International Journal of Analysis and Applications ISSN 2291-8639 Volume 4, Number 1 (2014), 45-57 http://www.etamaths.com CONVERGENCE THEOREMS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE TYPE MAPPINGS IN CONVEX METRIC SPACES G. S. SALUJA Abstract. The aim of this paper to study a Noor-type iteration process with errors for approximating common fixed point of a finite family of uniformly L-Lipschitzian asymptotically quasi-nonexpansive type mappings in the frame- work of convex metric spaces. We give a necessary and sufficient condition for strong convergence of said iteration scheme involving a finite family of above said mappings and also establish a strong convergence theorem by using condi- tion (A). The results presented in this paper extend, improve and unify some existing results in the previous work. 1. Introduction and Preliminaries Throughout this paper, we assume that E is a metric space, F(Ti) = {x ∈ E : Tix = x} be the set of all fixed points of the mappings Ti (i = 1, 2, . . . ,N), D(T) be the domain of T and N is the set of all positive integers. The set of common fixed points of Ti (i = 1, 2, . . . ,N) denoted by F , that is, F = ∩Ni=1F(Ti). Definition 1.1. (See [1]) Let T : D(T) ⊂ E → E be a mapping. (1) The mapping T is said to be L-Lipschitzian if there exists a constant L > 0 such that d(Tx,Ty) ≤ Ld(x,y), ∀x, y ∈ D(T).(1.1) (2) The mapping T is said to be nonexpansive if d(Tx,Ty) ≤ d(x,y), ∀x, y ∈ D(T).(1.2) (3) The mapping T is said to be quasi-nonexpansive if F(T) 6= ∅ and d(Tx,p) ≤ d(x,p), ∀x ∈ D(T), ∀p ∈ F(T).(1.3) 2010 Mathematics Subject Classification. 47H05, 47H09, 47H10. Key words and phrases. Asymptotically quasi-nonexpansive type mapping, Noor-type iteration process with errors, common fixed point, strong convergence, convex metric space. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 45 46 SALUJA (4) The mapping T is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂ [1,∞) with limn→∞kn = 1 such that d(Tnx,Tny) ≤ knd(x,y), ∀x, y ∈ D(T), ∀n ∈ N.(1.4) (5) The mapping T is said to be asymptotically quasi-nonexpansive if F(T) 6= ∅ and there exists a sequence {kn}⊂ [1,∞) with limn→∞kn = 1 such that d(Tnx,p) ≤ knd(x,p), ∀x ∈ D(T), ∀p ∈ F(T), ∀n ∈ N.(1.5) (6) The mapping T is said to be asymptotically nonexpansive type, if lim sup n→∞ { sup x,y∈D(T) ( d(Tnx,Tny) −d(x,y) )} ≤ 0.(1.6) (7) The mapping T is said to be asymptotically quasi-nonexpansive type, if F(T) 6= ∅ and lim sup n→∞ { sup x∈D(T),p∈F(T) ( d(Tnx,p) −d(x,p) )} ≤ 0.(1.7) Remark 1.2. It is easy to see that if F(T) is nonempty, then nonexpansive mapping, quasi-nonexpansive mapping, asymptotically nonexpansive mapping, asymptotical- ly quasi-nonexpansive mapping and asymptotically nonexpansive type mapping all are the special cases of asymptotically quasi-nonexpansive type mappings. In recent years, the problem concerning convergence of iterative sequences (and sequences with errors) for asymptotically nonexpansive mappings or asymptotically quasi-nonexpansive mappings converging to some fixed points in Hilbert spaces or Banach spaces have been considered by many authors. In 1973, Petryshyn and Williamson [13] obtained a necessary and sufficient con- dition for Picard iterative sequences and Mann iterative sequences to converge to a fixed point for quasi-nonexpansive mappings. In 1994, Tan and Xu [16] also proved some convergence theorems of Ishikawa iterative sequences satisfies Opi- al’s condition or has a Frechet differential norm. In 1997, Ghosh and Debnath [5] extended the result of Petryshyn and Williamson [13] and gave a necessary and sufficient condition for Ishikawa iterative sequences to converge to a fixed point of quasi-nonexpansive mappings. Also in 2001 and 2002, Liu [10, 11, 12] obtained some necessary and sufficient conditions for Ishikawa iterative sequences or Ishikawa iterative sequences with errors to converge to a fixed point for asymptotically quasi- nonexpansive mappings. In 2004, Chang et al. [1] extended and improved the result of Liu [12] in convex metric space. Further in the same year, Kim et al. [8] gave the necessary and suffi- cient conditions for asymptotically quasi-nonexpansive mappings in convex metric CONVERGENCE THEOREMS 47 spaces which generalized and improved some previous known results. Very recently, Tian and Yang [18] gave some necessary and sufficient conditions for a new Noor-type iterative sequences with errors to approximate a common fixed point for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. The purpose of this paper is to give some necessary and sufficient conditions for Noor-type iteration process with errors to approximate a common fixed point for a finite family of uniformly L-Lipschitzian asymptotically quasi-nonexpansive type mappings in convex metric spaces. The results presented in this paper gen- eralize, improve and unify some main results of [1]-[3], [5]-[8], [10]-[17], [19] and [21]. Let T be a given self mapping of a nonempty convex subset C of an arbitrary normed space. The sequence {xn}∞n=0 defined by x0 ∈ C and xn+1 = αnxn + βnTyn + γnun, n ≥ 0, yn = anxn + bnTzn + cnvn, zn = dnxn + enTxn + fnwn,(1.8) is called the Noor-type iterative procedure with errors [2], where αn,βn,γn,an,bn,cn, dn,en and fn are appropriate sequences in [0, 1] with αn +βn +γn = an +bn +cn = dn + en + fn = 1, n ≥ 0 and {un}, {vn} and {wn} are bounded sequences in C. If dn = 1(en = fn = 0), n ≥ 0, then (1.8) reduces to the Ishikawa iterative procedure with errors [20] defined as follows: x0 ∈ C and xn+1 = αnxn + βnTyn + γnun, n ≥ 0, yn = anxn + bnTxn + cnvn.(1.9) If an = 1(bn = cn = 0), then (1.9) reduces to the following Mann type iterative procedure with errors [20]: x0 ∈ C and xn+1 = αnxn + βnTyn + γnun, n ≥ 0.(1.10) For the sake of convenience, we first recall some definitions and notations. Definition 1.3. (See [1]) Let (E,d) be a metric space and I = [0, 1]. A mapping W : E3 × I3 → E is said to be a convex structure on E if it satisfies the following condition: d(u,W(x,y,z; α,β,γ)) ≤ αd(u,x) + βd(u,y) + γd(u,z), for any u,x,y,z ∈ E and for any α,β,γ ∈ I with α + β + γ = 1. If (E,d) is a metric space with a convex structure W, then (E,d) is called a convex metric space and denotes it by (E,d,W). Let (E,d) be a convex metric space, a nonempty subset C of E is said to be convex if W(x,y,z,λ1,λ2,λ3) ∈ C, ∀(x,y,z,λ1,λ2,λ3) ∈ C3 × I3. 48 SALUJA Remark 1.4. It is easy to prove that every linear normed space is a convex metric space with a convex structure W(x,y,z; α,β,γ) = αx + βy + γz, for all x,y,z ∈ E and α,β,γ ∈ I with α + β + γ = 1. But there exist some convex metric spaces which can not be embedded into any linear normed spaces (see, Takahashi [15]). Definition 1.5. Let (E,d,W) be a convex metric space and Ti : E → E be a finite family of asymptotically quasi-nonexpansive type mappings with i = 1, 2, . . . ,N. Let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en} and {fn} be nine sequences in [0, 1] with αn + βn + γn = an + bn + cn = dn + en + fn = 1, n = 0, 1, 2, . . . .(1.11) For a given x0 ∈ E, define a sequence {xn} as follows: xn+1 = W(xn,T n n yn,un; αn,βn,γn), n ≥ 0, yn = W(f(xn),T n n zn,vn; an,bn,cn), zn = W(f(xn),T n n xn,wn; dn,en,fn),(1.12) where Tnn = T n n(mod N) , f : E → E is a Lipschitz continuous mapping with a Lip- schitz constant ξ > 0 and {un}, {vn}, {wn} are any given three sequences in E. Then {xn} is called the Noor-type iterative sequence with errors for a finite family of asymptotically quasi-nonexpansive type mappings {Ti}Ni=1. If f = I (the identity mapping on E) in (1.12), then the sequence {xn} defined by (1.12) can be written as follows: xn+1 = W(xn,T n n yn,un; αn,βn,γn), n ≥ 0, yn = W(xn,T n n zn,vn; an,bn,cn), zn = W(xn,T n n xn,wn; dn,en,fn),(1.13) If dn = 1(en = fn = 0) for all n ≥ 0 in (1.12), then zn = xn for all n ≥ 0 and the sequence {xn} defined by (1.12) can be written as follows: xn+1 = W(xn,T n n yn,un; αn,βn,γn), n ≥ 0, yn = W(f(xn),T n n xn,vn; an,bn,cn).(1.14) If f = I and dn = 1(en = fn = 0) for all n ≥ 0, then the sequence {xn} defined by (1.12) can be written as follows: xn+1 = W(xn,T n n yn,un; αn,βn,γn), n ≥ 0, yn = W(xn,T n n xn,vn; an,bn,cn),(1.15) which is the Ishikawa type iterative sequence with errors considered in [17]. Fur- ther, if f = I and dn = an = 1(en = fn = bn = cn = 0) for all n ≥ 0, then zn = yn = xn for all n ≥ 0 and (1.12) reduces to the following Mann type iterative sequence with errors [17]: xn+1 = W(xn,T n n yn,un; αn,βn,γn), n ≥ 0.(1.16) CONVERGENCE THEOREMS 49 Recall that a family {Ti : i ∈ N = 1, 2, . . . ,N} of N asymptotically quasi- nonexpansive type self mappings of C with F = ∩Ni=1F(Ti) 6= ∅ is said to satisfy condition (A) ([4]) if there exists a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(t) > 0 for all t ∈ (0,∞) such that ‖x−Tx‖ ≥ f(d(x,F)) for all x ∈ C holds for at least one T ∈{Ti : i ∈N}. In the sequel, we shall need the following lemmas. Lemma 1.6. (See [11]) Let {pn}, {qn}, {rn} be three nonnegative sequences of real numbers satisfying the following conditions: pn+1 ≤ (1 + qn)pn + rn, n ≥ 0, ∞∑ n=0 qn < ∞, ∞∑ n=0 rn < ∞.(1.17) Then (1) limn→∞pn exists. (2) In addition, if lim infn→∞pn = 0, then limn→∞pn = 0. Lemma 1.7. Let (E,d,W) be a complete convex metric space and C be a nonemp- ty closed convex subset of E. Let Ti : C → C be a finite family of uniformly L- Lipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,N such that F = ∩Ni=1F(Ti) 6= ∅ and f : C → C be a contractive mapping with a contractive constant ξ ∈ (0, 1). Let {xn} be the iterative sequence with errors de- fined by (1.12) and {un}, {vn}, {wn} be three bounded sequences in C. Let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. Then the following conclusions hold: (1) for all p ∈ F and n ≥ 0, d(xn+1,p) ≤ (1 + 3βn)d(xn,p) + 3Kσn + Mσn,(1.18) where σn = βn + γn for all n ≥ 0 and M = sup p∈F, n≥0 { d(un,p) + d(vn,p) + d(wn,p) + 2d(f(p),p) } , (2) there exists a constant M1 > 0 such that d(xn+m,p) ≤ M1d(xn,p) + 3KM1 n+m−1∑ k=n σk + MM1 n+m−1∑ k=n σk, ∀p ∈ F,(1.19) for all n,m ≥ 0. 50 SALUJA Proof. (1) Let p ∈ F. It follows from (1.7) that lim sup n→∞ { sup x∈E, p∈F ( d(Tnx,p) −d(x,p) )} ≤ 0. This implies that for any given K > 0, there exists a positive integer n0 such that for n ≥ n0 we have sup x∈E, p∈F ( d(Tnx,p) −d(x,p) ) < K.(1.20) Since {xn},{yn},{zn}⊂ E, we have d(Tnn xn,p) −d(xn,p) < K, ∀p ∈ F, ∀n ≥ n0 d(Tnn yn,p) −d(yn,p) < K, ∀p ∈ F, ∀n ≥ n0 d(Tnn zn,p) −d(zn,p) < K, ∀p ∈ F, ∀n ≥ n0.(1.21) Thus for each n ≥ 0 and for any p ∈ F, using (1.12), and (1.21), we have d(xn+1,p) = d(W(xn,T n n yn,un; αn,βn,γn),p) ≤ αnd(xn,p) + βnd(Tnn yn,p) + γnd(un,p) ≤ αnd(xn,p) + βn[d(yn,p) + K] + γnd(un,p) ≤ αnd(xn,p) + βnd(yn,p) + βnK + γnd(un,p),(1.22) and d(yn,p) = d(W(f(xn),T n n zn,vn; an,bn,cn),p) ≤ and(f(xn),p) + bnd(Tnn zn,p) + cnd(vn,p) ≤ and(f(xn),f(p)) + and(f(p),p) +bn[d(zn,p) + K] + cnd(vn,p) ≤ anξd(xn,p) + and(f(p),p) + bnd(zn,p) +bnK + cnd(vn,p),(1.23) and d(zn,p) = d(W(f(xn),T n n xn,wn; dn,en,fn),p) ≤ dnd(f(xn),p) + end(Tnn xn,p) + fnd(wn,p) ≤ dnd(f(xn),f(p)) + dnd(f(p),p) +en[d(xn,p) + K] + fnd(wn,p) ≤ dnξd(xn,p) + dnd(f(p),p) + end(xn,p) +enK + fnd(wn,p) ≤ (dnξ + en)d(xn,p) + dnd(f(p),p) +enK + fnd(wn,p).(1.24) CONVERGENCE THEOREMS 51 Substituting (1.23) into (1.22) and simplifying it, we have d(xn+1,p) ≤ αnd(xn,p) + βn [ anξd(xn,p) + and(f(p),p) +bnd(zn,p) + bnK + cnd(vn,p) ] + βnK + γnd(un,p) ≤ (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + bnβnK +bnβnd(zn,p) + cnβnd(vn,p) + βnK + γnd(un,p) = (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + (1 + bn)βnK +bnβnd(zn,p) + cnβnd(vn,p) + γnd(un,p) ≤ (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + 2βnK +bnβnd(zn,p) + cnβnd(vn,p) + γnd(un,p).(1.25) Substituting (1.24) into (1.25) and simplifying it, we have d(xn+1,p) ≤ (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + 2βnK +bnβn [ (dnξ + en)d(xn,p) + dnd(f(p),p) + enK +fnd(wn,p) ] + cnβnd(vn,p) + γnd(un,p) ≤ [ αn + anβnξ + bnβn(dnξ + en) ] d(xn,p) +βn(an + bndn)d(f(p),p) + βnK(2 + bnen) +bnβnfnd(wn,p) + cnβnd(vn,p) + γnd(un,p) ≤ [ αn + βn(anξ + bndnξ + bnen) ] d(xn,p) +2βnd(f(p),p) + 3βnK + βnd(wn,p) +βnd(vn,p) + γnd(un,p) ≤ (1 + 3βn)d(xn,p) + 2βnd(f(p),p) +2γnd(f(p),p) + 3βnK + 3γnK + βnd(wn,p) +γnd(wn,p) + βnd(vn,p) + γnd(vn,p) +βnd(un,p) + γnd(un,p) = (1 + 3βn)d(xn,p) + 3K(βn + γn) + 2(βn + γn)d(f(p),p) +(βn + γn) [ d(un,p) + d(vn,p) + d(wn,p) ] = (1 + 3βn)d(xn,p) + 3K(βn + γn) + (βn + γn) [ d(un,p) +d(vn,p) + d(wn,p) + 2d(f(p),p) ] = (1 + 3βn)d(xn,p) + 3Kσn + Mσn, ∀n ≥ 0, p ∈ F,(1.26) where M = sup p∈F n≥0 { d(un,p) + d(vn,p) + d(wn,p) + 2d(f(p),p) } , σn = βn + γn. This completes the proof of part (1). 52 SALUJA (2) Since 1 + x ≤ ex for all x ≥ 0, it follows from (1.26) that, for n,m ≥ 0 and p ∈ F , we have d(xn+m,p) ≤ (1 + 3βn+m−1)d(xn+m−1,p) + 3Kσn+m−1 + Mσn+m−1 ≤ e3βn+m−1d(xn+m−1,p) + 3Kσn+m−1 + Mσn+m−1 ≤ e3βn+m−1 [ e3βn+m−2d(xn+m−2,p) + 3Kσn+m−2 + Mσn+m−2 ] +3Kσn+m−1 + Mσn+m−1 ≤ e3(βn+m−1+βn+m−2)d(xn+m−2,p) + 3K [ e3βn+m−1σn+m−2 +σn+m−1 ] + M [ e3βn+m−1σn+m−2 + σn+m−1 ] ≤ . . . ≤ . . . ≤ M1d(xn,p) + 3KM1 n+m−1∑ k=n σk + MM1 n+m−1∑ k=n σk, = M1d(xn,p) + (3K + M)M1 n+m−1∑ k=n σk,(1.27) where M1 = e 3 ∑∞ k=0 βk. This completes the proof of part (2). � 2. Main Results Theorem 2.1. Let (E,d,W) be a complete convex metric space and C be a nonemp- ty closed convex subset of E. Let Ti : C → C be a finite family of uniformly L- Lipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,N such that F = ∩Ni=1F(Ti) 6= ∅ and f : C → C be a contractive mapping with a con- tractive constant ξ ∈ (0, 1). Let {xn} be the iterative sequence with errors defined by (1.12) and {un}, {vn}, {wn} be three bounded sequences in C. Let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be nine sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. Then the sequence {xn} converges to a common fixed point p in F if and only if lim infn→∞d(xn,F) = 0, where d(x,F) = infp∈F d(x,p). Proof. The necessity is obvious. Now, we prove the sufficiency. In fact, from Lemma 1.7, we have d(xn+1,F) ≤ (1 + 3βn)d(xn,F) + (3K + M)σn, ∀n ≥ 0,(2.1) where σn = βn + γn. By conditions (i) and (ii), we know that ∞∑ n=0 σn < ∞, ∞∑ n=0 βn < ∞.(2.2) CONVERGENCE THEOREMS 53 It follows from Lemma 1.6 that limn→∞d(xn,F) exists. Since lim infn→∞d(xn,F) = 0, we have lim n→∞ d(xn,F) = 0.(2.3) Next, we prove that {xn} is a Cauchy sequence in C. In fact, for any given ε > 0, there exists a positive integer n1 ≥ n0 (where n0 is the positive integer appeared in Lemma 1.7) such that for any n ≥ n1, we have d(xn,F) < ε 8M1 , ∞∑ n=n1 σn < ε 12(K + M)M1 , ∀n ≥ 0.(2.4) From (2.4), there exists p1 ∈ F and positive integer n2 ≥ n1 such that d(xn2,p1) < ε 4M1 .(2.5) Thus Lemma 1.7(2) implies that, for any positive integer n,m with n ≥ n2, we have d(xn+m,xn) ≤ d(xn+m,p1) + d(xn,p1) ≤ M1d(xn2,p1) + 3(K + M)M1 n+m−1∑ k=n2 σk +M1d(xn2,p1) + 3(K + M)M1 n+m−1∑ k=n2 σk ≤ 2M1d(xn2,p1) + 6(K + M)M1 n+m−1∑ k=n2 σk < 2M1. ε 4M1 + 6(K + M)M1. ε 12(K + M)M1 < ε.(2.6) This shows that {xn} is a Cauchy sequence in a nonempty closed convex subset C of a complete convex metric space E. Without loss of generality, we can assume that limn→∞xn = q ∈ E. Now we will prove that q ∈ F. Since xn → q and d(xn,F) → 0 as n → ∞, for any given ε1 > 0, there exists a positive integer n2 ≥ n1 ≥ n0 such that for n ≥ n2, we have d(xn,q) < ε1, d(xn,F) < ε1.(2.7) Again from (2.7), there exists q1 ∈ F and positive integer n3 ≥ n2 such that d(xn3,q1) < 2ε1.(2.8) Moreover, it follows from (1.20) that for any n ≥ n3, we have d(Tnq,q1) −d(q,q1) < K.(2.9) 54 SALUJA Thus for any i = 1, 2, . . . ,N, from (2.7) - (2.9) and for any n ≥ n3, we have d(Tni q,q) ≤ d(T n i q,q1) + d(q1,q) ≤ d(q,q1) + K + d(q1,q) = K + 2d(q,q1) ≤ K + 2[d(q,xn3 ) + d(xn3,q1)] < K + 2(ε1 + 2ε1) = K + 6ε1 = ε ′,(2.10) where ε′ = K + 6ε1, since K > 0 and ε1 > 0, it follows that ε ′ > 0. By the arbitrariness of ε′ > 0, we know that Tni q = q for all i = 1, 2, . . . ,N. Again since for any n ≥ n3, we have d(Tni q,Tiq) ≤ d(T n i q,q1) + d(Tiq,q1) ≤ d(q,q1) + K + d(Tiq,q1) ≤ d(q,q1) + K + Ld(q,q1) = (1 + L)d(q,q1) + K ≤ (1 + L)[d(q,xn3 ) + d(xn3,q1)] + K < (1 + L)[ε1 + 2ε1] + K = 3(1 + L)ε1 + K = ε ′′,(2.11) where ε′′ = 3(1 + L)ε1 + K, since K > 0 and ε1 > 0, it follows that ε ′′ > 0. By the arbitrariness of ε′′ > 0, we know that Tni q = Tiq for all i = 1, 2, . . . ,N. From the uniqueness of limit, we have q = Tiq for all i = 1, 2, . . . ,N, that is, q ∈ F. This shows that q is a common fixed point of the mappings {Ti}Ni=1. This completes the proof. � Taking f = I in Theorem 2.1, then we have the following result. Theorem 2.2. Let (E,d,W) be a complete convex metric space and C be a nonemp- ty closed convex subset of E. Let Ti : C → C be a finite family of uniformly L- Lipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,N such that F = ∩Ni=1F(Ti) 6= ∅. Let {xn} be the iterative sequence with errors de- fined by (1.13) and {un}, {vn}, {wn} be three bounded sequences in C. Let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be nine sequences in [0, 1] satisfy- ing the conditions (i) and (ii) of Theorem 2.1. Then the sequence {xn} converges to a common fixed point p in F if and only if lim inf n→∞ d(xn,F) = 0,(2.12) where d(x,F) = infp∈F d(x,p). Taking dn = 1(en = fn = 0) for all n ≥ 0 in Theorem 2.1, then we have the following result. Theorem 2.3. Let (E,d,W) be a complete convex metric space and C be a nonemp- ty closed convex subset of E. Let Ti : C → C be a finite family of uniformly L- Lipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,N such that F = ∩Ni=1F(Ti) 6= ∅ and f : C → C be a contractive mapping with a con- tractive constant ξ ∈ (0, 1). Let {xn} be the iterative sequence with errors defined CONVERGENCE THEOREMS 55 by (1.14) and {un}, {vn} be two bounded sequences in C. Let {αn}, {βn}, {γn}, {an}, {bn}, {cn} be six sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = 1, for all n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. Then the sequence {xn} converges to a common fixed point p in F if and only if lim inf n→∞ d(xn,F) = 0,(2.13) where d(x,F) = infp∈F d(x,p). As an application of Theorem 2.1, we establish another strong convergence result as follows. Theorem 2.4. Let (E,d,W) be a complete convex metric space and C be a nonemp- ty closed convex subset of E. Let Ti : C → C be a finite family of uniformly L- Lipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,N such that F = ∩Ni=1F(Ti) 6= ∅ and f : C → C be a contractive mapping with a con- tractive constant ξ ∈ (0, 1). Let {xn} be the iterative sequence with errors defined by (1.12) and {un}, {vn}, {wn} be three bounded sequences in C. Let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be nine sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. Assume that limn→∞d(xn,Tlxn) = 0 for l = 1, 2, . . . ,N. If {Ti : i ∈N} satisfies condition (A), then the sequence {xn} converges strongly to a point in F . Proof. As in the proof of Theorem 2.1, we have that limn→∞d(xn,F) exists. Again by hypothesis of the theorem limn→∞d(xn,Tlxn) = 0 for l = 1, 2, . . . ,N. So condition (A) guarantees that limn→∞f(d(xn,F)) = 0. Since f is a non-decreasing function and f(0) = 0, it follows that limn→∞d(xn,F) = 0. Therefore, Theorem 2.1 implies that {xn} converges strongly to a point in F. This completes the proof. � Remark 2.5. Theorems 2.1 - 2.3 generalize, improve and unify some corresponding result in [1]-[3], [5]-[8], [10]-[17], [19] and [21]. Remark 2.6. Our results also extend the corresponding results of [18] to the case of more general class of uniformly quasi-Lipschitzian mappings considered in this paper. Example 2.7. Let E = [−π, π] and let T be defined by Tx = xcosx for each x ∈ E. Clearly F(T) = {0}. T is a quasi-nonexpansive mapping since if x ∈ E and z = 0, then d(Tx,z) = d(Tx, 0) = |x||cosx| ≤ |x| = |x−z| = d(x,z), 56 SALUJA and T is asymptotically quasi-nonexpansive mapping with constant sequence {kn} = {1}. Hence by remark 1.1, T is asymptotically quasi-nonexpansive type mapping. But it is not a nonexpansive mapping and hence asymptotically nonexpansive map- ping. In fact, if we take x = π 2 and y = π, then d(Tx,Ty) = ∣∣∣π 2 cos π 2 −π cosπ ∣∣∣ = π, whereas d(x,y) = ∣∣∣π 2 −π ∣∣∣ = π 2 . Example 2.8. Let E = R and let T be defined by T(x) = { x 2 cos 1 x , if x 6= 0, 0, if x = 0. If x 6= 0 and Tx = x, then x = x 2 cos 1 x . Thus 2 = cos 1 x . This is impossible. T is a quasi-nonexpansive mapping since if x ∈ E and z = 0, then d(Tx,z) = d(Tx, 0) = | x 2 ||cos 1 x | ≤ |x| 2 < |x| = |x−z| = d(x,z), and T is asymptotically quasi-nonexpansive mapping with constant sequence {kn} = {1}. Hence by remark 1.1, T is asymptotically quasi-nonexpansive type mapping. But it is not a nonexpansive mapping and hence asymptotically nonexpansive map- ping. In fact, if we take x = 2 3π and y = 1 π , then d(Tx,Ty) = ∣∣∣ 1 3π cos 3π 2 − 1 2π cosπ ∣∣∣ = 1 2π , whereas d(x,y) = ∣∣∣ 2 3π − 1 π ∣∣∣ = 1 3π . 3. Conclusion If F(T) is nonempty, then asymptotically nonexpansive mapping and asymp- totically quasi-nonexpansive mappings are asymptotically quasi-nonexpansive type mappings by Remark 1.2, thus our results are good improvement and extension of some previous work from the existing literature (see, e.g., [1]-[3], [5]-[8], [10]-[21] and many others). References [1] S.S. Chang, J.K. Kim and D.S. Jin, Iterative sequences with errors for asymptotically quasi- nonexpansive type mappings in convex metric spaces, Archives of Inequality and Applications 2(2004), 365-374. [2] Y.J. Cho, H. Zhou and G. Guo, Weak and strong convergence theorems for three step itera- tions with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47(2004), 707-717. [3] H. Fukhar-ud-din and S.H. Khan, Convergence of iterates with errors of asymptotically quasi- nonexpansive mappings and applications, J. Math. Anal. Appl. 328(2)(2007), 821-829. [4] H. Fukhar-ud-din, A.R. Khan and M.A.A. Khan, A new implicit algorithm of asymptotically quasi-nonexpansive maps in uniformly convex Banach spaces, IAENG Int. J. Appl. Maths. 42(3)(2008). [5] M.K. Ghosh and L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive map- pings, J. Math. Anal. Appl. 207(1997), 96-103. [6] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35(1972), 171-174. CONVERGENCE THEOREMS 57 [7] J.U. Jeong and S.H. Kim, Weak and strong convergence of the Ishikawa iteration process with errors for two asymptotically nonexpansive mappings, Appl. Math. Comput. 181(2) (2006), 1394-1401. [8] J.K. Kim, K.H. Kim and K.S. Kim, Three-step iterative sequences with errors for asymptot- ically quasi-nonexpansive mappings in convex metric spaces, Nonlinear Anal. Convex Anal. RIMS Vol. 1365(2004), 156-165. [9] W.A. Kirk, Fixed point theorems for non-lipschitzian mappings of asymptotically nonexpan- sive type, Israel J. Math. 17(1974), 339-346. [10] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 259(2001), 1-7. [11] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl. 259(2001), 18-24. [12] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member of uniformly convex Banach spaces, J. Math. Anal. Appl. 266(2002), 468-471. [13] W.V. Petryshyn and T.E. Williamson, Strong and weak convergence of the sequence of suc- cessive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43(1973), 459-497. [14] G.S. Saluja, Convergence of fixed point of asymptotically quasi-nonexpansive type mappings in convex metric spaces, J. Nonlinear Sci. Appl. 1(3)(2008), 132-144. [15] W. Takahashi, A convexity in metric space and nonexpansive mappings I, Kodai Math. Sem. Rep. 22(1970), 142-149. [16] K.K. Tan and H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122(1994), 733-739. [17] Y.-X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi- nonexpansive mappings, Compt. Math. Appl. 49(11-12)(2005), 1905-1912. [18] Y.-X. Tian and Chun-de Yang, Convergence theorems of three-step iterative scheme for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Fixed Point Theory and Applications, Vol. 2009, Article ID 891965, 12 pages, doi:10.1155/2009/891965. [19] B.L. Xu and M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267(2002), no.2, 444-453. [20] Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224(1)(1998), 91-101. [21] H. Zhou; J.I. Kang; S.M. Kang and Y.J. Cho, Convergence theorems for uniformly quasi- lipschitzian mappings, Int. J. Math. Math. Sci. 15(2004), 763-775. Department of Mathematics, Govt. Nagarjuna P.G. College of Science, Raipur - 492010 (C.G.), India