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

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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.

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,zX:

  1. (i)

    d(x,y)=d(y,x)=0x=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):xX,ε>0}, where B d (x,ε)={yX:d(x,y)<ε} for all xX 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,yX, 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,yX, 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 zX 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,zX;

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

(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,yX, with εd(x,y)<ε+δ we have d(fx,fy)<ε, then f has a unique fixed point zX and f n xz for all xX.

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

εt<ε+δimpliesϕ(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(fx,fy)<d(x,y) for xy, and d(fx,fy)ϕ(d(x,y)) for all x,yX, 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+δimpliesϕ(t)ε.

Then f has a unique fixed point zX and f n xz for all xX.

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(fx,fy)ϕ ( q ( x , y ) ) ,
(1)

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

Proof Fix x 0 X. For each nω 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 nω.

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 nω.

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 nω. Then { r n } n ω converges to some r R + . Of course, r< r n for all nω.

If r>0 there exists δ=δ(r) such that

r<t<r+δϕ(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 nN, 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 nN. 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,jN with i,j>k. Then i=n+k and j=m+k for some n,mN. Hence, by (3),

q( x k , x i )=q( x k , x n + k )< δ 2 +μ<δandq( 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 zX 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 nN. 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)<2ε for all k k 0 . We deduce that q( x n ,z)0.

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

q(z,z)=q(fz,fz)ϕ ( q ( z , z ) ) <q(z,z),

a contradiction.

Finally, let uX such that u=fu and uz. If q(u,z)>0 we deduce that

q(u,z)=q(fu,fz)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(fx,fy)ϕ ( q ( x , y ) ) ,

for all x,yX, then f has a unique fixed point zX. 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(fx,fy)ϕ ( q ( x , y ) ) ,

for all x,yX, then f has a unique fixed point zX. 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(fx,fy)ϕ ( q ( x , y ) ) ,

for all x,yX, then f has a unique fixed point zX. 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 nN. Since εϕ(ε)>0, it follows from right upper semicontinuity of ϕ that ϕ( t n )ϕ(ε)<εϕ(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(fx,fy)ϕ ( q ( x , y ) ) ,
(4)

for all x,yX, then f has a unique fixed point zX. 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 nN. Since ϕ is non-decreasing we deduce that ϕ(t)ε whenever tε. Hence ϕ n (t)ε whenever tε, 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 xX and d(x,y)=1 whenever xy. Let f:XX defined as f0=1 and f1=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(fx,fy)ϕ ( d ( x , y ) ) ,

for all x,yX. 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,yX.

Now define f:XX as f0=0 and fn=n1 for all nN, and ϕ: R + R + as ϕ(0)=0 and ϕ(t)=n1 where t(n1,n], nN.

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(fx,f0)=0 for all xX, and for each n,mX with m0, we have

q(fn,fm)=fm=m1=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(f0,fn)=d(0,n1)=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{yx,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=1ε, and thus, from εt<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=2ε, and thus, from εt<ε+δ=2, it follows ϕ(t)=t/2<1<ε;

  4. (4)

    if ε2, we fix δ>0, and thus, from εt<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(fx,fy)ϕ(q(x,y)) for all x,yX, because

q(fx,fy)=fy=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<y2 we have

d(f1,fy)=d ( 0 , y 2 ) = y 2 >0=ϕ(y1)=ϕ ( d ( 1 , y ) ) .

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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.

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Correspondence to Salvador Romaguera.

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The three authors contributed equally in writing this article. They read and approved the final manuscript.

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Alegre, C., Marín, J. & Romaguera, S. A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces. Fixed Point Theory Appl 2014, 40 (2014) doi:10.1186/1687-1812-2014-40

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Keywords

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