Remarks on Multivalued Variants of Ekeland Principle with Applications

Abstract

This review provides an extended discussion on multivalued variants of Ekeland’s variational principle, with the aim of highlighting the importance of this special type of result. After a presentation of the background containing general problems which appear in relation to this extended and current subject, three series of results are developed to describe three multiple-valued versions of this important principle. In this part of the work, some different or improved proofs have been given, along with several special remarks from the author, together with their demonstrations. The discussion section contains an overview of many applications of such multivalued variants in solving problems issuing from real phenomena modeling.

1. Introduction

The Ekeland variational principle, discovered in 1972 [1], has, over the last 50 years, become the foundation of modern variational calculus. Its applications are current, numerous, and varied, and include optimization and optimal control, mathematical programming, fixed-point theorems, dynamical systems, geometry of Banach spaces, nonlinear analysis, differential equations and partial differential equations, global analysis, probabilistic analysis, differential geometry, and nonlinear semigroups.

In the context of variational calculus, the elementary proposition, “If φ: X → R, X real normed space, has in $x_0$ a local minimum point (hence in particular a global minimum point) and it is Gâteaux differentiable at $x_0$, then ${\mathrm{\phi }\mathrm{\prime }}_w$($x_0$) = 0” ($x_0$ is critical point) ${\mathrm{\phi }\mathrm{\prime }}_w$ is called variational principle ([1]; w comes from weak). This is the reason why the neighbor propositions, for instance, those in which X is replaced by a metric space, or in which the statement “${\mathrm{\phi }\mathrm{\prime }}_w$($x_0$) = 0” is replaced by “$‖{{\mathrm{\phi }}^{\mathrm{\prime }}}_w(x_0)‖\le$ε, ε > 0 any”, etc., are called variational principles (perturbed). The adjective “perturbed” is imposed by the fact that the function φ is not minimized, but a function of the form φ + ε$‖\left( · \right)-x_{\epsilon }‖$ (Ekeland), or of a more general form. Roughly speaking, the Ekeland variational principle guarantees the existence of an almost-best solution that is stable against small changes—a powerful tool for proving the existence of solutions in nonsmooth or infinite-dimensional problems. Intuitively, imagine we are standing in a very large and uneven landscape—with hills and valleys everywhere. We want to find the lowest valley (the global minimum of a function), but maybe the landscape is messy (the function is not smooth or has no classical minimum). The Ekeland principle affirms the following: Even if we cannot find the true lowest valley, we can always find a point that is almost the lowest and that also has a very useful stability property: we cannot move away from it without going noticeably uphill.

Multivalued variants of the Ekeland principle represent special kinds of versions of this important result, which are involved in optimization (with multiobjective problems, fuzzy optimization, robust, interval-valued control), equilibrium problems (economic models with incomplete or uncertain preferences, variational inequalities with uncertainty), mechanics (contact with friction, hemivariational inequalities, nonsmooth behavior in solid mechanics), partial differential equations (reaction–diffusion systems with set-valued or discontinuous sources, nonsmooth porous media dynamics), artificial intelligence and learning (training fuzzy neural networks, learning under uncertain or interval-valued loss functions). Such a wide range of applicability makes this subject both very topical and important.

Many multivalued variants of the Ekeland principle, as well as equivalent theorems like Takahashi and Caristi’s results, are represented as the target of various papers, such as [2,3,4,5]. Regarding their application, one can mention interesting papers such as [6], where the existence of optimal solutions for multiobjective problems and the necessary conditions for optimality and suboptimality in constrained multiobjective problems are determined involving set-valued variants of the Ekeland principle. In the work [7], a multivalued version of the Ekeland principle was used to obtain a fixed-point theorem for generalized set-valued contractions in partial metric spaces. The existence results of error bounds for inequality systems defined by finitely many lower semicontinuous functions are obtained via several multivalued function versions of Ekeland’s variational principle in [8]. Applications for optimization theory also for a fixed-point theory have been obtained by using a variant of the Ekeland principle through fuzzy quasi-metric spaces, as presented in [9]. A special kind of multivalued Ekeland principle has been involved to sustain the capability theory of well-being via variational rationality in paper [10].

In connection with the relationship between set-valued versions of the Ekeland principle and symmetry, one can mention that this is rather less common. A multiple-valued principle was obtained under the setting that the functional under study is invariant to the action of a group, as the recent and interesting work [11] has proven. Multivalued Ekeland-type principles are often applied in settings where the underlying function space or the variational problem possesses symmetry—such as invariance under group actions (these are rotations, reflections, or permutations). Multivalued variants of the Ekeland variational principle are related, directly or indirectly, with concepts of asymmetry, unsymmetry, and quasi-metric spaces, having interesting applications in optimization, where the cost of passage depends on direction (e.g., traffic, networks, asymmetric cost behaviors), and disputes in economic or dynamic systems, where distance/losses are not reciprocal.

Set-valued versions in asymmetrically ordered spaces are studied in [12], with implication in cases where the order or preferences are not reciprocal (e.g., uncertain preferences, non-symmetric rules in optimization); the principle still provides ε-optimal solutions robust to asymmetry. Another multivalued variant is proposed in [13], and is applied for important results in duopoly markets, such as for the empty intersection of production sets, profit maximization over a longer period, different types of strategic behaviors, leader–follower relations, etc. Otherwise, these variants are used in optimization, economic equilibrium, and systems with asymmetric preferences/structures. Generalizations of the Ekeland principle in special spaces (without standard metrics) appear in [14], with applications in multiobjective optimization, inclusively set-valued optimization involved in nonlinear equilibrium systems, problems of fuzzy control, and mathematical economics.

Symmetry or asymmetry is a central point where Ekeland variational principle variants show their real depth. Concerning the multiplicity of solutions, symmetry leads to families of solutions (by group invariance), and Ekeland principle variants are a systematic way to extract them. A selection of physically relevant solutions is related to asymmetry forces solution branches to bifurcate or concentrate in preferred regions; this principle ensures that we can still find approximate minimizers that reflect these asymmetries. Considering its robustness, the flexibility of the Ekeland variational principle (since it works for nonsmooth, nonconvex, and set-valued maps) makes it the “bridge” that survives whether the problem has symmetry or not. Symmetry-adapted Ekeland principle variants allow us to exploit invariances to obtain structured families of solutions, while asymmetry-tolerant variants of the principle ensure existence and stability when symmetry is broken. This balance is central in nonlinear partial differential equations, variational inequalities, and applications like contact mechanics.

Taking into account the importance of results of this kind, this paper discusses several multivalued variants of the Ekeland principle. These are preceded by an extended theoretical framework and, in certain key places, include alternative (original) proofs and interesting applications. The aim of this work is not only to present new or improved proofs of the selected multivalued variants, but also to explore their interesting connections. Such contributions gain particular value in a review paper when the overall picture of their applications and possibilities for numerical implementation is highlighted, and their relevance is clearly demonstrated.

The structure of the work is the following: in the next section, the necessary notions and basic results are displayed; the third section contains the multivalued versions under consideration; the fourth section is dedicated to discussion, while the last section is devoted to conclusions.

2. Materials and Methods: Theoretical Background and Conceptual Framework

In this section, the basic elements are presented to provide a comprehensive theoretical framework for developing the considered multivalued variants that provide the subject matter for the consideration and discussion performed in this paper.

2.1. Basic Aspects for the First Considered Multivalued Variant of Ekeland Principle

2.1.1. Preliminaries of Affine Geometry

Definitions. 

Let X, Y be nonempty sets, and f : X → 2^Y.

dom f : = {x ∈ X: f (x) ≠ ∅}.

f is called proper if dom f ≠ ∅.

Suppose Y is a topological space. f has compact values when f (x) is compact ∀x ∈ X.

Assume X and Y are topological spaces, and let $x_0$ be from X and f : X → Y. f is upper semicontinuous (u.s.c.) in $x_0$ if, for any neighborhood V of f ($x_0$), there exists U neighborhood of $x_0$, such that ∀x ∈ U,

f (x) ⊂ V.

(2.1)

Let E be a real vector space. The nonempty subset C of E is named cone when, for any λ > 0, we have λC ⊂ C, which is equivalent to the following: for any μ > 0, we have μC = C.

The cone C is, by definition, a cone with vertex when 0 ∈ C. This is obviously characterized by the following: for any λ ≥ 0, we have λC ⊂ C.

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

Here, a few results are introduced relative to the cones which will be further used.

Proposition 2.1.

Let C be a cone. C is connected iff C + C ⊂ C.

Proposition 2.2.

Let C be a convex cone with vertex. C is pointed iff C ∩ (−C) = {0}.

Let X be a set, Y be a real vector space, and C be a cone from Y. f is C-lower bounded if there is $y_f$ in X such that

$$f(x) \subset y_f+C\forall x\in X.$$

(2.2)

When dom f = ∅, f is, by definition, C-lower bounded.

Finally, let X be a set, Y be a real locally convex space, C be a closed pointed convex cone with vertex in Y, and $k_0$ from int C. x* from X is ε-solution of f, where ε > 0, if there is y* in f (x*) such that

[f (x*) − y*] ∩ [−C\{0}] = ∅

(2.3)

and

$$[f (x)-y*+\mathrm{\epsilon }{k}_0]\cap [-C\backslash \{0\}]=\varnothing \forall x\in X\backslash \{x*\}.$$

(2.4)

Denote the set of ε-solutions of f by ε − S(f).

Remark 2.1.

For ε = 0, (2.3) defines x* as an efficient solution of f (H.W. Corely). The adjective “efficient” is in accordance with the definition of the efficient point, i.e., for E, B being proper subsets of Y, the element $y_1$ from E is an efficient point of E with respect to B when

y₁ ∉ y + (B\{0}) ∀y ∈ E,

(2.5)

since this can be written under the form

y − y₁ ∉ − B\{0} ∀ y ∈ E.

(2.6)

In the following, some considerations are presented, for which we appeal to [15].

2.1.2. Other Basic Elements

Proposition 2.3.

Let E be real vector space.

• $1^0$ If E is preordered, then the set

  C : = {x ∈ E: x ≥ 0}

  (2.8)

  is a convex cone with vertex.
• $2^0$ Let C be a convex cone with vertex in E. The binary relation

  $$x \stackrel{C}{\le }y\stackrel{\mathit{def}}{\iff }y-x\in C$$

  (2.9)

  is a preordered relation in E compatible with the structure of vector space, and

  $$C=\{x \in E: x \stackrel{C}{\ge }0\}.$$

  (2.10)

  $\stackrel{C}{\le }$is an order relation iff C is pointed.

Definition. 

C from the above statement is called the positive cone of E.

Explanation. 

Preorder relation means reflexivity + transitivity.

Definition. 

Let Y be a real topological vector space and C be a convex cone in Y. The nonempty subset M of Y has the dominant property (DP) if, for each y from M ∃ z in Eff (M, C), (Eff (E, B) = the set of the points from M which verify (2.6); here, it is not required M and C proper) such that

$$z \stackrel{C}{\le }y$$

(2.11)

Obviously,

M has the property (DP) ⇔ M ⊂ Eff (M, C) + C.

(2.12)

The following was affirmed:

Proposition 2.4.

If Y is locally convex and C is a closed pointed convex cone with vertex, any nonempty compact subset of Y has the (DP) property.

Proof. 

Let M be the same as in the above statement and x be any in M.

M_x: = M ∩ (x − C)

(2.13)

being compact, it is C-complete ([15], Ch. 2, Lemma 3.5, 2)); apply Theorem 4.3, i), Ch. 2 [15], correctly, since $\overline{C}+C$\{0} ⊂ C, because C is closed and convex (Proposition 2.1). □

And now, let X be a set, Y be a real locally convex space, C be a closed pointed convex cone with a vertex, and $k_0$ be from int C.

Proposition 2.5.

If f : X → 2^Y is proper, with compact values and C-bounded from below, then, for any ε > 0, there exists x* in dom f and y* in f (x*) such that

[f (x*) − y*] ∩ [−C\{0}] = ∅

(2.14)

and

$$[f (x)-y* + \mathrm{\epsilon }{k}_{0 }]\cap (-C)=\varnothing \forall x\in \mathrm{dom}f$$

(2.15)

Ref. [16].

Proof. 

We begin by observing the following:

“∀x* ∈ dom f ∃ y* ∈ f (x*) such that (2.14) is verified”.

Indeed, with f (x*) being nonempty compact, it follows, via Proposition 2.4, that in particular ∃y* ∈ Eff (f (x*), C), i.e., (see (2.6))

y − y* ∉ − C\{0} ∀ y ∈ f (x*),

which is equivalent to (2.14).

Pass to the proof of the statement and suppose ad absurdum the contrary, that is, taking into account the remark from the beginning, that ∃${\epsilon }_0$ > 0 such that ∀x ∈ dom f and ∀y ∈ f (x) ∃ z ∈ dom f, having the property

$$[f (z)-y+{\epsilon }_0{k}_0]\cap (-C)\ne \varnothing$$

(2.16)

Let $x_1$ be from dom f and $y_1^{\prime }$ be from Eff (f ($x_1$), C) (see Proposition 2.4). According to (2.16), ∃$z_1$ ∈ dom f and ∃$y_2$ ∈ f ($z_1$) such that

$$y_2{-y}_1^{\prime }+{\epsilon }_0{k}_0\in -C.$$

(2.17)

f ($z_1$) being nonempty compact, for $y_2$ ∃$y_2^{\prime }$ in Eff (f ($z_1$), C) such that (Proposition 2.4)

$$y_2^{\prime } - y_2 \in - C.$$

(2.18)

By adding (2.17) and (2.18), we find (see Proposition 2.1)

$$y_2^{\prime }-y_1^{\prime }+{\epsilon }_0{k}_{0 }\in - C.$$

(2.19)

Repeating the reasoning with $y_2^{\prime }$ and f ($z_1$), ∃$z_2$ ∈ dom f and $y_3$ ∈ f ($z_0$) such that $y_3$ − $y_{2 }^{\prime }$ + ${\epsilon }_0{k}_{0 }$ ∈ − C. Take in addition $y_3^{\prime }$ from Eff (f ($z_2$), C) such that $y_3^{\prime }$ − $y_3$ ∈ − C and add, etc. So, we obtain a sequence $(y_i^{\prime }{)}_{i\ge 1}$ having the property $y_{i }^{\prime }-y_{i-1}^{\prime }+{\epsilon }_0{k}_{0 }$ ∈ − C ∀i ≥ 2, and hence ${\sum }_{i=2}^n(y_{i }^{\prime }-y_{i-1 }^{\prime }+{\epsilon }_0{k}_0)$ = $y_{n }^{\prime }-y_{1 }^{\prime }$ + (n − 1)${\mathrm{\epsilon }}_0{k}_{0 }$ ∈ − C ∀n ≥ 2,

$${\displaystyle \frac{1}{n-1}}\left(y_{n }^{\prime }-y_{1 }^{\prime }\right)+{\epsilon }_0{k}_{0 }\in -C \forall n\ge 2.$$

(2.20)

But f being C-lower bounded, ∃$y_f$ in Y such that $y_{n }^{\prime }-y_f\in C\forall n\ge 1$, so $y_f-y_{n }^{\prime }\in -C$ and taking into account that ${\displaystyle \frac{1}{n-1}}\left(-C\right)\subset -C$, consequently, by adding ${\displaystyle \frac{1}{n-1}}(y_f-y_{n }^{\prime })$ and (2.20),

$${\displaystyle \frac{1}{n-1}} (y_f-y_1^{\prime })+{\epsilon }_0{k}_{0 }\in -C\forall n\ge 2$$

(2.21)

(involving Proposition 2.1). □

In passing to the limit for n → ∞, one obtains, as C is closed, ${\epsilon }_0{k}_{0 }$ ∈ − C. But ${\epsilon }_0{k}_{0 }$ ∈ C, and consequently ${\mathrm{\epsilon }}_0{k}_{0 }$ = 0, C being pointed (Proposition 2.2). As ${\epsilon }_0$ > 0, a fortiori $k_{0 }$ = 0, a contradiction, since $k_{0 }$∈ int C. □

Remark 2.2.

(relative to Theorem 2.1, [16]). In the second part of Proposition 2.5, the true assertion is ∀x ∈ dom F not ∀x ∈ X\{x*}. Additionally and moreover, the proof was completed by the author.

Remark 2.3.

The reference to [15] for Lemma 2.1, [16] (here Proposition 2.4) is completed by combining several statements from [15] in order to obtain this result.

Proposition 2.5 may be reformulated, X, Y, and C being as in this statement.

Proposition 2.6.

If f : X → 2^Y is proper, with compact values and C-lower bounded, then, for any ε > 0, we have

$$\mathrm{\epsilon }-S(f)\ne \varnothing .$$

(2.22)

2.1.3. Luc Functional

Definitions. 

Let E be a vector space on K, K = R or K = C, and x, y from E, x ≠ y. The closed interval [x, y] is defined by the relation:

[x, y] = {(1 − λ)x + λy: λ ∈ [0,1]}.

We obtain other intervals denoted [x, y), (x, y] and (x, y) (open interval) when λ ∈ [0,1), λ ∈ (0,1] or λ ∈ (0,1), respectively. Also, the straight line d which passes through x and y is defined as follows:

d = {(1 − λ)x + λy: λ ∈ R}.

When $x_1$ ∈ d, one obtains d = {$x_1$ + λ(y − x): λ ∈ R}, since $x_1$ ∈ d ⇒ $x_1$ = (1 − ${\mathrm{\lambda }}_1$)x + ${\mathrm{\lambda }}_1$y, and we replace.

For M, a nonempty subset of E, $x_0$ in M is, by definition, an algebraic interior point if on each straight line passing through $x_0$ there exists an open interval contained in M to which $x_0$ belongs. The set of these points is called the algebraic interior of M, and it is denoted $M^i$ or aint M. The element $y_0$ from E is, by definition, an algebraic adherent point to M if there exists x in M such that [x, $y_0$) ⊂ M. The set of these points is named the algebraic closure of M, and it is denoted $M^a$. $M^a\backslash M^i$ is, by definition, the algebraic boundary of M.

One can immediately observe that $M^i$ ⊂ M, and M convex ⇒ M ⊂ $M^a$. When M is convex and $M^i$ ≠ ∅, M is called an algebraic convex field.

The following assertion holds:

When C is a convex cone with vertex and $C^i$ ≠ ∅, it results in the following:

$$C + C^i= C^i.$$

(2.23)

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

Therefore,

k₀ ∈ C\(−C).

(2.24)

For each y from Y, consider the set

$$M_y:=\{t \in \mathbf{R}: y \in t{k}_{0 }-C\}.$$

(2.25)

Proposition 2.6.

If M_y ≠ ∅, then there is t_y in R such that

M_y = [t_y, + ∞).

Consider the Luc functional L: Y → $\overline{R}$ ([15]),

$$L(y)=\left\{\begin{array}{c}\inf M_y,\hspace{0.33em}{M}_y\ne \varnothing \\ +\infty ,\hspace{0.33em}{M}_y=\varnothing .\end{array}\right.$$

(2.26)

Proposition 2.7.

$1^0$ L is lower semicontinuous, proper, sublinear, and C-increasing.

$$2^0 \{y \in Y: L(y) \le t\}=t{k}_{0 }-C;$$

L(y + λk₀) = L(y) + λ ∀y ∈ Y, ∀λ ∈ R;

L(−y) ≥ − L(y) ∀y ∈ Y;

$$L(0)=0, L(\mathrm{\lambda }{k}_{0 })=\lambda \forall \lambda \in \mathbf{R}.$$

$3^0$ Moreover, if $k_0$ ∈ aint C, L takes its values in R, and

$$\{y \in Y: L(y) < t\}=t{k}_{0 }-\mathrm{aint}C,$$

$$y_{2 } -y_1\in \mathrm{aint} C \Rightarrow L(y_1) < L(y_2)$$

Remark 2.4.

For many details related to the above results, one can consult [17], §8, 8.10.

Remark 2.5.

When $k_{0 }$ ∈ $C^i$ and int C ≠ ∅, L is continuous, since on the $k_{0 }$ − C neighborhood of 0, we have L(y) ≤ 1.

Remark 2.6.

The careful analysis of the proofs of Propositions 2.6 and 2.7, which can be followed in [17], §8, 8.9 and 8.10, shows that, if Y is only a topological vector space and if (2.24) is replaced by $k_{0 }$ ∈ C\(−$\overline{C}$), C is a convex cone with a vertex, and, in the definition (2.25) of M_y , C is replaced by $\overline{C}$ (convex cone with vertex!), then Proposition 2.6 remains true, as well as the properties $1^0$ and $2^0$ of the Luc functional from Proposition 2.7, when C is replaced by $\overline{C}$. This remark is useful in the next Section 3.

2.1.4. Classical Variants of Ekeland Principle and Caristi–Kirk Theorem

Ekeland Principle

([18,19]). Let (X, d) be a complete metric space and φ: X → (−∞, +∞] bounded from below, lower semicontinuous and proper. For any ε > 0 and u of X with

φ(u) ≤ inf φ(X) + ε

and for any λ > 0, there exists $v_{\epsilon }$ in X such that

$$\mathrm{\phi }(v_{\epsilon }) < \mathrm{\phi }(w)+{\displaystyle \frac{\epsilon }{\lambda }}d(v_{\epsilon }, w) \forall w \in X\backslash \{v_{\epsilon }\}$$

(2.27)

and

$$\mathrm{\phi }(v_{\epsilon })\le \mathrm{\phi }(u),$$

(2.28)

$$d(v_{\epsilon },u)\le \mathrm{\lambda }.$$

(2.29)

Remark 2.7.

In the context of variational calculus, the above theorem was raised up to the rank of a principle. Nevertheless, it is not justified to assess this title to the other statements.

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 ∃ y in T(x) such that

d(x, y) ≤ φ(x) − φ(y),

then ∃$x_0$ in X with

$$x_0\in T(x_0).$$

${\mathrm{x}}_0$ is called a fixed point of T.

Remark 2.8.

If T is single valued, we are in the case of the Caristi theorem in the form (2.1)′, [20], a form suggested by F.E. Browder.

Remark 2.9.

A generalization of Caristi–Kirk theorem is found in [20], and also extensively debated in [17], §9, I, 9.59 and [22].

Note. 

In a paper by W.A. Kirk [23], the Caristi–Kirk theorem is rightly named “Carsti’s theorem”.

Theorem. 

The abbreviated Ekeland principle and Caristi–Kirk theorem are equivalent [17].

Explanation. 

The abbreviated Ekeland principle means the Ekeland principle (displayed above) with ε = λ = 1 and reduced to the fundamental inequality (2.27).

2.2. Theoretical Elements for the Second Type of Analyzed Multivalued Variants of Ekeland Principle

These variants involve the Clarke normal cone to the graph of a multivalued function. Some preparations are necessary.

Clarke tangent cone. For this reason, X and Y are real Banach spaces.

Definitions. 

Let X be a real normed space, E ⊂ X, and $x_0\in \overline{E}$. The vector v of X is, by definition, a possible direction for E according to $x_0$, if there is ρ > 0 such that $x_0$ + tv ∈ E ∀t in (0, ρ). In addition, let f : E → (−∞, +∞] and $x_0$ ∈ dom f. If

$$\underset{\mathrm{t}\to 0+}{\lim }{\displaystyle \frac{f(x_0+tv)-f(x_0)}{t}}$$

exists and is finite, it is designated by $f\mathrm{\prime }\left(x_0;v\right),$ the derivative of f at $x_0$ according to the vector v or according to the direction v (directional derivative). Thus,

$$f\mathrm{\prime }(x_0;v)=\underset{t\to 0+}{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \frac{f(x_0+tv)-f(x_0)}{t}}$$

(2.30)

Suppose E convex and f convex and finite on a neighborhood V of $x_0$. Then, there exists f ′ ($x_0$; v).

Let X be a real normed space, E ⊂ X, f : E → R, $x_0\in \stackrel{o}{E}$, and v ∈ X. We set

$$f^0\left(x_0;v\right):=\overline{\underset{\begin{array}{c}x\to x_0\\ t\to 0+\end{array}}{\lim }}{\displaystyle \frac{f\left(x+tv\right)-f\left(x\right)}{t}}.$$

The upper limit obviously exists. f ⁰ ($x_0$; v) is, by definition, Clarke derivative (or the generalized directional derivative) of the function f at $x_0$ in the direction v.

Thus, by definition,

$$f^0\left(x_0;v\right)=\underset{\begin{array}{l}V\in \mathcal{V}(x_0)\\ r\in (0,+\infty )\end{array}}{\inf }\underset{\begin{array}{l}x\in V\\ t\in (0,\mathrm{r})\end{array}}{\sup }{\displaystyle \frac{f\left(x+tv\right)-f\left(x\right)}{t}}.$$

(2.31)

Let X be a real normed space, E ⊂ X, f : E → R, and $x_0\in \stackrel{\mathrm{o}}{E}$. The functional ξ from X* is, by definition, Clarke subderivative (or generalized gradient) of f in $x_0$ if

$$f^0(x_0; v)\ge \xi (v)\forall v\in X.$$

(2.32)

The comparison of the definitions (2.32) and that of the subderivative (below explanation) justifies the name “Clarke subderivative”. The set of these generalized gradients is designated by

$$\partial f (x_0).$$

Some even call ∂f (x₀) the Clarke subderivative at $x_0$.

Explanation. 

Let $x_0$ be a point of dom f_. The functional ξ from X* is, by definition, a subderivative of f in $x_0$, if f (x) − f ($x_0$) ≥ ξ(x − $x_0$) ∀x ∈ E. Sometimes, it accepts the name ξ subgradient of f in $x_0$. The set of subderivatives of f in $x_0$ is denoted by ∂ f ($x_0$). Sometimes, ∂ f ($x_0$) is even called the subderivative in $x_0$. When ∂ f ($x_0$) ≠ ∅, f is, by definition, subdifferentiable in $x_0$

Let X be a real normed space, E ⊂ X, f : E → R, $x_0$ ∈ $\stackrel{\mathrm{o}}{E}$. f is, by definition, regular at $x_0$, if, for every v of X, there exists f′($x_0$; v) and

$$f^{\prime }(x_0; v)=f^0(x_0;v).$$

(2.33)

Let M be a nonempty subset of X, $d_M$: X → R, the function x → d(x, M) (d the distance afferent to the norm), and $x_0$ from M. The vector v from M is tangent to M in $x_0$ if

$$d_M^0(x_0; v)=0$$

(the Clarke derivative of $d_M$ at $x_0$ in the direction v; the function $d_M$ is Lipschitz of constant 1 and convex). The set of tangent vectors to M at $x_0$ is denoted T($x_0$; M), the Clarke tangent cone to M at $x_0$. T($x_0$; M) is a closed convex cone with a vertex (see in Section 2.1.1).

Proposition 2.8.

Let M be convex. Then,

$$v \in T(x_0; M) \iff d_M^0({\mathrm{x}}_0; v) =d_M^{\prime }(x_0;v)=0.$$

Proposition 2.9.

(Intrinsic characterization). v is tangent to M at $x_0$ ⇔ ∀ sequence ${(x}_n{)}_{n\ge 1}$ from M with $x_n$ →$x_0$ and ∀ sequence ${(t}_n{)}_{n\ge 1}$ from (0, +∞) with $t_n$ ↓ 0 ∃ a sequence ${(v}_n{)}_{n\ge 1}$ from X with $v_n$ → v such that $x_n$ +$t_n{v}_n$ ∈ M ∀n ≥ 1 ([24], 2.4.5).

Remark. 

For the explicit proofs of the above two results, one can consult [17].

Corollary. 

Let X = $X_1$ × $X_2$, $X_1$and $X_2$ Banach spaces, and x = ($x_1$, $x_2$) from $M_1$ × $M_2$, $M_i$ subset of $X_i$, i = 1, 2. Then

$$T(x; M_1\times M_2)=T(x_1; M) \times T(x_2;M),$$

$$N(x; M_1\times M_2)=N(x_1; M_1) \times N(x_2; M_2)$$

(for the second relation, see the next relation (2.34)).

Clarke normal cone. Let M be a nonempty subset of X and $x_0$ from M. The Clarke normal cone to M at $x_0$, N($x_0$; M), is

$$N(x_0; M)=\{\mathrm{\xi } \in X*: \mathrm{\xi }(v) \le 0 \forall v \in T(x_0;M)\}.$$

(2.34)

We have (Clarke subderivative, Proposition A6, $d_M$ is convex and Lipschitz)

$$N\left(x_0; M\right)=\mathrm{c}\mathrm{l}\left\{\bigcup _{\lambda >0}\mathrm{\lambda }\partial d_M\left(x_0\right)\right\},$$

(2.35)

where cl denotes the *-weak closure ([24], 2.4.2).

Clarke contingent cone and regular sets. Let M be a nonempty subset of X and $x_0$ from M. The Clarke contingent cone K($x_0$; M) is ($S_1$(0) = {x ∈ X: ||x|| ≤ 1}):

$$K(x_0; M) = \{v \in X: \forall \mathrm{\epsilon } > 0 \exists t \in (0, \mathrm{\epsilon }) \mathrm{and} \exists w \in v+ \mathrm{\epsilon }{S}_1(0), \mathrm{such} \mathrm{that} x_0+tw\in M\}.$$

(2.36)

Obviously, $x_0$ ∈ $\overline{M}$. We have the following:

$$T(x_0; M) \subset K(x_0;M)$$

(2.37)

(use Proposition 2.9).

Definition. 

M is regular at $x_0$ when

$$T(x_0; M)=K(x_0; M)$$

(2.38)

Any convex nonempty set is regular at each of its points (use Proposition 2.8).

Proposition 2.10.

Let f : X → R be Lipschitz around $x_0$ and 0 ∉ ∂ f ($x_0$). Then

$$\{v \in X: f^0 (x_0; v) \le 0\} \subset T(x_0; M),$$

(2.39)

where M = {x ∈ X: f (x) ≤ f ($x_0$)}.

If f is regular at $x_0$ (see relation (2.33)), then the equality holds, and M is regular at $x_0$ (Ref. [24], 2.4.7).

Corollary. 

Let f : X → R be Lipschitz around $x_0$, 0 ∉ ∂ f ($x_0$) and M as in Proposition 2.10. Then

$$N\left(x_0; M\right)\subset \bigcup _{\lambda >0}\lambda \partial f\left(x_0\right).$$

If f is regular at $x_0$, then the equality holds.

The epigraph intervention. The epigraph of the function f : X → R, epi f, is the set

{(x, r) ∈ X × R: r ≥ f (x)}.

Proprosition 2.11.

Let f: X → R be Lipschitz around $x_0$. Then

a) epi f ⁰ ($x_0$; ·) = T(($x_0$, f ($x_0$)); epi f);

b) f is regular at $x_0$ ⇔ epi f is regular at ($x_0$, f ($x_0$)) ([24], 2.4.9).

Corollary. 

Let f : X → R be Lipschitz around $x_0$. Then

$$\mathrm{\xi } \in \partial f (x_0) \iff (\mathrm{\xi }, -1) \in N((x_0, f (x_0)); \mathrm{epi} f).$$

Generalized Clarke subderivative. This will be accepted also for subderivation functions with infinite values. The generalization is made via the above corollary.

Definition. 

Let f : X → R ∪ {±∞} and $x_0$ from X with f ($x_0$) finite. The generalized Clarke subderivative of f at $x_0$, ∂ f ($x_0$), is

$$\partial f(x_0)=\{\mathrm{\xi } \in X*: (\mathrm{\xi }, -1) \in N((x_0, f (x_0)); \mathrm{epi} f)\}.$$

(2.40)

Keeping the notation ∂ f ($x_0$) for Clarke subderivative is justified by above corollary.

For instance, for the function x → ||x||, we have

∂ || · || (0) = S*, where S* = {ξ ∈ ∈ X*: ||ξ|| ≤ 1}.

(2.41)

f is, by definition, regular at $x_0$, f ($x_0$) ∈ R, provided epi f is regular at ($x_0$, f ($x_0$)) (the epigraph definition remains the same for the functions with values in $\overline{\mathbf{R}}$).

The following includes some properties of the generalized Clarke subderivative.

Proposition 2.12.

Let f : X → R ∪ {±∞} be with f ($x_0$) finite. If f (x) ≥ f ($x_0$) on a neighborhood of $x_0$, then

$$0 \in \partial f (x_0).$$

Proposition 2.13.

(Corollary to the Rockafellar theorem, [24], 2.9.8). Let $f_1$, $f_2$: X → R ∪ {±∞}. If $f_1$ is finite at $x_0$ and $f_2$ is Lipschitz around $x_0$, then

$$\partial (f_1 + f_2)(x_0) \subset \partial f_1(x_0) + \partial f_2(x_0).$$

when $f_1$, $f_2$ are also regular at $x_0$, we have equality.

Definition. 

The indicator of a subset M of X is the function ${\mathrm{\psi }}_M$: X → R ∪ {+∞},

$${\mathrm{\psi }}_M=\left\{\hspace{0.33em}\begin{array}{c}0,\hspace{1em} x\in M\\ +\infty ,\hspace{0.33em}x\in X\backslash M\end{array}\right. .$$

Proposition 2.14.

Let $x_0$ be a point from M. Then

a) ∂ ${\mathrm{\psi }}_M$($x_0$) = N($x_0$; M),

b) ${\mathrm{\psi }}_M$ is regular at $x_0$ ⇔ M is regular at $x_0$.

The coderivative.

Let f : X → $2_{*}^Y$. The graph of f, Gr f, is Gr f = {(x, y) ∈ X × Y: y ∈ f (x)}. Let ($x_0$, $y_0$) be from Gr f. The coderivative of f at ($x_0$, $y_0$) is the set-valued map

$$D* f(x_0, y_0): Y^{*} \to 2^{{\mathrm{X}}^{*}}, D* f(x_0, y_0)(\mathrm{\eta })=\{\mathrm{\xi } \in X*: (\mathrm{\xi },-\mathrm{\eta }) \in N((x_0, y_0); \mathrm{Gr} f)\}.$$

(2.42)

In the following,

C is closed pointed convex cone with vertex from Y.

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 set the following:

C⁺: = {ξ ∈ Y*: ξ(y) ≥ 0 ∀y ∈ C},

C⁺⁺: = {ξ∈Y*: ξ(y) > 0 ∀y ∈C\{0}}.

Consider the order relation:

$$y_1\stackrel{C}{\le }{y}_2\stackrel{\mathrm{def}}{\iff }y-x\in C.$$

Definitions. 

Let A be a nonempty subset of Y and $a_0$ from A. Begin with the remark:

$$a_0\mathrm{is} \mathrm{a} C-\mathrm{minimal} \mathrm{element} \mathrm{of} A \iff a \notin a_0-(C\backslash \{0\}) \forall a \in A$$

(2.43)

The set of the C-minimal elements of A is denoted by ${\mathrm{\mu }}_C$(A).

a₀ is a C-properly positive minimal element of A if there exists ξ in C⁺⁺ such that
ξ(a) ≥ ξ(a₀) ∀a ∈ A.

(2.44)

The set of these is denoted by π${\mathrm{\mu }}_C$(A).

We have the following:

$$\mathrm{\pi }{\mathrm{\mu }}_C(A) \subset {\mathrm{\mu }}_C(A)$$

(2.45)

and, in the case A compact,

$${\mathrm{\mu }}_C(A) \ne \varnothing , A \subset {\mathrm{\mu }}_C(A)+\mathrm{C}, \mathrm{\pi }{\mathrm{\mu }}_C(A)\ne \varnothing$$

(2.46)

([15]).

$$A \mathrm{is} C-\mathit{bounded}\mathit{from}\mathit{below}\mathrm{if} \exists v \in Y \mathrm{such} \mathrm{that} v \stackrel{C}{\le } a,\forall a\in A, \mathrm{i}.\mathrm{e}., A\supset V+C.$$

(2.47)

A is C-weak bounded from below if ∃ M bounded subset of Y such that A ⊂ M + C. Every set C-bounded from below is C-weak bounded from below, but there are sets which are C-weak bounded from below but not C-bounded from below. These concepts coincide when int C ≠ ∅.

f is C-lower bounded if there is $y_f$ in X such that

$$f (x) \subset y_f+C \forall x\in X.$$

(2.48)

When dom f = ∅, f is, by definition, C-lower bounded.

Let be f : X → $2_{\mathrm{*}}^Y$. f is C-bounded from below resp. C-weak bounded from below when

$$f\left(X\right):=\bigcup _{x\in X}f(x)$$

(2.49)

is C-bounded from below (see also above (2.48)) resp. C-weak bounded from below.

Proposition 2.15.

Let f : X → $2_{*}^Y$ and ($x_0$, $y_0$) from Gr f.

a) ${(x}_0$, $y_0$) is a C-minimizer of f if $y_0$ ∈ ${\mathrm{\mu }}_C$(f ($x_0$));

b) ($x_0$, $y_0$) is a C-properly positive minimizer of f if $y_0$ ∈ π${\mathrm{\mu }}_C$(f ($x_0$)).

Proposition 2.16.

Let f : X → $2_{*}^Y$, $(x_0$, $y_0$) from Gr f and ε > 0, $k_0$ from C\{0}.

a) ${(x}_0$, $y_0$) is an ε$k_0$-minimizer of f if $y_0$ ∈ ${\mathrm{\mu }}_C$(f ($x_0$)) and y + ε$k_0$ $\stackrel{C}{\nleqq }$ $y_0$ ∀y ∈ f (X);

b) ${(x}_0$, $y_0$) is a C-properly positive ε$k_0$-minimizer of f if there is ξ in C⁺⁺ with ξ($k_0$) = 1 and ξ($y_0$) ≤ ξ(y) ∀y ∈ f ${(x}_0$) and ξ(y) + ε > ξ($y_0$) ∀y ∈ f (X).

The Hiriart-Urruty functional ([25]). Let A be a proper closed subset of Y. The Hiriart-Urruty functional ${\mathrm{\Delta }}_A$: Y→R is

Δ_A(y) = d(y, A) − d(y, Y\A),

where d is the distance on Y afferent to the norm.

One can affirm the following ([25]):

Proposition 2.17.

(properties of ${\mathrm{\Delta }}_A$). $1^0$ ${\mathrm{\Delta }}_A$ is Lipschitz of constant 1;

• $2^0 {\mathrm{\Delta }}_A$ is convex when A is convex and ${\mathrm{\Delta }}_A$is positive homogeneous when A is a cone with vertex;
• $3^0 {\mathrm{\Delta }}_A$(y) > 0 ⇔ y ∉ A (from definition, A is closed);
• $4^0$∂${\mathrm{\Delta }}_A$(a) = N(a; A) ∩ $S_Y^{*}$\{0} when a ∈ Fr A and int A ≠ ∅ ($S_Y^{*}$ = {ξ ∈ Y¬^*: ||ξ|| ≤ 1}).
• $5^0$We have the following:

  $$\left\{\hspace{0.33em}\hspace{0.33em}\begin{array}{c}{\mathrm{\Delta }}_{-C}(k)=d(k,-C)>0\hspace{0.33em}\forall k\in C\backslash \left\{0\right\}(k\notin -C\hspace{0.33em}\mathrm{because}\hspace{0.33em}C\hspace{0.33em}\mathrm{is}\hspace{0.33em}\mathrm{pointed});\\ \\ \partial {\mathrm{\Delta }}_{-C}\left(0\right)=C^{+}\cap S_Y^{*}\backslash \left\{0\right\}\hspace{0.33em}\mathrm{when}\hspace{0.33em}\mathrm{int}C\ne (\mathrm{take}\hspace{0.33em}a=0\hspace{0.33em}\mathrm{in}\hspace{0.33em}{4}^0);\\ \begin{array}{c}\\ {\mathrm{\Delta }}_{-C}(y_1+y_2)\le {\mathrm{\Delta }}_{-C}(y_1)+{\mathrm{\Delta }}_{-C}(y_2)\hspace{0.33em}\forall y_1,y_2\in Y\end{array}\end{array}\right.$$

  (consider the cases $y_1$, $y_2$ ∈ − C (use then Proposition 2.1), $y_1$, $y_2$ ∉ − C, $y_1$ ∈ − C, and $y_{2 }$∉ − C).
• $6^0$ The minimal element may be characterized with the Hiriart-Urruty functional:

  $$a_0\in {\mu }_C(A) \iff {\mathrm{\Delta }}_{-C}(a - a_0) > 0 \forall a \in A\backslash \{a_0\}$$

  [26].

Two other propositions are necessary ([27], lemmas 3.1 and 3.2).

Proposition 2.18.

Let $k_0$ be from C\{0} with d ($k_0$, − C) = 1 and ε > 0. Then

$$y+\mathrm{\epsilon }{k}_0\stackrel{C}{\nleqq }{\mathrm{y}}^{\prime } \Rightarrow 0 < {\mathrm{\Delta }}_{-C}(y-y^{\prime })+\epsilon .$$

Proposition 2.19.

Let f : X → $2_{*}^Y$be with compact values, upper semicontinuous and C-bounded from below. For any ε > 0 and $k_0$ from C\{0} and for any $\overline{y}$ from f (X), there exists an ε$k_0$-minimizer ($x_0$, $y_0$) of f such that

$$y_0\le \overline{y}.$$

Explanation. 

Suppose X and Y topological spaces and let $x_0$ be from X. f is upper semicontinuous (u.s.c.) in $x_0$ if for any neighborhood V of f ($x_0$) there exists U neighborhood of $x_0$ such that f (x) ⊂ V ∀x ∈ U.

2.3. Theoretical Elements for Menger PM-Space

2.3.1. Other Preliminaries

Distribution function

A function δ: R → [0,1] increasing continuously to the left with inf δ(R) = 0, sup δ(R) = 1 is called a distribution function. $\mathcal{D}$ will denote the set of these distribution functions having the property

δ(0) = 0,

which imposes a fortiori

t ≤ 0 ⇒ δ(t) = 0.

The function H: R → [0, 1],

$$H(t)=\left\{\hspace{0.33em}\begin{array}{c}0,\hspace{0.33em}t\le 0\\ 1,\hspace{0.33em}t>0\hspace{0.33em}\end{array}\right.$$

(2.50)

is, obviously, in $\mathcal{D}$, the specific distribution function.

t-Norm

A map T: [0,1] × [0,1] → [0,1] commutative, associative (T(T(x, y), z)) = T(x, T(y, z)), increasing in every variable and

T(a,1) = a ∀a ∈ [0,1],

(2.51)

T(0,0) = 0

is, by definition, t-norm (K. Menger, B. Schweizer - A. Sklar).

2.3.2. Menger PM-Space

A triplet (X, F, T) is, by definition, Menger probabilistic metric space (Menger PM-space) if X is a nonempty set, T t-norm and

$$\mathit{F}:\mathit{X}\times \mathit{X}\to \mathcal{D}$$

has the properties (F(x, y) is denoted F_xy)

F_xy = H ⇔ x = y,

(PM₁)

F_xy = F_yx ∀x, y ∈ X,

(PM₂)

F_xz (t₁ + t₂) ≥ T(F_xy (t₁), F_yz (t₂)) ∀x, y, z ∈ X, ∀t₁, t₂ ∈ R

(PM₃)

(B. Schweizer - A. Sklar).

Suppose that F_xy (t₁) = 1, F_yz (t₂) = 1. Then, F_xz (t₁ + t₂)$\stackrel{(\mathrm{P}{\mathrm{M}}_3)}{\ge }$T (1,1)$\stackrel{(2.51)}{=}$1, and since F_xz takes values in [0,1], one obtains

F_xy (t₁) = 1 and F_yz (t₂) = 1 ⇒ F_xz (t₁ + t₂) = 1.

(2.52)

Now let d: X × X → [0, + ∞), having the property

F_xy (t) = H(t − d (x, y)) ∀x, y ∈X, ∀t ∈ R.

(2.53)

Then

d is distance on X.

(2.54)

Indeed, let x = y and, ad absurdum, d (x, y) > 0. Taking t, 0 < t < d (x, y), we have on one side, F_xx (t) $\stackrel{(\mathrm{P}{\mathrm{M}}_1),(2.50)}{=}$1 and, on the other side, F_xx (t)$\stackrel{(2.53)}{=}$0, a contradiction. Reciprocally, d (x, y) = 0 $\stackrel{(2.53)}{\Rightarrow }$F_xy (t) = H(t) ∀t ∈ R$\stackrel{(\mathrm{P}{\mathrm{M}}_1)}{\Rightarrow }$x = y. Obviously d (x, y)$\stackrel{(\mathrm{P}{\mathrm{M}}_2)}{=}$d (y, x) (take t = d (x, y), t = d (y, x)). Finally, let x, y, z be any in X and t₁, t₂ be any with t₁ > d (x, y), t₂ > d (y, z). Then

$$F_{xy} (t_1) \stackrel{(2.53)}{=}H(t_1-d (x, y))=1, F_{yz}(t_2)=H(t_2-d(y, z))=1,$$

and consequently F_xz (t₁ + t₂)$\stackrel{(2.52)}{=}$1 = H(t₁ + t₂ − d (x, z)). Hence, t₁ + t₂$\stackrel{(2.50)}{>}$d (x, z) and hence d (x, y) + d (y, z) ≥ d (x, z), (2.54) is proved.

On the other hand, when d is distance on the set X, the map F defined on X × X by

F_xy (t) = H(t − d (x, y)) ∀t ∈ R

(2.55)

takes values in D and verifies (PM₁) (x = y $\stackrel{(2.55)}{\Rightarrow }$F_xy = H, F_xy = H $\stackrel{(2.55)}{\Rightarrow }$x = y (ad absurdum, take t with 0 < t < d (x, y))), (PM₂) and F_xy (t₁) = 1, F_yz (t₂) = 1 ⇒ F_xz (t₁ + t₂) = 1 [t₁ $\stackrel{(2.55)}{>}$ d (x, y), t₂ > d (y, z) ⇒ t₁ + t₂ > d (x, y) + d (y, z) ≥ d (x, z)].

Thus, via (2.54), we have the guarantee that the metric spaces theory may be included in the Menger PM-spaces theory.

Let (X, F, T) be Menger PM-space. Suppose that T has the property

$$\underset{t\in [0,1]}{\mathrm{s}\mathrm{u}\mathrm{p}}T(t, t)=1$$

(2.56)

Consider for every x from X and for every ε > 0, λ ∈ (0,1) the set

U_x(ε, λ): = {y ∈X: F_xy (ε) > 1 − λ}

(B. Schweizer - A. Sklar).

There is a Hausdorff topology T₀ on X such that, for any x from X, the sets U_x(ε, λ), ε > 0 and λ ∈ (0, 1), form a basis of neighborhoods of x. T₀ is metrizable ([28]).

From here on

Any Menger PM-space (X, F, T) will satisfy the condition

$$\underset{t\to 1-}{\mathrm{l}\mathrm{i}\mathrm{m}}T(t, a)=a \forall a \in [0, 1].$$

(2.57)

The condition (2.57) implies the condition (2.56) ([29]).

Proposition 2.20.

Let (X, F, T) be Menger PM-space and ${(x_n)}_{n\ge 1}$, ${(y_n)}_{n\ge 1}$ sequences from X. If $x_n$ $\stackrel{{\mathrm{T}}_{\mathit{0}}}{\to }$x and $y_n$$\stackrel{{\mathrm{T}}_{\mathit{0}}}{\to }$y, then

a) $\underset{n\to \infty }{\underset{\_}{\mathrm{l}\mathrm{i}\mathrm{m}}}{F}_{x_n{y}_n}$(t) ≥ F_xy (t) ∀t ∈ R;

b) When F_xy is continuous in t, we have

$$\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{F}_{x_n{y}_n}(t)=F_{xy}(t)$$

(Refs. [29,30]).

Proposition 2.21.

Let (X, F, T) be T₀-complete Menger PM-space, φ: X → R bounded from below lower semicontinuous and μ > 0. Then the binary relation in X

$$x \le y \stackrel{\mathrm{def}}{\iff } F_{xy}(t)\ge H(t-\mu (\phi (x)-\phi (y))) \forall t>0$$

(∗)

is an order relation. This has at least one maximal element [29].

3. Presentation of the Results Subjected to Analysis

In this section, the multivalued variants of the Ekeland principle, which represent the aim of this paper, are presented.

3.1. First Variant

Pass to a multivalued variant of the Ekeland principle. In the following, (X, d) is a complete metric space, Y is metrizable real locally convex space, C is a closed pointed convex cone with a vertex, and $k_{0 }$∈ int C. The order relation $\stackrel{C}{\le }$ is that from Section 2.

Theorem 3.1.

[16] Let f : X → 2^Y be proper, with compact values, C-bounded from below and u.s.c. on X. For any ε > 0 and $x_1$ from dom f, $y_1$ from f ($x_1$) which verify

$$[f(x_1)-y_1]\cap [-C\backslash \{0\}]=\varnothing$$

(3.1)

and

$$[f(x)-y_1+\mathrm{\epsilon }{k}_0]\cap [-C\backslash \{0\}]=\varnothing \forall x\in \mathrm{d}\mathrm{o}\mathrm{m} f$$

(3.2)

and for any λ > 0, there exist $x_2$ in dom f and $y_2$ in f ($x_2$) such that

$$y_2\stackrel{C}{\le }{y}_1;$$

(3.3)

$$d(x_1,x_2)\le \mathrm{\lambda };$$

(3.4)

$$[f(x_2)-y_2]\cap [-C\backslash \{0\}]=\varnothing ;$$

(3.5)

$$[f(x)-y_2+\mathrm{\epsilon }{k}_{0 }]\cap [-C\backslash \{0\}]=\varnothing \forall x\in \mathrm{d}\mathrm{o}\mathrm{m} f;$$

(3.6)

$$[f (x)- y_2+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x)k_0] \cap (-C)=\varnothing \forall x \in \mathrm{dom} f \backslash \{x_2\}.$$

(3.7)

Proof. 

The statement has consistency: for any ε > 0 ∃ $x_{1 }$ ∈ dom f, ∃ $y_1$ ∈ f ($x_1$) such that (3.1) and (3.2) are verified (see Proposition 2.6).

Consider the auxiliary multivalued function $f_1$: X →$2^Y$,

$$f_{1 }(x)=\{y\in f\left(x\right):y \stackrel{C}{\le }{y}_1\}=f(x)\cap (y_1-C).$$

(3.8)

$f_{1 }$ is proper ($y_1$ ∈ f ($x_1$), $f_{1 }$($x_1$) ≠ ∅), with compact values ($y_{1 }$− C is closed since C is closed) and C-bounded from below ($f_{1 }$(x) ⊂ f (x)!).

Now, consider the function φ: X → (−∞, +∞], where L is the Luc functional (see in Section 2.1.3):

$$\mathrm{\phi }(x)=\left\{\begin{array}{c}\underset{y\in f_1(x)}{\mathrm{i}\mathrm{n}\mathrm{f}}L(y-y_1),\hspace{0.33em}\mathrm{when} x\in \mathrm{dom}{f}_1\\ +\infty ,\hspace{0.33em}\mathrm{when} x\notin \mathrm{dom}{f}_1.\end{array}\right.$$

(3.9)

Underline that L is continuous, since $k_0$ ∈ int C (Remark 2.5). Prove that φ is lower bounded, l.s.c., and proper, in order to apply the Ekeland principle (Section 2.1.4).

$f_1$ is proper and let x ∈ dom $f_1$. Then φ(x) ∈ R as L is continuous on the nonempty compact set $f_1$(x) − $y_{1 }$. Here, there is one more argument. We have the following:

$$\mathrm{\phi }(x_1)=0.$$

(3.10)

Indeed, (3.1) gives

$$y-y_1\notin -C\backslash \left\{0\right\}\forall y\in f(x_1)$$

and, on the other side,

$$y-y_1\in -C \forall y \in f_1(x_1).$$

Consequently, y − $y_1$ = 0 on $f_1$($x_1$), and hence L(y − $y_1$) = 0 on $f_1$($x_1$) (Proposition 2.7).

Let $y_f$ be from Y such that $f_1$(x) ⊂ $y_f$ + C on X, i.e., y ∈ $y_f$ + C ∀y ∈ $f_1$(x), ∀x ∈ dom $f_{1 }$; consequently, L being C-increasing (Proposition 2.7), L(y) ≥ L($y_f$) ∀y ∈ $f_1$(x), ∀x ∈ dom $f_1$, φ is bounded from below.

Pass to the lower semicontinuity of φ. Let t be any from R and M : = {x ∈ X: φ(x) ≤ t}. One must prove that M is closed. Let $(a_n{)}_{n\ge 1}$ be a sequence from M with $a_n$ → x*. φ($a_n$) ≤ t implies (see (3.9), L attains its infimum on $f_1$($a_n$)) ∃ $b_n$∈$f_1$($a_n$) such that

$$b_n\stackrel{C}{\le }{y}_1, L(b_n-y_1)\le t.$$

(3.11)

Let δ > 0. Take a covering of f (x*) with open balls with diameter ${\displaystyle \frac{\mathrm{\delta }}{2}}$ such that each of these to cut f (x*) (Y is metrizable). For the union V of these balls, there exists U neighborhood of x* having the property relative to the upper semicontinuity of f in x*; let $a_{k_1}$ be from U ($a_n$ → x*!). Then f ($a_{k_1}$) ⊂ V and hence $b_{k_1}$ is in a ball S of the covering ($f_1$($a_{k_1}$) ⊂ f ($a_{k_1}$)!). Take an element $c_{k_1}$ from S ∩ f (x*), and then dist ($b_{k_1}$, $c_{k_1}$) < ${\displaystyle \frac{\mathrm{\delta }}{2}}$. By repetition, we find $(b_{k_n}{)}_{n\ge 1}$ subsequence of ${(b}_n{)}_{n\ge 1}$ and ${(c}_{k_n}{)}_{n\ge 1}$ a sequence from f (x*) such that dist($b_{k_n}$,$c_{k_n}$) <${\displaystyle \frac{\mathrm{\delta }}{2^n}}$. So,

$$b_{k_n}-{ c}_{k_n}\to 0.$$

(3.12)

But ${(c}_{k_n}{)}_{n\ge 1}$ has a convergent subsequence $(c_{l_{k_n}}{)}_{n\ge 1}$ (f (x*) is compact), $c_{l_{k_n}}$→ y* ∈ f (x*), and consequently $b_{l_{k_n}}$→ y* (relation (3.12)). Then, from (3.13), y* $\stackrel{C}{\le }{y}_1$ (C is closed), i.e., y* ∈ $f_{1 }$(x*). Moreover, once again via (3.11), L(y* − $y_1$) ≤ t (L is continuous), and consequently φ(x*) ≤ t, x* ∈ M, M is closed.

In the following, show that

φ(x) + ε ≥ 0 ∀x ∈ X.

(3.13)

Supposing that ad absurdum ∃ x′ such that φ(x′) + ε < 0, then ∃ y′ ∈ $f_{1 }$(x′) such that L(y′ − $y_1$) + ε $\stackrel{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 2.7}{=}$L(y′ − $y_1$ + ε$k_0$) < 0 $\stackrel{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 2.7}{=}$L(0), which implies y′ − $y_1$ + ε$k_0$ ∈ − C\ {0} (see Proposition 2.6), in contradiction with (3.2).

So, taking into account (3.13) and (3.10),

$$\mathrm{\phi }(x_1)\le \inf \phi (X)+\epsilon ,$$

apply the Ekeland principle, ∃$x_2$ in X such that

$$\mathrm{\phi }(x_2)\le 0,$$

(3.14)

$$d(x_1,x_2)\le \lambda ,$$

(3.15)

$$\mathrm{\phi }(x_2) < \mathrm{\phi }(x)+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_{2 }, x), \forall x \in X\backslash \{x_2\}.$$

(3.16)

(3.14) gives $f_1$($x_2$) ≠ ∅ (see (3.9)). ∃$y_2^{\prime }$ in $f_1$($x_2$) such that

$$\mathrm{\phi }(x_2)=L(y_2^{\prime }-y_1),$$

(3.17)

hence L($y_2^{\prime }$ − y₁) $\stackrel{(3.14)}{\le }$ 0 and ∃$y_2$ in Eff ($f_1$($x_2$), C) such that, in accordance with Proposition 2.4,

$$y_2\stackrel{C}{\le }{y}_2^{\prime }.$$

(3.18)

Consequently, $y_2-y_1$ $\stackrel{C}{\le }{y}_2^{\mathrm{\prime }}-y_1$, L($y_2$ − $y_1)\stackrel{\mathrm{Proposition}2.7}{\le }$L($y_2^{\mathrm{\prime }}-y_1$) ≤ 0. Then, according to Proposition 2.6, $y_2-y_1$ = L($y_2-y_1$)$k_0$ − u, u ∈ C and as L($y_2-y_1$)$k_0$ ∈ − C, it results $y_2-y_1$ ∈ − C − C $\stackrel{\mathrm{Proposition}2.1}{\subset }$− C, $y_2$ $\stackrel{C}{\le }$ $y_1$.

So, (3.5), (3.6), and (3.7) remain to be verified. Begin with (3.5), which means

$$y_2\notin y+(C\backslash \{0\}) \forall y \in f (x_2).$$

(3.19)

(3.19) is true ∀y ∈$f_1$($x_2$), because $y_2$ ∈ Eff ($f_1$($x_2$), C). Now, let y ∈ f ($x_2$)\$f_{1 }$($x_2$), i.e., y ∈ f ($x_2$) and y $\stackrel{(3.8)}{\notin }{y}_1$ – C, and suppose ad absurdum $y_2$ ∈ y + (C\{0}). But then $y_1\stackrel{\underset{C}{(3.3)}}{\ge }{y}_2$, $y_2\stackrel{C}{\ge }y$, hence $y_1\stackrel{C}{\ge }y$, that is y ∈ $y_1$ – C, a contradiction, and (3.19) is proved.

Pass to (3.6). Suppose $y_2$ ≠ $y_1$; for the contrary case, we send to (3.2), and suppose ad absurdum ∃ x′ ∈ dom f, y′ ∈ f (x′) such that

$$y\prime -y_2+\mathrm{\epsilon }{k}_0 \in -C\backslash \left\{0\right\}.$$

(3.20)

(3.3) gives $y_1$ = $y_2$ + u, u ∈ C\{0}, and combining with (3.20), y′ − $y_1$ + ε$k_0$ ∈ − C\{0}, which contradicts (3.2).

Finally, pass to (3.7) and let x be from [(dom f \dom $f_1$) ∪ dom $f_1$] \ {$x_2$}, for instance, x ∈ dom f \dom $f_1$. Suppose ad absurdum ∃ y′ ∈ f (x)\$f_1$(x), that is (see (3.8)),

$$y\prime \notin y_1–C$$

(3.21)

such that

$$y\prime -y_2+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x)k_0\in -C.$$

(3.22)

But –$y_2\stackrel{(3.3)}{=}-y_1$ + u, u ∈ C and, combining with (3.22), one finds, taking into account Proposition 2.1, y′ ∈ $y_1$– C, in contradiction with (3.21). The remaining case is x ∈ dom $f_{1 }$\{$x_2$}. Suppose ad absurdum

$$\exists y^{\prime } \in f_1(x) \mathrm{such} \mathrm{that} y\prime -y_2+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x)k_0\in -C.$$

(3.23)

On the other side, (3.16) and (3.17) give the following (see also (3.9)):

$$\underset{y\in f_1(x)}{\mathrm{i}\mathrm{n}\mathrm{f}}L(y-y_1) - L(y_2^{\prime }-y_1) + {\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x)>0,$$

$$L(y-y_1)-L(y_2^{\prime }-y_1)+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x) > 0, \forall y \in f_1(x).$$

(3.24)

As $y_2^{\mathrm{\prime }}-y_1\stackrel{C}{\ge }{y}_2-y_1$ (see (3.18)), we have L($y_2^{\mathrm{\prime }}-y_1$)$\stackrel{\mathrm{Proposition}2.7}{\ge }$L($y_2-y_1$), and then, via (3.24),

$$L(y-y_1)-L(y_2-y_1)+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x) > 0 \forall y \in f_1(x),$$

$$L(y-y_2)+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x) > 0 \forall y \in f_1(x),$$

as L(y − $y_1$) $\stackrel{\mathrm{Proposition}2.7}{\le }$ L(y − $y_2$) + L($y_2$ − $y_1$). Consequently (see (2.26)),

$$y-y_2+{\displaystyle \frac{\mathrm{\epsilon }}{\mathrm{\lambda }}} d(x_2, x)k_0\notin -C \forall y \in f_1(x),$$

in contradiction to (3.23). □

The following remarks are related to Theorem 3.1, which has the same name in [16].

Remark 3.1.

The topological vector space Y must be metrizable since it is involved in the assertion “any sequence which is in a compact part of Y has a convergent subsequence”.

Remark 3.2.

Demand VI must be replaced by “∀x ∈ dom F” and the assertions X and XI are true ∀x ∈ dom F, respectively, ∀x ∈ dom F\{$x_2$}.

Remark 3.3.

The assertion (4) (proof of Theorem 3.1, [16]) is not true; it is facilitated by the absence of the justification. Actually, it is true for a subsequence $({\overline{y}}_{l_{k_n}}{)}_{n\ge 1}$ of a subse-quence (${\overline{y}}_{k_n}{)}_{n\ge 1}$ of the sequence (${\overline{y}}_n{)}_{n\ge 1}$.

Remark 3.4.

Relation (3.7), i.e., the last assertion of the above theorem, actually affirms the existence of an ε$k_0$-minimizer, an idea developed in the proof of the next Proposition 3.1.

3.2. Second Type of Multivalued Variants of Ekeland Principle ([27])

We pass to multivalued variants of the Ekeland principle.

Proposition 3.1.

(Variant of Ekeland principle). Let f : X →$2_{*}^Y$ be with compact values, upper semicontinuous, C-bounded from below, and int C ≠ ∅. For any ε > 0 and $k_0$ from C\{0} with d($k_0$, − C) = 1 and for any $\overline{y}$ from f (X), there exists an ε$k_0$-minimizer (x_ε, y_ε) of f such that

1) y_ε $\stackrel{C}{\le }\overline{y}$;

2) y + ε||x − $x_0||k_0\stackrel{C}{\nleqq }$y_ε ∀(x, y) ∈ Gr f \{(x_ε, y_ε)};

3) there exists (ξ_ε, − η_ε) in N((x_ε, y_ε); Gr f) with η_ε ∈ (C⁺ ∩ $S_Y^{*}$)\{0} and ||ξ_ε|| ≤ ε ([27]).

• The assertion 3) may be expressed using the coderivative ((2.42)), $S_X^{*}$ = {ξ ∈ X*: ||ξ|| ≤ 1}):
• ∃ η_ε in (C⁺ ∩ $S_Y^{*}$)\{0} such that D* f (x_ε, y_ε)(η_ε) ∩ ε$S_X^{*}$≠ ∅.

Proof. 

Note from the beginning that Gr f is closed.

Let (x₀, y₀) be an εk₀-minimizer of f with (Proposition 2.19)

$$y_0\stackrel{C}{\le }\overline{y}.$$

(3.25)

Applying (3.7) from Theorem 3.1, one finds an ε$k_0$-minimizer (x_ε, y_ε) of f which verifies assertion (2), and also

$$y_{\mathrm{\epsilon }}\stackrel{C}{\le }{y}_0.$$

(3.26)

From (3.25) and (3.26), assertion 1) also follows.

Now, combine assertion 2) with Proposition 2.18, and one obtains

Δ_−C(y − y_ε) + ε||x − x_ε|| > 0 ∀(x, y) ∈ Gr f \{(x_ε, y_ε)}.

(3.27)

(3.27) expresses that (x_ε, y_ε) is a unique minimizer of the function g: X × Y → R ∪ {+∞},

$$g (x, y)={\mathrm{\Delta }}_{-C}(y-y_{\mathrm{\epsilon }})+\mathrm{\epsilon }\left|\right|x-x\mathrm{\epsilon } \left|\right|+{\mathrm{\psi }}_{\mathrm{G}\mathrm{r}f}(x, y)$$

(the indicator of Gr f ) and hence, applying successively Proposition 2.12, Proposition 2.13, Proposition 2.14 a), (2.41) and Proposition 2.17, $5^0$, one obtains

0 ∈ ∂ g (x_ε, y_ε)

and

$$\begin{gathered} \partial \mathrm{g} (x_{\mathrm{\epsilon }}, y_{\mathrm{\epsilon }}) \subset \left\{0\right\} \times \partial {\mathrm{\Delta }}_{-C}(0)+\mathrm{\epsilon }{S}_X^{*}\times \left\{0\right\}+N((x_{\mathrm{\epsilon }}, y_{\mathrm{\epsilon }}); \mathrm{Gr} f )=\left\{0\right\} \times (C^{+} \cap S_Y^{*}\backslash \{0\})+ \\ \mathrm{\epsilon }{S}_X^{*} \times \left\{0\right\}+N((x_{\epsilon }, y_{\epsilon }); \mathrm{Gr} f). \end{gathered}$$

Consequently, ∃ ξ_ε ∈ ε$S_X^{*}$ and ∃ η_ε ∈ (C ⁺ ∩ $S_Y^{*}$)\{0} such that (ξ_ε, − η_ε) ∈ N((x_ε, y_ε); Gr f ), that is assertion 3). □

Proposition 3.2.

(Variant of Ekeland principle). Let f : X → $2_{*}^Y$ be with compact values, upper semicontinuous and C-weak bounded from below. For any ε > 0, $k_0$ from C\{0} and η from C⁺⁺ with η($k_0$) = 1, there exists a properly positive ε$k_0$-minimizer (x_ε, y_ε) of f such that

1) y + ε ||x − x_ε ||$k_0\stackrel{C}{\nleqq }$ y_ε ∀(x, y) ∈ Gr f \{(x_ε, y_ε)};

2) ∃ ξ_ε ∈ X* such that (ξ_ε, − η) ∈ N((x_ε, y_ε); Gr f) and ||ξ_ε|| ≤ ε ([27]).

The assertion 2) may be expressed using the coderivative:

$$D* f (x_{\mathrm{\epsilon }}, y_{\mathrm{\epsilon }})(\mathrm{\eta }) \cap \mathrm{\epsilon } S_X^{* }\ne \varnothing .$$

Proof. 

We have the following:

inf {η(y): y ∈ f (X)} > − ∞.

(3.28)

Indeed, let y be any from f (X). But f (X) ⊂ M + C, M bounded, hence y = u + k, u ∈ M and k ∈ C. η being linear and continuous, it is bounded on M, so ∃ $m_0$ such that η(v) ≥ $m_0$, ∀v ∈ M, and then, as η ∈ C⁺, η(y) = η(u) + η(k) ≥ η(u) ≥ $m_0$, i.e., (3.28).

Because η is bounded from below and attains its infimum on any f (x), x ∈ X, as this is compact, one can consider the function φ: X → R, φ(x) = $\underset{y\in f(x)}{\min }$η(y). φ is lower semiconti-nuous (use f is compact-valued and u.s.c.) and, in addition, (3.28) gives φ is lower bounded. Apply the Ekeland principle (Section 2.1.4) with λ = 1, ∃ x_ε in X, such that

φ(x_ε) < φ(x) + ε ∀x ∈ X

(3.29)

and

φ(x) + ε||x − x_ε|| > φ(x_ε) ∀x ∈ X\{x_ε}.

(3.30)

Let y_ε be from f (x_ε) such that φ(x_ε) = η(y_ε) (≤ η(y) ∀y ∈ f (x_ε)). (3.29) and (3.30) give, respectively,

η(y_ε) < η(y) + ε ∀y ∈ f (X)

(3.31)

and

η(y) − η(y_ε) + ε||x − x_ε|| > 0 ∀(x, y) ∈ Gr f \{(x_ε, y_ε)}.

(3.32)

The last bracket and (3.31) tell us that (x_ε, y_ε) is a properly positive ε$k_0$-minimizer of f (Proposition 2.16, b)).

Prove 1). Suppose ad absurdum that ∃ (x′, y′) in Gr f \{(x_ε, y_ε)} such that

$$y\prime +\mathrm{\epsilon }\left|\right|x\prime -x_{\epsilon } \left|\right|k_0-y_{\epsilon } \stackrel{C}{\le } 0.$$

η being from C⁺ with η($k_0$) = 1, we have then

η(y′) − η(y_ε) + ε||x′ − x_ε || ≤ 0,

in contradiction with (3.32).

Prove 2). (3.32) expresses that (x_ε, y_ε) is a minimizer unique for the function g: X × Y → R ∪ {+∞},

$$g (x, y)=\mathrm{\eta }(y-y_{\mathrm{\epsilon }})+\mathrm{\epsilon }\left|\right|x-x_{\epsilon } \left|\right|+{\mathrm{\psi }}_{\mathrm{G}\mathrm{r} f}(x, y).$$

In the following, the proof repeats this procedure for assertion 3) from Proposition 3.1. □

3.3. Ekeland Principle and Caristi-Kirk Theorem in Probabilistic Metric Space

3.3.1. Caristi-Kirk Theorem in Probabilistic Metric Spaces

Let (X, F, T) be a $\mathcal{T}$₀ - complete Menger PM-space having the property (2.57).

Theorem 3.2.

Let f : E → X be surjective, E ⊂ X, φ: X → R bounded from below lower semicontinuous, (Φ_α)_α∈I, Φ_α : E → $2_{*}^X$ and μ > 0.

Suppose ∀x ∈ E with f (x) ∉${\bigcap }_{\mathrm{\alpha }\in I}{\mathrm{\Phi }}_{\mathrm{\alpha }}\left(x\right)$, there exist α₀ in I and y in ${\mathrm{\Phi }}_{{\mathrm{\alpha }}_0}$(x)\{ f (x)} with the property

F_f(x),y(t) ≥ H(t − μ(φ(f (x)) − φ(y))) ∀t > 0.

(3.33)

Then ∃ u in E such that

$$f(u)\in \bigcap _{\mathrm{\alpha }\in I}{\mathrm{\Phi }}_{\mathrm{\alpha }}(u)$$

(3.34)

(Ref. [31]).

Explanation. 

$2_{*}^X$ is the set of nonempty subsets of X.

Proof. 

Consider the order relation in X (see Proposition 2.21, (∗)):

$$x \le y\stackrel{\mathrm{def}}{\iff } F_{xy}(t)\ge H(t-\mu (x)-\phi (y)))\forall t>0.$$

X has a maximal element v (Proposition 2.21). f being surjective, ∃ u in X so that

v = f (u).

(3.35)

Suppose ad absurdum

$$f\left(u\right)\notin \bigcap _{\alpha \in I}{\Phi }_{\alpha }\left(u\right).$$

(3.36)

Then, according to the hypothesis, there exist ${\alpha }_0$ in I and y in ${\mathrm{\Phi }}_{{\mathrm{\alpha }}_0}$(u)\{f (u)} such that

F_{f(u), y} (t) ≥ H(t − μ(φ(f (u)) − φ(y))) ∀t > 0,

i.e., f (u) ≤ y. (3.35) imposes f (u) = y and hence f (u) ∈ ${\mathrm{\Phi }}_{{\mathrm{\alpha }}_0}$(u)\{ f (u)}, a contradiction, and (3.36) is false. □

Corollary 1.

([31]). Let f, φ, μ be as in Theorem 3.2 and Φ: E → $2_{*}^X$. Suppose that ∀x from E with f (x) ∉ Φ(x) there exists y in Φ(x)\{f (x)} with the property

F_{f(x), y} (t) ≥ H(t − μ(φ(f (x)) − φ(y))) ∀t > 0.

Then there is u in E such that

f (u) ∈ Φ(u).

Proof. 

Take in Theorem 3.2 Φ_α = Φ ∀α ∈ I. □

Remark 3.5.

In Corollary 3.2, [31], we must have y ∈ S(x)\{ f (x)} (notation from the original).

Corollary 2.

Let φ: X → R be bounded from below l.s.c. If Φ: X → X satisfies

F_{x, Φ(x)} (t) ≥ H(t − μ(φ(x) − φ(Φ(x)))) ∀x ∈ X, ∀t > 0,

(3.37)

then Φ has a fixed point.

Proof. 

Demonstration of Cantorian type. Suppose ad absurdum

x ≠ Φ(x) ∀x ∈ X.

(3.38)

Take f = Id_X and apply Corollary 1; this is correct according to (3.38) and (3.37). One obtains that Φ has a fixed point, a contradiction. □

Corollary 3.

Let (X, d) be complete metric space, φ: X → R bounded from below l.s.c., Φ: X → $2_{*}^X$, f : X → X surjective and μ > 0. Suppose ∀x from X ∃ y in Φ(x)\{ f (x)}, having the property

d (f (x), y) ≤ μ(φ(f (x)) − φ(y)).

(3.39)

Then there exists u in X such that

f (u) ∈ Φ(u).

(Ref. [31])

Proof. 

The map T₁: [0, 1] × [0, 1] → [0, 1], T₁ (x, y) = min{x, y} is t-norm, and the map F: X × X → D,

F_xy (t) = H(t − d (x, y)) ∀t ∈ R

(3.40)

verifies (PM₁), (PM₂) (see (2.55)), but also (PM₃) relative to T₁, that is, ∀ x, y, z ∈ X and ∀ t₁, t₂ ∈ R

H(t₁ + t₂ − d (x, z)) ≥ min{H(t₁ − d (x, y)), H(t₂ − d (y, z))}.

(3.41)

When the first member takes the value 1, (3.41) is verified. Suppose that it takes the value 0, i.e.,

t₁ + t₂ ≤ d (x, z).

(3.42)

If the second member is equal to 0, (3.41) is verified. Suppose ad absurdum that it is equal to 1, that is t₁ > d (x, y) and t₂ > d (y, z). We add t₁ + t₂ > d (x, y) + d (y, z) ≥ d (x, z), in contradiction with (3.42).

So, (X, F, T₁) is a $\mathcal{T}$₀ - complete Menger PM-space (attention to d!). φ remains l.s.c. But (3.39) and (3.40) give ∀x of X ∃ y in Φ(x)\{ f (x)} such that

F_{f (x), y} (t) ≥ H(t − μ(φ( f (x)) − φ(y))) ∀t > 0.

Apply Corollary 1 with E = X and one obtains the conclusion. □

Remark 3.6.

In [31]: φ must be bounded from below; y ∈ S(x)\{ f (x)}, not only y ∈ S(x).

Proposition 3.3.

Caristi-Kirk theorem (classical variant) is consequence of Corollary 3.

Proof. 

Preserve the notations from Section 2.1.4. Suppose ad absurdum x ∉ T(x), i.e., T(x) = T(x)\{x}, ∀x ∈ X; consequently, the hypothesis affirms, in fact, that ∀x of X ∃ y in T(x)\{x}, so that

d (x, y) ≤ φ(x) − φ(y).

Apply Corollary 3 with f = Id_X and μ = 1, ∃ u in X so that u ∈ T(u), a contradiction, and hence the conclusion. □

Remark 3.7.

The proof for Proposition 3.3, and also for Corollary 2, is of Cantorian type.

Remark 3.8.

Proposition 3.3 allows for the assertion that Theorem 3.2 is a probabilistic variant of the Caristi-Kirk theorem.

3.3.2. Ekeland Principle in Probabilistic Metric Spaces

Let (X, F, T) be a $\mathcal{T}$₀ - complete Menger PM-space having the property (2.57).

Theorem 3.3

(Probabilistic variant of Ekeland principle) [31]. Let φ: X → R be bounded from below lower semicontinuous and μ > 0.

For any ε > 0 and u from X with

φ(u) ≤ inf φ(X) + ε,

(3.43)

there exists v_ε ∗-maximal between the ∗-majorants of u such that

$$F_{u{v}_{\epsilon }}(t)\ge H(t-\epsilon \mu )\forall t>0$$

(3.44)

and ∀x from X\{v_ε} there exists t_x > 0 such that

$$F_{v_{\epsilon }x}(t_x)<H(t_x-\mu (v_{\epsilon })-\phi (x))).$$

(3.45)

Proof. 

First assertion. Consider the set

X_u : = {x ∈ X: F_ux (t) ≥ H(t − μ(φ(u) − φ(x))) ∀t > 0}.

(3.46)

X_u ≠ ∅ since u ∈ X_u ((PM₁)). Prove the following:

X_u is closed.

(3.47)

Let ${(x_n)}_{n\ge 1}$ be a sequence of points from X_u with

$$x_{n }\stackrel{{\mathcal{T}}_{\mathit{0}}}{\to }x*.$$

(3.48)

To obtain (3.47), we must show x* ∈ X_u, that is

F_ux* (t) ≥ H(t − μ(φ(u) − φ(x^*))) ∀t > 0.

(3.49)

Fix t > 0.

Suppose F_ux* continuous in t. We have, ∀n ≥ 1,

$$F_{u{x}_n}(t) \ge H(t-\mathrm{\mu }(\mathrm{\phi }(u)-\mathrm{\phi }(x_n))).$$

Pass to the limit for n → ∞, taking into account (3.48), Proposition 2.20, and the lower semicontinuity of φ (H is increasing),

$$F_{\mathit{ux}}* (t) \ge \underset{n\to \infty }{\underset{\_}{\mathrm{l}\mathrm{i}\mathrm{m}}}H(t-\mathrm{\mu }(\mathrm{\phi }(u)-\mathrm{\phi }(x_n))) \ge H(\underset{n\to \infty }{\underset{\_}{\mathrm{l}\mathrm{i}\mathrm{m}}}(t-\mathrm{\mu }(\mathrm{\phi }(u)-\mathrm{\phi }(x_n))))\ge H(t-\mu (\phi (u)-\phi (x*)))$$

and hence (3.49) [justify the relation $\underset{n\to \infty }{\underset{\_}{\mathrm{l}\mathrm{i}\mathrm{m}}}$H($y_n$) ≥ H($\underset{n\to \infty }{\underset{\_}{\mathrm{l}\mathrm{i}\mathrm{m}}}{y}_n$): l : = $\underset{n\to \infty }{\underset{\_}{\mathrm{l}\mathrm{i}\mathrm{m}}} y_n$, l > 0 (H(l) = 1, n ≥ N ⇒ $y_n$ > 0 ⇒ H($y_n$) = 1 → 1), l ≤ 0 (H(l) = 0, H($y_n$) ≥ 0 ∀n ≥ 1, $\underset{n\to \infty }{\underset{\_}{\mathrm{l}\mathrm{i}\mathrm{m}}}$H($y_n$) ≥ 0)].

Suppose F_ux* discontinuous in t. Taking a strictly increasing sequence ${(t_n)}_{n\ge 1}$ of continuity points of F_ux* with $t_n$ → t and working as for Proposition 2.21, (3.47) is proved.

So, (X_u, F, T) is a $\mathcal{T}$₀ - complete Menger PM-space. Consequently, according to Proposition 2.21, the binary relation in X_u

$$x \le y \stackrel{\mathrm{def}}{\iff }{F}_{xy}(t)\ge H(t-\mu (\phi (x)-\phi (y)))\forall t>0,$$

obvious restriction to X_u of the order relation (∗) (Proposition 2.21) in X, is an order relation, and it has a maximal element, let it be $v_{\epsilon }$, and the elements from X_u are ∗-majorants of u. Moreover, according to (3.43),

0 ≤ φ(u) − φ(v_ε) ≤ ε

(3.50)

And, combining (3.46) with (3.50), one obtains (3.44), H being increasing.

Second assertion. Suppose ad absurdum

$$\exists x_0\in X\backslash \{v_{\epsilon }\} \mathrm{such} \mathrm{that} F_{v_{\epsilon }{x}_0}(t) \ge H(t-\mathrm{\mu }(\mathrm{\phi }(v_{\epsilon })-\mathrm{\phi }(x_0)))\forall t>0.$$

(3.51)

But (3.51) means x₀ ≥ v_ε, and because $v_{\epsilon }$ ≥ u, it results in $x_0\stackrel{(3.46)}{\in }$X_u, which imposes $x_0$ = $v_{\epsilon }$, since $v_{\epsilon }$ is maximal in X_u, in contradiction with (3.51). □

Remark 3.9.

Theorem 3.3 is Theorem 4.1 from [31] with the statement and the demonstration changed by the author: (1) and (2) are without value; for u = $x_0$ both relations are verified because H is increasing; the justification for (3) is not true, since the opposite of the assertion (3) is not what one affirms it to be.

Retain an abbreviated statement for Theorem 3.3, (X, F, T) is $\mathcal{T}$₀ - complete Menger PM-space.

Theorem 3.4.

If φ: X → R is bounded from below l.s.c. and μ > 0, there exists v in X with the property

∀x ∈ X\{v} ∃ t_x > 0 such that F_vx (t_x) < H(t_x − μ(φ(v) − φ(x))).

Proof. 

Suppose ad absurdum the contrary, i.e., ∀x of X ∃ y in X\{x} such that ∀t > 0 we have F_xy (t) ≥ H(t − μ(φ(x) − φ(y))). Consider the map Φ: X → X, Φ(x) = y ≠ x, where y verifies the last relation. Apply Corollary 2; Φ has a fixed point $x_0$, hence Φ($x_0$) = $x_0$ ≠ $x_0$, a contradiction. □

Proposition 3.4.

Theorem 3.4 and Theorem 3.3 are equivalent.

Proof. 

Th. 3.4 ⇒ Th. 3.3. The hypothesis supplies v in X having the property ∀x ∈ X\{v} ∃t_x > 0 such that

F_vx (t_x) < H(t_x − μ(φ(v) − φ(x))).

(3.52)

f being surjective, ∃$x_0$ in E with f ($x_0$) = v. If f ($x_0$) ∈ ${\bigcap }_{\mathrm{\alpha }\in I}{\mathrm{\Phi }}_{\mathrm{\alpha }}(x_0)$, the proof is finished. Suppose f ($x_0$) ∉ ${\bigcap }_{\mathrm{\alpha }\in I}{\mathrm{\Phi }}_{\mathrm{\alpha }}(x_0)$. The hypothesis of Theorem 3.2 supplies ${\mathrm{\alpha }}_0$ and $y_0$ as follows:

$$y_0\in {\mathrm{\Phi }}_{\mathrm{\alpha }}(x_0)\backslash \{ f (x_0)\} = {\mathrm{\Phi }}_{\mathrm{\alpha }}(x_0)\backslash \{v\},$$

(3.53)

such that (see (3.33), f (x₀) = v)

$$F_{v{y}_0}(t) \ge H(t-\mathrm{\mu }(\mathrm{\phi }(v)-\mathrm{\phi }(y_0)))\forall t>0.$$

(3.54)

But $y_0\stackrel{(3.52)}{\ne }$v, and comparing (3.52), where x = $y_0$, with (3.54), where t = t_x, the contradiction is obtained and f ($x_0$) ∈ ${\bigcap }_{\mathrm{\alpha }\in I}{\mathrm{\Phi }}_{\mathrm{\alpha }}(x_0)$.

Th. 3.3 ⇒ Th. 3.4. Indeed, since Theorem 3.3 ⇒ Corollary 1 ⇒ Corollary 2, and Corollary 2 has been used to prove Theorem 3.4. □

Remark 3.10.

The equivalence Theorem 3.4 ⇔ Theorem 3.2 was observed by the author. The equivalence Theorem 5.1, from [31], i.e., 3.1 ⇔ 4.1, is not true (see Remark 3.8).

Remark 3.11.

There are some other things about [31]. The presence of the function k: (0,1) → (0, +∞), in Lemma 2.2 and in Section 3 and Section 4, is not necessary; it appears nowhere; M is simply a real strictly positive number.

The following is a sketch which visualizes the established interdependences between the statements.

One still deduces the following from the above sketch:

Proposition 3.5.

Theorem 3.2, Corollary 1, Corollary 2, and Theorem 3.4 are equivalent.

We finish the section with

Proposition 3.6.

Theorem 3.3 is, indeed, a variant of the abbreviated Ekeland principle for Menger PM-spaces.

Proof. 

The abbreviated Ekeland principle is equivalent to the Caristi-Kirk theorem (see the theorem in Section 2.1.4); for the rest — consult the above sketch. □

The differences among these three types of multivalued variants, discussed in this section, are summarized and briefly presented in the following comparative

Table 1. Differences among the three multivalued variants presented above.

Differences	First Variant	Second Variant	Third Variant
Frame	(X, d) is a complete metric space	X and Y are real Banach spaces	$\mathcal{T}$0 - complete Menger PM-space
	Y metrizable real locally convex	Clarke tangent cone to the graph of a multivalued function
	C closed pointed convex cone with vertex	Hiriart-Urruty functional	property (2.57)
Multivalued map properties	f proper, with compact values	f with compact values	bounded from below
	C-bounded from below	C-bounded from below	lower semicontinuous
	upper semicontinuous	upper semicontinuous	other specific properties
	other specific properties of f	other specific properties of f
Assertions	$\exists x_2$ in dom f	$\exists \mathrm{an} \mathrm{\epsilon }{k}_0$-minimizer (xε, yε) of f	∃vε
	$\exists y_2$$\mathit{in} f (x_2$)	specific properties	∗-maximal among ∗-majorants
	specific properties	expression with coderivative	specific properties
	a specific expression	a specific expression	a specific expression

.

4. Discussion

4.1. Applications of Multivalued Versions of Ekeland Principle in Variational Methods Results

The most important applications of the Ekeland principle can be considered the major theorems of minimax, mountain pass, and saddle point types. In this direction, ref. [32] presents an interval-valued mountain pass theorem based on set-valued versions of the Ekeland variational principle, along with related results such as fixed-point theorems, solving interval-valued optimization problems, noncooperative interval-valued games, and interval-valued optimal control problems described by interval-valued differential equations.

4.2. On Several Applications of Various Multivalued Versions of Ekeland Principle

These three series of results are presented side by side in this paper to highlight and compare the methods used to obtain them, to reveal links and connections, to provide new or refined proofs in certain places and to emphasize their permanent relevance. To support this assertion, some applications of multivalued versions of the Ekeland variational principle in modeling real phenomena are presented below. Set-valued variants of the principle have been applied in a range of real-world contexts, particularly where discontinuities, lack of smoothness, or uncertainty are present. These applications span mechanics, economics, biology, materials science, and control theory.

When the governing laws are not smooth, these appear in contact mechanics and elasto-plastic phenomena like modeling unilateral contact, friction, and material damage. In [33], one can see the multivalued forces (such as Coulomb friction, for instance) lead to differential inclusions, and the Ekeland variational principle is used to prove the existence of weak solutions or equilibrium points.

Set-valued constitutive laws describing solid–liquid transitions, magnetic hysteresis, and other phenomena appear in phase transition models [34], where the energy functionals are nonconvex, and often discontinuous and multivalued versions of the Ekeland principle are applied to prove the existence of solutions and equilibrium configurations.

In population dynamics and ecology, modeling growth with environmental constraints, or harvesting strategies, one uses inclusion-type models to account for multiple possible outcomes, treating such problems by using multivalued variants of the considered principle to find the existence of stable populations or equilibrium distributions under uncertainty. In [35,36], one can see where multivalued differential inclusions in population dynamics are involved.

Multiple-valued versions of the Ekeland variational principle are also used to derive generalized Nash equilibria, even in noncooperative differential games involved in economics and game theory with equilibrium in markets with nonsmooth utilities, nonconvex preferences, or discontinuous demand, as [37,38] show. The paper [37] presents generalizations of the Ekeland variational principle applicable to behavioral science modeling, addresses quasi-metric spaces and the concept of behavioral traps being a central theme in variational rationality, and serves as a solid foundation for mathematical formulations linking dynamic decision processes and variational methods.

In smart materials and hysteresis systems, multivalued variants of the Ekeland principle are used to study energetic solutions and variational stability. When modeling piezoelectric materials, shape memory alloys, and magnetic systems often involving rate-independent hysteresis, these materials follow nonunique evolution laws, modeled with multivalued operators as is shown in [39].

Concerning the field of control systems with uncertainty, one can mention the control of dynamical systems with state-dependent constraints or uncertain actuator behaviors, where multivalued versions of the Ekeland principle allow for relaxed controls and proofs of optimality in set-valued differential inclusions [40].

4.3. Formulations of Some Real-World Problems

Regarding the above-mentioned applications, one can cite interesting problems in contact mechanics with set-valued laws (friction or contact) [33] of the form:

$$\left\{\begin{array}{c}-\mathrm{div}\left(\sigma \left(u\right)\right)=f\\ u=0 \mathrm{on} {\mathrm{\Gamma }}_{\mathrm{D}}\\ \sigma \left(u\right)·\nu \in \partial j\left(u\right) \mathrm{on} {\mathrm{\Gamma }}_C\end{array},\right.$$

where Ω ⊂ ${\mathbf{R}}^n$ is a bounded domain (an elastic body, for instance), u: Ω → ${\mathbf{R}}^n$ is the displacement vector field, representing the unknown deformation of the elastic body, and u is searched in $W^{1,p}$(Ω; ${\mathbf{R}}^n$), the problem being of linear elasticity for p = 2. $\mathrm{\sigma }\left(u\right)$: Ω → ${\mathbf{R}}^{n\times n}$ is the stress tensor describing the internal forces per unit area, which is symmetric and defined by $\mathrm{\sigma }\left(u\right)$ = $\mathbf{C}\mathrm{\epsilon }(u)$, where $\mathbf{C}: {\mathbf{R}}_{sym}^{n\times n}\to {\mathbf{R}}_{sym}^{n\times n}$ is the fourth order elasticity tensor, specific to the material satisfying symmetry and coercivity conditions. $\mathrm{\epsilon }\left(u\right)$ = ${\displaystyle \frac{1}{2}}$ (∇u + ∇u^T) is the strain tensor which measures the symmetric part of the gradient of displacement and the deformation. f : Ω → ${\mathbf{R}}^n$ represents the volume force, for instance, gravity or external body force. ${\mathrm{\Gamma }}_D$ and ${\mathrm{\Gamma }}_{\mathrm{C}}$ are parts of the boundary $\partial$Ω of the considered domain, the first one for the Dirichlet boundary problem (i.e., fixed displacement) and the second one for the contact boundary (possibly frictional or unilateral), i.e., the von Newmann problem. ν is the outward unit vector normal to $\partial$Ω. j: ${\mathbf{R}}^n$ → R ∪ {+∞} is the potential which should be locally Lipschitz, modeling nonsmooth phenomena like friction, unilateral contact (Signorini conditions, for instance), adhesion, etc.; it is nonconvex and may involve constraints, while $\partial j(u)$ is the Clarke subderivative (Section 2.2). Regarding the second boundary condition, it is nonmonotone and multivalued, modeling the contact/friction law where the boundary stress is not uniquely determined but belongs to the Clarke subderivative of j [41]. In such a case, a multivalued version of the Ekeland principle is used to find a critical point of the nonconvex, nonsmooth energy functional. One can exemplify the involvement of some multivalued versions of the Ekeland variational principle for the above-presented mixed problem. To construct the energy functional, consider firstly that its definition set V is a subset of $W^{1,p}$(Ω; ${\mathbf{R}}^n$) containing those functions u verifying $u=0 \mathrm{on} {\mathrm{\Gamma }}_D$ with the norm ${‖\nabla u‖}_{L^p(\mathrm{\Omega })}.$ The elastic energy is W: ${\mathbf{R}}^{n\times n}$ → R satisfying standard p-growth and coercivity, while the potential j follows a linear growth, considering f ∈ $W^{-1,p\prime }$(Ω; ${\mathbf{R}}^n$), ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{p\prime }}=1$. So, the functional is Φ: V → R as follows:

$$\Phi \left(u\right)=\underset{\mathrm{\Omega }}{\int }W(\nabla u)dx+\underset{{\mathrm{\Gamma }}_C}{\int }j(u)ds-〈f,u〉.$$

To obtain the existence of the solution of the above-mentioned contact problem, some properties of Φ such as coercivity and sequentially lower semicontinuity, as also locally Lipschitz, should be involved. Use a multivalued version of the Ekeland principle to build ε-minimizers with near-optimality inequalities and then pass to the limit to obtain 0 ∈ ∂Φ(u). Then u is the necessary solution.

Another interesting problem, which appears in phase transition modeling, can be formulated as follows:

$$\left\{\begin{array}{c}{g\left(x,t\right) \in \partial }_{t}u\left(x,t\right)-∆u\left(x,t\right)+\mathrm{\xi }\left(x,t\right), \mathrm{in} \Omega \times \left(0,T\right)\\ \xi \left(x,t\right) \in \mathrm{\partial }\varphi \left(x,t\right) \mathrm{a}.\mathrm{e}.\mathrm{in}\Omega \times \left(0,T\right)\end{array},\right.$$

where u is the unknown state function which could represent temperature, order parameter, concentration, etc., u: $\Omega \times [0,T]$ → R its partial derivative on time t representing the evolution of the state (how it changes over time), introducing the dynamical (evolutionary) character of the problem. $∆u$ models the diffusion (spatial spreading of u). $\partial \mathrm{\varphi }$ represents the Clarke subderivative of the nonconvex potential $\mathrm{\varphi }$, while ξ captures the effect of hysteresis or nonsmooth energy barriers. The source term g represents external influence, such as heat sources, applied fields, external forces, and some given data. The Ekeland variational principle is here involved in constructing approximate minimizers of a functional related to the energy, proving the existence of solutions to the inclusion problem and handling nonsmooth and nonconvex functionals.

When considering a problem of economic equilibrium with discontinuous preferences, i.e., a nonsmooth economic equilibrium problem where consumer preferences or demands may be discontinuous or exhibit set-valued behavior (for instance, due to threshold effects, uncertainty, or discontinuities in utility functions), the following formulation can be given:

$$Z(p) \in \partial J\left(p\right), p \in X \subset {\mathbf{R}}^n,$$

p* ∈ X is searched such that

0 ∈ Z(p*).

p ∈ X is a price vector for n goods, and X is the feasible set of price vectors, often assumed to be convex and compact. Z(p) represents the excess demand function, representing the aggregate excess demand at price vector p. It may be set-valued or discontinuous, especially in models with nonconvex preferences, nonsmooth utility functions, rationing constraints, or shocks. J (p) is the potential or welfare function, i.e., a potential function such that Z(p) ∈ $\partial J\left(p\right)$; J is possibly non-differentiable, meaning that Z(p) belongs to the subderivative of J. This representation links the equilibrium condition 0 ∈ Z(p*) to a variational inequality or subdifferential inclusion. The relation 0 ∈ Z(p*) ⊂ $\partial J$(p*) implies that p* is a minimizer of J. Hence, an equilibrium price corresponds to a variational minimum.

4.4. The Importance of Set-Valued Versions of Ekeland Principle as Reflected in Numerical Applications

Multivalued versions of the Ekeland variational principle provide a wide palette of approximation methods, consolidating the passage to numerical applications. In recent papers, such as [42], variational tools have been enhanced via beta-weighted Kantorovich operators. Wavelet-based operators for function approximation were studied in [43], and an improved approximation for variational methods was obtained in [44].

Appell polynomials in discretization schemes were presented in [45], where a sequence of positive linear operators using generalized Appell polynomials in integral form was introduced. These operators approximate functions defined on a Lebesgue measurable space called Szasz-integral type operators and estimates to establish the rate of convergence and accuracy of approximation are given. Local and global approximation results are obtained using these operators, and thus improved approximation tools in different functional spaces are determined.

Enhanced beta-type operators for approximation were capitalized in [44], while statistical methods for variational sequences were studied in [46]. Such approaches refine and deepen approximation methods.

An algorithm was developed in [47], and it was used to solve minimization problems involving the fixed-point set of a non-expansive mapping. In addition, numerical solutions derived from computer simulations were presented in order to locate the common solution for a variational multivalued problem, in new inclusion problems with multivalued maximal monotone mapping.

Regarding the complete treatment involving both mathematical models and numerical solutions in contact mechanics and hemivariational instruments, one can cite [33,41,48,49,50].

4.5. Final Comments

The presentation of several multivalued variants of the Ekeland principle is justified by the wide range of applications in which they can be involved, particularly in modeling various real phenomena. It is interesting to note the equivalences highlighted among many results, as they allow for a wide range of approaches to obtain and/or characterize the solutions to real-world problems. In this manner, using appropriate numerical methods, the solution can be constructed and visualized.

With regard to asymmetric characteristics, they are typical of multivalued versions of the Ekeland principle, as shown in several studies such as [51,52,53,54].

In the next

Table 2. Single-valued versus multivalued versions of Ekeland variational principle together with their possible applications.

Type of Variant	Map Types	Results	Applications
Single-valued Ekeland principle	smooth functions	ε-minimizers near the true minimum of a function	convex optimization
			existence results for PDEs
	Lipschitz or convex nonsmooth	existence of stable ε-minimizers close to the true minimum	machine learning
			hemivariational inequalities
Multivalued Ekeland principle	mappings where each input may have multiple possible outputs	existence of ε-solutions within the “value sets”	optimization and control
			contact mechanics (friction laws)
			economics and equilibrium theory
			game theory, multiple responses
			population dynamics
			PDEs with weights

, a concise comparison is provided between the single-valued and multivalued versions, including their fields of application, types of results, and potential applicability in modeling real-world phenomena.

5. Conclusions

This review aims to provide an extended discussion of multivalued versions of the Ekeland variational principle, given their importance in solving problems arising from the modeling of real phenomena.

Three series of results have been presented to provide the theoretical framework for constructing three groups of multiple-valued variants of this important principle. Comparisons and discussions of equivalences have been included as original contributions of this work. The equivalence of the Theorems 3.2 and 3.4 was proven by the author.

The remarks on the theoretical tools are completed by comments on some applications to models in mechanics, phase transition, economic equilibrium, and control systems subject to uncertainty.