Strong convergence of multi-step iterates with errors for generalized asymptotically quasi-nonexpansive mappings

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1 Palestine Journal of Mathematics Vol. 1 01, Palestine Polytechnic University-PPU 01 Strong convergence of multi-step iterates with errors for generalized asymptotically quasi-nonexpansive mappings Gurucharan Singh Saluja Communicated by Ayman Badawi MSC 010 Classifications: 47H09, 47H10. Keywords and phrases: Generalized asymptotically quasi-nonexpansive mapping, common fixed point, multi-step iteration with errors, uniformly convex Banach space, uniformly L, α-lipschitz mapping, strong convergence, wea convergence. Abstract. In this paper, we study multi-step iteration with errors and give the necessary and sufficient condition to converge to common fixed points for a finite family of generalized asymptotically quasi-nonexpansive mappings in the framewor of Banach spaces. Also we establish some strong convergence theorems to converge to common fixed points for a finite family of said mappings and scheme in a uniformly convex Banach spaces. Our results extend and improve the corresponding results of [1,, 5, 7, 8, 9, 11, 1, 17,. 1 Introduction Let K be a subset of normed space E and T : K K be a mapping. Then 1 T is said to be an asymptotically nonexpansive mapping [, if there exists a sequence {r n } [0, with lim r n = 0 such that for all x, y K. T n x T n y 1 + r n x y, 1.1 T is said to be L, α-uniformly Lipschitz [9 if there are constants L > 0 and α > 0 such that T n x T n y L x y α, n 1, 1. for all x, y K. Every asymptotically nonexpansive mapping is L, 1-uniformly Lipschitz mapping. T is said to be an asymptotically quasi-nonexpansive mapping, if F T and there exists a sequence {r n } [0, with lim r n = 0 such that T n x p 1 + r n x p, x K and p F T T is said to be generalized asymptotically quasi-nonexpansive [18 if there exist sequences {r n }, {s n } in [0, with lim r n = 0 = lim s n such that for all x K, p F T and n 1. T n x p 1 + r n x p + s n, 1.4 If s n = 0 for all n 1, then T is nown as an asymptotically quasi-nonexpansive mapping. From the above definitions, it follows that if F T is nonempty, then asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings are all special cases of generalized asymptotically quasi-nonexpansive mappings. But the converse does not hold in general. In 197, Petryshyn and Williamson [11 gave the necessary and sufficient condition for Mann iterative sequence cf.[10 to converge to fixed points of quasi-nonexpansive mappings. In 1997, Ghosh and Debnath [ extended the results of Petryshyn and Williamson [11 and gave the necessary and sufficient condition for Ishiawa iterative sequence to converge to fixed points for

2 Strong convergence of multi-step iterates quasi-nonexpansive mappings. Liu [8 extended the results of [, 11 and gave the necessary and sufficient condition for Ishiawa iterative sequence with errors to converge to fixed points of asymptotically quasinonexpansive mappings. Iterative techniques for approximating fixed points of asymptotically nonexpansive and asymptotically quasi nonexpansive mappings in Banach spaces have been studied by many authors; see [, 7, 8, 17, 19, 0, 1 and the references therein. Related wor can be found in [1, 5, 1, 1, 14, 15, and many others. Recently, Tang and Peng [ studied the following iteration scheme in Banach space: Let {T i : i = 1,,..., }: K K, where K is a nonempty subset of a Banach space E, be a finite family of uniformly quasi-lipschitzian mappings. For a given x 1 K, then the sequence {x n } is defined by x n+1 = a n x n + b n T n y 1n + c n u n, y 1n = a 1n x n + b 1n T 1y n n + c 1n u 1n, y n = a n x n + b n T y n n + c n u n,. y n = a n x n + b n T n y 1n + c n u n y 1n = a 1n x n + b 1n T1 n x n + c 1n u 1n, n 1, 1.5 where {a in }, {b in }, {c in } are sequences in [0, 1 with a in + b in + c in = 1 for all i = 1,,..., and n 1, {u in, i = 1,,...,, n 1} are bounded sequences in K. Also, they gave the necessary and sufficient condition to converge to common fixed points for a finite family of said class of mappings. Remar 1.1. The iterative algorithm 1.5 is called multi-step iterative algorithm with errors. It contains well nown iterations as special case. Such as, the modified Mann iteration see, [19, the modified Ishiawa iteration see, [1, the three-step iteration see, [, the multistep iteration see, [5. The purpose of this paper is to study the multi-step iterative algorithm with bounded errors 1.5 for a finite family of generalized asymptotically quasi-nonexpansive mappings to converge to common fixed points in Banach spaces. The results obtained in this paper extend and improve the corresponding results of [1,, 5, 7, 8, 9, 11, 17, and many others. Preliminaries The following lemmas will be used to prove the main results of this paper: Lemma.1. see [0 Let {a n }, {b n } and {δ n } be sequences of nonnegative real numbers satisfying the inequality a n δ n a n + b n, n 1. If n=1 δ n < and n=1 b n <, then lim a n exists. In particular, if {a n } has a subsequence converging to zero, then lim a n = 0. Lemma.. Schu [19 Let E be a uniformly convex Banach space and 0 < a t n b < 1 for all n 1. Suppose that {x n } and {y n } are sequences in E satisfying x n r, y n r and lim t n x n + 1 t n y n = r for some r 0. Then lim x n y n = 0. Recall that the following:

3 5 Gurucharan Singh Saluja A family {T i : i = 1,,..., } of self-mappings of K with F = i=1 F T i is said to satisfy the following conditions. 1 Condition A [1. If there is a nondecreasing function f : [0, [0, with f0 = 0 and fr > 0 for all r 0. such that 1/ i=1 x T ix fdx, F for all x K, where dx, F = inf{ x p : p F}. Condition B [1. If there is a nondecreasing function f : [0, [0, with f0 = 0 and fr > 0 for all r 0. such that max 1i { x T i x } fdx, F for all x K. Condition C [1. If there is a nondecreasing function f : [0, [0, with f0 = 0 and fr > 0 for all r 0. such that x T l x } fdx, F for all x K and for at least one T l, l = 1,,...,. Note that condition B and C are equivalent, condition B reduces to condition A [16 when all but one T l s are identities, and in addition, it also condition A. It is well nown that every continuous and demicompact mapping must satisfy condition A see [16. Since every completely continuous mapping T : K K is continuous and demicompact so that it satisfies condition A. Thus we will use condition C instead of the demicompactness and complete continuity of a family {T i : i = 1,,..., }. Let K be a nonempty closed convex subset of a Banach space E. Then I T is demiclosed at zero if, for any sequence {x n } in K, condition x n x wealy and lim x n T x n = 0 implies I T x = 0. Main Results In this section, we prove strong convergence theorems of multi-step iterative algorithm with bounded errors for a finite family of generalized asymptotically quasi-nonexpansive mappings in a real Banach space. Theorem.1. Let E be a real arbitrary Banach space, K be a nonempty closed convex subset of E. Let {T i : i = 1,,..., }: K K be a finite family of generalized asymptotically quasi-nonexpansive mappings. Let {x n } be the sequence defined by 1.5 with n=1 r in <, n=1 s in < and n=1 c in < for all i = 1,,...,. If F = i=1 F T i. Then the sequence {x n } converges strongly to a common fixed point of {T i : i = 1,,..., } if and only if lim inf dx n, F = 0, where dx, F denotes the distance between x and the set F. Proof. The necessity is obvious and it is omitted. Now we prove the sufficiency. Since {u in, i = 1,,...,, n 1} are bounded sequences in K, therefore there exists a M > 0, such that M = max { sup u in p, i = 1,,..., n 1 }. Let p F, r n = max{r in : i = 1,,..., } and s n = max{s in : i = 1,,..., } for all n. Since n=1 r in < and n=1 s in <, for all i = 1,,...,, therefore n=1 r n < and

4 Strong convergence of multi-step iterates... 5 n=1 s n <. For each n 1, from 1.4 and 1.5, we note that y 1n p = a 1n x n + b 1n T n 1 x n + c 1n u 1n p a 1n x n p + b 1n T1 n x n p + c 1n u 1n p [ a 1n x n p + b 1n 1 + r 1n x n p + s 1n +c 1n u 1n p a 1n x n p + b 1n [ 1 + r n x n p + s n = +c 1n u 1n p a 1n + b 1n 1 + r n x n p + b 1n s n + c 1n M 1 c 1n 1 + r n x n p + b 1n s n + c 1n M 1 + r n x n p + s n + c 1n M = 1 + r n x n p + A 1n.1 where A 1n = s n + c 1n M, since by assumption n=1 s n < and n=1 c 1n <, it follows that n=1 A 1n <. Furthermore, from 1.5 and.1, we obtain y n p = a n x n + b n T n y 1n + c n u n p a n x n p + b n T n y 1n p + c n u n p [ a n x n p + b n 1 + r n y 1n p + s n +c n u n p a n x n p + b n [ 1 + r n y 1n p + s n +c n u n p a n x n p + b n 1 + r n y 1n p + b n s n + c n M a n x n p + b n 1 + r n [1 + r n x n p + A 1n +b n s n + c n M a n + b n 1 + rn x n p + b n 1 + r n A 1n +b n s n + c n M = 1 c n 1 + rn x n p + b n 1 + r n A 1n +b n s n + c n M 1 + r n x n p r n A 1n + s n + c n M 1 + r n x n p + A n. where A n = 1 + r n A 1n + s n + c n M, since by assumption n=1 r n <, n=1 s n <, n=1 c n < and n=1 A 1n <, it follows that n=1 A n <. Similarly, using 1.5 and

5 54 Gurucharan Singh Saluja., we see that y n p = a n x n p + b n T n y n p + c n u n p a n x n p + b n T n y n p + c n u n p [ a n x n p + b n 1 + r n y n p + s n +c n u n p a n x n p + b n [ 1 + r n y n p + s n +c n u n p a n x n p + b n 1 + r n y n p + b n s n + c n M a n x n p + b n 1 + r n [1 + r n x n p + A n +b n s n + c n M a n + b n 1 + rn x n p + b n 1 + r n A n +b n s n + c n M = 1 c n 1 + rn x n p + b n 1 + r n A n +b n s n + c n M 1 + r n x n p r n A n + s n + c n M 1 + r n x n p + A n. where A n = 1 + r n A n + s n + c n M, since by assumption n=1 r n <, n=1 s n <, n=1 c n < and n=1 A n <, it follows that n=1 A n <. By continuing the above process, there are nonnegative real sequences {A in } in [0, such that n=1 A in < and y in p 1 + r n i x n p + A in, i = 1,,...,..4 For the case i =, from 1.5 and.4, we have x n+1 p 1 + r n x n p + A n, n 1 and p F,.5 where A n = 1+r n A 1n +s n +c n M, since by assumption n=1 r n <, n=1 s n <, n=1 c n < and n=1 A 1n <, it follows that n=1 A n <. This implies that dx n+1, F 1 + r n dx n, F + A n = 1... t rn t dx n, F t! t=1 +A n..6 Since n=1 r n <, it follows that n=1 t= t + 1/t!rt n < and n=1 A n <. Therefore, applying Lemma.1 to the inequality.6, we conclude that lim dx n, F exists. Since by hypothesis lim inf dx n, F = 0, so by Lemma.1, we have lim dx n, F = 0..7 Next, we will prove that {x n } is a Cauchy sequence. If x 0, then 1 + x e x and so,

6 Strong convergence of multi-step iterates x e x, for = 1,,.... Thus, from.5, it follows that x n+m p 1 + r n+m 1 x n+m 1 p + A n+m exp { r n+m 1 } xn+m 1 p + A n+m 1 exp { n+m 1 n+m 1 } r i xn p + i=n exp { } r i xn p + i=1 Q x n p + i=n i=n A i A i A i.8 where Q = exp { i=1 r i}, for all p F and m, n N. Since lim dx n, F = 0, for each ε > 0, there exists a natural number n 1 such that for n n 1, i=n dx n, F < ε 41 + Q and n+m 1 i=n 1 A i < ε..9 Hence, there exists a point q F such that x n1 q < By.8,.9 and.10, for all n n 1 and m 1, we have ε 1 + Q..10 x n+m x n x n+m q + x n q Q x n1 q + A i + x n1 q i=n Q x n1 q + i=n 1 A i ε < 1 + Q. 1 + Q + ε = ε..11 This implies that {x n } is a Cauchy sequence. Since E is complete, there exists a p 1 E such that x n p 1 as n. Now we have to prove that p 1 is a common fixed point of {T i : i = 1,,..., }, that is, p 1 F. By contradiction, we assume that p 1 is not in F. Since F = i=1 F T i is closed in Banach spaces, dp 1, F > 0. So for all p F, we have By the arbitrary of p F, we now that p 1 p p 1 x n + x n p..1 dp 1, F p 1 x n + dx n, F..1 By lim dx n, F = 0, above inequality and x n p 1 as n, we have dp 1, F = 0,.14 which contradicts dp 1, F > 0. Thus p 1 is a common fixed point of the mappings {T i : i = 1,,..., }. This completes the proof. Theorem.. Let K be a nonempty compact convex subset of a uniformly convex Banach space E and for i = 1,,...,, let T i : K K be a finite family of uniformly L i, α i -Lipschitz

7 56 Gurucharan Singh Saluja and generalized asymptotically quasi-nonexpansive mappings. Let {x n } be the sequence defined by 1.5 with n=1 r in <, n=1 s in <, n=1 c in < and 0 < α b in β < 1 for all i = 1,,...,. If F = i=1 F T i. Then the sequence {x n } converges strongly to a common fixed point of the mappings {T i : i = 1,,..., }. Proof. Let p F, r n = max{r in : i = 1,,..., } and s n = max{s in : i = 1,,..., } for all n. From Theorem.1, we have that lim x n p exists for all p F. Let lim x n p = R for some R > 0. Then, from.1, we note that y 1n p 1 + r n x n p + A 1n x n p = R,.15 and T1 n x n p 1 + r 1n x n p + s 1n 1 + r n x n p + s n x n p = R,.16 and lim y 1n p = lim a 1nx n + b 1n T1 n x n + c 1n u 1n p = lim 1 b 1n c 1n x n + b 1n T n 1 x n + c 1n u 1n p = lim 1 b 1nx n p + c 1n u 1n x n + b 1n T n 1 x n p + c 1n u 1n x n =R..17 Again since lim x n p exists, so {x n } is a bounded sequence in K. By virtue of condition n=1 c in < for all i = 1,,..., and the boundedness of the sequence {x n } and {u 1n }, we have It follows from.16 that x n p + c 1n u 1n x n x n p T1 n x n p + c 1n u 1n x n T1 n x n p c 1n u 1n x n + R, p F Therefore, from and Lemma. we now that c 1n u 1n x n 1 + r 1n x n p + s 1n c 1n u 1n x n 1 + r n x n p + s n c 1n u 1n x n R, p F..19 lim T 1 n x n x n = 0..0

8 Strong convergence of multi-step iterates Again from., we note that and from.15, we note that Next, consider Also, and y n p 1 + r n x n p + A n T n y 1n p x n p = R, r n y 1n p + s n 1 + r n y 1n p + s n y 1n p = R.. T n y 1n p + c n u n x n T n y 1n p c n u n x n 1 + r n y 1n p + s n c n u n x n 1 + r n y 1n p + s n c n u n x n x n p + c n u n x n x n p R, p F.. lim y n p = lim a nx n + b n T n y 1n + c n u n p c n u n x n + R, p F,.4 = lim 1 b n c n x n + b n T n y 1n + c n u n p = lim 1 b nx n p + c n u n x n + b n T n y 1n p + c n u n x n =R..5 Therefore, from. -.5 and Lemma. we now that lim T n y 1n x n = 0..6 Now, we shall show that lim T n y n x n = 0. For each n 1, x n p T n y 1n x n + T n y 1n p T n y 1n x n r n y 1n p + s n T n y 1n x n r n y 1n p + s n..7

9 58 Gurucharan Singh Saluja Using.6, we have It follows from.15 that This implies that R = lim x n p lim inf y 1n p. R = lim x n p lim inf y 1n p y 1n p R..8 lim y 1n p = R..9 On the other hand, we have y n p 1 + r n x n p + A n, n 1, where n=1 A n <. Therefore y n p 1 + r n x n p + A n, R,.0 and hence T n y n p 1 + r n y n p + s n 1 + r n y n p + s n x n p = R..1 Next, consider Also, T n y n p + c n u n x n T n y n p c n u n x n 1 + r n y n p + s n c n u n x n 1 + r n y n p + s n c n u n x n x n p + c n u n x n x n p R, p F.. c n u n x n + R, p F,.

10 Strong convergence of multi-step iterates and lim y n p = lim a nx n + b n T n y n + c n u n p = lim 1 b n c n x n + b n T n y n + c n u n p = lim 1 b nx n p + c n u n x n + b n T n y n p + c n u n x n =R..4 Therefore, from. -.4 and Lemma. we now that lim T n y n x n = 0..5 Similarly, by using the same argument as in the proof above, we have lim Ti n y i 1n x n = 0,.6 for all i =,,...,. Since K is compact, {x n } n=1 has a convergent subsequence {x n j } j=1. Let Then from 1.5 and.6, we have lim x n j = p..7 xnj+1 x nj bnj T n j y 1n j x nj + c nj unj x nj From 1.5 and.0, we have Again from.19 and.7, we have Since lim x nj+1 = p, we have From.8,.40 and.41, we have From.6 and.7, we have 0, as j..8 y 1n x n b 1n T n 1 x n x n + c 1n u 1n x n 0, as n..9 lim T nj 1 x n j = p..40 lim T nj+1 1 x nj+1 = p p T 1 p p T nj+1 1 x nj+1 + T nj+1 1 x nj+1 T nj+1 1 x nj + T nj+1 1 x nj T 1 p p T nj+1 + L 1 x nj+1 x nj+1 1 x nj+1 +L 1 T n j 1 x n j p α 1 0 as j..4 α 1 lim T nj y 1n j = p..4

11 60 Gurucharan Singh Saluja Since lim x nj+1 = p, we have From.8,.9,.4 and.44, we have Now, from 1.5 and.6, we have Again from.5 and.7, we have Since lim x nj+1 = p, we have lim T nj+1 y 1nj+1 = p p T p p T nj+1 y 1nj+1 + T nj+1 y 1nj+1 T nj+1 x nj+1 + T nj+1 x nj+1 T nj+1 x nj + T nj+1 x nj T nj+1 y 1nj + T nj+1 y 1nj T p p T nj+1 + L y1nj+1 x nj+1 y 1nj+1 +L xnj+1 x nj α + L xnj y 1nj α +L T n j y 1n j p α 0 as j..45 α y n x n b n T n y 1n x n + c n u n x n From.8,.46,.47 and.48, we have 0, as n..46 lim T nj y n j = p..47 lim T nj+1 y nj+1 = p p T p p T nj+1 y nj+1 + T nj+1 y nj+1 T nj+1 x nj+1 + T nj+1 x nj+1 T nj+1 x nj + T nj+1 x nj T nj+1 y nj + T nj+1 y nj T p p T nj+1 + L ynj+1 x nj+1 y nj+1 +L x nj+1 x nj α + L x nj y nj α +L T n j y n j p α 0 as j..49 Similarly, from 1.5 and.6, we have y 1n x n b 1n T n 1 y n x n + c 1n u 1n x n 0, as n..50 α

12 Strong convergence of multi-step iterates Again from.6 and.7, we have Since lim x nj+1 = p, we have From.8,.50,.51 and.5, we have Hence lim T nj y 1n j = p..51 lim T nj+1 y 1nj+1 = p..5 0 p T p p T nj+1 y 1nj+1 + T nj+1 y 1nj+1 T nj+1 x nj+1 + T nj+1 x nj+1 T nj+1 x nj + T nj+1 x nj T nj+1 y 1nj + T nj+1 y 1nj T p p T nj+1 + L y 1nj+1 x nj+1 y 1nj+1 +L xnj+1 x nj α + L xnj y 1nj α +L T n j y 1n j p α 0 as j..5 lim p T ip = 0 i = 1,,...,..54 Thus p is a common fixed point of the mappings {T i : i = 1,,..., }. Since the subsequence {x nj } j=1 of {x n} n=1 converges to p and lim x n p exists, we conclude that lim x n = p. This completes the proof. Theorem.. Let K be a nonempty closed convex subset of a uniformly convex Banach space E and for i = 1,,...,, let T i : K K be a finite family of uniformly L i, α i -Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Let {x n } be the sequence defined by 1.5 with n=1 r in <, n=1 s in <, n=1 c in < and 0 < α b in β < 1 for all i = 1,,...,. If F = i=1 F T i. Then lim T i x n x n = 0 for all i = 1,,...,. Proof. From Theorem. equation.6, we have lim T n i y i 1n x n = 0,.55 for all i =,,...,. In the case i = 1 that lim T n 1 x n x n = 0, where y 0n = x n. For i =,,...,, we obtain from.55 that Therefore T n i x n x n T n i x n T n i y i 1n + T n i y i 1n x n L i xn y i 1n α i + T n i y i 1n x n α L i a i 1n T n i 1 y i n x n + ci 1n ui 1n x n αi + T n i y i 1n x n 0 as n..56 lim T i n x n x n = 0, i = 1,,...,..57

13 6 Gurucharan Singh Saluja This completes the proof. Remar.1. Theorem.1 extend and improve the corresponding results of Khan et al. [5 and Tang and Peng [ to the case of more general class of asymptotically quasi-nonexpansive or uniformly quasi-lipschitzian mappings considered in this paper. Remar.. Theorem.1 also extend and improve the corresponding results of [1,, 7, 8, 11, 17. Especially Theorem.1 extends and improves Theorem 1 and in [8, Theorem 1 in [7 and Theorem. in [17 in the following ways: 1 The asymptotically quasi-nonexpansive mapping in [7, [8 and [17 is replaced by finite family of generalized asymptotically quasi-nonexpansive mappings. The usual Ishiawa [4 iteration scheme in [7, the usual modified Ishiawa iteration scheme with errors in [8 and the usual modified Ishiawa iteration scheme with errors for two mappings in [17 are extended to the multi-step iteration scheme with errors for a finite family of mappings. Remar.. Theorem.1 also extends and improves Theorem.0. in [1 in the following aspects: 1 Two asymptotically quasi-nonexpansive mappings in [1 is replaced by finite family of generalized asymptotically quasi-nonexpansive mappings. The usual modified Ishiawa iteration scheme with errors in the sense of Liu [6 for two mappings in [1 is extended to the multi-step iteration scheme with errors in the sense of Xu [4 for a finite family of mappings. Remar.4. Theorem. extends and improves the corresponding result of [9 in the following aspects: 1 The asymptotically quasi-nonexpansive mapping in [9 is replaced by finite family of generalized asymptotically quasi-nonexpansive mappings. The usual modified Ishiawa iteration scheme with errors in [9 is extended to the multistep iteration scheme with errors for a finite family of mappings. Remar.5. Theorem.1 also extends the corresponding result of [ to the case of more general class of asymptotically nonexpansive mappings and multi-step iteration scheme with errors for a finite family of mappings considered in this paper. 4 Application In this section we give an application of the convergence criteria established in Theorem.1 is given below to obtain yet another strong convergence result in our setting. Theorem 4.1. Let K be a nonempty closed convex subset of a uniformly convex Banach space E and for i = 1,,...,, let T i : K K be a finite family of uniformly L i, α i -Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Let {x n } be the sequence defined by 1.5 with n=1 r in <, n=1 s in <, n=1 c in < and 0 < α b in β < 1 for all i = 1,,...,. Assume that F = i=1 F T i and the family {T i : i = 1,,..., } satisfies condition C. Then the sequence {x n } converges strongly to a common fixed point of the family of mappings {T i : i = 1,,..., }. Proof. From Theorem. we have lim T n i x n x n = 0 for all i = 1,,..., and the family {T i : i = 1,,..., } satisfying condition C, we have that lim inf fdx n, F = 0. Since f is a nondecreasing function with f0 = 0 and fr > 0 for all r 0,, it follows that lim inf dx n, F = 0. Now by Theorem.1, x n F, i.e., {x n } converges strongly to a common fixed point of the family of mappings {T i : i = 1,,..., }. This completes the proof. Example 1. Let E be the real line with the usual norm. and K = [0, 1. Define T : K K

14 Strong convergence of multi-step iterates... 6 by T x = { x/, if x 0, 0, if x = 0. Obviously T 0 = 0, i.e., 0 is a fixed point of the mapping T. Thus, T is quasi-nonexpansive. It follows that T is uniformly quasi-1 Lipschitzian and asymptotically quasi-nonexpansive with constant sequence { n } = {1} for each n 1 and hence it is generalized asymptotically quasinonexpansive mapping with constant sequences { n } = {1} and {s n } = {0} for each n 1 but the converse is not true in general. References [1 C.E. Chidume abd Bashir Ali, Convergence theorems for finite families of asymptotically quasi-nonexpansive mappings, J. Inequalities and Applications, Article ID 68616, Vol.007, 10 pages, 007. [ M.K. Ghosh and L. Debnath, Convergence of Ishiawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. 07, [ K. Goebel and W.A. Kir, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 5, [4 S. Ishiawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44, [5 A.R. Khan, A.A. Domlo and H. Fuhar-ud-din, Common fixed points of Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 41, [6 L.S. Liu, Ishiawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 1941, [7 Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 59, [8 Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl. 59, [9 Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member of uniformly convex Banach spaces, J. Math. Anal. Appl. 66, [10 W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4, [11 W.V. Petryshyn and T.E. Williamson, Strong and wea convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 4, [1 G.S. Saluja, Strong convergence theorem for two asymptotically quasi-nonexpansive mappings with errors in Banach space, Tamang J. Math. 81, [1 G.S. Saluja, Approximating common fixed point of three-step iterative sequence with errors for asymptotically nonexpansive non-self mappings, JP J. Fixed Point Theory and Applications, [14 G.S. Saluja, Approximating common fixed points of finite family of asymptotically nonexpansive non-self mappings, Filomat, [15 G.S. Saluja and H.K. Nashine, Common fixed point of multi-step iteration scheme with errors for finite family of asymptotically quasi-nonexpansive mappings, Nonlinear Funct. Anal. Appl. 144, [16 H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44, [17 N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, Article ID 18909, Vol.006, Pages [18 N. Shahzad and H. Zegeye, Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps, Appl. Math. Comp. 189,

15 64 Gurucharan Singh Saluja [19 J. Schu, Wea and strong convergence theorems to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 4, [0 K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishiawa iteration process, J. Math. Anal. Appl. 178, [1 K.K. Tan and H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 1, [ Y.C. Tang and J.G. Peng, Approximation of common fixed points for a finite family of uniformly quasi-lipschitzian mappings in Banach spaces, Thai. J. Maths. 81, [ B. Xu and M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67, [4 Y. Xu, Ishiawa and Mann iterative process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 41, Author information Gurucharan Singh Saluja, Department of Mathematics and Information Technology, Govt. Nagarjuna P.G. College of Science, Raipur C.G.,. saluja_196@rediffmail.com, saluja196@gmail.com Received: March, 01 Accepted: July, 01