Common Stationary Points of Multivalued Mappings on Bounded Metric Spaces

Necessary and sufficient conditions for the existence of common stationary points of two multivalued mappings and common stationary point theorems for multival-ued mappings on bounded metric spaces are given. Our results extend the theorems due 1. Introduction. Let (X, d) be a metric space and B(X) denote the set of all nonempty bounded subsets of X. For A, B ∈ X, let δ(A, B) = sup{d(a, b) : a ∈ A, b ∈ B} and δ(A) = δ(A, A). If A consists of a single point a, we write δ(A, B) = δ(a, B). If B also consists of a single point b, we write δ(A, B) = δ(a, b) = d(a, b). Let N and ω denote the sets of positive integers and nonnegative integers, respectively. Let Φ denote a family of mappings such that for each φ ∈ Φ, φ : [0, ∞) → [0, ∞) is upper semicontin-uous, nondecreasing and φ(t) < t for t > 0. The following definitions and lemmas were introduced by Fisher [3] and Singh and Meade [6].

1. Introduction.Let (X, d) be a metric space and B(X) denote the set of all nonempty bounded subsets of X.For A, B ∈ X, let δ(A, B) = sup{d(a, b) : a ∈ A, b ∈ B} and δ(A) = δ(A, A).If A consists of a single point a, we write δ(A, B) = δ(a, B).If B also consists of a single point b, we write δ(A, B) = δ(a, b) = d(a, b).Let N and ω denote the sets of positive integers and nonnegative integers, respectively.Let Φ denote a family of mappings such that for each φ ∈ Φ, φ : [0, ∞) → [0, ∞) is upper semicontinuous, nondecreasing and φ(t) < t for t > 0.
The following definitions and lemmas were introduced by Fisher [3] and Singh and Meade [6].
Definition 1.1 [3].Let {A n } be a sequence of sets in B(X) and A ∈ B(X).The sequence {A n } is said to converge to the set A if (i) each point a ∈ A is the limit of some convergent sequence {a n }, where a n ∈ A n for n ∈ N; (ii) for arbitrary > 0, there exists k ∈ N such that A n ⊆ A , for n > k, where A is the union of all open spheres with centres in A and radius .
Definition 1.2 [3].Let F be a multivalued mapping of (X, d) into B(X).The mapping F is called continuous in X if whenever {x n } is a sequence of points in X converging to x ∈ X, the sequence {Fx n } in B(X) converges to Fx ∈ B(X).
Lemma 1.3 [3].If {A n } and {B n } are sequences of bounded subsets of a complete metric space (X, d) which converge to the bounded subsets A and B, respectively, then the sequence {δ(A n ,B n )} converges to δ(A, B).Lemma 1.4 [6].Let φ ∈ Φ.Then φ(t) < t for each t > 0 if and only if where φ n denotes the n-times composition of φ.
Let F and G be mappings of (X, d) into B(X).A point x ∈ X is called a common stationary point of F and G if Fx = Gx = {x}.For A ⊆ X, let FA = ∪ a∈A Fa and GF A = G(F A).The mappings F and G are said to commute if FGx = GF x for x ∈ X. Define C F = {T : T is a mapping of X into B(X) and T and F commute} and Throughout the rest of the paper, we assume that (X, d) is a complete bounded metric space.
In 1979, Fisher [1] proved a common fixed point theorem for commuting mappings f and g of (X, d) into itself satisfying for all x, y ∈ X, where 0 ≤ c < 1.
In 1983, Fisher [4] established a common fixed point theorem for continuous, commuting mappings F and G of (X, d) into B(X) satisfying for all x, y ∈ X, where 0 ≤ c < 1 and p is a fixed positive integer.
In 1994, Ohta and Nikaido [5] obtained the existence of fixed point for a continuous self mapping f of (X, d) satisfying for all x, y ∈ X, where 0 ≤ c < 1 and k is a fixed positive integer.
The first purpose of the paper is to establish criteria for the existence of common stationary points of commuting mappings F and G of (X, d) into B(X).The second purpose of the paper is to prove common stationary point theorems for commuting mappings F and G of (X, d) into B(X) satisfying one of the following: for all x, y ∈ X, where φ ∈ Φ and p, q are fixed positive integers; for all x, y ∈ X, where φ ∈ Φ and p, q are fixed positive integers; for all x, y ∈ X, where φ ∈ Φ.

Common stationary points.
Our main results are as follows.
Theorem 2.1.Let F and G be continuous and commuting mappings of (X, d) into B(X).Then the following statements are equivalent: (i) F and G have a common stationary point; Proof.We shall verify the following implications: (i)⇒(ii)⇒(iii)⇒(iv)⇒(i). Suppose, first of all, that F and G have a common stationary point z.Define mappings S and for all x, y ∈ X, φ ∈ Φ, that is, (ii) holds.Note that CC F ⊆ C F and C S ∩ C T ⊆ C ST .Therefore (ii)⇒(iii)⇒(iv) are clear.We now assume that (iv) holds.Then for any A, B ∈ B(X), we have for n ∈ N. We now will prove by induction that Note that S and T commute.From (2.5), we have that is, (2.6) holds for n = 1.Assume now that (2.6) holds for some n ∈ N. It follows from (2.5) that (2.8) by our assumption.Hence (2.6) follows by induction.Choose x n ∈ X n for n ∈ N.Then, by (2.6), we get (2.9) Consequently, {x n } is a Cauchy sequence by Lemma 1.4.Since X is complete, there exists a point z in X such that x n → z as n → ∞.From (2.6), we have for m, n ∈ N with m > n.Letting m tend to infinity, we obtain Since F is continuous and x n → z, then {Fx n } converges to {Fz}.Note that Letting n tend to infinity, we have δ(z, F z) ≤ 0 by (2.11) and Lemmas 1.3 and 1.4, that is, Fz = {z}.Similarly, we have Gz = {z}.This completes the proof.
Theorem 2.2.Let F and G be continuous and commuting mappings of (X, d) into B(X) satisfying (1.6) or (1.7).Then F and G have a unique common stationary point z and the sequence {F n G n x} converges to {z} for all x ∈ X.
Proof.Let M = δ(X), k = p + q, X n = F n G n X and x n ∈ X n for n ∈ N. Note that every n ∈ N can be written as where j ∈ ω and 0 ≤ i < k.Now we claim that δ X n ≤ φ j (M). (2.14) ≤ φ δ X k(j−1) . (2.17) (2.20) Letting n tend to infinity, by Lemma 1.4 and Definition 1.1 and the above inequalities, we conclude that {F n G n x} converges to {z}.This completes the proof.
As a consequence of Theorem 2.2, we have the following corollary.
Corollary 2.3.Let F and G be continuous and commuting mappings of (X, d) into B(X) satisfying one of the following: where φ ∈ Φ and p, q are fixed positive integers; where φ ∈ Φ and p, q are fixed positive integers.Then F and G have a unique common stationary point z and the sequence {F n G n x} converges to {z} for all x ∈ X.
From Corollary 2.3, we have the following.
Corollary 2.4 [4,Theorem 1].Let F and G be continuous and commuting mappings of (X, d) into B(X) satisfying (1.3).Then F and G have a unique common stationary point z and the sequence {F n G n x} converges to {z} for all x ∈ X. Corollary 2.5 [5,Theorem 3].Let f be a continuous mapping of (X, d) into itself satisfying (1.5).Then f has a unique fixed point z and for each x ∈ X, f n x → z as n → ∞.Theorem 2.6.Let F and G be commuting mappings of (X, d) into B(X) satisfying (1.8).Then F and G have a unique common stationary point z and the sequence {F n G n x} converges to {z} for all x ∈ X.Further, {z} = Dz for all D ∈ CC FG .
From Theorem 2.6, we have the following corollary.
Corollary 2.7 [2, Theorem 2].Let F and G be commuting mappings of (X, d) into B(X) satisfying (1.3).Then F and G have a unique common stationary point z and the sequence {F n G n x} converges to {z} for all x ∈ X. Corollary 2.8 [1,Theorem 4].Let f and g be commuting mappings of (X, d) into itself satisfying (1.2).Then f and g have a unique common fixed point z and for each x ∈ X, f n g n → z as n → ∞.
The following example shows that Theorem 2.6 extends properly Corollaries 2.7 and 2.8.Example 2.9.Let X = {1, 2, 5, 8} with the usual metric.Define self mappings f and g of (X, d) by Similarly,(2.16)holdsalso.It follows from(2.16)and(2.17)that(2.14)holds.Given x n ∈ X n for all n ∈ N.For any m > n > k, by (2.13) and (2.14) we haved x n ,x m ≤ δ X n ≤ φ j (M).(2.18)As in the proof of Theorem 2.1, we conclude that F and G have a common stationary point z and that x n → z as n → ∞.Suppose that F and G have a second common stationary point w.Then {u} = F n G n u ⊆ X n for u ∈ {z, w} and n ∈ N.Letting n tend to infinity we have d(z, w) ≤ 0 by Lemma 1.4, that is, z = w.Hence F and G have a unique common stationary point z.For x ∈ X and n ∈ N, choose y n ∈ F n G n x.Using (2.13) and (2.14), we have .28) Set Fx = {f x} and Gx = {gx} for x ∈ X.Let φ(t) = (1/2)t for t ≥ 0. It is easy to check that F and G satisfy the conditions of Theorem 2.6.But Corollaries 2.7 and 2.8 are not applicable sinced(f 1,g1) = 4 = max d(1, 1), d(1,f 1), d(1,g1), d(1,g1), d(1,f 1) , (2.29)that is, f and g do not satisfy (1.2).Similarly F and G do not satisfy (1.3).