Electronic Journal: Southwest Journal of Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 1083-0464 Vol. No. 01, 1996, pp. 6–10. Submitted: March 1, 1996. Published: July 29, 1996.

A COMMON FIXED POINT THEOREM

IN COMPLETE METRIC SPACES

R.A.RASHWAN AND A.M.Sadeek

Abstract. In this note we establish a common fixed point theorem in complete metric spaces which generalizes some results in [2].

(1991) A.M.S. Subject Classification Codes. 47H10.

Key Words and Phrases. Common fixed points, complete metric spaces-generalized contractive conditions.

1. INTRODUCTION

Delbosco [1] and Skof [3]have established fixed point theorems for self - maps of complete metric spaces by altering the distances between the points with the use of a function ϕ : [0, ∞) → [0, ∞) satisfying the following properties:

(i) ϕ : [0, ∞) → [0, ∞) is continuous and strictly increasing in R^{+};

(ii) ϕ(t) = 0 iff t = 0 ;

(iii) ϕ(t) ≥ Mt^{u} for every t > 0, u > 0 are constants.

In 1977, F. Skof [3] gave the following theorem

THEOREM A. Let T be a self-map of a complete metric space (X, d) and ϕ ∈ Φ such that for every x, y ∈ X,

(I) ϕ(d(T x, T y)) ≤ aϕ(d(x, y)) + bϕ(d(x, T x)) + cϕ(d(y, T y)),

where a, b and c are three nonnegative constants satisfying a + b + c < 1. Then T has a unique fixed point.

In [1], the author has considered functions ϕ ∈ Φ such that ϕ(t) = t^{n}, n ∈ N for every t ≥ 0. Clearly, ϕ is not a metric since ϕ has only a argument.

In 1984, M.S.Khan et al. [2] generalized Theorem (A) by using a much extensive condition than (I) and removed the hypothesis (iii) which seems superfluous. They proved the following theorem:

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

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

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

THEOREM B. Let T be a self-map of a complete metric space (X, d) and ϕ satisfying (i) and (ii). Furthermore, let a, b, c be three decreasing functions from R^{+} 0 into [0, 1) such that

a(t) + 2b(t) + c(t) < 1

for every t > 0.

Suppose that T satisfies the following condition:

(II) ϕ(d(T x, T y)) ≤ a(d(x, y))ϕ(d(x, y)) + b(d(x, y))[ϕ(d(x, T x)) + ϕ(d(y, T y))]
+ c(d(x, y)) min{ϕ(d(x, T y)), ϕ(d(y, T x))},

where x, y ∈ X and x ≠ y. Then T has a unique fixed point.

In this note we shall establish a common fixed point theorem for a pair of mappings S and T which satisfying the contractive defintion corresponding to (II), i.e., for all x, y ∈ X with x ≠ y

(III) ϕ(d(Sx, T y)) ≤ a(d(x, y))ϕ(d(x, y)) + b(d(x, y))[ϕ(d(x, Sx)) + ϕ(d(y, T y))]

+ c(d(x, y)) min{ϕ(d(x, T y)), ϕ(d(y, Sx))},

A COMMON FIXED POINT THEOREM

We establish the following:

THEOREM 2.1. Let S and T be self-mappings on a complete metric space (X, d) and ϕ satisfying the conditions (i) and (ii). Furthermore, let a, b, c be three decreasing functions from R^{+} into [0, 1) such that

a(t) + 2b(t) + c(t) < 1

for every t > 0.

Suppose that S and T satisfy (III). Then S and T have a unique fixed point in X.

Proof. Let x_{0} be some point in X, define

x_{2n} = Sx_{2n+1}, n = 0, 1, 2, ...
x_{2n+1} = T x_{2n}, n = 0, 1, 2, ...

Let τ_{n} = d(x_{n}, x_{n+1}). We assume that τ_{n} > 0 for every n. Now, for an even integer n, we have

ϕ(τ_{n}) = ϕ(d(x_{n}, x_{n+1})) = ϕ(d(Sx_{n−1}, T x_{n}))
≤ A_{1} + A_{2} + A_{3}

where

A_{1} = a(d(x_{n−1}, x_{n}))ϕ(d(x_{n−1}, x_{n})),
A_{2} = b(d(x_{n−1}, x_{n}))[ϕ(d(x_{n−1}, Sx_{n−1})) + ϕ(d(x_{n}, T x_{n}))],
A_{3} = c(d(x_{n−1}, x_{n})) min{ϕ(d(x_{n−1}, T x_{n})), ϕ(d(x_{n}, Sx_{n−1}))}.

So

ϕ(τ_{n}) =≤ a(τ_{n−1})ϕ(τ_{n−1}) + b(τ_{n−1})[ϕ(τ_{n−1}) + ϕ(τ_{n})].

Hence we obtain:

ϕ(τ_{n}) ≤ \frac{a(τ_{n−1}) + b(τ_{n−1})}{1 − b(τ_{n−1})}ϕ(τ_{n−1}) < ϕ(τ_{n−1}).  (1)

Since ϕ is increasing, {τ_{n}} is a decreasing sequence. Put lim_{n} τ_{n} = τ with τ > 0. By (1), then τ_{n} ≥ τ implies that

ϕ(τ_{n}) ≤ \frac{a(τ) + b(τ)}{1 − b(τ)}ϕ(τ_{n−1}).

By letting n → ∞, Since ϕ is continuous, we have:

ϕ(τ) ≤ \frac{a(τ) + b(τ)}{1 − b(τ)}ϕ(τ) < ϕ(τ),

which is a contradiction, hence τ = 0.

Similarly for an odd integer n one can show that τ = 0 . Now, we wish to show that {x_{n}} is a Cauchy sequence. Since τ = 0 it suffices to prove that {x_{2n}} is a Cauchy sequence we proceed by contradiction. Then there is an ϵ > 0 such that for each even integer 2k, k = 0, 1, 2, ..., there exists two sequences {2p(k)} , {2q(k)} with 2k ≤ 2q(k) < 2p(k) for which

d(x_{2p(k)}, x_{2q(k)}) > ϵ.  (2)

For each even integer 2k, let 2p(k) be the least even integer exceeding 2q(k) and satisfy (2). Then

d(x_{2q(k)}, x_{2p(k)−2}) ≤ ϵ and d(x_{2q(k)}, x_{2p(k)}) > ϵ.

Put d_{2k} = d(x_{2q(k)}, x_{2p(k)}), then we have

ϵ < d_{2k} = d(x_{2q(k)}, x_{2p(k)})
≤ d(x_{2q(k)}, x_{2p(k)}) + d(x_{2p(k)−2}, x_{2p(k)−1}) + d(x_{2p(k)−1}, x_{2p(k)})
≤ ϵ + τ_{2p(k)−2} + τ_{2p(k)−1}.  (3)

Since {τ_{n}} converges to 0, d_{2k} → ϵ. Furthermore, by triangular inequality, it follows that

−τ_{2p(k)} − τ_{2q(k)} + d_{2k} ≤ d(x_{2q(k)+1}, x_{2p(k)+1})
≤ τ_{2p(k)} + τ_{2q(k)} + d_{2k},

and therefore the sequence {d(x_{2q(k)+1}, x_{2p(k)+1})} converges to ϵ.

Moreover, by triangle inequality we have

d(x_{2p(k)+1}, x_{2q(k)}) ≤ d(x_{2p(k)+1}, x_{2p(k)}) + d(x_{2p(k)}, x_{2q(k)})
≤ τ_{2p(k)} + d_{2k} → ϵ, as k → ∞

and

d(x_{2p(k)+2}, x_{2q(k)+1}) ≤ d(x_{2p(k)+2}, x_{2p(k)+1}) + d(x_{2p(k)+1}, x_{2q(k)+1}) ≤ A_{4}

So

A_{4} = d(x_{2p(k)+2}, x_{2q(k)+1}) ≤ τ_{2p(k)+1} + d(x_{2p(k)+1,x_{2q(k)+1})}

Hence

A_{4} → ϵ as k → ∞

From (III), we also deduce:

ϕ(d(x_{2p(k)+2}, x_{2q(k)+1})) = ϕ(d(Sx_{2p(k)+1}, T x_{2q(k)}))
≤ L_{1} + L_{2} + L_{3} ≤ L_{4} + L_{5} + L_{6}

where

L_{1} = a(d(x_{2p(k)+1}, x_{2q(k)}))ϕ(d(x_{2p(k)+1}, x_{2q(k)})),

L_{2} = b(d(x_{2p(k)+1}, x_{2q(k)}))[ϕ(d(x_{2p(k)+1}, Sx_{2p(k)+1})) + ϕ(d(x_{2q(k)}, T x_{2q(k)}))],

L_{3} = c(d(x_{2p(k)+1}, x_{2q(k)})) min{ϕ(d(x_{2p(k)+1}, T x_{2q(k)})), ϕ(d(x_{2q(k)}, Sx_{2p(k)+1}))},

L_{4} = a(τ_{2p(k)} + d_{2k})ϕ(τ_{2p(k)} + d_{2k}),
L_{5} = b(τ_{2p(k)} + d_{2k})[ϕ(τ_{2p(k)+1}) + ϕ(τ_{2q(k)})],

L_{6} = c(τ_{2p(k)} + d_{2k}) min{ϕ(d(x_{2p(k)+1}, x_{2q(k)}) + τ_{2q(k)}),
ϕ(τ_{2q(k)} + d(x_{2q(k)+1}, x_{2p(k)+2}))}.

As k → ∞, we have

ϕ(ϵ) ≤ a(ϵ)ϕ(ϵ) + c(ϵ)ϕ(ϵ)
= {a(ϵ) + b(ϵ)}ϕ(ϵ) < ϕ(ϵ).

This is a contradiction. Hence {x_{n}} is a Cauchy sequence in a complete metric space X, then there exists a point z ∈ X such that x_{n} → z as n → ∞.

Now, we shall show that Sz = z = T z.

Hence using (III), we obtain:

ϕ(d(x_{2p(k)+2}, T z)) = ϕ(d(Sx_{2p(k)+1}, T z))
≤ a(d(x_{2p(k)+1}, z))ϕ(d(x_{2p(k)+1}, z))
+ b(d(x_{2p(k)+1}, z))[ϕ(d(x_{2p(k)+1}, x_{2p(k)+2})) + ϕ(d(z, T z))]
+ c(d(x_{2p(k)+1}, z)) min{ϕ(d(x_{2p(k)+1}, T z)), ϕ(d(z, x_{2p(k)+2}))}.  (4)

Letting k → ∞, and use b < \frac{1}{2} in (4), we have

ϕ(d(z, T z)) < \frac{1}{2}ϕ(d(z, T z)).

That is, ϕ(d(z, T z)) = 0 and therefore d(z, T z) = 0, i.e., z = T z. Similarly, it can be shown that Sz = z. Hence S and T have a common fixed point z ∈ X. We claim that z is the unique common fixed point of S and T. For this, we suppose that w(w ≠ z) is another fixed point of S and T. Then

ϕ(d(z, w)) = ϕ(d(Sz, T w))
≤ a(d(z, w))ϕ(d(z, w)) + b(d(z, w))[ϕ(d(z, Sz)) + ϕ(d(w, T w))]
+ c(d(z, w)) min{ϕ(d(z, T w)), ϕ(d(w, Sz))}
≤ [a(d(z, w)) + c(d(z, w))]ϕ(d(z, Sz))
< ϕ(d(z, T w))

a contradiction. This completes the proof of the theorem.

Finally, we give an example to discuss the validity of the hypothesis of Theorem 2.1.

EXAMPLE 2.1

Let X = {1, 2, 3} be with the Euclidean metric induced by d which is defined by

d(1, 2) = d(2, 3) = 2, d(1, 3) = 1.

Suppose S and T are maps on X defined by

S1 = S2 = S3 = 1, T 1 = 1, T 2 = 3, T 3 = 1.

Then it is clear that X is complete with respect to d and S, T satisfy all the conditions of Theorem 2.1. Now setting

x = 1, y = 2, a(t) = \frac{2}{5}, b(t) = \frac{1}{5}, c(t) = \frac{1}{6},

ϕ(t) = t^{2}, t ∈ R^{+}.

We observe that S and T satisfy condition (III), and 1 is the unique common fixed point of S and T.

REFERENCES

1. D. Delbosco, Un estensione di un teorema sul punto fisso di S. Reich, Rend. Sem. Mat. Univers. Politecn. Torino 35 (1976-77), 233-239.

2. M.S.Khan, Fixed point theorems by altering distance between the points, Bull. Austral. Math. Soc. 30 (1984), 1-9.

3. F. Skof, Teorema di punti fisso per applicazioni negli spazi metrici, Atti. Accad. Aci. Torino 111 (1977), 323-329.