CUBO A Mathemati al Journal Vol.15, N o 03, (71�87). O tober 2013 Convergen e theorems for generalized asymptoti ally quasi-nonexpansive mappings in one metri spa es G. S. Saluja Department of Mathemati s and Information Te hnology, Govt. Nagarjuna P.G. College of S ien e, Raipur - 492010 (C.G.), India. saluja_1963�rediffmail. om, saluja1963�gmail. om ABSTRACT The purpose of this paper is to study an Ishikawa type iteration pro ess with er- rors to approximate the ommon �xed point of two generalized asymptoti ally quasi- nonexpansive mappings in the framework of one metri spa es. Our results extend and generalize many known results from the existing literature. RESUMEN El propósito de este artí ulo es estudiar el pro eso de itera ión del tipo Ishikawa on errores para aproximar el puto �jo omún de dos apli a iones uasi-expansivas asin- tóti amente generalizadas en el mar o de espa ios métri os óni os. Nuestro resultado extiende y generaliza mu hos resultados de la literatura existente. Keywords and Phrases: Generalized asymptoti ally quasi-nonexpansive mapping, ommon �xed point, Ishikawa type iteration pro ess with errors, one metri spa e, normal and non-normal one. 2010 AMS Mathemati s Subje t Classi� ation: 47H10, 54H25. 72 G. S. Saluja CUBO 15, 3 (2013) 1 Introdu tion and Preliminaries The Well-known Bana h ontra tion prin iple and its several generalization in the setting of met- ri spa es play a entral role for solving many problems of nonlinear analysis. For example, see [2,6,7,20,21℄. In 1980, Rzepe ki [23℄ introdu ed a generalized metri by repla ing the set of real numbers with normal one of the Bana h spa e. In 1987, Lin [17℄ introdu ed the notion of K-metri spa es by repla ing the set of real numbers with one in the metri fun tion. Zabrejko [33℄ studied new revised version of the �xed point theory in K-metri and K-normed linear spa es by repla ing an ordered Bana h spa e instead of the set of real numbers, as the o-domain for a metri . Ordered normed spa es and ones have appli ations in applied mathemati s, for instan e, in using Newton's approximations method [28,33℄, and in optimization theory [7℄. Re ently, Huang and Zhang [10℄ used the notion of one metri spa es as a generalization of metri spa es. They have repla ed the real numbers (as the o-domain of a "metri ") by an ordered Bana h spa e. The authors des ribed the onvergen e in one metri spa es and introdu ed their ompleteness. Then they proved some �xed point theorems for ontra tive single-valued mappings in su h spa es. In their theorems one is normal. For more �xed point results in one metri spa es, see [1,3,11,24,25,32℄. Most re ently, Duki et al. [8℄ studied an Ishikawa type iteration pro ess with errors for two uniformly quasi-Lips hitzian mappings in omplete onvex one metri spa es and they gave a ne - essary and su� ient ondition to approximate the ommon �xed point for said mappings. Their results extended and generalized many known results from the literature. The main goal of this paper is to study an Ishikawa type iteration pro ess with errors for two generalized asymptoti ally quasi-nonexpansive mappings in the setting of omplete onvex one metri spa es and also give a ne essary and su� ient ondition to approximate the ommon �xed point for said mappings. The results presented in this paper extend and generalize many known results from the literature. Consistent with [7℄ and [10℄, the following de�nitions and results will be needed in the sequel. Let E be a real Bana h spa e. A subset P of E is alled a one whenever the following ondi- tions hold: (C1) P is losed, nonempty and P 6= {0}; CUBO 15, 3 (2013) Convergen e theorems for generalized . . . 73 (C2) a, b ∈ R, a, b ≥ 0 and x, y ∈ P imply ax + by ∈ P; (C3) P ∩ (−P) = {0}. Given a one P ⊂ E, we de�ne a partial ordering � with respe t to P by x � y if and only if y − x ∈ P. We shall write x ≺ y to indi ate that x � y but x 6= y, while x ≪ y will stand for y − x ∈ intP (interior of P). If intP 6= ∅ then P is alled a solid one (see [28℄). There exist two kinds of ones: normal (with the normal onstant k) and non-normal ones [7℄). Let E be a real Bana h spa e, P ⊂ E a one and � partial ordering de�ned by P. Then P is alled normal if there is a number k > 0 su h that for all x, y ∈ P, 0 � x � y imply ‖x‖ ≤ k ‖y‖ , (1) or equivalently, if (∀n) xn � yn � zn and lim n→∞ xn = lim n→∞ zn = x imply lim n→∞ yn = x. (2) The least positive number k satisfying (1) is alled the normal onstant of P. It is lear that k ≥ 1. Example 1.1. (see [28℄) Let E = C1 R [0, 1] with ‖x‖ = ‖x‖∞ + ‖x ′‖∞ on P = {x ∈ E : x(t) ≥ 0}. This one is not normal. Consider, for example, xn(t) = t n n and yn(t) = 1 n . Then 0 � xn � yn, and limn→∞ yn = 0, but ‖xn‖ = maxt∈[0,1] | t n n | + maxt∈[0,1] |t n−1| = 1 n + 1 > 1; hen e xn does not onverge to zero. It follows by (2) that P is a non-normal one. De�nition 1.1. (see [10,33℄) Let X be a nonempty set. Suppose that the mapping d: X×X → E satis�es: (d1) 0 � d(x, y) for all x, y ∈ X and d(x, y) = 0 if and only if x = y; (d2) d(x, y) = d(y, x) for all x, y ∈ X; (d3) d(x, y) � d(x, z) + d(z, y) x, y, z ∈ X. 74 G. S. Saluja CUBO 15, 3 (2013) Then d is alled a one metri [10℄ or K-metri [33℄ on X and (X, d) is alled a one metri [10℄ or K-metri spa e [33℄ (we shall use the �rst term). The on ept of a one metri spa e is more general than that of a metri spa e, be ause ea h metri spa e is a one metri spa e where E = R and P = [0, +∞). Example 1.2. (see [10℄) Let E = R2, P = {(x, y) ∈ R2 : x ≥ 0, y ≥ 0}, X = R and d: X×X → E de�ned by d(x, y) = (|x − y|, α|x − y|), where α ≥ 0 is a onstant. Then (X, d) is a one metri spa e with normal one P where k = 1. Example 1.3. (see [24℄) Let E = ℓ2, P = {{xn}n≥1 ∈ E : xn ≥ 0, for all n}, (X, ρ) a metri spa e, and d: X × X → E de�ned by d(x, y) = {ρ(x, y)/2n}n≥1. Then (X, d) is a one metri spa e. Clearly, the above examples show that the lass of one metri spa es ontains the lass of metri spa es. De�nition 1.2. (see [10℄) Let (X, d) be a one metri spa e. We say that {xn} is: (i) a Cau hy sequen e if for every ε in E with 0 ≪ ε, then there is an N su h that for all n, m > N, d(xn, xm) ≪ ε; (ii) a onvergent sequen e if for every ε in E with 0 ≪ ε, then there is an N su h that for all n > N, d(xn, x) ≪ ε for some �xed x in X. A one metri spa e X is said to be omplete if every Cau hy sequen e in X is onvergent in X. Let us re all ( [10℄) that if P is a normal solid one, then xn ∈ X is a Cau hy sequen e if and only if ‖d(xn, xm)‖ → 0 as n, m → ∞. Further, xn ∈ X onverges to x ∈ X if and only if ‖d(xn, x)‖ → 0 as n → ∞. In the sequel we assume that E is a real Bana h spa e and that P is a normal solid one in E, that is, normal one with intP 6= ∅. The last assumption is ne essary in order to obtain reasonable results onne ted with onvergen e and ontinuity. The partial ordering indu ed by the one P will be denoted by �. CUBO 15, 3 (2013) Convergen e theorems for generalized . . . 75 2 Convexity in one metri spa e Let (X, d) be a one metri spa e with solid one P and T : X → X a given mapping. Let F(T) denote the set of �xed points of T. De�nition 2.1. (1) The mapping T is said to be nonexpansive if d(Tx, Ty) � d(x, y) (3) for all x, y ∈ X. (2) The mapping T is said to be quasi-nonexpansive if F(T) 6= ∅ and d(Tx, p) � d(x, p) (4) for all x ∈ X and p ∈ F(T). (3) The mapping T is said to be asymptoti ally nonexpansive if there exists a sequen e {rn} ∈ [0, ∞) with rn → 0 as n → ∞ su h that d(Tnx, Tny) � (1 + rn)d(x, y) (5) for all x, y ∈ X. (4) The mapping T is said to be asymptoti ally quasi-nonexpansive if F(T) 6= ∅ and there exists a sequen e {rn} ∈ [0, ∞) with rn → 0 as n → ∞ su h that d(Tnx, p) � (1 + rn)d(x, p) (6) for all x ∈ X and p ∈ F(T). (5) The mapping T is said to be generalized asymptoti ally quasi-nonexpansive [12℄ if F(T) 6= ∅ and there exist two sequen es of real numbers {rn} and {sn} ∈ [0, ∞) with rn → 0 and sn → 0 as n → ∞ su h that d(Tnx, p) � (1 + rn)d(x, p) + sn, (7) 76 G. S. Saluja CUBO 15, 3 (2013) for all x ∈ X and p ∈ F(T). (6) The mapping T is said to be uniformly L-Lips hitzian if there exists a onstant L > 0 su h that d(Tnx, Tny) � L d(x, y), (8) for all x, y ∈ X. Remark 2.1. (i) It is lear that the nonexpansive mappings with the nonempty �xed point set F(T) are quasi-nonexpansive. (ii) The linear quasi-nonexpansive mappings are nonexpansive, but it is easily seen that there exist nonlinear ontinuous quasi-nonexpansive mappings whi h are not nonexpansive; for example, de�ne T(x) = (x/2)sin(1/x) for all x 6= 0 and T(0) = 0 in R. (iii) It is obvious that if T is nonexpansive, then it is asymptoti ally nonexpansive with the onstant sequen e {1}. (iv) If T is asymptoti ally nonexpansive, then it is uniformly Lips hitzian with the uniform Lips hitz onstant L = sup{1 + rn : n ≥ 1}. However, the onverse of this laim is not true. (v) If in de�nition (5), sn = 0 for all n ∈ N, then T be omes asymptoti ally quasi-nonexpansive, and hen e the lass of generalized asymptoti ally quasi-nonexpansive maps in ludes the lass of asymptoti ally quasi-nonexpansive maps. In re ent years, asymptoti ally nonexpansive mappings, asymptoti ally nonexpansive type mappings, asymptoti ally quasi-nonexpansive mappings and asymptoti ally quasi-nonexpansive type mappings have been studied extensively in the setting of onvex metri spa es (see e.g. [5,9,14�16,18,19,27℄). In 1970, Takahashi [26℄ introdu ed the on ept of onvexity in a metri spa e and the prop- erties of the spa e. De�nition 2.2. (see [26℄) Let (X, d) be a metri spa e and I = [0, 1]. A mapping W : X × X × I → X is said to be a onvex stru ture on X if for ea h (x, y, λ) ∈ X × X × I and u ∈ X, d(u, W(x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y). CUBO 15, 3 (2013) Convergen e theorems for generalized . . . 77 X together with a onvex stru ture W is alled a onvex metri spa e, denoted by (X, d, W). A nonempty subset K of X is said to be onvex if W(x, y, λ) ∈ K for all (x, y, λ) ∈ K × K × I. Remark 2.2. Every normed spa e is a onvex metri spa e, where a onvex stru ture 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. In fa t, d(u, W(x, y, z; α, β, γ)) = ‖u − (αx + βy + γz)‖ ≤ α ‖u − x‖ + β ‖u − y‖ + γ ‖u − z‖ = αd(u, x) + βd(u, y) + γd(u, z), ∀u ∈ X. But there exists some onvex metri spa es whi h an not be embedded into normed spa e. Now we de�ne the following: De�nition 2.3. Let (X, d) be a one metri spa e and I = [0, 1]. A mapping W : X×X×I → X is said to be a onvex stru ture on X if for any (x, y, λ) ∈ X × X × I and u ∈ X, the following in- equality holds: d(u, W(x, y, λ)) � λd(u, x) + (1 − λ)d(u, y). If (X, d) be a one metri spa e with a onvex stru ture W, then (X, d) is alled a onvex abstra t metri spa e or onvex one metri spa e (see also [13℄, [22℄). Moreover, a nonempty subset C of X is said to be onvex if W(x, y, λ) ∈ C, for all (x, y, λ) ∈ C × C × I. De�nition 2.4. Let (X, d) be a one metri spa e, I = [0, 1] and {an}, {bn}, {cn} are real sequen es in [0,1℄ with an + bn + cn = 1. A mapping W : X 3 × I3 → X is said to be a onvex stru ture on X if for any (x, y, z, an, bn, cn) ∈ X 3 × I3 and u ∈ X, the following inequality holds: d(u, W(x, y, z, an, bn, cn)) � and(u, x) + bnd(u, y) + cnd(u, z). If (X, d) be a one metri spa e with a onvex stru ture W, then (X, d) is alled a gener- alized onvex one metri spa e. Moreover, a nonempty subset C of X is said to be onvex if W(x, y, z, an, bn, cn) ∈ C, for all (x, y, z, an, bn, cn) ∈ C 3 × I3. Remark 2.3. If E = R, P = [0, +∞), ‖.‖ = |.|, then (X, d) is a onvex metri spa e, i.e., generalized onvex metri spa e as in [30℄. 78 G. S. Saluja CUBO 15, 3 (2013) Example 2.1. Let (X, d) be a one metri spa e as in Example (1.2)(1). If W(x, y, λ) =: λx + (1 − λ)y, then (X, d) is a one metri spa e. Hen e, this notion is more general than that of a onvex metri spa e. De�nition 2.5. Let (X, d) be a one metri spa e with a onvex stru ture W : X3 × I3 → X, S, T : X → X be two generalized asymptoti ally quasi-nonexpansive mappings with sequen es of real numbers {rn} and {sn} ∈ [0, ∞) su h that rn → 0 and sn → 0 as n → ∞ and {an}, {bn}, {cn}, {a′n}, {b ′ n}, {c ′ n} are six sequen es in [0,1℄ with an + bn + cn = a ′ n + b ′ n + c ′ n = 1, n = 1, 2, . . . . For any given x1 ∈ X, de�ne a sequen e {xn} as follows: yn = W(xn, S nxn, vn, a ′ n, b ′ n, c ′ n), xn+1 = W(xn, T nyn, un, an, bn, cn), (9) where {un}, {vn} are two sequen es in X satisfying the following ondition: for any nonnegative integers n, m, 1 ≤ n < m, if δ(Anm) > 0, then max n≤i,j≤m { ‖d(x, y)‖ : x ∈ {ui, vi}, y ∈ {xj, yj, Tyj, Sxj, uj, vj} } < δ(Anm), (10) where Anm = {xi, yi, Tyi, Sxi, ui, vi : n ≤ i ≤ m}, δ(Anm) = sup x,y∈Anm ‖d(x, y)‖ . Then {xn} is alled the Ishikawa type iteration pro ess with errors for two generalized asymp- toti ally quasi-nonexpansive mappings S and T in onvex one metri spa e (X, d). Remark 2.4. Note that some iteration pro esses onsidered in [9,14,19,27℄ an be obtained from the above pro ess (9) as spe ial ases by hoosing suitable spa es and mappings. In the sequel, we shall need the following lemma. Lemma 2.1. (see [19℄) Let {an}, {bn} and {αn} be sequen es of nonnegative real numbers satisfying the inequality an+1 ≤ (1 + αn)an + bn, n ≥ 1. If ∑∞ n=1 bn < ∞ and ∑∞ n=1 αn < ∞. Then (a) limn→∞ an exists. (b) If lim infn→∞ an = 0, then limn→∞ an = 0. CUBO 15, 3 (2013) Convergen e theorems for generalized . . . 79 3 Main Results Now we give our main results of this paper. Lemma 3.1. Let C be a nonempty losed onvex subset of a omplete onvex one met- ri spa e X, S, T : C → C be two generalized asymptoti ally quasi-nonexpansive mappings with sequen es {rn} and {sn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞ and ∑∞ n=1 sn < ∞. Assume that F = F(S) ∩ F(T) 6= ∅. Let {xn} be the Ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. Then (i) there exists a onstant ve tor v ∈ P\ {0} su h that ‖d(xn+1, p)‖ ≤ k. (1 + An). ‖d(xn, p)‖ + k. Bn + k. ‖v‖ . Cn, where An = r 2 n + 2rn, Bn = (2 + rn)sn and Cn = (1 + rn)d(un, p) + d(vn, p), for all n ∈ N and for all p ∈ F, where k is the normal onstant of a one P; (ii) there exists a real onstant M > 0 su h that ‖d(xn+m, p)‖ ≤ k. M. ‖d(xn, p)‖ + k. M. n+m−1∑ j=n Bj + k. M. ‖v‖ . n+m−1∑ j=n Cj, for all n, m ∈ N and for all p ∈ F, where k is the normal onstant of a one P. proof. For any p ∈ F, from (7) and (9), we have d(xn+1, p) = d(W(xn, T nyn, un, an, bn, cn), p) � and(xn, p) + bnd(T nyn, p) + cnd(un, p) � and(xn, p) + bn[(1 + rn)d(yn, p) + sn] + cnd(un, p) = and(xn, p) + bn(1 + rn)d(yn, p) + bnsn + cnd(un, p) � and(xn, p) + bn(1 + rn)d(yn, p) + sn + cnd(un, p) (11) and d(yn, p) = d(W(xn, S nxn, vn, a ′ n, b ′ n, c ′ n), p) � a′nd(xn, p) + b ′ nd(S nxn, p) + c ′ nd(vn, p) � a′nd(xn, p) + b ′ n[(1 + rn)d(xn, p) + sn] + c ′ nd(vn, p) � (a′n + b ′ n)(1 + rn)d(xn, p) + b ′ nsn + c ′ nd(vn, p) = (1 − c′n)(1 + rn)d(xn, p) + b ′ nsn + c ′ nd(vn, p) � (1 + rn)d(xn, p) + sn + c ′ nd(vn, p). (12) 80 G. S. Saluja CUBO 15, 3 (2013) Substituting (12) into (11), it an be obtained that d(xn+1, p) � and(xn, p) + bn(1 + rn)[(1 + rn)d(xn, p) + sn +c′nd(vn, p)] + sn + cnd(un, p) � (an + bn)(1 + rn) 2d(xn, p) + bn(1 + rn)sn + sn +bnc ′ n(1 + rn)d(vn, p) + cnd(un, p) = (1 − cn)(1 + rn) 2d(xn, p) + bn(1 + rn)sn + sn +bnc ′ n(1 + rn)d(vn, p) + cnd(un, p) � (1 + rn) 2d(xn, p) + (2 + rn)sn + cnd(un, p) +c′n(1 + rn)d(vn, p) � (1 + An)d(xn, p) + Bn + vCn, (13) where An = r 2 n + 2rn, Bn = (2 + rn)sn, Cn = cn + c ′ n and v = (1 + rn)d(vn, p) + d(un, p). Now, (i) follows from (1), where k is a normal onstant of the one P. (ii) It is well known that 1 + x ≤ ex for all x ≥ 0. Using this, it follows from on lusion (i) that for all n, m ∈ N and p ∈ F, we have d(xn+m, p) � (1 + An+m−1)d(xn+m−1, p) + Bn+m−1 + v.Cn+m−1 � eAn+m−1d(xn+m−1, p) + Bn+m−1 + v.Cn+m−1 � eAn+m−1[eAn+m−2d(xn+m−2, p) + Bn+m−2 +v.Cn+m−2] + Bn+m−1 + v.Cn+m−1 � eAn+m−1+An+m−2d(xn+m−2, p) + e An+m−1+An+m−2. [Bn+m−1 + Bn+m−1] + e An+m−1+An+m−2. [Cn+m−1 + Cn+m−1].v . . . � M. d(xn, p) + M. n+m−1∑ j=n Bj + M. ( n+m−1∑ j=n Cj ) . v, (14) where M = e ∑ ∞ j=1 Aj . Further, (ii) follows from (1), be ause P is a normal one with the normal onstant k. Theorem 3.1. Let C be a nonempty losed onvex subset of a omplete onvex one metri spa e X, S, T : C → C be two generalized asymptoti ally quasi-nonexpansive mappings with se- quen es {rn} and {sn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞ and ∑∞ n=1 sn < ∞. Assume that F = F(S)∩F(T) 6= ∅. Let {xn} be the Ishikawa type iteration pro ess with errors de�ned by (9) {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. Then {xn} onverges to a ommon �xed CUBO 15, 3 (2013) Convergen e theorems for generalized . . . 81 point of S and T if and only if lim infn→∞ ‖d(xn, F)‖ = 0, where ‖d(x, F)‖ = inf{‖d(x, q)‖ : q ∈ F}. proof. The ne essity of ondition is obvious. Thus, we will only prove the su� ien y. From Lemma 3.1(i), we have ‖d(xn+1, p)‖ ≤ k. (1 + An). ‖d(xn, p)‖ + k. Bn + k. ‖v‖ . Cn, where An = r 2 n + 2rn, Bn = (2 + rn)sn, Cn = cn + c ′ n and v = (1 + rn)d(un, p) + d(vn, p) with∑∞ n=1 An < ∞, ∑∞ n=1 Bn < ∞ and ∑∞ n=1 Cn < ∞. Sin e ∑∞ n=1 An < ∞, ∑∞ n=1 Bn < ∞ and ∑∞ n=1 Cn < ∞, it follows from Lemma 2.1 that limn→∞ ‖d(xn, F)‖ exists. A ording to the hypothesis, lim infn→∞ ‖d(xn, F)‖ = 0, hen e we have that limn→∞ ‖d(xn, F)‖ = 0. Next, we show that {xn} is a Cau hy sequen e. Let ε > 0 be given. There exists an integer n0 su h that for all n > n0, we have ‖d(xn, F)‖ < ε 6k2M , ∞∑ n=n0+1 Bn < ε 6k2M , and ∞∑ n=n0+1 Cn < ε 6k2 ‖v‖ M . In parti ular, there exists a q ∈ F and an integer n1 > n0 su h that ‖d(xn1, q)‖ < ε 6k2M . It follows from Lemma 3.1(ii) that when n > n1, we get ‖d(xn+m, q)‖ = ∥ ∥d ( xn1+(n+m−n1), q ) ∥ ∥ ≤ k. M. . ‖d(xn1, q)‖ + k. M. ( n+m−1∑ j=n1 Bj ) +k. M. ‖v‖ . ( n+m−1∑ j=n1 Cj ) (15) and 82 G. S. Saluja CUBO 15, 3 (2013) ‖d(xn, q)‖ = ∥ ∥d ( xn1+(n−n1), q ) ∥ ∥ ≤ k. M. ‖d(xn1, q)‖ + k. M. ( n−1∑ j=n1 Bj ) +k. M. ‖v‖ . ( n−1∑ j=n1 Cj ) . (16) Therefore from (1), (15) and (16), we obtain that ‖d(xn+m, xn)‖ ≤ k. ‖d(xn+m, q) + d(q, xn)‖ ≤ k. ‖d(xn+m, q)‖ + k. ‖d(q, xn)‖ ≤ 2k2. M. ‖d(xn1, q)‖ + 2k 2. M. ( n+m−1∑ j=n1 Bj + n−1∑ j=n1 Bj ) +2k2. M. ‖v‖ . ( n+m−1∑ j=n1 Cj + n−1∑ j=n1 Cj ) ≤ 2k2. M. ‖d(xn1, q)‖ + 2k 2. M. ( n+m−1∑ j=n1 Bj ) +2k2. M. ‖v‖ . ( n+m−1∑ j=n1 Cj ) ≤ 2k2. M. ε 6k2M + 2k2. M. ε 6k2M +2k2. M. ‖v‖ . ε 6k2 ‖v‖ M = ε. (17) Hen e {xn} is a Cau hy sequen e in losed onvex subset C of a omplete one metri spa e X. Therefore, it must be onvergent to a point in C. Suppose limn→∞ xn = p. We will prove that p ∈ F. For a given ε > 0, there exists an integer n2 su h that for all n ≥ n2, we have ‖d(xn, p)‖ < ε 2k(2 + r1) and ‖d(xn, F)‖ < ε 4k(2 + r1) (18) In parti ular, there exists a p1 ∈ F and an integer n3 > n2 su h that ‖d(xn3, p1)‖ < ε 2k(2 + r1) (19) CUBO 15, 3 (2013) Convergen e theorems for generalized . . . 83 Then, we have d(Tp, p) � d(Tp, p1) + d(p1, xn3) + d(xn3, p) � (1 + r1)d(p, p1) + s1 + d(p1, xn3) + d(xn3, p) � (1 + r1)d(p, p1) + d(p1, xn3) + d(xn3, p) � (1 + r1)d(p, xn3) + (1 + r1)d(xn3, p1) +d(p1, xn3) + d(xn3, p) = (2 + r1)d(xn3, p) + (2 + r1)d(xn3, p1) (20) Now using (1), (18) and (19), we obtain ‖d(Tp, p)‖ ≤ k(2 + r1) ‖d(xn3, p)‖ + k(2 + r1) ‖d(xn3, p1)‖ < k(2 + r1). ε 2k(2 + r1) + k(2 + r1). ε 2k(2 + r1) = ε. (21) Similarly, we an also have ‖d(Sp, p)‖ < ε. Sin e ε is arbitrary, it follows that d(Tp, p) = d(Sp, p) = 0, that is, p is a ommon �xed point of S and T. This ompletes the proof of Theorem 3.1. We dedu e some results from Theorem 3.1 as follows. Corollary 3.1. Let C be a nonempty losed onvex subset of a omplete onvex one metri spa e X, S, T : C → C be asymptoti ally quasi-nonexpansive mappings with sequen e {rn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞. Assume that F = F(S) ∩ F(T) 6= ∅. Let {xn} be the Ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. Then {xn} onverges to a ommon �xed point of S and T if and only if lim infn→∞ ‖d(xn, F)‖ = 0, where ‖d(x, F)‖ = inf{‖d(x, q)‖ : q ∈ F}. proof. It follows from Theorem 3.1 with sn = 0 for all n ≥ 1. Corollary 3.2. Let C be a nonempty losed onvex subset of a omplete onvex one met- ri spa e X, S, T : C → C be uniformly quasi-Lips hitzian mappings with L > 0. Assume that F = F(S) ∩ F(T) 6= ∅. Let {xn} be the Ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. Then {xn} onverges to a ommon �xed point of S and T if and only if lim infn→∞ ‖d(xn, F)‖ = 0, where ‖d(x, F)‖ = inf{‖d(x, q)‖ : q ∈ F}. proof. Sin e {rn} ∈ [0, ∞) with rn → 0 as n → ∞, then there exists L > 0 su h that L = sup{1 + rn : n ≥ 1}. In this ase S and T are uniformly quasi-Lips hitzian mappings with 84 G. S. Saluja CUBO 15, 3 (2013) L > 0. Hen e, Corollary 3.2 an be proven by Corollary 3.1. Corollary 3.3. Let C be a nonempty losed onvex subset of a omplete onvex one met- ri spa e X, S, T : C → C be asymptoti ally nonexpansive mappings with sequen e {rn} ∈ [0, ∞) su h that ∑∞ n=1 rn < ∞. Assume that F = F(S) ∩ F(T) 6= ∅. Let {xn} be the Ishikawa type iteration pro ess with errors de�ned by (9) and {un}, {vn} satisfying (10) with the restri tion ∑∞ n=1 (cn + c ′ n) < ∞. Then {xn} onverges to a ommon �xed point of S and T if and only if lim infn→∞ ‖d(xn, F)‖ = 0, where ‖d(x, F)‖ = inf{‖d(x, q)‖ : q ∈ F}. proof. It is lear that an asymptoti ally nonexpansive mapping must be asymptoti ally quasi- nonexpansive. Therefore, Corollary 3.3 an be proven by Corollary 3.1. Corollary 3.4. In Corollary 3.1 by setting E = R, P = [0, ∞), d(x, y) = |x−y|, x, y ∈ R (that is ‖.‖ = |.|), we get Lemma 2 and Theorem 1, 2, 3 of [29℄. Corollary 3.5. If we set in Corollary 3.1 E = R, P = [0, ∞), d(x, y) = |x − y|, x, y ∈ R (that is ‖.‖ = |.|), we obtain the main result of [4℄, Theorem 2.1 and 2.2 of [30℄ and Theorem 2.1 and Corollary 2.3 of [31℄. Remark 3.1. Our results extend the orresponding results of Duki et al. [8℄ to the ase of more general lass of uniformly quasi-Lips hitzian mappings onsidered in this paper. Remark 3.2. Our results also extend, improve and generalize many known results from the existing literature. Example 3.1. Let E be the real line with the usual norm |.| and K = [0, 1]. De�ne S, T : K → K by T(x) = x 2 sin ( 1 x ) , x ∈ [0, 1], and S(x) = x 3 , x ∈ [0, 1], for x ∈ K. Obviously S(0) = 0 and T(0) = 0. Hen e, F = F(S) ∩ F(T) = {0}, that is, 0 is a ommon �xed point of S and T. Now we he k that S and T are generalized asymptoti ally quasi- nonexpansive mappings. In fa t, if x ∈ [0, 1] and p = 0 ∈ [0, 1], then |T(x) − p| = |T(x) − 0| = |(x/2) sin (1/x) − 0| = |(x/2) sin (1/x)| CUBO 15, 3 (2013) Convergen e theorems for generalized . . . 85 ≤ |x/2| ≤ |x| = |x − 0| = |x − p|, that is, |T(x) − p| ≤ |x − p|. Thus, T is quasi-nonexpansive. It follows that T is asymptoti ally quasi-nonexpansive with the onstant sequen e {kn} = {1} for ea h n ≥ 1 and hen e it is generalized asymptoti ally quasi- nonexpansive mapping with onstant sequen es {kn} = {1} and {sn} = {0} for ea h n ≥ 1. Similarly, we an show that S is also generalized asymptoti ally quasi-nonexpansive mapping with onstant sequen es {kn} = {1} and {sn} = {0} for ea h n ≥ 1. But the onverse does not hold in general. Re eived: Mar h 2012. A epted: February 2013. Referen es [1℄ M. Abbas and B.E. 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