Pseudo-metric space and fixed point theorem

Abstract

The aim of this paper is to give a generalized version of Caristi fixed point theorems in pseudo-metric spaces. Our results generalize and improve many of well-known theorems. As an application of our results, we give a new existence theorem to the generalized nonlinear complementarity problem and a solution of differential inclusion in the distributions setting.

1 Introduction

It is well known that the Ekeland variational principle [1] and Caristi-Kirk fixed point theorem are both equivalent. Many authors [2–7] have established a generalized version of these two results in different settings, that is, in vector-valued generalized metric space with respect to a convex cone $\mathbb{K}$ in a Banach space. Recall that a subset $\mathbb{K} \subset\mathbb{Y}$ is called a convex cone on a topological vector space $\mathbb{Y}$ if:

• 1. $\mathbb{K}+\mathbb{K}\subset\mathbb{K}$;

• 2. for every $\lambda>0$, $\lambda\mathbb{K}\subset\mathbb{K}$;

• 3. $\mathbb{K}\cap (-\mathbb{K} )= \{ \theta \}$, where θ denotes the zero of $\mathbb{Y}$.

A convex cone $\mathbb{K}\subset\mathbb{Y}$ generates a partial ordering on $\mathbb{Y}$ (i.e. a reflexive, antisymmetric, and transitive relation) by

$$ x\preceq y\quad \Longleftrightarrow \quad y-x\in\mathbb{K} . $$

Thereby, since its appearance, the Brezis-Browder ordering principle [8] seems to be a strong tool to prove fixed point or minimal point theorems in an ordered set. Zermelo’s theorem [9] shows that there is an equivalency between the existence of a fixed point of such a map and the monotonicity of the map. By the way, Hamel [10] studied existence theorems, namely minimal point, Caristi fixed point, and Ekeland variational principle in the topological product space $\mathbb{X}\times\mathbb{Y}$ where $\mathbb{X}$ is a separated uniform space, and $\mathbb{Y}$ is a topological vector space.

Fang [11] introduced the concept of ‘F-type topological spaces’ generating the topology by families of quasi-metrics and gave a generalization of Ekeland’s variational principle.

Furthermore, Isac [12] proved an interesting Caristi-type theorem in the framework of locally convex space, which led him to derive an existence result of a nonlinear equation.

Hence, the aim of this paper is to generalize some of the well-known fixed point theorems [11, 13–15] for a pseudo-metric space $\mathbb{X}$. This paper is divided into three sections after showing some basic results in preliminaries. Using in Section 3 the Brezis-Browder principle, we give generalized Caristi’s fixed point theorems for set-valued maps and derive some corollaries. Section 4 is devoted to an Ekeland-type variational principle in more applied general setting, namely pseudo-metric spaces, and also discuss the relationships of our main results. Finally, following investigations by Isac, Section 5 is devoted to applications.

2 Preliminaries

Over this section, $\mathbb{Y}$ is a locally convex space, and $\mathbb {K}$ is a convex cone in $\mathbb{Y}$. A set Λ is said to be a directed set if ‘≺’ is a preorder and every pair of elements of Λ has an upper bound.

Definition 2.1

Let $\mathbb{X}$ be a nonempty set, and $(\Lambda,\prec )$ a directed set. A family of cone pseudo-metrics on $\mathbb{X}$ is a system $\{ d_{\alpha} \} _{\alpha\in\Lambda}$ of mappings $d_{\alpha}:\mathbb{X}\times\mathbb{X}\rightarrow\mathbb{K}$ satisfying the following conditions for each $\alpha\in\Lambda$ and $x,y,z\in\mathbb{X}$:

(A1):

$\theta\preceq d_{\alpha} (x,y )$, and $d_{\alpha} (x,x )=\theta$;

(A2):

$d_{\alpha} (x,y )=d_{\alpha} (y,x )$;

(A3):

$d_{\alpha} (x,z )\preceq d_{\alpha} (x,y )+d_{\alpha} (y,z )$;

(A4):

If $\alpha\prec\beta$ then $d_{\alpha} (x,y )\preceq d_{\beta} (x,y )$.

Then the pair $(\mathbb{X}, \{ d_{\alpha} \} _{\alpha\in \Lambda} )$ is called a cone pseudo-metric space. Additionally, if

(A5):

for all $\alpha\in\Lambda$ and $x,y\in\mathbb{X}$, $d_{\alpha } (x,y )=\theta$ implies $x=y$,

then the family of cone pseudo-metrics is said to be separating.

The concept of a cone pseudo-metric space was already defined by Włodarczyk et al. [16], who called it a Hausdorff cone pseudo-metric space. In this paper, we use a locally convex space as a target set for a cone pseudo-metric, which is more general that a normed space. If $(\mathbb{Y},\tau )$ is a locally convex space, then it is known that the topology τ can be generated by a family of seminorms $\{ p_{i} \} _{i\in I}$ [17]. A subset B of $\{ p_{i} \} _{i\in I}$ is called a basis for $\{ p_{i} \} _{i\in I}$ if for every $i\in I$, there exist $q\in B$ and $\lambda>0$ such that $p_{i}\leqslant \lambda q$.

We say that a family of seminorms $\{ p_{i} \} _{i\in I}$ is separating if $\ker \{ p_{i} \} _{i\in I}=\{\theta\}$ or has a Hausdorff basis B if $\ker B=\{\theta\}$, where

$$ \ker B= \bigl\{ x\in\mathbb{Y} : p (x )=0, \forall p\in B \bigr\} . $$

The most useful class of cones in topological vector space is the class of normal cones. For more details, we refer the reader to [18].

Definition 2.2

[13]

If $(\mathbb{Y}, \{ p_{i} \} _{i\in I} )$ is a locally convex space, then a convex cone $\mathbb{K}\subset\mathbb{Y}$ is said to be normal if there exists a basis B of $\{ p_{i} \} _{i\in I}$ such that, for each $p\in B$ and all $x,y\in\mathbb{K}$,

$$ \theta\preceq x\preceq y\quad \Longrightarrow\quad p (x )\leqslant p (y ) . $$

Throughout this paper, we assume that the topology defined on $\mathbb {Y}$ is generated by the basis B [13], and we simply write $B= \{ p_{i} \} _{i\in I}$.

Proposition 2.3

Let $(\mathbb{X}, \{ d_{\alpha} \} _{\alpha\in \Lambda} )$ be a cone pseudo-metric space over a normal cone $\mathbb{K}$.

Then the mappings $\delta_{\alpha i} : \mathbb{X}\times\mathbb {X}\rightarrow [0,\infty [$ defined for each $(\alpha,i ) \in\Lambda\times I$ by $\delta_{\alpha i}=p_{i}\circ d_{\alpha}$ is a family of pseudo-metrics on $\mathbb{X}$.

Proof

By (A1) and (A2) we have immediately $\delta_{\alpha i} (x,x )=0$ and $\delta_{\alpha i} (x,y )=\delta_{\alpha i} (y,x )$ for every $x,y\in\mathbb{X}$.

Since for each $\alpha\in\Lambda$ and all $x,y,z\in\mathbb{X}$, we have $d_{\alpha} (x,y )\in\mathbb{K}$ and

$$ \theta\preceq d_{\alpha} (x,z )\preceq d_{\alpha} (x,y )+d_{\alpha} (y,x ) $$

and since $\mathbb{K}$ is a normal cone, we get, for each $i \in I$,

$$ p_{i} \bigl(d_{\alpha} (x,z ) \bigr)\leqslant p_{i} \bigl(d_{\alpha} (x,y )+d_{\alpha} (y,x ) \bigr)\leqslant p_{i} \bigl(d_{\alpha} (x,y ) \bigr)+p_{i} \bigl(d_{\alpha} (y,x ) \bigr). $$

Then $\delta_{\alpha i}$ satisfies the triangle inequality. If we assume that $\{ d_{\alpha} \} _{\alpha\in\Lambda}$ is a separating family, so is $\{ \delta_{\alpha i} \} _{ (\alpha,i ) \in\Lambda\times I}$. □

If the convex cone $\mathbb{K}$ is solid ($\operatorname{int} \mathbb {K}\neq\emptyset$) and not normal and if $\mathbb{Y}$ is a locally convex space, then the Gerstewitz functional [19] $\xi_{e} :\mathbb{Y}\rightarrow\mathbb{R}$, where $e\in\operatorname{int} {\mathbb{K}}$, is defined as

$$ \xi_{e} (x )=\inf \{ \lambda\in\mathbb{R}: x\in\lambda e-\mathbb{K} \} $$

for each $x\in\mathbb{Y}$.

We have the following result.

Lemma 2.4

For all $\lambda\in\mathbb{R}$ and $x\in\mathbb{Y}$, we have the following statements:

• (i) $\xi_{e} (x )\leqslant\lambda\Longleftrightarrow x\in \lambda e-\mathbb{K}$;

• (ii) $\xi_{e} (x )>\lambda\Longleftrightarrow x\notin\lambda e-\mathbb{K}$;

• (iii) $\xi_{e} (x )\geqslant\lambda\Longleftrightarrow x\notin \lambda e-\operatorname{int} \mathbb{K}$;

• (iv) $\xi_{e} (x )<\lambda\Longleftrightarrow x\in\lambda e-\operatorname{int}\mathbb{K}$;

• (v) $\xi_{e} (\cdot )$ is positively homogeneous and continuous on $\mathbb{Y}$;

• (vi) if $x_{1}\in x_{2}+\mathbb{K}$, then $\xi_{e} (x_{2} )\leqslant\xi_{e} (x_{1} )$;

• (vii) $\xi_{e} (x_{1}+x_{2} )\leqslant\xi_{e} (x_{1} )+\xi_{e} (x_{2} )$ for all $x_{1},x_{2}\in\mathbb{Y}$.

Proof

See, for instance, [7, 20–23]. □

The following result is Theorem 2.1 of Du [24].

Proposition 2.5

Let $(\mathbb{X}, \{ d_{\alpha} \} _{\alpha} )$ be a cone pseudo-metric space over a solid cone $\mathbb{K}$. Then the family of mappings $\delta_{\alpha} : \mathbb{X}\times\mathbb {X}\rightarrow [0,\infty [$ defined by $\delta_{\alpha}=\xi_{e}\circ d_{\alpha}$ is a family of pseudo-metrics on $\mathbb{X}$.

Proof

Since $\xi_{e} (\cdot )$ is a seminorm on $\mathbb{Y}$ by Lemma 2.4, Proposition 2.3 gives the result. □

If the cone $\mathbb{K}$ is normal and solid, then $\xi_{e} (\cdot )$ is a norm over $\mathbb{Y}$, and we have the following proposition.

Proposition 2.6

If $(\mathbb{Y},\tau )$ is a Hausdorff topological space ordered by a normal solid cone $\mathbb{K}$, then $(\mathbb{Y},\tau )$ is a normable space.

Proof

See Proposition 1.10 in [18], Chapter 2. □

Next, we discuss some convergence properties of cone pseudo-metric spaces. We note that $x\ll y$ if and only if $y-x \in\operatorname{int} \mathbb{K}$, where the ‘int’ is the interior.

Definition 2.7

Let $(\mathbb{X}, \{ d_{\alpha} \} _{\alpha} )$ be a cone pseudo-metric space over a solid convex cone $\mathbb{K}\subset \mathbb{Y}$, where $\mathbb{Y}$ is a locally convex space, $x \in \mathbb{X}$, and $\{ x_{n} \} _{n}$ a sequence in $\mathbb{X}$.

• 1. $\{ x_{n} \} _{n}$ is Cauchy sequence whenever for every $\alpha\in\Lambda$ and $c\in\mathbb{Y}$ with $\theta\ll c$, there is a natural number $N_{0}$ such that

  $$ d_{\alpha} (x_{n},x_{m} )\ll c, \quad \forall n,m \geq N_{0} . $$

• 2. $\{ x_{n} \} _{n}$ converges to x whenever for every $\alpha\in\Lambda$ and $c\in\mathbb{Y}$ with $\theta\ll c$, there is a natural number $N_{0}$ such that

  $$ d_{\alpha} (x_{n},x )\ll c,\quad \forall n\geq N_{0} . $$

• 3. $(\mathbb{X}, \{ d_{\alpha} \} _{\alpha} )$ is complete if each Cauchy sequence converges in $\mathbb{X}$.

Proposition 2.8

Let $(\mathbb{X}, \{ d_{\alpha} \} _{\alpha} )$ be a cone pseudo-metric space over a solid convex cone $\mathbb{K}\subset \mathbb{Y}$, where $\mathbb{Y}$ is a locally convex space.

Then, for each $\alpha\in\Lambda$, we get

$$ d_{\alpha} (x_{n},x ) \longrightarrow \theta\quad\iff\quad \delta_{\alpha} (x_{n},x )=\xi_{e} \bigl( d_{\alpha} (x_{n},x ) \bigr) \longrightarrow 0 . $$

Proof

It is similar to the proof of Theorem 3.2 in [25]. □

Using this pseudo-metric $\delta_{\alpha}$, we keep saying that $(\mathbb{X}, \{ \delta_{\alpha} \} _{\alpha} )$ is a pseudo-metric space over a solid convex cone $\mathbb{K}$.

3 Fixed point theorems

Recall that the most famous ordering principle.

Theorem 3.1

Brezis-Browder

Let $(W,\precsim )$ be a quasi-ordered set (i.e. ≾ is a reflexive and transitive relation), and let $\varPsi:W\longrightarrow\mathbb{R}$ be a function satisfying the following conditions:

(B1):

Ψ is bounded below;

(B2):

$w_{1}\precsim w_{2}\Longrightarrow\varPsi (w_{1} )\leqslant\varPsi (w_{2} )$;

(B3):

For every decreasing sequence $\{ w_{n} \} _{n\in\mathbb {N}}\subset W$ with respect to ‘≾’, there exists $w\in W$ such that $w\leqslant w_{n}$ for all $n\in\mathbb{N}$.

Then, for every $w_{0}\in W$, there exists $\bar{w}\in W$ such that

(i):

$\bar{w}\precsim w_{0}$;

(ii):

$\hat{w}\precsim\bar{w}\Longrightarrow\varPsi (\hat{w} )=\varPsi (\bar{w} )$.

In particular, if we strengthen (B2) to

(B2′):

$(w_{1}\precsim w_{2}, w_{1}\neq w_{2} )\Longrightarrow\varPsi (w_{1} )<\varPsi (w_{2} )$,

then

(ii′):

$\hat{w}\precsim\bar{w}\Longrightarrow\hat{w}=\bar{w}$, that is, w̄ is minimal in W with respect to ‘≾’.

Proof

See Corollary 1 in [8]. □

Now we are able to give the main result of this section.

Theorem 3.2

Let $(\mathbb{\mathbb{Y}}, \{ p_{i} \} _{i\in I} )$ a complete separated locally convex space, $(\mathbb{X}, \{ \delta_{\alpha} \} _{\alpha\in\Lambda } )$ be a complete Hausdorff pseudo-metric space over a solid convex cone $\mathbb{K}$, $T:\mathbb{X}\longrightarrow2^{\mathbb{X}}$ and $S:\mathbb{X}\longrightarrow2^{\mathbb{Y}}$ two set-valued maps with nonempty values.

Suppose that, for every $(\alpha,i )\in\Lambda\times I$ and two constants $c_{\alpha},c_{i}>0$, there exist lower semicontinuous functions $\varphi_{\alpha i}:\mathbb{\mathbb{Y}}\longrightarrow [0,\infty )$, and for each $(x,y )\in G_{S}$, there exist $u\in Tx$ and $v\in Su$ such that

$$ \max \bigl\{ c_{\alpha}\delta_{\alpha} (x,u ),c_{i}p_{i} (y-v ) \bigr\} \leqslant\varphi_{\alpha i} (y )-\varphi _{\alpha i} (v ). $$

(1)

Then T has a fixed point in $\mathbb{X}$.

Proof

Put

$$ W_{0}= \bigl\{ (x,y )\in G_{S} ; \forall (\alpha,i )\in \Lambda\times I, \max \bigl\{ c_{\alpha}\delta_{\alpha} (x_{0},x ),c_{i}p_{i} (y_{0}-y ) \bigr\} +\varphi_{\alpha i} (y )\leqslant\varphi_{\alpha i} (y_{0} ) \bigr\} $$

for some $(x_{0},y_{0} )\in G_{S}$. Then $W_{0}$ is a nonempty closed subset of $G_{S}$. Indeed, let $(x_{n},y_{n} )_{n}$ be a sequence in $W_{0}$ that converges to $(x,y )$, that is, $\lim_{n\rightarrow\infty} p_{i} (y_{n}-y )=0$. Since for each $(\alpha,i )\in\Lambda\times I$, the function $\varphi_{\alpha i}$ is lower semicontinuous, that is,

$$ \varphi_{\alpha i} (y )\leqslant \liminf_{n\rightarrow\infty} \varphi_{\alpha i} (y_{n} ) , $$

we have

$$\begin{aligned} c_{i}p_{i} (y_{0}-y ) \leqslant& c_{i}p_{i} (y_{0}-y_{n} )+c_{i}p_{i} (y_{n}-y ) \\ \leqslant& \varphi_{\alpha i} (y_{0} )-\varphi_{\alpha i} (y_{n} )+c_{i}p_{i} (y_{n}-y ) \\ \leqslant& \varphi_{\alpha i} (y_{0} )- \liminf _{k\rightarrow\infty}\varphi_{\alpha i} (y_{k} )+c_{i}p_{i} (y_{n}-y ) \\ \leqslant& \varphi_{\alpha i} (y_{0} )-\varphi_{\alpha i} (y )+c_{i}p_{i} (y_{n}-y ). \end{aligned}$$

So, taking the limit with respect to n, we get $c_{i}p_{i} (y_{0}-y ) \leqslant\varphi_{\alpha i} (y_{0} )-\varphi _{\alpha i} (y )$, and by similar arguments we get

$$ c_{\alpha}\delta_{\alpha} (x_{0},x )\leqslant \varphi_{\alpha i} (y_{0} )-\varphi_{\alpha i} (y ) . $$

Hence, $\max \{ c_{\alpha}\delta_{\alpha} (x_{0},x ),c_{i}p_{i} (y_{0}-y ) \} +\varphi_{\alpha i} (y )\leqslant\varphi_{\alpha i} (y_{0} )$, so that $(x,y )\in W_{0}$.

Now we define a binary relation in $W_{0}$ as follows: for every $(x_{1},y_{1} )$ and $(x_{2},y_{2} )$ in $W_{0}$,

$$ (x_{1},y_{1} )\precsim (x_{2},y_{2} ) \quad \Longleftrightarrow\quad \max \bigl\{ c_{\alpha}\delta_{\alpha} (x_{1},x_{2} ),c_{i}p_{i} (y_{1}-y_{2} ) \bigr\} \leqslant \varphi_{\alpha i} (y_{2} )-\varphi_{\alpha i} (y_{1} ) $$

for each $(\alpha,i )\in\Lambda\times I$. We can show that the relation ≾ is an ordering on $W_{0}$.

Next, we show that, for every decreasing sequence $(x_{n},y_{n} )_{n\in\mathbb{N}}\subset W_{0}$ with respect to ‘≾’, there exists $(x^{*},y^{*} )\in W_{0}$ such that $(x^{*},y^{*} )\precsim (x_{n},y_{n} )$ for all $n\in\mathbb{N}$. Let $(x_{n},y_{n} )_{n\in\mathbb{N}}$ be a ≾-decreasing sequence in $W_{0}$. Then, for any $m,n\in \mathbb{N}$ such that $m\geqslant n$, we have

$$\begin{aligned}& (x_{m},y_{m} )\precsim (x_{n},y_{n} )\quad \Longleftrightarrow\quad \max \bigl\{ c_{\alpha} \delta_{\alpha} (x_{m},x_{n} ),c_{i}p_{i} (y_{m}-y_{n} ) \bigr\} \leqslant \varphi_{\alpha i} (y_{n} )-\varphi_{\alpha i} (y_{m} ) \\& \quad \mbox{for each } (\alpha,i )\in\Lambda\times I , \end{aligned}$$

which gives that the positive sequence $\{ \varphi_{\alpha i} (y_{n} ) \} _{n}$ is decreasing (for α and i fixed). Hence, there exists $r_{\alpha i}$ such that $\lim\varphi_{\alpha i} (y_{n} )=r_{\alpha i}$. Let $\varepsilon>0$ and $(\alpha,i )\in\Lambda\times I$. There exists $N_{0}\in\mathbb{N}^{*}$ such that, for any $n\geqslant N_{0}$, we have

$$ r_{\alpha i}\leqslant\varphi_{\alpha i} (y_{n} )\leqslant r_{\alpha i}+\min (c_{\alpha},c_{i} )\cdot\varepsilon $$

and then, for every $m\geqslant n\geqslant N_{0}$,

$$\begin{aligned} c_{i}p_{i} (y_{m}-y_{n} ) \leqslant& \varphi_{\alpha i} (y_{n} )-\varphi_{\alpha i} (y_{m} ) \\ \leqslant& r_{\alpha i}+\min (c_{\alpha},c_{i} )\cdot \varepsilon-r_{\alpha i}. \end{aligned}$$

Thus,

$$ c_{i}p_{i} (y_{m}-y_{n} )\leqslant \min (c_{\alpha },c_{i} )\cdot\varepsilon\leqslant c_{i}\varepsilon . $$

Also, we get

$$\begin{aligned} c_{\alpha}\delta_{\alpha} (x_{m},x_{n} ) \leqslant& \varphi _{\alpha i} (y_{n} )-\varphi_{\alpha i} (y_{m} ) \\ \leqslant& r_{\alpha i}+\min (c_{\alpha},c_{i} )\cdot \varepsilon-r_{\alpha i} \end{aligned}$$

and thus

$$ c_{\alpha}\delta_{\alpha} (x_{m},x_{n} ) \leqslant c_{\alpha }\varepsilon . $$

Repeating the last computation for every $(\alpha,i )\in \Lambda\times I$ and using the fact that $\{ \delta_{\alpha} \} _{\alpha\in \Lambda}$ and $\{ p_{i} \} _{i\in I}$ are separated families, we obtain that $\{ x_{n} \} _{n}$ and $\{ y_{n} \} _{n}$ are Cauchy sequences in the complete spaces $\mathbb{X}$ and $\mathbb{Y}$, respectively. Therefore, there exist $x^{*}\in\mathbb{X}$ and $y^{*}\in\mathbb{Y}$ such that

$$ x_{n} \longrightarrow x^{*}\quad \mbox{and}\quad y_{n} \longrightarrow y^{*} . $$

Since $W_{0}$ is closed, we have that $(x^{*},y^{*} )\in W_{0}$ and $y^{*}\in Sx^{*}$ by the definition of $W_{0}$.

Also, for all $(n,m)\in\mathbb{N}^{2}$ such that $m\geqslant n$, we have $(x_{m},y_{m} )\precsim (x_{n},y_{n} )$, so that for all $(\alpha,i )\in\Lambda\times I$,

$$\begin{aligned} \max \bigl\{ c_{\alpha}\delta_{\alpha} (x_{m},x_{n} ),c_{i}p_{i} (y_{m}-y_{n} ) \bigr\} \leqslant& \varphi_{\alpha i} (y_{n} )-\varphi_{\alpha i} (y_{m} ) \\ \leqslant& \varphi_{\alpha i} (y_{n} )-\liminf _{k\rightarrow\infty}\varphi_{\alpha i} (y_{k} ) \\ \leqslant& \varphi_{\alpha i} (y_{n} )-\varphi_{\alpha i} \bigl(y^{*} \bigr). \end{aligned}$$

Taking the limit with respect to m and using the fact that $\delta_{\alpha}$ and $p_{i}$ are continuous, we get

$$ \max \bigl\{ c_{\alpha}\delta_{\alpha} \bigl(x^{*},x_{n} \bigr),c_{i}p_{i} \bigl(y^{*}-y_{n} \bigr) \bigr\} \leqslant\varphi_{\alpha i} (y_{n} )-\varphi_{\alpha i} \bigl(y^{*} \bigr)\quad \mbox{for all } (\alpha,i )\in\Lambda\times I. $$

Thus, for each $n\in\mathbb{N}$,

$$ \bigl(x^{*},y^{*} \bigr)\precsim (x_{n},y_{n} ) . $$

Let $(\alpha,i )\in\Lambda\times I$ be fixed and choose $\varPsi:W_{0}\longrightarrow\mathbb{R}$ as follows: $\varPsi (x,y )=\varphi_{\alpha i} (y )$ for each $(x,y )\in W_{0}$. Condition (B1) from Theorem 3.1 holds since $\varphi_{\alpha i} (y )\geq0$. We also have

$$ (x_{1},y_{1} )\precsim (x_{2},y_{2} ) \quad \Longrightarrow\quad \varphi_{\alpha i} (y_{1} )\leqslant \varphi_{\alpha i} (y_{2} )\quad \mbox{for each } (\alpha,i )\in \Lambda\times I. $$

So $\varPsi (x_{1},y_{1} )\leqslant\varPsi (x_{2},y_{2} )$, and thus (B2) also holds. Then all assumptions of the Brezis-Browder principle are satisfied. Hence, for each $(x_{0},y_{0} )\in W_{0}$, there exists $(\bar{x},\bar{y} )\in W_{0}$ such that:

• (i) $(\bar{x},\bar{y} )\precsim (x_{0},y_{0} )$;

• (ii) if $(\hat{x},\hat{y} )\precsim (\bar{x},\bar{y} )$, then $\varPsi (\hat{x},\hat{y} )=\varPsi (\bar{x},\bar {y} )$.

We claim that x̄ is a fixed point for T. For this $(\bar {x},\bar{y} )\in W_{0} \subset G_{S}$, there exists $(u,v )\in\mathbb{X}\times\mathbb{Y}$ such that $u\in T\bar{x}$ and $v\in S\bar{u}$ satisfy the following inequality for each $(\alpha,i )\in \Lambda\times I$:

$$ \max \bigl\{ c_{\alpha}\delta_{\alpha} (u,\bar{x} ),c_{i}p_{i} (v-\bar{y} ) \bigr\} \leqslant \varphi_{\alpha i} (\bar{y} )-\varphi_{\alpha i} (v ) . $$

Given $(u,v )\precsim (\bar{x},\bar{y} )$, we have $\varPsi (u,v )=\varPsi (\bar{x},\bar{y} )$; hence, $x=\bar{x}$, and thus $\bar{x}\in T\bar{x}$, which completes the proof. □

Theorem 3.3

Under the hypotheses of Theorem  3.2, suppose that the condition ‘for each $(x,y )\in G_{S}$, there exist $u\in Tx$ and $v\in Su$ ’ is replaced by ‘for each $(x,y )\in G_{S}$ and for every $u\in Tx$, there exists $v\in Su$ ’.

Then T has a critical point, that is, there exists $\bar{x}\in\mathbb{X}$ such that $\{ \bar{x} \} =T\bar{x}$.

Proof

By Theorem 3.2, T has a fixed point x̄ in $\mathbb{X}$. We claim that it is a critical point. For this, let us show that assumption (B2′) of Brezis-Browder holds, and so we have (ii′). Let $(\alpha ,i )\in\Lambda\times I$ be fixed and choose $\varPsi:W_{0}\longrightarrow\mathbb{R}$ as in the above proof: $\varPsi (x,y )=\varphi_{\alpha i} (y )$ for each $(x,y )\in W_{0}$. Then

$$ (x_{1},y_{1} )\precsim (x_{2},y_{2} ), \quad (x_{1},y_{1} )\neq (x_{2},y_{2} ) \quad \Longrightarrow\quad \varPsi (x_{1},y_{1} )< \varPsi (x_{2},y_{2} ) . $$

Indeed, suppose that $x_{1}\neq x_{2}$. Then, for each $\alpha\in \Lambda$, we get

$$ \delta_{\alpha} (x_{1},x_{2} )\neq0\quad \implies \quad \delta_{\alpha } (x_{1},x_{2} )>0 . $$

Then

$$ 0< c_{\alpha}\delta_{\alpha} (x_{1},x_{2} ) \leqslant\varphi _{\alpha i} (y_{2} )-\varphi_{\alpha i} (y_{1} ) , $$

and hence $\varphi_{\alpha i} (y_{1} )<\varphi_{\alpha i} (y_{2} )\Longleftrightarrow\varPsi (x_{1},y_{1} )<\varPsi (x_{2},y_{2} )$.

Otherwise, if $x_{1}=x_{2}$, then by the assumption $(x_{1},y_{1} )\neq (x_{2},y_{2} )$ we must have $y_{1}\neq y_{2}$, and then $\varphi_{\alpha i} (y_{1} )<\varphi_{\alpha i} (y_{2} )$. Therefore, assumption (B2′) in Theorem 3.1 is satisfied. Then $(\bar{x},\bar{y} )$ is minimal point in $W_{0}$ by (ii′) of the Brezis-Browder principle.

Now we claim that x̄ is a critical point for T. By inequality (1) we have

$$ \max \bigl\{ c_{\alpha}\delta_{\alpha} (u,\bar{x} ),c_{i}p_{i} (v-\bar{y} ) \bigr\} \leqslant \varphi_{\alpha i} (\bar{y} )-\varphi_{\alpha i} (v ) $$

for each $u\in T\bar{x}$ and $(\alpha,i )\in\Lambda\times I$, and then $(u,v )\precsim (\bar{x},\bar{y} )$. Since $(\bar{x},\bar{y} )$ is a minimal point in $W_{0}$, it follows that $u=\bar{x}$, and thus $T\bar{x}= \{ \bar{x} \}$, which completes the proof. □

By the same process as before we can also get the same results if we replace the cone pseudo-distance $\{ \delta_{\alpha} \} _{\alpha\in\Lambda}$ with respect to the solid cone with the real-valued pseudo-distance $\{ d_{\alpha} \}_{\alpha\in \Lambda}$.

Proposition 3.4

Let $(\mathbb{X}, \{ d _{\alpha} \} _{\alpha\in\Lambda } )$ be a complete Hausdorff pseudo-metric space, $(\mathbb{\mathbb {Y}}, \{ p_{i} \} _{i\in I} )$ a complete separated locally convex space, and $T:\mathbb {X}\longrightarrow2^{\mathbb{X}}$ and $S:\mathbb{X}\longrightarrow2^{\mathbb{Y}}$ two set-valued maps with nonempty values.

Suppose that, for every $(\alpha,i )\in\Lambda\times I$ and two constants $c_{\alpha},c_{i}>0$, there exist lower semicontinuous functions $\varphi_{\alpha i}:\mathbb{\mathbb{Y}}\longrightarrow [0,\infty )$ and, for each $(x,y )\in G_{S}$, there exist $u\in Tx$ and $v\in Su$ (resp., for every $u\in Tx$, there exists $v\in Su$) such that:

$$ \max \bigl\{ c_{\alpha}d_{\alpha} (x,u ),c_{i}p_{i} (y-v ) \bigr\} \leqslant\varphi_{\alpha i} (y )-\varphi_{\alpha i} (v ). $$

(2)

Then T has a fixed point (resp. critical point) in $\mathbb{X}$.

If the set-valued map S in Proposition 3.4 is only a single-valued map, then we have the following:

Corollary 3.5

Isac [12]

Let $(\mathbb{X}, \{ p_{\alpha} \} _{\alpha\in\Lambda } )$ be a Hausdorff locally convex space, and $M\subset\mathbb{X}$ be a nonempty set. The set-valued map $T:\mathbb{X}\longrightarrow 2^{\mathbb{X}}$ has a critical point if and only if there exist a complete Hausdorff locally convex space $(\mathbb{Y}, \{ q_{i} \} _{i\in I} )$, a subset $M_{0}\subseteq M$, $S:M_{0}\longrightarrow\mathbb{Y}$, for every couple $(\alpha,i )\in\Lambda \times I$, a function $\varphi_{\alpha i}:\overline{S (M_{0} )}\longrightarrow [0,\infty )$, and two constants $c_{\alpha},c_{i}>0$ such that:

• (i) $T (M_{0} )\subset M_{0}$, and $M_{0} \subset M$ is closed;

• (ii) S is closed, and $\overline{S (M_{0} )}$ is complete;

• (iii) $\varphi_{\alpha i}$ is lower semicontinuous for each $(\alpha,i )\in\Lambda \times I$;

• (iv) $\max \{ c_{\alpha}p_{\alpha} (x-y ),c_{i}q_{i} (S (x )-S (y ) ) \} \leqslant\varphi _{\alpha i} (S (x ) )-\varphi_{\alpha i} (S (y ) )$ for all $x\in M_{0}$ and all $y\in Tx$.

Proof

If T has a critical point $\bar{x}\in M$, then the assumptions of Isac’s theorem are satisfied if we put $M_{0}= \{ \bar {x} \}$, $\mathbb{X}=\mathbb{Y}$, $\{ p_{\alpha} \} _{\alpha\in\Lambda}= \{ q_{i} \} _{i\in I}$, $S=I_{M_{0}}$, and for each $(\alpha,i )\in\Lambda\times I$, $c_{\alpha }=c_{i}=1$ and $\varphi_{\alpha i}=0$.

Conversely, $\{ p_{\alpha} \} _{\alpha\in\Lambda}$ is generating family of separated seminorms on $\mathbb{X}$, and if we set

$$ p_{\alpha} (x-y )=d_{\alpha} (x,y ) $$

for each $\alpha\in\Lambda$, then $(M_{0}, \{ d_{\alpha} \} _{\alpha\in\Lambda} )$ is a complete Hausdorff pseudo-metric subspace of $\mathbb{X}$. Also, by (ii) we get that $(\overline {S (M_{0} )}, \{ q_{i} \} _{i\in I} )$ is a complete Hausdorff locally convex subspace of $\mathbb{Y}$, and since $T (M_{0} )\subset M_{0}$, all assumptions of Proposition 3.4 are satisfied, so that we get the result. □

Remark 3.6

Our main result does not involve any assumptions about closeness of intermediary set-valued map S, contrary to the result of Isac [12].

Corollary 3.7

Fang [11]

Let $T:\mathbb{X}\longrightarrow\mathbb{X}$ be a map of a complete Hausdorff locally convex space $(\mathbb{X}, \{ p_{\alpha} \} _{\alpha\in\Lambda} )$. Suppose that there exists a lower semicontinuous function $\varphi :\mathbb{X}\longrightarrow [0,\infty )$ such that, for each $x\in\mathbb{X}$ and for each $\alpha\in\Lambda$,

$$ p_{\alpha} (x-Tx )\leqslant\varphi (x )-\varphi (Tx ). $$

(3)

Then T has a fixed point.

Proof

For every $x,y\in\mathbb{X}$, we even replace $p_{\alpha} (x-y )=d _{\alpha} (x,y )$ and take single-valued maps $T'$ and S with $Sx=\{x\}$ and $T'x=\{Tx\}$ for all $x\in\mathbb{X}$. Then inequality (3) implies inequality (2) of Proposition 3.4, and the result follows. □

We get the next obvious two corollaries.

Corollary 3.8

Downing and Kirk [15]

Let $\mathbb{X}$ and $\mathbb{Y}$ be complete metric spaces, and $T:\mathbb{X}\longrightarrow\mathbb{X}$ an arbitrary mapping. Suppose that there exist a closed mapping $S:\mathbb{X}\longrightarrow\mathbb{Y}$, a lower semicontinuous mapping $\varphi:S (\mathbb{X} )\longrightarrow [0,\infty )$, and a constant $c>0$ such that, for each $x\in\mathbb{X}$,

$$ \max \bigl\{ d_{\mathbb{X}} (x,Tx ),cd_{\mathbb{Y}} \bigl(S (x ),S (Tx ) \bigr) \bigr\} \leqslant\varphi \bigl(S (x ) \bigr)-\varphi \bigl(S (Tx ) \bigr) . $$

Then there exists $x\in\mathbb{X}$ such that $Tx=x$.

Corollary 3.9

Caristi [14]

Let $(\mathbb{X},d )$ be a complete metric space, and let $\varphi:\mathbb{X}\longrightarrow [0,\infty )$ be a lower semicontinuous function. If a mapping $T:\mathbb {X}\longrightarrow\mathbb{X}$ satisfies for each $x\in\mathbb{X}$ the condition

$$ d (x,Tx )\leqslant\varphi (x )-\varphi (Tx ) , $$

then T has a fixed point in $\mathbb{X}$.

We conclude this section with an application of Theorem 3.2.

Theorem 3.10

Let $(\mathbb{Y}, \{ p_{i} \} _{i\in I} )$ be a complete separated locally convex space, $(\mathbb{X}, \{ \delta_{\alpha} \} _{\alpha\in\Lambda } )$ be a complete Hausdorff pseudo-metric space over a solid cone $\mathbb {K}$, $S:\mathbb{X}\longrightarrow2^{\mathbb{Y}}$ be set-valued map, and for every $(\alpha,i )\in\Lambda \times I$, $\varphi_{\alpha i}:\mathbb{\mathbb{Y}}\longrightarrow [0,\infty )$ be lower semicontinuous function.

Suppose that, for each $(x,y )\in G_{S}$, there exists $(x_{0},y_{0} )\in G_{S}$ such that

• 1. $x_{0}\neq x$;

• 2. $\varphi_{\alpha i} (y_{0} )+\max \{ c_{\alpha}\delta_{\alpha} (x,x_{0} ),c_{i}p_{i} (y-y_{0} ) \} \leqslant\varphi_{\alpha i} (y )$ for every $(\alpha,i )\in\Lambda\times I$.

Then there exist $(\bar{x},\bar{y} )\in G_{S}$ and $(\alpha_{0},i_{0} )\in\Lambda\times I$ such that $\varphi_{\alpha _{0} i_{0}} (\bar{y} )=\inf_{t\in\mathbb{Y}}\varphi _{\alpha_{0} i_{0}} (t )$.

Proof

By contradiction suppose that, for each $(x,y )\in G_{S}$ and for every $(\alpha,i )\in\Lambda\times I$, we have

$$ \varphi_{\alpha i} (y )>\inf_{t\in\mathbb{Y}}\varphi _{\alpha i} (t ) . $$

By assumptions, there exists $(x_{0},y_{0} )\in G_{S}$ such that 1 and 2 hold. Set

$$\begin{aligned} E (x,y ) =& \bigl\{ (z,t )\in G_{S}:z\neq x, \mbox{and } \forall ( \alpha,i )\in\Lambda\times I, \\ &\varphi_{\alpha i} (t )+\max \bigl\{ c_{\alpha}\delta_{\alpha} (x,z ),c_{i}p_{i} (y-t ) \bigr\} \leqslant\varphi_{\alpha i} (y ) \bigr\} . \end{aligned}$$

For all $(x,y )\in G_{S}$, we have $(x_{0},y_{0} )\in E (x,y )$ and $(x,y )\notin E (x,y )$. For all $x\in\mathbb{X}$, we put $G_{S}(x)=\{y\in\mathbb{Y} : (x,y )\in G_{S}\}$. Define the set-valued map T by

$$ Tx=\bigcup_{y\in G_{S}(x)} \bigl\{ z\in\mathbb{X} : \exists t \in Sz \mbox{ such that } (z,t )\in E (x,y ) \bigr\} $$

for $x\in\mathbb{X}$. For all $(x,y )\in G_{S}$ and $(\alpha,i )\in\Lambda \times I$, there exist $z\in Tx$ and $t\in Sz$ such that

$$ \max \bigl\{ c_{\alpha}\delta_{\alpha} (x,z ),c_{i}p_{i} (y-t ) \bigr\} \leqslant\varphi_{\alpha i} (y )-\varphi _{\alpha i} (t ) . $$

Then by Theorem 3.2, T admits a point x̄ such that $\bar{x}\in T\bar{x}$. For this x̄, we get that, for some $\bar{y}_{1},\bar{y}_{2}\in\mathbb{Y}$, $(\bar {x},\bar{y}_{1} )\in E (\bar{x},\bar{y}_{2} )$, which is absurd. □

4 Variational principle

Theorem 4.1

Let $(\mathbb{Y}, \{ p_{i} \} _{i\in I} )$ be a complete separated locally convex space, $(\mathbb{X}, \{ \delta_{\alpha} \} _{\alpha\in\Lambda } )$ be a complete Hausdorff pseudo-metric space over a solid cone $\mathbb {K}$, $S:\mathbb{X}\longrightarrow2^{\mathbb{Y}}$ be a set-valued map, and, for every $(\alpha,i )\in\Lambda \times I$, $\varphi_{\alpha i}:\mathbb{\mathbb{Y}}\longrightarrow [0,\infty )$ be a lower semicontinuous function.

Then, for each $\varepsilon>0$ and $(x_{0},y_{0} )\in G_{S}$ satisfying

$$ \varphi_{\alpha i} (y_{0} )\leqslant\inf\varphi_{\alpha i}+ \varepsilon, \quad \forall (\alpha,i )\in\Lambda\times I , $$

there exists $(\bar{x},\bar{y} )\in G_{S}$ such that:

• (i) for each $(\alpha,i )\in\Lambda\times I$, $\varphi _{\alpha i} (\bar{y} )\leqslant\varphi_{\alpha i} (y_{0} )$;

• (ii) for each $(x,y )\in G_{S}$ with $x\neq\bar{x}$, there exist $(\alpha,i )\in\Lambda\times I$ and two constants $c_{\alpha},c_{i}>0$ such that

  $$ \varphi_{\alpha i} (\bar{y} )< \varphi_{\alpha i} (y )+\varepsilon\max \bigl\{ c_{\alpha}\delta_{\alpha} (x,\bar{x} ),c_{i}p_{i} (y-\bar{y} ) \bigr\} . $$

Proof

Let $\varepsilon>0$ and $(x_{0},y_{0} )\in G_{S}$. Put

$$ W_{0}= \bigl\{ (x,y )\in G_{S}; \forall (\alpha,i )\in \Lambda\times I, \varphi_{\alpha i} (y )+\varepsilon\max \bigl\{ c_{\alpha}\delta_{\alpha} (x,x_{0} ),c_{i}p_{i} (y-y_{0} ) \bigr\} \leqslant\varphi_{\alpha i} (y_{0} ) \bigr\} . $$

It is a nonempty and closed subset of $G_{S}$ since the family $\{ \varphi_{\alpha i} \}_{\alpha i}$ is lower semicontinuous.

For all $x\in\mathbb{X}$, we put $W_{0}(x)=\{y\in\mathbb{Y} : (x,y )\in W_{0}\}$. Next, we define the set-valued map $T:\mathbb{X}\longrightarrow 2^{\mathbb{X}}$ by

$$\begin{aligned} Tx =&\bigcup_{y\in W_{0}(x)} \bigl\{ \hat{x}\in\mathbb{X}; \exists\hat{y}\in S\hat{x}, \forall (\alpha,i )\in\Lambda\times I , \\ & \varphi_{\alpha i} (\hat{y} )+\varepsilon\max \bigl\{ c_{\alpha} \delta_{\alpha } (\hat{x},x ),c_{i}p_{i} (\hat{y}-y ) \bigr\} \leqslant\varphi_{\alpha i} (y ) \bigr\} . \end{aligned}$$

Obviously, T satisfies inequality (1) of Theorem 3.2 with $\phi_{\alpha i}=\frac{1}{\varepsilon}\varphi_{\alpha i}$ so that T has a fixed point, that is, there exists $(\bar {x},\bar{y} )\in W_{0}$ such that $\bar{x}\in T\bar {x}$ with

$$ (\bar{x},\bar{y} )\in W_{0}\quad \Longrightarrow\quad \varphi _{\alpha i} (\bar{y} )\leqslant\varphi_{\alpha i} (y_{0} ) , $$

and if $(\hat{x},\hat{y} )\in G_{S}$ with $(\hat{x},\hat {y} )\precsim (\bar{x},\bar{y} )$, then $\hat {x}=\bar{x}$, which is equivalent to the assertion that, for each $(x,y )\in G_{S}$ with $x\neq\bar{x}$, there exist $(\alpha,i )\in\Lambda\times I$ and two constants $c_{\alpha},c_{i}>0$ such that

$$ \varphi_{\alpha i} (\bar{y} )< \varphi_{\alpha i} (y )+\varepsilon\max \bigl\{ c_{\alpha}\delta_{\alpha} (x,\bar{x} ),c_{i}p_{i} (y-\bar{y} ) \bigr\} . $$

The proof is complete. □

Remark 4.2

We claim that Theorem 4.1 implies Theorem 3.2. Indeed, let $(x_{0},y_{0} )\in G_{S}$ be given and take $\varepsilon=1$. By Theorem 4.1 there exists $(\bar{x},\bar{y} )\in G_{S}$ such that assertions (i) and (ii) hold. Since (i), we have $(\bar {x},\bar{y} )\in W_{0}$. We claim that x̄ is a fixed point of T. Assuming the contrary, by inequality (1) we get the existence of some $(x,y )\in G_{S}$ such that $x\in T\bar{x}$, $x\neq\bar{x}$, and

$$ \max \bigl\{ c_{\alpha}\delta_{\alpha} (x,\bar{x} ),c_{i}p_{i} (y-\bar{y} ) \bigr\} \leqslant \varphi_{\alpha i} (\bar{y} )-\varphi_{\alpha i} (y ) \quad \mbox{for every } (\alpha,i )\in\Lambda\times I . $$

This contradicts (ii). Hence, x̄ is a fixed point.

The above considerations show that Theorem 4.1 and Theorem 3.2 are equivalent.

Since the Caristi theorem (Corollary 3.9) is a particular case of our main result and the Ekeland variational principle is equivalent to Caristi’s theorem, Theorem 4.1 is a generalization of the variational principle of Ekeland:

Corollary 4.3

Ekeland [1]

Let $(\mathbb{X},d )$ be a complete metric space, and $\varphi:\mathbb{X}\longrightarrow [0,\infty )$ be a lower semicontinuous function. Let $\varepsilon>0$, and let a point $u\in\mathbb{X}$ be such that $\varphi (u )\leqslant\inf\varphi +\varepsilon$. Then there exists a point $v\in\mathbb{X}$ such that:

• (i) $\varphi (v )\leqslant\varphi (u )$;

• (ii) $\varphi (v )<\varphi (w )+\varepsilon d (w;v )$ for any $w\in\mathbb{X}$; $w\neq v$.

5 Applications

In this section, we propose two applications.

5.1 General nonlinear complementarity problem

In a Hilbert space $(\mathbb{X},\langle\cdot,\cdot\rangle )$, the dual cone $\mathbb{K}'$ of a convex cone $\mathbb{K}$ with respect to the duality $\langle\mathbb{X}^{\prime},\mathbb{X}\rangle$ is defined by

$$ \mathbb{K}'= \bigl\{ y\in\mathbb{X}:\langle y,x\rangle\geqslant 0, \forall x\in\mathbb{K} \bigr\} , $$

and the polar of $\mathbb{K}$ is $\mathbb{K}^{0}=-\mathbb{K}'$.

Next, we suppose that $\mathbb{K}$ is a closed convex cone in $\mathbb{X}$. It is shown in [26] that the projection operator onto $\mathbb{K}$, denoted by $P_{\mathbb{K}}$, is well defined and satisfies, for all $x\in\mathbb {X}$,

$$ \bigl\lVert x - P_{\mathbb{K}} (x ) \bigr\rVert =\min_{y\in\mathbb {K}} \lVert x - y \rVert . $$

The next two results can be found in [26].

Theorem 5.1

For every $x\in\mathbb{X}$, $P_{\mathbb{K}}$ has the following properties:

• 1. $\langle P_{\mathbb{K}} (x )-x,y\rangle\geq0$ for every $y\in\mathbb{K}$;

• 2. $\langle P_{\mathbb{K}} (x )-x, P_{\mathbb{K}} (x ) \rangle= 0$.

Theorem 5.2

For all $x,y,z\in\mathbb{X}$, the following statements are equivalent:

• 1. $z=x+y$, $x\in\mathbb{K}$, $y\in\mathbb{K}^{0}$, and $\langle x,y\rangle=0$;

• 2. $x= P_{\mathbb{K}} (z )$ and $y= P_{\mathbb {K}^{0}} (z )$.

Following Isac [26, 27], we give a new application of our main result to the so called general nonlinear complementarity problem (GNCP).

Let $S:\mathbb{K} \rightarrow2^{\mathbb{X}}$ be a set-valued mapping. As is known [28], the GNCP with S and $\mathbb{K}$, denoted by $\operatorname{GNCP} (S,\mathbb{K} )$, is

$$ \operatorname{GNCP} (S,\mathbb{K} ) \mbox{:}\quad \textstyle\begin{cases} \mbox{find } (\hat{x},\hat{y} )\in\mathbb {K}\times\mathbb{X} \\ \mbox{s.t. } \hat{y}\in S (\hat{x} )\cap\mathbb{K}' \mbox{ and } \langle\hat{x},\hat{y}\rangle=0. \end{cases} $$

Before we obtain some existence results for $\operatorname{GNCP} (S,\mathbb {K} )$ by using existence results obtained in the previous sections, we give a useful theorem, which improves Theorem 4 in [26].

Theorem 5.3

The problem $\operatorname{GNCP} (S,\mathbb{K} )$ has a solution if and only if the set-valued map defined, for all $x\in \mathbb{X}$, by

$$ Tx= \bigl\{ z\in\mathbb{X}, z\in P_{\mathbb{K}} (x ) - S \bigl(P_{\mathbb{K}} (x ) \bigr) \bigr\} $$

has a fixed point in $\mathbb{X}$. Moreover, if $x_{0}$ is a fixed point of T, then $\hat{x}=P_{\mathbb{K}} (x_{0} )$ is a solution of the problem $\operatorname{GNCP} (S,\mathbb{K} )$.

Proof

Suppose that T has a fixed point $x_{0}$, that is,

$$ x_{0}\in P_{\mathbb{K}} (x_{0} ) - S \bigl(P_{\mathbb{K}} (x_{0} ) \bigr) . $$

Then there exists $\hat{y}\in S (P_{\mathbb{K}} (x_{0} ) )$ such that

$$ x_{0}= P_{\mathbb{K}} (x_{0} )-\hat{y} . $$

Then if we denote by $\hat{x}=P_{\mathbb{K}} (x_{0} )$, then it is clear that $\hat{x}\in\mathbb{K}$, and by item 1 of Theorem 5.1 we get for all $x\in\mathbb{K}$,

$$ \langle\hat{y},x\rangle=\langle\hat{x}-x_{0},x\rangle\geq0 , $$

then $\hat{y}\in\mathbb{K}'$. Therefore, by item 2 of Theorem 5.1 $\langle\hat{y},\hat{x}\rangle=\langle\hat{x}-x_{0},\hat {x}\rangle=0$, which implies that $(\hat{x},\hat{y} )$ is a solution of $\operatorname{GNCP} (S,\mathbb{K} )$.

Conversely, if $(\hat{x},\hat{y} )$ is a solution of $\operatorname {GNCP} (S,\mathbb{K} )$, then denoting

$$ x_{0}=\hat{x}-\hat{y} , $$

by Theorem 5.2 we get $\hat{x}=P_{\mathbb{K}} (x_{0} )$, and since $\hat{y}\in S (\hat{x} )\cap\mathbb{K}'$, we get $\hat {y}\in S (P_{\mathbb{K}} (x_{0} ) )$. Hence, $x_{0}\in P_{\mathbb{K}} (x_{0} )-S (P_{\mathbb{K}} (x_{0} ) )$, and thus $x_{0}\in Tx_{0}$. This completes the proof. □

Now we formulate an existence result for the $\operatorname{GNCP} (S,\mathbb {K} )$ problem.

Theorem 5.4

Let $(\mathbb{X},\langle\cdot,\cdot\rangle )$ be a Hilbert space, and $\mathbb{K}$ be a closed convex cone in $\mathbb{X}$. Let $\{\varphi_{i} \}_{i\in I}$ be a family of lower semicontinuous functions from $\mathbb{X}$ to $\mathbb{R}_{+}$, and $a_{i}>0$ and $b_{i}>0$ be two families of positive real numbers. Suppose that the set-valued maps T and S defined before satisfy the supplementary condition:

For all $i\in I$ and $(x,y )\in G_{S}$, there exist $z\in Tx\cap\mathbb{K}$ and $t\in S (z )$ such that

$$ \max \bigl\{ a_{i}\lVert x-z \rVert_{\mathbb{X}} ,b_{i}\lVert y-t\rVert _{\mathbb{X}} \bigr\} \leqslant \varphi_{i} (y )-\varphi_{i} (t ) . $$

Then $\operatorname{GNCP} (S,\mathbb{K} )$ has a solution.

Proof

It suffices to replace T by $T^{\prime}$ defined from $\mathbb{K}$ into $2^{\mathbb{K}}$ as $T^{\prime} (x )=T (x )\cap \mathbb{K}$ and apply Theorem 5.3 and Proposition 3.4. □

Example 5.5

Let $\mathbb{X}=\mathbb{R}$, $\mathbb{K}=\mathbb{R_{+}}$, and, for all $i\in I$, $a_{i}=b_{i}=1$, $\varphi_{i} (x )=\lvert x\rvert$ for $x\in\mathbb{X}$, and $S (x )=[0,x]$ for all $x\in\mathbb {K}$. Then the GNCP problem becomes:

$$ \operatorname{GNCP} (S,\mathbb{R}_{+} ) \mbox{:}\quad \textstyle\begin{cases} \mbox{find } (\hat{x},\hat{y} )\in\mathbb {R}_{+}\times\mathbb{R} \\ \mbox{s.t. } \hat{y}\in[0,\hat{x}] \mbox{ and } \hat{x}\hat{y}=0. \end{cases} $$

It is obvious that $T (x )=[0,x]$ for each $(x,y )\in G_{S}$. It is clear that, for all $x\geq0$ and $y\in[0,x]$, we get

$$ \lvert x-y\rvert+\lvert y\rvert\leq\lvert x \rvert\quad \Leftrightarrow\quad \lvert x-y\rvert\leqslant\varphi_{i} (x )-\varphi_{i} (y ) , $$

and choosing $z\in T (x )$ and $t\in S (z )$, we have:

• 1. for $x=y$, we choose $z=0$ and $t=0$, and then we have

  $$ \max \bigl\{ \lvert x\rvert,\lvert y\rvert \bigr\} \leqslant\varphi _{i} (y ) ; $$

• 2. for $y< x$, we choose $z=x-y+t$ and $t\leq\min\{x-y,y\}$, so that $\lvert x-z\rvert=\lvert y-t\rvert$, and then we get

  $$ \lvert y-t\rvert\leqslant\varphi_{i} (y )-\varphi_{i} (t ) . $$

Finally, by 1 and 2 we get

$$ \max \bigl\{ a_{i}\lvert x-z \rvert,b_{i}\lvert y-t\rvert \bigr\} \leqslant\varphi_{i} (y )-\varphi_{i} (t ) . $$

Then all assumptions of Theorem 5.4 hold, and hence problem $\operatorname {GNCP} (S,\mathbb{R}_{+} )$ has a solution, and the set of solutions is

$$ \operatorname{Sol}\bigl(\operatorname{GNCP} (S,\mathbb{R}_{+} ) \bigr)=\bigl\{ ( x,0) ; x\geq0 \bigr\} . $$

5.2 Differential inclusion in a nuclear space

Let $\mathbb{R}^{d}$ (with fixed $d\in\mathbb{N}^{\ast}$), set $\mathcal{D} (\mathbb{R}^{d} )$ to be the space of all complex-valued infinitely differentiable functions on $\mathbb{R}^{d}$ with compact support, and define the differential operator for each multiindex $\alpha\in\mathbb{N}^{d}$ with $\alpha= (\alpha_{1},\alpha _{2},\ldots,\alpha_{d} )$ by

$$ D^{\alpha}=\cfrac{\partial^{\vert \alpha \vert }}{\partial x_{1}^{\alpha_{1}} \, \partial x_{2}^{\alpha_{2}}\cdots\partial x_{d}^{\alpha_{d}}}, $$

where $\vert \alpha \vert =\alpha_{1}+\cdots+\alpha_{d}$. The space $\mathcal{D} (\mathbb{R}^{d} )$ is endowed by a locally convex topology defined by the family of separated seminorms

$$ \Vert \varphi \Vert _{N}=\sup \bigl\{ \bigl\vert D^{\alpha}\varphi (x )\bigr\vert ; x\in\mathbb{R}^{d} \mbox{ and }\vert \alpha \vert \leq N \bigr\} . $$

Recall that a subset $B\subset\mathcal{D} (\mathbb{R}^{d} )$ is bounded if for some compact $K\subset\mathbb{R}^{d}$, we have $B\subset\mathcal{D} (K )$ and there are numbers $M_{N}<\infty$ such that every $\varphi\in B$ satisfies the inequalities

$$ \Vert \varphi \Vert _{N}\leq M_{N}, \quad N=0,1,2, \ldots . $$

It is worth noting that $\mathcal{D} (\mathbb{R}^{d} )$ endowed with the limit inductive topology of $\{ \mathcal{D} (K_{n} ) \} _{n}$ is a complete nonmetric space, where $(K_{n} )_{n\in\mathbb{N}}$ is an exhaustive sequence of compact subsets, that is, for every $n\in \mathbb{N}$, $K_{n}$ included in the interior of $K_{n+1}$, and $\mathbb{R}^{d}=\cup _{n}K_{n}$; for more details, see [29].

Now, let $\mathcal{D}^{\prime} (\mathbb{R}^{d} )$ be the strong dual of $\mathcal{D} (\mathbb{R}^{d} )$, also endowed with the locally convex topology generated by an uncountable separated family of seminorms over the bounded subset of $\mathcal{D} (\mathbb {R}^{d} )$ denoted by τ, that is,

$$ p_{B} (f )=\sup_{\varphi\in B}\bigl\vert \langle f,\varphi \rangle\bigr\vert , \quad B\subset\mathcal{D} \bigl(\mathbb{R}^{d} \bigr) \quad \mbox{bounded} . $$

Definition 5.6

In a Hausdorff locally convex space $(\mathbb{X}, \{ p_{i} \} _{i\in\Lambda} )$, a convex cone $\mathbb{K}\subset\mathbb{X}$ is supernormal [13] if for each $i\in\Lambda$, there exists a continuous linear form $f_{i}\in\mathbb{K}^{\prime}$ (dual cone) such that, for each $x\in\mathbb{K}$, we have

$$ p_{i} (x )\leq f_{i} (x ) . $$

$\mathcal{D}^{\prime} (\mathbb{R}^{d} )$ endowed with τ-topology is a nuclear space [17], and we have the following:

Proposition 5.7

In a nuclear space $\mathbb{X}$, a convex cone $\mathbb {K}\subset\mathbb{X}$ is τ-supernormal if and only if it is τ-normal.

It is shown in [17] that the cone $\mathbb{K}$ defined by

$$ \mathbb{K}= \bigl\{ \Lambda\in\mathcal{D}^{\prime} \bigl(\mathbb {R}^{d} \bigr); \langle\Lambda,\varphi \rangle\geq0, \forall \varphi\in \mathcal{C} \bigr\} $$

is τ-normal cone, where $\mathcal{C}= \{ \varphi\in\mathcal {D} (\mathbb{R}^{d} ); \varphi (x )\geq0, \forall x\in \mathbb{R}^{d} \}$, and hence $\mathbb{K}$ is τ-supernormal.

Next, we propose to solve the partial differential inclusion problem;

$$ (\mathcal{P} ) \mbox{:}\quad \textstyle\begin{cases} \mbox{find a locally integrable function }u\in L_{\mathrm{loc}}^{1} (\mathbb {R}^{d} ) \mbox{ such that} \\ D^{\alpha}u\in F (u ) \mbox{ a.e. on }\mathbb{R}^{d}, \end{cases} $$

where $\alpha\in\mathbb{N}^{d}$ a multiindex, and $F:L_{\mathrm{loc}}^{1} (\mathbb{R}^{d} )\longrightarrow2^{L_{\mathrm{loc}}^{1} (\mathbb {R}^{d} )}$.

Given $u\in L_{\mathrm{loc}}^{1} (\mathbb{R}^{d} )$, it is shown in [29] that u defines a regular distribution, denoted $\Lambda_{u}\in\mathcal{D}^{\prime} (\mathbb{R}^{d} )$, as follows:

$$ \Lambda_{u} (\varphi )= \int_{\mathbb{R}^{d}}u (x )\varphi (x )\, dx $$

for all $\varphi\in\mathcal{D} (\mathbb{R}^{d} )$.

Also, if $u\in L_{\mathrm{loc}}^{1} (\mathbb{R}^{d} )$, we know that $\Lambda_{D^{\alpha}u}=D^{\alpha}\Lambda_{u}$, and hence we propose to solve problem ($\mathcal{P}$) in regular distributions setting and consider the differentiability in the weak sense. Problem ($\mathcal{P}$) is transformed by the canonical isomorphism

$$ \mathcal{G}:L_{\mathrm{loc}}^{1} \bigl(\mathbb{R}^{d} \bigr)\longrightarrow \mathcal{G} \bigl(\mathcal{D}^{\prime} \bigl(\mathbb{R}^{d} \bigr) \bigr) $$

to

$$ \bigl(\mathcal{P}^{\prime} \bigr) \mbox{:}\quad \textstyle\begin{cases} \mbox{find a regular distribution }\Lambda_{u}\in\mathcal{D}^{\prime } (\mathbb{R}^{d} ) \mbox{ such that} \\ D^{\alpha}\Lambda_{u}\in\mathcal{F} (\Lambda_{u} ) \mbox{ a.e. on }\mathbb{R}^{d}, \end{cases} $$

where $\mathcal{F}$ is the set-valued map defined from $\mathcal {D}^{\prime} (\mathbb{R}^{d} )$ into $2^{\mathcal{D}^{\prime } (\mathbb{R}^{d} )}$ by

$$ \Lambda_{v}\in\mathcal{F}(\Lambda_{u})\quad \Leftrightarrow\quad v\in F(u) . $$

Now, passing to the second part of our developments, there is no chance that problem ($\mathcal{P^{\prime}}$) has a solution, so we will give a sufficient condition on the set-valued map F in order that the problem has at least one solution. For this, we define two subsets $\mathcal{I}$ and $\mathcal{J}$ of $\mathcal{D}^{\prime} (\mathbb{R}^{d} )$ by

$$\begin{aligned}& \mathcal{I}= \biggl\{ \Lambda_{f}; f\in L_{\mathrm{loc}}^{1} \bigl(\mathbb {R}^{d} \bigr), \Lambda_{f}(\varphi)= \int_{\mathbb{R}^{d}}f (x )D^{\alpha}\varphi (x )\,dx \mbox{ for each } \varphi\in\mathcal{D} \bigl(\mathbb{R}^{d} \bigr) \biggr\} ; \\& \forall\Lambda_{u}\in\mathcal{D}^{\prime} \bigl( \mathbb{R}^{d} \bigr)\mbox{:} \quad \mathcal{J} (\Lambda_{u} )= \bigl\{ \Lambda_{f}\in\mathcal {I}; u (x )\geq (-1 )^{\vert \alpha \vert }f (x ), \forall x\in\mathbb{R}^{d} \bigr\} , \end{aligned}$$

and for each regular distribution $\Lambda_{u}\in\mathcal{D}^{\prime } (\mathbb{R}^{d} )$, we define the set-valued maps $\mathcal{R}$ and $\mathcal{T}$ as follows:

$$\begin{aligned}& \mathcal{R} (\Lambda_{u} )= \bigl\{ \Lambda_{v}\in\mathcal {D}^{\prime} \bigl(\mathbb{R}^{d} \bigr); \forall\varphi\in \mathcal{C}, \langle\Lambda_{u}-\Lambda_{v},\varphi \rangle \geq0 \bigr\} ; \\& \mathcal{T} (\Lambda_{u} )= \bigl\{ \Lambda_{v}\in\mathcal {R} (\Lambda_{u} ); D^{\alpha}v\in F (u ) \mbox{ a.e. on } \mathbb{R}^{d} \bigr\} . \end{aligned}$$

It is obvious that $\mathcal{R} (\Lambda_{u} )$ is nonempty since $\Lambda_{u}\in\mathcal{R} (\Lambda_{u} )$, and for $\mathcal{T} (\Lambda_{u} )$, we need the next lemma.

Lemma 5.8

If for each $\Lambda_{u}\in\mathcal{D}^{\prime} (\mathbb {R}^{d} )$, $\mathcal{F} (\Lambda_{u} )\cap\mathcal{J} (\Lambda _{u} )\neq\emptyset$, then $\mathcal{T} (\Lambda_{u} )$ is a nonempty subset of $\mathcal{D}^{\prime} (\mathbb{R}^{d} )$.

Proof

Let f be a locally integrable function, and let $\Lambda_{u}\in \mathcal{D}^{\prime} (\mathbb{R}^{d} )$. Then the function

$$ \varphi\mapsto \int_{\mathbb{R}^{d}}f (x )D^{\alpha}\varphi (x )\,dx\quad \mbox{is an element of } \mathcal{F} (\Lambda_{u} ) , $$

and a simple calculation leads to

$$\begin{aligned} \int_{\mathbb{R}^{d}}f (x )D^{\alpha}\varphi (x )\,dx & = (-1 )^{\vert \alpha \vert } \int_{\mathbb{R}^{d}}D^{\alpha }f (x )\varphi (x )\,dx \\ & = \int_{\mathbb{R}^{d}}D^{\alpha} \bigl[ (-1 )^{\vert \alpha \vert }f (x ) \bigr]\varphi (x )\,dx. \end{aligned}$$

Put $v (x )= (-1 )^{\vert \alpha \vert }f (x )$ for $x\in\mathbb{R}^{d}$. Then $v\in L_{\mathrm{loc}}^{1} (\mathbb {R}^{d} )$, and

$$ \int_{\mathbb{R}^{d}}D^{\alpha}v (x )\varphi (x )\,dx= \Lambda_{D^{\alpha}v} (\varphi ) , $$

which leads to $\Lambda_{D^{\alpha}v}\in\mathcal{F} (\Lambda _{u} )$. Thus, $D^{\alpha}v\in F(u)$.

For each $\varphi\in\mathcal{C}$, we have

$$\begin{aligned} \Lambda_{u} (\varphi )-\Lambda_{v} (\varphi ) & = \Lambda_{u} (\varphi )- (-1 )^{\vert \alpha \vert }\Lambda_{f} ( \varphi ) \\ & = \int_{\mathbb{R}^{d}} \bigl[u (x )- (-1 )^{ \vert \alpha \vert }f (x ) \bigr]\geq0. \end{aligned}$$

Hence, $\Lambda_{v}\in\mathcal{T} (\Lambda_{u} )$. □

As an interesting application of the main result, we can state and prove the following existence theorem.

Theorem 5.9

If $\mathbb{K}$ and $\mathcal{R}$ are as before and $\mathcal{T}$ satisfies the assumption in the previous lemma, then problem ($\mathcal{P^{\prime}}$) has a solution.

Proof

By assumption, for each $\Lambda_{u}\in\mathcal{D}^{\prime} (\mathbb {R}^{d} )$, there exists $\Lambda_{v}\in\mathcal{T} (\Lambda_{u} )$ such that:

• (i) $D^{\alpha}\Lambda_{v}\in\mathcal{F} (\Lambda_{u} )$, and

• (ii) $\Lambda_{v}\in\mathcal{R} (\Lambda_{u} )$.

Then, for every $\varphi\in\mathcal{D} (\mathbb{R}^{d} )$, we have

$$ \langle\Lambda_{u}-\Lambda_{v},\varphi \rangle\geq 0\quad \Longleftrightarrow\quad (\Lambda_{u}-\Lambda_{v} ) (\varphi )\geq0 , $$

which implies that $(\Lambda_{u}-\Lambda_{v} )\in\mathbb{K}$; since $\mathbb{K}$ is a supernormal cone, for each bounded subset B of $\mathcal{D} (\mathbb{R}^{d} )$, there exists $f_{B}\in \mathbb{K^{\prime}}$ such that

$$ p_{B} (\Lambda_{u}-\Lambda_{v} )\leq f_{B} (\Lambda _{u}-\Lambda_{v} )\quad \Longleftrightarrow\quad p_{B} (\Lambda _{u}- \Lambda_{v} )\leq f_{B} (\Lambda_{u} )-f_{B} (\Lambda_{v} ) . $$

All assumptions of our former result in Proposition 3.4 hold. Therefore, $\mathcal{T}$ has a fixed point $\Lambda_{u^{\star}}\in\mathcal{D}^{\prime} (\mathbb {R}^{d} )$, that is,

$$ \Lambda_{u^{\star}}\in\mathcal{T} (\Lambda_{u^{\star}} )\quad \Leftrightarrow\quad D^{\alpha}u^{\star}\in F \bigl(u^{\star}\bigr) . $$

 □