Open Access

Pseudo-metric space and fixed point theorem

Fixed Point Theory and Applications20172017:3

DOI: 10.1186/s13663-017-0594-z

Received: 6 August 2016

Accepted: 13 January 2017

Published: 1 February 2017

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.

Keywords

Caristi-type fixed point Brezis-Browder principle nonlinear complementarity problem supernormal cone differential inclusion

1 Introduction

It is well known that the Ekeland variational principle [1] and Caristi-Kirk fixed point theorem are both equivalent. Many authors [27] 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. 1.

    \(\mathbb{K}+\mathbb{K}\subset\mathbb{K}\);

     
  2. 2.

    for every \(\lambda>0\), \(\lambda\mathbb{K}\subset\mathbb{K}\);

     
  3. 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, 1315] 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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (v)

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

     
  6. (vi)

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

     
  7. (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, 2023]. □

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. 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. 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. 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, 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:
  1. (i)

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

     
  2. (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 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 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 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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (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. 1.

    \(x_{0}\neq x\);

     
  2. 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 such that \(\bar{x}\in T\bar{x}\). For this , 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:
  1. (i)

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

     
  2. (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 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, 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:
  1. (i)

    \(\varphi (v )\leqslant\varphi (u )\);

     
  2. (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. 1.

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

     
  2. 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. 1.

    \(z=x+y\), \(x\in\mathbb{K}\), \(y\in\mathbb{K}^{0}\), and \(\langle x,y\rangle=0\);

     
  2. 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. 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. 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:
  1. (i)

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

     
  2. (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,
$$ \Lambda_{u^{\star}}\in\mathcal{T} (\Lambda_{u^{\star}} )\quad \Leftrightarrow\quad D^{\alpha}u^{\star}\in F \bigl(u^{\star}\bigr) . $$
 □

Declarations

Acknowledgements

The authors are grateful to the anonymous referees for their helpful comments and remarks.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Laboratory of Algebra Analysis and Applications (L3A), Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University of Casablanca

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