Remarks on some variants of minimal point theorem and Ekeland variational principle with applications

Irina Meghea; Cristina Stefania Stamin

doi:10.1515/dema-2022-0024

Article Open Access

Remarks on some variants of minimal point theorem and Ekeland variational principle with applications

Irina Meghea and Cristina Stefania Stamin

Published/Copyright: August 5, 2022

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

This paper presents some variants of minimal point theorem together with corresponding variants of Ekeland variational principle. In the second part of this article, there is a discussion on Ekeland variational principle and minimal point theorem relative to it in uniform spaces. The aim of these series of important results is to highlight relations between them, some improvements and specific applications.

Keywords: minimal point; Ekeland variational principle; Luc functional; convex cone; uniform space; equilibrium problem; location problem; multicriteria control problem; multicriteria fractional programming problem; stochastic efficiency

MSC 2010: 49J27; 49J35; 49J40

1 Introduction

Ekeland principle is a perturbed variational principle discovered in 1972 [1] and nowadays, after more than 40 years, it was proved to be the foundation of the modern variational calculus (see, for instance, the minimax theorems in Banach spaces or in the Finsler manifolds, in which the key step of the proof is made by the application of the Ekeland principle).

As referring to the applications, these are numerous and various: the geometry of Banach spaces, nonlinear analysis, differential equations and partial differential equations, global analysis, probabilistic analysis, differential geometry, fixed point theorems, nonlinear semigroups, dynamical systems, optimization, mathematical programming, and optimal control.

Minimal points are of special interest in multiple objective optimization problems. Vector equilibrium problem provides a unified model of several problems, such as the vector optimization problem, vector variational inequality, and vector complementarity problem. The stability analysis of the solution mappings for vector equilibrium problem is an important topic in the vector equilibrium theory [2]. As important applications of minimal point results, we can mention: equilibrium problems, location problems in town planning [3], multicriteria control problems, multicriteria fractional programming problems, stochastic efficiency in a set by usage of functional analysis over cones, optimization in partially ordered spaces, involving well-posedness of vector optimization problems [4], sensitivity of vector optimization problem, existence by scalarization of vector equilibrium problems, mixed vector equilibrium problems, applications to vector variational inequalities (vector variational-like inequalities, perturbed vector variational inequalities, hemivariational inequality systems with applications to vector optimization problems, minimax theorems for vector-valued mappings, ε -saddle point results) [3] and approximation problems as, for instance, the inverse Stefan problem, solutions for real-valued control approximation problems, proximal point algorithm, vector control approximation problem, relaxation and penalization, location problems, multicriteria fractional programming, multicriteria control problems, and stochastic efficiency in a set [3].

The first author has substantial contributions on variational principles in [5], discussions on theoretical results for variational principles and related statements [6,7], applications of variational principles [8,9, 10,11,12, 13,14,15], and also in collaboration [9,15].

In this paper, the authors start from some series of results from [16] and [17] for the first part, [18] and [19] for the second part aiming to make remarks on statements of minimal points in relation to vector variants of Ekeland variational principle and similar assertions in uniform spaces, to compare them and to discuss on their links, implications, and applications in models evolved from real phenomena. From our point of view, the value of any theoretical result resides in a real possibility to be applied in solving problems of real phenomena modeling.

2 Preliminaries

2.1 Preliminaries of affine geometry

Start this study with a series of definitions and results on cones. The proofs of the following assertions can be found, for instance, in [5]. Let E be real vector space.

The nonempty subset C of E is named cone when

(1) for any λ > 0 we have λ C ⊂ C ,

which is equivalent to

(2) for any μ > 0 we have μ C = C .

The cone C is said to be a cone with vertex when 0 ∈ C . This is obviously characterized by

(3) for any λ ≥ 0 we have λ C ⊂ C .

A convex cone with vertex is by definition pointed when the cone C ⧹ { 0 } is convex.

Proposition 1

Let C be a cone. C is convex iff

C + C ⊂ C .

Proposition 2

Let C be a convex cone with vertex. C is pointed iff

C ∩ ( − C ) = { 0 } .

Proposition 3

Let E be a real vector space.

If E is a preordered, then the set
C ≔ { x ∈ E : x ≥ 0 }
is a convex cone with vertex.
Let C be a convex cone with vertex in E. The binary relation
x ≤ C y ⇔ def y − x ∈ C
is a preordered relation in E compatible with the structure of vector space and
C = { x ∈ E : x ≥ C 0 } .
≤ C is an order relation iff C is pointed.

C is said to be the positive cone of E .

Explanation. Preorder relation means reflexivity + transitivity.

Remark 1

x ≤ C y ⇔ y ∈ x + C .

The preorder relation ≤ k 0

Let Y be a real locally convex space. For any C convex cone with vertex in Y ( Y ∗ is the topological dual of Y ), we put

(4) C + ≔ { ξ ∈ Y ∗ : ξ ( y ) ≥ 0 ∀ y ∈ C } ,

(5) C + + ≔ { ξ ∈ Y ∗ : ξ ( y ) > 0 ∀ y ∈ C ⧹ { 0 } } .

C + is a convex cone with vertex in Y ∗ (use (3) and Proposition 1). Let be

(6) k 0 ∈ C ⧹ ( − C ¯ ) .

Retain that − C ¯ = − C ¯ . { k 0 } being compact and − C ¯ closed, there is a closed hyperplane which separates them very strongly, therefore there is ξ 0 in C + so that

(7) ξ 0 ( k 0 ) = 1 .

Associate with Y the metric space ( X , d ) and consider the set X × Y . One introduces on this last set the binary relation ≤ k 0 :

(8) ( x 1 , y 1 ) ≤ k 0 ( x 2 , y 2 ) ⇔ def y 1 + k 0 d ( x 1 , x 2 ) ≤ C y 2 .

This definition is clearly inspired (see below) by the order relation established on epi φ by the Phelps cone (see in the following).

≤ k 0 is a preorder relation, being reflexive (as ≤ C is reflexive) but transitive also: let be y 2 − [ y 1 + k 0 d ( x 1 , x 2 ) ] ∈ C and y 3 − [ y 2 + k 0 d ( x 2 , x 3 ) ] ∈ C , then by addition one obtains (take into account Proposition 1)

y 1 + k 0 [ d ( x 1 , x 2 ) + d ( x 2 , x 3 ) ] ≤ C y 3 ,

but the first member is C -minorized by y 1 + k 0 d ( x 1 , x 3 ) (the triangle inequality, k 0 ∈ ( 6 ) C ), and hence the conclusion, ≤ C being transitive. If C is pointed, ≤ k 0 is even an order relation in X × Y : ( x 1 , y 1 ) ≤ k 0 ( x 2 , y 2 ) and ( x 2 , y 2 ) ≤ k 0 ( x 1 , y 1 ) ⇒

(9) y 2 − y 1 − k 0 d ( x 1 , x 2 ) ∈ C and y 1 − y 2 − k 0 d ( x 1 , x 2 ) ∈ C

add them (see Proposition 1), − 2 k 0 d ( x 1 , x 2 ) ∈ C . If, ad absurdum, d ( x 1 , x 2 ) ≠ 0 , it results k 0 ∈ − C , this contradicts (6) and consequently x 1 = x 2 . Then, according to (9), y 2 − y 1 ∈ C ∩ ( − C ) , that imposes y 2 = y 1 (Proposition 2).

Retain from the last proof (the hypothesis “ C pointed” has been used only at the end)

(10) ( x 1 , y 1 ) ≤ k 0 ( x 2 , y 2 ) and ( x 2 , y 2 ) ≤ k 0 ( x 1 , y 1 ) ⇒ x 1 = x 2 .

If X is a real normed space, ≤ k 0 coincides with ≤ C k 0 , C k 0 ≔ { ( x , y ) ∈ X × Y : ‖ x ‖ k 0 ≤ C y } , convex cone with vertex ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ∈ C k 0 ⇒ ( x 1 , y 1 ) + ( x 2 , y 2 ) ∈ C k 0 as ‖ x 1 + x 2 ‖ k 0 ≤ C y 1 + y 2 since k 0 ‖ x 1 + x 2 ‖ ≤ C ‖ x 1 ‖ k 0 + ‖ x 2 ‖ k 0 ≤ C y 1 + y 2 ) . Indeed, ( x 1 , y 1 ) ≤ C k 0 ( x 2 , y 2 ) ⇔ ( x 2 − x 1 , y 2 − y 1 ) ∈ C k 0 ⇔ k 0 ‖ x 2 − x 1 ‖ ≤ C y 2 − y 1 ⇔ y 2 − y 1 − k 0 ‖ x 2 − x 1 ‖ ∈ C ⇔ ( 8 ) ( x 1 , y 1 ) ≤ k 0 ( x 2 , y 2 ) .

Retain also

(11) C pointed ⇒ C k 0 pointed .

For (11) use Proposition 2. Let be ( x , y ) ∈ C k 0 ∩ ( − C k 0 ) , that is ‖ x ‖ k 0 ≤ C y and ‖ x ‖ k 0 ≤ C − y , hence, by adding, − 2 ‖ x ‖ k 0 ∈ C , i.e. ‖ x ‖ k 0 ∈ − C , but ‖ x ‖ k 0 ∈ C ((6)), hence ‖ x ‖ k 0 = 0 and x = 0 . Then also y ∈ C and − y ∈ C , i.e. y ∈ C ∩ ( − C ) and hence y = 0 .

Let A be a nonempty subset of the set X × Y . For every ( x , y ) from A consider

(12) A ( x , y ) ≔ { ( v , w ) ∈ A : ( v , w ) ≤ k 0 ( x , y ) } .

A ( x , y ) is the lower cut of A along of ( x , y ) with respect to ≤ k 0 . Obviously ( x , y ) ∈ A ( x , y ) .

Condition (H1). Let ( x n , y n ) n ≥ 1 be a k 0 -decreasing sequence of points from A . If x n ⟶ x , there is y in Y so that ( x , y ) ∈ A and

( x , y ) ≤ k 0 ( x n , y n ) ∀ n ≥ 1 .

Condition (H2). Let ( x n , y n ) n ≥ 1 be a sequence of points from A with x n ⟶ x and ( y n ) n ≥ 1 C -decreasing. Then there is y in Y so that ( x , y ) ∈ A and

y ≤ C y n ∀ n ≥ 1 .

When Y is separated, these two conditions are connected by

Proposition 4

If A verifies (H2) and C ∩ ( z − R + k 0 ) is closed for any z in C , then A verifies also (H1).

Finally, designate π X , π Y the canonical projections of X × Y , π X ( x , y ) = x , and π Y ( x , y ) = y .

2.2 Preliminaries for Ekeland principle

In this subsection, we give some variants of Ekeland variational principle in relation to cones.

Ekeland principle [1,20]. Let ( X , d ) be a complete metric space and φ : X → ( − ∞ , + ∞ ] bounded from below, lower semicontinuous and proper. For any ε > 0 and u in X with

φ ( u ) ≤ inf φ ( X ) + ε ,

and for any λ > 0 , there exists v ε in X such that

(13) φ ( v ε ) < φ ( w ) + ε λ d ( v ε , w ) ∀ w ∈ X ⧹ { v ε } ,

and

(14) φ ( v ε ) ≤ φ ( u ) ,

(15) d ( v ε , u ) ≤ λ .

Geometrical interpretation of the fundamental inequality (13)of the Ekeland principle. This was given by Phelps ([21], p. 45; [22], p. 256).

Let X be a real normed space. In the product vector space X × R is considered, for every λ > 0 , the set

C λ ≔ { ( x , α ) ∈ X × R : λ ‖ x ‖ ≤ − α } .

C λ is a convex cone with vertex (see above 2.1, λ ‖ x + x ′ ‖ ≤ λ ‖ x ‖ + λ ‖ x ′ ‖ ≤ − ( α + α ′ ) ), we call it the cone of Phelps.

Suppose verified inequality (13) of Ekeland principle with λ = 1 . Then we have

(16) [ ( v ε , φ ( v ε ) ) + C ε ] ∩ epi φ = ( v ε , φ ( v ε ) ) .

The inclusion ⊃ being obvius as ( 0 , 0 ) ∈ C ε , pass to ⊂ . Let ( x , α ) be in C ε , that is ε ‖ x ‖ ≤ − α and ( v ε , φ ( v ε ) ) + ( x , α ) = ( v ε + x , φ ( v ε ) + α ) ∈ epi φ , that is

φ ( v ε + x ) ≤ φ ( v ε ) + α .

But φ ( v ε ) + α ≤ φ ( v ε ) − ε ‖ x ‖ , hence φ ( v ε ) ≥ φ ( v ε + x ) + ε ‖ x ‖ , therefore, setting y = v ε + x , φ ( v ε ) ≥ φ ( y ) + ε ‖ v ε − y ‖ . This inequality compared with (13) gives y = v ε , hence x = 0 , that leads to φ ( v ε ) ≤ φ ( v ε ) + α ≤ φ ( v ε ) , hence α = 0 and thus (16).

Reciprocally,

( 16 ) ⇒ ( 13 ) ( λ = 1 ) .

Indeed, suppose

φ ( v ε ) ≥ φ ( y ) + ε ‖ v ε − y ‖ .

It has to prove y = v ε . For y = v ε + x , the inequality becomes

φ ( v ε ) ≥ φ ( v ε + x ) + ε ‖ x ‖ .

But ( x , − ε ‖ x ‖ ) ∈ C ε , hence, according to the assumption, if ( v ε , φ ( v ε ) ) + ( x , − ε ‖ x ‖ ) = ( v ε + x , φ ( v ε ) − ε ‖ x ‖ ) ∈ epi φ , that is φ ( v ε + x ) ≤ φ ( v ε ) − ε ‖ x ‖ , then x = 0 . But the last inequality has been accepted, therefore x = 0 , y = v ε .

(17) ( 13 ) ( λ = 1 ) ⇔ ( 16 ) .

Come back to C λ . This is pointed, that is C λ ∩ ( − C λ ) = { 0 } (see 2.2): ( x , α ) , ( − x , − α ) ∈ C λ ⇒ λ ‖ x ‖ ≤ − α and λ ‖ x ‖ ≤ α hence λ ‖ x ‖ ≤ α ≤ − λ ‖ x ‖ , and it results x = 0 and λ = 0 .

Consider in X × R the binary relation ≤ C λ : ( x 1 , y 1 ) ≤ C λ ( x 2 , y 2 ) ⇔ def ( x 2 , y 2 ) ∈ ( x 1 , y 1 ) + C λ . This is an order relation (Proposition 3). Suppose (16) checked and let ( x , α ) be whatever element of epi φ , ( x , α ) ≥ C ε ( v ε , φ ( v ε ) ) , that is ( x , α ) ∈ ( v ε , φ ( v ε ) ) + C ε . But this implies ( x , α ) = ( v ε , φ ( v ε ) ) , hence ( v ε , φ ( v ε ) ) is a maximal element of epi φ . Reciprocally, this property of ( v ε , φ ( v ε ) ) implies (16): let ( x , α ) be in epi φ and ( x , α ) ∈ ( v ε , φ ( v ε ) ) + C ε , that is ( x , α ) ≥ C ε ( v ε , φ ( v ε ) ) , but then ( x , α ) = ( v ε , φ ( v ε ) ) , this being maximal, (16) is checked. So, taking also into account for (17), it can affirm:

The fundamental inequality (13) ( λ = 1 ) of the Ekeland principle is equivalent to the property: ( v ε , φ ( v ε ) ) is a maximal element in epi φ ordered by the Phelps cone C ε .

From now on, Y is a real separated locally convex space and, except for the last section, X is a complete metric space.

3 Some variants of minimal point theorem

In this section, we present four variants of minimal point theorem from [16] together with corresponding variants of Ekeland variational principle [16] in order to make some observations, to highlight links and relations with other forms of minimal point assertions and Ekeland variational principle. At subsection 3.3, we gave the proof of the Lemma 1 from there since this one is made by the first author and we did not consider the proof from [16].

3.1 Minimal point theorem I

Theorem 1

(Minimal point theorem I; [16], Theorem 1). Let A be a nonempty subset of the set X × Y which verifies (H1) and for which there is y A in Y having the property

(18) π Y ( A ) ⊂ y A + C .

Then for each ( x 0 , y 0 ) from A , there is ( v , w ) in A so that

( v , w ) ≤ k 0 ( x 0 , y 0 )

and if ( x , y ) ∈ A and ( x , y ) ≤ k 0 ( v , w ) then

x = v .

Remark 2

Despite of its name, ( v , w ) is not, as one observes, an authentic minimal point. Such a point will be provided by Theorem 4.

As consequences of Theorem 1, we will see in the following two vector variants of Ekeland principle. But firstly consider the disjoint union Y • ≔ Y ∪ { + ∞ } . Take by definition y ≤ C ∞ ∀ y ∈ Y . Let be φ : X → Y • . The domain of φ , dom φ , is { x ∈ X : φ ( x ) ≠ + ∞ } , the epigraph of φ , epi φ , is { ( x , y ) ∈ X × Y : y ≤ C φ ( x ) } , the graph of φ , Gr φ , is { ( x , φ ( x ) ) : x ∈ X , φ ( x ) ≠ + ∞ } . φ is proper if dom φ ≠ ∅ .

Theorem 2

(Vector variant of Ekeland principle; [16]). Let φ : X → Y • be C -bounded from below and proper which verifies the condition

( H 3 ) { x ′ ∈ X : φ ( x ′ ) + k 0 d ( x ′ , x ) ≤ C φ ( x ) } is closed ∀ x ∈ X .

Then for any x 0 from dom φ there is v in X so that

(19) φ ( v ) + k 0 d ( v , x 0 ) ≤ C φ ( x 0 )

and

(20) φ ( x ) + k 0 d ( v , x ) ≤ φ ( v ) ⇒ x = v .

Remark 3

From Theorem 2 it can indeed obtain a form of Ekeland principle: under the conditions of Ekeland principle, there exists v ε in X verifying (13) and (15), while (14) is replaced by

(21) φ ( v ε ) ≤ φ ( u ) − ε λ d ( v ε , u ) .

The above form of Ekeland principle justifies the name of Theorem 2 as “vector variant” of this principle.

Let ( X , d ) be a complete metric space and φ : X → ( − ∞ , + ∞ ] bounded from below, lower semicontinuous and proper. For any ε > 0 and x 0 of X , there is v ε in X such that

φ ( v ε ) < φ ( x ) + ε d ( v ε , x ) ∀ x ∈ X \ { v ε }

and

φ ( v ε ) < φ ( x 0 ) − ε d ( v ε , x 0 ) .

Indeed, take in Theorem 2 Y = R , Y • = ( − ∞ , + ∞ ] , C = [ 0 , + ∞ ) , k 0 = ε ( C \ ( − C ¯ ) = ( 0 , + ∞ ) ) . The preorder relation ≤ C coincides with the order relation on R which is total, and so (15) gives the fundamental inequality and (14) the second one. The application of Theorem 2 is correct since (H3) is ensured by the lower semicontinuity of φ .

Remark 4

If at the hypotheses of Theorem 2 one adds

(22) φ ( X ) ∩ [ φ ( x 0 ) − λ k 0 − ( C \ { 0 } ) ] = ∅ ,

where λ ∈ ( 0 , + ∞ ) , then at the conclusions of (22) one adds

(23) d ( v , x 0 ) ≤ λ .

Indeed, suppose ad absurdum

(24) d ( v , x 0 ) > λ .

Then, for a u from C we have (see (14)), φ ( v ) = φ ( x 0 ) − k 0 d ( v , x 0 ) − u = φ ( x 0 ) − λ k 0 − [ d ( v , x 0 ) − λ ] k 0 − u ∈ φ ( x 0 ) − λ k 0 − ( C \ { 0 } ) , because [ d ( v , x 0 ) − λ ] k 0 + u ∈ C \ { 0 } (ad absurdum k 0 d ( v , x 0 ) + u = λ k 0 , one gets φ ( v ) = φ ( x 0 ) − λ k 0 and, from (14), k 0 [ d ( v , x 0 ) − λ ] ≤ C 0 , contradiction if one takes into account (24) and (6)), and we come in collision with (22).

Remark 5

A particular case of Theorem 2 is obtained in [23], Theorem 4.

Here is a second vector variant of the Ekeland principle.

Theorem 3

(Vector variant of the Ekeland principle; [16]). Let φ : X → Y • be lower C-bounded and proper verifying

(H4) for any sequence ( x n ) n ≥ 1 from dom φ with x n → x and ( φ ( x n ) ) n ≥ 1 C -decreasing, we have

φ ( x ) ≤ φ ( x n ) ∀ n ≥ 1 .

If C ∩ ( z − R + k 0 ) is closed for any z from C, then are valid the conclusions from Theorem 2.

Remark 6

A particular case of Theorem 3 is found in [23], Proposition 1.

Attention. For Y = R , Y • = ( − ∞ , + ∞ ] and C = [ 0 , + ∞ ) , Theorem 3 gives the form of the Ekeland principle from the above remark 3 without the demand φ lower semicontinuous but with the demand (H4) verified.

For example, this variant can be applied to the function φ : R → R , φ ( x ) = e − ∣ x ∣ , x ≠ 0 , φ ( 0 ) = 2 ; φ is not lower semicontinuous in 0, verifies the demands from Theorem 3, and [ 0 , + ∞ ) ∩ ( z − [ 0 , + ∞ ) k 0 ) is closed for any z from [ 0 , + ∞ ) ( k 0 ∈ 2.1 , ( 6 ) ( 0 , + ∞ ) ) .

3.2 Minimal point theorem II

Consider on X × Y the binary relation ( ξ 0 is the functional from (7))

(25) ( x 1 , y 1 ) ≤ k 0 , ξ 0 ( x 2 , y 2 ) ⇔ ( x 1 , y 1 ) = ( x 2 , y 2 ) or ( x 1 , y 1 ) ≤ k 0 ( x 2 , y 2 ) and ξ 0 ( y 1 ) < ξ 0 ( y 2 ) .

This is an order relation. The transitivity: let be ( x 1 , y 1 ) ≤ k 0 , ξ 0 ( x 2 , y 2 ) ≤ k 0 , ξ 0 ( x 3 , y 3 ) ; if ( x 1 , y 1 ) = ( x 2 , y 2 ) – nothing to prove; in the contrary case, we have ( x 1 , y 1 ) ≤ k 0 ( x 2 , y 2 ) and ξ 0 ( y 1 ) < ξ 0 ( y 2 ) ; if ( x 2 , y 2 ) = ( x 3 , y 3 ) – nothing to prove again; in the contrary case, we have ( x 2 , y 2 ) ≤ k 0 ( x 3 , y 3 ) and ξ 0 ( y 2 ) < ξ 0 ( y 3 ) ; as “ ≤ k 0 ” and “ < ” are transitive, the conclusion is imposed. The reflexivity is obvious. ( x 1 , y 1 ) ≤ k 0 , ξ 0 ( x 2 , y 2 ) and ( x 2 , y 2 ) ≤ k 0 , ξ 0 ( x 1 , y 1 ) ⇒ ( x 1 , y 1 ) = ( x 2 , y 2 ) : ad absurdum, one obtains ξ 0 ( y 1 ) < ξ 0 ( y 2 ) and ξ 0 ( y 2 ) < ξ 0 ( y 1 ) contradiction.

Theorem 4

(Minimal theorem II; [25]) Let A be a nonempty subset of the set X × Y which verifies (H1) and ξ 0 is lower bounded on π Y ( A ) . Then for each ( x 0 , y 0 ) from A there is ( v , w ) in A so that

( v , w ) ≤ k 0 , ξ 0 ( x 0 , y 0 )

and if ( x , y ) ∈ A and ( x , y ) ≤ k 0 , ξ 0 ( v , w ) , then ( x , y ) = ( v , w ) .

Remark 7

Theorem 1 is an immediate consequence of Theorem 4, since if, π Y ( A ) ⊂ y A + C , that is for any y of π Y ( A ) we have y = y A + u with u ∈ C , then ξ 0 ( y − y A ) ≥ 2.1 , ( 4 ) 0 , ξ 0 ( y ) ≥ ξ 0 ( y A ) , ξ 0 is bounded from below on π Y ( A ) .

3.3 Minimal point theorem III

Preliminaries for this result. Let E be a vector space on K , K = R , or K = C , and x , y in E , x ≠ y . The closed interval [ x , y ] is [ x , y ] = { ( 1 − λ ) x + λ y : λ ∈ [ 0 , 1 ] } . If λ ∈ [ 0 , 1 ) , λ ∈ ( 0 , 1 ] , or λ ∈ ( 0 , 1 ) , one obtains [ x , y ) , ( x , y ] , and ( x , y ) (open interval), respectively. The straight line d which passes through x and y is

d = { ( 1 − λ ) x + λ y ≔ λ ∈ R } .

If x 1 ∈ d , we have d = { x 1 + λ ( y − x ) : λ ∈ R } ( x 1 ∈ d ⇒ x 1 = ( 1 − λ 1 ) x + λ 1 y , replace ) .

Let M be a nonempty subset of E . x 0 from M is by definition algebraic interior point, if on each straight line that passes through x 0 there is an open interval included in M which contains x 0 . The set of these points is denoted M i or aint M , the algebraic interior of M . y 0 from E is by definition algebraic adherent point to M if ∃ x in M so that [ x , y 0 ) ⊂ M . The set of these points, the algebraic closure of M , is denoted M a . M a ⧹ M i is the algebraic boundary of M .

Obviously M i ⊂ M , M convex ⇒ M ⊂ M a . If M is convex and M i ≠ ∅ , M is called algebraic convex field.

One can affirm

(26) If C is a convex cone with vertex and C i ≠ ∅ , then C + C i = C i .

Return to the subject of this work.

Here, the convex cone C with vertex is assumed closed. So

(27) k 0 ∈ C ⧹ ( − C ) ( see ( 6 ) ) .

For each y from Y , consider the set

(28) M y ≔ { t ∈ R : y ∈ t k 0 − C } .

Proposition 5

If M y ≠ ∅ , then there is t y in R so that

M y = [ t y , + ∞ ) .

Consider the Luc functional L : Y → R ¯ [25],

(29) L ( y ) = inf M y M y ≠ ∅ + ∞ , M y = ∅ .

Lemma 1

L is lower semicontinuous, proper, sublinear, and C-increasing.
{ y ∈ Y : L ( y ) ≤ t } = t k 0 − C ;
L ( y + λ k 0 ) = L ( y ) + λ , ∀ y ∈ Y , ∀ λ ∈ R ; L ( − y ) ≥ − L ( y ) , ∀ y ∈ Y ; L ( 0 ) = 0 , L ( λ k 0 ) = λ , ∀ λ ∈ R .
Moreover, if k 0 ∈ aint C , L takes its values in R and
{ y ∈ Y : L ( y ) < t } = t k 0 − aint C ,

y 2 − y 1 ∈ aint C ⇒ L ( y 1 ) < L ( y 2 ) .

Proof

(i) Let y be arbitrary from Y . Assume L ( y ) < + ∞ . Then for any t from R , we have

(30) L − 1 ( ( − ∞ , t ] ) = t k 0 − C ,

as, for instance, L ( y ) ≤ t ⇒ 5 y ∈ t k 0 − C . Since t k 0 − C is closed, one concludes L l.s.c. in y . Suppose L ( y ) = + ∞ , i.e. y ∉ R k 0 − C , as R k 0 − C is closed since R k 0 has a finite dimension. There is V a neighborhood of y so that V ∩ ( R k 0 − C ) = ∅ , that is L ( z ) = + ∞ on V , L is l.s.c. in y .

From Proposition 5 and (29), one obtains

(31) epi L ( in Y × R ) = T − 1 ( C ) , T : Y × R → Y , T ( y , t ) = t k 0 − y .

As T − 1 ( C ) ≠ ∅ ( k 0 = T ( 0 , 1 ) ! ) , L is proper. In addition, T being linear and continuous, (31) imposes epi L (in Y × R ) is a convex cone with vertex and consequently L is sublinear.

Finally, let be y 1 ≤ C y 2 and L ( y 2 ) < + ∞ (when L ( y 2 ) = + ∞ – nothing to prove). According to Proposition 5, L ( y 2 ) ∈ M y 2 , i.e. y 2 = L ( y 2 ) k 0 − u , u ∈ C . But y 2 − y 1 = v , v ∈ C , hence y 1 = L ( y 2 ) k 0 − ( u + v ) , that is y 1 ∈ L ( y 2 ) k 0 − C and hence L ( y 1 ) ≤ L ( y 2 ) ((29), Proposition 5), L is C -increasing.

(ii) The first formula has been already obtained at (30). Go to the second formula. We have t ∈ M y + λ k 0 ⇔ ( 28 ) y + λ k 0 ∈ t k 0 − C ⇔ y ∈ ( t − λ ) k 0 − C ⇔ t − λ ∈ M y ⇔ t ∈ λ + M y , hence M y + λ k 0 = λ + M y , and hence the second formula. Pass to the third relation. Suppose L ( y ) < + ∞ , otherwise there is nothing to prove. − L ( y ) = − inf { t ∈ R : y ∈ t k 0 − C } = sup { − t ∈ R : y ∈ t k 0 − C } = sup { τ ∈ R : y ∈ − τ k 0 − C } = sup A , A ≔ { τ ∈ R : − y ∈ t k 0 + C } . L ( − y ) = inf B , B ≔ { t ∈ R : − y ∈ t k 0 − C } . Let τ , t be any from A and B , respectively. − y = τ k 0 + c 1 = t k 0 − c 2 , c 1 and c 2 from C , hence ( t − τ ) k 0 = c 1 + c 2 ∈ 2.1 C and hence t − τ ≥ 0 (use (27)), that imposes inf B ≥ sup A . Pass to the last two formulae. L ( 0 ) = inf { t ∈ R : 0 ∈ t k 0 − C } = inf { t ∈ R : t k 0 ∈ C } = 0 , because if, ad absurdum, there is t ′ ≔ − t , t > 0 so that t ′ k 0 ∈ C , then t k 0 ∈ − C , hence k 0 ∈ − C , in contradiction with (27). Finally, L ( λ k 0 ) = L ( 0 + λ k 0 ) = L ( 0 ) + λ = λ .

(iii) In this case, we have

(32) dom L = Y

that is

(33) M y ≠ ∅ , ∀ y from Y .

Indeed (see the preliminaries), the straight line { t k 0 + ( 1 − t ) y : t ∈ R } passing through k 0 has an open interval contained in C in which is k 0 , otherwise there expressed, for δ > 0 sufficiently small t k 0 + ( 1 − t ) y = u t , u t ∈ C , t ∈ ( 1 − δ , 1 + δ ) , hence y = t t − 1 k 0 − u t t − 1 for 1 < t < 1 + δ and so (33). Moreover,

(34) inf L ( Y ) = − ∞ .

Indeed, take t < 0 and y ∈ t k 0 − C , multiply by n from N , n y ∈ n t k 0 − C , therefore L ( n y ) ≤ n t and pass to the limit for n → ∞ .

Pass to the second assertion of (iii). We have { y ∈ Y : L ( y ) < t } ⊂ { y ∈ Y : L ( y ) ≤ t } i : let be L ( y ) < t and { y + λ z : λ ∈ R } any straight line passing through y , we have

(35) L ( y + λ z ) ≤ ( i ) L ( y ) + L ( λ z ) , but L ( λ z ) = λ L ( z ) , λ ≥ 0 − λ L ( − z ) , λ < 0 .

because lim λ → 0 + λ L ( z ) = 0 , lim λ → 0 − − λ L ( − z ) = 0 ( L ( − z ) ∈ ( 32 ) R ! ) , from this it results lim λ → 0 L ( λ z ) = 0 , and hence there is δ > 0 so that λ ∈ ( − δ , δ ) ⇒ L ( λ z ) < t − L ( y ) , consequently, taking into account (35), λ ∈ ( − δ , δ ) ⇒ L ( y + λ z ) < t . Reciprocally, { y ∈ Y : L ( y ) ≤ t } i ⊂ { y ∈ Y : L ( y ) < t } : let y be an element from the first member, the straight line { y + λ k 0 : λ ∈ R } and λ ∈ ( − δ , δ ) ⇒ L ( y + λ k 0 ) ≤ t , then y + λ k 0 ∈ t k 0 − C , y ∈ ( t − λ ) k 0 − C for any λ ∈ ( − δ , δ ) , take λ from ( 0 , δ ) so that t − λ < t , hence L ( y ) ≤ t − λ < t , y is an element from the second member. So,

{ y ∈ Y : L ( y ) < t } = { y ∈ Y : L ( y ) ≤ t } i = ( i i ) ( t k 0 − C ) i = t k 0 − C i .

Finally, pass at the third assertion of ( i i i ) . y 2 = L ( y 2 ) k 0 − u , u ∈ C , y 1 = y 2 − v , v ∈ C i , hence y = L ( y 2 ) k 0 − ( u + v ) with u + v ∈ ( 26 ) C i and the conclusion is imposed by considering the straight line through u + v and k 0 , both from C i (already L ( y 1 ) ≤ L ( y 2 ) ).□

Remark 8

In [16], the proof of the Lemma 7, here Lemma 1, is full of errors. For this reason, the authors presented here the recovered proof.

Remark 9

When k 0 ∈ C i and int C ≠ ∅ , L is continuous, because on the k 0 − C neighborhood of 0 we have L ( y ) ≤ 1 .

Remark 10

The detailed analysis of the proofs (for the first one, see, for instance, in [5]) of Proposition 5 and Lemma 1 shows that, if Y is only a vector topological space and if (27) is replaced by k 0 ∈ C ⧹ ( − C ¯ ) , C convex cone with vertex, and in the definition of M y (28), C is replaced by C ¯ (convex cone with vertex!), then remain true Proposition 5 and also the properties (i) and (ii) of the Luc functional from the Lemma 1, where C is replaced by C ¯ .

This remark will be used in the second part of this article (Section 4). For any nonempty subset A of X × Y and for each x from X and y from Y consider the sets

(36) A x ≔ { y ∈ Y : ( x , y ) ∈ A } , A y ≔ { x ∈ X : ( x , y ) ∈ A } .

Obviously

(37) π X ( A ) = { x ∈ X : A x ≠ ∅ } , π Y ( A ) = { y ∈ Y : A y ∈ ∅ } .

And now

Theorem 5

(Minimal point theorem; [16], Theorem 8). Let ( x 0 , y 0 ) be from A. Suppose:

There is t 0 in R so that π Y ( A ) ∩ ( y 0 − t 0 k 0 − C ) = ∅ ;
For any r from R
A r ≔ { x ∈ X : ∃ y ≤ C y 0 + r k 0 and ( x , y ) ∈ A } is closed ;
Any function Y → R ¯ lower semicontinuous, proper, sublinear and C -increasing attains its infimum on A x − y 0 ∀ x ∈ π X ( A ) .
Then there is ( v , w ) in A so that
(38) w + k 0 d ( v , x 0 ) ≤ C y 0
and
(39) ( x , y ) ∈ A , y + k 0 d ( v , x ) ≤ C w ⇒ x = v and y ∈ ( w − C ) ⧹ ( w − ( 0 , + ∞ ) k 0 − C ) .
If k 0 ∈ C i (respectively k 0 ∈ int C ), (c) may be replaced by:
Any lower semicontinuous (respectively continuous) functional Y → R , sublinear and C -increasing attains its infimum on A x − y 0 ∀ x ∈ π X ( A ) and (39) by:
(40) ( x , y ) ∈ A , y + k 0 d ( v , x ) ≤ C w ⇒ x = v and y ∉ w − C i ( respectively y ∉ w − int C ) .

Remark 11

The point ( v , w ) supplied by Theorem 5 is not, like in Theorem 4, an authentic minimal point.

And now

Theorem 6

(Vector variant of Ekeland principle; [16]). Let be φ : X → Y • proper, x 0 ∈ dom φ and ε > 0 . If

(41) for any r from R , { x ∈ X : φ ( x ) ≤ C φ ( x 0 ) + r k 0 } is closed

and

(42) φ ( X ) ∩ ( φ ( x 0 ) − ε k 0 − C ) = ∅ ,

then there is v in dom φ so that

(43) φ ( v ) + ε k 0 d ( v , x 0 ) ≤ C φ ( x 0 ) , d ( v , x 0 ) ≤ ε

and

(44) φ ( x ) + ε k 0 d ( v , x ) ≤ C φ ( v ) ⇒ x = v .

Remark 12

The variant – Theorem 6 does not require “ φ lower C -bounded” as Theorems 2 and 3 do this.

Taking in Theorem 6 Y = R , Y • = ( − ∞ , + ∞ ] , C = [ 0 , + ∞ ) and k 0 = 1 , we find ( φ l.s.c. ⇒ (41)):

Let φ : X → ( − ∞ , + ∞ ] be lower semicontinuous and proper. For any x 0 in dom φ and ε > 0 , if

φ ( X ) ∩ ( φ ( x 0 ) − ε − [ 0 , + ∞ ) ) = ∅ ,

then there is v in X so that

φ ( v ) < φ ( x ) + ε d ( v , x ) ∀ x ∈ X ⧹ { v }

and

φ ( v ) ≤ φ ( x 0 ) − ε d ( v , x 0 ) , d ( v , x 0 ) ≤ ε .

3.4 Minimal point theorem IV

Here it is presented a minimal point theorem in a product space X × Y that uses “distances” on the set X with values into a cone C , C -valued metrics. It is inspired by a vector variant of Ekeland principle, obtained by Németh, where such metrics appear [17].

Definition 1

Let X be a nonempty set, Y real separated locally convex space and C a convex cone with vertex in Y . C is named normal if 0 has a basis of full neighborhoods with respect to C (the subset M of Y is full with respect to C if

(45) M = ( M + C ) ∩ ( M − C ) ) .

Definition 2

The map r : X × X → C is, by definition, C -valued metric on X or C -metric on X if it verifies the usual distance axioms:

r ( x 1 , x 2 ) = 0 ⇔ x 1 = x 2 , r ( x 1 , x 2 ) = r ( x 2 , x 1 ) ,

r ( x 1 , x 2 ) ≤ C r ( x 1 , x 3 ) + r ( x 2 , x 3 ) .

Recapitulation. Let A be a nonempty set and “ ≤ ” a binary relation in A . This is called preorder relation if it is reflexive and transitive. In this case, A becomes preordered. “ ≤ ” is filtrable to the right when ∀ x , y from A ∃ z in A so that x ≤ z , y ≤ z .

Let E be a set and A filtrable to the right preordered set. The family ( x α ) α ∈ A , x α ∈ E is named generalized sequence. Suppose E a topological space. The generalized sequence ( x α ) α ∈ A from E converges to x from E , lim α ∈ A x α = x or x α → α ∈ A x , if for any V neighborhood of x ∃ α V in A so that

α ≥ α V ⇒ x α ∈ V (Moore-Smith) .

Return to the subject of this section and let be ( X , r ) , r C -metric on X , and ( x i ) i ∈ I generalized sequence in X . ( x i ) i ∈ I r-converges to x ,

x i → i ∈ I r x ,

if r ( x i , x ) → i ∈ I 0 ( ( r ( x i , x ) ) i ∈ I is generalized sequence in Y ). The sequence ( x i ) i ∈ I is r -fundamental or r -Cauchy if for any V neighborhood of 0 in Y there is i V in I so that

i , j ≥ i V ⇒ r ( x i , x j ) ∈ V .

( X , r ) is named complete when every generalized sequence r -fundamental from X is r -convergent.

Suppose C normal. Then

(46) the r -limit of any generalized sequence is unique

and

(47) any r -convergent generalized sequence is r -fundamental .

Indeed, denote V ( 0 ) the set of neighborhoods of 0 from Y and let be r ( x i , x ) → i ∈ I 0 and r ( x i , y ) → i ∈ I 0 , V from V ( 0 ) which verifies (45), W symmetrical from V ( 0 ) with W ⊂ V , W 1 , W 2 from V ( 0 ) with W 1 + W 2 ⊂ W . ∃ i 1 , i 2 in I so that i ≥ i 1 ⇒ r ( x i , x ) ∈ W 1 , i ≥ i 2 ⇒ r ( x i , y ) ∈ W 2 , so if i 3 ≥ i 1 , i 2 ( I is filtrable to right) then i ≥ i 3 ⇒ r ( x i , x ) + r ( x i , y ) ∈ W . Suppose i ≥ i 3 . r ( x , y ) ≤ C r ( x i , x ) + r ( x i , y ) , i.e. r ( x , y ) − ( r ( x i , x ) + r ( x i , y ) ) ∈ − C , hence r ( x , y ) ∈ W − C ⊂ V − C . Likewise, as r ( x , y ) + r ( x i , x ) − r ( x i , y ) ∈ C , r ( x , y ) ∈ C + r ( x i , y ) − r ( x i , x ) ⊂ 1 C − r ( x i , x ) ⊂ C − W = C + W , hence r ( x , y ) ∈ C + W ⊂ C + V , which implies r ( x , y ) ∈ ( 45 ) V . Consequently r ( x , y ) ∈ ⋂ U ∈ V ( 0 ) U , therefore r ( x , y ) = 0 because Y is separated, x = y , i.e. (46).

The proof for (47) is identical: let be r ( x i , x ) → 0 , use as above the relations r ( x i , x ) + r ( x j , x ) − r ( x i , x j ) ∈ C and r ( x i , x j ) + r ( x j , x ) − r ( x i , x ) ∈ C .

Add to these

(48) x i → r i ∈ I x ⇒ r ( x i , y ) → i ∈ I r ( x , y ) .

Indeed, let V be from V ( 0 ) full with respect to C and W symmetrical in V ( 0 ) with W ⊂ V . ∃ i W in I so that i ≥ i W ⇒ r ( x i , x ) ∈ W . But r ( x i , y ) − r ( x , y ) ∈ C − r ( x i , x ) , hence i ≥ i W ⇒ r ( x i , y ) − r ( x , y ) ∈ C − W = W + C ⊂ V + C . On the other hand, i ≥ i W ⇒ r ( x i , y ) − r ( x , y ) = r ( x i , y ) + r ( x i , x ) − r ( x i , x ) − r ( x , y ) ∈ W − C ⊂ V − C , hence i ≥ i W ⇒ r ( x i , y ) − r ( x , y ) ∈ V and consequently (48).

Example. Let d be a distance on the set X and y 0 from Y ⧹ { 0 } . C ≔ [ 0 , + ∞ ) y 0 is a convex cone with vertex. The map r : X × X → C , r ( x 1 , x 2 ) = d ( x 1 , x 2 ) y 0 is C -metric on X because, for instance, r ( x 1 , x 2 ) ≤ C r ( x 1 , x 3 ) + r ( x 2 , x 3 ) and ( x 1 , x 3 ) + d ( x 2 , x 3 ) − d ( x 1 , x 2 ) ≥ 0 . Observe that a generalized sequence from X is r -convergent or r -fundamental iff it is convergent or fundamental, respectively.

Definition 3

Let be C 0 and C , C 0 ⊂ C convex cones with vertex from Y . C 0 is sequentially C -bounded, C -seq- b , if any sequence from C 0 which is C 0 -increasing and C -bounded is fundamental. In this case

(49) C 0 ∩ ( − C ) = { 0 } .

In particular C 0 is pointed (Proposition 2.2).

Let now be C 0 and C , C 0 ⊂ C , convex cones with vertex from Y and r is C 0 -metric on X . Consider the binary relation ≤ r in X × Y ,

(50) ( x 1 , y 1 ) ≤ r ( x 2 , y 2 ) ⇔ def y 1 + r ( x 1 , x 2 ) ≤ r y 2 .

“ ≤ r ” is a preorder relation and, in the case C pointed, even an order relation.

Condition (H5). Let A be a nonempty subset of X × Y . For any generalized sequence ( x i , y i ) i ∈ I from A r ≤ -decreasing and with x i → i ∈ I r x , there is y in Y so that

(51) ( x , y ) ∈ A and ( x , y ) ≤ r ( x i , y i ) ∀ i ∈ I .

Condition (H6). Let A be a nonempty subset of X × Y . For any generalized sequence ( x i , y i ) i ∈ I from A with x i → i ∈ I r x and ( y i ) i ∈ I C -decreasing, there is y in Y so that

(52) ( x , y ) ∈ A and y ≤ C y i ∀ i ∈ I .

Proposition 6

If C 0 is normal and C ∩ ( y − C 0 ) closed for any y from C, then ( H 6 ) ⇒ ( H 5 ) .

Theorem 7

(Minimal point theorem IV; [16], Theorem 10). Let C be a pointed convex cone with vertex in Y, C 0 a convex cone with vertex normal C-seq-b, r a C 0 -metric on X with ( X , r ) complete and A a nonempty subset of X × Y with π Y ( A ) lower C -bounded and which verifies (H5). Then, for any ( x 0 , y 0 ) from A , there is ( v , w ) in A ≤ R -minimal point so that

( v , w ) ≤ r ( x 0 , y 0 ) .

Theorem 7 easily supplies the following:

Theorem 8

(Vector variant of Ekeland principle; [16]). Let be φ : X → Y • , C a pointed convex cone with vertex in Y , C a normal convex cone with vertex C -seq- b with C ∩ ( y − C 0 ) closed ∀ y ∈ C and r a C 0 -metric on X , ( X , r ) complete.

Suppose φ C -bounded from below and for any ( x i ) i ∈ I a generalized sequence from X with x i → i ∈ I r x and ( φ ( x i ) ) i ∈ I C -decreasing, we have

φ ( x ) ≤ φ ( x i ) ∀ i ∈ I .

Then for any x 0 from dom φ , there is v in dom φ so that

(53) φ ( v ) + r ( v , x 0 ) ≤ C φ ( x 0 )

and

(54) φ ( x ) + r ( v , x ) ≤ C φ ( v ) ⇒ x = v .

4 Ekeland principle in uniform spaces

In this section, two theorems of the mininimal element are exposed, Theorem 10 and Theorem 12, and then with the aid of these two variants of the Ekeland principle, Proposition 16 and Proposition 22, respectively, are obtained in a uniform space defined by a generalized family of quasi-distances. In the first case, the aim function φ takes values in ( − ∞ , + ∞ ] , and in the second case in a topological vector space.

4.1 Theoretical frame

Let X be a nonempty set. For M subset of X × X set

(55) M − 1 ≔ { ( x , y ) ∈ X × X : ( y , x ) ∈ M } .

M is called symmetrical if M = M − 1 (it is enough M ⊂ M − 1 ; for instance, P ∪ P − 1 and P ∩ P − 1 , with P ⊂ X × X , are symmetrical).

Let M , N be subsets of X × X . Denote

(56) M o N ≔ { ( x , y ) ∈ X × X : ∃ z ∈ X so that ( x , z ) ∈ M , ( z , y ) ∈ N } .

In particular, M 2 ≔ M o M , M 3 ≔ M o M o M , etc. (we have associativity). Set M 1 ≔ M . We have

(57) M o N ︷ − 1 = N − 1 o M − 1 .

Also

(58) M ⊂ N and P ⊂ Q ⇒ M o P ⊂ N o Q .

In particular

(59) M ⊂ N ⇒ M n ⊂ N n ∀ n .

Let be A ⊂ X , M ⊂ X × X . Denote

M ( A ) ≔ { y ∈ X : ∃ x ∈ A so that ( x , y ) ∈ M } .

In particular, for A = { a }

M ( a ) = { y ∈ X : ( a , y ) ∈ M } − the cut along a .

Obviously M ⊂ N ⇒ M ( a ) ⊂ N ( a ) .

The diagonal Δ of the set X × X is the subset of this

Δ = { ( x , x ) : x ∈ X } .

We have, ∀ M ⊂ X × X ,

(60) M o Δ = Δ o M = M

( ( x , y ) ∈ M o Δ ⇒ ∃ z so that ( x , z ) ∈ M and ( z , y ) ∈ Δ , hence z = y , ( x , y ) ∈ M etc.).

UNIFORM SPACE

Let X be a nonempty set. The set U of nonempty subsets of X × X is by definition a uniform structure on X if

( F I ) M ∈ U and N ⊃ M ⇒ N ∈ U ( F I I ) M , N ∈ U ⇒ M ∩ N ∈ U (H. Cartan axioms)

( U I ) The diagonal Δ of X × X is included in each set from U ;

( U I I ) U ∈ U ⇒ U − 1 ∈ U ;

( U I I I ) ∀ U ∈ U ∃ V ∈ U so that V 2 ⊂ U (A. Weil axiom).

The set X with the uniform structure U define a uniform space ( X , U ) (A. Weil). The sets from U are the entourages of ( X , U ) . Retain:

U entourage ⇒ U n ⊂ U n + 1 ∀ n .

Indeed, U n ∘ Δ ⊂ U n ∘ U ( ( U I ) and (58)) and U n ∘ Δ = ( 60 ) U n .

Definition 4

Let ( X , U ) be uniform space. A subset B of U is by definition basis of U if any entourage includes a set from B .

In this case

( B I ) the intersection of two sets from B includes a set from B ;

( U I ′ ) Δ is included in any set from B ;

( U I I ′ ) ∀ U from B ∃ V in B so that V ⊂ U − 1 ;

( U I I I ′ ) ∀ U from B ∃ V in B so that V 2 ⊂ U ;

Reciprocally, let X be a nonempty set and B a set of nonempty subsets of X which verifies ( B I ) , ( U I ′ ) , ( U I I ′ ) and ( U I I I ′ ) . In this case, there exists U uniform structure unique on X with the property that B is basis of U . Indeed, U is the set of the parts of X ⊂ X which include a set from B .

Examples.

The set of the symmetrical entourages of a uniform structure form a basis for this. Indeed, let U be any entourage, then U − 1 is an entourage ( ( U I I ) ) and also U ∩ U − 1 ( ( F I I ) ) and U ∩ U − 1 ⊂ U .
For each real number r > 0 , let be U r ≔ { ( x , y ) ∈ R 2 : ∣ x − y ∣ < r } . The sets U r , r > 0 verify ( B I ) − ( U I I I ′ ) ( U r 1 ∩ U r 2 = U min ( r 1 , r 2 ) , U r = U − 1 r , U 2 ⊂ U 2 r ) , hence they form a basis for a uniform structure on R.
Let ( X , d ) be a metric space and, for each real number r > 0 , U r ≔ { ( x , y ) ∈ X × X : d ( x , y ) < r } . The sets U r , r > 0 form a basis for a uniform structure on X .

Definition 5

The subset S of a uniform structure U is prebasis of this when the set of finite intersections of the sets from S forms a basis for U .

Let be X ≠ ∅ . The set S of nonempty subsets of X × X forms a prebasis for a uniform structure on X iff

( U I ″ ) Δ is included in each set from S ;

( U I I ″ ) For any U from S , U − 1 includes a set from S ;

( U I I I ″ ) ∀ U from U ∃ V in B so that V 2 ⊂ U .

So the union of a family of uniform structures on X is prebasis for a uniform structure on X .

Remark 13

( U I ″ ) , ( U I I ″ ) and ( U I I I ″ ) are even the basis axioms, but without ( B I ) .

THE TOPOLOGY OF A UNIFORM SPACE

Let ( X , U ) be a uniform space. Consider, for each x from X ,

V ( x ) ≔ { U ( x ) : U ∈ U } .

V ( x ) fulfils the properties of neighborhoods of a point in a topological space. The topology associated to a uniformity is defined through the neighborhood system V ( x ) .

So, there exists a unique topology on X with the property that V ( x ) is, for any x from X , the set of neighborhoods of x (Hausdorff neighborhoods theorem). This is noted τ ( U ) . Retain: B basis for U ⇒ B ( x ) basis for V ( x ) .

Each uniform space ( X , U ) is considered endowed with the topology τ ( U ) and any topological property of this will be expressed by an added adjective to “uniform space,” for instance separated uniform space, compact uniform space, etc.

Proposition 7

( X , U ) is separated ⇔ ⋂ U ∈ U U = Δ .

Proposition 8

(Main result). Let ( X , U ) be a uniform space and M a nonempty subset of the topological space X × X . For every entourage U , U ∘ M ∘ U is a neighborhood of M and

(61) M ¯ = ⋂ U ∈ U U ∘ M ∘ U .

A sequence of consequences follows.

(62) U entourage ⇒ U 3 entourage and neighborhood of U .

(63) U entourage ⇒ U n entourage ∀ n .

(64) The open entourages form a basis.

(65) Let B be a basis of entourages and n from N . Then { U n : U ∈ B } is a basis of entourages.

(66) U entourage ⇒ U ∘ entourage .

(67) Let B be basis of entourages. Then { U ∘ : U ∈ B } and { U ¯ : U ∈ B } are basis of entourages .

(68) The set of the closed symmetrical entourages forms a basis of entourages.

Proposition 9

Let ( X , U ) be a uniform space and A a nonempty subset of X. For any entourage U, U ( A ) is a neighborhood of A and

(69) A ¯ = ⋂ U ∈ U U ( A ) .

Proposition 10

Any uniform space verifies the axiom:

“The set of closed neighborhoods of an arbitrary point forms a basis of neighborhoods of this.”

Every separated uniform space is regular.

PSEUDO-DISTANCE

Let X be a nonempty set. A mapping q : X × X → [ 0 , + ∞ ) is called pseudo-distance on X if

(P1) q ( x , x ) = 0 ∀ x ∈ X ;

(P2) q ( x , y ) = q ( y , x ) ∀ x , y ∈ X ;

(P3) q ( x , y ) ≤ q ( x , z ) + q ( z , y ) ∀ x , y , z ∈ X .

A set endowed with a pseudo-distance is called pseudo-metric space. q ( x , y ) is the pseudo-distance between x and y .

Let ( X , q ) be a pseudo-metric space. For a a point from X and r > 0 the open ball B ( a , r ) , respectively, the closed ball B r ( a ) with a center and r radius is the set { x ∈ X : q ( a , x ) < r } and { x ∈ X : q ( a , x ) ≤ r } , respectively. Every point from the intersection of two open balls is the center of an open ball included in this: x ∈ B ( a , r 1 ) ∩ B ( b , r 2 ) ⇒ B ( x , r ) ⊂ B ( a , r 1 ) ∩ B ( b , r 2 ) , where r = min ( r 1 − q ( a , x ) , r 2 − q ( b , x ) ) . So being, the set of open balls forms a basis of a topology on X − the topology defined by q . In this topology, the closed balls are closed sets.

Let ( X , q ) be a pseudo-metric space. The pseudo-distance q ( x , A ) beween the point x and the nonempty subset A is q ( x , A ) = inf { q ( x , y ) : y ∈ A } . The function x → q ( x , A ) is continuous on X . We have A ¯ = { x ∈ X : q ( x , A ) = 0 } .

Every pseudo-metric space X is normal and verifies the first axiom of countability. The second axiom of countability is verified ⇔ X is separable.

Let ( X , q ) be a pseudo-metric space. Consider, for each r > 0 , the subset of X × X , U r ≔ { ( x , y ) ∈ X × X : q ( x , y ) < r } . The sets U r verify the axioms ( B I ), ( U I ′ ), ( U I I ′ ) and ( U I I I ′ ) : U r 1 ∩ U r 2 = U r , r ≔ min ( r 1 , r 2 ) ; Δ ⊂ U r ∀ r > 0 ; U − 1 r = U r ∀ r > 0 ; U 2 r 2 ⊂ U r ∀ r > 0 . So being, there is a unique uniform structure U on X having B as basis. U is by definition the uniform structure defined by q .

A uniform space ( X , U ) is by definition pseudo-metrizable if U is defined by a pseudo-distance on X . A uniform space X is pseudo-metrizable ⇔ the uniform structure of X has a countable basis ([26], Chapter 6, 13).

Let X be a nonempty set and π a set of pseudo-distances on X . Consider, for each q from π and for each r > 0 , the subset of X × X .

U q , r ≔ { ( x , y ) ∈ X × X : q ( x , y ) < r } .

The sets U q , r , q ∈ π and r > 0 form a prebasis of a uniform structure U on X .

Proposition 11

( X , U ) is separated ⇔ ( q ( x , y ) = 0 ∀ q ∈ π ⇒ x = y ) .

Proposition 11 justifies the fact that we called π separating used when π has the property:

q ( x , y ) = 0 ∀ q ∈ π ⇒ x = y .

Actually

Theorem 9

Every uniform structure on a set X is defined by a set of pseudo-distances on X ([26], Chapter 6, 15).

FAMILY OF QUASI-DISTANCES

Let X be a nonempty set and I an ordered set. The family of functions ( q i ) i ∈ I , q i : X × X → [ 0 , + ∞ ) is called family of quasi-distances on X if

(Q1) q i ( x , x ) = 0 ∀ i ∈ I , ∀ x ∈ X ;

(Q2) q i ( x , y ) = q i ( y , x ) ∀ i ∈ I , ∀ x , y ∈ X ;

(Q3) ∀ i ∈ I , ∃ j ∈ I , i ≤ j so that q i ( x , y ) ≤ q j ( x , z ) + q j ( z , y ) ∀ x , y , z ∈ X .

If in addition

(Q4) q i ( x , y ) = 0 ∀ i ∈ I ⇒ x = y ,

then the family ( q i ) i ∈ I is by definition separating ([27]).

Remark 14

The authors remove from the axioms system from [27] (p. 399), the axiom

i ≤ j ⇒ q i ( x , y ) ≤ q j ( x , y ) ∀ x , y ∈ X

being without connection with the subject of this research.

Remark 15

Apart from the “pseudo-distance,” the concept “quasi-distance” was not defined, but that of “family of quasi-distances” was.

Let X be a nonempty set and ( q i ) i ∈ I a family of quasi-distances on X . Consider, for each i from I and for each r from the interval ( 0 , + ∞ ) , the subset of X × X

(70) U i , r ≔ { ( x , y ) ∈ X × X : q i ( x , y ) < r } .

The sets U i , r form, when i goes over I and r over ( 0 , + ∞ ) , a prebasis of a uniform structure U on X , because they verify ( U I ′ ), ( U I I ′ ), ( U I I I ′ ): Δ ⊂ U i , r , ∀ i ∈ I , ∀ r > 0 (take into account (Q1)); U − 1 i , r = U i , r ∀ i ∈ I , ∀ r > 0 (take into account (Q2)); let be U i , r , ∃ j ∈ I so that q i ( x , y ) ≤ q j ( x , z ) + q j ( z , x ) ∀ x , y , z ∈ X ((Q3)), which implies U j , r 2 ∘ U j , r 2 ⊂ U i , r . U is, by definition, the uniform structure defined by the family ( q i ) i ∈ I . In the topology τ ( U ) , for each point x from X , the sets ⋂ k = 1 n U i k , r k ( x ) , n ∈ N , i 1 , … , i n ∈ I , r 1 , … , r n > 0 form a basis of neighborhoods of x .

Similar to Proposition 11, we have

Proposition 12

( X , U ) is separated ⇔ ( q i ) i ∈ I is separating.

Actually,

Proposition 13

Any uniform structure on a set is defined by a quasi-distance family on this set.

4.2 Ekeland principle in uniform spaces (families of quasi-distances)

Definition 6

Let X be a uniform space. The sequence ( x n ) n ≥ 1 from X is a Cauchy sequence if for any symmetrical entourage V of X there is N in N so that

m , n ≥ N ⇒ ( x m , x n ) ∈ V .

X is sequentially complete if any Cauchy sequence from X is convergent.

Definition 7

Let ≼ be a preorder relation in X (i.e. the relation is reflexive and transitive). The sequence ( x n ) n ≥ 1 from X is decreasing (with respect to ≼ ) if x n + 1 ≼ x n ∀ n ≥ 1 . The interval I ( x ) ≔ { y ∈ X : y ≼ x } is sequentially lower closed when for any decreasing sequence ( x n ) n ≥ 1 from I ( x ) , if x n → x 0 then x 0 ∈ I ( x ) . The preorder relation ≼ is sequentially lower closed if, for any x from X , the interval I ( x ) is sequentially lower closed.

The following statement has the principal role in this subsection.

Theorem 10

(Minimal element; [28]) Let X be a sequentially complete separated uniform space, ≼ sequentially lower closed preorder relation in X and φ : X → ( − ∞ , + ∞ ] proper, bounded from below and increasing with respect to ≼ . In addition, for every entourage U there exists δ > 0 so that

(71) x ≼ y and φ ( y ) − φ ( x ) < δ ⇒ ( x , y ) ∈ U .

Then for any I ( x 0 ) , x 0 from dom φ , there is x ∗ in I ( x 0 ) so that

(72) { x ∗ } = I ( x ∗ )

Theorem 10 may be completed, namely:

Proposition 14

Consider, under the assumptions of Theorem 10, the multivalued function T : X → 2 ∗ X . If for any x from dom φ there is y in T ( x ) so that y ≼ x , then T has a fixed point.

If for any x from dom φ and for any y from T ( x ) we have y ≼ x , then T has an invariant point ([18], Theorem 2).

Look now at another formulation of Theorem 10.

Proposition 15

Under the assumptions of Theorem 10, for any x₀ from dom φ , if the subset M satisfies

∀ x ∈ I ( x 0 ) ⧹ M ∃ y ∈ I ( x ) ⧹ { x } ,

then I ( x 0 ) ∩ M ≠ ∅ ([18], th. 3).

Proposition 15 is indeed another formulation of Theorem 10. If we accept Proposition 15 and taking I ( x 0 ) with x 0 ∈ dom φ , the set M ≔ { x ∈ X : { x } = I ( x ) } verifies the condition from the hypothesis. Hence I ( x 0 ) ∩ M ≠ ∅ , and every element from this intersection verifies the demand from Theorem 10.

Remark 16

Proposition 15 is a variant for uniform spaces of the Oettli-Théra theorem:

Let M be a subset of X with the property: for every v ¯ from S 0 ^[1] ⧹ M , there is v in X such that v ≠ v ¯ and f ( v ¯ , v ) + d ( v ¯ , v ) ≤ 0 . Then S 0 ∩ M ≠ ∅ .

Definition 8

A function f : X → ( − ∞ , + ∞ ] , X a topological space, is called sequentially lower monotone, if for any sequence ( x n ) n ≥ 1 from X convergent to x 0 and satisfying f ( x n + 1 ) ≤ f ( x n ) for all n ≥ 1 , we have f ( x 0 ) ≤ f ( x n ) for all n ≥ 1 .

Sequential lower monotonicity is weaker than sequential lower semicontinuity.

And now

Proposition 16

(Uniform generalization of Ekeland principle). Let X be a sequentially complete separated uniform space, ( q i ) i ∈ I an afferent family of quasi-distances (61)and φ : X → ( − ∞ , + ∞ ] lower bounded, proper, sequentially lower monotone. For any ε : I → ( 0 , + ∞ ) increasing and x 0 from dom φ , there exists v ∗ in dom φ so that

(73) ∀ x from X ⧹ { v ∗ } ∃ j in I for which φ ( v ∗ ) < φ ( x ) + ε ( j ) q j ( x , v ∗ )

and

(74) φ ( v ∗ ) + ε ( i ) q i ( v ∗ , x 0 ) ≤ φ ( x 0 ) ∀ i ∈ I

[18,27].

Explanation. We present the proof of this statement, since our version from below recovers Theorem 4 from [18].

Proof

Use Theorem 10. Define on X the binary relation ≼ ,

(75) x ≼ y ⇔ φ ( x ) + ε ( i ) q i ( x , y ) ≤ φ ( y ) ∀ i ∈ I .

≼ is obviously reflexive, show that it is also transitive, i.e. a preorder relation. Let be x ≼ y , y ≼ z . If φ ( z ) = + ∞ , (75) tells that x ≼ z . Suppose φ ( z ) < + ∞ . A fortiori φ ( x ) , φ ( y ) < + ∞ . According to (75),

(76) φ ( x ) + ε ( i ) q i ( x , y ) ≤ φ ( y ) ∀ i ∈ I ,

(77) φ ( y ) + ε ( i ) q i ( y , z ) ≤ φ ( z ) ∀ i ∈ I .

By applying (Q3), one finds j ≥ i (the order relation from I ) with a certain property and then for every i we have, combining with (76) and (77),

φ ( x ) + ε ( i ) q i ( x , z ) ≤ φ ( x ) + ε ( j ) [ q j ( x , y ) + q j ( y , z ) ] ≤ φ ( x ) + φ ( y ) − φ ( x ) + φ ( z ) − φ ( y ) = φ ( z ) ,

i.e.

x ≼ z .

Prove in the following that ≼ is sequentially lower closed. Let ( x n ) n ≥ 1 be decreasing sequence with x n → x ′ . It is enough to prove

(78) ∃ N in N so that x ′ ≼ x N .

According to (75), we have

(79) φ ( x n + 1 ) + ε ( i ) q i ( x n + 1 , x n ) ≤ φ ( x n ) ∀ n ≥ 1 .

(79) gives φ ( x n + 1 ) ≤ φ ( x n ) ∀ n ≥ 1 , hence, according to the hypothesis,

(80) φ ( x ′ ) ≤ φ ( x n ) ∀ n ≥ 1 .

There are the possible cases: φ ( x n ) = + ∞ ∀ n ≥ 1 and there exists a rank from which φ ( x n ) < + ∞ . In the first case it results x ′ ≼ x n ∀ n ≥ 1 . Pass to the second case. It can suppose φ ( x n ) < + ∞ ∀ n ≥ 1 . Fix N in N . The transitivity gives m > N ⇒ x m ≼ x N , i.e.

(81) m > N ⇒ q i ( x m , x N ) ≤ 1 ε ( i ) [ φ ( x N ) − φ ( x m ) ] ∀ i ∈ I .

Let i be arbitrary fixed. ∃ j ≥ i so that q i ( x ′ , x N ) ≤ q j ( x ′ , x m ) + q j ( x m , x N ) , hence m > N ⇒ q i ( x ′ , x N ) ≤ ( 81 ) q j ( x ′ , x m ) + 1 ε ( j ) [ φ ( x N ) − φ ( x m ) ] ≤ ( 81 ) q j ( x ′ , x m ) + 1 ε ( i ) [ φ ( x N ) − φ ( x ′ ) ] ( ε is increasing).

Taking the first and the third term and then the limit for m → + ∞ , it results

q i ( x ′ , x N ) ≤ 1 ε ( i ) [ φ ( x N ) − φ ( x ′ ) ] .

As i is arbitrary, (78) is obtained.

Finally, verify (71). Let U be any entourage of X . ∃ U i k , r k , k = 1 , n ¯ (see (70)) so that ⋂ k = 1 n U i k , r k ⊂ U . Taking δ = min 1 ≤ k ≤ n r k ε ( i k ) , if x ≼ y and φ ( y ) − φ ( x ) < δ , then in particular ε ( i k ) q i k ( x , y ) ≤ φ ( y ) − φ ( x ) < r k ε ( i k ) , k = 1 , n ¯ and hence ( x , y ) ∈ ⋂ k = 1 n U i k , r k ⊂ U .

Apply Theorem 10, ∃ v ∗ in dom φ so that

v ∗ ∈ I ( x 0 ) and { v ∗ } = I ( v ∗ ) .

The first relation gives v ∗ ≼ x 0 , i.e. (74). The second relation gives (73), because x ≠ v ∗ ⇒ x ⊀ – v ∗ .□

Remark 17

For ( X , U ) = ( X , d ) , complete metric space, and ε i = ε ∀ i ∈ I in Proposition 16, one recovers the following form of Ekeland principle^[2]:

Let X be a complete metric space and φ : X → ( − ∞ , + ∞ ] bounded from below, lower semicontinuous and proper. For each u from dom φ and each ε > 0 there exists v ε in X so that

(82) φ ( v ε ) ≤ φ ( u ) − ε d ( v ε , u ) ,

(83) φ ( v ε ) < φ ( x ) + ε d ( v ε , x ) ∀ x ∈ X ⧹ { v ε } .

Remark 18

In [18], page 918, there are two mistakes:

In the statement of Theorem 4 (here Proposition 16) it must be “for every x 0 ∈ dom f ” instead of “for x 0 ∈ X .”
The proof for Theorem 4 was made for f with values only in R .

Other minor mistakes are tacitly corrected. The authors were able to recover everything.

Two statements obtained with Proposition 16 are following.

Proposition 17

(Uniform generalization of Caristi-Kirk theorem). Let X be a sequentially complete separated uniform space, ( q i ) i ∈ I an afferent family of quasi-distances (61)and T : X → 2 ∗ X . If there is φ : X → ( − ∞ , + ∞ ] lower bounded proper sequentially lower monotone and there exists α : I → ( 0 , + ∞ ) decreasing so that

(84) ∀ i ∈ I , ∀ x ∈ X ∃ y ∈ T ( x ) so that q i ( x , y ) ≤ α ( i ) [ φ ( x ) − φ ( y ) ] ,

respectively,

(85) q i ( x , y ) ≤ α ( i ) [ φ ( x ) − φ ( y ) ] ∀ i ∈ I , ∀ x ∈ X , ∀ y ∈ T ( x ) ,

then T has a fixed point respectively T has an invariant point [18].

Remark 19

For ( X , U ) = ( X , d ) , complete metric space, and α ( i ) = 1 ∀ i ∈ I , (84) gives indeed the Caristi–Kirk theorem^[3]:

Theorem 11

(Caristi-Kirk theorem). Let X be a complete metric space and T : X → 2 ∗ X a multivalued function. If there exists a lower bounded and lower semicontinuous function φ : X → R with the property: for every x from X there is y in T ( x ) such that d ( x , y ) ≤ φ ( x ) − φ ( y ) , then there exists x 0 in X with x 0 ∈ T ( x 0 ) ^[4], but with φ even with values in ( − ∞ , + ∞ ] , obviously proper.

Proposition 18

Let X be a sequentially complete separated uniform space, ( q i ) i ∈ I an afferent family of quasi-distances (61)and φ : X → R lower bounded, sequentially lower monotone. If ∃ ε : I → ( 0 , + ∞ ) increasing with the property for every x from X with φ ( x ) > inf φ ( X ) there exists y ∈ X ⧹ { x } so that φ ( y ) + ε ( i ) q i ( y , x ) ≤ φ ( x ) ∀ i ∈ I , then there is x ∗ in X so that φ ( x ∗ ) = inf φ ( X ) [18].

Proof

It can obviously suppose φ not constant. Consider the multivalued function T : X → 2 ∗ X ,

T ( x ) = { y ∈ X : φ ( y ) + ε ( i ) q i ( y , x ) ≤ φ ( x ) ∀ i ∈ I } .

T ( x ) is, according to the hypothesis, indeed nonempty. T verifies (85) with α = 1 ε (here occurs the hypothesis “ φ with values in R ”), apply Proposition 17, there is x ∗ in X so that { x ∗ } = T ( x ∗ ) . We have φ ( x ∗ ) = inf φ ( X ) . Indeed, if ad absurdum φ ( x ∗ ) > inf φ ( X ) , then, according to the hypothesis, there exists y ∈ T ( x ∗ ) ⧹ { x ∗ } , a contradiction.□

Remark 20

In Theorem 6 from [18], the hypothesis f : X → R ∪ { + ∞ } is wrong; the proof presented there does not permit to f infinite values.

4.3 Ekeland principle in uniform spaces (generalized families of quasi-distances)

Let X be a nonempty set. The function family ( q i ) i ∈ I , q i : X × X → [ 0 , + ∞ ) is called generalized family of quasi-distances if

(Q’1) q i ( x , x ) = 0 ∀ i ∈ I , ∀ x ∈ X ;

(Q’2) q i ( x , y ) = q i ( y , x ) ∀ i ∈ I , ∀ x , y ∈ X ;

(Q’3) There exists ( j 1 ( i ) ) i ∈ I , ( j 2 ( i ) ) i ∈ I subfamilies of I so that

z ≠ y and z ≠ x ⇒ q i ( x , y ) ≤ q j 1 ( i ) ( x , z ) + q j 2 ( i ) ( z , y ) i ∈ I .

If in addition

(Q’4) q i ( x , y ) = 0 ∀ i ∈ I ⇒ x = y , the family ( q i ) i ∈ I is by definition separating.

Let X be a nonempty set and ( q i ) i ∈ I a generalized family of quasi-distances on X . Consider, for each i 1 , … , i n from I and r from the interval ( 0 , + ∞ ) , the subset of X × X

(86) U ( i 1 , … , i n , r ) ≔ { ( x , y ) ∈ X × X : q i k ( x , y ) < r , k = 1 , n ¯ } .

The sets U ( i 1 , … , i n , r ) , where n ∈ N , i 1 , … , i n ∈ I and r ∈ ( 0 , + ∞ ) , form a basis of a uniform structure U on X . Indeed, because ( B I ), ( U I ′ ), ( U I I ′ ) and ( U I I I ′ ) are verified: let be U ( i 1 , … , i m , r ) , U ( j 1 , … , j n , s ) , then if t = min ( r , s ) , we have U ( i 1 , … , i m , j 1 , … , j n , t ) ⊂ U ( i 1 , … , i m , r ) ∩ U ( j 1 , … , j n , s ) ; for ( U I ′ ) take into account (Q’1); U ( i 1 , … , i n , r ) ¯ − 1 = U ( i 1 , … , i n , r ) -use (Q’2); let be U ≔ U ( i 1 , … , i n , r ) , for

V ≔ U i 1 , … , i n , j 1 ( i 1 ) , j 2 ( i 1 ) , … , j 1 ( i n ) , j 2 ( i n ) , r 2

(see (Q’3)) we have

(87) V ∘ V ∈ U .

Indeed, let be ( x , y ) ∈ V 2 , ∃ z ∈ X so that ( x , z ) , ( z , y ) ∈ V . If either z = y , or z = x , we have ( x , y ) ∈ V , hence in particular q i k ( x , y ) < r 2 < r , k = 1 , n ¯ and hence ( x , y ) ∈ U . In the contrary case, q i k ( x , y ) ≤ q j 1 ( i k ) ( x , z ) + q j 2 ( i k ) ( z , y ) < r 2 + r 2 = r , k = 1 , n ¯ , so again ( x , y ) ∈ U and hence (87).

U is by definition the uniform structure defined by ( q i ) i ∈ I . In the topology τ ( U ) , for each point x from X the sets U ( i 1 , … , i n , r ) ( x ) , n ∈ N , i 1 , … , i n ∈ I and r ∈ ( 0 , + ∞ ) , form a basis of neighborhoods of x .

With the same proof as for Proposition 11 one obtains

Proposition 19

( X , U ) is separated ⇔ ( q i ) i ∈ I is separating.

Reciprocally,

Proposition 20

Every uniform structure on a set is defined by a generalized family of quasi-distances on this set.

MINIMAL POINT THEOREM

Let Y be a real vector space and C a convex cone with vertex in Y . The binary relation y 1 ≤ C y 2 ⇔ y 2 − y 1 ∈ C is a preorder relation in Y . ≤ C is an order relation ⇔ C is pointed (Proposition 3).

Let X be a separated uniform space, ( q i ) i ∈ I an afferent generalized family of quasi-distances on X (69) and

(88) k 0 ∈ C ⧹ ( − C ¯ ) .

Put on the set X × Y the binary relation (inspired by Subsection 2.1, (8)).

(89) ( x 1 , y 1 ) ≤ k 0 ( x 2 , y 2 ) ⇔ def y 1 + q i ( x 1 , x 2 ) k 0 ≤ C y 2 ∀ i ∈ I .

Proposition 21

≤ k 0 is a preorder relation. If in addition C is pointed, then this is an order relation.

Theorem 12

(Minimal point; [19]). Let X be a separated uniform space, ( q i ) i ∈ I an afferent generalized family of quasi-distances on X (69), Y real topological vector space, C a convex cone with vertex in Y and k 0 from C ⧹ ( − C ¯ ) .

Let A be a nonempty subset of X × Y and ( x 0 , y 0 ) from A for which A 0 ≔ { ( x , y ) ∈ A : ( x , y ) ≤ k 0 ( x 0 , y 0 ) } has the properties:

π Y ( A 0 ) is ≤ C - lower bounded;
Each ≤ k 0 -decreasing sequence from A 0 has a k 0 -minorant.

Then there exists ( x ¯ , y ¯ ) in A so that

1 ∘ ( x ¯ , y ¯ ) ≤ k 0 ( x 0 , y 0 ) ,

2 ∘ ( x , y ) ∈ A and ( x , y ) ≤ k 0 ( x ¯ , y ¯ ) ⇒ x = x ¯ .

Explanation. See the notations π X ( A ) and π Y ( A ) from (37).

Proposition 22

(Uniform variant of Ekeland principle; [19], Theorem 5). Let X be a sequentially complete separated uniform space, ( q i ) i ∈ I an afferent generalized family of quasi-distances on X (69), Y a real topological vector space, C a convex cone with vertex in Y , k 0 ∈ C ⧹ ( − C ¯ ) and φ : X → Y • ≤ C – lower bounded proper.

If ∀ x from dom φ the set

S ( x ) ≔ { u ∈ X : φ ( u ) + q i ( u , x ) k 0 ≤ C φ ( x ) ∀ i ∈ I }

is sequentially closed, then ∀ x 0 from dom φ ∃ x ∗ in dom φ so that

(90) x ∈ X ⧹ { x ∗ } ⇒ ∃ j ∈ I so that φ ( x ) + q j ( x , x ∗ ) k 0 ≰ C ( x ∗ )

and

(91) φ ( x ∗ ) + q i ( x ∗ , x 0 ) k 0 ≤ C φ ( x 0 ) ∀ i ∈ I .

Explanation. For Y • see Section 3, before 3.2.

Proof

Let be A ≔ { ( x , y ) ∈ X × Y : y = φ ( x ) } , a set obviously nonempty because φ is proper, ( x 0 , y 0 ) from A and A 0 ≔ { ( x , y ) ∈ A : ( x , y ) ≤ k 0 ( x 0 , y 0 ) } . Verify (a) and (b) from Theorem 12. (a) is ensured as φ is ≤ C -lower bounded. Pass to verify b) and let ( x n , y n ) n ≥ 1 be a ≤ C -decreasing sequence from A 0 . Prove

(92) ( x n ) n ≥ 1 is Cauchy sequence in X .

Taking into account the definition and (86), it follows to prove: ∀ U i 1 , … , i n , r ∃ N in N so that m , n ≥ N ⇒ q i k ( x m , x n ) < r , k = 1 , n ¯ and for this it is sufficient to prove (16) ∀ i ∈ I and ∀ ε > 0 ∃ N in N so that m , n ≥ N ⇒ q i ( x m , x n ) ≤ ε . Fix i and let be m ≥ n . We have ( x m , y m ) ≤ k 0 ( x n , y n ) , so

(93) y m − y 0 + q i ( x m , x n ) k 0 ≤ C y n − y 0 .

Apply the Luc functional to (93) (see the proof for Theorem 12) and denoting a p ≔ L ( y p − y 0 ) , p ≥ 1 one obtains

(94) a m + q i ( x m , x n ) ≤ a n .

But, for p any, we have y p + q i ( x p , x 0 ) k 0 ≤ C y 0 , hence y p − y 0 ≤ C 0 and hence a p ≤ 0 (Lemma 1, ( i ) and ( i i ) ), so ( a p ) p ≥ 1 is a sequence of real numbers. Moreover, this is decreasing (94) and lower bounded because φ is C -lower bounded and L C -increasing. So being, ( a p ) is convergent, hence Cauchy sequence, this imposes, via (94), (16) and hence (92). Let be x ′ = lim n → ∞ x n and n arbitrary. As m ≥ n ⇒ x m ∈ S ( x n ) (remove y 0 from (93)!), passing to the limit one obtains x ′ ∈ S ( x n ) , i.e. ( x ′ , φ ( x ′ ) ) ≤ k 0 ( x n , φ ( x n ) ) ∀ n ≥ 1 , b ) is verified.

Apply Theorem 12, ∃ ( x ∗ , y ∗ ) in A so that

(95) ( x ∗ , y ∗ ) ≤ k 0 ( x 0 , y 0 ) ,

(96) ( x , y ) ∈ A and ( x , y ) ≤ k 0 ( x ∗ , y ∗ ) ⇒ x = x ∗ .

(95) is even (91). Let be x ∈ X ⧹ { x ∗ } . Then, from (96), ( x , φ ( x ) ) ≰ k 0 ( x ∗ , φ ( x ∗ ) ) , hence ∃ j ∈ I so that φ ( x ) + q j ( x , x ∗ ) k 0 ≰ C φ ( x ∗ ) , i.e. (90).□

Remark 21

Proposition 22 is indeed a variant of the Ekeland principle, as show (90) and (91) in the particular case ( X , U ) = ( X , d ) , complete metric space, Y = R and C = [ 0 , + ∞ ) , when therefore k 0 ∈ ( 0 , + ∞ ) and ≤ C is even the order relation ≤ between the real numbers: (90) becomes x ≠ x ∗ ⇒ φ ( x ) + k 0 d ( x , x ∗ ) ≰ φ ( x ∗ ) , hence φ ( x ∗ ) < φ ( x ) + k 0 d ( x , x ∗ ) and (91) becomes φ ( x ∗ ) + k 0 d ( x ∗ , x 0 ) ≤ φ ( x 0 ) , that is (83) respectively (82) from the variant of the Ekeland principle (after Proposition 16).

Remark 22

We point out the deficiencies in the article [19]:

In the statement of Theorem 5 it was forgotten the condition “ X sequentially complete”;
In the proof of Theorem 5 the consideration of the multivalued function F is useless.

5 Applications

Ekeland variational principle has many fundamental applications to nonlinear semigroups [29], optimization [20], control theory [20], geometry of Banach spaces [30].

For optimization with regular constraints, actually to solve the problem:

Let X be a real normed space, φ : X → R with Fréchet derivative and G i : X → R , i = 1 , m ¯ of C 1 class. Consider the constrained optimization problem

inf φ ( v ) G i ( v ) = 0 , i = 1 , p ¯ G i ( v ) ≥ 0 , i = p + 1 , m ¯

the solution has been given in [20] in a theorem proven by direct application of Ekeland principle in its original form.

In control theory, we often have to minimize more than one objective function, for instance, a cost functional as well as the distance between the final state and a given point. To realize this task, we usually take as objective function a weighted sum of different objectives. The more natural way would be to study the set of efficient points of a vector optimization problem with the given objective functions.

Variants of Ekeland variational principle are very usefull results in optimization theory. They highlight that there exists an exact solution of a slightly perturbed optimization problem in a neighborhood of an approximate solution of the original problem. In [3] is derived an ε -minimum principle for suboptimal controls of multicriteria control problems from the vector-valued variational principle (see 3.2). There the objective function takes values in R m . For multicriteria stochastic control problem, one similarly obtains some results by using 3.8 [3]. For stochastic efficiency in a set, 4.12 is at the basis of determining algorithm.

6 Conclusions

This paper presents a series of statements on minimal point related to Ekeland variational principle, some variants and generalizations of them, in order to highlight links, relations, and applications, taking into account their importance.

We start with preliminaries on cones, continue with four variants of minimal point theorem followed by corresponding vector variants of Ekeland variational principle. In Section 4, other variant of minimal point theorem followed by Ekeland principle in uniform spaces (with families of quasi-distances) and another one once again followed by Ekeland principle in uniform spaces (with generalized families of quasi-distances).

The last section shows some applications of minimal point (element) theorem, especially corresponding variants of Ekeland variational principle in obtaining solutions (via numerical algorithms) in optimization theory for vector optimization problem, for suboptimal controls of multicriteria control problems from the vector-valued variational principle, for stochastic efficiency in a set.

The novelty of this work consists in juxtaposing these results, to make observations, remarks, and comments, and finally to give two improved proofs for two of the given statements and to highlight appropriate applications.

Funding information: The open access was supported by University POLITEHNICA of Bucharest.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The author states no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] I. Ekeland, On the Variational Principle, Cahiers de mathématique de la décision, No. 7217, Université Paris, 1972. Search in Google Scholar

[2] Y. Han and X. H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett. 28 (2014), 38–41. 10.1016/j.aml.2013.09.006Search in Google Scholar

[3] A. Göpfert, C. Tammer, H. Riahi, and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York Inc, 2003. Search in Google Scholar

[4] J. P. Dauer and O. A. Saleh, A characterization of proper minimal points as solutions of sublinear optimization problems, J. Math. Anal. Appl. 178 (1993), 227–246. 10.1006/jmaa.1993.1303Search in Google Scholar

[5] I. Meghea, Ekeland Variational Principle: With Generalizations and Variants, Old City Publishing Philadelphia, Éditions des Archives Contemporaines, Paris, 2009. Search in Google Scholar

[6] I. Meghea, On a theorem of variational calculus with applications, BSG Proc. 19 (2012), 66–81. Search in Google Scholar

[7] I. Meghea, Ekeland variational principle in differential geometry, in: D. Andrica, S. Moroianu (eds), Contemporary Geometry and Topology and Related Topics, Cluj University Press, Cluj, 2008, pp. 209–218. Search in Google Scholar

[8] I. Meghea, Solutions for mathematical physics problems involving the p-Laplacian and the p-pseudo-Laplacian for issues evolved from modeling some real phenomena, in print. Search in Google Scholar

[9] I. Meghea and C. S. Stamin, On a problem of mathematical physics equations, Bull. Transilv. Univ. Brasov Ser. III. Math. Comput. Sci. Phys. 11(60) (2018), no. 2, 169–180. Search in Google Scholar

[10] I. Meghea, Variational approaches to characterize weak solutions for some problems of mathematical physics equations, Abstr. Appl. Anal. 2016 (2016), 2071926. 10.1155/2016/2071926Search in Google Scholar

[11] I. Meghea, Two solutions for a problem of mathematical physics equations, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 3, 41–58. Search in Google Scholar

[12] I. Meghea, Weak solutions for the pseudo-Laplacian using a perturbed variational principle, BSG Proc. 17 GPB 2010 (2010), 140–150. Search in Google Scholar

[13] I. Meghea, Some results obtained in dynamical systems using a variational calculus theorem, BGS Proc. 15 GPB 2009 (2009), 91–98. Search in Google Scholar

[14] I. Meghea, On some perturbed variational principles: connexions and applications, Rev. Roumaine Math. Pures Appl. 54 (2009), no. 5–6, 493–511. Search in Google Scholar

[15] I. Meghea and V. Stanciu, Existence of the solutions of forced pendulum equation by variational methods, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 71 (2009), no. 4, 115–124. Search in Google Scholar

[16] A. Göpfert, C. Tammer, and C. Zălinescu, On the vectorial Ekeland ’s variational principle and minimal points in product spaces, Nonlinear Anal. 39 (2000), 909–922. 10.1016/S0362-546X(98)00255-7Search in Google Scholar

[17] A. B. Németh, A nonconvex vector-minimization problem, Nonlinear Anal. 10 (1986), no. 7, 669–678. 10.1016/0362-546X(86)90126-4Search in Google Scholar

[18] A. Hamel, Equivalents to Ekelandas variational principle in uniform spaces, Nonlin. Anal. 62 (2005), 913–924. 10.1016/j.na.2005.04.011Search in Google Scholar

[19] A. Benbrik, A. Mbarki, S. Lahrech, and A. Ouhab, Ekelandas principle for vector-valued maps based on the characterization of uniform spaces via families of generalized quasi-metrics, Lobachevskii J. Math. 21 (2006), 33–44. Search in Google Scholar

[20] I. Meghea, On some perturbed variational principles: connexions and applications, Rev. Roumaine Math. Pures Appl. 54 (2009), no. 5–6, 493–511. Search in Google Scholar

[21] R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, 2nd edition, Springer Verlag, Berlin Heidelberg, 1993. 10.1007/978-3-662-21569-2Search in Google Scholar

[22] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York Chichester Brisbane Toronto Singapore, 1984. Search in Google Scholar

[23] G. Isac, The Ekeland principle and the Pareto epsilon-efficiency, Lecture Notes in Economics and Mathematical Systems, vol. 432, Springer Verlag, Berlin, 1996, pp. 148–163. 10.1007/978-3-642-87561-8_12Search in Google Scholar

[24] A. B. Németh, The nonconvex minimization principle in ordered regular Banach spaces, Mathematica (Cluj) 23 (1981), 43–46. Search in Google Scholar

[25] D. T. Luc, Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems 319, Springer Verlag, Berlin Heidelberg New York, 1989. 10.1007/978-3-642-50280-4Search in Google Scholar

[26] J. L. Kelley, General Topology, D. Van Nostrand Company, Inc. Princeton, New Jersey Toronto London New York, 1957. Search in Google Scholar

[27] J. X. Fang, The variational principle and fixed point theorem in certain topological spaces, J. Math. Anal. Appl. 202 (1996), 398–412. 10.1006/jmaa.1996.0323Search in Google Scholar

[28] A. Hamel, Phelp’s lemma, Daneš drop theorems and Ekeland ’s principle in locally convex spaces, Proc. Amer. Math. Soc. 131 (2003), 3025–3038. 10.1090/S0002-9939-03-07066-7Search in Google Scholar

[29] H. Brezis and F. Browder, A general ordering principle in nonlinear functional analysis, Adv. Math. 21 (1976), 355–364. 10.1016/S0001-8708(76)80004-7Search in Google Scholar

[30] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–474. 10.1090/S0273-0979-1979-14595-6Search in Google Scholar

Received: 2021-07-23

Revised: 2022-05-05

Accepted: 2022-05-30

Published Online: 2022-08-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0024

Keywords for this article

minimal point; Ekeland variational principle; Luc functional; convex cone; uniform space; equilibrium problem; location problem; multicriteria control problem; multicriteria fractional programming problem; stochastic efficiency

Creative Commons

BY 4.0