International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 (2015), 39-52 http://www.etamaths.com STRONG AND ∆-CONVERGENCE OF MODIFIED TWO-STEP ITERATIONS FOR NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN HYPERBOLIC SPACES G. S. SALUJA Abstract. The aim of this article is to establish a ∆-convergence and some strong convergence theorems of modified two-step iterations for two nearly asymptotically nonexpansive mappings in the setting of hyperbolic spaces. Our results extend and generalize the previous work from the current existing literature. 1. Introduction The class of asymptotically nonexpansive mapping, introduced by Goebel and Kirk [7] in 1972, is an important generalization of the class of nonexpansive map- ping. They proved that if C is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self mapping of C has a fixed point. There are number of papers dealing with the approximation of fixed points / common fixed points of asymptotically nonexpansive and asymptotically quasi- nonexpansive mappings in uniformly convex Banach spaces using modified Mannn and Ishikawa iteration processes and have been studied by many authors (see, e.g., [17, 18, 24, 28, 29, 31, 34, 35]). The concept of ∆-convergence in a general metric space was introduced by Lim [16]. In 2008, Kirk and Panyanak [14] used the notion of ∆-convergence introduced by Lim [16] to prove in the CAT(0) space and analogous of some Banach space results which involve weak convergence. Further, Dhompongsa and Panyanak [6] obtained ∆-convergence theorems for the Picard, Mann and Ishikawa iterations in a CAT(0) space. Since then, the existence problem and the ∆-convergence problem of iterative sequences to a fixed point for nonexpansive mapping, asymptotically nonexpansive mapping, nearly asymptotically nonexpansive mapping, asymptoti- cally quasi-nonexpansive mapping in the intermediate sense, total asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping through Picard, Mann [19], Ishikawa[10], modified Agarwal et al. [2] have been rapidly de- veloped in the framework of CAT(0) space and many papers have appeared in this 2010 Mathematics Subject Classification. 47H10. Key words and phrases. Nearly asymptotically nonexpansive mapping; modified two-step it- eration scheme; common fixed point; strong convergence; ∆-convergence; hyperbolic space. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 39 40 SALUJA direction (see, e.g., [1, 5, 6, 11, 20, 25]). The purpose of this paper is to establish some strong convergence theorems of modified two-step iteration process for two nearly asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces which include both uniformly con- vex Banach spaces and CAT(0) spaces. Our results extend and improve the previous work from the current existing literature. 2. Preliminaries Let F(T) = {x ∈ K : Tx = x} denotes the set of fixed points of the mapping T. We begin with the following definitions. Definition 2.1. Let (X,d) be a metric space and K be its nonempty subset. Then T : K → K said to be (1) nonexpansive if d(Tx,Ty) ≤ d(x,y) for all x,y ∈ K; (2) asymptotically nonexpansive if there exists a sequence {un} ⊂ [0,∞) with limn→∞un = 0 such that d(T nx,Tny) ≤ (1 + un)d(x,y) for all x,y ∈ K and n ≥ 1; (3) asymptotically quasi-nonexpansive if F(T) 6= ∅ and there exists a sequence {un} ⊂ [0,∞) with limn→∞un = 0 such that d(Tnx,p) ≤ (1 + un)d(x,p) for all x ∈ K, p ∈ F(T) and n ≥ 1; (4) uniformly L-Lipschitzian if there exists a constant L > 0 such that d(Tnx,Tny) ≤ Ld(x,y) for all x,y ∈ K and n ≥ 1; (5) semi-compact if for a sequence {xn} in K with limn→∞d(xn,Txn) = 0, there exists a subsequence {xnk} of {xn} such that xnk → p ∈ K as k →∞. (6) a sequence {xn} in K is called approximate fixed point sequence for T (AFPS, in short) if limn→∞d(xn,Txn) = 0. The class of nearly Lipschitzian mappings is an important generalization of the class of Lipschitzian mappings and was introduced by Sahu [26] (see, also [27]). Definition 2.2. Let K be a nonempty subset of a metric space (X,d) and fix a sequence {an} ⊂ [0,∞) with limn→∞an = 0. A mapping T : K → K said to be nearly Lipschitzian with respect to {an} if for all n ≥ 1, there exists a constant kn ≥ 0 such that d(Tnx,Tny) ≤ kn[d(x,y) + an] for all x, y ∈ K. The infimum of the constants kn for which the above inequality holds is denoted by η(Tn) and is called nearly Lipschitz constant of Tn. A nearly Lipschitzian mapping T with sequence {an,η(Tn)} is said to be: (i) nearly nonexpansive if η(Tn) = 1 for all n ≥ 1; STRONG AND ∆-CONVERGENCE OF MODIFIED TWO-STEP ITERATIONS 41 (i) nearly asymptotically nonexpansive if η(Tn) ≥ 1 for all n ≥ 1 and limn→∞η(T n) = 1. (ii) nearly uniformly k-Lipschitzian if η(Tn) ≤ k for all n ≥ 1. Throughout this paper, we work in the setting of hyperbolic space introduced by Kohlenbach [15]. It is worth noting that they are different from Gromov hyper- bolic space [4] or from other notions of hyperbolic space that can be found in the literature (see for example [8, 13, 23]. A hyperbolic space [15] is a triple (X,d,W) where (X,d) is a metric space and W : X2 × [0, 1] → X is such that (i) d(u,W(x,y,α)) ≤ αd(u,x) + (1 −α)d(u,y) (ii) d ( W(x,y,α),W(x,y,β) ) = |α−β|d(x,y) (iii) W(x,y,α) = W(x,y, (1 −α)) (iv) d ( W(x,z,α),W(y,w,β) ) ≤ αd(x,y) + (1 −α) d(z,w) for all x,y,z,w ∈ X and α,β ∈ [0, 1]. The class of hyperbolic spaces in the sense of Kohlenbach [15] contains all normed linear spaces and convex subsets thereof as well as Hadamard manifolds and CAT(0) spaces in the sense of Gromov [9]. An important example of a hyperbolic space is the open unit ball BH in a real Hilbert space H is as follows. Let BH be the open unit ball in H. Then kBH (x,y) = arg tanh(1 −σ(x,y)) 1/2, where σ(x,y) = ( 1 −‖x‖2 )( 1 −‖y‖2 ) |1 −〈x,y 〉|2 for all x, y ∈ BH, defines a metric on BH (also known as Kobayashi distance). A metric space (X,d) is called a convex metric space introduced by Takahashi in [33] if it satisfies only (i). A subset K of a hyperbolic space X is convex if W(x,y,α) ∈ K for all x,y ∈ K and α ∈ [0, 1]. A hyperbolic space (X,d,W) is said to be uniformly convex [32] if for all u,x,y ∈ X, r > 0 and ε ∈ (0, 2], there exists a δ ∈ (0, 1] such that d(W(x,y, 1 2 ),u) ≤ (1−δ)r whenever d(x,u) ≤ r, d(y,u) ≤ r and d(x,y) ≥ εr. A mapping η : (0,∞)×(0, 2] → (0, 1] which provides such a δ = η(r,ε) for given r > 0 and ε ∈ (0, 2], is known as modulus of uniform convexity. We call η monotone if it decreases with r (for a fixed ε). 42 SALUJA Let K be a nonempty subset of hyperbolic space X. Let {xn} be a bounded sequence in a hyperbolic space X. For x ∈ X, define a continuous functional r(., {xn}) : X → [0,∞) by r(x, {xn}) = lim supn→∞d(x,xn). The asymptotic radius ρ = r({xn}) of {xn} is given by ρ = inf{r(x,{xn}) : x ∈ X}. The asymptotic center AK({xn}) of a bounded sequence {xn} with respect to a subset K of X is defined as follows: AK({xn}) = { x ∈ X : r(x, {xn}) ≤ r(y, {xn}) } for any y ∈ K. The set of all asymptotic center of {xn} is denoted by A({xn}). It has been shown in [32] that bounded sequences have unique asymptotic center with respect to closed convex subsets in a complete and uniformly hyperbolic space with monotone modulus of uniform convexity. A sequence {xn} in X is said to ∆-converge to x ∈ X if x is the unique asymp- totic center of {un} for every subsequence {un} of {xn} [14]. In this case, we write ∆-limn xn = x and call x is the ∆-limit of {xn}. Recall that ∆-convergence coincides with weak convergence in Banach space with Opial’s property [21]. In the sequel we need the following lemmas. Lemma 2.3. [12] Let (X,d,W) be a uniformly convex hyperbolic space with mono- tone modulus of uniform convexity η. Let x ∈ X and {αn} be a sequence in [b,c] for some b, c ∈ (0, 1). If {xn} and {yn} are sequences in X such that lim supn→∞d(xn,x) ≤ r, lim supn→∞d(yn,x) ≤ r and limn→∞d(W(xn,yn,αn),x) = r for some r ≥ 0, then limn→∞d(xn,yn) = 0. Lemma 2.4. [12] Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space X and {xn} a bounded sequence in K such that A({xn}) = {y} and r({xn}) = ρ. If {ym} is another sequence in K such that limm→∞r(ym,{xn}) = ρ, then limm→∞ym = y. Lemma 2.5. (See [34]) Let {pn}∞n=1, {qn}∞n=1 and {rn}∞n=1 be sequences of non- negative numbers satisfying the inequality pn+1 ≤ (1 + qn)pn + rn, ∀n ≥ 1. If ∑∞ n=1 qn < ∞ and ∑∞ n=1 rn < ∞, then limn→∞pn exists. First, we define the modified two-step iteration scheme in hyperbolic space as follows. Let K be a nonempty closed convex subset of a hyperbolic space X and S, T : K → K be two nearly asymptotically nonexpansive mappings. Then, for an arbitrary chosen x1 ∈ K, we construct the sequence {xn} in K such that (2.1) { xn+1 = W(T nxn,S nyn,αn), yn = W(S nxn,T nxn,βn), n ≥ 1, STRONG AND ∆-CONVERGENCE OF MODIFIED TWO-STEP ITERATIONS 43 where {αn} and {βn} are appropriate sequences in (0,1) is called modified two-step iteration scheme. Iteration scheme (2.1) is independent of modified Ishikawa itera- tion and modified Mann iteration schemes. If βn = 0 for all n ≥ 1 and S = I, where I is the identity mapping, then iteration scheme (2.1) reduces to the following. (2.2) { xn+1 = W(T nxn,xn,αn), n ≥ 1, where {αn} is an appropriate sequence in (0,1) is called modified Mann iteration scheme in hyperbolic space. 3. Main Results Lemma 3.1. Let K be a nonempty convex subset of a hyperbolic space X and let S, T : K → K be two nearly asymptotically nonexpansive mappings with sequences {(a′n,η(Sn))} and {(a′′n,η(Tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)2η(Tn)2− 1 ) < ∞. Let {xn} be a sequence in K defined by (2.1). Then limn→∞d(xn,p) exists for each p ∈ F = F(S) ∩F(T). Proof. Let p ∈ F = F(S) ∩ F(T), ρ = supn∈N η(Sn) ∨ supn∈N η(Tn) and an = max{a′n,a′′n} for all n. From (2.1), we have d(yn,p) = d(W(S nxn,T nxn,βn),p) ≤ (1 −βn)d(Snxn,p) + βnd(Tnxn,p) ≤ (1 −βn)[η(Sn)(d(xn,p) + a′n)] + βn[η(T n)(d(xn,p) + a ′′ n)] ≤ (1 −βn)[η(Sn)(d(xn,p) + an)] + βn[η(Tn)(d(xn,p) + an)] = (1 −βn)η(Sn) d(xn,p) + βnη(Tn) d(xn,p) + ( η(Sn) + η(Tn) ) an ≤ η(Sn)η(Tn)[(1 −βn)d(xn,p) + βnd(xn,p)] + 2ρan = η(Sn)η(Tn)d(xn,p) + 2ρan.(3.1) Again, using (2.1) and (3.1), we get d(xn+1,p) = d(W(T nxn,S nyn,αn),p) ≤ (1 −αn)d(Tnxn,p) + αnd(Snyn,p) ≤ (1 −αn)[η(Tn)(d(xn,p) + a′′n)] + αn[η(S n)(d(yn,p) + a ′ n)] ≤ (1 −αn)[η(Tn)(d(xn,p) + an)] + αn[η(Sn)(d(yn,p) + an)] = (1 −αn)η(Tn)d(xn,p) + αnη(Sn)d(yn,p) + ( η(Sn) + η(Tn) ) an ≤ (1 −αn)η(Tn)d(xn,p) + 2ρan +αnη(S n)[η(Sn)η(Tn)d(xn,p) + 2ρan] ≤ η(Sn)2η(Tn)2d(xn,p) + (1 + η(Sn))2ρan ≤ η(Sn)2η(Tn)2d(xn,p) + 2ρ(1 + ρ)an = (1 + µn)d(xn,p) + νn(3.2) where µn = ( η(Sn)2η(Tn)2 − 1 ) and νn = 2ρ(1 + ρ)an. Since ∑∞ n=1 ( η(Sn)2 η(Tn)2−1 ) < ∞ and ∑∞ n=1 an < ∞, it follows that ∑∞ n=1 µn < ∞ and ∑∞ n=1 νn < 44 SALUJA ∞. Hence by Lemma 2.5, we get that limn→∞d(xn,p) exists. This completes the proof. � Lemma 3.2. Let K be a nonempty closed convex subset of a uniformly con- vex hyperbolic space X with monotone modulus of uniform convexity η and let S, T : K → K be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(Sn))} and {(a′′n,η(Tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)2η(Tn)2 − 1 ) < ∞. Let {xn} be a sequence in K defined by (2.1). Assume that F = F(S) ∩ F(T) 6= ∅. Suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). If d(x,Tnx) ≤ d(Snx,Tnx) and d(x,Snx) ≤ d(Tnx,Snx) for all x ∈ K, then limn→∞d(xn,Sxn) = 0 and limn→∞d(xn,Txn) = 0. Proof. From Lemma 3.1, we obtain limn→∞d(xn,p) exists for each p ∈ F. Suppose that limn→∞d(xn,p) = r ≥ 0. Since d(Snxn,p) ≤ η(Sn)(d(xn,p) + an) for all n ≥ 1, we have lim sup n→∞ d(Snxn,p) ≤ r. Also, since d(Tnxn,p) ≤ η(Tn)(d(xn,p) + an) for all n ≥ 1, we have lim sup n→∞ d(Tnxn,p) ≤ r. Also (3.1) yields lim sup n→∞ d(yn,p) ≤ r.(3.3) Hence lim sup n→∞ d(Snyn,p) ≤ lim sup n→∞ η(Sn)(d(yn,p) + an) ≤ r.(3.4) Since r = lim n→∞ d(xn+1,p) = lim n→∞ d(W(Tnxn,S nyn,αn),p), it follows from Lemma 2.3 that lim n→∞ d(Tnxn,S nyn) = 0.(3.5) From (2.1) and (3.5), we have d(xn+1,T nxn) = d(W(T nxn,S nyn,αn),T nxn) ≤ αn d(Tnxn,Snyn) ≤ d(Tnxn,Snyn) → 0 as n →∞.(3.6) Hence from (3.5) and (3.6), we have d(xn+1,S nyn) ≤ d(xn+1,Tnxn) + d(Tnxn,Snyn) → 0 as n →∞.(3.7) Now using (3.7), we have d(xn+1,p) ≤ d(xn+1,Snyn) + d(Snyn,p) ≤ d(xn+1,Snyn) + η(Sn)(d(yn,p) + an).(3.8) STRONG AND ∆-CONVERGENCE OF MODIFIED TWO-STEP ITERATIONS 45 The inequality (3.8) gives r ≤ lim inf n→∞ d(yn,p).(3.9) From (3.3) and (3.9), we get r = lim n→∞ d(yn,p) = lim n→∞ d(W(Snxn,T nxn,βn),p).(3.10) Applying Lemma 2.3 in (3.10), we obtain lim n→∞ d(Snxn,T nxn) = 0.(3.11) Now using (3.11) and hypothesis of the theorem d(x,Tnx) ≤ d(Snx,Tnx) for all x ∈ K, we get d(xn,S nxn) ≤ d(xn,Tnxn) + d(Tnxn,Snxn) ≤ d(Snxn,Tnxn) + d(Tnxn,Snxn) = 2 d(Snxn,T nxn) → 0 as n →∞.(3.12) Again using (3.11) and hypothesis of the theorem d(x,Snx) ≤ d(Tnx,Snx) for all x ∈ K, we get d(xn,T nxn) ≤ d(xn,Snxn) + d(Snxn,Tnxn) ≤ d(Tnxn,Snxn) + d(Snxn,Tnxn) = 2 d(Snxn,T nxn) → 0 as n →∞.(3.13) By uniform continuity of S and T , limn→∞d(xn,S nxn) = 0 implies that limn→∞d(Sxn,S n+1xn) = 0 and limn→∞d(xn,T nxn) = 0 implies that limn→∞d(Txn,T n+1xn) = 0. Note that d(xn+1,xn) = d(W(T nxn,S nyn,αn),xn) ≤ (1 −αn)d(xn,Tnxn) + αn d(Snyn,xn) ≤ (1 −αn)d(xn,Tnxn) + αn d(Snyn,xn+1) + αn d(xn+1,xn) ≤ (1 −αn)d(xn,Tnxn) + αn d(Snyn,xn+1) + md(xn+1,xn) This implies that (1 −m) d(xn+1,xn) ≤ (1 −αn)d(xn,Tnxn) + αn d(Snyn,xn+1). (3.14) Since (1 −m) > 0, using (3.7) and (3.13) in (3.14), we get lim n→∞ d(xn+1,xn) = 0.(3.15) Also d(xn,Sxn) ≤ d(xn,xn+1) + d(xn+1,Sn+1xn+1) +d(Sn+1xn+1,S n+1xn) + d(S n+1xn,Sxn) ≤ ( 1 + η(Sn+1) ) d(xn,xn+1) + d(xn+1,S n+1xn+1) +d(Sn+1xn,Sxn) + an+1.(3.16) Using (3.12), (3.15) and uniform continuity of S, equation (3.16) gives lim n→∞ d(xn,Sxn) = 0.(3.17) 46 SALUJA Similarly d(xn,Txn) ≤ d(xn,xn+1) + d(xn+1,Tn+1xn+1) +d(Tn+1xn+1,T n+1xn) + d(T n+1xn,Txn) ≤ ( 1 + η(Tn+1) ) d(xn,xn+1) + d(xn+1,T n+1xn+1) +d(Tn+1xn,Txn) + an+1.(3.18) The above inequality gives lim n→∞ d(xn,Txn) = 0.(3.19) This completes the proof. � We now establish a ∆-convergence and some strong convergence theorems of modified two-step iteration scheme for non-Lipschitzian mappings in the framework of uniformly convex hyperbolic spaces. Theorem 3.3. Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let S, T : K → K be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(Sn))} and {(a′′n,η(Tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)2η(Tn)2−1 ) < ∞. Let {xn} be a sequence in K defined by (2.1). Assume that F = F(S)∩F(T) 6= ∅. Suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). Then {xn} is ∆-convergent to an element of F . Proof. It follows from Lemma 3.1 that {xn} is bounded, therefore {xn} has a unique asymptotic center (see, [32]), that is, A({xn}) = {x} (say). Let A({yn}) = {v}. Then by Lemma 3.2, limn→∞d(yn,Syn) = 0 and limn→∞d(yn,Tyn) = 0. S and T are nearly asymptotically nonexpansive mappings with sequences {(a′n,η(Sn))} and {(a′′n,η(Tn))}. By uniform continuity of S and T , we have lim n→∞ d(Siyn,S i+1yn) = 0 for i = 1, 2, . . . .(3.20) and lim n→∞ d(Tjyn,T j+1yn) = 0 for j = 1, 2, . . . .(3.21) Now we claim that v is a common fixed point of S and T. For this, we define a sequence {zn} in K by zm = Smv and zm = Tmv, m ∈ N. for integers m,n ∈ N, we have d(zm,yn) ≤ d(Smv,Smyn) + d(Smyn,Sm−1yn) + · · · + d(Syn,yn) ≤ η(Sn)(d(v,yn) + a′m) + m−1∑ i=0 d(Siyn,S i+1yn).(3.22) Then from (3.20) and (3.22), we have r(zm,{yn}) = lim sup m→∞ d(zm,yn) ≤ η(Sm)[r(v,{yn}) + a′m]. Hence lim sup m→∞ r(zm,{yn}) ≤ r(v,{yn}).(3.23) STRONG AND ∆-CONVERGENCE OF MODIFIED TWO-STEP ITERATIONS 47 Since AK({yn}) = {v}, by definition of asymptotic center AK({yn}) of a bounded sequence {yn} with respect to K ⊂ X, we have r(v,{yn}) ≤ r(y,{yn}), ∀ y ∈ K. This implies that lim inf m→∞ r(zm,{yn}) ≥ r(v,{yn}),(3.24) therefore, from (3.23) and (3.24), we have lim m→∞ r(zm,{yn}) = r(v,{yn}). It follows from Lemma 2.4 that Smv → v. By uniform continuity of S, we have Sv = S( lim m→∞ Smv) = Sm+1v = v, which implies that v is a fixed point of S, that is, v ∈ F(S). Similarly, we can show that v ∈ F(T). Thus v ∈ F = F(S) ∩F(T). Next, we claim that v is the unique asymptotic center for each subsequence {yn} of {xn}. Assume contrarily, that is, x 6= v. Since limn→∞d(xn,v) exists by Lemma 3.1, therefore, by the uniqueness of asymptotic centers, we have lim sup n→∞ d(yn,v) < lim sup n→∞ d(yn,x) ≤ lim sup n→∞ d(xn,x) < lim sup n→∞ d(xn,v) = lim sup n→∞ d(yn,v), a contradiction and hence x = v. Since {yn} is an arbitrary subsequence of {xn}, therefore, AK({yn}) = {v} for all subsequence {yn} of {xn}. This proves that {xn} ∆-converges to an element of F. This completes the proof. � Theorem 3.4. Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let S, T : K → K be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(Sn))} and {(a′′n,η(Tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)2η(Tn)2−1 ) < ∞. Let {xn} be a sequence in K defined by (2.1). Assume that F = F(S) ∩F(T) 6= ∅ is a closed set. Then {xn} converges strongly to a point in F if and only if lim infn→∞d(xn,F) = 0. Proof. Necessity is obvious. Conversely, suppose that lim infn→∞d(xn,F) = 0. As proved in Lemma 3.1, for all p ∈ F, limn→∞d(xn,F) exists. Thus by hypothesis limn→∞d(xn,F) = 0. 48 SALUJA Next, we show that {xn} is a Cauchy sequence in K. With the help of inequality 1 + x ≤ ex, x ≥ 0. For any integer m ≥ 1, we have from (3.2) d(xn+m,p) ≤ (1 + µn+m−1)d(xn+m−1,p) + νn+m−1 ≤ eµn+m−1d(xn+m−1,p) + νn+m−1 ≤ eµn+m−1 [eµn+m−2d(xn+m−2,p) + νn+m−2] +νn+m−1 ≤ e(µn+m−1+µn+m−2)d(xn+m−2,p) + e(µn+m−1+µn+m−2) × [νn+m−1 + νn+m−2] ≤ . . . ≤ ( e ∑n+m−1 k=n µk ) d(xn,p) + ( e ∑n+m−1 k=n µk )n+m−1∑ k=n νk = W d(xn,p) + W n+m−1∑ k=n νk,(3.25) where W = e ∑∞ n=1 µn < ∞. Since limn→∞d(xn,F) = 0, without loss of generality, we may assume that a subsequence {xnk} of {xn} and a sequence {pnk} ⊂ F such that d(xnk,pnk ) → 0 as k →∞. Then for any ε > 0, there exists kε > 0 such that d(xnk,pnk ) < ε 4W and ∞∑ k=nkε νk < ε 4W ,(3.26) for all k ≥ kε. For any m ≥ 1 and for all n ≥ nkε , by (3.25) and (3.26), we have d(xn+m,xn) ≤ d(xn+m,pnk ) + d(xn,pnk ) ≤ W d(xn,pnk ) + W ∞∑ k=nkε νk +W d(xn,pnk ) + W ∞∑ k=nkε νk = 2W d(xn,pnk ) + 2W ∞∑ k=nkε νk < 2W. ε 4W + 2W. ε 4W = ε.(3.27) This proves that {xn} is a Cauchy sequence in closed subset K of a complete hyperbolic space X and so it must converge to a point z in K, that is, limn→∞xn = z. Now, limn→∞d(xn,F) = 0 gives d(z,F) = 0. Since F is closed, we have z ∈ F. Thus {xn} converges strongly to a point in F . This completes the proof. � Theorem 3.5. Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let S, T : K → K be two uniformly continuous nearly asymptotically nonexpansive STRONG AND ∆-CONVERGENCE OF MODIFIED TWO-STEP ITERATIONS 49 mappings with sequences {(a′n,η(Sn))} and {(a′′n,η(Tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)2η(Tn)2−1 ) < ∞. Let {xn} be a sequence in K defined by (2.1). Assume that F = F(S)∩F(T) 6= ∅. Suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). If either Sm or Tm for some m ≥ 1 is semi-compact, then {xn} converges strongly to a point in F . Proof. Suppose Tm for some m ≥ 1 is semi-compact. By Lemma 3.2, we have limn→∞d(xn,Txn) = 0. By the uniform continuity of T , we get d(xn,Txn) → 0 ⇒ d(Txn,T 2xn) → 0 ⇒ ···⇒ d(Tixn,Ti+1xn) → 0 for all i = 1, 2, 3, . . . , it follows that d(xn,T mxn) ≤ m−1∑ i=0 d(Tixn,T i+1xn) → 0 as n →∞. Since d(xn,T mxn) → 0 and Tm is semi-compact, there exists a subsequence {xnj} of {xn} such that limj→∞Tmxnj = x ∈ K. Note that d(xnj,x) ≤ d(xnj,T mxnj ) + d(T mxnj,x) → 0 as j →∞. Since limn→∞d(xn,Txn) = 0, we get x ∈ F(T). Similarly, we can show that x ∈ F(S). Thus x ∈ F = F(S) ∩ F(T). Since limn→∞d(xn,x) exists by Lemma 3.1 and limj→∞d(xnj,x) = 0, we conclude that xn → x ∈ F. This shows that the sequence {xn} converges strongly to a point in F. This completes the proof. � Senter and Dotson [30] introduced the concept of condition (A) as follows. Definition 3.6. (See [30]) A mapping T : K → K is said to satisfy condition (A) if there exists a non-decreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(r) > 0 for all r > 0 such that d(x,Tx) ≥ f(d(x,F(T))), for all x ∈ K. We modify this definition for two mappings. Definition 3.7. Two mappings S,T : K → K, where K is a subset of a metric space (X,d), is said to satisfy condition (A′) if there exists a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(t) > 0 for all t ∈ (0,∞) such that ad(x,Sx) + bd(x,Tx) ≥ f(d(x,F)) for all x ∈ K where d(x,F) = inf{d(x,p) : p ∈ F = F(S) ∩ F(T) 6= ∅} and a and b are two nonnegative real numbers such that a + b = 1. It is to be noted that condition (A′) is weaker than compactness of the domain K. Remark 3.8. Condition (A′) reduces to condition (A) when S = T. As an application of Theorem 3.3, we establish another strong convergence result employing condition (A′). Theorem 3.9. Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η and let S, T : K → K be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(Sn))} and {(a′′n,η(Tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(Sn)2η(Tn)2−1 ) < ∞. Let {xn} be a sequence in K defined by (2.1). 50 SALUJA Assume that F = F(S)∩F(T) 6= ∅. Suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). Suppose that S and T satisfy the condition (A′). Then {xn} converges strongly to a point in F . Proof. By Lemma 3.2, we know that lim n→∞ d(xn,Sxn) = 0 and lim n→∞ d(xn,Txn) = 0.(3.28) From condition (A′) and (3.28), we get lim n→∞ f(d(xn,F)) ≤ a. lim n→∞ d(xn,Sxn) + b. lim n→∞ d(xn,Txn) = 0, that is, lim n→∞ f(d(xn,F)) = 0. Since f : [0,∞) → [0,∞) is a nondecreasing function satisfying f(0) = 0, f(t) > 0 for all t ∈ (0,∞), therefore we obtain lim n→∞ d(xn,F) = 0. The conclusion now follows from Theorem 3.4. This completes the proof. � Example 3.10. (See [26]) Let E = R, K = [0, 1] and T : K → K be a mapping defined by T(x) = { 1 2 , if x ∈ [0, 1 2 ], 0, if x ∈ ( 1 2 , 1]. Here F(T) = {1 2 }. Clearly, T is discontinuous and a non-Lipschitzian mapping. However, it is a nearly nonexpansive mapping and hence nearly asymptotically nonexpansive mapping with sequence {an,η(Tn)} = { 12n , 1}. Indeed, for a sequence {an} with a1 = 12 and an → 0, we have d(Tx,Ty) ≤ d(x,y) + a1 for all x, y ∈ K and d(Tnx,Tny) ≤ d(x,y) + an for all x, y ∈ K and n ≥ 2, since Tnx = 1 2 for all x ∈ [0, 1] and n ≥ 2. 4. Conclusion 1. We prove a ∆-convergence and some strong convergence theorems of modified two-step iteration process which contains modified Mann iteration process in the framework of uniformly convex hyperbolic spaces. 2. 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