + bn)2kxn pk + cn2 i + n(k 2)dn + n(k 2)(1 + bn) [(1 + bn) kxn pk + cn1] + n(k 2)dn+ n(k 2) un(k 2) p n(k 2)+ n(k 2)+ n(k 2) (1 + bn)3kxn pk + n(k 2)(1 + bn) cn2 + n(k 2)dn+ n(k 2)(1 + bn) cn1+ n(k 2)dn+ n(k 2) un(k 2) p (1 + bn)3kxn pk + cn3; where cn3 = n(k 2)(1 + bn) cn2+ n(k 2)dn+ n(k 2)(1 + bn) cn1+ n(k 2)dn+
n(k 2) un(k 2) p : Since un(k 2) ; fbng are bounded, 1 P n=1 cn2 < 1; 1 P n=1 dn< 1; and P1 n=1 n(k 2) < 1; it follows that 1 P n=1 cn3 < 1: By continuing the above method, there are nonnegative real sequences fcnig in [0; 1) such that
1 P n=1
cni< 1 and
(10) kyn+j pk (1 + bn)j+1kxn pk + cn(j+1); j = 0; 1; : : : ; k 2: This completes the proof of (a).
(b)It follows from (5), (10) that (11) kxn+1 pk k n1xn+ n1T1nyn+k 2+ n1T1nyn+k 3+ n1un1 pk n1kxn pk + n1kT1nyn+k 2 pk + n1kT1nyn+k 3 pk + n1kun1 pk n1kxn pk + n1[(1 + bn) kyn+k 2 pk + dn] + n1[(1 + bn) kyn+k 3 pk + dn] + n1kun1 pk n1kxn pk + n1(1 + bn) kyn+k 2 pk + n1dn + n1(1 + bn) kyn+k 3 pk + n1dn+ n1kun1 pk n1(1 + bn)kkxn pk + n1(1 + bn) h (1 + bn)k 1kxn pk + cn(k 1) i + n1dn + n1(1 + bn) h (1 + bn)k 2kxn pk + cn(k 2) i + n1dn+ n1kun1 pk ( n1+ n1+ n1) (1 + bn)kkxn pk + n1(1 + bn) cn(k 1)+ n1dn + n1(1 + bn) cn(k 2)+ n1dn+ n1kun1 pk (1 + bn)kkxn pk + cnk where cnk= n1(1 + bn) cn(k 1)+ n1dn+ n1(1 + bn) cn(k 2)+ n1dn+ n1kun1 pk : Since fun1g ; fbng are bounded,
1 P n=1 cn(k 1)< 1; 1 P n=1 cn(k 2)< 1; 1 P n=1 dn< 1; and P1 n=1 n1 < 1; it follows that 1 P n=1 cnk< 1: We have (12) kxn+1 pk (1 + bn)kkxn pk + cnk: It follows from Lemma 1 (i) that lim
n!1kxn pk exists, for all p 2 F: (c)If t 0; then 1 + t etand so, (1 + t)k
(12), it follows that (13) kxn+m pk (1 + bm+n 1)kkxn+m 1 pk + c(m+n 1)k exp fkbm+n 1g kxn+m 1 pk + c(m+n 1)k .. . exp k n+m 1P i=n bi kxn pk + n+m 1P i=n cin exp kP1 i=1 bi kxn pk + 1 P i=n cin M kxn pk + 1 P i=n si; where M = exp kP1 i=n bi and si= cin:
Theorem 1. Let E be a Banach space and K a nonempty closed and convex subset of E and fTi: i = 1; 2; : : : ; kg a …nite family of generalized asymptotically quasi-nonexpansive self-mappings of K with the sequences frnig ; flnig [0; 1) such that P1 n=1 rni< 1 and 1 P n=1
lni< 1 for all i = 1; 2; : : : ; k: Assume that F 6= ? is closed and P1
n=1 ni< 1 for each i = 1; 2; : : : ; k: Then the iterative sequence fxng ; fyng ; : : : ; fyn+k 2g de…ned by (5) converges strongly to a common …xed point of the family of mappings if and only if lim infn!1d(xn; F) = 0:
Proof. We prove only the su¢ ciency because the necessity is obvious. From (12), we have kxn+1 pk (1 + bn)kkxn pk + cnk; for all n and all p 2 F: Hence, we have (14) d(xn+1; F) (1 + bn)kd(xn; F) + cnk = 1 + k P r=1 k(k 1):::(k r+1) r! b r n d(xn; F) + cnk: Since P1 n=1 bn < 1; it follows that 1 P n=1 k P r=1 k(k 1):::(k r+1) r! b r n < 1: Since 1 P n=1
cnk < 1 and lim infn!1d(xn; F) = 0; it follows from Lemma 1 (ii) that limn!1d(xn; F) = 0: Next, we prove that fxng is a Cauchy sequence. From Lemma 2 (c), we have (15) kxn+m pk M kxn pk + 1 X i=n si, 8p 2 F; n; m 2 N:
Since limn!1d(xn; F) = 0 and 1 P i=1
8n02 N such that (16) d(xn; F) < " 4M; 1 X i=n0 si< " 4; 8n n0: Therefore, there exists q in F such that
(17) kxn0 qk <
" 4M: From (15) to (17), for all n n0 and m 1; we have
(18) kxn+m xnk kxn+m qk + kxn qk M kxn0 qk + 1 P i=n0 si+ M kxn0 qk + 1 P i=n0 si < M4M" +"4+ M4M" +"4 = ":
This shows that fxng is a Cauchy sequence, hence xn ! q 2 K: It remains to show that q 2 F: Notice that
(19) jd(q; F) d(xn; F)j kq xnk ; 8n: Since limn!1d(xn; F) = 0;we obtain that q 2 F:
Since an asymptotically quasi-nonexpansive mapping is generalized asymptoti-cally quasi-nonexpansive mapping, so we have the following result.
Corollary 1. Let E be a Banach space and K a nonempty closed and convex subset of E and fTi: i = 1; 2; : : : ; kg a …nite family of asymptotically quasi-nonexpansive self-mappings of K with the sequences frnig ; flnig [0; 1) such that P1 n=1 rni < 1 and 1 P n=1
lni < 1 for all i = 1; 2; : : : ; k: Assume that F 6= ? is closed and P1
n=1 ni< 1 for each i = 1; 2; : : : ; k: Then the iterative sequence fxng ; fyng ; : : : ; fyn+k 2g de…ned by (5) converges strongly to a common …xed point of the family of mappings if and only if lim infn!1d(xn; F) = 0:
Theorem 2. Let E be a Banach space and K a nonempty closed and convex subset of E and fTi: i = 1; 2; : : : ; kg a …nite family of generalized asymptoti-cally quasi-nonexpansive self-mappings of K with the sequences frnig ; flnig [0; 1) such that P1 n=1 rni < 1 and 1 P n=1
lni < 1 for all i = 1; 2; : : : ; k: Sup-pose that F 6= ? is closed. Let x1 2 K and fxng be the sequence de…ned by (5). If P1
n=1 ni < 1; lim
n!1kxn Tixnk = 0 for all i = 1; 2; : : : ; k and fTi: i = 1; 2; : : : ; kg satis…es Condition C , then fxng converges strongly to a common …xed point of the family of mappings.
Proof. From limn!1kxn Tixnk = 0 for all i = 1; 2; : : : ; k and fTi: i = 1; 2; : : : ; kg satisfying Condition C ; there is a nondecreasing func-tion f : [0; 1) ! [0; 1) with f(0) = 0 and f(t) > 0 for all t 2 (0; 1) such that kxn Ti0xnk f (d (xn; F)) for some i0 2 f1; 2; : : : ; kg, it follows that
limn!1d(xn; F) = 0: From Theorem 1, we obtain that fxng converges strongly to a common …xed point of the family of mappings.
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