New fixed point results in partial quasi-metric spaces

In 1970, D.S. Scott introduced a fixed point technique in order to describe the meaning of recursive denotational specifications in programming languages ([15]). In order to state the aforementioned fixed point result let us recollect some germane concepts about partially ordered sets.

As usual ([4]), a partial order on a nonempty set X is a reflexive, antisymmetric and transitive binary relation ≤ on X. A partially ordered set is a pair (X, ≤) such that X is nonempty set and ≤ is a partial order on X. Moreover, given a subset Y ⊆ X, an upper bound for Y in (X, ≤) is an element x ∈ X such that y ≤ x for all y ∈ Y. The supremum for Y in (X, ≤), if exists, is an element z ∈ X which is an upper bound for Y and, in addition, satisfies that z ≤ x provided that x ∈ X is an upper bound for Y. We will denote by ↑_≤x (↓_≤x), with x ∈ X, the set {y ∈ X: x ≤ y} ({y ∈ X: y ≤ x}). Furthermore, a sequence (xn)n∈ℕ $(x_{n})_{n\in \mathbb{N}}$ in (X, ≤) is increasing if x_n ≤ x_n+1 for all n ∈ ℕ, where ℕ denotes the set of positive integer numbers.

Following [3], a partially ordered set (X, ≤) is said to be chain complete provided that every increasing sequence has a least upper bound, where a sequence (xn)n∈ℕ $(x_{n})_{n\in \mathbb{N}}$ is said to be increasing whenever x_n ≤ x_n+1 for all n ∈ ℕ.

Of course, a mapping f from a partially ordered set (X, ≤) into itself will be called monotone if f(x) ≤ f(y) whenever x ≤ y. Besides, a mapping from a partially ordered set (X, ≤) into itself is said to be ≤ -continuous if the least upper bound of the sequence (f(x_n))_{n ∈ ℕ} is f(x) for every increasing sequence (x_n)_{n ∈ ℕ} whose least upper bound exists and is x. Notice that every ≤ -continuous function is always monotone.

In the light of the above notions the celebrated Kleene’s fixed point theorem can be stated as follows (see, for instance, [3]):

Since S. Banach proved his celebrated fixed point technique for self-mappings defined between complete metric spaces in 1922, a few extensions to different generalized metric frameworks have been provided in the literature in order to yield quantitative counterparts of the Kleene fixed point theorem which allow to apply metric fixed point techniques to denotational semantics in the spirit of Scott. Thus, in 1994, S.G. Matthews introduced the notion of partial metric space and developed a partial metric fixed point technique in order to develop suitable mathematical tools for describing the meaning of recursive specifications in denotational semantics for programming languages ([8]). With the aim of stating the fixed point result in which such a technique is based on, let us recall a few pertinent notions about partial metric spaces.

From now on, ℝ⁺ will denote the set of nonnegative real numbers. Let us recall that, according to [8], a partial metric on a nonempty set X is a function p : X × X → ℝ⁺ such that for all x, y, z ∈ X:

p(x;x) = p(x;y) = p(y;y),x = y;

Clearly a metric on a nonempty set X is a partial metric p on X satisfying, in addition, the following condition for all x ∈ X: (v) p(x, x) = 0.

Of course, a partial metric space is a pair (X, p) such that X is a nonempty set and p is a partial metric on X.

On account of [8], each partial metric p on X generates a T₀ topology 𝒯(p) on X which has as a base 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. Thus, a sequence (x_n)_{n ∈ ℕ} in a partial metric space (X, p) converges to a point x ∈ X with respect to T(p)⇔p(x,x)=limn→∞p(x,xn) $\mathscr{T}(p)\Leftrightarrow p(x,x)=\lim_{n\rightarrow \infty}p(x,x_{n})$ . Moreover, a sequence (x_n)_{n ∈ ℕ} in a partial metric space (X, p) is called a Cauchy sequence if lim_{n, m → ∞}p(x_n, x_m) exists and it is finite. Besides, a partial metric space (X, p) is said to be complete if every Cauchy sequence (x_n)_{n ∈ ℕ} in X converges, with respect to 𝒯(p), to a point x ∈ X such that p(x, x) = lim_{n, m → ∞}p(x_n, x_m).

Following [8], in every partial metric space (X, p) a partial order, the specialization order, ≤_p can be induced as follows: x ≤_py ⇔ p(x, y) = p(x, x).

In the light of the exposed notions, the fixed point result which provided the basis for the aforesaid Matthews fixed point technique can be stated as follows (see [8]).

Later on, in 1995, M.P. Schellekens introduced a new extension of the classical Banach fixed point theorem. In particualr, he developed a quasi-metric fixed point technique which turned out to be useful in asymptotic complexity analysis of algorithms ([14]). In order to recall it, let us fix some notions about quasi-metric spaces that will be useful later on.

On account on [5 Chapter 11], a quasi-metric on a nonempty set X is a function d : X × X → ℝ⁺ such that for all x, y, z ∈ X:

d(x;y) = d(y;x) = 0 ⇔ x = y;

It is clear that a metric d on a nonempty set X is a quasi-metric which holds additionally the property (iv) d(x, y) = d(y, x) for all x, y ∈ X.

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

According to [5. Chapter 11], each quasi-metric d on X generates a T₀-topology 𝒯(d) on X which has as a base the family of open d-balls {B_d(x, ε):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, then the function d^s : X × X → ℕ⁺ is a metric, where d^s(x, y) = max(d(x, y), d(y, x)) for all x, y ∈ X.

Following [12], a sequence (x_n)_{n ∈ ℕ} in a quasi-metric space (X, d) is said to be left K-Cauchy if, given ε > 0, there exists n₀ ∈;ℕ such that d(x_n, x_m) < ε for all m = n = n₀. Moreover, on account of [7], a quasi-metric space (X, d) is Smyth complete provided that every left K-Cauchy sequence is convergent with respect to τ(d^s).

Similar to the partial metric case, in each quasi-metric space (X, d) a partial order, the specialization order, ≤_d can be defined as follows: x ≤_dy ⇔ d(x, y) = 0.

Taking into account the preceding notions, the aforementioned fixed point result of Schellekens can be stated as follows ([14]):

In [6], H.P.A. Künzi, H. Pajooshesh and M.P. Schellekens introduced the concept of partial quasi-metric space with the aim of providing a framework which generalizes both the partial metric space and the quasi-metric space.

Following [6], a partial quasi-metric on a nonempty set X is a function q : X × X → ℕ⁺ such that for all x, y, z ∈ X:

q(x;x) = q(x;y) and q(y;y) = q(y;x) ⇔ x = y;

It must be stressed that a partial quasi-metric satisfying the preceding assertions is called a lopsided partial quasi-metric in [6].

Notice that a partial metric on a set X is a partial quasi-metric q satisfying additionally the condition: (iv) q(x, y) = q(y, x) for all x, y ∈ X. Besides a quasi-metric on a set X is a partial quasi-metric q satisfying in addition the condition: (iv) q(x, x) = 0 for all x ∈ X.

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

A partial quasi-metric q induces a T₀-topology 𝒯(q) and a partial order ≤_q on X in the same way how partial metrics make it. The completeness can be also literally adapted from the partial metric context to the partial quasi-metric one. A few fixed point theorems for self-mappings between complete partial quasi-metric spaces and applications to Computer Science were given in [2] and [9] recently.

In 2005, J.J. Nieto and R. Rodríguez-López made an in-depth study of how to reconcile order-theoretic and metric fixed point techniques in the classical metric case with the aim of providing the existence and uniqueness of solutions to first-order differential equations admitting only the existence of a lower solution [10] (see also [11]). Let us recall the fixed point results in which the aforementioned techniques are based on:

In the preceding result the continuity of the mapping can be interchanged by a kind of chain-completeness of the partially ordered set obtaining as a consequence the next result.

Motivated by Theorems 4 and 5, our propose is prove counterparts of such results in the context of partial quasi-metric spaces, when the specialization order is considered, in such a way that Theorems 2 and 3 can be retrieved as a particular case. Thus the remainder of the paper is organized in the following way. In Section 2 we introduce a new notion of completeness for partial quasi-metric spaces that generalizes at the same time the completeness used in Theorems 2 and 3 and, in addition, we prove existence of fixed point results for self-mappings satisfying a new contractive condition which is inspired in the contractive condition given in Theorems 4 and 5. Moreover, examples that show that the assumptions in the news results cannot be weakened are given. Section 3 is devoted to discuss the uniqueness of the fixed point guaranteed by the results yielded in Section 2. Hence sufficient conditions that ensure the uniqueness, local and global, of fixed point are provided.

In this section our aim is to prove a partial quasi-metric version of Theorems 4 and 5 in such a way Theorems 2 and 3 can be retrieved as a corollary. To this end, we introduce a new natural notion, different from those used in [2] and [9], of completeness for partial quasi-metric spaces.

According to [6], every partial quasi-metric q on a nonempty set X induces a quasi-metric d_q defined by

dq(x,y)=q(x,y)−q(x,x)$$ \begin{equation*}\nonumber d_{q}(x,y)=q(x,y)-q(x,x) \end{equation*} $$

for all x, y ∈ X.

From now on, a partial quasi-metric space (X, q) will be said to be Smyth complete provided that the induced quasi-metric space (X, d_q) is Smyth complete. Note that the new notion of completeness coincides with the Smyth completeness and completeness when the partial quasi-metric is exactly a quasi-metric and a partial metric, respectively.

In order to prove a partial quasi-metric version of Theorems 4 and 5 we need to manage a contractive notion that involves the order. Taking this into account we introduce the concept of order-contraction.

In the following, given a partial quasi-metric space (Xq) and x, y ∈ X, we will write x⋈_qy whenever either x ≤_qy or y ≤_qx.

In the sequel, we will say that a mapping f from a partial quasi-metric space (X, q) into itself is order-contractive provided that there exists c ∈ [0, 1[ such that

q(f(x),f(y))≤cq(x,y),$$ \begin{equation} q(f(x),f(y))\leq c q(x,y), \end{equation} $$(4)

for all x⋈_qy.

q(fn(x0),fm(x0))=q(fn(x0),fn(x0))$$ \begin{equation*}\nonumber q(f^{n}(x_{0}),f^{m}(x_{0}))=q(f^n(x_0),f^n(x_0)) \end{equation*} $$

for all n, m ∈ ℕ with m = n. Hence (fⁿ(x₀))_{n ∈ ℕ} is left K-Cauchy sequence in (X, d_q), since

dq(fn(x0),fm(x0))=q(fn(x0),fm(x0))−q(fn(x0),fn(x0))=0$$ \begin{equation*}\nonumber d_q(f^{n}(x_{0}),f^{m}(x_{0}))=q(f^{n}(x_{0}),f^{m}(x_{0}))-q(f^{n}(x_{0}),f^{n}(x_{0}))=0 \end{equation*} $$

for all n, m ∈ ℕ with m≥n. The Smyth completeness of (X, q) provides the existence of x* ∈ X such that (fⁿ(x₀))_n ∈ ℕ converges to x* in (X,dqs) $(X,d^s_q)$ . Whence

limn→∞q(x∗,fn(x0))=q(x∗,x∗)$$ \begin{equation*}\nonumber \lim_{n\rightarrow \infty}q(x^{\ast},f^{n}(x_{0}))=q(x^\ast,x^\ast) \end{equation*} $$

and

limn→∞q(fn(x0),x∗)=limn→∞q(fn(x0),fn(x0))$$ \begin{equation*}\nonumber \lim_{n\rightarrow \infty}q(f^{n}(x_{0}),x^{\ast})=\lim_{n\rightarrow \infty}q(f^{n}(x_{0}),f^{n}(x_{0})) \end{equation*} $$

By (4) we deduce that

q(fn(x0),fn(x0))≤cnq(x0,x0)$$ \begin{equation*}\nonumber q(f^{n}(x_{0}),f^{n}(x_{0})) \leq c^{n}q(x_{0},x_{0}) \end{equation*} $$

for all n ∈ ℕ. Hence we obtain that lim_{n → ∞}q(fⁿ(x₀), fⁿ(x₀)) = 0. Thus we have that lim_{n → ∞}q(fⁿ(x₀), x*) = 0.

Next we show that in x* ∈ ↑ x₀. Since x₀ ≤_qfⁿ(x₀) for all n ∈ ℕ we have that q(x₀, fⁿ(x₀)) = q(x₀, x₀) for all n ∈ ℕ. Now, let ε > 0. Then there exists n₀ ∈ ℕ such that q(fⁿ(x₀), x*) − q(fⁿ(x₀), fⁿ(x₀)) < ε for all n = n₀, since lim_{n → ∞}q(fⁿ(x₀), x*) = lim_{n → ∞}q(fⁿ(x₀), fⁿ(x₀)) = 0. Thus

q(x0,x∗)−q(x0,x0)≤q(x0,fn(x0))+q(fn(x0),x∗)−q(fn(x0),fn(x0))−q(x0,x0)<ε$$ \begin{equation}\begin{array}{ll} q(x_{0},x^{\ast})-q(x_{0},x_{0})\leq & \\ \\ q(x_{0},f^{n}(x_{0}))+q(f^{n}(x_{0}),x^{\ast})-q(f^{n}(x_{0}),f^{n}(x_{0}))-q(x_{0},x_{0})\lt\varepsilon& \end{array} \end{equation} $$

for all n = n₀. Whence we obtain that q(x₀, x*) − q(x₀, x₀) = 0 and, hence, that x₀ ≤_qx*, i.e., x* ∈ ↑ x₀.

Next we show that x* is the supremum of (fⁿ(x₀))_{n ∈ ℕ} in (X, ≤_q).

First of all we prove that x* is an upper bound of (fⁿ(x₀))_{n ∈ ℕ}. Since

q(fn(x0),x∗)−q(fn(x0),fn(x0))≤q(fm(x0),x∗)$$ \begin{equation*}\nonumber q(f^{n}(x_{0}),x^{\ast})-q(f^{n}(x_{0}),f^{n}(x_{0}))\leq q(f^{m}(x_{0}),x^{\ast}) \end{equation*} $$

for all m,n ∈ ℕ with n ≤ m and lim_{m → ∞}q(f^m(x₀), x*) = 0, we deduce that q(fⁿ(x₀), x*) − q(fⁿ(x₀), fⁿ(x₀)) = 0 for all n ∈ ℕ. Consequently, fⁿ(x₀) ≤_qx* for all n ∈ ℕ.

Assume that there exists y ∈ X such that fⁿ(x₀) ≤_qy for all n ∈ ℕ. Then q(fⁿ(x₀), y) = q(fⁿ(x₀), fⁿ(x₀)) for all n ∈ ℕ. Hence

q(x∗,y)−q(x∗,x∗)≤q(x*,fn(x0))+q(fn(x0),y)−q(fn(x0),fn(x0))−q(x∗,x∗)=q(x∗,fn(x0))−q(x∗,x∗).$$ \begin{equation}\begin{array}{ll} q(x^{\ast},y)-q(x^{\ast},x^{\ast})\leq & \\ \\ q(x^{*},f^{n}(x_{0}))+q(f^{n}(x_{0}),y)-q(f^{n}(x_{0}),f^{n}(x_{0}))-q(x^{\ast},x^{\ast})=& \\ \\ q(x^{\ast},f^{n}(x_{0}))-q(x^{\ast},x^{\ast}).&\\ \end{array} \end{equation} $$

It follows, from lim_{n → ∞}q(x*, fⁿ(x₀)) = q(x*, x*), that q(x*, y) = q(x*, x*) and, hence, that x* ≤_qy. Whence x* is the supremum of (fⁿ(x₀))_n∈ℕ.

Next we assume that q(x*, x*) = 0. In the following our aim is to show that x* ∈ Fix (f). To this end we note that

q(x∗,f(x∗))=q(x∗,x∗)=0,$$ \begin{equation*}\nonumber q(x^{\ast},f(x^{\ast}))=q(x^{\ast},x^{\ast})=0, \end{equation*} $$

since

q(x∗,f(x∗))≤q(x∗,fn(x0))+q(fn(x0),f(x∗))−q(fn(x0),fn(x0))$$ q(x^{\ast},f(x^{\ast}))\leq q(x^{\ast},f^n(x_0))+q(f^n(x_0),f(x^{\ast}))-q(f^n(x_0),f^n(x_0)) $$

and, in addition, we have, on one hand, by (4) that

q(fn(x0),f(x*))≤cq(fn−1(x0),x*)$$ \begin{equation} q(f^n(x_0),f(x*)) \le cq(f^{n-1}(x_0),x*) \end{equation} $$

for all n ∈ ℕ and, on the other hand, that

limn→∞q(fn(x0),x∗)=limn→∞q(fn(x0),fn(x0))=0.$$ \begin{equation*} \lim_{n\rightarrow \infty}q(f^n(x_0),x^{\ast})=\lim_{n\rightarrow \infty}q(f^n(x_0),f^n(x_0))=0. \end{equation*} $$

Thus we deduce that x*≤_qf(x*).

Next we prove thatf(x*)≤_qx*. On one hand, we have thatq(fⁿ(x₀),x*) = q(fⁿ(x₀),fⁿ(x₀)) for alln ∈ ℕ.

On the other hand, we have that

limn→∞q(x∗,fn(x0))=q(x∗,x∗)=0.$$ \begin{equation*}\nonumber \lim_{n\rightarrow \infty}q(x^{\ast},f^{n}(x_{0}))=q(x^\ast,x^\ast)=0. \end{equation*} $$

Thus we deduce that q(f(x*),x*) = q(f(x*),f(x*)) because

q(f(x∗),x∗)−q(f(x∗),f(x∗))≤q(f(x∗),fn(x0))+q(fn(x0),x∗)−q(fn(x0),fn(x0))−q(f(x∗),f(x∗))≤cq(x∗,fn(x0)).$$ \begin{equation}\begin{array}{cc} q(f(x^{\ast}),x^{\ast})-q(f(x^{\ast}),f(x^{\ast}))\leq & \\ \\ q(f(x^{\ast}),f^{n}(x_{0}))+q(f^{n}(x_{0}),x^{\ast})-q(f^{n}(x_{0}),f^{n}(x_{0}))-q(f(x^{\ast}),f(x^{\ast})) \leq & \\ \\ cq(x^{\ast},f^{n}(x_{0})).& \\ \end{array} \end{equation} $$

Consequently x*=f(x*) and, thus, x*∈ Fix(f).

Suppose that x*∈ Fix(f). Next we show that q(x*,x*)=0. Assume for the purpose of contradiction that 0<q(x*,x*). Then, by (4), we have that

q(f(x∗),f(x∗))≤cq(x∗,x∗).$$ \begin{equation*}\nonumber q(f(x^\ast),f(x^\ast))\leq c q(x^\ast,x^\ast). \end{equation*} $$

Whence we deduce that 1 ≤ c which is a contradiction. So q(x*,x*)=0

In the next example we show that the Smyth completeness of the partial quasi-metric space cannot be deleted in Theorem 6 in order to guarantee the existence of fixed point.

The next example evidences that the order-contractive condition of the self-mapping cannot be deleted in the statement of Theorem 6 in order to assure the existence of fixed point.

When the order-contractive mapping is, in addition, monotone we obtain the next result.

The next result can be stated when one considers a quasi-metric space, a particular instance of partial quasi-metric spaces, in the statement of Theorem 6.

The below result is obtianed immediately from Corollary 11.

When the order-contractivity is replaced by the quasi-metric version of the classical Banach contractivity, Theorem 6 provides the following result.

d(x,y)+wd(x)=d(y,x)+wd(y) $$d(x, y)+w_{d}(x)=d(y,x)+w_{d}(y)$$

for all x,y ∈ X. On account of [7], every bicomplete weightable quasi-metric space is always Smyth complete. Following [13], every partial metric space (X,q) induces a weightable quasi-metric space (X,d_q), where the quasi-metric d_q is defined by

dq(x,y)=q(x,y)−q(x,x) $$d_{q}(x, y)=q(x, y)-q(x,x)$$

for all x,y ∈ X and the weight wdq $w_{d_{q}}$ is given by wdq(x)=q(x,x) $w_{d_{q}}(x)=q(x,x)$ for all x ∈ X.

The next result can be stated when one considers a partial metric space, a particular instance of partial quasi-metric spaces, in the statement of Theorem 6.

As a particular case of Corollary 14 we get the following result.

We end this subsection noting that dual versions of our main results, Theorem 6 and Corollary 10, can be obtained following similar arguments to those given in the proof of the aforesaid resutls.

for all n ∈ ℕ. It follows that

q(fn(x0),fn+k(x0))≤q(fn(x0),fn+1(x0))+…+q(fm+k−1(x0),fn+k(x0))≤(cn+…+cn+k−1)q(f(x0),x0)≤cn1−cq(f(x0),x0)$$ \begin{array}{lll} q(f^{n}(x_{0}),f^{n+k}(x_{0}))&\leq &q(f^{n}(x_{0}),f^{n+1}(x_{0}))+\ldots +q(f^{m+k-1}(x_{0}),f^{n+k}(x_{0}))\\ \\ &\leq & \left(c^{n}+\ldots+c^{n+k-1} \right)q(f(x_{0}),x_{0})\\ \\ &\leq & \frac{c^{n}}{1-c} q(f(x_{0}),x_{0}) \end{array} $$

for all k ∈ ℕ. Whence we obtain that {fⁿ(x₀)}_{n ∈ ℕ} is a left K-Cauchy sequence in (X, q), since

q(fn(x0),fn+k(x0))−q(fn(x0),fn(x0))≤q(fn(x0),fn+k(x0)) $$q(f^{n}(x_{0}),f^{n+k}(x_{0}))-q(f^{n}(x_{0}),f^{n}(x_{0}))\leq q(f^{n}(x_{0}),f^{n+k}(x_{0}))$$

for all k,n ∈ ℕ. The Smyth completeness of (X,q) guarantees the existence of x^* ∈ X such that lim_n → ∞fⁿ(x₀) = x^* with respect to T(dqs)$ \mathscr{T}(d^s_q) $ .

The proof of the remainder assertions in statement of the result runs as in the proof of Theorem 6.

We end this section discussing whether the main conclusions of our main results, Theorems 6 and 18, can be retrieved as a particular case of Kleene's theorem (Theorem 1):

Fortunately, the answer to the posed question is negative. Although every partially ordered set (X,≤_q) induced by a Smyth complete partial quasi-metric space (X,q) is always countably chain complete, there exist order-contractive mappings that are not ≤_q-continuous. The following example illustrates the preceding fact.

A natural question that one can wonders is whether Theorem 6 guarantees the uniqueness of fixed point in general. However, the next example gives a negative answer to the posed question .

for all (x,y),(w,r) ∈ X with (x,y) ⋈_q (w,r). However, Fix(f) = {(0,1),(1,0)}.

Inspired by the above example we discuss conditions that guarantee the uniqueness of fixed point in the following results. With this aim we observe that Example 7 shows that there are order-contractive mappings f such that Fix(f) = ∅.

In the next result we give sufficient conditions in order to guarantee a local uniqueness.

Again Example 7 gives an instance of a monotone order-contractive mapping without fixed points defined from a partial quasi-metric. Moreover, observe that in the same example the induced partially ordered set is not directed.