Mathematica Aeterna, Vol. 1, 2011, no. 07, 491 - 507
A Unique Common 3-tupled fixed point theorem
for ψ − φ contractions in partial metric spaces
K. P. R. Rao
Department of Applied Mathematics,
A.N.U.-Dr.M.R.Appa Row Campus,
Nuzvid-521 201, Andhra Pradesh, India
[email protected]
G.N.V.Kishore
Department of Mathematics,
Swarnandhra Institute of Engineering and Technology, Seetharampuram,
Narspur- 534 280 , West Godavari District, Andhra Pradesh, India
[email protected]
N.Srinivasa Rao
Department of Applied Sciences and Humanities,
Tirumala Engineering College,Jonnalgadda,
Narasaraopeta-522601,A.P.,India.
[email protected]
Abstract
In this paper, we obtain a unique common 3-tupled fixed point theorem in partial metric spaces and also mention an example to support
our theorem .
Mathematics Subject Classification: 54H25, 47H10.
Keywords: partial metric, weakly compatible maps, 3-tupled fixed point,
complete space.
1
Introduction
The notion of partial metric space was introduced by Matthews [17] as a part of
the study of denotational semantics of data flow networks. In fact, it is widely
recognized that partial metric spaces play an important role in constructing
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K.P.R.Rao, G.N.V.Kishore and N.Srinivasa Rao
models in the theory of computation and domain theory in computer science
(see e.g. [23, 26, 27, 8, 14, 20, 15, 21, 24]).
Matthews [17, 18], Oltra and Valero [19] and Romaguera [22] and Altun et
al. [2] proved some fixed point theorems in partial metric spaces for a single
map (see also [1, 9, 10, 11, 12, 13, 3, 4, 5, 25, 20]).
Recently, the concept of coupled fixed points was introduced by Bhaskar
and Lakshmikantham [7]. In [4], Aydi proved some coupled fixed point theorems for the mappings satisfying contractive conditions in partial metric
spaces.
In this paper, we introduce a 3-tupled fixed point and obtain a unique
common 3-tupled fixed point theorem for four self mappings satisfying a ψ − φ
contractive condition in partial metric spaces.
First we recall some basic definitions and lemmas which play crucial role
in the theory of partial metric spaces.
2
Preliminary Notes
Definition 2.1 (See [17, 18]) A partial metric on a nonempty set X is a
function p : X × X → R+ such that for all x, y, z ∈ X:
(p1 ) x = y ⇔ p(x, x) = p(x, y) = p(y, y),
(p2 ) p(x, x) ≤ p(x, y), p(y, y) ≤ p(x, y),
(p3 ) p(x, y) = p(y, x),
(p4 ) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z).
The pair (X, p) is called a partial metric space (PMS).
If p is a partial metric on X, then the function ps : X × X → R+ given by
ps (x, y) = 2p(x, y) − p(x, x) − p(y, y),
(1)
is a metric on X.
Example 2.2 (See e.g. [18, 10, 11, 1]) Consider X = [0, ∞) with p(x, y) =
max{x, y}. Then (X, p) is a partial metric space. It is clear that p is not a
(usual) metric. Note that in this case ps (x, y) = |x − y|.
Example 2.3 (See [9]) Let X = {[a, b] : a, b, ∈ R, a ≤ b} and define
p([a, b], [c, d]) = max{b, d} − min{a, c}. Then (X, p) is a partial metric space.
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Example 2.4 (See [9]) Let X := [0, 1]∪[2, 3] and define p : X ×X → [0, ∞)
by
p(x, y) =
(
max{x, y} if {x, y} ∩ [2, 3] 6= ∅,
|x − y| if {x, y} ⊂ [0, 1].
Then (X, p) is a complete partial metric space.
Each partial metric p on X generates a T0 topology τp on X which has as
a base the family of open p-balls {Bp (x, ε), x ∈ X, ε > 0}, where Bp (x, ε) =
{y ∈ X : p(x, y) < p(x, x) + ε} for all x ∈ X and ε > 0.
We now state some basic topological notions (such as convergence, completeness, continuity) on partial metric spaces (see e.g. [17, 18, 2, 1, 10, 13].)
Definition 2.5
1. A sequence {xn } in the PMS (X, p) converges to the
limit x if and only if p(x, x) = n→∞
lim p(x, xn ).
2. A sequence {xn } in the PMS (X, p) is called a Cauchy sequence if
lim p(xn , xm ) exists and is finite.
n,m→∞
3. A PMS (X, p) is called complete if every Cauchy sequence {xn } in X
converges with respect to τp , to a point x ∈ X such that
p(x, x) = lim p(xn , xm ).
n,m→∞
4. A mapping F : X → X is said to be continuous at x0 ∈ X if for every
ǫ > 0, there exists δ > 0 such that F (Bp (x0 , δ)) ⊆ Bp (F x0 , ǫ).
The following lemma is one of the basic results in PMS([17, 18, 1, 2, 10, 13]).
Lemma 2.6
1. A sequence {xn } is a Cauchy sequence in the PMS (X, p) if and only if
it is a Cauchy sequence in the metric space (X, ps ).
2. A PMS (X, p) is complete if and only if the metric space (X, ps ) is complete. Moreover
lim ps (x, xn ) = 0 ⇔ p(x, x) = lim p(x, xn ) = lim p(xn , xm )
n→∞
n→∞
n,m→∞
(2)
Next, we give two lemmas which will be used in the proof of our main
result. For the proofs we refer to e.g. [1, 10, 11].
Lemma 2.7 Assume xn → z as n → ∞ in a PMS (X, p) such that p(z, z) =
0. Then limn→∞ p(xn , y) = p(z, y) for every y ∈ X.
Lemma 2.8 Let (X, p) be a PMS. Then
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K.P.R.Rao, G.N.V.Kishore and N.Srinivasa Rao
(A) If p(x, y) = 0 then x = y,
(B) If x 6= y, then p(x, y) > 0,
(C) If x = y, p(x, y) may not be 0.
Lemma 2.9 If {xn } is converges to x in (X, p), then limn→∞ p(xn , y) ≤
p(x, y) for all y ∈ X.
Definition 2.10 [7]. An element (x, y) ∈ X × X is called a coupled fixed
point of mapping F : X × X → X if x = F (x, y) and y = F (y, x).
Very recently Berinde and Borcut [6] introduced the notion of tripled fixed
point of a mapping as follows
Definition 2.11 [6]. An element (x, y, z) ∈ X × X × X is called a tripled
fixed point of mapping F : X × X × X → X if x = F (x, y, z), y = F (y, x, y)
and z = F (z, y, x) .
Based on these difinitions, we give the following definitions.
Definition 2.12 An element (x, y, z) ∈ X ×X ×X is called a 3-tupled fixed
point of the mapping T : X × X × X → X if x = T (x, y, z), y = T (y, z, x) and
z = T (z, y, x).
Definition 2.13 An element (x, y, z) ∈ X × X × X is called
(i) a 3-tupled coincident point of mappings T : X ×X ×X → X and f : X → X
if f x = T (x, y, z), f y = T (y, z, x) and f z = T (z, x, y);
(ii) a common 3-tupled fixed point of mappings T : X × X × X → X and f :
X → X if x = f x = T (x, y, z), y = f y = T (y, z, x) and z = f z = T (z, x, y).
Definition 2.14 The mappings T : X × X × X → X and f : X → X
are called w - compatible if f (T (x, y, z)) = T (f x, f y, f z), f (T (y, z, x)) =
T (f y, f z, f x) and f (T (z, x, y)) = T (f z, f x, f y) whenever f x = T (x, y, z),
f y = T (y, z, x) and f z = T (z, x, y).
Now we prove our main result.
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495
Main Results
Let Ψ denotes the set of all functions ψ : [0, ∞) → [0, ∞) satisfying
(ψ1 ) ψ is continuous and non-decreasing,
(ψ2 ) ψ(t) = 0 if and only if t = 0,
(ψ3 ) ψ(t + s) ≤ ψ(t) + ψ(s), for all t, s ∈ [0, ∞),
while Φ denotes the set of all functions φ : [0, ∞) → [0, ∞) satisfying
(φ1 ) φ is continuous,
(φ2 ) φ(t) = 0 if and only if t = 0.
Theorem 3.1 Let (X, p) be a partial metric space and let S, T : X × X ×
X → X and f, g : X → X be mappings satisfying
(i) ψ(p (S(x, y, z), T (u, v, w))) ≤ 13 ψ(p(f x, gu) + p(f y, gv) + p(f z, gw))
−φ(p(f x, gu) + p(f y, gv) + p(f z, gw))
∀x, y, z, u, v, w ∈ X where ψ ∈ Ψ and φ ∈ Φ,
(ii) S(X × X × X) ⊆ g(X), T (X × X × X) ⊆ f (X),
(iii) either f (X) or g(X) is a complete subspace of X,
(iv) the pairs (f, S) and (g, T ) are w - compatible.
Then S, T, f and g have a unique common 3-tupled fixed point in X × X × X
of the form (α, α, α).
Proof 3.2 Let x0 , y0 , z0 be arbitrary points in X. From (ii), there exist
sequences {xn }, {yn }, {zn }, {un }, {vn } and {wn } in X such that
u2n = gx2n+1 = S(x2n , y2n , z2n ),
v2n = gy2n+1 = S(y2n , z2n , x2n ),
w2n = gz2n+1 = S(z2n , x2n , y2n ),
u2n+1 = f x2n+2 = T (x2n+1 , y2n+1, z2n+1 ),
v2n+1 = f y2n+2 = T (y2n+1 , z2n+1 , x2n+1 ),
w2n+1 = f z2n+2 = T (z2n+1 , x2n+1 , y2n+1 ),
(3)
n = 0, 1, 2, · · ·
Then by (i), we have
ψ(p(u2 , u1 )) = ψ(p(S(x2 , y2, z2 ), T (x1 , y1 , z1 )))
≤ 31 ψ(p(f x2 , gx1 ) + p(f y2 , gy1) + p(f z2 , gz1 ))
−φ(p(f x2 , gx1) + p(f y2 , gy1) + p(f z2 , gz1))
1
= 3 ψ(p(u1 , u0 ) + p(v1 , v0 ) + p(w1 , w0 ))
−φ(p(u1 , u0 ) + p(v1 , v0 ) + p(w1 , w0)).
ψ(p(v2 , v1 )) = ψ(p(S(y2, z2 , x2 ), T (y1 , z1 , x1 )))
≤ 31 ψ(p(f y2, gy1) + p(f z2 , gz1) + p(f x2 , gx1 ))
−φ(p(f y2, gy1) + p(f z2 , gz1 ) + p(f x2 , gx1 ))
1
= 3 ψ(p(u1 , u0 ) + p(v1 , v0 ) + p(w1 , w0))
−φ(p(u1 , u0) + p(v1 , v0 ) + p(w1 , w0 )).
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And
ψ(p(w2 , w1 )) = ψ(p(S(z2 , x2 , y2 ), T (z1 , x1 , y1)))
≤ 31 ψ(p(f z2 , gz1) + p(f x2 , gx1 ) + p(f y2, gy1))
−φ(p(f z2 , gz1) + p(f x2 , gx1 ) + p(f y2 , gy1))
1
= 3 ψ(p(u1, u0 ) + p(v1 , v0 ) + p(w1 , w0 ))
−φ(p(u1 , u0 ) + p(v1 , v0 ) + p(w1 , w0 )).
By definition of ψ, we have
ψ(p(u2 , u1) + p(v2 , v1 ) + p(w2 , w1 )) ≤ ψ(p(u2 , u1 )) + ψ(p(v2 , v1 )) + ψ(p(w2 , w1 ))
≤ ψ(p(u1 , u0 ) + p(v1 , v0 ) + p(w1 , w0))
−3φ(p(u1 , u0 ) + p(v1 , v0 ) + p(w1 , w0 ))
≤ ψ(p(u1 , u0 ) + p(v1 , v0 ) + p(w1 , w0)).
Since ψ is non - decreasing, we have
p(u2 , u1 ) + p(v2 , v1 ) + p(w2 , w1 ) ≤ p(u1, u0 ) + p(v1 , v0 ) + p(w1 , w0 ).
Similarly
ψ(p(u3 , u2) + p(v3 , v2 ) + p(w3 , w2 )) ≤ ψ(p(u2 , u1 ) + p(v2 , v1 ) + p(w2 , w1))
−3φ(p(u2 , u1 ) + p(v2 , v1 ) + p(w2 , w1 )).
Hence by induction, we have


p(un , un+1)


ψ  +p(vn , vn+1 )  ≤ ψ (p(un , un−1) + p(vn , vn−1 ) + p(wn , wn−1 ))
+p(wn , wn+1)
−3φ (p(un , un−1) + p(vn , vn−1 ) + p(wn , wn−1 )) (4)
≤ ψ(p(un , un−1 ) + p(vn , vn−1 ) + p(wn , wn−1)).
Since ψ is non - decreasing, we have
p(un , un+1)+p(vn , vn+1 )+p(wn , wn+1 ) ≤ p(un , un−1)+p(vn , vn−1 )+p(wn , wn−1).
Put Rn = p(un , un+1) + p(vn , vn+1 ) + p(wn , wn+1 ). Then {Rn } is a non increasing sequence of real numbers and must converge to r ≥ 0.
Suppose r > 0.
Letting n → ∞ in (4), we have that
ψ(r) ≤ ψ(r) − φ(r)
< ψ(r).
It is a contradiction. Hence r = 0.
Thus
lim [p(un , un+1) + p(vn , vn+1 ) + p(wn , wn+1 )] = 0.
n→∞
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It follows that
lim p(un , un+1) = lim p(vn , vn+1 ) = lim p(wn , wn+1 ) = 0.
n→∞
n→∞
n→∞
(5)
From (p2 ), we have
lim p(un , un ) = n→∞
lim p(vn , vn ) = n→∞
lim p(wn , wn ) = 0.
n→∞
(6)
From definition of ps , (2.3) and (2.4), we have
lim ps (un , un+1 ) = 0.
(7)
lim ps (vn , vn+1 ) = 0
(8)
lim ps (wn , wn+1 ) = 0.
(9)
n→∞
Similarly
n→∞
and
n→∞
Now we prove that {u2n }, {v2n } and {w2n } are Cauchy sequences.
On contrary, suppose that {u2n } or {v2n } or {w2n } is not Cauchy.
Then there exists an ǫ > 0 and monotone increasing sequences of natural
numbers {2mk } and {2nk } such that nk > mk ,
ps (u2mk , u2nk ) + ps (v2mk , v2nk ) + ps (w2mk , w2nk ) ≥ ǫ
(10)
ps (u2mk , u2nk −2 ) + ps (v2mk , v2nk −2 ) + ps (w2m , w2nk −2 ) < ǫ.
(11)
and
From (10) and (11), we have
ǫ ≤ ps (u2mk , u2nk ) + ps (v2mk , v2nk ) + ps (w2mk , w2nk )
≤ ps (u2mk , u2nk −2 ) + ps (v2mk , v2nk −2 ) + ps (w2mk , w2nk −2 )
+ps (u2nk −2 , u2nk −1 ) + ps (v2nk −2 , v2nk −1 ) + ps (w2nk −2 , w2nk −1 )
+ps (u2nk −1 , u2nk ) + ps (v2nk −1 , v2nk ) + ps (w2nk −1 , w2nk )
< ǫ + ps (u2nk −2 , u2nk −1 ) + ps (v2nk −2 , v2nk −1 ) + ps (w2nk −2 , w2nk −1 )
+ps (u2nk −1 , u2nk ) + ps (v2nk −1 , v2nk ) + ps (w2nk −1 , w2nk ).
Letting k → ∞ and using (7), (8) and (9) we have
lim [ps (u2mk , u2nk ) + ps (v2mk , v2nk ) + ps (w2mk , w2nk )] = ǫ.
k→∞
(12)
By definition of ps and (6), we have
ǫ
lim [p(u2mk , u2nk ) + p(v2mk , v2nk ) + p(w2mk , w2nk )] = .
k→∞
2
(13)
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Also,
ǫ ≤ ps (u2mk , u2nk ) + ps (v2mk , v2nk ) + ps (w2mk , w2nk )
≤ ps (u2mk , u2mk −1 ) + ps (v2mk , v2mk −1 ) + ps (w2mk , w2mk −1 )
+ps (u2mk −1 , u2nk ) + ps (v2mk −1 , v2nk ) + ps (w2mk −1 , w2nk )
≤ 2[ps (u2mk , u2mk −1 ) + ps (v2mk , v2mk −1 ) + ps (w2mk , w2mk −1 )]
+[ps (u2mk , u2nk ) + ps (v2mk , v2nk ) + ps (w2mk , w2nk )].
(14)
Letting k → ∞ and using (7),(8), (9), (12) and (14), we have
lim [ps (u2mk −1 , u2nk ) + ps (v2mk −1 , v2nk ) + ps (w2mk −1 , w2nk )] = ǫ.
(15)
k→∞
By definition of ps and (6), we have
ǫ
lim [p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk )] = .
k→∞
2
(16)
On other hand we have
ps (u2mk , u2nk ) + ps (v2mk , v2nk ) + ps (w2mk , w2nk )
≤ ps (u2mk , u2nk +1 ) + ps (v2mk , v2nk +1 ) + ps (w2mk , w2nk +1 )
+ps (u2nk +1 , u2nk ) + ps (v2nk +1 , v2nk ) + ps (w2nk +1 , w2nk ).
Letting k → ∞ and using (6), (7),(8)(9) and (12), we have
ǫ ≤ lim [ps (u2mk , u2nk +1 ) + ps (v2mk , v2nk +1 ) + ps (w2mk , w2nk +1 )] + 0
k→∞
= 2 lim [p(u2mk , u2nk +1 ) + p(v2mk , v2nk +1 ) + p(w2mk , w2nk +1 )].
k→∞
Thus,
ǫ
≤ lim [p(u2mk , u2nk +1 ) + p(v2mk , v2nk +1 ) + p(w2mk , w2nk +1 )].
2 k→∞
By the properties of ψ, we have
ψ
ǫ
2
≤ ψ
≤
lim [p(u2mk , u2nk +1 ) + p(v2mk , v2nk +1 ) + p(w2mk , w2nk +1 )]
k→∞
lim
k→∞
ψ(p(u2mk , u2nk +1 )) + ψ(p(v2mk , v2nk +1 ))
+ψ(p(w2mk , w2nk +1 ))
!
.
(17)
Now
ψ(p(u2mk , u2nk +1 )) = ψ (p(S(x2mk , y2mk , z2mk ), T (x2nk +1 , y2nk +1 , z2nk +1 )))
≤ 31 ψ (p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk ))
−φ (p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk )) .
Similarly
ψ(p(v2mk , v2nk +1 )) ≤ 13 ψ (p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk ))
−φ (p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk ))
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and
ψ(p(w2mk , w2nk +1 )) ≤ 13 ψ (p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk ))
−φ (p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk )) .
Hence from (16) and (17), we have
ψ
ǫ
2
ψ (p(u2mk −1 , u2nk ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk ))
≤ lim
−3φ
(p(u
, u ) + p(v2mk −1 , v2nk ) + p(w2mk −1 , w2nk ))
k→∞
2mk −1 2nk
ǫ
ǫ
= ψ 2 − 3φ 2
< ψ 2ǫ .
!
It is a contradiction.
Hence {u2n }, {v2n } and {w2n } are Cauchy sequences in the metric space
(X, ps ).
Letting n, m → ∞ in
|ps (u2n+1 , u2m+1 ) − ps (u2n , u2m )| ≤ ps (u2n+1 , u2n ) + ps (u2m+1 , u2m )
we get lim ps (u2n+1 , u2m+1 ) = 0.
n,m→∞
Letting n, m → ∞ in
|ps (v2n+1 , v2m+1 ) − ps (v2n , v2m )| ≤ ps (v2n+1 , v2n ) + ps (v2m+1 , v2m )
we get lim ps (v2n+1 , v2m+1 ) = 0.
n,m→∞
Letting n, m → ∞ in
|ps (w2n+1 , w2m+1 ) − ps (w2n , w2m )| ≤ ps (w2n+1 , w2n ) + ps (w2m+1 , w2m )
we get lim ps (w2n+1 , w2m+1 ) = 0.
n,m→∞
Thus {u2n+1 }, {v2n+1 } and {w2n+1 } are Cauchy sequences in the metric space
(X, ps ).
Hence {un }, {vn } and {wn } are Cauchy sequences in the metric space (X, ps ).
Hence we have that
lim ps (um , un ) = lim ps (vm , vn ) = lim ps (wm , wn ) = 0.
n,m→∞
n,m→∞
n,m→∞
(18)
Now from definition of ps and from (6), we have
lim p(um , un ) = lim p(vm , vn ) = lim p(wm , wn ) = 0.
n,m→∞
n,m→∞
n,m→∞
(19)
Suppose f (X) is complete.
Since {u2n+1 } ⊆ f (X), {v2n+1} ⊆ f (X) and {w2n+1 } ⊆ f (X) are Cauchy
sequences in the complete metric space (f (X), ps ), it follows that the sequences
{u2n+1 }, {v2n+1 } and {w2n+1 } are convergent in (f (X), ps).
Thus lim ps (u2n+1 , α) = 0, lim ps (v2n+1 , β) = 0 and lim ps (w2n+1 , γ) = 0 for
n→∞
n→∞
n→∞
some α, β and γ in f (X).
There exist x, y, z ∈ X such that α = f x, β = f y and γ = f z.
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Since {un }, {vn } and {wn } are Cauchy sequences in X and {u2n+1 } → α,
{v2n+1 } → β and {w2n+1 } → γ , it follows that {u2n } → α, {v2n } → β and
{w2n } → γ.
From Lemma 2.6(2), we have
p(α, α) = lim p(u2n , α) = lim p(u2n+1 , α) =
n→∞
n→∞
p(β, β) = lim p(v2n , β) = lim p(v2n+1 , β) =
n→∞
n→∞
lim
n, m→∞
lim
p(un , um ).
n, m→∞
p(vn , vm )
(20)
(21)
and
p(γ, γ) = n→∞
lim p(w2n , γ) = n→∞
lim p(w2n+1 , γ) = n, lim
p(wn , wm ).
m→∞
(22)
From (19), (20), (21) and (22) we have
p(α, α) = lim p(u2n , α) = lim p(u2n+1 , α) = 0.
(23)
p(β, β) = lim p(v2n , β) = lim p(v2n+1 , β) = 0
(24)
p(γ, γ) = lim p(w2n , γ) = lim p(w2n+1 , γ) = 0.
(25)
n→∞
n→∞
n→∞
n→∞
and
n→∞
n→∞
Now
p(S(x, y, z), α) ≤ p(S(x, y, z), u2n+1) + p(u2n+1 , α) − p(u2n+1 , u2n+1 )
≤ p(S(x, y, z), T (x2n+1 , y2n+1 , z2n+1 )) + p(u2n+1 , α).
Letting n → ∞, we have
p(S(x, y, z), α) ≤ lim p(S(x, y, z), T (x2n+1, y2n+1 , z2n+1 )) + 0.
n→∞
Since by the property of ψ, we have
ψ(p(S(x, y, z), α)) ≤ lim ψ(p(S(x, y, z), T (x2n+1, y2n+1 , z2n+1 ))
n→∞ (
)
1
ψ(p(f
x,
gx
)
+
p(f
y,
gy
)
+
p(f
z,
gz
))
2n+1
2n+1
2n+1
3
≤ n→∞
lim
−φ(p(f
x, gx2n+1 ) + p(f y, gy2n+1) + p(f z,)gz2n+1 ))
(
1
ψ(p(α, u2n ) + p(β, v2n ) + p(γ, w2n ))
3
= n→∞
lim
−φ(p(α, u2n ) + p(β, v2n ) + p(γ, w2n ))
= ψ(0) − φ(0) = 0.
It follows that S(x, y, z) = α.
Similarly S(y, z, x) = β and S(z, x, y) = γ.
Thus
α = f x = S(x, y, z), β = f y = S(y, z, x) and γ = f z = S(z, x, y).
A Unique Common 3-tupled fixed point theorem · · · in pms
501
Since (f, S) is w-compatible we have
S(α, β, γ) = f α, S(β, γ, α) = f β and S(γ, α, β) = f γ.
By definition
ps (f α, u2n ) = 2p(f α, u2n ) − p(f α, f α) − p(u2n , u2n ).
Letting n → ∞, we get
ps (f α, α) = 2 lim p(f α, u2n ) − p(f α, f α) − 0.
n→∞
Thus we have
2p(f α, α) − p(f α, f α) − p(α, α) = 2 lim p(f α, u2n ) − p(f α, f α) − 0.
n→∞
Hence lim p(f α, u2n ) = p(f α, α). By the same analogy, we have
n→∞
lim p(f β, v2n ) = p(f β, β) and n→∞
lim p(f γ, w2n ) = p(f γ, γ).
n→∞
Now, we shall prove that f α = α, f β = β and f γ = γ.
By the triangle inequality (p4 ), we have
p(f α, α) ≤ p(f α, u2n+1) + p(u2n+1 , α) − p(u2n+1 , u2n+1)
≤ p(S(α, β, γ), T (x2n+1, y2n+1 , z2n+1 )) + p(u2n+1 , α).
Letting n → ∞ in the equality above , we get
p(f α, α) ≤ lim p(S(α, β, γ), T (x2n+1, y2n+1, z2n+1 )) + 0.
n→∞
Since ψ is continuous and non - decreasing, we have
ψ(p(f α, α)) ≤ lim ψ(p(S(α, β, γ), T (x2n+1, y2n+1 , z2n+1 )))
n→∞
!
1
ψ(p(f
α,
gx
)
+
p(f
β,
gy
)
+
p(f
γ,
gz
))
2n+1
2n+1
2n+1
3
≤ lim
n→∞
−φ(p(f α, gx2n+1) + p(f β, gy2n+1) + p(f γ, gz!
2n+1 ))
1
ψ(p(f α, u2n ) + p(f β, v2n ) + p(f γ, w2n ))
3
=
lim
n→∞
−φ(p(f α, u2n ) + p(f β, v2n ) + p(f γ, w2n ))
= 31 ψ(p(f α, α) + p(f β, β) + p(f γ, γ)) − φ(p(f α, α) + p(f β, β) + p(f γ, γ)).
Similarly
ψ(p(f β, β)) ≤ 13 ψ(p(f α, α) + p(f β, β) + p(f γ, γ)) − φ(p(f α, α) + p(f β, β) + p(f γ, γ))
and
ψ(p(f γ, γ)) ≤ 31 ψ(p(f α, α) + p(f β, β) + p(f γ, γ)) − φ(p(f α, α) + p(f β, β) + p(f γ, γ)).
502
K.P.R.Rao, G.N.V.Kishore and N.Srinivasa Rao
By definition of ψ, we have
ψ(p(f α, α) + p(f β, β) + p(f γ, γ))
≤ ψ(p(f α, α)) + ψ(p(f β, β)) + ψ(p(f γ, γ))
≤ ψ(p(f α, α) + p(f β, β) + p(f γ, γ)) − 3φ(p(f α, α) + p(f β, β) + p(f γ, γ)).
It follows that
φ(p(f α, α) + p(f β, β) + p(f γ, γ)) = 0.
Thus
f α = α, f β = β and f γ = γ.
Hence
α = f α = S(α, β, γ), β = f β = S(β, γ, α) and γ = f γ = S(γ, α, β).
(26)
Since S(X × X × X) ⊆ g(X), there exist r, s, t ∈ X such that
α = S(α, β, γ) = gr, β = S(β, γ, α) = gs and γ = S(γ, α, β) = gt.
Consider
ψ(α, T (r, s, t)) = ψ(S(α, β, γ), T (r, s, t)))
≤ 31 ψ(p(f α, gr) + p(f β, gs) + p(f γ, gt))
−φ(p(f α, gr) + p(f β, gs) + p(f γ, gt))
= ψ(p(α, α) + p(β, β) + p(γ, γ))
−φ(p(α, α) + p(β, β) + p(γ, γ))
= ψ(0) − φ(0) = 0,
from (23), (24) and (25).
It follows that T (r, s, t) = α = gr.
Similarly T (s, t, r) = β = gs and T (t, r, s) = γ = gt.
Since (g, T ) is w - compatible, we have
T (α, β, γ) = gα, T (β, γ, α) = gβ and T (γ, α, β) = gγ.
Now
ψ(p(α, gα)) = ψ(p(S(α, β, γ), T (α, β, γ)))
≤ 31 ψ(p(f α, gα) + p(f β, gβ) + p(f γ, gγ))
−φ(p(f α, gα) + p(f β, gβ) + p(f γ, gγ))
1
≤ 3 ψ(p(α, gα) + p(β, gβ) + p(γ, gγ)) − φ(p(α, gα) + p(β, gβ) + p(γ, gγ)).
Similarly
ψ(p(β, gβ)) ≤ 13 ψ(p(α, gα) + p(β, gβ) + p(γ, gγ)) − φ(p(α, gα) + p(β, gβ) + p(γ, gγ))
and
ψ(p(γ, gγ)) ≤ 13 ψ(p(α, gα) + p(β, gβ) + p(γ, gγ)) − φ(p(α, gα) + p(β, gβ) + p(γ, gγ)).
A Unique Common 3-tupled fixed point theorem · · · in pms
503
By definition of ψ, we have
ψ(p(α, gα) + p(β, gβ) + p(γ, gγ))
≤ ψ(p(α, gα)) + ψ(p(β, gβ)) + ψ(p(γ, gγ))
≤ ψ(p(α, gα) + p(β, gβ) + p(γ, gγ)) − 3φ(p(α, gα) + p(β, gβ) + p(γ, gγ)).
It follows that
α = gα, β = gβ and γ = gγ.
Thus
α = gα = T (α, β, γ), β = gβ = T (β, γ, α) and γ = gγ = T (γ, α, β).
(27)
Hence from (26) and (27),
(α, β, γ) is common 3-tupled fixed point of S, T, f and g.
To prove uniqueness,
let (α∗ , β ∗ , γ ∗ ) is another common 3-tupled fixed point of S, T, f and g.
Consider
ψ(p(α, α∗)) = ψ(p(S(α, β, γ), T (α∗, β ∗ , γ ∗ )))
≤ 13 ψ(p(f α, gα∗) + p(f β, gβ ∗) + p(f γ, gγ ∗))
−φ(p(f α, gα∗) + p(f β, gβ ∗) + p(f γ, gγ ∗))
1
≤ 3 ψ(p(α, α∗ ) + p(β, β ∗) + p(γ, γ ∗ )) − φ(p(α, α∗) + p(β, β ∗) + p(γ, γ ∗ )).
Similarly
ψ(p(β, β ∗)) ≤ 31 ψ(p(α, α∗) + p(β, β ∗) + p(γ, γ ∗ )) − φ(p(α, α∗) + p(β, β ∗ ) + p(γ, γ ∗ ))
and
ψ(p(γ, γ ∗ )) ≤ 31 ψ(p(α, α∗) + p(β, β ∗) + p(γ, γ ∗ )) − φ(p(α, α∗) + p(β, β ∗) + p(γ, γ ∗ )).
By definition of ψ, we have
ψ(p(α, α∗) + p(β, β ∗) + p(γ, γ ∗ ))
≤ ψ(p(α, α∗)) + ψ(p(β, β ∗)) + ψ(p(γ, γ ∗ ))
≤ ψ(p(α, α∗) + p(β, β ∗) + p(γ, γ ∗ )) − 3φ(p(α, α∗) + p(β, β ∗) + p(γ, γ ∗ )).
It follows that
α = α∗ , β = β ∗ and γ = γ ∗ .
Therfore (α, β, γ) is unique common 3-tupled fixed point of S, T, f and g.
Now we claim that α = β = γ.
ψ(p(α, β)) = ψ(p(S(α, β, γ), T (β, γ, α)))
≤ 13 ψ(p(f α, gβ) + p(f β, gγ) + p(f γ, gα))
−φ(p(f α, gβ) + p(f β, gγ) + p(f γ, gα))
1
= 3 ψ(p(α, β) + p(β, γ) + p(γ, α))
−φ(p(α, β) + p(β, γ) + p(γ, α)).
504
K.P.R.Rao, G.N.V.Kishore and N.Srinivasa Rao
ψ(p(β, γ)) = ψ(p(S(β, γ, α), T (γ, α, β)))
≤ 31 ψ(p(f β, gγ) + p(f γ, gα) + p(f α, gβ))
−φ(p(f β, gγ) + p(f γ, gα) + p(f α, gβ))
= 31 ψ(p(α, β) + p(β, γ) + p(γ, α))
−φ(p(α, β) + p(β, γ) + p(γ, α))
and
ψ(p(γ, α)) = ψ(p(S(γ, α, β), T (α, β, γ)))
≤ 31 ψ(p(f γ, gα) + p(f α, gβ) + p(f β, gγ))
−φ(p(f γ, gα) + p(f α, gβ) + p(f β, gγ))
= 13 ψ(p(α, β) + p(β, γ) + p(γ, α))
−φ(p(α, β) + p(β, γ) + p(γ, α)).
By definition of ψ, we have
ψ(p(α, β) + p(β, γ) + p(γ, α))
≤ ψ(p(α, β)) + ψ(p(β, γ)) + ψ(p(γ, α))
= ψ(p(α, β) + p(β, γ) + p(γ, α)) − 3φ(p(α, β) + p(β, γ) + p(γ, α)).
It follows that α = β = γ.
Thus S, T, f and g have unique common 3-tupled fixed point of the form
(α, α, α) in X × X × X.
Example 3.3 Let X = [0, 1], the mappings S, T : X × X × X → X
2
2
2
and f, g : X → X be defined by S(x, y, z) = x +y6 +z , T (x, y, z) = x+y+z
,
12
x
2
f (x) = x and g(x) = 2 respectively and p : X × X → [0, ∞) by
p(x, y) = max{x, y}. Let ψ, φ : [0, ∞) → [0, ∞) be defined by ψ(x) = x and
φ(x) = x6 .
Clearly the conditions (ii),(iii) and (iv) are satisfied.
Now
n
2
2
2
o
p(S(x, y, z), T (u, v, w)) = max x +y6 +z , u+v+w
12
n 2
2
2
u
= max x6 + y6 + z6 , 12
+
≤
=
=
=
=
o
v
+ w
n 2
o
n 2 12 o 12
n 2
o
u
v
w
max x6 , 12
+ max y6 , 12
+ max z6 , 12
h
n
o
n
o
n
oi
1
max x2 , u2 + max y 2, v2 + max z 2 , w2
6
1
[p(f x, gu) + p(f y, gv) + p(f z, gw)]
6
1
[p(f x, gu) + p(f y, gv) + p(f z, gw)]
3
− 16 [p(f x, gu) + p(f y, gv) + p(f z, gw)]
1
ψ(p(f x, gu) + p(f y, gv) + p(f z, gw))
3
−φ(p(f x, gu) + p(f y, gv) + p(f z, gw)).
Hence all conditions of Theorem 3.1 are satisfied and (0, 0, 0) is unique common
3-tupled fixed point of S, T, f and g.
A Unique Common 3-tupled fixed point theorem · · · in pms
505
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Received: October 21, 2011