Common fixed point for weakly compatible mappings satisfying a general
contractive condition of integral type in D-metric spaces
UNIVERSITATEA DIN BACĂU
STUDII ŞI CERCETĂRI ŞTIINŢIFICE
Seria: MATEMATICĂ
Nr. 15(2005), pag.39-42
COMMON FIXED POINT FOR WEAKLY COMPATIBLE MAPPINGS
SATISFYING A GENERAL CONTRACTIVE CONDITION OF
INTEGRAL TYPE IN D-METRIC SPACES
by
RENU CHUGH*, RAMESH KUMAR VATS**
and SANJAY KUMAR***
Abstract. In this paper, we analysis the existence of fixed points for weakly
compatible mappings or (coincidentally commuting mappings) defined on a
D-metric space satisfying a general contractive condition of integral type.
1.Introduction
In 1984, Dhage introduced the concept of D-metric and proved various results
using bounded D-metric space. For futher detail, see [1-4]. In 1999, Dhage
introduced the concept of coincidentally commuting mappings (weakly
compatible maps) for a pair of maps as follows
A pair of maps f,g:(X,d)→(X,d) is weakly compatible pair if they commute
at coincidence points i.e., fx = gx iff fgx = gfx.
Example 1.1.
Let X = [0, 3] be equipped with the usual metric space
d(x, y) = |x−y| . Define f, g : [0, 3] → [0, 3] by
x if x ε [0,1)
3−x
if x ε [0,1)
f(x) =
and
g(x) =
3 if x ε [1,3]
3
if x ε [1,3]
then for any x∞[1,3], fgx = gfx, showing that f, g are weakly compatible
maps on [0,3].
Key words and phrases: D-metric space, weakly compatible mappings, fixed
point.
(2000) Mathematics Subject Classification: 54H25
39
Common fixed point for weakly compatible mappings satisfying a general
contractive condition of integral type in D-metric spaces
Now we establish a fixed point theorem for a pair of weakly
compatible maps satisfying a general contractive condition of integral type in
D-metric spaces as follows:
2.Main Results.
Theorem 2.1. Let (X,D) be a complete D-metric space, f and g are weakly
compatible self maps of X satisfying the following conditions:
(2.1) f(X) ⊂ g(X)
(2.2) any one of f(X) or g(X) is complete
D ( fx, fy , fz )
∫
(2.3)
D( x, y , z )
∫
φ(t) dt ≤ c
0
φ(t) dt, for each x,y ε X, c ε [0,1)
0
where φ: R+→ R+ is a Lebesque integral mapping which is a summable, non∈
negative, and such that for each ∈ >0, ∫ φ(t) dt > 0. Then f and g have a
0
unique common fixed point.
Proof: Let xεX and, for brevity, define xn+1 = fn(x) for all n ≥ 1.
Now, for any positive integer m > n, we have
D ( xn , xn +1, xm )
∫
D ( xn −1, xn , xm −1 )
∫
φ(t) dt ≤ k
0
D ( fxn − 2 , fxn −1, fxm− 2 )
∫
φ(t) dt = k
0
φ(t) dt
0
≤k
2
D ( xn − 2 , xn −1, xm − 2 )
∫
φ(t) dt
0
...
≤k
n
D ( x0 , x1, xm − n )
∫
φ(t) dt ,
0
Proceeding limit as n→∞, we have
D ( xn , xn +1, xm )
∫
φ(t)
dt
≤
0
,
this
0
(2.4)
40
implies
D(xn,xn+1,xm)
=
0.
Common fixed point for weakly compatible mappings satisfying a general
contractive condition of integral type in D-metric spaces
Now we show that {xn} is a Cauchy sequence.Suppose that it is not.
Then there exists an ∈ > 0 and subsequence {l(p)}, {m(p)}, {n(p)} such that
l(p) < n(p) < m(p) < l(p+1) with
D(xl(p),xm(p)-1,xn(p)-1) < ∈
and
D(xn(p),xl(p),xm(p)) ≥ ∈ .
(2.5)
Using (2.4) and (2.5), we have
D(xl(p)-1,xm(p)-1,xn(p)-1) ≤D(xl(p)-1,xm(p)-1,xl(p))+D(xl(p)-1,xn(p)-1,xl(p))+D(xl(p),xm(p)1,xn(p)-1)
<∈
Now ,
∈
D ( xl ( p )−1, xm ( p )−1, xn ( q )−1)
∫
limp→∞
∫
φ(t) dt ≤
Also from (2.5),
D ( xn ( p ) , xl ( p ) , xm ( p ) )
∈
φ(t) dt ≤ k
∫ φ(t) dt ≤
∫
0
φ(t) dt
0
0
D ( xn ( p )−1, xl ( p )−1, xm ( p )−1 )
∫
0
φ(t) dt
0
∈
≤k
∫
φ(t) dt,
0
which is a contradiction. Therefore {xn} is a Cauchy sequence.
Let x0 ε X.Since f(X) ⊂ g(X), choose x1εX such that gx1 = fx0. In general,
choose xn+1 such that yn = gxn+1= fxn.
Since g(X) is complete, so there exists a point uεg(X) such that limn→∞yn
=limn→∞gxn = u.
Now we show that u is a common fixed point of f and g. Since uεg(X) so there
exists a point pεX such that gp = u.
D ( fp , fxn , fxn )
D ( fp, gp, gp)
Now from(2.3),
∫
φ(t) dt = limn→∞
0
∫
φ(t) dt
0
d ( gp , gxn , gxn )
≤ c limn→∞
∫
0
dt
which implies that fp = gp.
41
D ( fp,u ,u )
φ(t)dt =
∫
0
φ(t)
Common fixed point for weakly compatible mappings satisfying a general
contractive condition of integral type in D-metric spaces
Now we show that u = fp =gp is a common fixed point of f and g.
Since fp = gp and f and g are weakly compatible, therefore , it follows that fu
= fgp =gfp =gu.Now we show that u is a common fixed point of f and g.
D ( fu ,u ,u )
∫
D ( fu , fgp, fgp )
φ(t) dt
0
=
∫
D( gu , ggp, ggp)
φ(t) dt
0
≤ c
∫
φ(t) dt =
0
D ( gu , ggp, ggp )
c
∫
φ(t) dt,
0
this implies fu = u. Hence fu = gu = u.
Hence u is a common fixed point of f and g. Uniqueness follows easily.
References
[1] B.C.,Dhage, A study of some fixed point theorems, Ph.D. Thesis,
Marathwada University, Aurangabad (1984), India.
[2] B.C.,Dhage, Generalized metric spaces and mappings with fixed point,
Bull. Calcutta Math. Soc., 84(1992), 329-336.
[3] B.C.,Dhage, On generalized metric spaces and topological structure II,
Pure and Applied Mathematics Sciences, 40(1-2),(1994),37-41.
[4] B.C.,Dhage, On continuity of mappings in D-metric spaces, Bull.
Calcutta Math. Soc., 86(1994), 503-508.
[5] B.C.Dhage, On common fixed point of coincidentally commuting
mappings in D-metric spaces, Indian J. Pure & Appl. Math., 30(3),(1999),
395-406.
*Department of Mathematics, M.D.University, Rohtak-124002
INDIA
**BRCM College of Engg. & Tech., Bahal, Bhiwani-127028
INDIA
([email protected])
***CIET, NCERT, New Delhi-110016.
([email protected])
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