Open Access

A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces

Fixed Point Theory and Applications20142014:40

https://doi.org/10.1186/1687-1812-2014-40

Received: 30 September 2013

Accepted: 4 February 2014

Published: 14 February 2014

Abstract

We obtain a fixed point theorem for generalized contractions on complete quasi-metric spaces, which involves w-distances and functions of Meir-Keeler and Jachymski type. Our result generalizes in various directions the celebrated fixed point theorems of Boyd and Wong, and Matkowski. Some illustrative examples are also given.

MSC:47H10, 54H25, 54E50.

Keywords

fixed point generalized contraction w-distance complete quasi-metric space

1 Introduction and preliminaries

In their celebrated paper [1], Kada, Suzuki and Takahashi introduced and studied the notion of a w-distance on a metric space. By using that notion they obtained, among other results, generalizations of the nonconvex minimization theorem of Takahashi [2], of Caristi’s fixed point theorem [3] and of Ekeland’s variational principle [4], as well as a general fixed point theorem that improves fixed point theorems of Subrahmanyam [5], Kannan [6] and Ćirić [7]. This study was continued by Suzuki and Takahashi [8], and by Park [9] who extended several results from [1] to quasi-metric spaces. Park’s approach was successful continued by Al-Homidan, Ansari and Yao [10], who obtained, among other interesting results, quasi-metric versions of Caristi-Kirk’s fixed point theorem and Nadler’s fixed point theorem by using Q-functions (a slight generalization of w-distances). More recently, Latif and Al-Mezel [11], and Marín et al. [1214] have proved some fixed point theorems both for single-valued and multi-valued mappings in complete quasi-metric spaces and preordered quasi-metric spaces by using Q-functions and w-distances, and generalizing in this way well-known fixed point theorems of Mizoguchi and Takahashi [15], Bianchini and Grandolfi [16], and Boyd and Wong [17], respectively.

In this paper we shall obtain a fixed point theorem for generalized contractions with respect to w-distances on complete quasi-metric spaces from which we deduce w-distance versions of Boyd and Wong’s fixed point theorem [17] and of Matkowski’s fixed point theorem [18]. Our approach uses a kind of functions considered by Jachymski in [[19], Corollary of Theorem 2] and that generalizes the notion of a function of Meir-Keeler type.

In the sequel the letters R + , and ω will denote the set of non-negative real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively.

By a quasi-metric on a set X we mean a function d : X × X R + such that for all x , y , z X :
  1. (i)

    d ( x , y ) = d ( y , x ) = 0 x = y , and

     
  2. (ii)

    d ( x , y ) d ( x , z ) + d ( z , y ) .

     

A quasi-metric space is a pair ( X , d ) such that X is a set and d is a quasi-metric on X.

Each quasi-metric d on a set X induces a topology τ d on X which has as a base the family of open balls { B d ( x , r ) : x X , ε > 0 } , where B d ( x , ε ) = { y X : d ( x , y ) < ε } for all x X and ε > 0 .

Given a quasi-metric d on X, the function d 1 defined by d 1 ( x , y ) = d ( y , x ) for all x , y X , is also a quasi-metric on X, and the function d s defined by d s ( x , y ) = max { d ( x , y ) , d ( y , x ) } for all x , y X , is a metric on X.

There exist several different notions of Cauchy sequence and of complete quasi-metric space in the literature (see e.g. [20]). In this paper we shall use the following general notion.

A quasi-metric space ( X , d ) is called complete if every Cauchy sequence ( x n ) n ω in the metric space ( X , d s ) converges with respect to the topology τ d 1 (i.e., there exists z X such that d ( x n , z ) 0 ).

Definition 1 ([9, 10])

A w-distance on a quasi-metric space ( X , d ) is a function q : X × X R + satisfying the following three conditions:

(W1) q ( x , y ) q ( x , z ) + q ( z , y ) for all x , y , z X ;

(W2) q ( x , ) : X R + is lower semicontinuous on ( X , τ d 1 ) for all x X ;

(W3) for each ε > 0 there exists δ > 0 such that q ( x , y ) δ and q ( x , z ) δ imply d ( y , z ) ε .

Several examples of w-distances on quasi-metric spaces may be found in [912].

Note that if d is a metric on X then it is a w-distance on ( X , d ) . Unfortunately, this does not hold for quasi-metric spaces, in general. Indeed, in [[12], Lemma 2.2] there was observed the following.

Lemma 1 If q is a w-distance on a quasi-metric space ( X , d ) , then for each ε > 0 there exists δ > 0 such that q ( x , y ) δ and q ( x , z ) δ imply d s ( y , z ) ε .

It follows from Lemma 1 (see [[12], Proposition 2.3]) that if a quasi-metric d on X is also a w-distance on ( X , d ) , then the topologies induced by d and by the metric d s coincide, so ( X , τ d ) is a metrizable topological space.

2 Results and examples

Meir and Keeler proved in [21] that if f is a self-map of a complete metric space ( X , d ) satisfying the condition that for each ε > 0 there is δ > 0 such that, for any x , y X , with ε d ( x , y ) < ε + δ we have d ( f x , f y ) < ε , then f has a unique fixed point z X and f n x z for all x X .

This well-known result suggests the notion of a Meir-Keeler function:

A function ϕ : R + R + is said to be a Meir-Keeler function if ϕ ( 0 ) = 0 , and satisfies the following condition:

(MK) For each ε > 0 there exists δ > 0 such that
ε t < ε + δ implies ϕ ( t ) < ε , for all  t R + .

Remark 1 It is obvious that if ϕ is a Meir-Keeler function then ϕ ( t ) < t for all t > 0 .

Later on, Jachymski proved in [19] the following interesting result and showed that both Boyd and Wong’s fixed point theorem and Matkowski’s fixed point theorem are easy consequences of it.

Theorem 1 ([[19], Corollary of Theorem 2])

Let f be a self-map of a complete metric space ( X , d ) such that d ( f x , f y ) < d ( x , y ) for x y , and d ( f x , f y ) ϕ ( d ( x , y ) ) for all x , y X , where ϕ : R + R + satisfies the condition

(Ja) for each ε > 0 there exists δ > 0 such that for any t R + ,
ε < t < ε + δ implies ϕ ( t ) ε .

Then f has a unique fixed point z X and f n x z for all x X .

Theorem 1 suggests the following notion:

A function ϕ : R + R + is said to be a Jachymski function if ϕ ( 0 ) = 0 and it satisfies condition (Ja) of Theorem 1.

Remark 2 Obviously, each Meir-Keeler function is a Jachymski function. However, the converse does not follow even in the case that ϕ ( t ) < t for all t > 0 : Indeed, let ϕ : R + R + defined as ϕ ( t ) = 0 for all t [ 0 , 1 ] and ϕ ( t ) = 1 otherwise. Clearly ϕ is a Jachymski function such that ϕ ( t ) < t for all t > 0 . Finally, for ε = 1 and any δ > 0 we have ϕ ( ε + δ / 2 ) = ε , so ϕ is not a Meir-Keeler function.

Now we establish the main result of this paper.

Theorem 2 Let f be a self-map of a complete quasi-metric space ( X , d ) . If there exist a w-distance q on ( X , d ) and a Jachymski function ϕ : R + R + such that ϕ ( t ) < t for all t > 0 , and
q ( f x , f y ) ϕ ( q ( x , y ) ) ,
(1)

for all x , y X , then f has a unique fixed point z X . Moreover q ( z , z ) = 0 .

Proof Fix x 0 X . For each n ω let x n = f n x 0 . Then
q ( x n + 1 , x n + 2 ) ϕ ( q ( x n , x n + 1 ) ) ,
(2)

for all n ω .

First, we shall prove that { x n } n ω is a Cauchy sequence in ( X , d s ) .

To this end put r n = q ( x n , x n + 1 ) for all n ω .

If there is n 0 ω such that r n 0 = 0 , then r n = 0 for all n n 0 by (2) and our assumption that ϕ ( 0 ) = 0 . Therefore q ( x n , x m ) = 0 whenever m > n n 0 by condition (W1), and consequently, d s ( x n , x m ) = 0 by Lemma 1. Thus x n = x n 0 + 1 for all n n 0 + 1 .

Otherwise, we assume, without loss of generality, that r n + 1 < r n for all n ω . Then { r n } n ω converges to some r R + . Of course, r < r n for all n ω .

If r > 0 there exists δ = δ ( r ) such that
r < t < r + δ ϕ ( t ) r .

Take n δ N such that r n < r + δ for all n n δ . Therefore ϕ ( r n ) r , so by condition (2), r n + 1 r for all n n δ , a contradiction. Consequently r = 0 .

Now choose an arbitrary ε > 0 . There exists δ = δ ( ε ) , with δ ( 0 , ε ) , for which conditions (W3) and (Ja) hold. Similarly, for δ / 2 there exists μ = μ ( δ / 2 ) , with μ ( 0 , δ / 2 ) for which conditions (W3) and (Ja) also hold, i.e.,

q ( x , y ) μ and q ( x , z ) μ , imply d ( y , z ) δ / 2 , and for any t > 0 , δ / 2 < t < δ / 2 + μ implies ϕ ( t ) δ / 2 .

Since r n 0 , there exists k 0 N such that r n < μ for all n k 0 .

By using a similar technique to the one given by Jachymski in [[19], Theorem 2] we shall prove, by induction, that for each k k 0 and each n N , we have
q ( x k , x n + k ) < δ 2 + μ .
(3)

Indeed, fix k k 0 . Since q ( x k , x k + 1 ) < μ , condition (3) follows for n = 1 .

Assume that (3) holds for some n N . We shall distinguish two cases.

  • Case 1: q ( x k , x n + k ) > δ / 2 . Then we deduce from the induction hypothesis and condition (Ja) that
    ϕ ( q ( x k , x n + k ) ) δ / 2 ,
so by (1), q ( x k + 1 , x n + k + 1 ) δ / 2 . Therefore
q ( x k , x n + k + 1 ) q ( x k , x k + 1 ) + q ( x k + 1 , x n + k + 1 ) < μ + δ 2 .
  • Case 2: q ( x k , x n + k ) δ / 2 .

If q ( x k , x n + k ) = 0 , we deduce that q ( x k + 1 , x n + k + 1 ) = 0 by (1). So, by (W1),
q ( x k , x n + k + 1 ) q ( x k , x k + 1 ) < μ < μ + δ 2 .
If q ( x k , x n + k ) > 0 , we deduce that ϕ ( q ( x k , x n + k ) ) < q ( x k , x n + k ) δ / 2 , so
q ( x k , x n + k + 1 ) q ( x k , x k + 1 ) + q ( x k + 1 , x n + k + 1 ) q ( x k , x k + 1 ) + ϕ ( q ( x k , x n + k ) ) < μ + δ 2 .
Now take i , j N with i , j > k . Then i = n + k and j = m + k for some n , m N . Hence, by (3),
q ( x k , x i ) = q ( x k , x n + k ) < δ 2 + μ < δ and q ( x k , x j ) = q ( x k , x m + k ) < δ 2 + μ < δ .

Now, from Lemma 1 it follows that d s ( x i , x j ) ε whenever i , j > k . We conclude that { x n } n N is a Cauchy sequence in ( X , d s ) .

Since ( X , d ) is complete, there exists z X such that d ( x n , z ) 0 .

Next we show that q ( x n , z ) 0 : Indeed, choose an arbitrary ε > 0 . We have proved (see (3)) that there is k 0 N such that q ( x k , x n + k ) < ε for all k k 0 and n N . Fix k k 0 . Since d ( x n , z ) 0 it follows from condition (W2) that, for n sufficiently large,
q ( x k , z ) < q ( x k , x n + k ) + ε .

Hence q ( x k , z ) < 2 ε for all k k 0 . We deduce that q ( x n , z ) 0 .

From (1) it follows that q ( x n + 1 , f z ) 0 . So d s ( z , f z ) = 0 by Lemma 1. Consequently z = f z , i.e., is a fixed point of f. Furthermore q ( z , z ) = 0 . In fact, otherwise we have
q ( z , z ) = q ( f z , f z ) ϕ ( q ( z , z ) ) < q ( z , z ) ,

a contradiction.

Finally, let u X such that u = f u and u z . If q ( u , z ) > 0 we deduce that
q ( u , z ) = q ( f u , f z ) ϕ ( q ( u , z ) ) < q ( u , z ) ,

a contradiction. So q ( u , z ) = 0 . Similarly we check that q ( u , u ) = 0 . Since q ( z , z ) = 0 , we deduce from Lemma 1 that d s ( u , z ) = 0 , i.e., u = z . We conclude that z is the unique fixed point of f. □

Corollary 1 Let f be a self-map of a complete metric space ( X , d ) . If there exist a w-distance q on ( X , d ) and a Jachymski function ϕ : R + R + such that ϕ ( t ) < t for all t > 0 , and
q ( f x , f y ) ϕ ( q ( x , y ) ) ,

for all x , y X , then f has a unique fixed point z X . Moreover q ( z , z ) = 0 .

Corollary 2 Let f be a self-map of a complete quasi-metric space ( X , d ) . If there exist a w-distance q on ( X , d ) and a Meir-Keeler function ϕ : R + R + such that
q ( f x , f y ) ϕ ( q ( x , y ) ) ,

for all x , y X , then f has a unique fixed point z X . Moreover q ( z , z ) = 0 .

Proof Apply Remarks 1 and 2, and Theorem 2. □

Corollary 3 [13]

Let f be a self-map of a complete quasi-metric space ( X , d ) . If there exist a w-distance q on ( X , d ) and a right upper semicontinuous function ϕ : R + R + such that ϕ ( 0 ) = 0 , ϕ ( t ) < t for all t > 0 , and
q ( f x , f y ) ϕ ( q ( x , y ) ) ,

for all x , y X , then f has a unique fixed point z X . Moreover q ( z , z ) = 0 .

Proof It suffices to show that ϕ is a Meir-Keeler function. Assume the contrary. Then there exist ε > 0 and a sequence { t n } n N of positive real numbers such that ε t n < ε + 1 / n but ϕ ( t n ) ε for all n N . Since ε ϕ ( ε ) > 0 , it follows from right upper semicontinuity of ϕ that ϕ ( t n ) ϕ ( ε ) < ε ϕ ( ε ) eventually, i.e., ϕ ( t n ) < ε , a contradiction. We conclude that f has a unique fixed point by Corollary 2. □

Corollary 4 Let f be a self-map of a complete quasi-metric space ( X , d ) . If there exist a w-distance q on ( X , d ) and a non-decreasing function ϕ : R + R + such that ϕ ( 0 ) = 0 , ϕ n ( t ) 0 for all t > 0 , and
q ( f x , f y ) ϕ ( q ( x , y ) ) ,
(4)

for all x , y X , then f has a unique fixed point z X . Moreover q ( z , z ) = 0 .

Proof Again it suffices to show that ϕ is a Meir-Keeler function. Assume the contrary. Then there exist ε > 0 and a sequence { t n } n N of positive real numbers such that ε t n < ε + 1 / n but ϕ ( t n ) ε for all n N . Since ϕ is non-decreasing we deduce that ϕ ( t ) ε whenever t ε . Hence ϕ n ( t ) ε whenever t ε , which contradicts the hypothesis that ϕ n ( t ) 0 for all t > 0 . We conclude that f has a unique fixed point by Corollary 2. □

Remark 3 In [22] the authors proved Corollary 2 for the case that ( X , d ) is a complete metric space. Note also that Boyd and Wong’s fixed point theorem [17] and Matkowski’s fixed point theorem [18] are special cases of Corollaries 3 and 4, respectively, when ( X , d ) is a complete metric space and q is the metric d.

We conclude the paper with some examples that illustrate and validate the obtained results.

The first example shows that condition ‘ ϕ ( t ) < t for all t > 0 ’ in Theorem 2 cannot be omitted.

Example 1 Let X = { 0 , 1 } and let d be the discrete metric on X, i.e., d ( x , x ) = 0 for all x X and d ( x , y ) = 1 whenever x y . Let f : X X defined as f 0 = 1 and f 1 = 0 , and ϕ : R + R + defined as ϕ ( 1 ) = 1 and ϕ ( t ) = 0 for all x R + { 1 } . It is clear that ϕ is a Jachysmki function such that
d ( f x , f y ) ϕ ( d ( x , y ) ) ,

for all x , y X . However, f has no fixed point.

The next is an example where we can apply Theorem 2 for an appropriate w-distance q on a complete quasi-metric space ( X , d ) but not for d. Moreover, Corollary 1 cannot be applied for any w-distance on the metric space ( X , d s ) .

Example 2 Let X = ω and let d be the quasi-metric on X defined as
d ( x , x ) = 0 for all  x X ; d ( n , 0 ) = 1 / n for all  n N ; d ( 0 , n ) = 1 for all  n N ; d ( n , m ) = | 1 / n 1 / m | for all  n , m N .

Clearly ( X , d ) is complete (observe that { n } n N is a Cauchy sequence in ( X , d s ) with d ( n , 0 ) 0 ).

Let q be the w-distance on ( X , d ) given by q ( x , y ) = y for all x , y X .

Now define f : X X as f 0 = 0 and f n = n 1 for all n N , and ϕ : R + R + as ϕ ( 0 ) = 0 and ϕ ( t ) = n 1 where t ( n 1 , n ] , n N .

It is routine to check that ϕ is a Jachymski function satisfying ϕ ( t ) < t for all t > 0 (in fact, it is a Meir-Keeler function).

Since q ( f x , f 0 ) = 0 for all x X , and for each n , m X with m 0 , we have
q ( f n , f m ) = f m = m 1 = ϕ ( m ) = ϕ ( q ( n , m ) ) ,

it follows that all conditions of Theorem 2 are satisfied. In fact z = 0 is the unique fixed point of f.

However, the contraction condition (1) is not satisfied for d. Indeed, for any n > 1 we have
d ( f 0 , f n ) = d ( 0 , n 1 ) = 1 > 0 = ϕ ( 1 ) = ϕ ( d ( 0 , n ) ) .

Finally, note that we cannot apply Corollary 1 because ( X , d s ) is not complete (observe that { n } n N is a Cauchy sequence in ( X , d s ) that does not converge in ( X , d s ) ).

We conclude with an example where we can apply Corollary 2 but not Corollaries 3 and 4.

Example 3 Let d be the quasi-metric on R + given by d ( x , y ) = max { y x , 0 } for all x , y R + . Since d s is the usual metric on R + it immediately follows that ( R + , d ) is complete.

Define q : R + × R + R + as q ( x , y ) = y . It is clear that q is a w-distance on ( R + , d ) .

Now let ϕ : R + R + , defined by ϕ ( t ) = t / 2 if t ( 1 , 2 ] , and ϕ ( t ) = 0 otherwise.

Then ϕ is a Meir-Keeler function: Indeed, we first note that ϕ ( 0 ) = 0 . Now, given ε > 0 we distinguish the following cases:
  1. (1)

    if 0 < ε < 1 , we take δ = 1 ε , and thus, from ε t < ε + δ = 1 , it follows ϕ ( t ) = 0 < ε ;

     
  2. (2)

    if ε = 1 , we take δ = 1 / 2 , and thus, from 1 < t < 3 / 2 , it follows ϕ ( t ) = t / 2 < 3 / 4 < ε , whereas ϕ ( 1 ) = 0 < ε ;

     
  3. (3)

    if 1 < ε < 2 , we take δ = 2 ε , and thus, from ε t < ε + δ = 2 , it follows ϕ ( t ) = t / 2 < 1 < ε ;

     
  4. (4)

    if ε 2 , we fix δ > 0 , and thus, from ε t < ε + δ , it follows ϕ ( t ) < ε because ϕ ( 2 ) = 1 and ϕ ( t ) = 0 for t > 2 .

     
Finally, taking f = ϕ , we obtain q ( f x , f y ) ϕ ( q ( x , y ) ) for all x , y X , because
q ( f x , f y ) = f y = ϕ ( y ) = ϕ ( q ( x , y ) ) .

Therefore, all conditions of Corollary 2 are satisfied. In fact, z = 0 is the unique fixed point of f.

However, ϕ is not right upper semicontinuous at t = 1 , so we cannot apply Corollary 3.

Similarly, we cannot apply Corollary 4 because ϕ is not a non-decreasing function.

Observe also that the w-distance q cannot be replaced by the quasi-metric d because for 1 < y 2 we have
d ( f 1 , f y ) = d ( 0 , y 2 ) = y 2 > 0 = ϕ ( y 1 ) = ϕ ( d ( 1 , y ) ) .

Declarations

Acknowledgements

The authors are grateful to the referees for several useful suggestions. They also thank the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.

Authors’ Affiliations

(1)
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València

References

  1. Kada O, Suzuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 1996, 44: 381–391.MathSciNetMATHGoogle Scholar
  2. Takahashi W: Existence theorems generalizing fixed point theorems for multivalued mappings. Pitman Research Notes in Mathematics Series 252. In Fixed Point Theory and Applications. Edited by: Théra MA, Baillon JB. Longman, Harlow; 1991:397–406.Google Scholar
  3. Caristi J: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.View ArticleMathSciNetMATHGoogle Scholar
  4. Ekeland I: Nonconvex minimization problems. Bull. Am. Math. Soc. 1979, 1: 443–474. 10.1090/S0273-0979-1979-14595-6View ArticleMathSciNetMATHGoogle Scholar
  5. Subrahmanyam PV: Remarks on some fixed point theorems related to Banach’s contraction principle. J. Math. Phys. Sci. 1974, 8: 445–457. Erratum 9, 195 (1975)MathSciNetMATHGoogle Scholar
  6. Kannan R: Some results on fixed points. II. Am. Math. Mon. 1969, 76: 405–408. 10.2307/2316437View ArticleMATHGoogle Scholar
  7. Ćirić L: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 1974, 45: 267–273.MATHGoogle Scholar
  8. Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. 1996, 8: 371–382.MathSciNetMATHGoogle Scholar
  9. Park S: On generalizations of the Ekeland-type variational principles. Nonlinear Anal. TMA 2000, 39: 881–889. 10.1016/S0362-546X(98)00253-3View ArticleMATHGoogle Scholar
  10. Al-Homidan S, Ansari QH, Yao JC: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. TMA 2008, 69: 126–139. 10.1016/j.na.2007.05.004View ArticleMathSciNetMATHGoogle Scholar
  11. Latif A, Al-Mezel SA: Fixed point results in quasimetric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 178306Google Scholar
  12. Marín J, Romaguera S, Tirado P: Q -functions on quasi-metric spaces and fixed points for multivalued maps. Fixed Point Theory Appl. 2011., 2011: Article ID 603861Google Scholar
  13. Marín J, Romaguera S, Tirado P: Weakly contractive multivalued maps and w -distances on complete quasi-metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 2Google Scholar
  14. Marín J, Romaguera S, Tirado P: Generalized contractive set-valued maps on complete preordered quasi-metric spaces. J. Funct. Spaces Appl. 2013., 2013: Article ID 269246Google Scholar
  15. Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 1989, 141: 177–188. 10.1016/0022-247X(89)90214-XView ArticleMathSciNetMATHGoogle Scholar
  16. Bianchini RM, Grandolfi M: Trasformazioni di tipo contrattivo generalizzato in uno spazio metrico. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8) 1968, 45: 212–216.MathSciNetMATHGoogle Scholar
  17. Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9View ArticleMathSciNetMATHGoogle Scholar
  18. Matkowski J: Integrable solutions of functional equations. Diss. Math. 1975, 127: 1–68.MathSciNetMATHGoogle Scholar
  19. Jachymski J: Equivalent conditions and the Meir-Keeler type theorems. J. Math. Anal. Appl. 1995, 194: 293–303. 10.1006/jmaa.1995.1299View ArticleMathSciNetMATHGoogle Scholar
  20. Künzi HPA: Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology. 3. In Handbook of the History of General Topology. Edited by: Aull CE, Lowen R. Kluwer Academic, Dordrecht; 2001:853–968.View ArticleGoogle Scholar
  21. Meir A, Keeler E: A theorem on contraction mappings. J. Math. Anal. Appl. 1969, 28: 326–329. 10.1016/0022-247X(69)90031-6View ArticleMathSciNetMATHGoogle Scholar
  22. Alegre C, Marín J, Romaguera S: Fixed points for generalized contractions with respect to w -distances and Meir-Keeler functions. Proceedings of the Conference in Applied Topology 2013, 53–58. WiAT’13 Bilbao, SpainGoogle Scholar

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