Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

TL;DR

This paper establishes the equivalence of Ekeland, Takahashi, and Caristi principles in sequentially right K-complete quasi-pseudometric spaces, linking these variational principles to the space's completeness using Picard sequences.

Contribution

It introduces a unified approach to these principles in asymmetric pseudometric spaces and proves their equivalence to the space's completeness.

Findings

Proves the equivalence of the principles in quasi-pseudometric spaces.

Shows the principles are equivalent to the space's completeness.

Uses Picard sequences for set-valued mappings to unify the principles.

Abstract

We prove versions of Ekeland, Takahashi and Caristi principles in sequentially right $K$-complete quasi-pseudometric spaces (meaning asymmetric pseudometric spaces), the equivalence between these principles, as well as their equivalence to the completeness of the underlying quasi-pseudometric space. The key tools are Picard sequences for some special set-valued mappings corresponding to a function $\varphi$ on a quasi-pseudometric space, allowing a unitary treatment of all these principles.

Equations

$$d(x,y)\ge 0,andd(x,x)=0;$$

$$d(x,z)\le d(x,y)+d(y,z),$$

$$\text{mbox}(QM3)d(x,y)=d(y,x)=0\Rightarrow x=y,$$

$$B_d(x,r)=$$

$$B_d[x,r]=$$

$$V\in {\mathcal{V}}_d(x)$$

$$⟺\exists r^{\prime }>0\text{mbox}suchthat{B}_d[x,r^{\prime }]\subseteq V.$$

$$x_n\stackrel{d}{\to }x⟺d(x,x_n)\to 0.$$

$$x_n\stackrel{\bar{d}}{\to }x⟺\bar{d}(x,x_n)\to 0⟺d(x_n,x)\to 0.$$

$$\mathbb{N}=\{1,2,\dots \}\text{mbox}--thesetofnaturalnumbers,$$

$${\mathbb{N}}_0=\mathbb{N}\cup \{0\}\text{mbox}--thesetofnon-negativeintegers.$$

$$\forall n,m,\text{mbox}with{n}_{\epsilon }\le n<m,d(x_n,x_m)<\epsilon$$

$$⟺$$

$$\forall n,m,\text{mbox}with{n}_{\epsilon }\le n<m,d(x_m,x_n)<\epsilon$$

$$⟺$$

$$x{\le }_{\tau }y$$

$$⟺\forall V\in \mathcal{V}(x),y\in V,$$

$$x\in \overline{\{y\}}\text{mbox}and\{y\}\subseteq \overline{\{z\}}\Rightarrow x\in \overline{\{y\}}\subseteq \overline{\overline{\{z\}}}=\overline{\{z\}},$$

$$x{\le }_{\tau }y\text{mbox}andy{\le }_{\tau }z\Rightarrow x{\le }_{\tau }z.$$

$$x{\le }_{\tau }y\text{mbox}andy{\le }_{\tau }x\Rightarrow x=y,$$

$$x\ne y\Rightarrow x{\nleqq }_{\tau }y\text{mbox}ory{\nleqq }_{\tau }x,$$

$$x{\nleqq }_{\tau }y\text{mbox}ory{\nleqq }_{\tau }x$$

$$⟺\exists V\in \mathcal{V}(x),y\notin V\text{mbox}or\exists U\in \mathcal{V}(y),x\notin U$$

$$⟺\tau \text{mbox}is{T}_0.$$

$$x{\le }_{\tau }y⟺x\in \overline{\{y\}}=\{y\}⟺x=y,$$

$$x{\le }_{\tau }y⟺x=y,$$

$$x\in \overline{\{y\}}$$

$$\uparrow A$$

$$\downarrow A$$

$$y{\le }_{\tau }x⟺y\in \overline{\{x\}}.$$

$$\uparrow A\subseteq \bigcap \left\{U\in \tau :A\subseteq U\right\}.$$

$$\complement \left(\uparrow A\right)\subseteq \complement \left(\bigcap \left\{U\in \tau :A\subseteq U\right\}\right)⟺\bigcap \left\{U\in \tau :A\subseteq U\right\}\subseteq \uparrow A.$$

$$x{\le }_{d}y⟺d(x,y)=0,$$

$$x{\le }_{d}y$$

$$⟺\forall r>0,y\in B_d(x,r)$$

$$⟺\forall r>0,d(x,y)<r$$

Full text

The key tools are Picard sequences for some special set-valued mappings corresponding to a function $\varphi$ on a quasi-pseudometric space, allowing a unitary treatment of all these principles.

Classification MSC 2010: 58E30 47H10 54E25 54E35 54E50

Key words: quasi-metric space; completeness in quasi-metric spaces; variational principles; Ekeland variational principle; Takahashi minimization principle; fixed point; Caristi fixed point theorem.

1. Introduction

Ivar Ekeland announced in 1972, [14] (the proof appeared in 1974, [15]) a theorem asserting the existence of the minimum of a small perturbation of a lower semicontinuous (lsc) function defined on a complete metric space. This result, known as Ekeland Variational Principle (EkVP), proved to be a very versatile tool in various areas of mathematics and applications - optimization theory, geometry of Banach spaces, optimal control theory, economics, social sciences, and others. Some of these applications are presented by Ekeland himself in [16].

At the same time, it turned out that this principle is equivalent to a lot of results in fixed point theory (Caristi fixed point theorem), geometry of Banach spaces (drop property), and others (see [24], for instance). Takahashi [28] (see also [29]) found a sufficient condition for the existence of the minimum of a lsc function on a complete metric space, known as Takahashi minimization principle, which also turned to be equivalent to EkVP (see [28] and [18]).

For convenience, we mention these three principles.

Theorem 1.1** (Ekeland, Takahashi and Caristi principles).**

Let $(X,d)$ be a complete metric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below lsc function. Then the following hold.

1. (wEk)

There exists $z\in X$ such that $\varphi(z)<\varphi(x)+d(x,z)$ for all $x\in X\smallsetminus\{z\}.$ 2. (Tak)

If for every $x\in X$ with $\varphi(x)>\inf\varphi(X)$ there exists an element $y\in X\smallsetminus\{x\}$ such that $\varphi(y)+d(x,y)\leq\varphi(x),$ then $\varphi$ attains its minimum on $X$, i.e., there exists $z\in X$ such that $\varphi(z)=\inf\varphi(X).$ 3. (Car)

If the mapping $T:X\to X$ satisfies $d(Tx,x)+\varphi(Tx)\leq\varphi(x)$ for all $x\in X,$ then $T$ has a fixed point in $X$, i.e., there exists $z\in X$ such that $Tz=z.$

The statement (wEk) is called the weak form of the Ekeland variational principle. Later, various versions and extensions of these principles appeared, a good record (up to 2009) being given in the book [22].

Some versions of EkVP and Takahashi minimum principles in $T_{1}$ quasi-metric spaces were proved in [9] and [1], respectively. In [9] the equivalence of the weak Ekeland variational principle to Caristi fixed point theorem was proved and the implications of the validity of Caristi fixed point theorem principle on the completeness of the underlying quasi-pseudometric space were studied as well. In [1] the same is done for the weak form of Ekeland variational principle and Takahashi minimization principle. The extension of wEk to arbitrary quasi-metric spaces was given in [19].

Other extensions, with applications to various areas of social sciences and psychology, were given in [2]–[5], [8], [25].

All these extensions are obtained by relaxing the conditions, else on the minimized function or on the underlying metric space, or both. In this paper we consider both approaches – we study the validity of EkVP in quasi-pseudometric spaces for functions satisfying a weaker semicontinuity condition, called near lower semicontinuity (see Subsection 2.4). Roughly speaking, a quasi-pseudometric is a function $d$ on $X\times X$, where $X$ is a set, satisfying all the axioms of a pseudometric but symmetry, that is, the possibility that $d(x,y)\neq d(y,x)$ for some $x,y\in X$ is not excluded (see Section 2). We also prove (Theorem 3.4) a full version of EkVP in quasi-pseudometric spaces for lsc functions.

The nearly lsc functions were introduced in [19]. We show that a nearly lsc function is lsc if and only if it is monotone with respect to the specialization order (see Subsection 2.3 and Proposition 2.11).

The main results of the paper are contained in Section 3, where one proves quasi-pseudometric versions of Ekeland, Takahashi and Caristi principles and their equivalence. One proves also that the validity of wEkVP implies the sequential right-$K$-completeness of the underlying quasi-pseudometric space. We conclude this section by showing that the $T_{1}$ versions of these principles are particular cases of those proved for general quasi-pseudometric spaces.

The key tools used in the proofs of these results are Picard sequences for some special set-valued mappings corresponding to a function $\varphi$ on a quasi-pseudometric space (see Subsection 2.5), which allow a unitary treatment of all these principles. The idea to use Picard sequences appeared in [13] and was subsequently exploited in [7] and [2].

2. Quasi-metric spaces

2.1. Topological properties

A quasi-pseudometric on an arbitrary set $X$ is a mapping $d:X\times X\to[0,\infty)$ satisfying the following conditions:

[TABLE]

for all $x,y,z\in X.$ If further

[TABLE]

for all $x,y\in X,$ then $d$ is called a quasi-metric. The pair $(X,d)$ is called a quasi-pseudometric space, respectively a quasi-metric space111In [11] the term “quasi-semimetric” is used instead of “quasi-pseudometric”. The conjugate of the quasi-pseudometric $d$ is the quasi-pseudometric $\bar{d}(x,y)=d(y,x),\,x,y\in X.$ The mapping $d^{s}(x,y)=\max\{d(x,y),\bar{d}(x,y)\},\,$ $x,y\in X,$ is a pseudometric on $X$ which is a metric if and only if $d$ is a quasi-metric.

If $(X,d)$ is a quasi-pseudometric space, then for $x\in X$ and $r>0$ we define the balls in $X$ by the formulae

[TABLE]

The topology $\tau_{d}$ (or $\tau(d)$) of a quasi-pseudometric space $(X,d)$ can be defined starting from the family $\mathcal{V}_{d}(x)$ of neighborhoods of an arbitrary point $x\in X$:

[TABLE]

The convergence of a sequence $(x_{n})$ to $x$ with respect to $\tau_{d},$ called $d$-convergence and denoted by $x_{n}\xrightarrow{d}x,$ can be characterized in the following way

[TABLE]

Also

[TABLE]

As a space equipped with two topologies, $\tau_{d}$ and $\tau_{\bar{d}}\,$, a quasi-pseudometric space can be viewed as a bitopological space in the sense of Kelly [20].

The following topological properties are true for quasi-pseudometric spaces.

Proposition 2.1** (see [11]).**

If $(X,d)$ is a quasi-pseudometric space, then the following hold.

The ball $B_{d}(x,r)$ is $\tau_{d}$-open and the ball $B_{d}[x,r]$ is $\tau_{\bar{d}}$-closed. The ball $B_{d}[x,r]$ need not be $\tau_{d}$-closed. 2. 2.

*The topology $\tau_{d}$ is $T_{0}$ if and only if $d$ is a quasi-metric. *

The topology $\tau_{d}$ is $T_{1}$ if and only if $d(x,y)>0$ for all $x\neq y$ in $X$. 3. 3.

*For every fixed $x\in X,$ the mapping $d(x,\cdot):X\to(\mathbb{R},|\cdot|)$ is $\tau_{d}$-usc and $\tau_{\bar{d}}$-lsc. *

For every fixed $y\in X,$ the mapping $d(\cdot,y):X\to(\mathbb{R},|\cdot|)$ is $\tau_{d}$-lsc and $\tau_{\bar{d}}$-usc.

The following remarks show that imposing too many conditions on a quasi-pseudometric space it becomes pseudometrizable.

Remark 2.2** ([20]).**

Let $(X,d)$ be a quasi-metric space. Then

1. (a)

if the mapping $d(x,\cdot):X\to(\mathbb{R},|\cdot|)$ is $\tau_{d}$-continuous for every $x\in X,$ then the topology $\tau_{d}$ is regular; 2. (b)

if $\tau_{d}\subseteq\tau_{\bar{d}}$, then the topology $\tau_{\bar{d}}$ is pseudometrizable; 3. (c)

if $d(x,\cdot):X\to(\mathbb{R},|\cdot|)$ is $\tau_{\bar{d}}$-continuous for every $x\in X,$ then the topology $\tau_{\bar{d}}$ is pseudometrizable.

Remark 2.3**.**

The characterization of Hausdorff property (or $T_{2}$) of quasi-metric spaces can be given in terms of uniqueness of the limits, as in the metric case. The topology of a quasi-pseudometric space $(X,d)$ is Hasudorff if and only if every sequence in $X$ has at most one $d$-limit if and only if every sequence in $X$ has at most one $\bar{d}$-limit (see [30]).

In the case of an asymmetric normed space there exists a characterization in terms of the quasi-norm (see [11], Propositions 1.1.40).

Recall that a topological space $(X,\tau)$ is called:

• •

$T_{0}$ if for every pair of distinct points in $X$, at least one of them has a neighborhood not containing the other;

• •

$T_{1}$ if every pair of distinct points in $X$, each of them has a neighborhood not containing the other;

• •

$T_{2}$ (or Hausdorff) if every two distinct points in $X$ admit disjoint neighborhoods;

• •

regular if for every point $x\in X$ and closed set $A$ not containing $x$ there exist the disjoint open sets $U,V$ such that $x\in U$ and $A\subseteq V.$

2.2. Completeness in quasi-metric spaces

The lack of symmetry in the definition of quasi-metric spaces causes a lot of troubles, mainly concerning completeness, compactness and total boundedness in such spaces. There are a lot of completeness notions in quasi-metric spaces, all agreeing with the usual notion of completeness in the metric case, each of them having its advantages and weaknesses (see [26], or [11]).

As in what follows we shall shall work only with two of these notions, we shall present only them, referring to [11] for others.

We use the notation

[TABLE]

Definition 2.4**.**

Let $(X,d)$ be a quasi-pseudometric space. A sequence $(x_{n})$ in $(X,d)$ is called:

• •

left $d$-$K$-Cauchy if for every $\varepsilon>0$ there exists $n_{\varepsilon}\in\mathbb{N}$ such that

[TABLE]

• •

right $d$-$K$-Cauchy if for every $\varepsilon>0$ there exists $n_{\varepsilon}\in\mathbb{N}$ such that

[TABLE]

The quasi-pseudometric space $(X,d)$ is called:

• •

sequentially left $d$-$K$-complete if every left $d$-$K$-Cauchy is $d$-convergent;

• •

sequentially right $d$-$K$-complete if every right $d$-$K$-Cauchy is $d$-convergent.

Remarks 2.5**.**

It is obvious that a sequence is left $d$-$K$-Cauchy if and only if it is right $\bar{d}$-$K$-Cauchy.

• 2.

There are examples showing that a $d$-convergent sequence need not be left $d$-$K$-Cauchy, showing that in the asymmetric case the situation is far more complicated than in the symmetric one (see [26]).

• 3.

If each convergent sequence in a regular quasi-metric space $(X,d)$ admits a left $K$-Cauchy subsequence, then $X$ is metrizable ([21]).

Proposition 2.6**.**

Let $(X,d)$ be a quasi-pseudometric space. If a right $K$-Cauchy sequence $(x_{n})$ contains a subsequence convergent to some $x\in X$, then the sequence $(x_{n})$ converges to $x$.

Remark 2.7**.**

One can define more general notions of completeness by replacing in Definition 2.4 the sequences with nets. Stoltenberg [27, Example 2.4] gave an example of a sequentially right $K$-complete $T_{1}$ quasi-metric space which is not right $K$-complete (i.e., not right $K$-complete by nets).

Convention. In the following, when speaking about metric or topological properties in a quasi-pseudometric space $(X,d)$ we shall always understand those corresponding to $d$ and we shall omit $d$ or $\tau_{d}$, i.e., we shall write “$(x_{n})$ is right $K$-Cauchy” instead of “$(x_{n})$ is right $d$-$K$-Cauchy”, $\overline{A}$ instead of $\overline{A}^{d}$, etc.

2.3. The specialization order in topological spaces

Let $(X,\tau)$ be a topological space. Denote by $\mathcal{V}(x)$ the family of all neighborhoods of a point $x\in X$. The specialization order in $X$ is the partial order defined by

[TABLE]

that is $y$ belongs to every open set containing $x$.

By a preorder on a set $X$ we understand a relation $\leq$ on $X$ such that

(O1) $x\leq x$ and

(O2) $((x\leq y)\wedge(y\leq z))\;\Rightarrow\;x\leq z,$

for all $x,y,z\in X.$ If further

(O3) $((x\leq y)\wedge(y\leq x))\;\Rightarrow\;x=y,$

then $\leq$ is called an order on $X$.

Proposition 2.8**.**

Let $(X,\tau)$ be a topological space. Then

1. (i)

the relation defined by (2.5) is a preorder on $X$; 2. (ii)

it is an order if and only if the topology $\tau$ is $T_{0}$; 3. (iii)

the topology $\tau$ is $T_{1}$ if and only if $\,\leq_{\tau}$ is the equality relation in $X$.

Proof.

(i)   Since $x\in\overline{\{x\}}$ it follows $x\leq_{\tau}x$.

The transitivity follows from the following implication

[TABLE]

that is

[TABLE]

(ii)  The antisymmetry means that

[TABLE]

or, equivalently,

[TABLE]

for all $x,y\in X.$

But

[TABLE]

(iii)  The topological space $X$ is $T_{1}$ if and only if $\,\overline{\{x\}}=\{x\}$ for every $x\in X$. Consequently,

[TABLE]

Conversely,

[TABLE]

is equivalent to

[TABLE]

hence $\overline{\{y\}}=\{y\}$ for all $y\in X$, that is, $\tau$ is $T_{1}.$ ∎

Let $(X,\leq)$ be an ordered set. For $A\subseteq X$ put

[TABLE]

In the following results the order notions are considered with respect to the specialization order $\leq_{\tau}$.

Proposition 2.9**.**

Let $(X,\tau)$ be a topological space and $A\subseteq X$.

If the set $A$ is open, then it is upward closed, i.e. $\uparrow\!\!A=A$. 2. 2.

If the set $A$ is closed, then it is downward closed, i.e. $\downarrow\!\!A=A$.

Proof.

1.  It is a direct consequence of definitions. Let $x\in A$ and $y\in X,\,x\leq_{\tau}y.$ Since $A$ is open, this inequality implies $y\in A.$

2.   Let $x\in A$ and $y\in X,\,y\leq_{\tau}x.$ Then $y\in\overline{\{x\}}\subseteq\overline{A}=A$. ∎

Let us define the saturation of a subset $A$ of $X$ as the intersection of all open subsets of $X$ containing $A$. The set $A$ is called saturated if equals its saturation.

Proposition 2.10**.**

Let $(X,\tau)$ be a topological space.

For every $x\in X$, $\downarrow\!\!x=\overline{\{x\}}$. 2. 2.

For any subset $A$ of $X$ the saturation of $A$ coincides with $\uparrow\!\!A$.

Proof.

1.  This follows from the equivalence

[TABLE]

2.  Since every open set is upward closed, $U\in\tau$ and $U\supset A$ implies $U\supset\uparrow\!\!A,$ that is

[TABLE]

If $y\notin\uparrow\!\!A$, then for every $x\in A$ there exists $U_{x}\in\tau$ such that $x\in U_{x}$ and $y\notin U_{x}.$ It follows $y\notin V:=\bigcup\{U_{x}:x\in A\}\in\tau$ and $A\subseteq V$, hence $y\notin\bigcap\{U\in\tau:A\subseteq U\}$, showing that

[TABLE]

∎

2.4. Lower semi-continuous functions on quasi-pseudometric spaces

Let $(X,d)$ be a quasi-pseudometric space. The specialization order $\leq_{d}$ in $X$ with respect to the topology $\tau_{d}$ takes the form

[TABLE]

for $x,y\in X.$

Indeed,

[TABLE]

For reader’s convenience, we present some remarks about $\liminf$ and $\limsup$ of sequences in $\mathbb{R}.$ Let $(a_{n})$ be a sequence in $\mathbb{R}.$ For $n\in\mathbb{N}$ let

[TABLE]

It follows $\overline{a}_{n+1}\leq\overline{a}_{n}$ and $\underline{a}_{n+1}\geq\underline{a}_{n}$.

By definition one puts

[TABLE]

Note that $\overline{l}=\liminf_{n}a_{n}$ and $\underline{l}=\liminf_{n}a_{n}$ always exist, $\underline{l}\leq\overline{l}$ and the sequence $(a_{n})$ has the limit $l$ if and only if

[TABLE]

A cluster point of the sequence $(a_{n})$ is a number $x\in\overline{\mathbb{R}}$ such that $\lim\limits_{k\to\infty}\,a_{n_{k}}=x$, for some subsequence $(a_{n_{k}})$ of $(a_{n})$. The numbers $\overline{l}$ and $\underline{l}$ are cluster points of the sequence $(a_{n})$ and any other cluster point $\lambda$ of $(a_{n})$ satisfies the inequalities

[TABLE]

A function $f:X\to\mathbb{R}\cup\{\infty\}$ is called:

• •

lower semi-continuous (lsc) at $x\in X$ if for every sequence $(x_{n})$ in $X$ converging to $x$,

[TABLE]

• •

nearly lower semi-continuous (nearly lsc) at $x\in X$ if (2.7) holds only for sequences $(x_{n})$ with distinct terms converging to $x$;

• •

lower semi-continuous (nearly lower semi-continuous) on $X$ if it is lsc (nearly lsc) at every $x\in X.$

Obviously, a lsc function is nearly lsc. The notion of nearly lsc function was introduced by Karapinar and Romaguera [19] who showed by an example that it is effectively more general than lsc. We call a function $f:X\to\mathbb{R}\cup\{\infty\}$ $d$-monotone if

[TABLE]

for all $x,y\in X.$

Proposition 2.11**.**

Let $(X,d)$ be a quasi-pseudometric space and $f:X\to\mathbb{R}\cup\{\infty\}$ a function.

The function $f$ is nearly lsc at $x\in X$ if and only if (2.7) holds for all sequences $(x_{n})$ without constant subsequences such that $\lim_{n\to\infty}x_{n}=x.$ 2. 2.

The function $f$ is lsc if and only if it is nearly lsc and $d$-monotone. 3. 3.

If the topology $\tau_{d}$ is $T_{1}$, then any nearly lsc function is lsc.

Proof.

1. Suppose that (2.7) holds for all sequences with distinct terms converging to $x$. Let $(x_{n})$ be a sequence without constant subsequences converging to $x$. Put $n_{1}=1$ and define inductively

[TABLE]

Since every term $x_{i}$ of the sequence $(x_{n})$ appears only finitely many times, the numbers $n_{k}$ are well defined,

[TABLE]

Let also

[TABLE]

Then

[TABLE]

for all $k\in\mathbb{N}.$ It follows

[TABLE]

so that

[TABLE]

for all $k\in\mathbb{N}.$ Since $\lim_{k\to\infty}m_{k}=\infty$,

[TABLE]

Consequently,

[TABLE]

2. We have remarked that a lsc function is nearly lsc. We show that it is also $d$-monotone. Indeed, if $x\leq_{d}y$, that is $d(x,y)=0$, then the constant sequence $x_{n}=y,\,n\in\mathbb{N},$ converges to $x$, so that, by the lsc of the function $f$,

[TABLE]

Suppose now that $f:X\to\mathbb{R}\cup\{\infty\}$ is $d$-monotone and nearly lsc. The proof of the fact that it is lsc will be based on the following remark.

*Claim *I. If $f:X\to\mathbb{R}\cup\{\infty\}$ is $d$-monotone and

[TABLE]

for every sequence $(x_{n})$ without constant subsequences, then (2.8) holds for arbitrary sequences in $X$.

Let $(x_{n})_{n\in\mathbb{N}}$ be an arbitrary sequence in $X$ such that $x_{n}\to x$ and $f(x_{n})\to\lambda$ as $n\to\infty.$ If $(x_{n})$ contains a constant subsequence, say $x_{n_{k}}=y,\,k\in\mathbb{N},$ then $\lim_{k\to\infty}f(x_{n_{k}})=\lambda$ implies $f(y)=f(x_{n_{k}})=\lambda$ for sufficiently large $k\in\mathbb{N}.$

Also

[TABLE]

implies $x\leq_{d}y$, so that, by the $d$-monotony of $f$,

[TABLE]

Let now $(x_{n})$ be an arbitrary sequence in $X$ converging to $x$ and $\lambda=\liminf_{n\to\infty}f(x_{n}).$ Then there exists a subsequence $(x_{n_{k}})_{k\in\mathbb{N}}$ of $(x_{n})$ such that $\lim_{k\to\infty}f(x_{n_{k}})=\lambda.$ By Claim I,

[TABLE]

showing that $f$ is lsc.

3. Suppose that $f:X\to\overline{\mathbb{R}}$ is nearly lsc. If the topology $\tau(d)$ is $T_{1}$, then, by Proposition 2.8.(iii), the specialization order is the equality on $X$, hence the function $f$ is $d$-monotone. By 2 this implies that $f$ is lsc. ∎

2.5. Picard sequences in quasi-pseudometric spaces

Let $(X,d)$ be a quasi-pseudometric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a function. For $x\in\operatorname{dom}\varphi:=\{x\in X:\varphi(x)<\infty\}$ define the set $S(x)$ by

[TABLE]

The function $\varphi$ is called proper if $\operatorname{dom}\varphi\neq\emptyset.$

Proposition 2.12**.**

The sets $S(x)$ have the following properties:

[TABLE]

Proof.

The relations (i) are immediate consequences of the definition of $S(x)$.

(ii)  If $y\in S(X)$, then $0\leq d(y,x)\leq\varphi(x)-\varphi(y)$ implies $\varphi(y)\leq\varphi(x).$ Let now $y\in S(x)$ and $z\in S(y)$. Then

[TABLE]

showing that $z\in S(x).$

(iii) Follows from the inequalities:

[TABLE]

(iv) If there exists $y\in S(x)\smallsetminus\overline{\{x\}}$, then, by (iii),

[TABLE]

(v) Follows from the lsc of the function $\varphi$ and the $d$-lsc of $d(\cdot,x)$. ∎

Remark 2.13**.**

If $\varphi(x)=\infty$ then $S(x)=X$, so it is natural to consider $S(x)$ only for points $x$ in the domain of $\varphi$. 2. 2.

Considering the order $\preceq$ on $X$ given by

[TABLE]

we have

[TABLE] 3. 3.

It is worth to mention that sets of this kind were used by Penot [23] as early as 1977 in a proof of Caristi fixed point theorem in complete metric spaces.

A Picard sequence corresponding to a set-valued mapping $F:X\rightrightarrows X$ is a sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ such that

[TABLE]

for a given initial point $x_{0}\in X.$ This notion was introduced in [13] (see also [7]).

Proposition 2.14** (Picard sequences).**

Let $(X,d)$ be a quasi-pseudometric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below function. For $x\in\operatorname{dom}\varphi$ let

[TABLE]

Let $x_{0}\in\operatorname{dom}\varphi.$ We distinguish two situations.

There exists $m\in\mathbb{N}_{0}$ such that

[TABLE]

for all $\,0\leq k\leq m-1.$

Putting $z=x_{m}$, the following conditions are satisfied:

[TABLE] 2. 2.

There exists a sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ such that

[TABLE]

for all $n\in\mathbb{N}_{0}\,.$

Then the sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ satisfies the conditions

[TABLE]

If the space $X$ is sequentially right $K$-complete and the function $\varphi$ is nearly lsc, then the sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ is convergent to a point $z\in X$ such that

[TABLE]

where $\alpha$ is given by (2.13).(ii).

Proof.

Suppose that we have found $x_{0},x_{1},\dots,x_{m}$ satisfying the conditions (i) and (ii) from (2.10). If $\varphi(x_{m})=J(x_{m})$ then $x_{0},x_{1},\dots,x_{m}$ satisfy (2.10).

If $\varphi(x_{m})>J(x_{m})$, then there exists $x_{m+1}\in S(x_{m})$ such that $\varphi(x_{m+1})<(\varphi(x_{n})+J(x_{n}))/2\,.$ Supposing that this procedure continues indefinitely, we find a sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ satisfying (2.12).

1. Suppose that $x_{0},x_{1},\dots,x_{m}$ satisfy (2.10) and let $z=x_{m}\,.$ Then, by (2.9).(ii), $x_{k+1}\in S(x_{k})$ implies $S(x_{k+1})\subseteq S(x_{k})$ for $k=0,1,\dots,m-1.$

If $y\in S(z),$ then, by (2.9).(ii),

[TABLE]

It follows $\varphi(y)=\varphi(z)=J(z)$ for all $y\in S(z).$

If $y\in S(z)$ and $x\in S(y)\subseteq S(z)$, then

[TABLE]

so that $x\in\overline{\{y\}}$ and $S(y)\subseteq\overline{\{y\}}.$

We have shown that $z$ satisfies (2.11).

2. Suppose now that the sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ satisfies (2.12).

By (2.9).(ii), the relation $x_{n+1}\in S(x_{n})$ implies

[TABLE]

Also, by (2.12),

[TABLE]

so (2.13).(i) holds.

Since $\varphi$ is bounded below and, by (i), $(\varphi(x_{n}))_{n\in\mathbb{N}_{0}}$ is strictly decreasing, there exists the limit

[TABLE]

By (2.12),

[TABLE]

for all $n\in\mathbb{N}_{0}.$ Letting $n\to\infty$, one obtains

[TABLE]

By (i),

[TABLE]

proving (iii).

By (iii),

[TABLE]

Since the sequence $(\varphi(x_{n}))_{n\in\mathbb{N}_{0}}$ is Cauchy, this implies that $(x_{n})_{n\in\mathbb{N}_{0}}$ is right $K$-Cauchy.

Suppose now that $X$ is sequentially right $K$-complete and $\varphi$ is nearly lsc.

Taking into account the fact that the function $d(\cdot,x_{n})$ is lsc (Proposition 2.1.3) and the sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ has pairwise distinct terms, the inequalities (2.16) yield

[TABLE]

which shows that $z\in S(x_{n})$ and so $S(z)\subseteq S(x_{n})$, for all $n\in\mathbb{N}_{0}$.

If $y\in S(z)\subseteq S(x_{n}),$ then

[TABLE]

that is

[TABLE]

for all $n\in\mathbb{N}_{0}\,.$

Letting $n\to\infty$ and taking into account (2.13).(ii), one obtains

[TABLE]

for all $y\in S(z).$

The proof of the inclusion (2.14).(iii) is similar to that of (2.11).(iii). ∎

Remark 2.15**.**

If the function $\varphi$ satisfies

[TABLE]

then, for every $x_{0}\in\operatorname{dom}\varphi$, there exists a Picard sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ satisfying (2.12) (and so (2.13) as well).

3. Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

Along this section we shall use the notation: for a quasi-pseudometric space $X$, a function $\varphi:X\to\mathbb{R}\cup\{\infty\}$ and $x\in X$ put

[TABLE]

Ekeland, Takahashi and Caristi principles (see Theorem 1.1) can be expressed in terms of the sets $S(x)$ in the following form.

Theorem 3.1**.**

Let $(X,d)$ be a complete metric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below lsc function. Then the following hold.

1. (wEk)

There exists $z\in X$ such that $S(z)=\{z\}.$ 2. (Tak)

If $S(x)\smallsetminus\{x\}\neq\emptyset$ whenever $\varphi(x)>\inf\varphi(X),$ then $\varphi$ attains its minimum on $X$, i.e., there exists $z\in X$ such that $\varphi(z)=\inf\varphi(X).$ 3. (Car)

If the mapping $T:X\to X$ satisfies $Tx\in S(x)$ for all $x\in X,$ then $T$ has a fixed point in $X$, i.e., there exists $z\in X$ such that $Tz=z.$

In the following we shall prove some quasi-pseudometric versions of these results.

3.1. Ekeland variational principle

We start by a version of weak Ekeland principle.

Theorem 3.2**.**

Let $(X,d)$ be a sequentially right $K$-complete quasi-pseudometric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below nearly lsc function. Then there exists $z\in X$ such that

[TABLE]

In this case it follows that, for every $y\in S(z)$,

[TABLE]

Proof.

By Proposition 2.14, (2.11).(iii) and (2.14).(ii), there exists $z\in X$ satisfying (3.2).

Let us prove now that (3.2) implies (3.3).

If $y\in S(z)$ and $x\in S(y)\subseteq S(z)$, then, by (3.2),

[TABLE]

so that, $x\in\overline{\{y\}},$ that is, $S(y)\subseteq\overline{\{y\}}.$

If $x\in X\smallsetminus S(y)$, then, by the definition of the set $S(y)$,

[TABLE]

∎

Remark 3.3**.**

In [19] the following form of the weak form of EkVP is proved: Under the hypotheses of Theorem 3.2 there exists $z\in X$ such that

[TABLE]

Since $S(z)\subseteq\overline{\{z\}}$, the relations (3.2) and (3.3) can be rewritten in the form

[TABLE]

i.e., (3.4).(i) splits into (i*′) and (i′′*).

Supposing that $\varphi$ is lsc one can obtain the full version of Ekeland variational principle.

Theorem 3.4**.**

Let $(X,d)$ be a sequentially right $K$-complete quasi-pseudometric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below lsc function. Let $\varepsilon,\lambda>0$ and let $x_{0}\in X$ be such that

[TABLE]

Then there exists $z\in X$ such that

[TABLE]

Proof.

For convenience, put $\gamma=\varepsilon/\lambda$ and $d_{\gamma}=\gamma d.$ Then $d_{\gamma}$ is a quasi-pseudometric on $X$ Lipschitz equivalent to $d$, so that $(X,d_{\gamma})$ is sequentially right $K$-complete too.

Let

[TABLE]

Claim I. The set $X_{0}$ is closed and $x_{0}\in X_{0}\,.$

Indeed, let $(x_{n})$ be a sequence in $X_{0}$, $d_{\gamma}$-convergent to some $x\in X$, i.e., $\lim_{n\to\infty}d_{\gamma}(x,x_{n})=0.$ Then

[TABLE]

for all $n\in\mathbb{N}.$ Taking into account the lsc of the function $\varphi$, one obtains

[TABLE]

which shows that $x\in X_{0}.$

It is obvious that $x_{0}\in X_{0}\,.$

For $y\in X_{0}$ put

[TABLE]

Claim II. For every $y\in X_{0}$ * and* $x\in X\smallsetminus X_{0},$

[TABLE]

Indeed,

[TABLE]

so that, taking into account that $y\in X_{0}$, we obtain

[TABLE]

The inequality (3.7).(i) implies that

[TABLE]

Applying Proposition 2.14 to $(X_{0},d_{\gamma})$ we find an element $z\in X_{0}$ such that

[TABLE]

This shows that $z$ satisfies (3.6).(iii).

Now, by (2.11).(ii)and (2.14).(i), $z\in S_{X_{0}}(x_{0})$ which is equivalent to (3.6).(i).

Also, (3.6).(i) and (3.5) imply

[TABLE]

so that

[TABLE]

i.e., (3.6).(ii) holds too.

The inequality (2.14).(iv) follows from the definition of the set $S(z).$ ∎

3.2. Takahashi principle

Theorem 3.5**.**

Let $(X,d)$ be a sequentially right $K$-complete quasi-metric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper, bounded below and nearly lsc function. Suppose that, for every $x\in X$,

[TABLE]

Then there exists $z\in X$ such that $\varphi(z)=\inf\varphi(X)$, i.e., the function $\varphi$ attains its minimum on $X$.

Proof.

Suppose, by contradiction, that

[TABLE]

for all $x\in X$. Then, by (3.9),

[TABLE]

or, equivalently,

[TABLE]

for every $x\in X$.

Let $x_{0}\in\operatorname{dom}\varphi.$ By (3.12), Remark 2.15 and Proposition 2.14.2, there exists a sequence $(x_{n})_{n\in\mathbb{N}_{0}}$ satisfying (2.13) and (2.14). If $z=\lim_{n\to\infty}x_{n}$, then, by (3.11), there exists $y\in S(z)$ such that

[TABLE]

By (2.12).(ii),

[TABLE]

for all $n\in\mathbb{N}$ (because $y\in S(x_{n-1})$, by (2.14).(i)).

Taking into account the nearly lsc of the function $\varphi$ it follows

[TABLE]

in contradiction to (3.13).

Consequently, the hypothesis (3.10) leads to a contradiction, so it must exist a point $z\in X$ such that $\varphi(z)=\inf\varphi(X).$ ∎

Corollary 3.6**.**

Suppose that $(X,d)$ and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ satisfy the hypotheses of Theorem 3.5. If, for every $x\in X$,

[TABLE]

then the function $\varphi$ attains its minimum on $X$.

Proof.

Condition (3.14) means that, for every $x\in X$,

[TABLE]

By (2.9).(iii),

[TABLE]

so we can apply Theorem 3.5 to conclude. ∎

3.3. Caristi fixed point theorem

We present both single-valued and set-valued versions of Caristi fixed point theorem

Theorem 3.7** (Caristi’s theorem).**

Let $(X,d)$ be a sequentially right $K$-complete quasi-pseudometric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below nearly lsc function.

If the mapping $T:X\to X$ satisfies

[TABLE]

for all $x\in X,$ then there exists $z\in X$ such that $\varphi(Tz)=\varphi(z).$ 2. 2.

If $T:X\rightrightarrows X$ is a set-valued mapping such that

[TABLE]

for every $x\in X,$ then there exists $z\in X$ such that $\varphi(z)\in\varphi(Tz).$

Proof.

Observe that condition (3.15) is equivalent to

[TABLE]

for all $x\in X.$ This shows that (3.16) is an extension of (3.15), so it suffices to prove 2.

By Proposition 2.14, (2.11)(iii) and (2.14)(ii), there exists $z\in X$ such that $\varphi(y)=\varphi(z)$ for all $y\in S(z)$. By (3.16), there exists $y\in S(z)\cap Tz.$ But then

[TABLE]

∎

3.4. The equivalence of principles and completeness

We prove the equivalence between Ekeland, Takahashi and Caristi principles.

Theorem 3.8**.**

Let $(X,d)$ be a quasi-pseudometric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below function. Then the following statements are equivalent.

1. (wEk)

The following holds

[TABLE] 2. (Tak)

The following holds

[TABLE] 3. (Car)

If the mapping $T:X\to X$ satisfies

[TABLE]

then there exists $z\in X$ such that $\varphi(Tz)=\varphi(z).$

Proof.

(wEK)$\iff$ (Tak).

The proof is based on the following rules from mathematical logic:

[TABLE]

so that

[TABLE]

Observe that

[TABLE]

Indeed, for every $x\in X$ take $y\in S(x)$ such that $\varphi(y)<\varphi(x).$ Then

[TABLE]

For convenience, denote by Ta1 the expression

[TABLE]

Based on (3.21) and (3.22), one obtains:

[TABLE]

Since, by (2.9).(ii), $\varphi(y)\leq\varphi(x)$ for every $y\in S(x)$, it follows that

[TABLE]

But then

[TABLE]

(wEk) $\;\Rightarrow\;$  (Car).

Suppose that $T:X\to X$ satisfies (3.20). By (wEk) there exists $z\in X$ such that $\varphi(x)=\varphi(z)$ for all $x\in S(z).$ Since, by hypothesis, $Tz\in S(z)$, it follows $\varphi(Tz)=\varphi(z).$

$\neg$(wEk)  $\;\Rightarrow\;$  $\neg$(Car).

By (3.23), $\neg$(wEk) is equivalent to

[TABLE]

Define $T:X\to X$ by $Tx=y_{x}$, where $y_{x}$ is given by (3.24), $x\in X.$ Then $Tx\in S(x)$ for every $x\in X$ but $\varphi(Tx)<\varphi(x)$ for all $x\in X$, i.e., the assertion (Car) fails.

∎

Finally we show that the validity of each of these principles is further equivalent to the sequential right $K$-completeness of the quasi-pseudometric space $X$.

Theorem 3.9**.**

For a quasi-pseudometric space $(X,d)$ the following are equivalent.

The space $(X,d)$ is sequentially right $K$-complete. 2. 2.

(Ekeland variational principle - weak form)* For every proper bounded below nearly lsc function $\varphi:X\to\mathbb{R}\cup\{\infty\}$ there exists $z\in X$ such that*

[TABLE] 3. 3.

(Takahashi minimization principle)* Every proper bounded below nearly lsc function $\varphi:X\to\mathbb{R}\cup\{\infty\}$ such that, for every $x\in X$,*

[TABLE]

attains its minimum on $X$. 4. 4.

(Caristi fixed point theorem)* For every proper bounded below nearly lsc function $\varphi:X\to\mathbb{R}\cup\{\infty\}$ and every mapping $T:X\to X$ such that*

[TABLE]

there exists $z\in X$ such that $\varphi(Tz)=\varphi(z).$

Proof.

The equivalences $2\iff 3\iff 4$ are contained in Theorem 3.8 (even in a stronger form - with the same function $\varphi$.)

The implication $1\;\;\Rightarrow\;2$ is contained in Theorem 3.2.

2 $\;\Rightarrow\;\;$1.

The proof is inspired from [19]. We proceed by contradiction. Suppose that there exists a right $K$-Cauchy sequence $(x_{n})_{n\in\mathbb{N}}$ in $X$ which does not converge. Then $(x_{n})$ has no cluster points (by Proposition 2.6), so it does not contain constant subsequences. Passing to a subsequence if necessary, we can suppose that further

[TABLE]

The set

[TABLE]

is closed. Indeed, if there exists $x\in\overline{B}\smallsetminus B,$ then $x$ will be a cluster point for $(x_{n})$ in contradiction to the hypothesis.

Define $\varphi:X\to\mathbb{R}$ by

[TABLE]

The function $\varphi$ is nearly lsc. Indeed, let $(y_{n})$ be a sequence with distinct terms converging to a point $x\in X.$

If $x\in X\setminus B,$ then, since $B$ is closed, the sequence $(y_{n})$ must be eventually in $X\setminus B,$ and so $\lim_{n}\varphi(y_{n})=\infty=\varphi(x).$

Suppose now that $x=x_{k}$ for some $k\in\mathbb{N}.$

If the set $\{n\in\mathbb{N}:y_{n}\in B\}$ is infinite, then $x_{k}$ will be a cluster point of the sequence $(x_{n})$. Consequently, only finitely many terms $y_{n}$ belong to $B$. This implies that there exists $n_{0}\in\mathbb{N}$ such that $y_{n}\in X\smallsetminus B$ for all $n\geq n_{0},$ so that $\varphi(x_{k})<\infty=\lim_{n}\varphi(y_{n})$.

We have $\operatorname{dom}\varphi=B.$ If $z=x_{k}\in B$, for some $k\in\mathbb{N}$, then, by (3.26) and the definition of the function $\varphi$,

[TABLE]

which shows that $x_{k+1}\in S(x_{k})$. Since $\varphi(x_{k})=2^{-k+1}>2^{-k}=\varphi(x_{k+1})$, it follows that (3.25) fails for every $z\in\operatorname{dom}\varphi.$ ∎

3.5. The case of $T_{1}$ quasi-metric spaces

Let us notice that a topological space $(X,\tau)$ is $T_{1}$ if and only if $\overline{\{x\}}=\{x\}$ for all $x\in X.$ A quasi-pseudometric space $(X,d)$ is $T_{1}$ (i.e., the topology $\tau_{d}$ is $T_{1}$) if and only if $d(x,y)>0$ for all $x\neq y$ in $X$. It follows that a $T_{1}$ quasi-pseudometric space is a quasi-metric space. By Proposition 2.11.3 a nearly lsc function on a $T_{1}$ quasi-metric space is lsc.

Taking into account these remarks, the results proved for arbitrary quasi-pseudometric spaces take the following form in the $T_{1}$ case.

Theorem 3.10**.**

Let $(X,d)$ be a sequentially right $K$-complete $T_{1}$ quasi-metric space and $\varphi:X\to\mathbb{R}\cup\{\infty\}$ a proper bounded below lsc function. The following are true.

(Ekeland variational principle -weak form, [9])* There exists $z\in X$ such that*

[TABLE] 2. 2.

(Takahashi principle, [1])* If for every $x\in X$,*

[TABLE]

then there exists $z\in X$ such that $\varphi(z)=\inf\varphi(X).$ 3. 3.

(Caristi fixed point theorem)

1. (a)

If the mapping $T:X\to X$ satisfies

[TABLE]

for all $x\in X,$ then there exists $z\in X$ such that $Tz=z.$ 2. (b)

If $T:X\rightrightarrows X$ is a set-valued mapping such that

[TABLE]

for every $x\in X,$ then there exists $z\in X$ such that $z\in Tz.$

Proof.

1. By Theorem 3.2 there exists $z\in X$ satisfying (3.3). Since $X$ is $T_{1}$, $\overline{\{z\}}=\{z\}$, so that $S(z)=\{z\}.$ Taking into account this equality, (3.27) is equivalent to (3.3).(iii) for $y=z.$

2. Condition (3.28) says that $S(x)\smallsetminus\{x\}\neq\emptyset$ whenever $\varphi(x)>\inf\varphi(X).$ Since $X$ is $T_{1}$, $d(y,x)>0$, so that

[TABLE]

for every $y\in S(x)\smallsetminus\{x\}.$ This shows that condition (3.9) is verified by $\varphi.$

3. As we have noticed, condition (3.29) means that $Tx\in S(x)$ for every $x\in X,$ so it suffices to give the proof only for set-valued $T$. The proof is the same as that of Theorem 3.7, taking into account that $S(z)=\{z\}$.

Indeed, $S(z)=\{z\}$ and $S(z)\cap Tz\neq\emptyset$ imply $z\in Tz.$ ∎