# A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application

## Abstract

We obtain a fixed point theorem for a type of generalized contractions on preordered complete fuzzy quasi-metric spaces which is applied to deduce, among other results, a procedure to show in a direct and easy fashion the existence of solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively.

MSC:47H10, 54H25, 06A06, 68Q25.

## 1 Introduction

In 1975, Matkowski [, Theorem 1.2] proved the following distinguished generalization of Banach’s contraction principle.

Theorem 1 

Let $\left(X,d\right)$ be a complete fuzzy metric space and $f:X\to X$ a self-map such that

$d\left(fx,fy\right)\le \phi \left(d\left(x,y\right)\right)$

for all $x,y\in X$, where $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is a nondecreasing function satisfying ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. Then f has a unique fixed point.

Remark 1 It is well known and easy to check that if $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is a nondecreasing function such that ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$, then $\phi \left(t\right) for all $t>0$.

Matkowski’s theorem has been generalized or extended in several directions (see e.g. ). In particular, Jachymski obtained in [, Theorem 1] the following nice fuzzy version of it.

Theorem 2 

Let $\left(X,M,\ast \right)$ be a complete fuzzy metric space, with a continuous t-norm of Hadžić type, and let $f:X\to X$ be a self-map such that

$M\left(fx,fy,\phi \left(t\right)\right)\ge M\left(x,y,t\right)$

for all $x,y\in X$ and $t>0$, where $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is a function satisfying $0<\phi \left(t\right) and ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. Then f has a unique fixed point.

Remark 2 See  or  for the notion of a t-norm of Hadžić type (or of h-type).

Recently, Ricarte and Romaguera [, Theorem 2.2] established the following new fuzzy version of Matowski’s theorem by using a type of contraction introduced in the fuzzy intuitionistic context by Huang et al. , and that generalizes C-contractions as defined by Hicks in .

Theorem 3 

Let $\left(X,M,\ast \right)$ be a complete fuzzy metric space and $f:X\to X$ a self-map such that

$M\left(x,y,t\right)>1-t\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}M\left(fx,fy,\phi \left(t\right)\right)>1-\phi \left(t\right)$

for all $x,y\in X$ and $t>0$, where $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is a nondecreasing function satisfying ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. Then f has a unique fixed point.

In this paper we obtain a generalization of Theorem 3 to preordered fuzzy quasi-metric spaces which is applied to deduce, among other results, a procedure to show in a direct and easy way the existence of solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively. The key for this application is the nice fact that, for the specialization order of a fuzzy quasi-metric space, the contraction condition of Theorem 3 is automatically satisfied whenever the self-map f is nondecreasing for the specialization order and $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ is any function verifying $\phi \left(t\right)>0$ for all $t>0$ (see Theorem 5 in Section 3).

## 2 Background

In this section we recall several notions and properties which will be useful in the rest of the paper. Our basic reference for quasi-metric spaces is  and for fuzzy (quasi-)metric spaces they are [12, 13].

The letters , and ω will denote the set of real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively.

A preorder on a nonempty set X is a reflexive and transitive binary relation on X.

The preorder is called a partial order, or simply an order, if it is antisymmetric (i.e., condition $x⪯y$ and $y⪯x$, implies $x=y$).

Note that for any nonempty set X, the binary relation ${⪯}^{t}$ defined by $x{⪯}^{t}y$ if and only if $x,y\in X$, is obviously a preorder on X, the so-called trivial preorder on X.

A quasi-metric on a set X is a function $d:X×X\to \left[0,\mathrm{\infty }\right)$ such that for all $x,y,z\in X$: (i) $d\left(x,y\right)=d\left(y,x\right)=0$ if and only if $x=y$, and (ii) $d\left(x,y\right)\le d\left(x,z\right)+d\left(z,y\right)$.

A quasi-metric space is a pair $\left(X,d\right)$ such that X is a nonempty set and d is a quasi-metric on X.

Given a quasi-metric d on X the function ${d}^{s}$ defined by ${d}^{s}\left(x,y\right)=max\left\{d\left(x,y\right),d\left(y,x\right)\right\}$ for all $x,y\in X$, is a metric on X.

If d is a quasi-metric on X, then the relation ${\le }_{d}$ on X given by

$x{\le }_{d}y\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}d\left(x,y\right)=0,$

is an order on X, called the specialization order of d.

According to [14, 15], a binary operation $\ast :\left[0,1\right]×\left[0,1\right]\to \left[0,1\right]$ is a continuous t-norm if satisfies the following conditions: (i) is associative and commutative; (ii) is continuous; (iii) $a\ast 1=a$ for every $a\in \left[0,1\right]$; (iv) $a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$, with $a,b,c,d\in \left[0,1\right]$.

Two interesting examples are , and ${\ast }_{L}$, where, for all $a,b\in \left[0,1\right]$, $a\wedge b=min\left\{a,b\right\}$, and ${\ast }_{L}$ is the well-known Lukasiewicz t-norm defined by $a{\ast }_{L}b=max\left\{a+b-1,0\right\}$.

It seems appropriate to point out that  ≤  for any continuous t-norm .

Definition 1 [12, 13]

A KM-fuzzy quasi-metric on a set X is a pair $\left(M,\ast \right)$ such that is a continuous t-norm and M is a fuzzy set in $X×X×\left[0,\mathrm{\infty }\right)$ (i.e., a function from $X×X×\left[0,\mathrm{\infty }\right)$ into $\left[0,1\right]$) such that for all $x,y,z\in X$:

(KM1) $M\left(x,y,0\right)=0$;

(KM2) $x=y$ if and only if $M\left(x,y,t\right)=M\left(y,x,t\right)=1$ for all $t>0$;

(KM3) $M\left(x,z,t+s\right)\ge M\left(x,y,t\right)\ast M\left(y,z,s\right)$ for all $t,s\ge 0$;

(KM4) $M\left(x,y,\mathrm{_}\right):\left[0,\mathrm{\infty }\right)\to \left[0,1\right]$ is left continuous.

A KM-fuzzy quasi-metric $\left(M,\ast \right)$ on X such that for each $x,y\in X$:

(KM5) $M\left(x,y,t\right)=M\left(y,x,t\right)$ for all $t>0$

is a fuzzy metric on X (in the sense of Kramosil and Michalek ).

A simple but useful fact (see e.g. [12, 13]) is that for each KM-fuzzy quasi-metric $\left(M,\ast \right)$ on a set X and each $x,y\in X$, the function $M\left(x,y,\mathrm{_}\right)$ is nondecreasing.

In the following, KM-fuzzy quasi-metrics will be simply called fuzzy quasi-metrics.

If $\left(M,\ast \right)$ is a fuzzy quasi-metric on a set X, then the pair $\left({M}^{i},\ast \right)$ is a fuzzy metric on X where ${M}^{i}$ is the fuzzy set in $X×X×\left[0,\mathrm{\infty }\right)$ defined by ${M}^{i}\left(x,y,t\right)=min\left\{M\left(x,y,t\right),M\left(y,x,t\right)\right\}$ for all $x,y\in X$ and $t\ge 0$.

As in the fuzzy metric case, each fuzzy quasi-metric $\left(M,\ast \right)$ on a set X induces a topology ${\tau }_{M}$ on X which has as a base the family of open balls $\left\{{B}_{M}\left(x,\epsilon ,t\right):x\in X,0<\epsilon <1,t>0\right\}$, where ${B}_{M}\left(x,\epsilon ,t\right):=\left\{y\in X:M\left(x,y,t\right)>1-\epsilon \right\}$.

It immediately follows that a sequence ${\left({x}_{n}\right)}_{n\in \omega }$ in a fuzzy quasi-metric space $\left(X,M,\ast \right)$ converges to a point $x\in X$ with respect to ${\tau }_{M}$ if and only if ${lim}_{n\to \mathrm{\infty }}M\left(x,{x}_{n},t\right)=1$ for all $t>0$.

Definition 2 A fuzzy (quasi-)metric space is a triple $\left(X,M,\ast \right)$ such that X is a set and $\left(M,\ast \right)$ is a fuzzy (quasi-)metric on X.

Definition 3 A preordered fuzzy (quasi-)metric space is 4-tuple $\left(X,M,⪯,\ast \right)$ such that $\left(X,M,\ast \right)$ is a fuzzy (quasi-)metric space and is a preorder on X.

The notion of an ordered fuzzy (quasi-)metric space is defined in the obvious manner.

Remark 3 If $\left(M,\ast \right)$ is a fuzzy quasi-metric on X, then the relation ${\le }_{M}$ on X given by

is an order on X, called the specialization order of $\left(M,\ast \right)$.

It is interesting to note that, however, the relation ${\le }_{{M}_{p}}$ given by

is a preorder on X (see e.g. [, Remark 3]).

We conclude this section with two typical examples of fuzzy quasi-metrics induced by a given quasi-metric space.

Example 1 Let $\left(X,d\right)$ be a quasi-metric space. Then the pair $\left({M}_{01},\ast \right)$ is a fuzzy quasi-metric on X where is any continuous t-norm and ${M}_{01}$ is the fuzzy set on $X×X×\left[0,\mathrm{\infty }\right)$ given by ${M}_{01}\left(x,y,t\right)=1$ if $t>0$ and $d\left(x,y\right), and ${M}_{01}\left(x,y,t\right)=0$ otherwise.

Let $\left(X,d\right)$ be a quasi-metric space. Then the pair $\left({M}_{d},\ast \right)$ is a fuzzy quasi-metric on X where is any continuous t-norm and ${M}_{d}$ is the fuzzy set on $X×X×\left[0,\mathrm{\infty }\right)$ given by ${M}_{d}\left(x,y,0\right)=0$ for all $x,y\in X$ and

${M}_{d}\left(x,y,t\right)=\frac{t}{t+d\left(x,y\right)}$

for all $x,y\in X$ and $t>0$.

Remark 4 Let $\left(X,d\right)$ be a quasi-metric space. It is clear that the specialization orders ${\le }_{d}$, ${\le }_{{M}_{01}}$ and ${\le }_{{M}_{d}}$ coincide, while the preorder ${\le }_{{\left({M}_{01}\right)}_{p}}$ coincides with the trivial preorder ${⪯}^{t}$ on X, and the preorder ${\le }_{{\left({M}_{d}\right)}_{p}}$ coincides with the specialization order ${\le }_{{M}_{d}}$.

## 3 The fixed point theorem and some of its consequences

We start this section with the notions of fuzzy quasi-metric completeness and continuity of self-maps which will be used in our main result (Theorem 4 below).

A left K-Cauchy sequence in a fuzzy quasi-metric space $\left(X,M,\ast \right)$ is a sequence ${\left({x}_{n}\right)}_{n\in \omega }$ in X such that for each $t>0$ and each $\epsilon \in \left(0,1\right)$ there is ${n}_{\epsilon }\in \mathbb{N}$ such that $M\left({x}_{n},{x}_{m},t\right)>1-\epsilon$ whenever $m\ge n\ge {n}_{\epsilon }$.

A preordered fuzzy quasi-metric space $\left(X,M,⪯,\ast \right)$ will be called -complete if for each nondecreasing left K-Cauchy sequence ${\left({x}_{n}\right)}_{n\in \omega }$ there is $x\in X$ such that ${\left({x}_{n}\right)}_{n\in \omega }$ converges to x with respect to ${\tau }_{{M}^{i}}$, and ${x}_{n}⪯x$ for all $n\in \omega$.

Observe that if $\left(X,M,{⪯}^{t},\ast \right)$ is a preordered fuzzy metric space which is ${⪯}^{t}$-complete, then $\left(X,M,\ast \right)$ is a complete fuzzy metric space in the usual sense (see e.g. ).

Let $\left(X,M,⪯,\ast \right)$ be a preordered fuzzy quasi-metric space. A self-map $f:X\to X$ is said to be -nondecreasing if condition $x⪯y$ implies $fx⪯fy$ for all $x,y\in X$, and it will be called -continuous if whenever ${\left({x}_{n}\right)}_{n\in \omega }$ is a nondecreasing sequence for , which converges with respect to ${\tau }_{{M}^{i}}$ to some $x\in X$ such that ${x}_{n}⪯x$ for all $n\in \omega$, then the sequence ${\left(f{x}_{n}\right)}_{n\in \omega }$ converges to fx with respect to ${\tau }_{{M}^{i}}$.

Theorem 4 Let $\left(X,M,⪯,\ast \right)$ be a preordered -complete fuzzy quasi-metric space and $f:X\to X$ a -continuous nondecreasing self-map such that

$M\left(x,y,t\right)>1-t\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}M\left(fx,fy,\phi \left(t\right)\right)>1-\phi \left(t\right)$
(1)

for all $x,y\in X$ with $x⪯y$, and $t>0$, where $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ satisfies ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. If there is ${x}_{0}\in X$ such that ${x}_{0}⪯f{x}_{0}$, then $Fix\left(f\right)\ne \mathrm{\varnothing }$.

If, in addition, condition (1) is satisfied for each $x,y\in Fix\left(f\right)$, then f has a unique fixed point.

Proof Take ${t}_{0}>1$. Let $x,y\in X$ with $x⪯y$. Since $M\left(x,y,{t}_{0}\right)>1-{t}_{0}$, it follows that

$M\left(fx,fy,\phi \left({t}_{0}\right)\right)>1-\phi \left({t}_{0}\right).$
(2)

Since f is nondecreasing, we deduce that ${f}^{n}x⪯{f}^{n}y$ for all $n\in \omega$, so, by (1) and (2), we immediately deduce that

$M\left({f}^{n}x,{f}^{n}y,{\phi }^{n}\left({t}_{0}\right)\right)>1-{\phi }^{n}\left({t}_{0}\right)$
(3)

for all $x,y\in X$ with $x⪯y$ and $n\in \omega$.

Now let ${x}_{0}\in X$ such that ${x}_{0}⪯f{x}_{0}$. Put ${x}_{n}={f}^{n}{x}_{0}$ for all $n\in \omega$. Since f is nondecreasing it follows that ${\left({x}_{n}\right)}_{n\in \omega }$ is a nondecreasing sequence for .

Similarly to the proof of [, Theorem 2.2], we show that ${\left({x}_{n}\right)}_{n\in \omega }$ is a left K-Cauchy sequence in $\left(X,M,\ast \right)$. Choose $\epsilon \in \left(0,1\right)$ and $t>0$. Then there exists ${n}_{\epsilon }\in \mathbb{N}$ such that ${\phi }^{n}\left({t}_{0}\right) for all $n\ge {n}_{\epsilon }$. Let $m>n\ge {n}_{\epsilon }$. Then $m=n+k$ for some $k\in \mathbb{N}$, so, by (3)

$\begin{array}{rcl}M\left({x}_{n},{x}_{m},t\right)& =& M\left({f}^{n}{x}_{0},{f}^{n}{f}^{k}{x}_{0},t\right)\ge M\left({f}^{n}{x}_{0},{f}^{n}{f}^{k}{x}_{0},{\phi }^{n}\left({t}_{0}\right)\right)\\ >& 1-{\phi }^{n}\left({t}_{0}\right)>1-\epsilon .\end{array}$

Therefore ${\left({x}_{n}\right)}_{n\in \omega }$ is a nondecreasing left K-Cauchy sequence in $\left(X,M,\ast \right)$. Since $\left(X,M,⪯,\ast \right)$ is -complete there exists $z\in X$ such that ${lim}_{n\to \mathrm{\infty }}{M}^{i}\left({x}_{n},z,t\right)=1$ for all $t>0$, and ${x}_{n}⪯z$ for all $n\in \omega$. So ${lim}_{n\to \mathrm{\infty }}{M}^{i}\left({x}_{n},fz,t\right)=1$ for all $t>0$, by -continuity of f. Hence $z=fz$.

Finally suppose that $u=fu$ for some $u\in X$. From our hypothesis it follows, exactly as in the first part of the proof, that

$M\left({f}^{n}z,{f}^{n}u,{\phi }^{n}\left({t}_{0}\right)\right)>1-{\phi }^{n}\left({t}_{0}\right)$

for all $n\in \omega$. Since ${f}^{n}z=z$ and ${f}^{n}u=u$ we deduce, for $t>0$, $\epsilon \in \left(0,t\right)$ and $n\in \mathbb{N}$ with ${\phi }^{n}\left({t}_{0}\right)<\epsilon$, that

$M\left(z,u,t\right)\ge M\left(z,u,\epsilon \right)\ge M\left(z,u,{\phi }^{n}\left({t}_{0}\right)\right)>1-{\phi }^{n}\left({t}_{0}\right)>1-\epsilon .$

Since ε is arbitrary, we conclude that $M\left(z,u,t\right)=1$ for all $t>0$. Similarly $M\left(u,z,t\right)=1$ for all $t>0$, so $u=z$. This completes the proof. □

Corollary 1 Let $\left(X,M,⪯,\ast \right)$ be a preordered -complete fuzzy quasi-metric space and $f:X\to X$ a -nondecreasing self-map such that

$M\left(x,y,t\right)>1-t\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}M\left(fx,fy,\phi \left(t\right)\right)>1-\phi \left(t\right)$
(4)

for all $x,y\in X$ with $x⪯y$, and $t>0$, where $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ satisfies $0<\phi \left(t\right) and ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. If there is ${x}_{0}\in X$ such that ${x}_{0}⪯f{x}_{0}$, then $Fix\left(f\right)\ne \mathrm{\varnothing }$.

If, in addition, condition (4) is satisfied for each $x,y\in Fix\left(f\right)$, then f has a unique fixed point.

Proof We shall show that f is -continuous in $\left(X,M,⪯,\ast \right)$. Let ${\left({x}_{n}\right)}_{n\in \omega }$ be a nondecreasing sequence for , convergent with respect to ${\tau }_{{M}^{i}}$ to some $x\in X$ such that ${x}_{n}⪯x$ for all $n\in \omega$. Given $\epsilon \in \left(0,1\right)$ there is ${n}_{\epsilon }\in \mathbb{N}$ such that ${M}^{i}\left(x,{x}_{n},\epsilon \right)>1-\epsilon$. By condition (4) and the fact that $0<\phi \left(\epsilon \right)<\epsilon$ it follows that ${M}^{i}\left(fx,f{x}_{n},\epsilon \right)>1-\epsilon$ for all $n\ge {n}_{\epsilon }$. Therefore f is -continuous. The conclusions follow from Theorem 4. □

As an immediate consequence of Corollary 1 we obtain the following improvement of Theorem 3.

Corollary 2 Let $\left(X,M,\ast \right)$ be a complete fuzzy metric space and $f:X\to X$ a self-map such that

$M\left(x,y,t\right)>1-t\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}M\left(fx,fy,\phi \left(t\right)\right)>1-\phi \left(t\right)$

for all $x,y\in X$ and $t>0$, where $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ satisfies $0<\phi \left(t\right) and ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. Then f has a unique fixed point.

Proof It is clear that $\left(X,M,{⪯}^{t},\ast \right)$ is a preordered ${⪯}^{t}$-complete fuzzy metric space. Now the result follows from Corollary 1. □

The contraction (4) is almost trivially satisfied in the case that the preorder is the specialization order ${\le }_{M}$ and f is nondecreasing for ${\le }_{M}$. This situation, which will be crucial in our application in Section 4, is described in the following result.

Theorem 5 If the ordered fuzzy quasi-metric space $\left(X,M,{\le }_{M},\ast \right)$ is ${\le }_{M}$-complete and $f:X\to X$ is a ${\le }_{M}$-nondecreasing self-map such that there is ${x}_{0}\in X$ satisfying ${x}_{0}{\le }_{M}f{x}_{0}$, then f has a fixed point.

Proof Let $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be any function satisfying $0<\phi \left(t\right) and ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. Then, for each $x,y\in X$ such that $x{\le }_{M}y$, we have $fx{\le }_{M}fy$, and thus

$M\left(fx,fy,\phi \left(t\right)\right)=1>1-\phi \left(t\right)$

for all $t>0$. This shows that condition (4) is satisfied, and thus f has a fixed point by Corollary 1. □

We conclude this section with two examples illustrating the obtained results.

Example 3 Let $X=\left[0,\mathrm{\infty }\right)$ and let M be the fuzzy set in $X×X×\left[0,\mathrm{\infty }\right)$ defined as

It is routine to check that $\left(M,\wedge \right)$ is a fuzzy quasi-metric. Note that the specialization order ${\le }_{M}$ coincides with the usual order ≤ on X. Moreover, a sequence in X is left K-Cauchy in $\left(X,M,\wedge \right)$ if and only it is eventually constant, so $\left(X,M,\le ,\wedge \right)$ is an ordered ≤-complete fuzzy quasi-metric space.

Now let $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ defined as

Clearly ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. We show that if $f:X\to X$ is any ≤-nondecreasing self-map, the contraction condition (1) is satisfied for $x,y\in X$ with $x and $t>0$. (Note that f is automatically continuous and hence ≤-continuous, because ${\tau }_{{M}^{i}}$ is the discrete topology on X.)

Let $x\le y$ and $t>0$. Then $fx\le fy$ and $\phi \left(t\right)>0$, so

$M\left(fx,fy,\phi \left(t\right)\right)=1>1-\phi \left(t\right),$

so condition (1) is trivially satisfied. If, in addition, there exists ${x}_{0}\in X$ such that ${x}_{0}\le f{x}_{0}$, then f has a fixed point by Theorem 4 or Theorem 5. Observe that in this example we cannot apply Corollary 1 because $\phi \left(t\right)>t$ for $1/2; in fact, it is not nondecreasing.

Example 4 Let $X=\left\{a,b,c\right\}$ and let $\left(M,\wedge \right)$ be the fuzzy metric on X defined as $M\left(x,x,t\right)=1$ for all $x\in X$, $M\left(a,b,t\right)=M\left(b,a,t\right)=1/2$ for all $t\in \left(0,1\right]$, $M\left(a,b,t\right)=M\left(b,a,t\right)=1$ for all $t>1$, and $M\left(x,y,t\right)=0$ otherwise. Now define a preorder on X as follows: $x⪯x$ for all $x\in X$, $a⪯b$, and $b⪯a$. Obviously is not an order on X. Clearly $\left(X,M,⪯,\ast \right)$ is -complete. Let $f:X\to X$ be such that $fa=a$, $fb=a$ and $fc=c$. Then f is a -nondecreasing self-map with $a⪯fa$ (also $c⪯fc$). Let $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ be any function such that $0<\phi \left(t\right) and ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$. Since $M\left(fa,fb,\phi \left(t\right)\right)=M\left(a,a,\phi \left(t\right)\right)=1$ for all $t>0$, we deduce that the conditions of Corollary 1, and hence of Theorem 4, for $x,y\in X$ with $x⪯y$, are satisfied. However, we cannot apply Corollary 2 to this example. Indeed, suppose that there exists a function $\phi :\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$ satisfying $0<\phi \left(t\right) and ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$ for all $t>0$, and such that

$M\left(a,c,t\right)>1-t\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}M\left(fa,fc,\phi \left(t\right)\right)>1-\phi \left(t\right)$

for all $t>0$. Since condition $M\left(a,c,t\right)>1-t$ holds for $t>1$, we deduce that, for $t>1$, $M\left(fa,fc,\phi \left(t\right)\right)>1-\phi \left(t\right)$, so $\phi \left(t\right)>1$ because $M\left(fa,fc,\phi \left(t\right)\right)=M\left(a,c,\phi \left(t\right)\right)=0$. Repeating this argument, we deduce that ${\phi }^{n}\left(t\right)>1$ for all $n\in \omega$, which contradicts the fact that ${lim}_{n\to \mathrm{\infty }}{\phi }^{n}\left(t\right)=0$.

## 4 An application

In this section we apply Theorem 5 to obtain a general procedure from which we can deduce in a fast and easy fashion the existence the solution for the recurrence equations that are typically associated to Quicksort and Divide and Conquer algorithms, respectively.

Let us recall that Schellekens introduced in  the so-called complexity quasi-metric space in order to construct a topological foundation for the complexity analysis of programs and algorithms. In that paper, he also applied his theory to show that the existence and uniqueness of solution for the recurrence associated to Divide and Conquer algorithms. Further contributions to the study of these spaces and of other related ones may be found in [12, 2125], etc.

The complexity (quasi-metric) space (see ) consists of the pair $\left(\mathcal{C},{d}_{\mathcal{C}}\right)$, where

$\mathcal{C}=\left\{f:\mathbb{N}\to \left(0,\mathrm{\infty }\right]:\sum _{n=1}^{\mathrm{\infty }}{2}^{-n}\frac{1}{f\left(n\right)}<\mathrm{\infty }\right\},$

and ${d}_{\mathcal{C}}$ is the quasi-metric on given by

${d}_{\mathcal{C}}\left(f,g\right)=\sum _{n=1}^{\mathrm{\infty }}{2}^{-n}\left(\left(\frac{1}{g\left(n\right)}-\frac{1}{f\left(n\right)}\right)\vee 0\right)$

for all $f,g\in \mathcal{C}$. (We adopt the convention that $\frac{1}{\mathrm{\infty }}=0$.) The elements of are called complexity functions and ${d}_{\mathcal{C}}$ is said to be the complexity quasi-metric. Observe that

and thus condition ${d}_{\mathcal{C}}\left(f,g\right)=0$, can be computationally interpreted as f to be ‘more efficient’ than g on all inputs (see [, Section 6]).

In our context we shall work on the subset ${\mathcal{C}}_{1}$ of defined as

and we shall use the function ${Q}_{\mathcal{C}}$ introduced in  and defined, for each $f,g\in \mathcal{C}$ and $t>0$, as

${Q}_{\mathcal{C}}\left(f,g,t\right)=\sum _{k=n}^{\mathrm{\infty }}{2}^{-k}\left(\left(\frac{1}{g\left(k\right)}-\frac{1}{f\left(k\right)}\right)\vee 0\right),$

where $t\in \left(n-1,n\right]$, $n\in \mathbb{N}$.

The following well-known facts will be useful.

Remark 5 [, Remark 1]

${d}_{\mathcal{C}}\left(f,g\right)={Q}_{\mathcal{C}}\left(f,g,t\right)$ for all $f,g\in \mathcal{C}$ and $t\in \left(0,1\right]$.

Remark 6 ${Q}_{\mathcal{C}}\left(f,g,t\right)<1$ whenever $f,g\in {\mathcal{C}}_{1}$ and $f\ne g$.

Remark 7 [, Lemma 2]

Let ${\left({f}_{n}\right)}_{k\in \omega }$ be a sequence in such that ${f}_{n}\le {f}_{n+1}$ for all $n\in \omega$, and let $F\in \mathcal{C}$ defined as $F\left(k\right)=sup\left\{{f}_{n}\left(k\right):n\in \omega \right\}$ for all $k\in \mathbb{N}$. Then ${lim}_{n\to \mathrm{\infty }}{\left({d}_{\mathcal{C}}\right)}^{s}\left(F,{f}_{n}\right)=0$.

In the following, for any $f,g\in \mathcal{C}$, by $f\le g$ we mean that $f\left(n\right)\le g\left(n\right)$ for all $n\in \mathbb{N}$.

Now we construct a fuzzy set ${M}_{1}$ in ${\mathcal{C}}_{1}×{\mathcal{C}}_{1}×\left[0,\mathrm{\infty }\right)$ as

Note that ${M}_{1}\left(f,g,t\right)=1$ for all $t>0$ if and only if $f\le g$.

Then we have:

Lemma 1 $\left({\mathcal{C}}_{1},{M}_{1},{\le }_{{M}_{1}},{\ast }_{L}\right)$ is a ${\le }_{{M}_{1}}$-complete fuzzy quasi-metric space.

Proof We first see that $\left({M}_{1},{\ast }_{L}\right)$ is a fuzzy quasi-metric on ${\mathcal{C}}_{1}$ (recall that $a{\ast }_{L}b=max\left\{a+b-1,0\right\}$ for all $a,b\in \left[0,1\right]$). Indeed, conditions (KM1), (KM2) and (KM4) of Definition 1 are almost trivially satisfied, while condition (KM3) follows immediately from the fact showed in [, Lemma 1] that

${Q}_{\mathcal{C}}\left(f,g,t+s\right)\le {Q}_{\mathcal{C}}\left(f,h,t\right)+{Q}_{\mathcal{C}}\left(h,g,s\right)$

for all $f,g\in \mathcal{C}$ and $t,s>0$. Hence $\left({\mathcal{C}}_{1},{M}_{1},{\ast }_{L}\right)$ is a fuzzy quasi-metric space.

Now let ${\left({f}_{n}\right)}_{n\in \omega }$ be a nondecreasing left K-Cauchy sequence in $\left({\mathcal{C}}_{1},{M}_{1},{\le }_{{M}_{1}},{\ast }_{L}\right)$. Then ${f}_{n}\le {f}_{n+1}$ for all $n\in \omega$, so, by Remark 7,

$\underset{n\to \mathrm{\infty }}{lim}{d}_{\mathcal{C}}\left(F,{f}_{n}\right)=\underset{n\to \mathrm{\infty }}{lim}{d}_{\mathcal{C}}\left({f}_{n},F\right)=0.$
(5)

Since ${f}_{n}\le F$ for all $n\in \omega$ (note that $F\in {\mathcal{C}}_{1}$), we deduce that

${M}_{1}\left({f}_{n},F,t\right)=1$
(6)

for all $n\in \omega$ and $t>0$. On the other hand, and assuming that $F\ne {f}_{n}$ for all $n\in \omega$, we deduce from Remark 5 and (5) that

$\underset{n\to \mathrm{\infty }}{lim}{M}_{1}\left(F,{f}_{n},t\right)=\underset{n\to \mathrm{\infty }}{lim}\left(1-{Q}_{\mathcal{C}}\left(F,{f}_{n},t\right)\right)=\underset{n\to \mathrm{\infty }}{lim}\left(1-{d}_{\mathcal{C}}\left(F,{f}_{n},t\right)\right)=1$
(7)

for all $t\in \left(0,1\right]$, and thus for all $t>0$. By (6) and (7) we deduce that ${\left({f}_{n}\right)}_{n\in \omega }$ converges to F with respect to ${\tau }_{{M}^{i}}$. We have shown that $\left({\mathcal{C}}_{1},{M}_{1},{\le }_{{M}_{1}},{\ast }_{L}\right)$ is a ${\le }_{{M}_{1}}$-complete fuzzy quasi-metric space. □

Taking into account the preceding constructions and results, we immediately obtain the following consequence of Theorem 5.

Theorem 6 If $\mathrm{\Phi }:{\mathcal{C}}_{1}\to {\mathcal{C}}_{1}$ is a ${\le }_{{M}_{1}}$-nondecreasing map and there is ${f}_{0}\in {\mathcal{C}}_{1}$ such that ${f}_{0}{\le }_{M}\mathrm{\Phi }{f}_{0}$, then Φ has a fixed point.

We finish the paper by applying Theorem 6 to show the existence of solution for the recurrence equations associated to Quicksort and Divide and Conquer algorithm, respectively.

Example 5 Consider the recurrence equation T given by $T\left(1\right)=0$, and

$T\left(n\right)=\frac{2\left(n-1\right)}{n}+\frac{n+1}{n}T\left(n-1\right)$

for $n>1$. (This recurrence was obtained by Kruse [, Section 4.8.4] in discussing the average case analysis of Quicksort algorithms.)

In this case, we define a functional $\mathrm{\Phi }:{\mathcal{C}}_{1}{\to \mathcal{C}}_{1}$ as

$\begin{array}{c}\mathrm{\Phi }f\left(1\right)=\mathrm{\infty },\phantom{\rule{2em}{0ex}}\mathrm{\Phi }f\left(2\right)=1,\hfill \\ \mathrm{\Phi }f\left(n\right)=\frac{2\left(n-1\right)}{n}+\frac{n+1}{n}f\left(n-1\right)\hfill \end{array}$

for all $n>2$, where $f\in {\mathcal{C}}_{1}$ (see for instance [29, 30]).

It is clear that if $f\le g$ then $\mathrm{\Phi }f\le \mathrm{\Phi }g$, so by the definition of ${M}_{1}$, Φ is nondecreasing for ${\le }_{{M}_{1}}$. Moreover, the complexity function ${f}_{0}\in {\mathcal{C}}_{1}$ defined as ${f}_{0}\left(n\right)=1$ for all $n\in \mathbb{N}$, satisfies ${f}_{0}\le \mathrm{\Phi }{f}_{0}$, i.e., ${f}_{0}{\le }_{{M}_{1}}\mathrm{\Phi }{f}_{0}$. Therefore, we can apply Theorem 6 and thus Φ has a fixed point ${g}_{0}\in {\mathcal{C}}_{1}$. Consequently, the function $h:\mathbb{N}\to \left[0,\mathrm{\infty }\right)$ given by $h\left(1\right)=0$ and $h\left(n\right)={g}_{0}\left(n\right)$ for all $n>1$ is solution of the recurrence equation T.

Example 6 It is well known that Divide and Conquer algorithms solve a problem by recursively splitting it into subproblems each of which is solved separately by the same algorithm, after which the results are combined into a solution of the original problem (see e.g. [20, 31]). Thus, the complexity of a Divide and Conquer algorithm typically is the solution of the recurrence equation given by

$\begin{array}{c}T\left(1\right)=c\phantom{\rule{1em}{0ex}}\text{and}\hfill \\ T\left(n\right)=aT\left(\frac{n}{b}\right)+h\left(n\right),\hfill \end{array}$

where $a,b,c\in \mathbb{N}$ with $a,b\ge 2$, n ranges over the set $\left\{{b}^{p}:p=0,1,2,\dots \right\}$ and $h\left(n\right)<\mathrm{\infty }$ if $n\in \left\{{b}^{p}:p=0,1,2,\dots \right\}$.

This recurrence equation induces, in a natural way the associated functional $\mathrm{\Phi }:{\mathcal{C}}_{1}{\to \mathcal{C}}_{1}$ defined by $\mathrm{\Phi }f\left(1\right)=c$, $\mathrm{\Phi }f\left(n\right)=af\left(\frac{n}{b}\right)+h\left(n\right)$, if $n\in \left\{{b}^{p}:p=0,1,2,\dots \right\}$ and $\mathrm{\Phi }f\left(n\right)=\mathrm{\infty }$ if $n\notin \left\{{b}^{p}:p=0,1,2,\dots \right\}$.

As in Example 5, Φ is ${\le }_{{M}_{1}}$-nondecreasing. Since the complexity function ${f}_{0}\in {\mathcal{C}}_{1}$ defined as ${f}_{0}\left(n\right)=1$ for all $n\in \mathbb{N}$, satisfies ${f}_{0}{\le }_{{M}_{1}}\mathrm{\Phi }{f}_{0}$, we can apply Theorem 6 and thus Φ has a fixed point ${g}_{0}\in {\mathcal{C}}_{1}$ which is solution of the recurrence equation T.

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## Acknowledgements

The second and third named authors thank the supports of the Universitat Politècnica de València, grant PAID-06-12-SP20120471, and the Ministry of Economy and Competitiveness of Spain, grant MTM2012-37894-C02-01.

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### Competing interests

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Castro-Company, F., Romaguera, S. & Tirado, P. A fixed point theorem for preordered complete fuzzy quasi-metric spaces and an application. J Inequal Appl 2014, 122 (2014). https://doi.org/10.1186/1029-242X-2014-122

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• DOI: https://doi.org/10.1186/1029-242X-2014-122

### Keywords

• fixed point
• preordered fuzzy quasi-metric space
• recurrence equation
• algorithm 