Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 01, 1996, pp. 51–61. Submitted: February 11, 1996. Published: June 15, 1996.

COMMON FIXED POINTS FOR δ-COMPATIBLE MAPPINGS

R.A.RASHWAN AND M.A.AHMED

Abstract. In this paper, using the condition of δ-compatibility between a set-valued mapping and a single-valued mapping due to Jungck and Rhoades [6], we generalize a common fixed point theorem of Khan and Kubiaczyk [7] in a complete metric space. Their Theorem extends well known results due to Chang [1], Khan et al. [8], and Singh-Whitfield [13]. We give an example to support our extension.

(1991) A.M.S. Subject Classification Code. 54H25.

Key words and phrases. Common fixed points, δ-compatible mappings, complete metric spaces.

1. Basic Preliminaries.

Let (X,d) be a metric space, and B(X) be the set of all nonempty, bounded subsets of X.

As in [3,4], let δ(A, B) be the function defined by

δ(A, B) = sup{d(a, b) : a ∈ A, b ∈ B},

for all A, B in B(X).

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) = d(a, b).

It follows immediately from the definition that

δ(A, B) = δ(B, A) ≥ 0,
δ(A, B) ≤ δ(A, C) + δ(C, B),
δ(A, B) = 0 iff A = B = {a},
δ(A, A) = diamA,

for all A, B, C in B(X).

Typeset by AMS-TEX

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

E-mail Address: Assiut@frcu.enu.eg

Copyright \begin{matrix}c \\ ◯ \end{matrix}1996 by Cameron University

Definition 1.1 [3]. A sequence {A_{n}} of subsets of X is said to be convergent to a subset A of X if

(i) given a ∈ A, there is a sequence {a_{n}} in X such that a_{n} ∈ A_{n} for n=1,2, ...., and a_{n} converges to a.

(ii) given ϵ > 0 there exists a positive integer N such that A_{n} ⊆ A_{ϵ} for n > N where A_{ϵ} is the the union of all open spheres with centers in A and radius ϵ.

Lemma 1.1 [3,4]. If {A_{n}} and {B_{n}} are sequences in B(X) converging to A and B in B(X) respectively, then the sequence {δ(A_{n}, B_{n})} converges to δ(A, B).

Lemma 1.2 [4]. Let {A_{n}} be a sequence in B(X) and y be a point of X such that δ(A_{n}, y) → 0. Then the sequence {A_{n}} converges to the set {y} in B(X).

Definition 1.2 [4,12]. A set-valued mapping F of X into B(X) is said to be continuous at x ∈ X if the sequence {F x_{n}} in B(X) converges to Fx whenever {x_{n}} is a sequence in X converging to x in X. F is said to be continuous on X if it is continuous at every point of X.

The following Lemma was proved in [4].

Lemma 1.3. Let {A_{n}} be a sequence of nonempty subsets of X and let z ∈ X such that

lim_{n→∞} a_{n} = z,

z being independent of the particular choice of each a_{n} ∈ A_{n}. If a selfmap I of X is continuous, then Iz is the limit of the sequence {IA_{n}}.

Definition 1.3 [12]. The mappings F : X → B(X) and I : X → X are said to be weakly commuting on X if IF x ∈ B(X) and

δ(F Ix, IF x) ≤ max{δ(Ix, F x), diamIF x}  (1.1)

for all x in X.

Note that if F is a single-valued mapping, then the set {IF x} consists of a single point and therefore diam IFx=0 for all x ∈ X and condition (1.1) therefore reduces to the condition given by Sessa [11], that is

d(F Ix, IF x) ≤ d(Ix, F x)  (1.2)

for all x in X.

Two commuting mappings F and I clearly commute but two weakly commuting F and I do not necessarily commute as shown in [12].

In a recent paper Jungck [5] made an extension of weak commutativity in the following way

Definition 1.4.

Two single-valued mappings f and g of a metric space (X, d) into itself. are compatible if lim_{n→∞} d(fgx_{n}, gfx_{n}) = 0 whenever {x_{n}} is a sequence in X such that

lim_{n→∞} fx_{n} = lim_{n→∞} gx_{n} = t

for some z in X.

It can be seen that two weakly commuting mappings are compatible but the converse is false. Examples supporting this fact can be found in [5].

In [6], Jungck and Rhoades extended the definition 1.4 of compatibility to a set-valued setting as follows.

Definition 1.5.

The mappings I : X → X and F : X → B(X) are weakly δ-compatible if they commute at coincidence points, i.e., for each point u such that Fu=Iu, we have FIu=IFu. ( Note that the equation Fu=Iu implies that Fu is a singleton. )

Definition 1.6. The mappings I : X → X and F : X → B(X) are δ-compatible if lim_{n→∞} δ(F Ix_{n}, IF x_{n}) = 0 whenever {x_{n}} is a sequence in X such that IF x_{n} ∈ B(X),

F x_{n} → {t} Ix_{n} → t

for some t in X.

Two weakly commuting mappings are δ-compatible but the converse is not true as is shown in the following example.

Example 1.1. Let X = [0, ∞) with the Euclidean metric d and define

Ix = x^{3} + 2x^{2}, F x = [0, x^{2}]

for all x in X. We have

Ix_{n} → 0, F x_{n} → {0} if x_{n} → 0

and

δ(F Ix_{n}, IF x_{n}) → 0 as x_{n} → 0.

Also IF X_{n} ∈ B(X), we see that I and F are δ-compatible. But I and F are not weakly commuting, hence are not commuting because for x=1, we have

δ(F I1, IF 1) = 9 > 3 = max{δ(I1, F 1), diamIF 1},

and

F I1 = [0, 9] ≠ [0, 3] = IF 1.

The following proposition due to Jungck and Rhoades [6] is useful in the sequel.

Proposition 1.1. Let (X,d) be a complete metric space. Suppose I : X → X and F : X → B(X), and (I,F) is δ-compatible,

(1) Suppose that the sequence {F x_{n}} converges to {t} and {Ix_{n}} converges to t. If I is continuous, then F Ix_{n} → {It}.

(2) If {It} = F t for some t ∈ X, then FIt=IFt.

Throughout this paper, (X,d) stands for a complete metric space. Further Φ is the set of functions ϕ : [0, ∞)^{5} ⟶ [0, ∞) which satisfies the following conditions

(i) ϕ is upper semi-continuous from the right and non-decreasing in each coordinate variable,

(ii) For each t > 0 ,

Ψ(t) = max{ϕ(t, t, t, t, t), ϕ(t, t, t, 2t, 0), ϕ(t, t, t, 0, 2t)} < t.

The following Lemma is due to Matkowski [9].

Lemma 1.4. For every t > 0,

Ψ(t) < t iff lim_{n→∞} Ψ^{n}(t) = 0.

M.S.Khan and I.Kubiaczyk [7] proved the following Theorem.

Theorem 1.1. Let I : X → X and F, G : X → B(X) such that (I) I, F and G are continuous,

F X ⊂ IX and GX ⊂ IX,  (1.3)
IF = F I and IG = GI,  (1.4)

For all x, y in X

δ(F x, Gy) ≤ ϕ(d(Ix, Iy), δ(Ix, F x), δ(Iy, Gy), D(Ix, Gy), D(Iy, F x)), (1.5)

where ϕ satisfies (i) and

ϕ(t, t, t, at, bt) < t,

for each t > 0 , and a ≥ 0 , b ≥ 0 with a + b ≤ 2. Then I, F and G have a unique common fixed point u such that u = Iu ∈ F u ∩ Gu.

Now there arises a natural question. Is it possible to replace the commutativity of mappings in Theorem 1.1 by the δ-compatibility of a pair of mappings (Definition 1.6) ? The answer is in the affirmative.

2.Main results.

As an improvement of Theorem 1.1 we have the following.

Theorem 2.1. Let I, J : X → X and F, G : X → B(X) be mappings such that

F X ⊂ JX and GX ⊂ IX,  (2.1)

δ(F x, Gy) ≤ ϕ(d(Ix, Jy), δ(Ix, F x), δ(Jy, Gy), D(Ix, Gy), D(Jy, F x)), (2.2)

for all x, y ∈ X and ϕ ∈ Φ , either

(II) {F, I} are δ -compatible, F or I is continuous and (G,J) are weakly δ - compatible, or

(III){G, J} are δ -compatible, G or J is continuous and {F, I} are weakly δ -compatible.

Then I, J, F and G have a unique common fixed point u. Moreover F u = Gu = {u} is the unique common fixed point of F and I and of G and J.

Proof. Let x_{0} ∈ X be an arbitrary point in X. By (2.1) we choose a point x_{1} in X such that Jx_{1} ∈ F x_{0} = Z_{0} and for this point x_{1} there exists a point x_{2} in X such that Ix_{2} ∈ Gx_{1} = Z_{1} , and so on. Continuin

g in this manner we can define a sequence {x_{n}} as follows:

Jx_{2n+1} ∈ F x_{2n} = Z_{2n+1}, Ix_{2n+2} ∈ Gx_{2n+1} = Z_{2n+2}.  (2.3)

For simplicity, we Put V_{n} = δ(Z_{n}, Z_{n+1}), for n=0, 1, 2 ... By (2.2) and (2.3), we have

V_{2n+1} = δ(Z_{2n+1}, Z_{2n+2}) = δ(F x_{2n}, Gx_{2n+1})
≤ ϕ(d(Ix_{2n}, Jx_{2n+1}), δ(Ix_{2n}, F x_{2n}), δ(Jx_{2n+1}, Gx_{2n+1}),
D(Ix_{2n}, Gx_{2n+1}), D(Jx_{2n+1}, F x_{2n}))
≤ ϕ(V_{2n}, V_{2n+1}, V_{2n}, V_{2n} + V_{2n+1}, 0).

If V_{2n+1} > V_{2n} , then we have

V_{2n+1} ≤ ϕ(V_{2n+1}, V_{2n+1}, V_{2n+1}, 2V_{2n+1}, 0) ≤ Ψ(V_{2n+1}) < V_{2n+1},

which ia a contradiction. Thus,

V_{2n+1} ≤ ϕ(V_{2n}, V_{2n}, V_{2n}, 2V_{2n}, 0) ≤ Ψ(V_{2n}).  (2.4)

Similarly, we have

V_{2n+2} ≤ Ψ(V_{2n+1}).  (2.5)

It follows from (2.4) and (2.5) that

V_{n} ≤ ψ(V_{n−1}) ≤ ... ≤ Ψ^{n}(V_{0}),  (2.6)

where

V_{0} = δ(Z_{0}, Z_{1}) = δ(F x_{0}, Gx_{1}) ≤ ϕ(V_{0}, V_{0}, V_{0}, 2V_{0}, 0).

By (2.6) and Lemma 1.4 we obtain

lim_{n→∞} V_{n} = 0.  (2.7)

Let z_{n} be an arbitrary point in Z_{n} for n=0, 1, 2, ... Then

lim_{n→∞} d(z_{n}, z_{n+1}) ≤ lim_{n→∞} δ(Z_{n}, Z_{n+1}) = 0.

In order to show that {z_{n}} is a Cauchy sequence, it is sufficient to show that {z_{2n}} is a Cauchy sequence. Suppose that {z_{2n}} is not a Cauchy sequence. Then there is an ϵ > 0 such that, for each even integer 2k, there exist even integers 2m(k) and 2n(k) such that

δ(Z_{2m(k)}, Z_{2n(k)}) > ϵ  (2.8)

for m(k) > n(k) ≥ k .

For each even integer 2k, let 2m(k) be the least even integer exceeding 2n(k) satisfying (2.8), that is

δ(Z_{2n(k)}, Z_{2m(k)−2}) ≤ ϵ and δ(Z_{2n(k)}, Z_{2m(k)}) > ϵ  (2.9)

Then for each even integer 2k,

ϵ < δ(Z_{2n(k)}, Z_{2m(k)}) ≤ δ(Z_{2n(k)}, Z_{2m(k)−2}) + V_{2m(k)−2} + V_{2m(k)−1}.

The foregoing inequality implies

lim_{n→∞} δ(Z_{2n(k)}, Z_{2m(k)}) = ϵ.  (2.10)

Also by the triangle inequality, we have

|δ(Z_{2n(k)}, Z_{2m(k)−1}) − δ(Z_{2n(k)}, Z_{2m(k)})| ≤ V_{2m(k)−1},

and

|δ(Z_{2n(k)+1}, Z_{2m(k)−1}) − δ(Z_{2n(k)}, Z_{2m(k)})| ≤ V_{2m(k)−1} + V_{2n(k)}.

Using (2.10) we get

δ(Z_{2n(k)}, Z_{2m(k)−1}) → ϵ  (2.11)

and

δ(Z_{2n(k)+1}, Z_{2m(k)−1}) → ϵ.  (2.12)

From (2.2) and (2.3), we have

δ(Z_{2n(k)}, Z_{2m(k)}) ≤ V_{2n(k)} + δ(F x_{2n(k)}, Gx_{2m(k)−1})

≤ V_{2n(k)} + ϕ(d(Ix_{2n(k)}, Jx_{2m(k)−1}), δ(Ix_{2n(k)}, F x_{2n(k)}), δ(Jx_{2m(k)−1}, Gx_{2m(k)−1}), D(Ix_{2n(k)}, Gx_{2m(k)−1}), D(Jx_{2m(k)−1}, F x_{2n(k)})) ≤ V_{2n(k)} + ϕ(δ(Z_{2n(k)}, Z_{2m(k)−1}), V_{2n(k)}, V_{2m(k)−1}, δ(Z_{2n(k)}, Z_{2m(k)}), δ(Z_{2m(k)−1}, Z_{2n(k)+1})).

Letting k → ∞ , we obtain

ϵ ≤ ϕ(ϵ, 0, 0, ϵ, ϵ) ≤ Ψ(ϵ) < ϵ,

which is a contradiction. Therefore the sequence {z_{n}} is a Cauchy sequence in X and it has a limit u in X. So the sequences {Ix_{2n}} and {Jx_{2n+1}} converge to u and further, the sequences of sets {F x_{2n}} and {Gx_{2n+1}} converge to the set {u} .

Now supose that I is continuous. Then

I^{2}x_{2n} → Iu, IF x_{2n} → {Iu}

by Lemma 1.3. But I and F are δ-compatible. Therefore F Ix_{2n} ⟶ {Iu} by proposition 1.1 (1).

Using inequality (2.2), we have

δ(F Ix_{2n}, Gx_{2n+1}) ≤ ϕ(d(I^{2}x_{2n}, Jx_{2n+1}), δ(I^{2}x_{2n}, F Ix_{2n}),
δ(Jx_{2n+1}, Gx_{2n+1}), D(I^{2}x_{2n}, Gx_{2n+1}), D(Jx_{2n+1}, F Ix_{2n})).

for n ∈ N . As n → ∞ we obtain by Lemma 1.1

d(Iu, u) ≤ ϕ(d(Iu, u), 0, 0, d(Iu, u), d(Iu, u)) ≤ Ψ(d(Iu, u)) < d(Iu, u)

which is a contradiction. Thus Iu=u.

Further

δ(F u, Gx_{2n+1}) ≤ ϕ(d(Iu, Jx_{2n+1}), δ(Iu, F u),
δ(Jx_{2n+1}, Gx_{2n+1}), D(Iu, Gx_{2n+1}), D(Jx_{2n+1}, F u)).

Letting n ⟶ ∞, it follows by Lemma 1.1 that

δ(F u, u) ≤ ϕ(0, δ(F u, u), 0, 0, δ(u, F u))
≤ Ψ(δ(u, F u)) < δ(u, F u),

which is a contradiction. So F u = {u}.

Since F (X) ⊂ J(X) , there exists a point w ∈ X such that {u} = F u = {Jw}. We shall now show that Gw = {Jw}. From (2.2),

δ(F u, Gw) ≤ ϕ(d(Iu, Jw), δ(Iu, F u), δ(Jw, Gw), D(Iu, Gw), D(Jw, F u)),

or

δ(Jw, Gw) ≤ ϕ(δ(Jw, Gw)),

which implies that Gw = {Jw}. Thus Gw is a singleton and w is a coincidence point for G and J.

Since they are weakly δ -compatible, GJw=JGw.

It then follows that {Ju} = JGw = GJw = Gu. Using (2.2),

δ(F u, Gu) ≤ ϕ(d(Iu, Ju), δ(Iu, F u), δ(Ju, Gu), D(Iu, Gu), D(Ju, F u))

or

δ(u, Ju) ≤ ϕ(δ(u, Ju)),

which implies that u=Ju. Since {Ju} = Gu, u is a common fixed point of F,G, I and J.

The proof assuming the continuity of J is similar, as is the proof of (III).

Now suppose that F and I have a second common fixed point v such that F v = {v} = {Iv}. Then, using inequality (2.2), we obtain

d(v, u) = δ(F v, Gu) ≤ ϕ(d(v, u), 0, 0, d(v, u), d(v, u)) ≤ Ψ(d(v, u)) < d(v, u),

which is a contradiction. So v=u, and u is the unique common fixed point of F and I such that F u = {u}.

Similarly, it can be shown that u is the unique common fixed point of G and J such that Gu = {u}.

Remark. In Theorem 2.1, if F and G are single-valued mappings, we obtain Theorem 2.2 of [10].

Now , we give an example to support our extension.

Example. Let X = [0, ∞) with the Euclidean metric d. Define F, G : X → B(X) and I, J : X → X by

F x = [0, \frac{x^{4}}{2} ], Gx = [0, \frac{x^{2}}{2} ]

Ix = x^{4} + 2x^{2}, Jx = x^{8} + x^{4} + x^{2}

for all x, y in X. Now:

F (X) ⊂ I(X), G(X) ⊂ J(X),

and F, G, I and J are continuous. Moreover, for any sequence {x_{n}} in X, we have

Ix_{n} → 0, F x_{n} → {0} as x_{n} ⟶ 0,

IF x_{n} ∈ B(X),

δ(F Ix_{n}, IF x_{n}) = max\{\frac{(x_{n}^{4} + 2x_{n}^{2})^{4}}{2} , (\frac{x_{n}^{4}}{2} )^{2} + 2(\frac{x_{n}^{4}}{2} )^{2}\} → 0 as x_{n} → 0,

thus F and I are δ -compatible . Similarly, one can show that G and J are δ - compatible.

Assuming ϕ ∈ Φ as

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

for every t_{i} ∈ R^{+}, i=1,2,3,4,5.

We have for any x, y ∈ X

δ(F x, Gy) = max\{\frac{x^{4}}{2} , \frac{y^{2}}{2} \}

= \frac{1}{2}max\{x^{4}, y^{2}\}

≤ \frac{1}{2}max\{x^{4} + 2x^{2}, y^{8} + y^{4} + y^{2}\}

= \frac{1}{2}max\{δ(Ix, F x), δ(Jy, Gy)\}

= ϕ(d(Ix, Jy), δ(Ix, F x), δ(Jy, Gy), δ(Ix, Gy), δ(Jy, F x)).

Since ϕ obviously satisfies the conditions (i) and (ii), all the assumptions of Theorem 2.1 are verified and 0 is the unique common fixed point of F, G, I and J.

Note that Theorem 1.1 is not applicable even if F=G and I=J because F and I are not weakly commuting mappings at x=2, for

δ(F I1, IF 1) = \frac{81}{2} > 3 = max\{δ(F 1, I1), diamIF 1\},

δ(GJ1, JG1) = \frac{9}{2} > 3 = max\{δ(G1, J1), diamJG1\},

F I1 ≠ IF 1, GJ1 ≠ JG1.

3. A common fixed point theorem in compact metric space.

Mativated by the paper [2], we have the following.

Theorem 3.1. Let (X,d) be a compact metric space and let I, J be functions from X into X, F, G : X → B(X) be two set-valued functions with F X ⊂ JX and GX ⊂ IX such that for all x, y in X and ϕ ∈ Φ , the inequality

δ(F x, Gy) < ϕ(d(Ix, Jy), δ(Ix, F x), δ(Jy, Gy), D(Ix, Gy), D(Jy, F x)),

holds whenever the right hand side of (3.1) is positive. If the pairs (F,I) and (G,J) are δ - compatible, and if the functions F and I are continuous, then there is a unique fixed point u in X such that

F u = Gu = {u} = {Iu} = {Ju}.

Proof. Let η = inf_{x∈X} δ(Ix, F x). Since X is compact metric space, there is a convergent sequence {x_{n}} with limit x_{0} in X such that

δ(Ix_{n}, F x_{n}) → η as n → ∞.

Since

δ(Ix_{0}, F x_{0}) ≤ d(Ix_{0}, Ix_{n}) + δ(Ix_{n}, F x_{n}) + δ(F x_{n}, F x_{0}),

then by the continuity of F and I and lim_{n→∞} x_{n} = x_{0} we get δ(Ix_{0}, F x_{0}) ≤ η and thus δ(Ix_{0}, F x_{0}) = η.

Since F X ⊂ JX , then there exists a point y_{0} in X with Jy_{0} ∈ F x_{0} and d(Ix_{0}, Jy_{0}) ≤ η.

If η > 0 then

δ(Jy_{0}, Gy_{0}) ≤ δ(F x_{0}, Gy_{0})
< ϕ(d(Ix_{0}, Jy_{0}), δ(Ix_{0}, F x_{0}),
δ(Jy_{0}, Gy_{0}), D(Ix_{0}, Gy_{0}), D(Jy_{0}, F x_{0}))
≤ ϕ(η, η, δ(Jy_{0}, Gy_{0}), η + δ(Jy_{0}, Gy_{0}), 0).

If δ(Jy_{0}, Gy_{0}) > η in the above inequality, then we have

δ(Jy_{0}, Gy_{0})
< ϕ(δ(Jy_{0}, Gy_{0}), δ(Jy_{0}, Gy_{0}), δ(Jy_{0}, Gy_{0}), 2δ(Jy_{0}, Gy_{0}), 0)
≤ Ψ(δ(Jy_{0}, Gy_{0})) < δ(Jy_{0}, Gy_{0}),

which is a contradiction. Thus δ(Jy_{0}, Gy_{0}) ≤ η.

Since GX ⊂ IX, then there is a point z_{0} in X such that Iz_{0} ∈ Gy_{0} and d(Iz_{0}, Jy_{0}) ≤ η. Hence we have

η ≤ δ(Iz_{0}, F z_{0}) ≤ δ(F z_{0}, Gy_{0})
< ϕ(d(Iz_{0}, Jy_{0}), δ(Iz_{0}, F z_{0}), δ(Jy_{0}, Gy_{0}), D(Iz_{0}, Gy_{0}), D(Jy_{0}, F z_{0}))
≤ ϕ(η, δ(Iz_{0}, F z_{0}), η, 0, η + δ(Iz_{0}, F z_{0})).  (3.1)

If δ(Iz_{0}, F z_{0}) > η in the last inequality, then we get

η ≤ δ(Iz_{0}, F z_{0})
< ϕ(δ(Iz_{0}, F z_{0}), δ(Iz_{0}, F z_{0}), δ(Iz_{0}, F z_{0}), 0, 2δ(Iz_{0}, F z_{0}))
≤ Ψ(δ(Iz_{0}, F z_{0})) < δ(Iz_{0}, F z_{0}))

a contradiction. Thus δ(Iz_{0}, F z_{0}) ≤ η.

Substituting into (3.1) yields

η ≤ ϕ(η, η, η, 0, 2η) ≤ Ψ(η) < η,

a contradiction. Therefore η = 0 and hence

Ix_{0} = Jy_{0} = Iz_{0} = {Gy_{0}} = {F z_{0}}.

If δ(Jy_{0}, Gy_{0}) > 0, then

δ(Jy_{0}, Gy_{0}) ≤ δ(F x_{0}, Gy_{0})
< ϕ(d(Ix_{0}, Jy_{0}), δ(Ix_{0}, F x_{0}), δ(Jy_{0}, Gy_{0}), D(Ix_{0}, Gy_{0}), D(Jy_{0}, F x_{0}))
= ϕ(0, 0, δ(Jy_{0}, Gy_{0}), 0, 0) ≤ Ψ(δ(Jy_{0}, Gy_{0})) < δ(Jy_{0}, Gy_{0}),

a contradiction. Therefore

{Gy_{0}} = {Jy_{0}} = {F x_{0}} = {Ix_{0}}.

Since F and I are δ -compatible and δ(Ix_{0}, F x_{0}) = 0 , we get δ(F Ix_{0}, IF x_{0}) = 0, and we get

δ(F^{2}x_{0}, I^{2}x_{0}) = δ(F Ix_{0}, I^{2}x_{0}) = δ(IF x_{0}, I^{2}x_{0}) = δ(I^{2}x_{0}, I^{2}x_{0}) = 0.

We have

F^{2}x_{0} = F Ix_{0} = IF x_{0} = {I^{2}x_{0}}.

If I^{2}x_{0} ≠ Ix_{0} then we have

d(I^{2}x_{0}, Ix_{0}) = δ(F^{2}x_{0}, I^{2}x_{0})

< ϕ(δ(IF x_{0}, Jy_{0}), δ(IF x_{0}, F^{2}x_{0}), δ(Jy_{0}, Gy_{0}), D(IF x_{0}, Gy_{0}), D(Jy_{0}, F^{2}x_{0})

= ϕ(d(I^{2}x_{0}, Ix_{0}), 0, 0, d(I^{2}x_{0}, Ix_{0}), d(I^{2}x_{0}, Ix_{0})

≤ Ψ(d(I^{2}x_{0}, Ix_{0}) < d(I^{2}x_{0}, Ix_{0}),

which is a contradiction, and so I^{2}x_{0} = Ix_{0} . Hence

F Ix_{0} = {Ix_{0}} = {I^{2}x_{0}}.

Similarly, we have

GJy_{0} = {Jy_{0}} = {J^{2}y_{0}}.

Let u = Ix_{0} = Jy_{0}. Then

F u = {u} = {Iu} = {Ju} = Gu.

Suppose that the point y in X is a common fixed point of F,G, I and J with y ≠ u

d(u, y) = δ(u, y) = δ(F u, Gy)
< ϕ(d(Iu, Jy), δ(Iu, F u), δ(Jy, Gy), D(Iu, Gy), D(Jy, F u))
= ϕ(d(u, y), 0, 0, d(u, y), d(u, y)) ≤ Ψ(d(u, y)) < d(u, y),

whence u is the unique common fixed point of F, G, I and J.

Ackowledgements. We wish to thank Prof. B. E. Rhoades for helpful comments and a critical reading of manuscript.

references

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