CONVERGENCE FOR GENERALIZED HYBRID MAPPINGS IN ... - EMIS

∆-CONVERGENCE FOR GENERALIZED HYBRID MAPPINGS IN CAT(0) SPACES RABIAN WANGKEEREE† AND PAKKAPON PREECHASILP Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Abstract. In this paper, we first give some properties for generalized hybrid mappings in complete CAT(0) spaces. Then, we prove ∆-convergence and strong convergence theorems of the proposed Picard, Mann and Ishikawa iterative schemes for such mappings in complete CAT(0) spaces. Keywords: Strong and ∆-convergence, CAT(0) space, common fixed points, generalized hybrid mapping.

1. Introduction Let (X, d) be a metric space and C be a nonempty subset of X. Then a mapping T of C into itself is called nonexpansive iff d(T x, T y) ≤ d(x, y) for all x, y ∈ C. The class of nonexpansive mapping was studied by many authors; see, for example, [4, 5, 7, 8, 11–13]. We denote by F (T ) the set of all fixed points of T , i.e., F (T ) := {x ∈ C : T x = x}. A mapping T from C into C is also called quasi-nonexpansive iff the set F (T ) of fixed points of T is nonempty and d(T x, p) ≤ d(x, p) for all x ∈ C and p ∈ F (T ). Obviously, a nonexpansive mapping T with F (T ) 6= ∅ is quasi-nonexpansive. The class of nonexpansive mapping was studied by many authors; see, for example, [125]. In 2008, Kohsaka and Takahashi [16], and Takahashi [20] introduced the following two nonlinear mappings in a Hilbert space. A mapping T : C → H is called nonspreading [16] if 2d2 (T x, T y) ≤ d2 (T x, y) + d2 (T y, x), for all x, y ∈ C.

(1.1)

A mapping T : C → H is called hybrid [20] if 3d2 (T x, T y) ≤ d2 (x, y) + d2 (T x, y) + d2 (T y, x), for all x, y ∈ C.

(1.2)

It is easy to see that both nonspreading and hybrid mappings with F (T ) 6= ∅ are quasi-nonexpansive. They proved fixed point theorems for such mappings; see also Kohsaka and Takahashi [17], Iemoto and Takahashi [11] and Takahashi and Yao [22]. Furthermore, Kocourek, Takahashi and Yao [15] introduced a more broad class of nonlinear mappings than the class of nonspreading and hybrid mappings. They called such a class the class of generalized hybrid mapping. A mapping T : C → C is called (α, β)-generalized hybrid mapping if there exist α, β ∈ R such that αd2 (T x, T y) + (1 − α)d2 (x, T y) ≤ βd2 (T x, y) + (1 − β)d2 (x, y), for all x, y ∈ C.

(1.3)

Such a class contains some classes of nonlinear mappings in a Hilbert space. For example, an (α, β)generalized hybrid mapping is nonexpansive for α = 1 and β = 0, nonspreading for α = 2 and β = 1, and hybrid for α = 32 and β = 21 . It is easy to see that (α, β)-generalized hybrid mapping with a fixed point is quasi-nonexpansive. They proved a demiclosed principle for this class of mapping, i.e., xn ⇀ z and xn − T xn → 0 imply T z = z. Also, they proved a sequence of Mann’s type for this class of mappings in Hilbert spaces: xn+1 = αn xn + (1 − αn )T xn , n ≥ 1

(1.4)

converges weakly to an element v of F (T ), where v = limn→∞ PF (T ) xn , PC is the mertic projection of H onto C. On the other hand, Kirk and Panyanak [13] specialized Lims concept [18] of ∆-convergence in a general metric space to CAT(0) spaces and showed that many Banach space results which involve † Corresponding author: Email address: [email protected](Rabian Wangkeeree) and [email protected](Pakkapon Preechasilp).

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2

R. WANGKEEREE AND P. PREECHASILP

weak convergence have precise analogs in this setting; for instance, the Opial property, the KadecKlee property and the demiclosedness principle for LANE mappings. In 2008, Dhompongsa and Panyanak [4] obtained the convergence theorems for the following Picard, Mann and Ishikawa iterations, resp., for nonexpansive mappings in a CAT(0) space: xn = T n x, n = 0, 1, 2, . . . ,

(P)

and xn+1 = tn T xn ⊕ (1 − tn )xn , n = 0, 1, 2, . . . , where {tn } is a sequence in [0, 1], and xn+1 = tn T (sn T xn ⊕ (1 − sn )xn ) ⊕ (1 − tn )xn , n = 0, 1, 2, . . . ,

(M) (I)

where sn and {tn } are sequence in [0, 1]. By using the concept of ∆-convergence, they prove that the sequence are generated by (P), (M), and (I) ∆-converge to an element of F (T ) in a CAT(0) space. Motivated and inspired from above results, the aim of this paper is to study the convergence results for generalized hybrid mappings in a CAT(0) space. After we review literature, we next recall some definitions and useful lemmas in Section 2. In Section 3, we give the ∆-convergence theorems of the Picard and Mann iterate sequence for a such mapping considered by Takahashi and Yao [21]. We give a ∆-convergence of the Ishikawa iterate sequence for a generalized hybrid mapping. Finally, we give an example of a (α, β)-generalized hybrid mapping in CAT(0) space which is not a nonexpansive mapping. 2. Preliminaries 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 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. When it is unique this geodesic segment is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset C of a CAT(0) space is convex if [x, y] ⊆ C for all x, y ∈ C. A geodesic triangle △(x1 , x2 , x3 ) in a geodesic metric space (X, d) consists of three points x1 , x2 , x3 in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle △(x1 , x2 , x3 ) in (X, d) is a triangle △(x1 , x2 , x3 ) := △(x1 , x2 , x3 ) in the Euclidean plane E2 such that dE2 (xi , xj ) = d(xi , xj ) for all i, j ∈ 1, 2, 3. A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. CAT(0) : Let △ be a geodesic triangle in X and let △ be a comparison triangle for △. Then △ is said to satisfy the CAT(0) inequality if for all x, y ∈ △ and all comparison points x, y ∈ △, d(x, y) ≤ dE2 (x, y). If x, y1 , y2 are points in a CAT(0) space and if y0 is the midpoint of the segment [y1 , y2 ], then the CAT(0) inequality implies 1 1 1 (CN) d2 (x, y0 ) ≤ d2 (x, y1 ) + d2 (x, y2 ) − d2 (y1 , y2 ). 2 2 4 This is the (CN) inequality of Bruhat and Tits [3]. In fact (cf. [1], p. 163), a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality. 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, R-trees (see [1]), Euclidean buildings (see [2]), the complex Hilbert ball with a hyperbolic metric (see [7]), and many others. Complete CAT(0) spaces are often called Hadamard spaces. Next, we collect some useful lemmas in CAT(0) spaces. Lemma 2.1. [1, Proposition 2.2] Let X be a CAT(0) space, p, q, r, s ∈ X and λ ∈ [0, 1]. Then d(λp ⊕ (1 − λ)q, λr ⊕ (1 − λ)s) ≤ λd(p, r) + (1 − λ)d(q, s). Lemma 2.2. [4, Lemma 2.4] Let X be a CAT(0) space, x, y, z ∈ X and λ ∈ [0, 1]. Then d(λx ⊕ (1 − λ)y, z) ≤ λd(x, z) + (1 − λ)d(y, z).

∆-CONVERGENCE FOR GENERALIZED HYBRID MAPPINGS IN CAT(0) SPACES

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Lemma 2.3. [4, Lemma 2.5] Let X be a CAT(0) space, x, y, z ∈ X and λ ∈ [0, 1]. Then d2 (λx ⊕ (1 − λ)y, z) ≤ λd2 (x, z) + (1 − λ)d2 (y, z) − λ(1 − λ)d2 (x, y). We give the concept of ∆-convergence and collect some basic properties. Let {xn } be a bounded sequence in a CAT(0) space X. For x ∈ X, we set r(x, {xn }) = lim sup d(x, xn ). n→∞

The asymptotic radius r({xn }) of {xn } is given by r({xn }) = inf{r(x, {xn }) : x ∈ X}, and the asymptotic center A({xn }) of {xn } is the set A({xn }) = {x ∈ X : r(x, {xn }) = r({xn })}. It is known from Proposition 7 of [6] that in a CAT(0) space, A({xn }) consists of exactly one point. A sequence {xn } ⊂ X is said to ∆-converge to x ∈ X if A({xnk }) = {x} for every subsequence {xnk } of {xn }. Uniqueness of asymptotic center implies that CAT(0) space X satisfies Opial’s property, i.e., for given {xn } ⊂ X such that {xn } ∆-converges to x and given y ∈ X with y 6= x, lim sup d(xn , x) < lim sup d(xn , y). n→∞

n→∞

Since it is not possible to formulate the concept of demiclosedness in a CAT(0) setting, as stated in linear spaces, let us formally say that ”I − T is demiclosed at zero” if the conditions, {xn } ⊆ C ∆converges to x and d(xn , T xn ) → 0 imply x ∈ F (T ). Lemma 2.4. [13] Every bounded sequence in a complete CAT(0) space always has a ∆-convergent subsequence. Lemma 2.5. [5] If C is a closed convex subset of a complete CAT(0) space and if {xn } is a bounded sequence in C, then the asymptotic center of {xn } is in C. 3. ∆-convergence Theorems for a (α, β)-generalized hybrid mapping Let (X, d) be a CAT(0) space and C a nonempty, closed and convex subset of X. A mapping T : C → C is called (α, β)-generalized hybrid mapping if there exist α, β ∈ R such that αd2 (T x, T y) + (1 − α)d2 (x, T y) ≤ βd2 (T x, y) + (1 − β)d2 (x, y).

(3.1)

In [15], Kocourek, Takahashi and Yao obtained the demiclosed principle of (α, β)-generalized hybrid mapping in a Hilbert space. In a similar way, we present the demiclosed principle of (α, β)-generalized hybrid mapping in a CAT(0) space. Proposition 3.1. Let (X, d) be a CAT(0) space and C be a nonempty, closed and convex subset of X. Let T : C → C be an (α, β)-generalized hybrid mapping such that α ≥ 1 and β ≥ 0. Let {xn } be a bounded sequence in C such that limn→∞ d(xn , T (xn )) = 0 and {xn } ∆-converges to w. Then T (w) = w. Proof. Notice that T : C → C is an (α, β)-generalized hybrid mapping, i.e., there exist α, β ∈ R such that αd2 (T x, T y) + (1 − α)d2 (x, T y) ≤ βd2 (T x, y) + (1 − β)d2 (x, y). (3.2) Since T is generalized hybrid mapping, we have that αd2 (T xn , T w) + (1 − α)d2 (xn , T w) ≤ βd2 (T xn , w) + (1 − β)d2 (xn , w). Since α ≥ 1, β ≥ 0 and (3.3), we get that αd2 (T xn , T w) ≤

β(d(T xn , xn ) + d(xn , w))2 + (1 − β)d2 (xn , w) +(α − 1)(d(xn , T xn ) + d(T xn , T w))2 .

Hence (α − (α − 1))d2 (T xn , T w) ≤ (β + (1 − β))d2 (xn , w) + (β + α − 1)d2 (xn , T xn ) +2(β + α − 1)(d(xn , w) + d(T xn , T w))d(T xn , xn ),

(3.3)

4


and so d2 (T xn , T w) ≤

d2 (xn , w) + (β + α − 1)d2 (xn , T xn ) +2(β + α − 1)(d(xn , w) + d(T xn , T w))d(T xn , xn ).

(3.4)

Since {xn } is bounded and limn→∞ d(T xn , xn ) = 0, we get that {T xn } is also bounded. Thus (3.4) reduces to d2 (T xn , T w) ≤ d2 (xn , w) + (β + α − 1)d2 (xn , T xn ) +2(β + α − 1)d(T xn , xn )M,

(3.5)

where M ≥ supn≥1 d(xn , w) + d(T xn , T w). Since {xn } ⇀ w, we then have that AC ({xn }) = {w} and also A({xn }) = {w}. Assume that T w 6= w. Then By Opial’s conditions, lim sup d2 (xn , w)

0 there exists N ≥ 1 such that ε d(xn , F (T )) < , ∀n ≥ N. 4 Therefore d(xN , F (T )) < 4ε . From the definition of d(xN , F (T )), there exists q ∈ F (T ) such that d(xN , q) < 2ε . For any n, m ≥ N ≥ 1, we have ε d(xn , xm ) ≤ d(xn , q) + d(xm , q) ≤ 2d(xN , q) < 2 = ε. 2 Hence {xn } is a Cauchy sequence in a closed subset of a complete CAT(0) space. Hence {xn } converges to some p∗ ∈ C. Since limn→∞ d(xn , F (T )) = 0, then p∗ ∈ F (T ). This completes the proof. Remark 3.7. Consider R2 with the usual Euclidean meter d(·, ·) and k · k are defined by p d(x, y) = kx − yk = (x1 − y1 )2 + (x2 − y2 )2 , where x = (x1 , x2 ) and y = (y1 , y2 ). We define the radial metric dr by d(x, y), if y = tx for some t ∈ R; dr (x, y) = d(x, 0) + d(y, 0), otherwise.

Then X := (R2 , dr ) is an R-tree with the radial meter dr (see [12] and [19, page 65]). We show that X is not inner product space. We prove this by showing that the norm does not satisfy the parallelogram equality. Indeed, if we take x = (0, 1), y1 = (−1, 0), y2 = (1, 0) and y0 = (0, 0), a midpoint of y1 and y2 , then 1 1 1 1 1 1 1 = d2r (x, y0 ) < d2r (x, y1 ) + d2r (x, y2 ) − d2r (y1 , y2 ) = 4 + 4 − 4 = 3. 2 2 4 2 2 4 Therefore X does not satisfy Parallelogram law, and so X is not an inner product space. The following is an example of a (α, β)-generalized hybrid mapping in CAT(0) space which is not a nonexpansive mapping. Example 3.8. Let X be an R-tree with the radial meter dr . We put 1 1 C = (t, 0) : t ∈ [0, 2] ∪ 4, 5 ∪ (0, t) : t ∈ [0, 2] ∪ 4, 5 ⊂ R2 2 2 and define T : C → C by ( (0, 0), if t ∈ [0, 2]; 2 T (t, 0) = , if t ∈ 4, 5 21 , 0, (t−4) 2

and T (0, t) =

Clearly, F (T ) = {(0, 0)}.

(

(0, 0), 2 (t−4) 2

Claim that T is (2,1)-generalized hybrid mapping, i.e., 2d2r (T x, T y) ≤ d2r (T x, y) + d2r (x, T y).

if t ∈ [0, 2]; , 0 , if t ∈ [4, 5 21 ].

(3.14)

8


Case 1. x = (s, 0), y = (t, 0). If s, t ∈ [0, 2], then we have done. 2 (t−4)2 If s, t ∈ 4, 5 21 , then T x = 0, (s−4) and T y = 0, , and so 2 2 2d2r (T x, T y)

(t − 4)2 (s − 4)2 − 2 2

2

2 2 9 9 + ≤ 8 8

=

2

≤

42 + 42 2 2 (t − 4)2 (s − 4)2 +t + s+ = d2r (T x, y) + d2r (x, T y). 2 2

≤

2

), and so If s ∈ [0, 2], t ∈ [4, 5 21 ], then T x = (0, 0) and T y = (0, (t−4) 2 2 2 2 (t − 4)2 (t − 4)2 (t − 4)2 2d2r (T x, T y) = 2 ≤ + 2 2 2 2 2 (t − 4) ≤ (t)2 + s + 2 =

d2r (T x, y) + d2r (x, T y)

2

If s ∈ [4, 5 21 ], t ∈ [0, 2], then T x = (0, (s−4) ) and T y = (0, 0), and so 2 2 2 2 (s − 4)2 (s − 4)2 (s − 4)2 2 = + 2dr (T x, T y) = 2 2 2 2 2 2 (s − 4) ≤ t+( ) + (s)2 = d2r (T x, y) + d2r (x, T y). 2 Case 2. x = (s, 0), y = (0, t). If s, t ∈ [0, 2], then we havedone.

If s, t ∈ [4, 5 21 ], then T x = 0, (s−4) 2

2

and T y =

(t−4)2 ,0 2

, and so

2 2 2 (t − 4)2 9 9 (s − 4)2 + + ≤ = 2 2 2 4 4 2 2 9 9 ≤ 4− + 4− 8 8 2 2 2 (t − 4)2 (s − 4) + 4− ≤ 4− 2 2 2 2 (t − 4)2 (s − 4)2 + s− = d2r (T x, y) + d2r (x, T y). ≤ t− 2 2 2 If s ∈ [0, 2], t ∈ [4, 5 21 ], then T x = (0, 0) and T y = (t−4) , 0 , and so 2 2d2r (T x, T y)

2d2r (T x, T y)

= ≤ ≤ ≤ ≤

2 2 2 (t − 4)2 (t − 4)2 (t − 4)2 = + 2 2 2 2 2 9 (t)2 + 8 2 9 2 (t) + 4 − 8 2 (t − 4)2 (t)2 + 4 − 2 2 (t − 4)2 2 (t) + s − = d2r (T x, y) + d2r (x, T y). 2

∆-CONVERGENCE FOR GENERALIZED HYBRID MAPPINGS IN CAT(0) SPACES

9

2 If s ∈ 4, 5 21 , t ∈ [0, 2], then T x = 0, (s−4) and T y = (0, 0), and so 2 2d2r (T x, T y) = ≤ ≤ ≤ ≤

2 2 2 (s − 4)2 (s − 4)2 (s − 4)2 = + 2 2 2 2 9 + (s)2 8 2 9 + (s)2 4− 8 2 (s − 4)2 4− + (s)2 2 2 (s − 4)2 + (s)2 = d2r (T x, y) + d2r (x, T y). t− 2

2

Case 3. x = (0, s), y = (0, t) the proof is similarly to case 1. Case 4. x = (0, s), y = (t, 0) the proof is similarly to case 2. Hence we have the claim. By the same proof, we can hybrid. conclude that T is also (1, 1)-generalized 9 But T is not nonexpansive. Indeed, if x = 0, 5 41 , y = 0, 5 21 , then x = 25 32 , 0 , y = 8 , 0 . Thus, we have 25 9 11 1 1 1 dr (T x, T y) = − = > = 5 − 5 = dr (x, y). 32 8 32 4 4 2

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