International Journal of Analysis and Applications ISSN 2291-8639 Volume 3, Number 1 (2013), 14-24 http://www.etamaths.com FIXED POINT THEOREMS FOR T-CIRIC QUASI-CONTRACTIVE OPERATOR IN CAT(0) SPACES G. S. SALUJA Abstract. The purpose of this paper to study a three-step iterative algorithm for T-Ciric quasi-contractive (TCQC) operator in the framework of CAT(0) spaces and establish strong convergence theorems for above said scheme and operator. Our results improve and extend the recent corresponding results from the existing literature (see, e.g., [28, 29, 30] and some others). 1. Introduction A metric space X is a CAT(0) space if it is geodesically connected and if every geo- desic triangle in X is at least as ”thin” as its comparison triangle in the Euclidean plane. The precise definition is given below. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces (see [3]), R-trees (see [18]), Euclidean buildings (see [4]), the complex Hilbert ball with a hyperbolic metric (see [12]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger [3]. Fixed point theory in CAT(0) spaces was first studied by Kirk (see [19, 20]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since, then the fixed point theory for single-valued and multi-valued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [1], [7], [9]-[11], [13], [16]-[17], [21]-[22], [27], [31]-[32] and references therein). It is worth mentioning that the results in CAT(0) spaces can be applied to any CAT(k) space with k ≤ 0 since any CAT(k) space is a CAT(k′) space for every k′ ≥ k (see,e.g., [3]). Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂R to X such that c(0) = x, c(l) = y, and let d(c(t), c(t′)) = |t−t′| for all t, t′ ∈ [0, l]. In particular, c is an isometry, and d(x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. We say X is (i) a geodesic space if any two points of X are joined by 2010 Mathematics Subject Classification. 54H25, 54E40. Key words and phrases. T-Ciric quasi-contractive operator, three-step iteration process, fixed point, strong convergence, CAT(0) space. c©2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 14 FIXED POINT THEOREMS 15 a geodesic and (ii) uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X, which we will denoted by [x, y], called the segment joining x to y. A geodesic triangle 4(x1, x2, x3) in a geodesic metric space (X, d) consists of three points in X (the vertices of 4) and a geodesic segment between each pair of ver- tices (the edges of 4). A comparison triangle for geodesic triangle 4(x1, x2, x3) in (X, d) is a triangle 4(x1, x2, x3) :=4(x1, x2, x3) in R2 such that dR2 (xi, x j) = d(xi, x j) for i, j ∈ {1, 2, 3}. Such a triangle always exists (see [3]). A geodesic metric space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following CAT(0) comparison axiom. CAT(0) space. Let 4 be a geodesic triangle in X, and let 4⊂R2 be a comparison triangle for 4. Then 4 is said to satisfy the CAT(0) inequality if for all x, y ∈4 and all comparison points x, y ∈4, d(x, y) ≤ d(x, y).(1.1) Complete CAT(0) spaces are often called Hadamard spaces (see [15]). If x, y1, y2 are points of a CAT(0) space and y0 is the mid point of the segment [y1, y2] which we will denote by (y1 ⊕ y2)/2, then the CAT(0) inequality implies d2 ( x, y1 ⊕ y2 2 ) ≤ 1 2 d2(x, y1) + 1 2 d2(x, y2) − 1 4 d2(y1, y2).(1.2) The inequality (1.2) is the (CN) inequality of Bruhat and Titz [5]. The above inequality has been extended in [10] as d2(z,αx ⊕ (1 −α)y) ≤ αd2(z, x) + (1 −α)d2(z, y) −α(1 −α)d2(x, y).(1.3) for any α ∈ [0, 1] and x, y, z ∈ X. Let us recall that a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality (see [3, page 163]). Moreover, if X is a CAT(0) metric space and x, y ∈ X, then for any α ∈ [0, 1], there exists a unique point αx ⊕ (1 − α)y ∈ [x, y] such that d(z,αx ⊕ (1 −α)y) ≤ αd(z, x) + (1 −α)d(z, y),(1.4) for any z ∈ X and [x, y] = {αx ⊕ (1 −α)y : α ∈ [0, 1]}. A subset C of a CAT(0) space X is convex if for any x, y ∈ C, we have [x, y] ⊂ C. We recall the following definitions in a metric space (X, d). A mapping T : X → X is called an a-contraction if d(Tx, Ty) ≤ a d(x, y), for all x, y ∈ X,(1.5) where a ∈ (0, 1). The mapping T is called Kannan mapping [14] if there exists b ∈ (0, 12 ) such that d(Tx, Ty) ≤ b [d(x, Tx) + d(y, Ty)], for all x, y ∈ X.(1.6) 16 SALUJA A similar definition is due to Chatterjea [8]: there exists c ∈ (0, 12 ) such that d(Tx, Ty) ≤ c [d(x, Ty) + d(y, Tx)], for all x, y ∈ X.(1.7) Combining these three definitions, Zamfirescu [34] proved the following impor- tant result. Theorem Z. Let (X, d) be a complete metric space and T : X → X a mapping for which there exist real number a, b and c satisfying a ∈ (0, 1), b, c ∈ (0, 12 ) such that for any pair x, y ∈ X, at least one of the following conditions holds: (z1) d(Tx, Ty) ≤ a d(x, y), (z2) d(Tx, Ty) ≤ b [d(x, Tx) + d(y, Ty)], (z3) d(Tx, Ty) ≤ c [d(x, Ty) + d(y, Tx)]. Then T has a unique fixed point p and the Picard iteration {xn}∞n=0 defined by xn+1 = Txn, n = 0, 1, 2, . . . converges to p for any arbitrary but fixed x0 ∈ X. The conditions (z1) − (z3) can be written in the following equivalent form d(Tx, Ty) ≤ h max { d(x, y), d(x, Tx) + d(y, Ty) 2 , d(x, Ty) + d(y, Tx) 2 } , (∗) ∀x, y ∈ X; 0 < h < 1, has been obtained by Ciric [6] in 1974. A mapping satisfying (∗) is called Ciric quasi-contraction. It is obvious that each of the conditions (z1) − (z3) implies (∗). An operator T satisfying the contractive conditions (z1) − (z3) in the theorem Z is called Z-operator. In 2009, Beiranvand et al. [2] introduced the concept of T-Banach contraction and T-contractive mappings and they extended Banach’s contraction principle and Edelstein fixed point theorem. Followed by this, Moradi [23] introduced T-Kannan contractive mappings, extending in the way, the well-known Kannan fixed point theorem [14]. FIXED POINT THEOREMS 17 Recently, Morales and Rojas [25], [26] have extended the concept of T-contraction mappings to cone metric space by proving fixed point theorems for T-Kannan, T- Zamfirescu and T-weakly contraction mappings. In [24], they studied the existence of fixed point for T-Zamfirescu operators in complete metric spaces and proved a convergence theorem of T-Picard iteration for the class of T-Zamfirescu operators. The result is as follows: Theorem 1.1. (See [24]) Let (M, d) be a complete metric space and T, S : M → M be two mappings such that T is continuous, one-to-one and subsequentially convergent. If S is a TZ operator, S has a unique fixed point. Moreover, if T is sequentially convergent, then for every x0 ∈ M the T-Picard iteration associated to S, TSnx0 converges to Tx∗, where x∗ is the fixed point of S. Here we recall the definitions of the following classes of generalized T-contraction type mappings as given by Morales and Rojas [24]. Definition 1.2. (See [24]) Let (X, d) be a metric space and S, T : X → X be two mappings. A mapping S is said be T-contraction, if there exists a real number a ∈ [0, 1) such that for all x, y ∈ X, d(TSx, TSy) ≤ a d(Tx, Ty). If we take T = I, the identity map, in the definition 1.2, then we obtain the definition of Banach’s contraction. The following example shows that a T-contraction mapping need not be a con- traction mapping. Example 1. Let X = [1,∞) be with the usual metric. Define two mappings T, S : X → X as Tx = 12x +2 and Sx = 3x. Obviously, S is not contraction but S is T-contraction which is seen from the following: |TSx − TSy|= ∣∣∣∣ 16x − 16y ∣∣∣∣ = 13 |Tx − Ty|. Definition 1.3. (See [24]) Let (X, d) be a metric space and S, T : X → X be two mappings. A mapping S is said be T-Kannan contraction, if there exists a real number b ∈ [0, 12 ) such that for all x, y ∈ X, d(TSx, TSy) ≤ b [d(Tx, TSx) + d(Ty, TSy)]. If we take T = I, the identity map, in the definition 1.3, then we obtain the definition of Kannan operator [14]. Definition 1.4. (See [24]) Let (X, d) be a metric space and S, T : X → X be two mappings. A mapping S is said be T-Chatterjea contraction, if there exists a real number c ∈ [0, 12 ) such that for all x, y ∈ X, d(TSx, TSy) ≤ c [d(Tx, TSy) + d(Ty, TSx)]. If we take T = I, the identity map, in the definition 1.4, then we obtain the definition of Chatterjea operator [8]. 18 SALUJA Definition 1.5. (See [24]) Let (X, d) be a metric space and S, T : X → X be two mappings. A mapping S is said be T-Zamfirescu operator (TZ-operator), if there are real numbers 0 ≤ a < 1, 0 ≤ b < 12 , 0 ≤ c < 1 2 such that for all x, y ∈ X at least one of the following conditions holds: (TZ1) d(TSx, TSy) ≤ a d(Tx, Ty), (TZ2) d(TSx, TSy) ≤ b [d(Tx, TSx) + d(Ty, TSy)], (TZ3) d(TSx, TSy) ≤ c [d(Tx, TSy) + d(Ty, TSx)]. If we take T = I, the identity map, in the definition 1.5, then we obtain the definition of Zamfirescu operator [34]. Definition 1.6. (See [29]) Let (X, d) be a metric space and T : X → X be a mapping. (1) A mapping T is said to be sequentially convergent, if we have for every sequence {yn}, {Tyn} is convergent implies that {yn} is also convergent. (2) A mapping T is said to be subsequentially convergent, if we have for every sequence {yn}, {Tyn} is convergent implies that {yn} has a convergent subsequence. In 2002, Xu and Noor [33] introduced a three-step iterative scheme as follows: x0 ∈ C, xn+1 = (1 −αn)xn +αnTn yn, yn = (1 −βn)xn +βnTnzn, zn = (1 −γn)xn +γnTnxn, n ≥ 0 where {αn}, {βn} and {γn} are real sequences in [0, 1]. Recently, Y. Niwongsa and B. Panyanak [27] studied the Noor iteration scheme in CAT(0) spaces and they proved some 4 and strong convergence theorems for asymptotically nonexpansive mappings which extend and improve some recent results from the literature. In this paper, inspired and motivated by [24, 28, 33, 34], we study a three-step iteration scheme and prove strong convergence theorem to approximate the fixed point for T-Ciric quasi-contractive (TCQC) operator in the setting of CAT(0) spaces. Three-step iteration scheme in CAT(0) space Let C be a nonempty closed convex subset of a complete CAT(0) space X. Let T : C → C and let S : C → C be a T-contractive operator. Then for a given FIXED POINT THEOREMS 19 x1 = x0 ∈ C, compute the sequence {xn} by the iterative scheme as follows: Tzn = γnTSxn ⊕ (1 −γn)Txn, Tyn = βnTSzn ⊕ (1 −βn)Txn, Txn+1 = αnSTyn ⊕ (1 −αn)Txn, n ≥ 0,(1.8) where {αn}∞n=0, {βn} ∞ n=0, {γn} ∞ n=0 are appropriate sequences in [0,1]. If we take T = I, the identity map, then (1.8) reduces to Noor [33] iteration scheme in CAT(0) space: zn = γnSxn ⊕ (1 −γn)xn, yn = βnSzn ⊕ (1 −βn)xn, xn+1 = αnSyn ⊕ (1 −αn)xn, n ≥ 0,(1.9) where {αn}∞n=0, {βn} ∞ n=0, {γn} ∞ n=0 are appropriate sequences in [0,1]. If γn = 0 for all n ≥ 0, then (1.9) reduces to Ishikawa iteration scheme in CAT(0) space: yn = βnTxn ⊕ (1 −βn)xn, xn+1 = αnTyn ⊕ (1 −αn)xn, n ≥ 0,(1.10) where {αn}∞n=0 and {βn} ∞ n=0 are appropriate sequences in [0,1]. We note that if βn = 0 for all n ≥ 0, then (1.10) reduces to Mann iteration scheme in CAT(0) space: xn+1 = αnTxn ⊕ (1 −αn)xn, n ≥ 0,(1.11) where {αn}∞n=0 is a sequence in (0,1). We need the following useful lemmas to prove our main results in this paper. Lemma 1.7. (see [27]) Let (X, d) be a CAT(0) space. (i) For x, y ∈ X and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = t d(x, y) and d(y, z) = (1 − t) d(x, y). (A) We use the notation (1 − t)x ⊕ ty for the unique point z satisfying (A). (ii) For x, y ∈ X and t ∈ [0, 1], we have d((1 − t)x ⊕ ty, z) ≤ (1 − t)d(x, z) + td(y, z). Lemma 1.8. (see [28]) Let {pn}, {qn}, {rn} and {sn} be sequences of nonnegative numbers satisfying the following conditions: pn+1 ≤ (1 − qn)pn + qnrn + sn, n ≥ 1. If ∑ ∞ n=1 qn =∞, limn→∞ rn = 0 and ∑ ∞ n=1 sn < ∞ hold, then limn→∞ pn = 0. 20 SALUJA T-Ciric Quasi Contraction Mapping Let X be a CAT(0) space and S, T : X → X be two mappings. Then S is called T-Ciric quasi contraction mapping if it satisfies the following condition: d(TSx, TSy) ≤ h max { d(Tx, Ty), d(Tx, TSx) + d(Ty, TSy) 2 , d(Tx, TSy) + d(Ty, TSx) 2 } (TCQC)(1.12) for all x, y ∈ X and 0 < h < 1. Remark 1.9. If we take T = I, then (1.3) reduces to quasi contraction mapping introduced by Ciric [6] in 1974 (Proc. Amer. Math. Soc. 45 (1974), 727-730). Remark 1.10. A mapping satisfying (TCQC) is called T-Ciric quasi-contraction map- ping. It is obvious that each of the conditions (TZ1) − (TZ3) implies (TCQC). 2. Strong convergence theorems in CAT(0) spaces In this section, we establish some strong convergence results of a three-step iteration scheme to converge to a fixed point for T-Ciric quasi-contractive operator in the framework of CAT(0) spaces. Theorem 2.1. Let C be a nonempty closed convex subset of a complete CAT(0) space. Let S, T : C → C be two commuting mappings such that T is continuous, one-to-one, sub- sequentially convergent and S : C → C is a T-Ciric quasi-contractive operator satisfying (1.12) with 0 < h < 1. Let {xn} be defined by the iteration scheme (1.8). If ∑ ∞ n=1 αn = ∞,∑ ∞ n=1 αnβn = ∞ and ∑ ∞ n=1 αnβnγn = ∞, then {Txn} converges strongly to Tu, where u is the fixed point of the operator S in C. Proof. From Theorem 1.1, we get that S has a unique fixed point in C, say u. Con- sider x, y ∈ C. Since S is a T-Ciric quasi-contractive operator satisfying (1.12), then if d(TSx, TSy) ≤ h 2 [d(Tx, TSx) + d(Ty, TSy)] ≤ h 2 [d(Tx, TSx) + d(Ty, Tx) + d(Tx, TSx) + d(TSx, TSy)], implies ( 1 − h 2 ) d(TSx, TSy) ≤ h 2 d(Tx, Ty) + h d(Tx, TSx), which yields (using the fact that 0 < h < 1) d(TSx, TSy) ≤ ( h/2 1 − h/2 ) d(Tx, Ty) + ( h 1 − h/2 ) d(Tx, TSx). FIXED POINT THEOREMS 21 If d(TSx, TSy) ≤ h 2 [d(Tx, TSy) + d(Ty, TSx)] ≤ h 2 [d(Tx, TSx) + d(TSx, TSy) + d(Ty, Tx) + d(Tx, TSx)], implies ( 1 − h 2 ) d(TSx, TSy) ≤ h 2 d(Tx, Ty) + h d(Tx, TSx), which also yields (using the fact that 0 < h < 1) d(TSx, TSy) ≤ ( h/2 1 − h/2 ) d(Tx, Ty) + ( h 1 − h/2 ) d(Tx, TSx).(2.1) Denote δ = max { h, h/2 1 − h/2 } = h, L = h 1 − h/2 . Thus, in all cases, d(TSx, TSy) ≤ δ d(Tx, Ty) + L d(Tx, TSx) = h d(Tx, Ty) + ( h 1 − h/2 ) d(Tx, TSx).(2.2) holds for all x, y ∈ C. Also from (TCQC) with y = u = Su, we have d(TSx, TSu) ≤ h max { d(Tx, Tu), d(Tx, TSx) 2 , d(Tx, TSu) + d(Tu, TSx) 2 } ≤ h max { d(Tx, Tu), d(Tx, Tu) + d(Tu, TSx) 2 , d(Tx, Tu) + d(Tu, TSx) 2 } = h max { d(Tx, Tu), d(Tx, Tu) + d(Tu, TSx) 2 } ≤ h d(Tx, Tu).(2.3) Now (2.3) gives d(TSxn, Tu) ≤ h d(Txn, Tu).(2.4) d(TSyn, Tu) ≤ h d(Tyn, Tu).(2.5) 22 SALUJA and d(TSzn, Tu) ≤ h d(Tzn, Tu).(2.6) Using (1.8), (2.6) and Lemma 1.1 (ii), we have d(Tzn, Tu) = d(γnTSxn ⊕ (1 −γn)Txn, Tu) ≤ γnd(TSxn, Tu) + (1 −γn)d(Txn, Tu) ≤ γnhd(Txn, Tu) + (1 −γn)d(Txn, Tu) ≤ [1 − (1 − h)γn]d(Txn, Tu).(2.7) Again using (1.8), (2.5), (2.7) and Lemma 1.1 (ii), we have d(Tyn, Tu) = d(βnTSzn ⊕ (1 −βn)Txn, Tu) ≤ βnd(TSzn, u) + (1 −βn)d(Txn, Tu) ≤ βnhd(Tzn, Tu) + (1 −βn)d(Txn, Tu) ≤ βnh[1 − (1 − h)γn]d(Txn, Tu) + (1 −βn)d(Txn, Tu) ≤ [1 − (1 − h)βn − h(1 − h)βnγn]d(Txn, Tu).(2.8) Now using (1.8), (2.4), (2.8), TS = ST (by assumption of the theorem) and Lemma 1.7(ii), we have d(Txn+1, Tu) = d(αnSTyn ⊕ (1 −αn)Txn, Tu) ≤ αnd(STyn, Tu) + (1 −αn)d(Txn, Tu) ≤ αnd(TSyn, Tu) + (1 −αn)d(Txn, Tu) ≤ αnhd(Tyn, Tu) + (1 −αn)d(Txn, Tu) ≤ αnh[1 − (1 − h)βn − h(1 − h)βnγn]d(Txn, Tu) +(1 −αn)d(Txn, Tu) ≤ [1 −{(1 − h)αn − h(1 − h)αnβn +h2(1 − h)αnβnγn}]d(Txn, Tu) = (1 − Bn)d(Txn, Tu),(2.9) where Bn = {(1 − h)αn − h(1 − h)αnβn + h2(1 − h)αnβnγn}, since 0 < h < 1 and by assumption of the theorem ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn = ∞ and ∑ ∞ n=1 αnβnγn = ∞, it follows that ∑ ∞ n=1 Bn =∞, therefore by Lemma 1.8, we get that limn→∞ d(Txn, Tu) = 0. Therefore {Txn} converges strongly to Tu, where u is the fixed point of the operator S in C. This completes the proof. � Since T-Kannan contraction and T-Chatterjea contraction are both included in the T-Ciric quasi-contractive operator, by Theorem 2.1, we obtain the correspond- ing convergence result of the iteration process defined by (1.8) for the above said class of operators as corollary: Corollary 2.2. Let C be a nonempty closed convex subset of a complete CAT(0) space. Let S, T : C → C be two commuting mappings such that T is continuous, one-to-one, FIXED POINT THEOREMS 23 subsequentially convergent and S : C → C is a T-Kannan contractive operator satisfying the condition d(TSx, TSy) ≤ b [ d(Tx, TSx) + d(Ty, TSy) 2 ] , ∀x, y ∈ X; b ∈ (0, 12 ). Let {Txn} be defined by the iteration scheme (1.8). If ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn =∞ and ∑ ∞ n=1 αnβnγn =∞, then {Txn} converges strongly to Tu, where u is the fixed point of the operator S in C. Corollary 2.3. Let C be a nonempty closed convex subset of a complete CAT(0) space. Let S, T : C → C be two commuting mappings such that T is continuous, one-to-one, subsequentially convergent and S : C → C is a T-Chatterjea contractive operator satisfying the condition d(TSx, TSy) ≤ c [ d(Tx, TSy) + d(Ty, TSx) 2 ] , ∀x, y ∈ X; c ∈ (0, 12 ). Let {Txn} be defined by the iteration scheme (1.8). If ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn =∞ and ∑ ∞ n=1 αnβnγn =∞, then {Txn} converges strongly to Tu, where u is the fixed point of the operator S in C. If we take T = I, the identity map, in equation (1.12) and (2.2), then we obtain the following results as corollary: Corollary 2.4. Let C be a nonempty closed convex subset of a complete CAT(0) space and let S : C → C be an operator satisfying (2.2). Let {xn} be defined by the iteration scheme (1.9). If ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn = ∞ and ∑ ∞ n=1 αnβnγn = ∞, then {xn} converges strongly to the unique fixed point of S. Corollary 2.5. Let C be a nonempty closed convex subset of a complete CAT(0) space and let S : C → C be an operator satisfying (2.2). Let {xn} be defined by the iteration scheme (1.10). If ∑ ∞ n=1 αn = ∞ and ∑ ∞ n=1 αnβn = ∞, then {xn} converges strongly to the unique fixed point of S. Corollary 2.6. Let C be a nonempty closed convex subset of a complete CAT(0) space and let S : C → C be an operator satisfying (2.2). Let {xn} be defined by the iteration scheme (1.11). If ∑ ∞ n=1 αn =∞, then {xn} converges strongly to the unique fixed point of S. Remark 2.7. Our results extend the corresponding results of Rafiq [28], Raphael and Pulickakunnel [29] and Rhoades [30] to the case of three-step iteration and from Banach space or convex metric space or uniformly convex Banach space to the setting of CAT(0) spaces and by using T-Ciric quasi-contractive operators. References [1] A. Akbar and M. 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