Open Access

A Meir-Keeler common type fixed point theorem on partial metric spaces

Fixed Point Theory and Applications20122012:26

https://doi.org/10.1186/1687-1812-2012-26

Received: 28 November 2011

Accepted: 24 February 2012

Published: 24 February 2012

Abstract

In this article, we prove a general common fixed point theorem for two pairs of weakly compatible self-mappings of a partial metric space satisfying a generalized Meir-Keeler type contractive condition. The presented theorem extends several well known results in literature.

1 Introduction

Partial metric spaces were introduced by Matthews [1] to study denotational semantics of dataflow networks. In fact, (complete) partial metric spaces constitute a suitable framework to model several distinguished examples of the theory of computation and also to model metric spaces via domain theory. For example, in the research area of computer domains and semantics, partial metric spaces have serious applications potentials (see for example, [25]). In 1994, Matthews [1] generalized the Banach contraction principle to the class of complete partial metric spaces: a self mapping T on a complete partial metric space (X, p) has a unique fixed point if there exists 0 ≤ k < 1 such that p(Tx, Ty) ≤ kp(x, y) for all x, y X. After the remarkable contribution of Matthews, many authors have studied on partial metric spaces and its topological properties (see for example, [128]).

In the sequel we recall the notion of a partial metric space and some of its properties which will be useful later on. A partial metric is a function p : X × X → [0, ∞) satisfying the following conditions

(P1) p(x, y) = p(y, x),

(P2) If p(x, x) = p(x, y) = p(y, y), then x = y,

(P3) p(x, x) ≤ p(x, y),

(P4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z),

for all x, y, z X. Then (X, p) is called a partial metric space. If p is a partial metric p on X, then the function d p : X × X → [0, ∞) given by
d p ( x , y ) = 2 p ( x , y ) - p ( x , x ) - p ( y , y )

is a metric on X. Also, each partial metric p on X generates a T0 topology τ p on X with a base of the family of open p-balls {B p (x, ε) : x X, ε > 0}, where B p (x, ε) = {y X : p(x, y) < p(x, x) + ε} for all x X and ε > 0. Similarly, closed p-ball is defined as B p [x, ε] = {y X : p(x, y) ≤ p(x, x) + ε}.

Definition 1.1. [1, 7] Let (X, p) be a partial metric space.

(i) A sequence {x n } in X converges to x X whenever lim n p ( x , x n ) = p ( x , x ) ,

(ii) A sequence {x n } in X is called Cauchy whenever lim n , m p ( x n , x m ) exists (and finite),

(iii) (X, p) is said to be complete if every Cauchy sequence {x n } in X converges, with respect to τ p , to a point x X, that is, lim n , m p ( x n , x m ) = p ( x , x ) .

(iv) A mapping f : XX is said to be continuous at x0 X if for each ε > 0 there exists δ > 0 such that f(B(x0, δ)) B(f(x0), ε).

Lemma 1.1. [1, 7] Let (X, p) be a partial metric space.

(a) A sequence {x n } is Cauchy if and only if {x n } is a Cauchy sequence in the metric space (X, d p ),

(b) (X, p) is complete if and only if the metric space (X, d p ) is complete. Moreover,
lim n d p ( x , x n ) = 0 lim n p ( x , x n ) = lim n , m p ( x n , x m ) = p ( x , x ) .
(1)

Definition 1.2. [29] Let X be a non empty set and f, g : XX. If w = fx = gx, for some x X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. If w = x, then x is a common fixed point of f and g.

Definition 1.3. [29] Let f and g be two self-maps defined on a non empty set X. Then f and g are said to be weakly compatible if they commute at every coincidence point.

Recently, Ćirić et al. [17] established a common fixed point result for two pairs of weakly compatible mappings satisfying generalized contractions on a partial metric space. For this, denote by Φ the set of non-decreasing continuous functions φ : satisfying:
  1. (a)

    0 < φ(t) < t for all t > 0,

     
  2. (b)

    the series ∑n≥1 φ n (t) converge for all t > 0.

     

The result [17] is the following.

Theorem 1.2. Suppose that A, B, S, andT are self-maps of a complete partial metric space (X, p) such that AX TX, BX SX and
p ( A x , B y ) φ ( M ( x , y ) )
(2)
for all x, y X, where φ Φ and
M ( x , y ) = max p ( S x , T y ) , p ( A x , S x ) , p ( B y , T y ) , 1 2 [ p ( S x , B y ) + p ( A x , T y ) ] .
If one of the ranges AX, BX, TX and SX is a closed subset of (X, p), then
  1. (i)

    A and S have a coincidence point,

     
  2. (ii)

    B and T have a coincidence point.

     

Moreover, if the pairs {A, S} and {B, T} are weakly compatible, then A, B, T, and S have a unique common fixed point.

In this manuscript, replacing (2) by some new weaker hypotheses we also establish a common fixed point result for four self maps satisfying a generalized Meir-Keeler type contraction on partial metric spaces. Our theorem generalizes several well known results in the literature.

2 Main results

The following lemmas will be frequently used in the proofs of the main results.

Lemma 2.1. [6, 19] Let (X, p) be a partial metric space. Then

(a) If p(x, y) = 0, then x = y,

(b) If xy, then p(x, y) > 0.

Lemma 2.2. [6, 19] Let (X, p) be a partial metric space and x n z with p(z, z) = 0. Then lim n p ( x n , y ) = p ( z , y ) for all y X.

Now, we are ready to state and prove our main result.

Theorem 2.3. Let A, B, S, and T be the self maps defined on a complete partial metric space (X, p) satisfying the following conditions:

(C1) AX TX and BX SX,

(C2) for all ε > 0, there exists δ > 0 such that for all x, y X
ε < M ( x , y ) < ε + δ p ( A x , B y ) ε ,
(3)

where M ( x , y ) = max { p ( S x , T y ) , p ( A x , S x ) , p ( B y , T y ) , 1 2 [ p ( S x , B y ) + p ( A x , T y ) ] } ,

(C3) for all x, y X with M(x, y) > 0 p(Ax, By) < M(x, y),

(C4) p(Ax, By) ≤ max{a[p(Sx, Ty) + p(Ax, Sx) + p(By, Ty)], b[p(Sx, By) + p(Ax, Ty]} for all x, y X, 0 a < 1 2 and 0 b < 1 2 .

If one of the ranges AX, BX, TX, and SX is a closed subset of (X, p), then

(I) A and S have a coincidence point,

(II) B and T have a coincidence point.

Moreover, if A and S, as well as, B and T are weakly compatible, then A, B, S, and T have a unique common fixed point.

Proof. Let x0 be an arbitrary point in X. Since AX TX, there exists x1 X such that Tx1 = Ax0. Since BX SX, there exists x2 X such that Sx2 = Bx1. Continuing this process, we can construct sequences {x n } and {y n } in X defined by
y 2 n = T x 2 n + 1 = A x 2 n , y 2 n + 1 = S x 2 n + 2 = B x 2 n + 1 n .
(4)

Suppose p(y2n, y2n+1) = 0 for some n. Then y2n= y2n+1implies that Ax2n= Tx2n+1= Bx2n+1= Sx2n+2, so T and B have a coincidence point. Further, if p(y2n+1, y2n+2) = 0 for some n then Ax2n+2= Tx2n+3= Bx2n+1= Sx2n+2, so A and S have a coincidence point. For the rest, assume that p(y n , yn+1) ≠ 0 for all n ≥ 0.

If for some x, y X, M(x, y) = 0, then we get that Ax = Sx and By = Ty, so we proved (I) and (II).

If M(x, y) > 0 for all x, y X, then by (C3),
p ( A x , B y ) < M ( x , y ) for all , x , y X .
(5)
Hence, we have
p ( y 2 p , y 2 p + 1 ) < M ( x 2 p , x 2 p + 1 ) = max { p ( S x 2 p , T x 2 p + 1 ) , p ( A x 2 p , S x 2 p ) , p ( B x 2 p + 1 , T x 2 p + 1 ) , 1 2 [ p ( S x 2 p , B x 2 p + 1 ) + p ( A x 2 p , T x 2 p + 1 ) ] } = max { p ( y 2 p 1 , y 2 p ) , p ( y 2 p , y 2 p 1 ) , p ( y 2 p + 1 , y 2 p ) , 1 2 [ p ( y 2 p 1 , y 2 p + 1 ) + p ( y 2 p , y 2 p ) ] } max { p ( y 2 p 1 , y 2 p ) , p ( y 2 p + 1 , y 2 p ) , 1 2 [ p ( y 2 p 1 , y 2 p ) + p ( y 2 p , y 2 p + 1 ) ] } = max { p ( y 2 p 1 , y 2 p ) , p ( y 2 p , y 2 p + 1 ) }
since
p ( y 2 p - 1 , y 2 p + 1 ) + p ( y 2 p , y 2 p ) p ( y 2 p - 1 , y 2 p ) + p ( y 2 p , y 2 p + 1 ) .
It is easy that max {p(y2p-1, y2p), p(y2p, y2p+1)} = p(y2p, y2p+1) is excluded. It follows that
p ( y 2 p , y 2 p + 1 ) < M ( x 2 p , x 2 p + 1 ) p ( y 2 p - 1 , y 2 p ) for all p 1 .
(6)
Similarly, one can find
p ( y 2 p + 2 , y 2 p + 1 ) < M ( x 2 p + 2 x 2 p + 1 ) p ( y 2 p + 1 , y 2 p ) for all p 0 .
(7)
We deduce that
p ( y n , y n + 1 ) < p ( y n - 1 , y n ) for all n 1 .
Thus, { p ( y n , y n + 1 ) } n = 0 is a decreasing sequence which is bounded below by 0. Hence, it converges to some L [0, ∞), that is,
lim n p ( y n , y n + 1 ) = L .
(8)
We claim that L = 0. If L > 0, then from (8), there exist δ > 0 and a natural number m ≥ 1 such that for each nm L < d(y n , yn+1) < L + δ. In particular, from this and (6)
L < M ( x 2 m , x 2 m + 1 ) < L + δ .
Now by using (3), we get that p(Ax2m, Bx2m+1) = p(y2m, y2m+1) ≤ L which is a contradiction. Thus, L = 0, that is,
lim n p ( y n , y n + 1 ) = 0 .
(9)
We claim that {y n } is a Cauchy sequence in the partial metric space (X, p). From Lemma 1.1, we need to prove that {y n } is Cauchy in the metric space (X, d p ). We argue by contradiction. Then there exist ε > 0 and a subsequence {yn(i)} of {y n } such that d p (yn(i), yn(i+1)) > 4ε. Select δ in (C2) as 0 < δε. By definition of the metric d p ,
d p ( x , y ) 2 p ( x , y ) for all x , y X ,
so p(yn(i), yn(i+1)) > 2ε. Since lim n p ( y n , y n + 1 ) = 0 , hence there exists N such that
p ( y n , y n + 1 ) < δ 6 whenever n N .
Let n(i) ≥ N. Then, there exist integers m(i) satisfying n(i) < m(i) < n(i + 1) such that
p ( y n ( i ) , y m ( i ) ) ε + δ 3 .
If not, then by triangle inequality (which holds even for partial metrics)
p ( y n ( i ) , y n ( i + 1 ) ) p ( y n ( i ) , y n ( i + 1 ) - 1 ) + p ( y n ( i + 1 ) - 1 , y n ( i + 1 ) ) < ε + δ 3 + δ 6 < 2 ε ,
it is a contradiction. Without loss of generality, we can assume n(i) to be odd. Let m(i) be the smallest even integer such that
p ( y n ( i ) , y m ( i ) ) ε + δ 3 .
(10)
Then
p ( y n ( i ) , y m ( i ) - 2 ) < ε + δ 3 ,
and
ε + δ 3 p ( y n ( i ) , y m ( i ) ) p ( y n ( i ) , y m ( i ) - 2 ) + p ( y m ( i ) - 2 , y m ( i ) - 1 ) + p ( y m ( i ) - 1 , y m ( i ) ) < ε + δ 3 + δ 6 + δ 6 = ε + 2 δ 3 .
(11)
Also, p ( y n ( i ) , y m ( i ) ) M ( x n ( i ) + 1 , x m ( i ) + 1 ) < ε + 2 δ 3 + δ 6 < ε + δ , that is,
ε < ε + δ 3 M ( x n ( i ) + 1 , x m ( i ) + 1 ) < ε + δ .
In view of (C2), this yields that p(yn(i)+1, ym(i)+1) ≤ ε. But then
p ( y n ( i ) , y m ( i ) ) p ( y n ( i ) , y n ( i ) + 1 ) + p ( y n ( i ) + 1 , y m ( i ) + 1 ) + p ( y m ( i ) + 1 , y m ( i ) ) < δ 6 + ε + δ 6 = ε + δ 3 ,
which contradicts (10). Hence {y n } is a Cauchy sequence in the metric space (X, d p ), so also in the partial metric (X, p) which is complete. Thus, there exists a point y in X such that from Lemmas 1.1, 2.2, and (9)
p ( y , y ) = lim n p ( y n , y ) = lim n p ( y n , y n ) = 0 .
(12)
This implies that
lim n + p ( y 2 n , y ) = lim n + p ( y 2 n - 1 , y ) = 0 .
(13)
Thus from (13) we have
lim n + p ( A x 2 n , y ) = lim n + p ( T x 2 n + 1 , y ) = 0
(14)
and
lim n + p ( B x 2 n - 1 , y ) = lim n + p ( S x 2 n , y ) = 0 .
(15)

Now we can suppose, without loss of generality, that SX is a closed subset of the partial metric space (X, p). From (15), there exists u X such that y = Su. We claim that p(Au, y) = 0. Suppose, to the contrary, that p(Au, y) > 0.

By (P4) and (C4) we get
p ( y , A u ) p ( y , B x 2 n + 1 ) + p ( A u , B x 2 n + 1 ) - p ( B x 2 n + 1 , B x 2 n + 1 ) p ( y , B x 2 n + 1 ) + p ( A u , B x 2 n + 1 ) p ( y , B x 2 n + 1 ) + max a [ p ( y , y 2 n ) + p ( A u , y ) + p ( y 2 n + 1 , y 2 n ) ] , b [ p ( y , y 2 n + 1 ) + p ( A u , y 2 n ) ] p ( y , B x 2 n + 1 ) + max a [ p ( y , y 2 n ) + p ( A u , y ) + p ( y 2 n + 1 , y 2 n ) ] , b [ p ( y , y 2 n + 1 ) + p ( A u , y ) + p ( y , y 2 n ) - p ( y , y ) ] .
Letting n → ∞ in the above inequality and using (12)-(15), we obtain
0 < p ( y , A u ) max { a p ( A u , y ) , b p ( A u , y ) } < p ( A u , y )
it is a contradiction since 0 a < 1 2 < 1 and 0 b < 1 2 < 1 . Thus, by Lemma 2.1, we deduce that
p ( A u , y ) = 0 and y = A u .
(16)

Since y = Su, then Au = Su, that is, u is a coincidence point of A and S. So we proved (I).

From AX TX and (16), we have y TX. Hence we deduce that there exists v X such that y = Tv. We claim that p(Bv, y) = 0. Suppose, to the contrary, that p(Bv, y) > 0. From (C4) and (16), we have
0 < p ( y , B v ) = p ( A u , B v ) max a [ p ( S u , T v ) + p ( A u , S u ) + p ( B v , T v ) ] , b [ p ( S u , B v ) + p ( A u , T v ) ] = max a [ p ( y , y ) + p ( y , y ) + p ( B v , y ) ] , b [ p ( y , B v ) + p ( y , y ) ] = max { a p ( B v , y ) , b p ( B v , y ) }
as y = Su = Au = Tv and p(y, y) = 0. Since 0 ≤ a < 1 and 0 ≤ b < 1, this implies that
p ( B v , y ) < p ( B v , y ) ,
which is a contradiction. Then, we deduce that
p ( B v , y ) = 0 and y = B v = T v ,
(17)

that is, v is a coincidence point of B and T, then (II) holds.

Since the pair {A, S} is weakly compatible, from (16), we have Ay = ASu = SAu = Sy. We claim that p(Ay, y) = 0. Suppose, to the contrary, that p(Ay, y) > 0. We have
p ( A y , y ) p ( A y , y 2 n + 1 ) + p ( y 2 n + 1 , y ) = p ( A y , B x 2 n + 1 ) + p ( y 2 n + 1 , y ) p ( y 2 n + 1 , y ) + max a [ p ( S y , T x 2 n + 1 ) + p ( A y , S y ) + p ( B x 2 n + 1 , T x 2 n + 1 ) ] , b [ p ( S y , B x 2 n + 1 ) + p ( A y , T x 2 n + 1 ) ] = p ( y 2 n + 1 , y ) + max a [ p ( A y , y 2 n ) + p ( A y , A y ) + p ( y 2 n + 1 , y 2 n ) ] , b [ p ( A y , y 2 n + 1 ) + p ( A y , y 2 n ) ] .
Using (12) and (p2), we get letting n → +∞
0 < p ( A y , y ) max 2 a p ( A y , y ) , 2 b p ( A y , y ) < p ( A y , y )
a contradiction. Then we deduce that
p ( A y , y ) = 0 and A y = S y = y .
(18)
Since the pair {B, T} is weakly compatible, from (17), we have By = BTv = TBv = Ty. We claim that p(By, y) = 0. Suppose, to the contrary, that p(By, y) > 0, then by (C4) and (18), we have
0 < p ( y , B y ) = p ( A y , B y ) max { a [ p ( S y , T y ) + p ( A y , S y ) + p ( B y , T y ) ] , b [ p ( S y , B y ) + p ( A y , T y ) ] } = max { a [ p ( y , B y ) + p ( y , y ) + p ( B y , B y ) ] , b [ p ( y , B y ) + p ( y , B y ) ] } max { 2 a , 2 b } p ( B y , y ) ,
since p(y, y) = 0. Thus, we get
p ( y , B y ) = 0 and B y = T y = y .
(19)
Now, combining (18) and (19), we obtain
y = A y = B y = S y = T y ,

that is, y is a common fixed point of A, B, S, and T with p(y, y) = 0.

Now we prove the uniqueness of a common fixed point. Let us suppose that z X is a common fixed point of A, B, S, and T such that p(z, y) > 0. Using (iv), we get
p ( y , z ) = p ( A y , B z ) max { a [ p ( A y , B z ) + p ( A y , A y ) + p ( B z , B z ) ] , b [ p ( A y , B z ) + p ( A z , B y ) ] } = max { a [ p ( y , z ) + p ( y , y ) + p ( z , z ) ] , 2 b p ( y , z ) } max { 2 a , 2 b } p ( y , z ) ) < p ( y , z ) ,

which is a contradiction. Then we deduce that z = y. Thus the uniqueness of the common fixed point is proved. The proof is completed.

Repeating the proof of Theorem 2.3, we get easily the following.

Corollary 2.4. Let A, B, S, and T be the self maps defined on a partial metric space (X, p) satisfying the following conditions:

(C1) AX TX and BX SX,

(C2) for all ε > 0, there exists δ > 0 such that for all x, y X
ε < M ( x , y ) < ε + δ p ( A x , B y ) ε ,

where M ( x , y ) = max { p ( S x , T y ) , p ( A x , S x ) , p ( B y , T y ) , 1 2 [ p ( S x , B y ) + p ( A x , T y ) ] } ,

(C3) for all x, y X with M(x, y) > 0 p(Ax, By) < M(x, y),

(C4) p(Ax, By) < k[p(Sx, Ty) + p(Ax, Sx) + p(By, Ty) + p(Sx, By) + p(Ax, Ty] for all x, y X and 0 k < 1 3 .

If one of AX, BX, SX, or TX is a complete subspace of X, then

(I) A and S have a coincidence point,

(II) B and T have a coincidence point.

Moreover, if A and S, as well as, B and T are weakly compatible, then A, B, S, and T have a unique common fixed point.

3 Some equivalence statements of Meir-Keeler contraction

Jachymski [30] proved the following important lemma.

Lemma 3.1. Let be a subset of [0, ∞) × [0, ∞). Then the following statements are equivalent:

(J 1) There exists a function δ : (0, ∞) → (0, ∞) such that for any ε > 0, δ(ε) > ε and

(J 1a) sup{δ(s) : s (0, ε)} ≥ δ(ε) and

(J 1b) ( s , t ) and 0 ≤ s < δ(ε) imply t < ε.

(J 2) There exist functions β, η : (0, ∞) → (0, ∞) such that, for any ε > 0, β(ε) > ε, η(ε) < ε, and ( s , t ) and 0 ≤ s < β(ε) imply t < η(ε).

(J 3) There exists an upper semi continuous function ϕ : [0, ∞) → [0, ∞) such that ϕ is non-decreasing, ϕ(s) < s for s > 0, and ( s , t ) implies tϕ(s).

(J 4) There exists a lower-semi continuous function δ : (0, ∞) → (0, ∞) such that for any δ is non-decreasing, for any ε > 0, δ(ε) > ε, and ( s , t ) and 0 ≤ s < δ(ε) imply t < ε.

(J 5) There exists a lower-semi continuous function ω : [0, ∞) → [0, ∞) such that for any ω is non-decreasing, ω(s) > s for s > 0 and ( s , t ) implies w(t) ≤ s.

Theorem 3.2. Let (X, p) be a partial metric space, and S, T, A i (i ) be self-mappings on X. For x, y X and for i, j , we define
M i j ( x , y ) = p ( S x , T y ) , p ( S x , A i x ) , p ( T y , A j y ) , [ p ( S x , A j y ) + p ( T y , A i x ) ] 2 .

Then the following statements are equivalent.

(JT 1) There exists a lower-semi continuous function δ : (0, ∞) → (0, ∞) such that, for any ε > 0, δ(ε) > ε and for any x, y X and distinct i, j
ε M i j ( x , y ) < δ ( ε ) i m p l i e s p ( A i x , A j y ) < ε .
(JT 2) There exists an upper-semi continuous function ϕ : [0, ∞) → [0, ∞) such that, ϕ is non-decreasing, ϕ(t) < t, and
p ( A i x , A j y ) ϕ ( M i j ( x , y ) ) .

for any x, y X and distinct i, j .

(JT 3) There exists a lower-semi continuous function ω : [0, ∞) → [0, ∞) such that, ω is non-decreasing, ω(s) > s for s > 0, and
ω ( p ( A i x , A j y ) ) M i j ( x , y )

for any x, y X and distinct i, j .

Proof. It follows immediately from Lemma 3.1.

Remark 3.1. In Theorem 1.2, Ćirić et al. assumed that the hypothesis p(Ax, By) ≤ ϕ(M(x, y)) is satisfied for all x, y X with ϕ Φ and obtained a common fixed point result.

In particular from the assumptions on that ϕ, (JT 2) holds for A1 = A and A2 = B. So, by Theorem 3.2, (JT 1) holds, that is; for all ε > 0, there exists δ > 0 such that for all x, y X
ε M ( x , y ) < ε + δ p ( A x , B y ) < ε ,
(20)

By Lemma 3.1 of Jachymski [31], (20) implies (as in metric cases) that the conditions (C2) and (C3) are satisfied, but nothing on the condition (C4). Conversely, in Theorem 2.3 we have assumed that (C2) and (C3) hold, but we added another condition which is (C4) in order to get a common fixed point result.

Remark 3.2. Theorem 2.3 is the analogous of Theorem 1 of Rana et al. [32] on partial metrics, except that the conditions (20) and the fact that a, b [0, 1], are replaced by the weaker conditions (C2), (C3) and a , b [ 0 , 1 2 ] . The condition on a and b is modified due to the fact that p(x, x) may not equal to 0 for x X. Also, Corollary 2.4 extends Theorem 2.1 of Bouhadjera and Djoudi [33] on partial metric cases. Note that Theorem 2.1 in [33] was improved recently by Akkouchi [[34], Corollary 4.4]. Indeed, the Lipschitz constant k is allowed to take values in the interval [ 0 , 1 2 ] instead of the case studied in [33], where the constant k belon gs to the smaller interval [ 0 , 1 3 ] .

Declarations

Authors’ Affiliations

(1)
Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse
(2)
Department of Mathematics, Atılım University

References

  1. Matthews SG: Partial Metric Topology. In Papers on General Topology and Applications, Eighth Summer Conference at Queens College, Annals New York Acad Sci Edited by: Susan J. Andima, Gerald Itzkowitz, T. Yung Kong. 1994, 183–197.Google Scholar
  2. Kopperman RD, Matthews SG, Pajoohesh H: What do partial metrics represent? In Spatial Representation: Discrete vs. Continuous Computational Models, Dagstuhl Seminar Proceedings Edited by: Kopperman, R, Smyth, MB, Spreen, D, Webster, J. 2005, 1–4.Google Scholar
  3. Künzi HPA, Pajoohesh H, Schellekens MP: Partial quasi-metrics. Theor Comput Sci 2006, 365(3):237–246. 10.1016/j.tcs.2006.07.050View ArticleGoogle Scholar
  4. Romaguera S, Schellekens M: Duality and quasi-normability for complexity spaces. Appl General Topol 2002, 3: 91–112.MathSciNetGoogle Scholar
  5. Schellekens MP: A characterization of partial metrizability: domains are quantifiable. Theor Comput Sci 2002, 3: 91–112.MathSciNetGoogle Scholar
  6. Abdeljawad T, Karapınar E, Tas K: Existence and uniqueness of common fixed point on partial metric spaces. Appl Math Lett 2011, 24(11):1900–1904. 10.1016/j.aml.2011.05.014MathSciNetView ArticleGoogle Scholar
  7. Altun I, Erduran A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl 2011, 1–10. 2011, (Article ID 508730)Google Scholar
  8. Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topol Appl 2010, 157(18):2778–2785. 10.1016/j.topol.2010.08.017MathSciNetView ArticleGoogle Scholar
  9. Altun I, Sadarangani K: Corrigendum to generalized contractions on partial metric spaces. Topol Appl 2010, 157: 2778–2785. Topol. Appl. 158(13), 1738–1740 (2011) 10.1016/j.topol.2010.08.017View ArticleGoogle Scholar
  10. Aydi H: Some fixed point results in ordered partial metric spaces. J Nonlinear Sci Appl 2011, 4(3):210–217.MathSciNetGoogle Scholar
  11. Aydi H: Some coupled fixed point results on partial metric spaces. Int J Math Math Sci 2011, 1–11. 2011, (Article ID 647091)Google Scholar
  12. Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J Adv Math Studies 2011, 4(2):1–12.MathSciNetGoogle Scholar
  13. Aydi H: Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces. J Nonlinear Anal Opt Theory Appl 2011, 2(2):33–48.MathSciNetGoogle Scholar
  14. Aydi H: Common fixed point results for mappings satisfying ( ψ,ϕ )-weak contractions in ordered partial metric spaces. Int J Math Stat 2012, 12(2):53–64.MathSciNetGoogle Scholar
  15. Aydi H: A common fixed point result by altering distances involving a contractive condition of integral type in partial metric spaces. Demonstratio Math, in press.Google Scholar
  16. Aydi H, Karapınar E, Shatanawi W: Coupled fixed point results for ( ψ, φ )-weakly contractive condition in ordered partial metric spaces. Comput Math Appl 2011, 62(12):4449–4460. 10.1016/j.camwa.2011.10.021MathSciNetView ArticleGoogle Scholar
  17. Ćirić Lj, Samet B, Aydi H, Vetro C: Common fixed points of generalized contractions on partial metric spaces and an application. Appl Math Comput 2011, 218(6):2398–2406. 10.1016/j.amc.2011.07.005MathSciNetView ArticleGoogle Scholar
  18. Karapınar E: Weak ϕ -contraction on partial metric spaces. J Comput Anal Appl 2012, 14(2):206–210.MathSciNetGoogle Scholar
  19. Karapınar E, Erhan IM: Fixed point theorems for operators on partial metric spaces. Appl Math Lett 2011, 24(11):1894–1899. 10.1016/j.aml.2011.05.013MathSciNetView ArticleGoogle Scholar
  20. Karapınar E: Generalizations of Caristi Kirk's theorem on partial metric spaces. Fixed Point Theory Appl 2011, 4. 2011:Google Scholar
  21. Karapınar E, Yuksel U: Some common fixed point theorems in partial metric spaces. J Appl Math 2011, 1–17. 2011, (Article ID 263621)Google Scholar
  22. Chi KP, Karapınar E, Thanh TD: A generalized contraction principle in partial metric spaces. Math Comput Model 2012, 55(5–6):1673–1681. 10.1016/j.mcm.2011.11.005View ArticleGoogle Scholar
  23. Karapınar E: A note on common fixed point theorems in partial metric spaces. Miskolc Math Notes 2011, 12(2):185–191.MathSciNetGoogle Scholar
  24. Oltra S, Valero O: Banach's fixed point theorem for partial metric spaces. Rend Istit Mat Univ Trieste 2004, 36(1–2):17–26.MathSciNetGoogle Scholar
  25. Romaguera S: Fixed point theorems for generalized contractions on partial metric spaces. Topl Appl 2012, 159: 164–199.Google Scholar
  26. Samet B, Rajović M, Lazović R, Stojiljković R: Common fixed-point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl 2011, 71. 2011:Google Scholar
  27. Shatanawi W, Samet B, Abbas M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math Comput Model 2012, 55(3–4):680–687. 10.1016/j.mcm.2011.08.042MathSciNetView ArticleGoogle Scholar
  28. Valero O: On Banach fixed point theorems for partial metric spaces. Appl General Topol 2005, 6(2):229–240.MathSciNetView ArticleGoogle Scholar
  29. Jungck G: Compatible mappings and common fixed points. Int J Math Math Sci 1986, 9: 771–779. 10.1155/S0161171286000935MathSciNetView ArticleGoogle Scholar
  30. Jachymski J: Equivalent conditions and the Meir-Keeler type theorems. J Math Anal Appl 1995, 194: 293–303. 10.1006/jmaa.1995.1299MathSciNetView ArticleGoogle Scholar
  31. Jachymski J: Common fixed point theorems for some families of maps. Ind J Pure Appl Math 1994, 25(9):925–937.MathSciNetGoogle Scholar
  32. Rana R, Dimri RC, Tomar A: Some fixed point theorems on Meir-Keeler type under strict contractions. Int J Comput Appl 2011, 17(3):24–30.Google Scholar
  33. Bouhadjera H, Djoudi A: On common fixed point theorems of Meir and Keeler type. An Şt Univ Ovidius Constanta 2008, 16(2):39–46.MathSciNetGoogle Scholar
  34. Akkouchi M: A Meir-Keeler type common fixed point theorem for four mappings. Opuscula Math 2011, 31: 5–14.MathSciNetView ArticleGoogle Scholar

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© Aydi and Karapinar; licensee Springer. 2012

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