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A characterization of Smyth complete quasimetric spaces via Caristi’s fixed point theorem
Fixed Point Theory and Applications volume 2015, Article number: 183 (2015)
Abstract
We obtain a quasimetric generalization of Caristi’s fixed point theorem for a kind of complete quasimetric spaces. With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasimetric spaces which provides a quasimetric generalization of the wellknown characterization of metric completeness due to Kirk. Some illustrative examples are also given. As an application, we deduce a procedure which allows to easily show the existence of solution for the recurrence equation of certain algorithms.
Introduction and preliminaries
We start by recalling several notions and properties of the theory of quasimetric spaces. Our basic references are [1] and [2].
By a quasimetric on set X we mean a function \(d:X\times X\rightarrow [0,\infty)\) such that for all \(x,y,z\in X\): (i) \(x=y\Leftrightarrow d(x,y)=d(y,x)=0\); (ii) \(d(x,z)\leq d(x,y)+d(y,z)\).
A quasimetric space is a pair \((X,d)\) such that X is a set and d is a quasimetric on X.
Given a quasimetric d on X, the function \(d^{1}\) defined by \(d^{1}(x,y)=d(y,x)\) is also a quasimetric on X, called the conjugate of d, and the function \(d^{s}\) defined by \(d^{s}(x,y)=\max \{d(x,y),d^{1}(x,y)\}\) is a metric on X.
Each quasimetric d on X induces a \(T_{0}\) topology \(\tau_{d}\) on X which has as a base the family of open balls \(\{B_{d}(x,r):x\in X, \varepsilon>0\}\), where \(B_{d}(x,\varepsilon )=\{y\in X:d(x,y)<\varepsilon\}\) for all \(x\in X\) and \(\varepsilon>0\).
If \(\tau_{d}\) is a \(T_{1}\) topology on X, we say that \((X,d)\) is a \(T_{1}\) quasimetric space.
Note that a quasimetric space \((X,d)\) is \(T_{1}\) if and only if for each \(x,y\in X\), condition \(d(x,y)=0\) implies \(x=y\).
There exist many different notions of Cauchy net, Cauchy sequence and quasimetric completeness in the literature (see, e.g., [1–3]). For our purposes, here we will consider the following ones.
A net \((x_{\alpha})_{\alpha\in\Lambda}\) in a quasimetric space \((X,d)\) is called left KCauchy if for each \(\varepsilon>0\) there is \(\alpha _{\varepsilon}\in\Lambda\) such that \(d(x_{\alpha},x_{\beta })<\varepsilon\) whenever \(\alpha_{\varepsilon}\leq\alpha\leq\beta\). The notion of a left KCauchy sequence is defined in the obvious manner.
We say that a quasimetric space \((X,d)\) is complete if every left KCauchy net is convergent for \(\tau_{d^{1}}\), and say that it is sequentially complete if every left KCauchy sequence is convergent for \(\tau_{d^{1}}\). (Note that our notion of (sequential) completeness of \((X,d)\) coincides with the usual notion of right K(sequential) completeness of \((X,d^{1})\).)
A quasimetric space \((X,d)\) is Smyth complete provided that every left KCauchy net in \((X,d)\) is convergent for \(\tau_{d^{s}}\) (compare Definition 8 in [4], [5], p.454, etc.).
The following wellknown result is a consequence of Definition 8 and Theorem 9 in [4] (see also [6], p.323, [7], p.347).
Proposition 1
A quasimetric space \((X,d)\) is Smyth complete if and only if every left KCauchy sequence in \((X,d)\) is convergent for \(\tau_{d^{s}}\).
The following implications are also known and easy to check:
However, the converse implications do not hold, in general. For instance, the Sorgenfrey quasimetric space (see, e.g., [5], p.463 or Example 1.1.6 in [1]) provides a distinguished example of a complete \(T_{1}\) quasimetric space which is not Smyth complete, while Stoltenberg presented in Example 2.4 of [8] an example of a sequentially complete \(T_{1}\) quasimetric space which is not complete.
On the other hand, Caristi proved in 1976 the following important and wellknown generalization of the Banach contraction principle.
Theorem 1
([9])
Let T be a selfmapping of a complete metric space \((X,d)\). If there is a lower semicontinuous function \(\varphi:X\rightarrow[0,\infty)\) satisfying
for all \(x\in X\), then T has a fixed point in X.
Kirk showed in [10] that the validity of Caristi’s fixed point theorem in a metric space characterizes its completeness. More exactly, he proved the following.
Theorem 2
([10])
For a metric space \((X,d)\), the following conditions are equivalent:

(1)
\((X,d)\) is complete.

(2)
If T is a selfmapping of X such that there is a lower semicontinuous function \(\varphi:X\rightarrow[ 0,\infty)\) satisfying \(d(x,Tx)\leq\varphi(x)\varphi(Tx)\) for all \(x\in X\), then T has a fixed point in X.
Extensions and generalizations of Theorems 1 and 2 to partial metric spaces, cone metric spaces, quasimetric spaces and probabilistic metric spaces have been obtained by several authors (see, e.g., [11–20]). In particular, Cobzaş ([15], Theorem 2.3) proved, among other interesting results, the following quasimetric generalization of Caristi’s fixed point theorem.
Theorem 3
([15])
Let T be a selfmapping of a sequentially complete \(T_{1}\) quasimetric space \((X,d)\). If there is a function \(\varphi:X\rightarrow[ 0,\infty)\) which is lower semicontinuous for \(\tau_{d^{1}}\) and satisfies
for all \(x\in X\), then T has a fixed point in X.
Since complete and Smyth complete non\(T_{1}\) quasimetric spaces provide efficient tools in several areas as asymmetric functional analysis, domain theory, theoretical computer science, complexity analysis of algorithms defined by recurrence equations, etc. (see, e.g., [1, 4, 5, 7, 21, 22] and their references), it seems natural to discuss the question of generalizing Theorem 3 to (nonnecessarily \(T_{1}\)) quasimetric spaces. In this direction, we shall give an example of a sequentially complete quasimetric space for which Theorem 3 does not hold. We shall show that, nevertheless, Theorem 3 remains valid for complete quasimetric spaces. A suitable and slight modification of the proof of that result will be used to deduce a characterization of Smyth complete quasimetric spaces which provides a generalization to the quasimetric framework of Kirk’s characterization of metric completeness. As an application, we obtain a procedure which allows to easily deduce the existence of solution for the recurrence equation of certain algorithms.
Results and examples
In order to simplify the terminology and the statements of our results, we shall use the following notions.
A selfmapping T of a quasimetric space \((X,d)\) will be called a dCaristi mapping (resp. a \(d^{s}\)Caristi mapping) on \((X,d)\) if there is a function \(\varphi:X\rightarrow[0,\infty)\) which is lower semicontinuous for \(\tau_{d^{1}}\) (resp. for \(\tau_{d^{s}}\)) and satisfies \(d(x,Tx)\leq \varphi(x)\varphi(Tx)\) for all \(x\in X\).
Clearly, every dCaristi mapping is a \(d^{s}\)Caristi mapping. The following example shows that the converse is not true in general.
Example 1
Let d be the quasimetric on the set \(\mathbb{N}\) of all positive integer numbers, given by \(d(x,x)=0\) for all \(x\in \mathbb{N}\) and \(d(x,y)=1/x\) for all \(x,y\in\mathbb{N}\) with \(x\neq y\). Clearly \((\mathbb {N},d)\) is a \(T_{1}\) quasimetric space such that \(\tau_{d}\), and hence \(\tau_{d^{s}}\) is the discrete topology on \(\mathbb{N}\). Define \(T:\mathbb{N}\rightarrow\mathbb{N}\) as \(Tx=2x\) for all \(x\in\mathbb{N}\). Then \(d(x,Tx)=1/x=\varphi(x)\varphi(Tx)\), where \(\varphi:\mathbb {N}\rightarrow [0,\infty)\) is defined as \(\varphi(x)=2/x\) for all \(x\in\mathbb{N}\). Since \(\tau_{d^{s}}\) is the discrete topology on \(\mathbb{N}\), φ is lower semicontinuous for \(\tau_{d^{s}}\) and thus T is a \(d^{s}\)Caristi mapping on \((\mathbb{N},d)\). Finally, suppose that T is also a dCaristi mapping. Then there exists a function \(\varphi:\mathbb{N}\rightarrow [0,\infty)\) which is lower semicontinuous for \(\tau_{d^{1}}\) and satisfies \(d(x,2x)=1/x\leq\varphi(x)\varphi(2x)\) for all \(x\in\mathbb{N}\). We easily deduce that \(\varphi(1)\geq1+\varphi(2^{x})\) for all \(x\in \mathbb{N}\), which contradicts that φ is a lower semicontinuous function for \(\tau_{d^{1}}\) because the sequence \((2^{n})_{n\in\mathbb{N}}\) converges to 1 for \(\tau_{d^{1}}\).
Our next example, based on Example 2.1 in [5], shows that condition \(T_{1}\) cannot be removed in Theorem 3.
Example 2
Let \((\mathcal{A},d)\) be the non\(T_{1}\) quasimetric space such that \(\mathcal{A}\) is the family of all nonempty countable subsets of the set \(\mathbb{R}\) of all real numbers, and d is the quasimetric on \(\mathcal{A}\) defined as \(d(A,B)=0\) if \(A\subseteq B\), and \(d(A,B)=1\) otherwise. Let \((A_{n})_{n\in\mathbb{N}}\) be a left KCauchy sequence in \((\mathcal{A},d)\). Assume, without loss of generality, that \(d(A_{n},A_{m})=0\) whenever \(n\leq m\), i.e., \(A_{n}\subseteq A_{m}\) whenever \(n\leq m\). Since \(\bigcup_{n\in\mathbb{N}}A_{n}\in\mathcal{A}\) and \(d(A_{n},\bigcup_{n\in\mathbb{N}}A_{n})=0\) for all \(n\in\mathbb{N}\), we deduce that \((\mathcal{A},d)\) is sequentially complete. Now let
ordered by inclusion. Then the net \((A)_{A\in\Lambda}\) is left KCauchy in \((\mathcal{A},d)\) (see Example 2.1 in [5]) but it does not converge for \(\tau_{d^{1}}\) because the elements of \(\mathcal{A}\) are countable subsets of \(\mathbb{R}\). We conclude that \((\mathcal{A},d)\) is not complete.
However, we have the following extension of Theorem 3 whose proof is based on a classical technique used by Kirk [10], which is inspired in the partial order of Brøndsted [23, 24].
Theorem 4
Every dCaristi mapping on a complete quasimetric space \((X,d)\) has a fixed point in X.
Proof
Let \((X,d)\) be a complete quasimetric space and let \(T:X\rightarrow X\) be a dCaristi mapping on \((X,d)\). Then there exists a function \(\varphi:X\rightarrow[0,\infty)\) which is lower semicontinuous for \(\tau_{d^{1}}\) and satisfies
for all \(x\in X\). As in the classical metric case, define a binary relation ⪯ on X by
for all \(x,y\in X\). Clearly ⪯ is a partial order on X. Note also that \(x\preceq Tx\) for all \(x\in X\).
We shall prove that every (nonempty) linearly ordered subset of the partially ordered set \((X,\preceq)\) has an upper bound. Indeed, let A be a (nonempty) linearly ordered subset of X. We show that the net \((x_{x})_{x\in A}\) is a left KCauchy net in \((X,d)\) where we have defined \(x_{x}:=x\) for all \(x\in A\). To this end, put \(r=\inf_{x\in A}\varphi(x)\). Given an arbitrary \(\varepsilon>0\), choose \(x\in A\) such that \(\varphi (x)< r+\varepsilon\). Thus, for any \(y,z\in A\) with \(x\preceq y\preceq z\), we obtain
Consequently, \((x_{x})_{x\in A}\) is a left KCauchy net in \((X,d)\), and hence it converges, for \(\tau_{d^{1}}\), to some \(p\in X\). Fix \(x\in A\) and let \(\varepsilon>0\) be arbitrary. Then there is \(y\in A\) such that \(d(z,p)<\varepsilon\) and \(\varphi(p)\varphi(z)<\varepsilon\) whenever \(z\in A\) and \(y\preceq z\). Choose \(z_{0}\in A\) with \(x\preceq z_{0}\) and \(y\preceq z_{0}\). Hence
Since ε is arbitrary, we deduce that \(d(x,p)\leq\varphi (x)\varphi(p)\), i.e., \(x\preceq p\), so p is an upper bound of A. It follows from Zorn’s lemma that \((X,\preceq)\) has a maximal element, say a. Since \(a\preceq Ta\), we conclude that \(a=Ta\), so a is a fixed point of T. The proof is finished. □
Of course, Caristi’s fixed point theorem is a consequence of Theorem 4 when \((X,d)\) is a metric space. Next we present two examples of complete quasimetric spaces \((X,d)\) with appropriate dCaristi mappings, for which Caristi’s fixed point theorem cannot be applied to the metric space \((X,d^{s})\).
Example 3
Let \(X=\mathbb{N}\cup\{\infty\}\). Define a nonnegative realvalued function d on \(X\times X\) by \(d(\infty ,\infty)=0\), \(d(x,y)=\vert 1/x1/y\vert \) if \(x,y\in\mathbb{N}\), \(d(x,\infty)=1/x\) and \(d(\infty,x)=1\) for all \(x\in\mathbb{N}\). It is easily seen that \((X,d)\) is a complete \(T_{1}\) quasimetric space (in fact, note \((X,\tau_{d^{1}})\) is a compact topological space). Define \(T:X\rightarrow X\) as \(T\infty =\infty\), and \(Tx=x^{2}\) for all \(x\in\mathbb{N}\). Now define \(\varphi:X\rightarrow [0,\infty)\) as \(\varphi(\infty)=0\), and \(\varphi(x)=1/x\) for all \(x\in \mathbb{N}\). Then φ is clearly a lower semicontinuous function for \(\tau_{d^{1}}\). Since \(d(\infty,T\infty)=d(1,T1)=0\), and for every \(x\in X\backslash\{1,\infty\}\),
we conclude that T is a dCaristi mapping on \((X,d)\). Hence, we can apply Theorem 4 to this case. In fact, T has 1 and ∞ as fixed points. However, we cannot apply Caristi’s fixed point theorem to the metric space \((X,d^{s})\) because it is not complete. Indeed, \((x)_{x\in \mathbb{N}}\) is a Cauchy sequence in \((X,d^{s})\) that does not converge for \(\tau_{d^{s}}\).
In the above example the metric space \((X,d^{s})\) is not complete. Now, we give an example of a complete quasimetric space \((X,d)\) where the metric space \((X,d^{s})\) is complete and there is a dCaristi mapping on \((X,d)\) which is not a Caristi mapping for the metric space \((X,d^{s})\).
Example 4
As in Example 3, let \(X=\mathbb{N}\cup\{ \infty\} \). Define a nonnegative realvalued function d on \(X\times X\) by \(d(x,y)=0 \) if \(x\leq y\), and \(d(x,y)=y\) if \(y< x\) (here, ≤ denotes the usual order on X). It is routine to check that \((X,d)\) is a complete quasimetric space (note that every net in X converges to ∞ for \(\tau_{d^{1}}\)). Define \(T:X\rightarrow X\) as \(Tx=x+1\) for all \(x\in \mathbb{N} \) and \(T\infty=\infty\). Then \(d(x,Tx)=0\) for all \(x\in X\), so that T is trivially a dCaristi mapping on \((X,d)\). Hence, we can apply Theorem 4. Finally, suppose that there exists a lower semicontinuous function, for \(\tau_{d^{s}}\), \(\varphi:X\rightarrow[0,\infty)\), such that \(d^{s}(x,Tx)\leq\varphi(x)\varphi(Tx)\) for all \(x\in X\). Then
for all \(x\in\mathbb{N}\). We deduce that \(\varphi(1)=\infty\), a contradiction. Hence, we cannot apply the classical Caristi fixed point theorem in this case.
Observe that the aforementioned example of Stoltenberg and Example 4 (or Example 3) above show that Theorems 3 and 4 are independent of each other.
Although we do not know whether the converse of Theorem 4 holds, i.e., if Kirk’s theorem can be generalized to complete quasimetric spaces, we are going to show that it is possible to obtain such a generalization for Smyth complete quasimetric spaces. To this end, the following essentially wellknown fact (see, e.g., Proposition 1.2.4 in [1]) will be useful.
Proposition 2
Let \((x_{n})_{n\in\mathbb{N}}\) be a left KCauchy sequence in a quasimetric space \((X,d)\). If \((x_{n})_{n\in\mathbb{N}}\) has a subsequence convergent to \(x\in X\) for \(\tau_{d^{s}}\), then \((x_{n})_{n\in\mathbb{N}}\) converges to x for \(\tau_{d^{s}}\).
Theorem 5
A quasimetric space \((X,d)\) is Smyth complete if and only if every \(d^{s}\)Caristi mapping on \((X,d)\) has a fixed point in X.
Proof
Suppose that \((X,d)\) is a Smyth complete quasimetric space, and let T be a \(d^{s}\)Caristi mapping on \((X,d)\). Then there exists a function \(\varphi:X\rightarrow[0,\infty)\) which is lower semicontinuous for \(\tau_{d^{s}}\) and satisfies \(d(x,Tx)\leq \varphi (x)\varphi(Tx)\) for all \(x\in X\). Exactly as in the proof of Theorem 4, we construct a left KCauchy net in \((X,d)\), which converges for \(\tau_{d^{s}}\) to an element \(p\in X\) by Smyth completeness of \((X,d)\). Finally, we deduce that p is a fixed point of T again as in the proof of Theorem 4 and taking into account that φ is now lower semicontinuous for \(\tau _{d^{s}}\).
Conversely, it will be enough to prove, by Proposition 1, that every left KCauchy sequence in \((X,d)\) converges for \(\tau_{d^{s}}\). Assume the contrary. Then there exists a left KCauchy sequence \((x_{n})_{n\in\mathbb {N}} \) in \((X,d)\) which is not convergent for \(\tau_{d^{s}}\). For each \(k\in \mathbb{N}\), there exists \(n_{k}\geq k\) such that \(d(x_{n_{k}},x_{n})<2^{(k+1)}\) for all \(n\geq n_{k}\). Therefore \(d(x_{n_{k}},x_{n_{k+1}})<2^{(k+1)}\) for all \(k\in\mathbb{N}\). Put \(y_{k}:=x_{n_{k}}\) for all \(k\in\mathbb{N}\). Then, by Proposition 2, we can suppose, without loss of generality, that \(y_{k}\neq y_{j}\) whenever \(k\neq j\), and that the sequence \(\{y_{k}:k\in\mathbb{N}\}\) does not have any convergent subsequence for \(\tau_{d^{s}}\).
We want to show that the selfmapping T of X given by \(Ty_{k}=y_{k+1}\) for all \(k\in\mathbb{N}\), and \(Tx=y_{1}\) for all \(x\notin\{y_{k}:k\in \mathbb{N}\}\), is a \(d^{s}\)Caristi mapping. To this end, construct a function \(\varphi:X\rightarrow[0,\infty)\) as follows: \(\varphi(y_{k})=2^{k}\) for all \(k\in\mathbb{N}\), and \(\varphi(x)=d^{s}(x,y_{1})+1/2\) whenever \(x\notin \{y_{k}:k\in\mathbb{N}\}\). Since, for each \(k\in\mathbb{N}\), \(\varphi (y_{k})<\varphi(x)\) whenever \(x \notin\{y_{k}:k\in\mathbb{N}\}\), and the function \(x\rightarrow d^{s}(x,y_{1})\) is continuous for \(\tau _{d^{s}}\), we immediately deduce that φ is lower semicontinuous for \(\tau _{d^{s}}\). Moreover, we have
for all \(k\in\mathbb{N}\), and
for all \(x \notin\{y_{k}:k\in\mathbb{N}\}\), so T is a \(d^{s}\)Caristi mapping on \((X,d)\). However, T has no fixed point. This contradiction concludes the proof. □
As in the metric case, we are going to deduce a multivalued version of Theorem 5.
Given a quasimetric space \((X,d)\), we denote by \(\mathcal{P}_{0}(X)\) the collection of all nonempty subsets of X. A multivalued mapping \(T:X\rightarrow\mathcal{P}_{0}(X)\) will be called \(d^{s}\)Caristi on \((X,d)\) if there is a function \(\varphi:X\rightarrow[0,\infty)\) which is lower semicontinuous for \(\tau_{d^{s}}\) and satisfies the following condition: For each \(x\in X\), there exists \(y_{x}\in Tx\) such that \(d(x,y_{x})\leq \varphi(x)\varphi(y_{x})\).
As usual, we say that a point \(z\in X\) is a fixed point of \(T:X\rightarrow \mathcal{P}_{0}(X)\) if \(z\in Tz\).
Corollary
A quasimetric space \((X,d)\) is Smyth complete if and only if every \(d^{s}\)Caristi multivalued mapping on \((X,d)\) has a fixed point.
Proof
Suppose that \((X,d)\) is Smyth complete, and let \(T:X\rightarrow\mathcal{P}_{0}(X)\) be a \(d^{s}\)Caristi multivalued mapping. Then there is a function \(\varphi:X\rightarrow[0,\infty)\) which is lower semicontinuous for \(\tau_{d^{s}}\) and satisfies that for each \(x\in X\) there exists \(y_{x}\in Tx\) such that \(d(x,y_{x})\leq\varphi(x)\varphi (y_{x})\). Define a selfmapping f on X as follows: \(fx=y_{x}\) for all \(x\in X\). Obviously f is a \(d^{s}\)Caristi mapping on \((X,d)\), so, by Theorem 5, there is \(z\in X\) such that \(z=fz\). Therefore \(z=y_{z}\). Since \(y_{z}\in Tz\), we conclude that z is a fixed point of T.
Conversely, suppose that every \(d^{s}\)Caristi multivalued mapping on \((X,d)\) has a fixed point. Then every \(d^{s}\)Caristi mapping on \((X,d)\) has a fixed point, so \((X,d)\) is Smyth complete by Theorem 5. □
Note that if \((X,d)\) is a quasimetric space and T is a selfmapping of X such that \(d(x,Tx)=0\) for all \(x\in X\), then T is a \(d^{s}\)Caristi mapping on \((X,d)\). If, in addition, \((X,d)\) is Smyth complete, then T has a fixed point by Theorem 5. Our next example illustrates this situation.
Example 5
Let Σ be a nonempty alphabet. Denote by \(\Sigma^{\infty}\) the set of all finite and infinite words (sequences) over Σ, and denote by ϕ the empty word. For each \(x,y\in \Sigma^{\infty}\), we define \(x\sqcap y\) as the longest common prefix of x and y, and for each \(x\in\Sigma^{\infty}\), we denote by \(\ell(x)\) the length of x. Then \(\ell(x)\in[1,\infty]\) whenever \(x\neq\phi \) and \(\ell(\phi)=0\). Now, for each \(x,y\in\Sigma^{\infty}\), let \(d(x,y)=0\) if x is a prefix of y, and \(d(x,y)=2^{\ell(x\sqcap y)}\) otherwise. Then d is a quasimetric on \(\Sigma^{\infty}\) [6, 25]. In fact, the quasimetric space \((\Sigma^{\infty},d)\) is Smyth complete [5], Example 3.1. Define \(T:\Sigma^{\infty}\rightarrow \Sigma ^{\infty}\) as follows: For each \(x\in\Sigma^{\infty}\), Tx is an element of \(\Sigma^{\infty}\) such that x is a prefix of Tx with \(\ell (Tx)=\ell(x)+1\). Then \(d(x,Tx)=0\) for all \(x\in\Sigma^{\infty}\). By Theorem 5, T has a fixed point. In fact, \(Tx=x\) if and only if \(\ell (x)=\infty\).
Observe that if \((X,d)\) is a nonSmyth complete quasimetric space such that \((X,d^{s})\) is complete, we can apply Caristi’s fixed point theorem to \((X,d^{s})\). However, by Theorem 5, there exists a \(d^{s}\)Caristi mapping on \((X,d)\) without fixed point. We conclude this section with an example illustrating this fact.
Example 6
Let d be the quasimetric on \(\mathbb {R}\) given by \(d(x,y)=yx\) if \(x\leq y\), and \(d(x,y)=1\) if \(x>y\). Then \((\mathbb{R},d)\) is the Sorgenfrey quasimetric space. Since \(d^{s}(x,y)\geq1\) for all \(x,y\in\mathbb{R}\) with \(x\neq y\), we deduce that the metric space \((\mathbb{R},d^{s})\) is complete and \(\tau_{d^{s}}\) is the discrete topology on \(\mathbb{R}\). As we indicated in Section 1, \((\mathbb{R},d)\) is not Smyth complete (indeed, note that the sequence \(((n1)/n)_{n\in\mathbb{N}}\) is left KCauchy but it does not converge for \(\tau_{d^{s}}\)). Define \(T:\mathbb {R}\rightarrow\mathbb{R}\) as \(Tx=0\) for all \(x>0\), \(T0=1\), and \(Tx=x/2\) for all \(x<0\). Although T has no fixed point, we show that it is a \(d^{s}\)Caristi mapping on \((\mathbb{R},d)\). To this end, define \(\varphi:\mathbb{R} \rightarrow[0,\infty)\) as \(\varphi(x)=3\) for all \(x>0\), \(\varphi (0)=2\), and \(\varphi(x)=x\) for all \(x<0\). Obviously φ is lower semicontinuous for \(\tau_{d^{s}}\). Moreover, for \(x>0\), we obtain
For \(x=0\), we obtain
and for \(x<0\),
Hence T is a \(d^{s}\)Caristi mapping on \((X,d)\) without fixed point. Finally, observe that for \(x=1\) one has
An application
In this section we shall apply Theorem 5 to obtaining a general fixed point theorem in the setting of the complexity space, from which we shall deduce, in a unified and fast way, the existence of solution for a large class of algorithms defined by recurrence equations that includes Hanoi, Largetwo (average case), and Quicksort (worst case), (see, e.g., [26] for a detailed study of these algorithms).
Let us recall that the socalled complexity space was introduced by Schellekens in [27] to the development of a topological foundation for the complexity analysis of algorithms and programs. Further contributions to the study of this space and its applications may be found in [7, 22, 28–30], etc.
The complexity space is the quasimetric space \((\mathcal{C},d_{\mathcal {C}})\), where
and \(d_{\mathcal{C}}\) is the quasimetric on \(\mathcal{C}\) given by
for all \(f,g\in\mathcal{C}\). (We adopt the convention that \(1/\infty=0\).)
The set \(\{f\in\mathcal{C}:f(n)<\infty\mbox{ for all }n\in\mathbb{N}\}\) is denoted by \(\mathcal{C}_{0}\).
The elements of \(\mathcal{C}\) are called complexity functions. According to Schellekens [27], p.540, given two complexity functions f and g, the numerical value \(d_{\mathcal{C}}(f,g)\) (the complexity distance from f to g) can be interpreted as the relative progress made in lowering the complexity by replacing any program P with complexity function f by any program Q with complexity function g. Therefore, condition \(d_{\mathcal{C}}(f,g)=0\), with \(f\neq g\), can be read as the program P is at least as efficient as the program Q because \(d_{C}(f,g)=0\) if and only if \(f(n)\leq g(n)\) for all \(n\in\mathbb{N}\). Obviously, the metric \((d_{\mathcal{C}})^{s}\) is not able to give this information since in the case that \(d_{\mathcal {C}}(f,g)=0\), with \(f\neq g\), we deduce that \(d_{\mathcal {C}}(g,f)=(d_{\mathcal{C}})^{s}(f,g)\), and thus the last measure does not indicate that program is more efficient. However, we know that the program with complexity function f is more efficient than the one with complexity function g (see [27], p.541).
Now let c and a be positive real constants and \(h\in\mathcal{C}_{0}\). Define
Observe that \(\mathcal{C}_{cah}\neq\emptyset\) since the complexity function \(f_{1}\) defined by \(f_{1}(1)=c\) and \(f_{1}(n)=\infty\) for all \(n\geq2\) clearly belongs to \(\mathcal{C}_{cah}\).
The restriction of the quasimetric \(d_{\mathcal{C}}\) to \(\mathcal{C}_{cah}\) will be denoted by \(d_{\mathcal{C}_{cah}}\).
The following auxiliary results will be useful in the proof of the main result of this section (Theorem 6 below).
Lemma 1
Let \((f_{k})_{k\in\mathbb {N}}\) be a sequence in \(\mathcal{C}\) such that \(\lim_{k\rightarrow \infty}(d_{\mathcal{C}})^{s}(f,f_{k})=0\) for some \(f\in \mathcal{C}\), and let \(m\in\mathbb{N}\).

(a)
If \(f(m)<\infty\), then \(f_{k}(m)<\infty\) eventually, and \(\lim_{k\rightarrow\infty}f_{k}(m)=f(m)\).

(b)
\(f(m)=\infty\) if and only if \(\lim_{k\rightarrow\infty }f_{k}(m)=\infty\).
Proof
Since \(\lim_{k\rightarrow\infty}(d_{\mathcal {C}})^{s}(f,f_{k})=0\), for each \(\varepsilon>0\), there is \(k_{\varepsilon }\in \mathbb{N}\) such that
for all \(k\geq k_{\varepsilon}\). In particular
for all \(k\geq k_{\varepsilon}\).
Suppose that \(f(m)<\infty\). Taking \(\varepsilon=2^{m}/f(m)\), it follows from (1) that \(f_{k}(m)<\infty\) for all \(k\geq k_{\varepsilon}\). Hence \(\lim_{k\rightarrow\infty}f_{k}(m)=f(m)\) by Proposition 2 of [7]. Thus, we have shown (a).
If \(f(m)=\infty\), relation (1) gives \(2^{m}/\varepsilon< f_{k}(m)\) for all \(k\geq k_{\varepsilon}\). Since \(\varepsilon>0\) is chosen arbitrarily, we deduce that \(\lim_{k\rightarrow\infty}f_{k}(m)=\infty\). Conversely, if \(\lim_{k\rightarrow\infty}f_{k}(m)=\infty\), again it follows from (1) that \(1/f(m)=0\), i.e., \(f(m)=\infty\). Thus, we have shown (b). □
Lemma 2
([22])
The quasimetric space \((\mathcal{C},d_{\mathcal{C}})\) is Smyth complete.
Lemma 3
Let c and a be positive real constants and \(h\in\mathcal{C}_{0}\). Then the quasimetric space \((\mathcal{C}_{cah},d_{\mathcal{C}_{cah}})\) is Smyth complete.
Proof
We first show that \(\mathcal{C}_{cah}\) is a closed subset of the metric space \((\mathcal{C},(d_{\mathcal{C}})^{s})\). Indeed, let \((f_{k})_{k\in\mathbb{N}}\) be a sequence in \(\mathcal{C}_{cah}\) and \(f\in \mathcal{C}\) such that \(\lim_{k\rightarrow\infty}(d_{\mathcal{C}})^{s}(f,f_{k})=0\). We shall show that \(f(1)=c\) and \(f(m)\geq af(m1)+h(m)\) whenever \(m\geq2\).
To this end, we distinguish the following cases.
Case 1. \(m=1\). Then \(f_{k}(1)=c\) for all \(k\in\mathbb{N}\), so by Lemma 1(b), \(f(1)<\infty\). Then \(f(1)=c\) by Lemma 1(a).
Case 2. \(m>1\) and \(f(m)=\infty\). Then \(f(m)\geq af(m1)+h(m)\), obviously.
Case 3. \(m>1\) and \(f(m)<\infty\). Then, by Lemma 1(a), there is \(k_{0}\in \mathbb{N}\) such that \(f_{k}(m)<\infty\) for all \(k\geq k_{0}\), and \(\lim_{k\rightarrow\infty}f_{k}(m)=f(m)\). From this equality and the fact that \(f_{k}\in\mathcal{C}_{cah}\), we deduce the existence of \(k_{1}\geq k_{0}\) such that for each \(k\geq k_{1}\),
Consequently, \(f_{k}(m1)<\infty\) for all \(k\geq k_{1}\), and by Lemma 1(b), \(f(m1)<\infty\) (otherwise, \(\lim_{k}f_{k}(m1)=\infty\), which contradicts (2)). Therefore, we also have \(\lim_{k\rightarrow\infty }f_{k}(m1)=f(m1)\), by Lemma 1(a).
Now choose an arbitrary \(\varepsilon>0\). Then there exists \(k_{\varepsilon }\in\mathbb{N}\) such that
for all \(k\geq k_{\varepsilon}\). Hence
for all \(k\geq k_{\varepsilon}\). Thus \(\varepsilon +f(m)>a(f(m1)+\varepsilon)+h(m)\) for any \(\varepsilon>0\), so \(f(m)\geq af(m1)+h(m)\). Consequently, \(f\in C_{cah}\), and hence \(\mathcal {C}_{cah}\) is closed in the metric space \((\mathcal{C},(d_{\mathcal{C}})^{s})\). Then \((\mathcal{C}_{cah},d_{\mathcal{C}_{cah}})\) is Smyth complete by Lemma 2. □
Theorem 6
Let c and a be positive real constants with \(a\geq1\), let \(h\in\mathcal{C}_{0}\), and let Ψ be the mapping on \(\mathcal{C}_{cah}\) defined as
Then the following hold:

(A)
Ψ is a selfmapping on \(\mathcal{C}_{cah}\).

(B)
For each \(f\in\mathcal{C}_{cah}\),
$$d_{\mathcal{C}_{cah}}(f,\Psi f)=\varphi(f)\varphi(\Psi f), $$where \(\varphi:\mathcal{C}_{cah}\rightarrow[0,\infty)\) is the lower semicontinuous function for \(\tau_{(d_{\mathcal{C}_{cah}})^{s}}\) given by
$$\varphi(f)=\frac{a+1}{2ac}\sum_{n=1}^{\infty}2^{n} \frac{1}{f(n)} $$for all \(f\in\mathcal{C}_{cah}\).

(C)
Ψ has a fixed point in \(\mathcal{C}_{cah}\).
Proof
(A) Let \(f\in\mathcal{C}_{cah}\). Then \(\Psi f(1)=c\) by definition of Ψ. We also have \(\Psi f(2)=af(1)+h(2)=a\Psi f(1)+h(2)\).
Now let \(n>2\). Then
We conclude that \(\Psi f\in\mathcal{C}_{cah}\).
(B) We first observe that, in fact, \(\varphi(f)\geq0\) for all \(f\in \mathcal{C}_{cah}\). Indeed, since \(a\geq1\), we have \(f(n)\geq f(n1)\) for all \(n\geq2\), and thus \(f(n)\geq f(2)\geq ac\) for all \(n\geq2\). Therefore
Let now \(f\in\mathcal{C}_{cah}\) and \((f_{k})_{k\in\mathbb{N}}\) be a sequence in \(\mathcal{C}_{cah}\) such that \(\lim_{k\rightarrow\infty }(d_{\mathcal{C}_{cah}})^{s}(f,f_{k})=0\). Since
we deduce that \(\varphi(f)\leq\lim\inf_{k\rightarrow\infty}\varphi (f_{k})\). Therefore φ is lower semicontinuous for \(\tau_{(d_{\mathcal{C}_{cah}})^{s}}\).
Furthermore, for each \(f\in\mathcal{C}_{cah}\), we have \(f\geq\Psi f\), and hence
(C) From (B) we deduce that Ψ is a \((d_{\mathcal {C}_{cah}})^{s}\)Caristi mapping on \((\mathcal{C}_{cah},d_{\mathcal{C}_{cah}})\). Then Ψ has a fixed point by Lemma 3 and Theorem 5. □
It follows from Theorem 6 that those algorithms defined by recurrence equations, whose associated functional is a mapping Ψ of type (3), admit a solution. We conclude the paper by applying this fact to deduce the existence of solution for the three algorithms mentioned at the beginning of this section.
Example 7
The algorithm Hanoi solves the celebrated Towers of Hanoi problem. The running time of computing of this algorithm is the solution of the recurrence equation \(S:\mathbb{N}\rightarrow(0,\infty)\) given by
with \(c,d>0\) (see, e.g., [26]). The functional \(\Psi _{S}\) naturally associated to S is defined as
Clearly \(\Psi_{S}\) is a mapping of type (3) for \(a=2\), and \(h\in \mathcal{C}_{0}\) defined as \(h(n)=d\) for all \(n\in\mathbb{N}\). By Theorem 6, there exists \(f_{S}\in\mathcal{C}_{cah}\) such that \(f_{S}=\Psi f_{S}\). Hence \(f_{S}\) is a solution of the recurrence equation S.
Example 8
The algorithm Largetwo is a typical example of average case behavior whose running time of computing is the solution of the recurrence equation \(S:\mathbb{N}\rightarrow(0,\infty)\) given by
with \(c>0\) (see, e.g., [26]). The functional \(\Psi _{S}\) naturally associated to S is defined as
Clearly \(\Psi_{S}\) is a mapping of type (3) for \(a=1\), and \(h\in \mathcal{C}_{0}\) defined as \(h(n)=21/n\) for all \(n\in\mathbb{N}\). By Theorem 6, there exists \(f_{S}\in\mathcal{C}_{cah}\) such that \(f_{S}=\Psi f_{S}\). Hence \(f_{S}\) is a solution of the recurrence equation S.
Example 9
The running time of computing of the wellknown algorithm Quicksort is, for the worst case, the solution of the recurrence equation \(S:\mathbb{N}\rightarrow(0,\infty)\) given by
with \(c,b>0\) (see, e.g., [26]). The functional \(\Psi _{S}\) naturally associated to S is defined as
Clearly \(\Psi_{S}\) is a mapping of type (3) for \(a=1\), and \(h\in \mathcal{C}_{0}\) defined as \(h(n)=bn\) for all \(n\in\mathbb{N}\). By Theorem 6, there exists \(f_{S}\in\mathcal{C}_{cah}\) such that \(f_{S}=\Psi f_{S}\). Hence \(f_{S}\) is a solution of the recurrence equation S.
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Acknowledgements
The authors are grateful to the reviewers for several suggestions which have allowed to improve the first version of the paper. This research is supported by the Ministry of Economy and Competitiveness of Spain, Grant MTM201237894C0201.
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Romaguera, S., Tirado, P. A characterization of Smyth complete quasimetric spaces via Caristi’s fixed point theorem. Fixed Point Theory Appl 2015, 183 (2015). https://doi.org/10.1186/s1366301504311
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DOI: https://doi.org/10.1186/s1366301504311
MSC
 54H25
 47H10
 54E50
 68Q25
Keywords
 fixed point
 quasimetric
 complete
 Smyth complete
 algorithm
 recurrence equation