Strong and Convergence Theorems for Multivalued Mappings in Spaces

Abstract

We show strong and convergence for Mann iteration of a multivalued nonexpansive mapping whose domain is a nonempty closed convex subset of a CAT(0) space. The results we obtain are analogs of Banach space results by Song and Wang [2009, 2008]. Strong convergence of Ishikawa iteration are also included.

1. Introduction

Let be a nonempty subset of a Banach space We shall denote by the family of nonempty closed bounded subsets of by the family of nonempty bounded proximinal subsets of and by the family of nonempty compact subsets of . Let be the Hausdorff distance on that is,

(1.1)

where is the distance from the point to the set

A multivalued mapping is said to be a nonexpansive if

(1.2)

A point is called a fixed point of if We denote by the set of all fixed points of

In 2005, Sastry and Babu [1] introduced the Mann and Ishikawa iterations for multivalued mappings as follows: let be a real Hilbert space and be a multivalued mapping for which . Fix and define

• (A) the sequence of Mann iterates by

  

  (1.3)

  where is such that

• (B) the sequence of Ishikawa iterates by

  

  (1.4)

where is such that and

(1.5)

where is such that

They proved the following results.

Theorem 1.1.

Let be a nonempty compact convex subset of a Hilbert space Suppose is nonexpansive and has a fixed point Assume that (i) and (ii)Then the sequence of Mann iterates defined by (A) converges to a fixed point of

Theorem 1.2.

Let be a nonempty compact convex subset of a Hilbert space Suppose that a nonexpansive map has a fixed point Assume that (i) (ii) and (iii) Then the sequence of Ishikawa iterates defined by (B) converges to a fixed point of

In 2007, Panyanak [2] extended Sastry-Babu's results to uniformly convex Banach spaces as the following results.

Theorem 1.3.

Let be a nonempty compact convex subset of a uniformly convex Banach spaces Suppose that a nonexpansive map has a fixed point Let be the sequence of Mann iterates defined by (A). Assume that (i) and (ii)Then the sequence converges to a fixed point of

Theorem 1.4.

Let be a nonempty compact convex subset of a uniformly convex Banach spaces Suppose that a nonexpansive map has a fixed point . Let be the sequence of Ishikawa iterates defined by (B). Assume that (i)(ii) and (iii) Then the sequence converges to a fixed point of

Recently, Song and Wang [3, 4] pointed out that the proof of Theorem 1.4 contains a gap. Namely, the iterative sequence defined by (B) depends on the fixed point Clearly, if and then the sequence defined by is different from the one defined by Thus, for defined by , we cannot obtain that is a decreasing sequence from the monotony of . Hence, the conclusion of Theorem 1.4 (also Theorem 1.3) is very dubious.

Motivated by solving the above gap, they defined the modified Mann and Ishikawa iterations as follows.

Let be a nonempty convex subset of a Banach space and be a multivalued mapping. The sequence of Mann iterates is defined as follows: let and such that Choose and Let

(1.6)

There exists such that (see [5, 6]). Take

(1.7)

Inductively, we have

(1.8)

where such that

The sequence of Ishikawa iterates is defined as follows: let , and such that Choose and Let

(1.9)

There exists such that Let

(1.10)

There is such that Take

(1.11)

There exists such that Let

(1.12)

Inductively, we have

(1.13)

where and such that and

They obtained the following results.

Theorem 1.5 (see [3, Theorem 2.3]).

Let be a nonempty compact convex subset of a Banach space Suppose that is a multivalued nonexpansive mapping for which and for each Let be the sequence of Mann iteration defined by (1.8). Assume that

(1.14)

Then the sequence strongly converges to a fixed point of

Recall that a multivalued mapping is said to satisfy Condition I ([7]) if there exists a nondecreasing function with and for all such that

(1.15)

Theorem 1.6 (see [3, Theorem 2.4]).

Let be a nonempty closed convex subset of a Banach space Suppose that is a multivalued nonexpansive mapping that satisfies Condition I. Let be the sequence of Mann iteration defined by (1.8). Assume that and satisfies for each and

(1.16)

Then the sequence strongly converges to a fixed point of

Theorem 1.7 (see [3, Theorem 2.5]).

Let be a Banach space satisfying Opial's condition and be a nonempty weakly compact convex subset of Suppose that is a multivalued nonexpansive mapping. Let be the sequence of Mann iteration defined by (1.8). Assume that and satisfies for each and

(1.17)

Then the sequence weakly converges to a fixed point of

Theorem 1.8 (see [4, Theorem  1]).

Let be a nonempty compact convex subset of a uniformly convex Banach space Suppose that is a multivalued nonexpansive mapping and satisfying for any fixed point Let be the sequence of Ishikawa iterates defined by (1.13). Assume that (i)  (ii) and (iii)Then the sequence strongly converges to a fixed point of

Theorem 1.9 (see [4, Theorem  2]).

Let be a nonempty closed convex subset of a uniformly convex Banach space Suppose that is a multivalued nonexpansive mapping that satisfy Condition I. Let be the sequence of Ishikawa iterates defined by (1.13). Assume that satisfying for any fixed point and Then the sequence strongly converges to a fixed point of

In this paper, we study the iteration processes defined by (1.8) and (1.13) in a CAT(0) space and give analogs of Theorems 1.5–1.9 in this setting.

2. Spaces

A metric space is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in 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 space. Other examples include Pre-Hilbert spaces, -trees (see [8]), Euclidean buildings (see [9]), the complex Hilbert ball with a hyperbolic metric (see [10]), and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry (see Bridson and Haefliger [8]). Burago, et al. [11] contains a somewhat more elementary treatment, and Gromov [12] a deeper study.

Fixed point theory in a CAT(0) space was first studied by Kirk (see [13] and [14]). 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 multivalued mappings in CAT(0) spaces has been rapidly developed and many of papers have appeared (see, e.g., [15–24]). It is worth mentioning that the results in CAT(0) spaces can be applied to any CAT() space with since any CAT() space is a CAT( ) space for every (see [8], page 165).

Let be a metric space. A geodesic path joining to (or, more briefly, a geodesic from to ) is a map from a closed interval to such that and for all In particular, is an isometry and The image of is called a geodesic (or metric) segment joining and . When it is unique this geodesic is denoted by . The space is said to be a geodesic space if every two points of are joined by a geodesic, and is said to be uniquely geodesic if there is exactly one geodesic joining and for each A subset is said to be convex if includes every geodesic segment joining any two of its points.

A geodesic triangle in a geodesic space consists of three points in (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A comparison triangle for geodesic triangle in is a triangle in the Euclidean plane such that for

A geodesic space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom.

CAT(0): let be a geodesic triangle in and let be a comparison triangle for . Then is said to satisfy the CAT(0) inequality if for all and all comparison points

(2.1)

Let by [24, Lemma 2.1(iv)] for each there exists a unique point such that

(2.2)

From now on we will use the notation for the unique point satisfying (2.2). By using this notation Dhompongsa and Panyanak [24] obtained the following lemma which will be used frequently in the proof of our main theorems.

Lemma 2.1.

Let be a CAT (0)space . Then

(2.3)

for all and

If are points in a CAT(0) space and if then the CAT(0) inequality implies

(2.4)

This is the (CN) inequality of Bruhat and Tits [25]. In fact (cf. [8, page 163]), a geodesic metric space is a CAT(0) space if and only if it satisfies (CN).

The following lemma is a generalization of the (CN) inequality which can be found in [24].

Lemma 2.2.

Let be a CAT(0) space. Then

(2.5)

for all and

The preceding facts yield the following result.

Proposition 2.3.

Let be a geodesic space. Then the following are equivalent:

• (i) is a CAT (0) space;

• (ii) satisfies (CN);

• (iii) satisfies (2.5).

The existence of fixed points for multivalued nonexpansive mappings in a CAT(0) space was proved by S. Dhompongsa et al. [17], as follows.

Theorem 2.4.

Let be a closed convex subset of a complete CAT(0) space , and let be a nonexpansive nonself-mapping. Suppose

(2.6)

for some bounded sequence in Then has a fixed point.

3. The Setting

Let be a Banach space, and let be a bounded sequence in for we let

(3.1)

The asymptotic radius of is given by

(3.2)

and the asymptotic center of is the set

(3.3)

The notion of asymptotic centers in a Banach space can be extended to a CAT(0) space as well, simply replacing with It is known (see, e.g., [18, Proposition 7]) that in a CAT(0) space, consists of exactly one point.

Next we provide the definition and collect some basic properties of -convergence.

Definition 3.1 (see [23]).

A sequence in a CAT(0) space is said to -converge to if is the unique asymptotic center of for every subsequence of . In this case one must write and call the -limit of

Remark 3.2.

In a CAT(0) space , strong convergence implies convergence and they are coincided when is a Hilbert space. Indeed, we prove a much more general result. Recall that a Banach space is said to satisfy Opial's condition ([26]) if given whenever converges weakly to

(3.4)

Proposition 3.3.

Let be a reflexive Banach space satisfying Opial's condition and let be a bounded sequence in and let Then converges weakly to if and only if for all subsequence of

Proof.

() Let be a subsequence of . Then converges weakly to By Opial's condition () Suppose for all subsequence of and assume that does not converge weakly to Then there exists a subsequence of such that for each is outside a weak neighborhood of Since is bounded, without loss of generality we may assume that converges weakly to By Opial's condition a contradiction.

Lemma 3.4.

(i) Every bounded sequence in has a convergent subsequence (see [23, page 3690]). (ii) If is a closed convex subset of and if is a bounded sequence in then the asymptotic center of is in (see [17, Proposition 2.1]).

Now, we define the sequences of Mann and Ishikawa iterates in a CAT(0) space which are analogs of the two defined in Banach spaces by Song and Wang [3, 4].

Definition 3.5.

Let be a nonempty convex subset of a CAT(0) space and be a multivalued mapping. The sequence of Mann iterates is defined as follows: let and such that Choose and Let

(3.5)

There exists such that Take

(3.6)

Inductively, we have

(3.7)

where such that

Definition 3.6.

Let be a nonempty convex subset of a CAT(0) space and be a multivalued mapping. The sequence of Ishikawa iterates is defined as follows: let , and such that Choose and Let

(3.8)

There exists such that Let

(3.9)

There is such that Take

(3.10)

There exists such that Let

(3.11)

Inductively, we have

(3.12)

where and such that and

Lemma 3.7.

Let be a nonempty compact convex subset of a complete CAT (0) space and let be a nonexpansive nonself-mapping. Suppose that

(3.13)

for some sequence in Then has a fixed point. Moreover, if converges for each , then strongly converges to a fixed point of

Proof.

By the compactness of there exists a subsequence of such that Thus

(3.14)

This implies that is a fixed point of Since the limit of exists and we have This show that the sequence strongly converges to

Before proving our main results we state a lemma which is an analog of Lemma 2.2 of [27]. The proof is metric in nature and carries over to the present setting without change.

Lemma 3.8.

Let and be bounded sequences in a CAT (0)space and let be a sequence in with Suppose that for all and

(3.15)

Then

4. Strong and Convergence of Mann Iteration

Theorem 4.1.

Let be a nonempty compact convex subset of a complete CAT (0)space Suppose that is a multivalued nonexpansive mapping and satisfying for any fixed point If is the sequence of Mann iterates defined by (3.7) such that one of the following two conditions is satisfied:

• (i) and

• (ii)
  

Then the sequence strongly converges to a fixed point of

Proof

Case 1.

Suppose that (i) is satisfied. Let by Lemma 2.2 and the nonexpansiveness of we have

(4.1)

This implies

(4.2)

(4.3)

It follows from (4.2) that for all This implies that is bounded and decreasing. Hence exists for all On the other hand, (4.3) implies

(4.4)

Since diverges, we have and hence Then there exists a subsequence of such that

(4.5)

This implies

(4.6)

By Lemma 3.7, converges to a point Since the limit of exists, it must be the case that and hence the conclusion follows.

Case 2.

If (ii) is satisfied. As in the Case 1, exists for each It follows from the definition of Mann iteration (3.7) that

(4.7)

Therefore,

(4.8)

By Lemma 3.8, we obtain

(4.9)

This implies

(4.10)

so the conclusion follows from Lemma 3.7.

Theorem 4.2.

Let be a nonempty closed convex subset of a complete CAT(0) space Suppose that is a multivalued nonexpansive mapping that satisfies Condition I. Let be the sequence of Mann iterates defined by (3.7). Assume that satisfying for any fixed point and Then the sequence strongly converges to a fixed point of

Proof.

It follows from the proof of the Case 1 in Theorem 4.1 that exists for each and

(4.11)

Then

(4.12)

This implies

(4.13)

Thus, and hence Since

(4.14)

Therefore, Furthermore Condition I implies

(4.15)

The proof of remaining part closely follows the proof of of [2, Theorem 3.8], simply replacing with .

Next we show a convergence theorem of Mann iteration in a CAT(0) space setting which is an analog of Theorem 1.7. For this we need more lemmas.

Lemma 4.3 (see [24, Lemma  2.8]).

If is a bounded sequence in a complete CAT (0)space with and is a subsequence of with and the sequence converges, then

Lemma 4.4.

Let be a nonempty closed convex subset of a complete CAT (0) space and let be a nonexpansive nonself-mapping. Suppose that is a sequence in which converges to in and

(4.16)

Then

Proof.

Notice from Lemma 3.4(ii) that Since is compact-valued, for each there exists and such that and It follows from (4.16) that

(4.17)

By the compactness of there exists a subsequence of such that Then

(4.18)

This implies

(4.19)

Since and hence by (4.19). Therefore is a fixed point of

Lemma 4.5.

Let be a closed convex subset of a complete CAT (0)spaceand let be a nonexpansive mapping. Supposeis a bounded sequence in such thatandconverges for all then Here where the union is taken over all subsequences of Moreover, consists of exactly one point.

Proof.

Let , then there exists a subsequence of such that By Lemma 3.4(i) and (ii) there exists a subsequence of such that . By Lemma 4.4, . By Lemma 4.3, This shows that Next, we show that consists of exactly one point. Let be a subsequence of with and let . Since is convergent by the assumption. By Lemma 4.3, This completes the proof.

Theorem 4.6.

Let be a nonempty closed convex subset of a complete CAT (0) space Suppose that is a multivalued nonexpansive mapping. Let be the sequence of Mann iterates defined by (3.7). Assume that satisfying for any fixed point and

(4.20)

Then the sequence converges to a fixed point of

Proof.

Let it follows from (4.2) in the proof of Theorem 4.1 that for all This implies that is bounded and decreasing. Hence exists for all Since

(4.21)

Thus by (4.9). By Lemma 4.5, consists of exactly one point and is contained in . This shows that converges to an element of

5. Strong Convergence of Ishikawa Iteration

The following lemma can be found in [2].

Lemma 5.1.

Let be two real sequences such that

• (i)

• (ii) as

• (iii)

Let be a nonnegative real sequence such that is bounded. Then has a subsequence which converges to zero.

The following theorem is an analog of Theorem 1.8.

Theorem 5.2.

Let be a nonempty compact convex subset of a complete CAT (0) space Suppose that is a multivalued nonexpansive mapping and satisfying for any fixed point Let be the sequence of Ishikawa iterates defined by (3.12). Assume that

• (i)
• (ii)
• (iii)

Then the sequence strongly converges to a fixed point of

Proof.

Let by Lemma 2.2 and the nonexpansiveness of we have

(5.1)

This implies

(5.2)

(5.3)

It follows from (5.2) that the sequence is decreasing and hence exists for each On the other hand, (5.3) implies

(5.4)

By Lemma 5.1, there exists a subsequence of such that

(5.5)

This implies

(5.6)

By Lemma 3.7, converges to a point Since the limit of exists, it must be the case that and hence the conclusion follows.

The following theorem is an analog of Theorem 1.9.

Theorem 5.3.

Let be a nonempty closed convex subset of a complete CAT(0) space Suppose that is a multivalued nonexpansive mapping that satisfies Condition I. Let be the sequence of Ishikawa iterates defined by (3.12). Assume that satisfying for any fixed point and Then the sequence strongly converges to a fixed point of

Proof.

Similar to the proof of Theorem 5.2, we obtain exists for each and

(5.7)

Then

(5.8)

This implies

(5.9)

Thus, and hence Since

(5.10)

Therefore, Furthermore Condition I implies

(5.11)

The proof of remaining part closely follows the proof of [2, Theorem 3.8], simply replacing with .