Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 02, 1997, pp. 20–28. Submitted: September 10, 1997. Published: December 30, 1997.

COINCIDENCE POINTS OF

COMPATIBLE MULTIVALUED MAPPINGS

R.A.RASHWAN AND M.A.AHMED

Abstract. In this paper we establish a coincidence theorem for a generalized contractive type multivalued mappings in complete metric spaces generalizing some results of Constantin [1]. Two examples are given to support our generalization.

1991 AMS Mathematics subject classification Code. 54H25..

Key words and phrases. Coincidence points, compatible mappings, complete metric spaces.

1. INTRODUCTION

In [2,3,4,5,9,11,12,13,15,16], the authors proved some coincidence point theorems for multivalued mappings in complete metric spaces under various contractive conditions.

Recently Constantin [1] proved some coincidence point theorems for multivalued mappings in complete metric spaces generalizing some results in [4, 5, 11, 12] and others.

In this paper we consider a generalized contraction condition involving two multivalued mappings which are compatible with two single-valued mappings and we establish a coincidence theorem which improves, extends some results of Constantin [1].

Let (X, d) be a metric space. We denote by CB(X) the set of all nonempty closed bounded subsets of (X, d) and by H the Hausdorff metric on CB(X)

H(A, B) = max{sup_{x∈A} d(x, B), sup_{y∈B} d(y, A)},

where A, B ∈ CB(X) and

d(x, A) = inf_{y∈A} d(x, y).

Our main theorem is prefaced with the following lemma due to Nadler [10].

Typeset by AMS-TEX

Department of Mathematics, Faculty of Science, Assiut University, Assiut, EGYPT

Department of mathematics, Faculty of science, Assiut University, Assiut. EGYPT

\begin{matrix}c \\ ◯ \end{matrix}1997 Cameron University

Lemma 1.1. Let A, B ∈ CB(X) and k > 1. Then for each a ∈ A, there is a point b ∈ B such that

d(a, b) ≤ kH(A, B).

For single-valued mappings the notion of the weak commutativity is introduced by S. Sessa [14].

Definition 1.1. The mappings S, I : X → X are said to be weakly commuting if

d(SIx, ISx) ≤ d(Ix, Sx)

for all x in X.

It is clear that two commuting mappings are weakly commuting but the converse is not true as is shown in [14].

G.Jungck [6] extended the concept of weak commutativity in the following way:

Definition 1.2. Let S and I be mappings from a metric space (X, d) into itself. The mappings S and I are said to be compatible if

lim_{n→∞} d(SIx_{n}, ISx_{n}) = 0,

whenever {x_{n}} is a sequence in X such that

lim_{n→∞} Sx_{n} = lim_{n→∞} Ix_{n} = z

for some z ∈ X.

It is obvious that two weakly commuting mappings are compatible, but the converse is not true as is shown in [6].

H. Kaneko and S. Sessa [8] extended the concept of compatibility for singlevalued mappings to the setting of single-valued and multi-valued mappings in the following way:

Definition 1.3. The mappings A : X → CB(X) and S : X → X are compatible iff SAx ∈ CB(X) for all x ∈ X and d(SAx_{n}, ASx_{n}) → 0 whenever {x_{n}} ⊂ X such that Ax_{n} → M ∈ CB(X) and Sx_{n} → t ∈ M.

Definition 1.3 extends Definition 1.2 above as well as the Definition 1.1 above, where one may replace I : X → X with A : X → CB(X), that is S : X → X and A : X → CB(X) are said to be weakly commuting if for all x ∈ X, SAx ∈ CB(X) and H(SAx, ASx) ≤ d(Sx, Ax), where H is the Hausdorff metric defined on CB(X) .

On the other hand, O. Hadzic [4] introduced the following definition:

Definition 1.4. The mappings A : X → CB(X) and S : X → X are said to be weakly commuting iff for every x ∈ X and y ∈ X such that y ∈ Ax we have

d(Sy, SAx) ≤ d(Sx, Ax).

It is obvious that a commutative pair of mappings (S, A) is also weakly commutative in the sense of Definition 1.4 but the converse is not true as is shown by Constantin [1].

We now introduce the following new definition which is slight variant of Kaneko-Sessa’s [8] definition in the light of Definition 1.4.

Definition 1.5. The mappings A : X → CB(X) and S : X → X are said to be compatible if

lim_{n→∞} d(Sy_{n}, ASx_{n}) = 0

whenever {x_{n}} and {y_{n}} are sequences in X such that

lim_{n→∞} Sx_{n} = lim_{n→∞} y_{n} = z

for some z in X where y_{n} ∈ Ax_{n} for n = 1, 2, ...

Clearly every weakly commuting pair of mappings (S, A) are compatible but the converse is not true. To see this, let {x_{n}} and {y_{n}} be two sequences in X such that y_{n} ∈ Ax_{n} for n = 1, 2, ... and

lim_{n→∞} Sx_{n} = lim_{n→∞} y_{n} = z

for some z in X . Suppose S and A are weakly commuting. Since d(Sx_{n}, Ax_{n}) ≤ d(Sx_{n}, y_{n}), it follows that lim_{n→∞} d(Sx_{n}, Ax_{n}) = 0. Since S and A are weakly commuting, we have

lim_{n→∞} H(SAx_{n}, ASx_{n}) = 0.

Again, since d(Sy_{n}, ASx_{n}) ≤ H(SAx_{n}, ASx_{n}), we have

lim_{n→∞} d(Sy_{n}, ASx_{n}) = 0,

proving that S and A are compatible.

Example 1.1. Let X = [1, ∞) with the Euclidean metric d and define

Sx = 4x^{4} − 3, Ax = [1, x^{2}]

for all x ≥ 1.

Clearly S and A are continuous and A(X) = S(X) = X. Let {x_{n}} and {y_{n}} be two sequences in X defined by x_{n} = 1 + \frac{1}{n}, y_{n} = 1 for n = 1, 2, ... , we then have

lim_{n→∞} Sx_{n} = lim_{n→∞} y_{n} = 1 ∈ X, y_{n} ∈ Ax_{n}.

Moreover,

lim_{n→∞} H(SAx_{n}, ASx_{n}) = 0 iff x_{n} → 1 as n → ∞,

and since

d(Sy_{n}, ASx_{n}) ≤ H(SAx_{n}, ASx_{n}),

we have

lim_{n→∞} d(Sy_{n}, ASx_{n}) = 0.

Therefore S and A are compatible, but S and A are not weakly commuting at x = 2.

A point z in X is said to be a coincidence point of S and A if Sz ∈ Az and is a fixed point of A if z ∈ Az .

2. A COINCIDENCE THEOREM

Following Constantin [1], we consider the set G of all continuous functions g : [0, ∞)^{5} ⟶ [0, ∞) satisfying the following conditions:

(i) g(1, 1, 1, 2, 0) = g(1, 1, 1, 0, 2) = h ∈ (0, 1);

(ii) g is subhomogenuous nondecreasing in each variable;

(iii) g(x, y, z, t, 0) < g(x_{1}, y_{1}, z_{1}, t, 0) and g(x, y, z, 0, t) < g(x_{1}, y_{1}, z_{1}, 0, t) if x ≤ x_{1}, y ≤ y_{1}, z ≤ z_{1}, max{x, y, z} < max{x_{1}, y_{1}, z_{1}} and t ≤ max{x, y, z}+min{x, y, z};

(iv) if u ≤ g(v, v, u, u + v, 0) or u ≤ g(v, v, u, 0, u + v) or u ≤ g(v, u, v, u + v, 0) or u ≤ g(v, u, v, 0, u + v) then u ≤ hv.

The following Lemma due to Constantin [1] is useful in the sequel.

Lemma 2.1. If g ∈ G and k > 1 is such that kh < 1 we have that kg ∈ G. Adrian Constantin [1] proved the following Theorem.

Theorem 2.1. Let (X, d) be a complete metric space, S and T continuous mappings from X into X, A and B H-continuous mappings from X into CB(SX ∩T X) such that the pairs (A, S) and (B, T) are weakly commutative in the sense of Definition 1.4 and

H(Ax, By) ≤ g(d(Sx, T y), d(Sx, Ax), d(T y, By), d(Sx, By), d(T y, Ax)), (2.1)

for every x, y ∈ X, where g ∈ G. Then there exists a sequence {x_{n}}_{n∈N} such that 1. For every n ∈ N, Sx_{2n+1} ∈ Bx_{2n}, T x_{2n} ∈ Ax_{2n−1};

2. There exists z = lim_{n→∞} T x_{2n} = lim_{n→∞} Sx_{2n+1};

3. Sz ∈ Az, T z ∈ Bz.

Now replacing weakly commuting mappings with compatible mappings (Definition 1.5) in Theorem 2.1, we prove the following generalization of Theorem 2.1.

Theorem 2.2. Let (X, d) be a complete metric space, S and T continuous mappings from X into X, A and B H-continuous mappings from X into CB(SX ∩T X) such that the pairs (A, S) and (B, T) are compatible in the sense of Definition 1.5 and condition (2.1) holds. Then there exists a sequence {x_{n}}_{n∈N} such that

1. For every n ∈ N, Sx_{2n+1} ∈ Bx_{2n}, T x_{2n} ∈ Ax_{2n−1};

2. There exists z = lim_{n→∞} T x_{2n} = lim_{n→∞} Sx_{2n+1};

3. Sz ∈ Az, T z ∈ Bz.

Proof. Let k > 1 be such that kh < 1. Let x_{0} ∈ X. Since Bx_{0} ⊂ SX there exists x_{1} ∈ X such that Sx_{1} ∈ Bx_{0}. Then there exists an element u_{1} ∈ Ax_{1} such that

d(u_{1}, Sx_{1}) ≤ kH(Ax_{1}, Bx_{0})

and u_{1} = T x_{2} for some x_{2} ∈ X because Ax_{1} ⊂ T X. Continuing in this fashion we define a sequence {x_{n}}_{n∈N} in X where T x_{2n} ∈ Ax_{2n−1} and Sx_{2n+1} ∈ Bx_{2n} are such that

d(T x_{2n}, Sx_{2n−1}) ≤ kH(Ax_{2n−1}, Bx_{2n−2})  (2.2)
d(Sx_{2n+1}, T x_{2n}) ≤ kH(Bx_{2n}, Ax_{2n−1})  (2.3)

for every n ≥ 1.

We observe that by (2.1) and (2.3) we have that

d(Sx_{2n+1}, T x_{2n}) ≤ kH(Bx_{2n}, Ax_{2n−1}) ≤ kg(d(Sx_{2n−1}, T x_{2n}), d(Sx_{2n−1}, Ax_{2n−1}), d(T x_{2n}, Bx_{2n}), d(Sx_{2n−1}, Bx_{2n}), d(T x_{2n}, Ax_{2n−1})) ≤ kL,

where

L = g(d(Sx_{2n−1}, T x_{2n}), d(Sx_{2n−1}, T x_{2n}), d(T x_{2n}, Sx_{2n+1}),
d(Sx_{2n−1}, T x_{2n}) + d(T x_{2n}, Sx_{2n+1}), 0).

By Lemma 2.1 we have that kg ∈ G and so we deduce that

d(T x_{2n}, Sx_{2n+1}) ≤ khd(Sx_{2n−1}, T x_{2n}).

In a similar way it follows that

d(T x_{2n}, Sx_{2n−1}) ≤ khd(Sx_{2n−1}, T x_{2n−2}),

and so we deduce that

d(T x_{2n}, Sx_{2n+1}) ≤ (kh)^{2n}d(Sx_{1}, T x_{0}).  (2.4)

and

d(T x_{2n}, Sx_{2n−1}) ≤ (kh)^{2n}d(Sx_{1}, T x_{0}).  (2.5)

for every n ≥ 1.

We deduce that {Sx_{2n−1}}_{n∈N} is a Cauchy sequence in the complete metric space X and so lim_{n→∞} Sx_{2n−1} = z for some z in X. By (2.5), we also have lim_{n→∞} T x_{2n} = z.

We shall now prove that Sz ∈ Az, i.e., z is a coincidence point of S and A. For every n ∈ N, we have

d(ST x_{2n}, Az) ≤ d(ST x_{2n}, ASx_{2n−1}) + H(ASx_{2n−1}, Az).  (2.6)

By H-continuity of A and the fact that lim_{n→∞} Sx_{2n−1} = z, we have

lim_{n→∞} H(ASx_{2n−1}, Az) = 0.

Since S and A are compatible mappings and lim_{n→∞} Sz_{n} = lim_{n→∞} y_{n} = z, where y_{n} = T x_{2n} ∈ Ax_{2n−1} and z_{n} = x_{2n−1}, we have

lim_{n→∞} d(ST x_{2n}, ASx_{2n−1}) = lim_{n→∞} d(Sy_{n}, ASz_{n}) = 0.

Thus the inequality (2.6) yields lim_{n→∞} d(ST x_{2n}, Az) = 0 and so from

d(Sz, Az) ≤ d(Sz, ST x_{2n}) + d(ST x_{2n}, Az),

as n → ∞ and the continuity of S, it follows that d(Sz, Az) = 0 . Hence Sz ∈ Az, Since Az is a closed subset of SX ∩ T X. We can similarly prove that T z ∈ Bz, that is z is a coincidence point of T and B. This completes the proof.

Now , we give two examples to support our generalization.

Example 2.1. Let X = [1, ∞) be with the Euclidean metric d. Define A, B : X → CB(SX ∩ T X) and S, T : X → X by

Ax = [1, x^{2}], Bx = [1, x^{4}]
Sx = 2x^{4} − 1, T x = 2x^{8} − 1

for all x ∈ X, n ≥ 1.

Clearly A, B, S and T are continuous and A(X) = B(X) = S(X) = T (X) = X. Let \{x_{n}\} and \{y_{n}\} be two sequences in X defined by x_{n} = 1 + \frac{1}{n}, y_{n} = 1 for n = 1, 2, ..., we then have

lim_{n→∞} Sx_{n} = lim_{n→∞} y_{n} = 1 ∈ X, y_{n} ∈ Ax_{n}.

Moreover,

lim_{n→∞} H(SAx_{n}, ASx_{n}) = lim_{n→∞} |(2x_{n}^{4} − 1)^{2} − 2x_{n}^{8} + 1|
= 0 iff x_{n} → 1 as n → ∞,

and since d(Sy_{n}, ASx_{n}) ≤ H(SAx_{n}, ASx_{n}), we have

lim_{n→∞} d(Sy_{n}, ASx_{n}) = 0.

Therefore S and A are compatible . Similarly, one can show that B and T are compatible.

Assuming g ∈ G as

g(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}) = \frac{t_{1}}{2}

for every t_{i} ∈ R^{+}, i = 1, 2, 3, 4, 5.

We have for any x, y ∈ X

H(Ax, By) = |x^{2} − y^{4}|

≤ |x^{2} − y^{4}||x^{2} + y^{4}|

= |x^{4} − y^{8}|

= \frac{1}{2}|(2x^{4} − 1) − (2y^{8} − 1)|

= \frac{d(Sx, T y)}{2}

= g(d(Sx, T y), d(Sx, Ax), d(T y, By), d(Sx, By), d(T y, Ax)).

All the assumptions of Theorem 2.2 are verified and 1 is the coincidence point of S and A and of T and B.

Note that Theorem 2.1 is not applicable even if A = B and S = T because A and S are not weakly commuting mappings at x = 2.

Ax = [0, x^{2}], Bx = [0, 3x^{2}]
Sx = 4x^{4}, T x = 36x^{4}

for all x ∈ X.

Clearly A, B, S and T are continuous and A(X) = B(X) = S(X) = T (X) = X. Let \{x_{n}\} and \{y_{n}\} be two sequences in X defined by x_{n} = \frac{1}{n^{2}} , y_{n} = \frac{1}{n^{3}} for n = 1, 2, ..., we then have

lim_{n→∞} Sx_{n} = lim_{n→∞} y_{n} = 0 ∈ X, y_{n} ∈ Ax_{n}.

Moreover,

lim_{n→∞} H(SAx_{n}, ASx_{n}) = lim_{n→∞} |4x_{n}^{4} − 16x_{n}^{4}|
= 0 iff x_{n} → 0 as n → ∞,

and since d(Sy_{n}, ASx_{n}) ≤ H(SAx_{n}, ASx_{n}), we have

lim_{n→∞} d(Sy_{n}, ASx_{n}) = 0.

Therefore S and A are compatible . Similarly, one can show that B and T are compatible.

Assuming g ∈ G as

g(t_{1}, t_{2}, t_{3}, t_{4}, t_{5}) = \frac{t_{1}}{4}

for every t_{i} ∈ R^{+}, i = 1, 2, 3, 4, 5.

We have for any x, y ∈ X

H(Ax, By) = |x^{2} − 3y^{2}|

≤ |x^{2} − 3y^{2}||x^{2} + 3y^{2}|

= |x^{4} − 9y^{4}|

= \frac{1}{4}|4x^{4} − 36y^{4}|

= \frac{d(Sx, T y)}{4}

= g(d(Sx, T y), d(Sx, Ax), d(T y, By), d(Sx, By), d(T y, Ax)).

All the assumptions of Theorem 2.2 are verified and 0 is the coincidence point of S and A and of T and B.

Note that Theorem 2.1 is not applicable even if A = B and S = T because A and S are not weakly commuting mappings at x = 2.

Corollary 2.1 ([9]). Let (X, d) be a complete metric space, S and T continuous mappings from X into X, A and B closed mappings from X into CB(SX ∩ T X) such that ASx = SAx, BT x = T Bx for every x ∈ X and

H(Ax, By) ≤ q max\{d(Sx, T y), d(Sx, Ax), d(T y, By), \frac{1}{2}[d(Sx, By) + d(T y, Ax)]\},

for every x, y ∈ X, where q ∈ (0, 1). Then there exists a sequence {x_{n}}_{n∈N} in X such that

1. For every n ∈ N, Sx_{2n+1} ∈ Bx_{2n}, T x_{2n} ∈ Ax_{2n−1};

2. There exists z = lim_{n→∞} T x_{2n} = lim_{n→∞} Sx_{2n+1};

3. Sz ∈ Az, T z ∈ Bz.

Corollary 2.2 ([3]). Let (X, d) be a complete metric space, S and T continuous mappings from X into X, A a closed mapping from X into CB(SX ∩ T X) such that ASx = SAx, AT x = T Ax for every x ∈ X and

H(Ax, Ay) ≤ qd(Sx, T y),

for every x, y ∈ X, where q ∈ (0, 1). Then there exists a sequence {x_{n}}_{n∈N} in X such that

1. For every n ∈ N, Sx_{2n+1} ∈ Ax_{2n}, T x_{2n} ∈ Ax_{2n−1};

2. There exists z = lim_{n→∞} T x_{2n} = lim_{n→∞} Sx_{2n+1};

3. Sz ∈ Az, T z ∈ Az.

Corollary 2.3 ([7]). Let (X, d) be a complete metric space, A : X → CB(X) and S a continuous mapping from X into X, and

H(Ax, Ay) ≤ hd(Sx, Sy),

for every x, y ∈ X, where 0 ≤ h < 1 and ASx = SAx for every x ∈ X. If A(X) ⊆ S(X), then there exists t ∈ X such that St ∈ At.

Corollary 2.4 ([11]). Let (X, d) be a complete metric space, S and T continuous mappings from X into X, A and B closed mappings from X into CB(SX ∩ T X) such that ASx = SAx, BT x = T Bx for every x ∈ X and

H^{2}(Ax, By) ≤ q max{d(Sx, Ax)d(T y, By), d(Sx, By)d(T y, Ax),

\frac{1}{2}[d(Sx, Ax)d(Sx, By) + d(T y, Ax)d(T y, By)], d^{2}(Sx, T y)\},

for every x, y ∈ X, where q ∈ (0, 1). Then there exists a sequence {x_{n}}_{n∈N} in X such that

1. For every n ∈ N, Sx_{2n+1} ∈ Bx_{2n}, T x_{2n} ∈ Ax_{2n−1};

2. There exists z = lim_{n→∞} T x_{2n} = lim_{n→∞} Sx_{2n+1};

3. Sz ∈ Az, T z ∈ Bz.

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