@ Appl. Gen. Topol. 18, no. 1 (2017), 117-129 doi:10.4995/agt.2017.6578 c© AGT, UPV, 2017 On the generalized asymptotically nonspreading mappings in convex metric spaces Withun Phuengrattana a,b a Department of Mathematics, Faculty of Science and Technology, Nakhon Pathom Rajabhat University, Nakhon Pathom 73000, Thailand (withun ph@yahoo.com) b Research Center for Pure and Applied Mathematics, Research and Development Institute, Nakhon Pathom Rajabhat University, Nakhon Pathom 73000, Thailand. (withun ph@yahoo.com) Communicated by S. Romaguera Abstract In this article, we propose a new class of nonlinear mappings, namely, generalized asymptotically nonspreading mapping, and prove the exis- tence of fixed points for such mapping in convex metric spaces. Fur- thermore, we also obtain the demiclosed principle and a ∆-convergence theorem of Mann iteration for generalized asymptotically nonspreading mappings in CAT(0) spaces. 2010 MSC: 47H09; 47H10. Keywords: asymptotically nonspreading mapping; convex metric spaces; CAT(0) spaces; demiclosed principle. 1. Introduction Throughout this paper, we denote the set of positive integers by N. Let T be a mapping on a nonempty subset C of a Banach space X. We denote by F(T) the set of fixed points of T , i.e., F(T) = {x ∈ C : Tx = x}. In 2008, Kohsaka and Takahashi [14] introduced a nonlinear mapping called nonspreading mapping in a smooth, strictly convex, and reflexive Banach space X as follows: Let C be a nonempty closed convex subset of X. A mapping Received 09 September 2016 – Accepted 24 December 2016 http://dx.doi.org/10.4995/agt.2017.6578 W. Phuengrattana T : C → C is said to be nonspreading if φ(Tx,Ty) + φ(Ty,Tx) ≤ φ(Tx,y) + φ(Ty,x)(1.1) for all x,y ∈ C, where φ(x,y) = ‖x‖2 − 2〈x,Jy〉 + ‖y‖2 for all x,y ∈ X and J is the duality mapping on C. Observe that if X is a real Hilbert space, then J is the identity mapping and φ(x,y) = ‖x−y‖2 for all x,y ∈ X. So, a nonspreading mapping T in a real Hilbert space X is defined as follows: 2‖Tx−Ty‖2 ≤‖Tx−y‖2 + ‖Ty −x‖2 for all x,y ∈ C. Since then, some fixed point theorems of such mapping has been studied by many researchers; see, for example, [9, 10, 11]. Later in 2013, Naraghirad [17] introduced a new class of nonspreading-type mappings in a real Banach space, called an asymptotically nonspreading map- ping, as follows: A mapping T : C → C is called asymptotically nonspreading if ‖Tnx−Tny‖2 ≤‖x−y‖2 + 2〈x−Tnx,J(y −Tny)〉(1.2) for all x,y ∈ C and n ∈ N, where J is the normalized duality mapping of C. In the case when X is a real Hilbert space, we know that J is the identity mapping. So, an asymptotically nonspreading mapping T in a real Hilbert space X is defined as follows: ‖Tnx−Tny‖2 ≤‖x−y‖2 + 2〈x−Tnx,y −Tny〉(1.3) for all x,y ∈ C and n ∈ N. In a real Hilbert space, it is easy to show that (1.3) is equivalent to 2‖Tnx−Tny‖2 ≤‖Tnx−y‖2 + ‖Tny −x‖2 for all x,y ∈ C and n ∈ N. Naraghirad [17] proved weak and strong convergence theorems of the itera- tive sequences generated by an asymptotically nonspreading mapping in a real Banach space. Motivated by the above works, we define a new class of nonlinear mappings which contains the class of asymptotically nonspreading mappings in convex metric spaces, called a generalized asymptotically nonspreading mapping, and prove some existence theorems for such mapping in convex metric spaces. Fur- thermore, we also obtain the demiclosed principle and a ∆-convergence theo- rem of Mann iteration for generalized asymptotically nonspreading mappings in CAT(0) spaces. 2. Preliminaries In the sequel, we recall some definitions, notations, and conclusions which will be needed in proving our main results. Let (X,d) be a metric space. A mapping W : X ×X × [0, 1] → X is said to be a convex structure [22] on X if for each x,y ∈ X and λ ∈ [0, 1], d(z,W(x,y,λ)) ≤ λd(z,x) + (1 −λ)d(z,y) c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 118 On the generalized asymptotically nonspreading mappings n convex metric spaces for all z ∈ X. A metric space (X,d) together with a convex structure W is called a convex metric space which will be denoted by (X,d,W). A nonempty subset C of X is said to be convex if W(x,y,λ) ∈ C for all x,y ∈ C and λ ∈ [0, 1]. It is easy to see that open and closed balls are convex and the intersection of a family of convex subsets of a convex metric space X is also convex, see [22]. Clearly, a normed space and each of its convex subsets are convex metric spaces, but the converse does not hold. In 1996, Shimizu and Takahashi [21] introduced the concept of uniform con- vexity in convex metric spaces and studied the properties of these spaces. Definition 2.1. A convex metric space (X,d,W) is said to be uniformly convex if for any ε > 0, there exists δ(ε) ∈ (0, 1] such that for all r > 0 and x,y,z ∈ X with d(z,x) ≤ r, d(z,y) ≤ r and d(x,y) ≥ rε, imply that d ( z,W ( x,y, 1 2 )) ≤ (1 − δ(ε)) r. Obviously, uniformly convex Banach spaces are uniformly convex metric spaces. One of the special spaces of uniformly convex metric spaces is a CAT(0) space (see more details in [3]). The useful inequality of CAT(0) space is (CN) inequality [4], that is, if z,x,y are points in a CAT(0) space and if m is the midpoint of the geodesic segment [x,y], then the CAT(0) inequality implies (CN) d(z,m)2 ≤ 1 2 d(z,x)2 + 1 2 d(z,y)2 − 1 4 d(x,y)2. By using the (CN) inequality, it is easy to see that CAT(0) spaces are uniformly convex. Moreover, if X is a CAT(0) space and x,y ∈ X, then for any λ ∈ [0, 1], there exists a unique point λx⊕ (1 −λ)y ∈ [x,y] such that d(z,λx⊕ (1 −λ)y) ≤ λd(z,x) + (1 −λ)d(z,y), for any z ∈ X. It follows that CAT(0) spaces have a convex structure W(x,y,λ) := λx ⊕ (1 − λ)y. Existence theorems and convergence theorems in convex metric spaces and CAT(0) spaces have been studied and investigated, see, for examples, [13, 5, 16, 12, 18, 15, 19, 1, 2]. The notion of the asymptotic center can be introduced in the general setting of a CAT(0) space X as follows: Let {xn} be a bounded sequence in X. For x ∈ X, we define a mapping r (·,{xn}) : X → [0,∞) by r (x,{xn}) = lim sup n→∞ d(x,xn). The asymptotic radius of {xn} is given by r ({xn}) = inf {r (x,{xn}) : x ∈ X} , and the asymptotic center of {xn} is the set A ({xn}) = {x ∈ X : r (x,{xn}) = r ({xn})} . It is known by [7] that in a CAT(0) space, the asymptotic center A ({xn}) consists of exactly one point. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 119 W. Phuengrattana We now give the definition and collect some basic properties of the ∆- convergence which will be used in the sequel. Definition 2.2 ([13]). A sequence {xn} in a CAT(0) space X is said to ∆- converge to x ∈ X if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case, we write ∆-limn→∞ xn = x and call x the ∆-limit of {xn}. We now collect some basic properties of the ∆-convergence which will be used in the sequel. Lemma 2.3 ([13]). Every bounded sequence in a CAT(0) space has a ∆- convergent subsequence. Lemma 2.4 ([6]). Let C be a nonempty closed convex subset of a CAT(0) space X. If {xn} is a bounded sequence in C, then the asymptotic center of {xn} is in C. Lemma 2.5 ([8]). Let {xn} be a sequence in a CAT(0) space X with A({xn}) = {x}. If {un} is a subsequence of {xn} with A({un}) = {u} and {d(xn,u)} converges, then x = u. The following lemma is a generalization of the (CN) inequality which can be found in [8]. Lemma 2.6. Let X be a CAT(0) space. Then d(z,λx⊕ (1 −λ)y)2 ≤ λd(z,x)2 + (1 −λ)d(z,y)2 −λ(1 −λ)d(x,y)2, for any λ ∈ [0, 1] and x,y,z ∈ X. 3. Main results In this section, we study the existence and convergence theorems for a gen- eralized asymptotically nonspreading mapping in both convex metric spaces and CAT(0) spaces. We first define a generalized asymptotically nonspreading mapping in convex metric spaces. Definition 3.1. Let C be a nonempty subset of a convex metric space (X,d,W). A mapping T : C → C is called generalized asymptotically nonspreading if there exist two functions f,g : C → [0,γ], γ < 1 such that (C1) d(Tnx,Tny)2 ≤ f(x)d(Tnx,y)2 + g(x)d(Tny,x)2 for all x,y ∈ C and n ∈ N; (C2) 0 < f(x) + g(x) ≤ 1 for all x ∈ C. Remark 3.2. The class of generalized asymptotically nonspreading mappings contains the class of asymptotically nonspreading mappings. Indeed, we know that if f(x) = g(x) = 1 2 for all x ∈ C, then T is an asymptotically nonspreading mapping. The next example shows that there is a generalized asymptotically non- spreading mapping which is not asymptotically nonspreading. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 120 On the generalized asymptotically nonspreading mappings n convex metric spaces Example 3.3. Define a mapping T : [0,∞) → [0,∞) by Tx = { 0.9, if x ≥ 1, 0, if x ∈ [0, 1). Then T is not an asymptotically nonspreading mapping. Indeed, if x = 1.2 and y = 0.7, then Tx = 0.9, Ty = 0, and 2d(Tx,Ty)2 = 1.62 > 1.48 = 0.04 + 1.44 = d(Tx,y)2 + d(Ty,x)2. However, T is a generalized asymptotically nonspreading mapping. Indeed, let f,g : [0,∞) → [0, 0.9) be defined by f(x) = { 0, if x ≥ 1, 0.81, if x ∈ [0, 1), and g(x) = { 0.81, if x ≥ 1, 0, if x ∈ [0, 1). Now, we only need to consider the following two cases: (i) If x ≥ 1 and y ∈ [0, 1), then Tx = 0.9,Ty = 0,Tnx = Tny = 0 ∀n ≥ 2, f(x) = 0, and g(x) = 0.81. So, we have d(Tx,Ty)2 = 0.81 ≤ g(x)x2 = f(x)d(Tx,y)2 + g(x)d(Ty,x)2. On the other hand, for any n ≥ 2, we have d(Tnx,Tny)2 = 0 ≤ f(x)d(Tnx,y)2 + g(x)d(Tny,x)2. (ii) If x ∈ [0, 1) and y ≥ 1, then Tx = 0,Ty = 0.9,Tnx = Tny = 0 ∀n ≥ 2, f(x) = 0.81, and g(x) = 0. So, we have d(Tx,Ty)2 = 0.81 ≤ f(x)y2 = f(x)d(Tx,y)2 + g(x)d(Ty,x)2. On the other hand, for any n ≥ 2, we have d(Tnx,Tny)2 = 0 ≤ f(x)d(Tnx,y)2 + g(x)d(Tny,x)2. Therefore, T is a generalized asymptotically nonspreading mapping. 3.1. Existence theorems. We now prove existence theorems for generalized asymptotically nonspreading mappings in complete convex metric spaces. Theorem 3.4. Let C be a nonempty closed convex subset of a complete convex metric space (X,d,W) such that A({xn}) is singleton for all bounded sequence {xn} in C and T : C → C be a generalized asymptotically nonspreading map- ping. Then the following assertions are equivalent: (i) F(T) is nonempty; (ii) there exists a bounded sequence {xn} in C such that lim inf n→∞ d(xn,Txn) = 0. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 121 W. Phuengrattana Proof. Since it is obvious that (i) implies (ii), we show that (ii) implies (i). Suppose that there exists a bounded sequence {xn} in C such that lim inf n→∞ d(xn,Txn) = 0. Consequently, there is a bounded subsequence {xnk} of {xn} such that lim k→∞ d(xnk,Txnk ) = 0. Suppose A({xnk}) = {z}. Since T is a generalized asymptotically nonspreading mapping, we have d(xnk,Tz) 2 ≤ (d(xnk,Txnk ) + d(Tz,Txnk )) 2 = d(xnk,Txnk ) 2 + d(Tz,Txnk ) 2 + 2d(xnk,Txnk )d(Txnk,Tz) ≤ d(xnk,Txnk ) 2 + f(z)d(Tz,xnk ) 2 + g(z)d(Txnk,z) 2 + 2d(xnk,Txnk )d(Txnk,Tz) ≤ d(xnk,Txnk ) 2 + f(z)d(Tz,xnk ) 2 + g(z)(d(Txnk,xnk ) + d(xnk,z)) 2 + 2d(xnk,Txnk )d(Txnk,Tz) = (1 + g(z))d(xnk,Txnk ) 2 + f(z)d(Tz,xnk ) 2 + g(z)d(xnk,z) 2 + 2g(z)d(Txnk,xnk )d(xnk,z) + 2d(xnk,Txnk )d(Txnk,Tz). This implies that (1 −f(z))d(xnk,Tz) 2 ≤ (1 + g(z))d(xnk,Txnk ) 2 + g(z)d(xnk,z) 2 + 2M(1 + g(z))d(xnk,Txnk ), where M = supk∈N{d(xnk,z),d(Txnk,Tz)}. Taking lim sup on both sides of the above inequality, we get (1 −f(z)) lim sup k→∞ d(xnk,Tz) 2 ≤ g(z) lim sup k→∞ d(xnk,z) 2 ≤ (1 −f(z)) lim sup k→∞ d(xnk,z) 2. So, we have lim sup k→∞ d(xnk,Tz) ≤ lim sup k→∞ d(xnk,z). It implies that r(Tz,{xnk}) = lim sup k→∞ d(xnk,Tz) ≤ lim sup k→∞ d(xnk,z) = r(z,{xnk}). This shows that Tz ∈ A({xnk}). By the uniqueness of asymptotic center, we conclude that Tz = z. � Theorem 3.5. Let C be a nonempty closed convex subset of a complete convex metric space (X,d,W) such that A({xn}) is singleton for all bounded sequence {xn} in C and T : C → C be a generalized asymptotically nonspreading map- ping. If limn→∞ d(T nx,Tn+1x) = 0 for all x ∈ C, then the following assertions are equivalent: c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 122 On the generalized asymptotically nonspreading mappings n convex metric spaces (i) F(T) is nonempty; (ii) there exists x ∈ C such that {Tnx} is bounded. Proof. Since it is obvious that (i) implies (ii), we show that (ii) implies (i). Suppose that there exists x ∈ C such that {Tnx} is bounded. Setting yn = Tnx for all n ∈ N. Then we have lim n→∞ d(Tyn,yn) = lim n→∞ d(Tn+1x,Tnx) = 0. Since {yn} is bounded, it implies by Theorem 3.4 that F(T) is nonempty. � It follows from the fact that, in a complete uniformly convex metric space, the asymptotic center of a bounded sequence with respect to a closed convex subset is singleton; see [20]. So, we have the following results. Theorem 3.6. Let C be a nonempty closed convex subset of a complete uni- formly convex metric space (X,d,W) and T : C → C be a generalized asymp- totically nonspreading mapping. Then the following assertions are equivalent: (i) F(T) is nonempty; (ii) there exists a bounded sequence {xn} in C such that lim inf n→∞ d(xn,Txn) = 0. Theorem 3.7. Let C be a nonempty closed convex subset of a complete uni- formly convex metric space (X,d,W) and T : C → C be a generalized asymp- totically nonspreading mapping. If limn→∞ d(T nx,Tn+1x) = 0 for all x ∈ C, then the following assertions are equivalent: (i) F(T) is nonempty; (ii) there exists x ∈ C such that {Tnx} is bounded. Since every CAT(0) space is a uniformly convex metric space, the following results can be obtained from Theorems 3.6 and 3.7 immediately. Theorem 3.8. Let C be a nonempty closed convex subset of a complete CAT(0) space (X,d) and T : C → C be a generalized asymptotically nonspreading mapping. Then the following assertions are equivalent: (i) F(T) is nonempty; (ii) there exists a bounded sequence {xn} in C such that lim inf n→∞ d(xn,Txn) = 0. Theorem 3.9. Let C be a nonempty closed convex subset of a complete CAT(0) space (X,d) and T : C → C be a generalized asymptotically nonspreading mapping. If limn→∞ d(T nx,Tn+1x) = 0 for all x ∈ C, then the following assertions are equivalent: (i) F(T) is nonempty; (ii) there exists x ∈ C such that {Tnx} is bounded. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 123 W. Phuengrattana Remark 3.10. Theorems 3.4-3.9 improve and extend the main results of Naraghi- rad [17] from an asymptotically nonspreading mapping to a generalized asymp- totically nonspreading mapping and from a Banach space to a complete convex metric space. 3.2. ∆-convergence theorems. In this section, we study ∆-convergence theo- rems for a generalized asymptotically nonspreading mapping in complete CAT(0) spaces. The following theorem show the demiclosed principle for a generalized asymp- totically nonspreading mapping on complete CAT(0) spaces. Theorem 3.11. Let C be a nonempty closed convex subset of a complete CAT(0) space (X,d) and T : C → C be a generalized asymptotically non- spreading mapping. If {xn} is a bounded sequence in C that ∆-converges to z and limn→∞ d(xn,Txn) = 0, then z ∈ F(T). Proof. Suppose that {xn} is a bounded sequence in C such that ∆-limn→∞ xn = z and limn→∞ d(xn,Txn) = 0. By the definition of T, we have d(xn,Tz) 2 ≤ (d(xn,Txn) + d(Tz,Txn)) 2 = d(xn,Txn) 2 + 2d(xn,Txn)d(Tz,Txn) + d(Tz,Txn) 2 ≤ d(xn,Txn)2 + 2d(xn,Txn)d(Tz,Txn) + f(z)d(Tz,xn)2 + g(z)d(Txn,z) 2 ≤ d(xn,Txn)2 + 2d(xn,Txn)d(Tz,Txn) + f(z)d(Tz,xn)2 + g(z)(d(Txn,xn) + d(xn,z)) 2 = (1 + g(z))d(xn,Txn) 2 + 2d(xn,Txn)d(Tz,Txn) + f(z)d(Tz,xn) 2 + 2g(z)d(Txn,xn)d(xn,z) + g(z)d(xn,z) 2. This implies that (1 −f(z))d(xn,Tz)2 ≤ (1 + g(z))d(xn,Txn)2 + 2M(1 + g(z))d(xn,Txn) + g(z)d(xn,z) 2, where M = supn∈N{d(xn,z),d(Txn,Tz)}. Taking lim sup on both sides of the above inequality, we get (1 −f(z)) lim sup n→∞ d(xn,Tz) 2 ≤ g(z) lim sup n→∞ d(xn,z) 2 ≤ (1 −f(z)) lim sup n→∞ d(xn,z) 2. Thus, we have lim sup n→∞ d(xn,Tz) ≤ lim sup n→∞ d(xn,z). By the uniqueness of asymptotic centers, we have Tz = z. Hence z ∈ F(T). � By using Theorem 3.11, we obtain the following ∆-convergence theorem for a generalized asymptotically nonspreading mapping in complete CAT(0) spaces. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 124 On the generalized asymptotically nonspreading mappings n convex metric spaces Theorem 3.12. Let C be a nonempty closed convex subset of a complete CAT(0) space (X,d) and T : C → C be a generalized asymptotically non- spreading mapping with F(T) is nonempty. Assume that {αn} is a sequence in (0, 1) such that 0 < a ≤ αn ≤ b < 1. Let {xn} be a sequence in C generated by xn+1 = (1 −αn)xn ⊕αnTnxn, for all n ∈ N.(3.1) Then the sequence {xn} ∆-converges to a fixed point of T . Proof. Let z ∈ F(T). Since T is generalized asymptotically nonspreading, we have d(z,Tnxn) 2 ≤ f(z)d(z,xn)2 + g(z)d(Tnxn,z)2. This implies that (1 −g(z))d(z,Tnxn)2 ≤ f(z)d(z,xn)2. Since 0 < f(z) + g(z) ≤ 1, we have d(z,Tnxn) ≤ d(z,xn).(3.2) In view of Lemma 2.6, (3.1), and (3.2), d(xn+1,z) 2 ≤ (1 −αn)d(xn,z)2 + αnd(Tnxn,z)2 −αn(1 −αn)d(xn,Tnxn)2 ≤ d(xn,z)2 −αn(1 −αn)d(xn,Tnxn)2 ≤ d(xn,z)2 −a(1 − b)d(xn,Tnxn)2.(3.3) Thus, we have d(xn+1,z) ≤ d(xn,z). This implies that limn→∞ d(xn,z) exists for all z ∈ F(T) and hence {xn} is bounded. It follows by (3.3) that a(1 − b)d(xn,Tnxn)2 ≤ d(xn,z)2 −d(xn+1,z)2, which yields that lim n→∞ d(xn,T nxn) = 0.(3.4) In view of (3.1), we see that d(xn+1,xn) ≤ αnd(Tnxn,xn) ≤ bd(Tnxn,xn). Then we have lim n→∞ d(xn+1,xn) = 0.(3.5) c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 125 W. Phuengrattana Since T is generalized asymptotically nonspreading, we obtain that d(Tn+1xn,T n+1xn+1) 2 ≤ f(xn)d(Tn+1xn,xn+1)2 + g(xn)d(Tn+1xn+1,xn)2 ≤ f(xn)(d(Tn+1xn,Tn+1xn+1) + d(Tn+1xn+1,xn+1))2 + g(xn)(d(T n+1xn+1,xn+1) + d(xn+1,xn)) 2 = f(xn)d(T n+1xn,T n+1xn+1) 2 + f(xn)d(T n+1xn+1,xn+1) 2 + 2f(xn)d(T n+1xn,T n+1xn+1)d(T n+1xn+1,xn+1) + g(xn)d(T n+1xn+1,xn+1) 2 + g(xn)d(xn+1,xn) 2 + 2g(xn)d(T n+1xn+1,xn+1)d(xn+1,xn) ≤ γd(Tn+1xn,Tn+1xn+1)2 + γd(Tn+1xn+1,xn+1)2 + 2γd(Tn+1xn,T n+1xn+1)d(T n+1xn+1,xn+1) + γd(Tn+1xn+1,xn+1) 2 + γd(xn+1,xn) 2 + 2γd(Tn+1xn+1,xn+1)d(xn+1,xn). Thus, we have (1 −γ)d(Tn+1xn,Tn+1xn+1)2 ≤ 2γd(Tn+1xn+1,xn+1)2 + 2γM1d(Tn+1xn+1,xn+1) + γd(xn+1,xn) 2 + 2γd(Tn+1xn+1,xn+1)d(xn+1,xn), where M1 = supn∈N{d(Tn+1xn,Tn+1xn+1)}. This implies from (3.4) and (3.5) that lim n→∞ d(Tn+1xn,T n+1xn+1) = 0.(3.6) Consider d(xn,T n+1xn) ≤ d(xn,xn+1) + d(xn+1,Tn+1xn+1) + d(Tn+1xn+1,Tn+1xn), then, by (3.4), (3.5), and (3.6), we have lim n→∞ d(xn,T n+1xn) = 0.(3.7) Since d(Tnxn,T n+1xn) ≤ d(Tnxn,xn) +d(xn,Tn+1xn), it implies by (3.6) and (3.7) that lim n→∞ d(Tnxn,T n+1xn) = 0.(3.8) c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 126 On the generalized asymptotically nonspreading mappings n convex metric spaces Since T is generalized asymptotically nonspreading, we have d(Txn,T n+1xn) 2 ≤ f(xn)d(Txn,Tnxn)2 + g(xn)d(Tn+1xn,xn)2 ≤ f(xn)(d(Txn,Tn+1xn) + d(Tn+1xn,Tnxn))2 + g(xn)d(T n+1xn,xn) 2 ≤ f(xn)d(Txn,Tn+1xn)2 + f(xn)d(Tn+1xn,Tnxn)2 + 2f(xn)d(Txn,T n+1xn)d(T n+1xn,T nxn) + g(xn)d(T n+1xn,xn) 2 ≤ γd(Txn,Tn+1xn)2 + γd(Tn+1xn,Tnxn)2 + 2γd(Txn,T n+1xn)d(T n+1xn,T nxn) + γd(T n+1xn,xn) 2. This implies that (1 −γ)d(Txn,Tn+1xn)2 ≤ γd(Tn+1xn,Tnxn)2 + 2γM2d(Tn+1xn,Tnxn) + γd(Tn+1xn,xn) 2, where M2 = supn∈N{d(Txn,Tn+1xn)}. Thus, by (3.7) and (3.8), we have lim n→∞ d(Txn,T n+1xn) = 0.(3.9) From d(Txn,xn) ≤ d(Txn,Tn+1xn) + d(Tn+1xn,xn), it implies by (3.7) and (3.9) that lim n→∞ d(Txn,xn) = 0.(3.10) We now let ω∆(xn) := ⋃ A({un}), where the union is taken over all subse- quences {un} of {xn}. We claim that ω∆(xn) ⊂ F(T). Let u ∈ ω∆(xn). Then there exists a subsequence {un} of {xn} such that A({un}) = {u}. Since {un} is bounded, it implies by Lemma 2.3 that there exists a subsequence {unk} of {un} such that ∆-limk→∞ unk = y ∈ C. By (3.10) and Theorem 3.11, we have y ∈ F(T). Then limn→∞ d(xn,y) exists. Suppose that u 6= y. By the uniqueness of asymptotic centers, we obtain that lim sup k→∞ d(unk,y) < lim sup k→∞ d(unk,u) ≤ lim sup n→∞ d(un,u) < lim sup n→∞ d(un,y) = lim sup n→∞ d(xn,y) = lim sup k→∞ d(unk,y). This is a contradiction, hence u = y ∈ F(T). This shows that ω∆(xn) ⊂ F(T). Next, we show that ω∆(xn) consists of exactly one point. Let {un} be a subsequence of {xn} with A({un}) = {u} and let A({xn}) = {z}. Since u ∈ ω∆(xn) ⊂ F(T), it implies that limn→∞ d(xn,u) exists. By Lemma 2.5, we get z = u. Hence, the sequence {xn} ∆-converges to a fixed point z of T. � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 127 W. Phuengrattana Remark 3.13. 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