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Related Fixed Point Theorem for Mappings on Three Metric Spaces

L. Bishwakumar, Yumnam Rohen*

National Institute of Technology Manipur, Imphal-795001, India

Abstract: In this note we obtained a new related fixed point theorem on three metric spaces of which one is compact. Here we consider three mappings, not all of which are necessarily continuous. Our result generalizes some earlier results.

Keywords: compact metric space, related fixed point

Received July 25, 2014; Revised August 10, 2014; Accepted August 13, 2014

Citation: L. Bishwakumar, Yumnam Rohen, Related Fixed Point Theorem for Mappings on Three Metric Spaces. American Journal of Applied Mathematics and Statistics, 2 (4). 244-245. January 2014. doi:10.12691/ajams-2-4-13

1. Introduction

Related fixed point theorems on three metric spaces have been studied by several authors [1-8]. Fisher and Rao [8] proved a related fixed point theorem for three mappings on three metric spaces of which one is a compact metric space. The aim of this paper is to improve the result of Rao et. al. [6] and Fisher and Rao [8].

2. Main Results

We prove the following theorem.

2.1. Theorem. Let (X, d), (Y, ρ) and (Z, σ) be three metric spaces and T: X→Y, S:Y→Z and R:Z→X be mappings satisfying the inequalities

[picture] | (1)

[picture] | (2)

[picture] | (3)

for all x, x΄ in X , y, y΄ in Y and z, z΄ in Z. Further assume one of the following conditions:

(i) (X, d) is compact and RST is continuous.

(ii) (Y, ρ) is compact and TRS is continuous.

(iii) (Z, σ) is compact and STR is continuous.

Then RST has a unique fixed point w in X, TRS has a unique fixed point u in Y and STR has a unique fixed point v in Z. Further Su = v, Rv = w and Tw = u.

Proof: Suppose (i) holds. Define Φ(x) = d(x, RSTx) for xX. Then there exists p in X such that

Suppose that RSTRSTRSTp ≠ RSTRSTp.

Then STRSTRSTp≠STRSTp, TRSTRSTp≠TRSTp, RSTRSTp≠RSTp, STRSTp≠STp, TRSTp≠Tp, RSTp≠p.

Using (1) with x=RSTp and x΄=RSTRSTp

so that

[picture] | (4)

Using (2) with y = Tp and y΄ = TRSTp

so that

[picture] | (5)

Using (3) with z = STp and z΄ = STRSTp

so that

[picture] | (6)

From (4), (5) and (6) it follows that Φ(RSTRSTp) < Φ(RSTp), contradicting the existence of p.

Hence RSTRSTRSTp = RSTRSTp

Putting RSTRSTp = w in X, we have

Now let Tw = u in Y and Su = v in Z. Then Rv = RSu = RSTw = w and it follows that

and

To prove uniqueness, suppose RST has a second distinct fixed point w΄ in X.

Then

Using (1) with x = w and x΄ = w΄

so that

[picture] | (7)

Using (2) with y = Tw and y΄ = Tw΄

so that

[picture] | (8)

Using (3) with z = STw and z΄ = STw΄

so that

[picture] | (9)

From (7), (8) and (9), it follows that

so that w = w΄, proving the uniqueness of w.

Similarly we can show that v is the unique fixed point of STR and u is the unique fixed point of TRS.

It follows similarly that the theorem holds if (ii) or (iii) holds instead of (i).

Now we give the following example to illustrate our theorem.

Example. Let X = [0, 1], Y = (1, 2], Z = (2, 3] and let d = ρ = σ be the usual metric for the real numbers. Define T: X→Y, S: Y→Z and R: Z→X by

Here Y and Z are not compact spaces and T and R not continuous. However all the conditions of theorem 2.1 are satisfied. Clearly,

References

[1]   | Jain, R.K., Sahu, H.K. and Fisher, B. Related fixed point theorems for three metric spaces, Novi Sad J. Math. 26, 11-17, 1996.
[2]   | Jain, R.K., Sahu, H.K. and Fisher, B. A related fixed point theorem on three metric spaces, Kyungpook Math. J. 36, 151-154, 1996.
[3]   | Jain, R.K., Shrivastava, A.K. and Fisher, B. Fixed point theorems on three complete metric spaces, Novi Sad J. Math. 27, 27-35, 1997.
[4]   | Jain, S. and Fisher, B. A related fixed point theorem for three metric spaces, Hacettepe J. Math. and Stat. 31, 19-24, 2002.
[5]   | Nung, N.P. A fixed point theorem in three metric spaces, Math. Sem. Notes, Kobe Univ. 11, 77-79, 1983.
[6]   | Rao, K.P.R., Srinivasa Rao, N. and Hari Prasad, B.V.S.N. Three fixed point results for three maps, J. Nat. Phy. Sci. 18, 41-48, 2004.
[7]   | Rao, K.P.R., Hari Prasad, B.V.S.N. and Srinivasa Rao, N. Generalizations of some fixed point throrems in complete metric spaces, Acta Ciencia Indica, Vol. XXIX, M. No. 1, 31-34, 2003.
[8]   | Rao, K.P.R. and Fisher, B. A related fixed point theorem on three metric spaces, Hacettepe J. Math. and Stat., Vol. 36(2), 143-146, 2007.