On the Equilibria of Generalized Dynamical Systems

Abstract

This research work presents original properties of the equilibrium critical (ideal) points sets for an important class of generalized dynamical systems. The existence and significant results regarding such points are specified. Strong connections with the Vector Optimization by the Efficiency and the Potential Theory together with its applications following Choquet’s boundaries are provided.

1. Introduction

In this research work we present properties of critical points for an important class of generalized dynamical systems through the connections with the efficiency and Choquet’s boundaries. Section 2 is therefore devoted to the study of the existence and the properties of the critical point sets. In Section 3, two important coincidence results between the (approximate) critical point sets and the Choquet’s boundaries are given. The last section contains a new approximate modality for this kind of equilibrium point sets. All the elements of ordered topological vector spaces used in this work are in accordance with [1].

2. Existence of the Equilibrium Points under Completeness

In this section we study the existence of the critical points for a class of generalized dynamical systems in separated locally convex spaces ordered by the (weak) supernormal cones introduced by G. Isac in [2] and published in [3], using the (weak) completeness. We note that the weakly complete cones are very important in the functional analysis [4,5] and in the potential theory, including their applications [6,7,8,9,10].

Let X be a real Hausdorff locally convex space with the topology induced by a family P = {p_α: α ∈ I} of seminorms, ordered by a convex cone K, its topological dual space X* and the origin denoted by θ.

Definition 2.1 [3] 

is called an Isac’s (nuclear or supernormal) cone if for every p_α ∈ P there exists f ∈ X* such that p_α(x) ≤ f(x) for all x ∈ K.

Remark 2.1 

We have named cones such as these “Isac’s cone” in memory of my friend, Professor Isac as described in [2,3,11]. Many examples of such convex cones, their importance for efficiency and the last extension by the full nuclearity are described in [2,3,11,12,13,14,15,16,17,18,19,20] and elsewhere. Thus, in a normed linear space a convex cone is an Isac’s cone iff it is well based, that is, generated by a non-empty, convex and bounded set which does not contain the origin in its closure. A convex cone is an Isac’s cone in a nuclear space [21] if and only if it is a normal cone. Both these properties are not valid in general Hausdorff locally convex spaces. We also note that this concept is not a simple generalization of the corresponding notion defined in the normed linear spaces by M. A. Krasnoselski [22], being specific to the ordered Hausdorff locally convex spaces [23].

Let A be a non-empty set in a Hausdorff locally convex space X ordered by a convex cone K.

Definition 2.2 

A set valued map Γ: A → 2^A is called a generalized dynamical system if Γ(a) is non-empty for every a ∈ A.

Let Γ: A → 2^A be the generalized dynamical system defined by Γ(a) = A ∩ (a − K), a ∈ A.

Definition 2.3 

We say that a₀ is a critical point for Γ or an efficient (minimum) point for A with respect to K, in notation, a₀ ∈ MIN_K(A) (a₀ ∈ eff(A, K)) if it satisfies one of the following equivalent conditions;

(i)

A ∩ (a₀ − K) ⊆ a₀ + K

(ii)

K ∩ (a₀ − A) ⊆ −K

(iii)

(A + K) ∩ (a₀ − K) ⊆ a₀ + K

(iv)

K ∩ (a₀ − A − K) ⊆ −K

It is clear that the finite dimensional version of this concept is represented by the Pareto optimality [20,24,25] and references therein. We recall that every Isac’s cone is pointed, that is, K ∩ (−K) = {θ} and in any such case as this a₀ ∈ MIN_K(A), if and only if, A ∩ (a₀ − K) = {a₀} or equivalently K ∩ (a₀ − A) = {θ}. A projection in vector optimization by the super efficiency was given in [26]. The largest class of convex cones ensuring the existence of the efficient points in non-empty, compact subsets of the linear topological vector spaces was presented in [27], completed by the subsequent studies described in [28,29] as was the last extension of the ordering convex cones to investigate the efficiency in Hausdorff locally convex spaces under completeness instead of compactness which is performed by Isac’s cones; also used in the study of the geometrical aspects for Ekeland’s principle [30]. Taking into account Corollary 3 of Theorem 1 in [3], we obtain:

Theorem 2.1 

If (X, P = {p_α: α ∈ I}) is a separated locally convex space with the topology generated by a family P of seminorms and A is a non-empty complete set in X such that for every p_α ∈ P there exists a lower semicontinuous function φ_α:A → R₊ with p_α(x − y) ≤ φ_α(x) − φ_α(y), ∀x ∈ A, y ∈ Γ(x), then MIN_K(A) ≠ ϕ.

The importance of Isac’s cones concerning the existence and the domination property of the critical points sets for the corresponding dynamical systems is also illustrated by the following results.

Theorem 2.2 [18] 

Let A ⊆ B ⊆ A + K. If K is an Isac’s cone and B ∩ (A₀ − K) is bounded and complete for some non-empty set A₀ ⊆ A, then MIN_K(A) ≠ ϕ.

Corollary 2.2.1 

Let A ⊆ B ⊆ A + K. If K is a weak Isac’s cone and B ∩ (A₀ − K) is bounded and weakly complete for some non-empty set A₀ ⊆ A, then MIN_K(A) ≠ ϕ.

In particular, if K is a weak Isac’s cone in X and A∩(a − K) or (A + K)∩(a − K) is bounded and weakly complete for some a ∈ A, then MIN_K(A) ≠ ϕ When the boundedness and weak completeness properties hold for every a ∈ A, then we have the following domination property: A ⊆ MIN_K(A) + K.

Since in every separated locally convex space any normal cone is a weak Isac’s cone (Proposition 2 of [14]), the conclusion of the above corollary remains valid whenever K is a normal cone. Using this remark, one obtains the next existence results for the efficient points.

Corollary 2.2.2 

MIN_K(A) ≠ ϕ if one of the following conditions holds:

(i)

K is closed, normal, weakly complete and is weakly closed such that A∩(a − K) is bounded for some a ∈ A We also have A ⊆ MIN_K(A) + K if, under the above hypotheses, A∩(a − K) is bounded for every a ∈ A,

(ii)

K is closed, normal and A is bounded and weakly complete. The domination property holds again,

(iii)

K is closed, normal, weakly complete and A + K is weakly closed such that (A + K)∩(a − K) is bounded for some a ∈ A.

The domination property A ⊆ MIN_K(A) + K holds if, in addition, (A + K)∩(a − K) is bounded for any a ∈ A.

Remark 2.2 

Taking into account Corollary of Proposition 2 in [14] it is clear that all the above results remain valid if one replaces the hypothesis of normality on K with the usual weak normality.

Corollary 2.2.3 

If A is a non-empty, bounded and closed subset of X and K is well based (that is, generated by a non-empty convex, bounded set which does not contain the origin of the space in its closure) by a complete set, then MIN_K(A) ≠ ϕ and A ⊆ MIN_K(A) + K.

Corollary 2.2.4 

If A is a non-empty, bounded and closed subset of a Banach space ordered by a convex cone K well based by a closed set, then MIN_K(A) ≠ ϕ and A ⊆ MIN_K(A) + K.

Remark 2.3 

Since in a normed linear space a convex cone is an Isac’s cone if it is well based, the above last theorem and its immediate corollaries offers useful conclusions for the efficiency whenever K is an Isac’s cone. Moreover, the existence results given in this section show also the possibility to use the (weakly) complete cones for the study of the existence and the properties of the solutions for the vector optimization problems in Hausdorff locally convex spaces, with appropriate numerical methods and they generalize similar conclusions to those indicated in [24].

3. Coincidence Results between Equilibrium Point Sets and Choquet’s Boundaries under Related Topics

In our further considerations we suppose that (E, τ) is a Hausdorff locally convex space having its origin θ and τ its topology, K is a closed, convex pointed cone in E and ε is an arbitrary element of K{θ}. On the vector space E we consider the usual order relation ≤_K associated with K as follows: for x,y ∈ E one defines x≤_k y, if y ∈ x + K.

Clearly, this order relation on E is topologically closed, that is, the set G_K given by G_K = {(x,y) ∈ E × E: x≤_k y} is a closed subset of E × E. Also, the set G_ε+K = {(x,y) ∈ E × E: y ∈ x + ε + K} is closed in E × E endowed with the usual product topology.

Definition 3.1 

If A is a non-empty subset of E, then a₀ ∈ A is called an ε-critical point (or, ε-minimal element, ε-efficient point, ε-near to minimum point) of A with respect to K if there exists no a ∈ A such that a₀ − a − ε ∈ K that is, A≤(a₀ − a − ε) = ϕ.

The ε-critical points set of A with respect to K will be denoted by ε-MIN_K(A) or ε-eff(A,K) following the analogy with [31,32,33].

Remark 3.1 

It is clear that the concept of the ε-efficient element does not include the notion of the efficient point, but the immediate connection between them is represented by the relations MIN_K(A) ⊆ ε − MIN_K(A), ∀ε ∈ K\{θ} and .

Remark 3.2 

A very interesting and important generalization of the approximate efficiency given by Definition 3.1 was considered in [33] by replacing with a non-empty subset of K\{θ}. In this way, it was shown that the existence of this new type of efficient points for the lower bounded sets characterizes the semi-Archimedian ordered vector spaces and the regular ordered locally convex spaces. In [32] we also find many pertinent examples and comments inside the usual Euclidean spaces.

Definition 3.2 

A real function f:E → ℜ is called ε + K—increasing if f(x₁) ≥ f(x₂) whenever x₁,x₂ ∈ E and x₁ ∈ x₂ + ε + K.

For a non-empty and compact subset X of E we recall some basic considerations in Potential Theory concerning the Choquet boundary of X with respect to a convex cone of continuous functions on X. Thus, we remember that if S is a convex cone of real continuous functions on X such that the constant function on X belong to S, it is min-stable (i.e., for every f₁,f₂ ∈ S it follows inf(f₁,f₂) ∈ S) and it separates the points of X, then on the set M₊(X) of all positive Radon measures on X one associates the following order relation:

If μ,v ∈ M₊(X) then μ ≤ sv, if μ(S) ≤ v(S) for all s ∈ S.

Following [5] a measure μ ∈ M₊(X) is minimal with respect to the above order relation if for any continuous function f:X → ℜ we have μ(Q_sf) = μ(f), where μ(Q_sf) = inf{s ∈ S: f ≤ s}. Particularly, if x ∈ X, then the Dirac measure ε_x is minimal, if ε_x(Q_sf) = ε_x(f), that is, Q_sf(x) = f(x) for every continuous function f:X → ℜ.

The set of all points x ∈ X such that ε_x is minimal measure with respect to ≤_s is named the Choquet boundary of X with respect to S and it is denoted by δ_sX Hence, if C(X) is the usual Banach space of all real continuous functions on X, then

A closed set A ⊆ X is called S-absorbent if x ∈ A and μ≤_s ε_x implies that μ(X\A) = 0. The trace on δ_sX of the topology on X in which the closed sets coincide with X or with the absorbent subsets of X contained in such a set as {x ∈ X : ∃_x ∈ S with s(x) < θ} is usually named the Choquet topology on δ_sX.

An important connection between the Vector Optimization and the Potential Theory is the next coincidence result on the efficient points sets and Choquet’s boundaries in Hausdorff separated locally convex spaces, which cannot be obtained as a consequence of the Axiomatic Potential Theory, with significant details in [8,9,10,34], respectively.

Theorem 3.1 [8] 

MIN_K(X) coincides with the Choquet boundary of X with respect to the convex cone of all real continuous functions which are increasing with respect to the order relation Consequently, the set MIN_K(X) endowed with the trace topology τ_x induced on X by τ is a Baire space. Moreover, if X is metrizable, then MIN_K(X) is a G_δ—set in (X,τ_x).

Taking into account Theorem 2.11 and Theorem 2.12 respectively in [5] we obtain

Corollary 3.1.1 

(i) MIN_K(X) and MIN_K(X)∩{x ∈ X :s(x) ≤ 0} (s ∈ S) are compact sets with respect to Choquet’s topology;(ii) MIN_K(X) is a compact subset of X.

Remark 3.3 

In the conditions of Theorem 3.1, let us consider MIN_K(X) endowed with the trace topology also denoted by τ_x, the following game between partners A and B. each partner successively chooses a non-empty set belonging to τ_x such that the player A makes the first choice and each player must choose a set in τ_x which should be included in the previously chosen set of the other player.

Let G₁, G′₁, G₂, G′₂, …, G_n, G′_n, … be the successive options of the two players, (G₁, G₂, …, G_n, …) represent the option expressed by A and (G′₁, G′₂, …, G′_n, …) the option made by B. One says that player B wins regardless of the way A plays, he is able to make an option so that

Theorem 3.1 together with Choquet’s results (see, for examples, Chapter 2 of [5]) concerning the properties of the Choquet boundary shows that the above game on MIN_K(X) is won by player B.

Remark 3.4 

As we have seen in [8] under the hypotheses of Theorem 3.1, the set eff(X,K) coincides with the Choquet boundary of X only with respect to the convex cone of all real, continuous and K-increasing functions on X. Thus, for example, if X is a non-empty, compact and convex subset of E, then, taking into account the Theorem 2.2 in the first paragraph of Chapter 2 [5], the Choquet boundary of X with respect to the convex cone of all real, continuous and concave functions on X coincides with the set of all extreme points for A, that is, with the set of elements x ∈ X such that if x₁,x₂ ∈ X, λ ∈ (0,1) and x = λx₁ + (1 − λ), then x = x₁ = x₂. But, it is clear that, even in infinite dimensional cases, an extreme point for a compact convex set is not necessarily an efficient point.

The following result extends Theorem 3.1 for ε-efficiency.

Theorem 3.2 [35] 

If X is a non-empty subset of E, then the set ε-eff(X,K) coincides with the Choquet boundary of X with respect to the convex cone all ε + K—increasing real continuous functions on X. Consequently, the set ε-eff(X,K) endowed with the trace topology is a Baire space and if (X,τ_x) is metrizable, then ε-eff(X,K) is a G_δ—subset of X.

4. A Generalized Modality for the Equilibrium Points Sets

Let X be a vector space with its origin denoted by θ ordered by a convex cone K,K₁ a non-void subset of K and A a non-empty subset of X. The following definition introduces a new concept of approximate efficiency in Hausdorff locally convex spaces which particularly leads to the well-known notion of Pareto type efficiency in the usual Euclidean spaces.

Definition 4.1 

We say that a₀ ∈ A is a K₁-efficient (minimal) point of A, in notation, a₀ ∈ eff(A,K,K₁) (or a₀ ∈ MIN_K+K1 (A)) if it satisfies one of the following equivalent conditions:

(i)

A∩(a₀ − K − K₁) ⊆ a₀ + K + K₁

(ii)

(K + K₁)∩(a₀ − A) ⊆ −K − K₁

In a similar manner one defines the maximal efficient points by replacing K + K₁ with −(K + K₁).

Remark 4.1 

a₀ ∈ eff(A,K,K₁), if it is a fixed point for the multifunction F : A → A defined by F(t) = {a ∈ A: A∩(a − K − K₁) ⊆ t + K + K₁}.

Remark 4.2 

In [33] it was shown that whenever K₁ ⊂ K\{θ} the existence of this new type of efficient points for the lower bounded sets characterizes the semi-Archimedian ordered vector spaces and the regular ordered locally convex spaces.

Remark 4.3 

When K is pointed, then a₀ ∈ eff(A,K,K₁) means that A∩(a₀ − K − K₁) = Ø or, equivalently, (K + K₁)∩(a₀ − A) = Ø for θ ≠ K₁ and A∩(a₀ − K − K₁) = {a₀} respectively, if θ ∈ K₁ Whenever K is pointed and K₁ = {θ} from Definition 4.1, one obtains the well-known notion of the efficient (minimal, optimal or admissible) point, abbreviated a₀ ∈ eff(A,K) (a₀ ∈ MIN_K(A)), that is, satisfying the next equivalent properties:

(i)

A∩(a₀ − K) = {a₀}

(ii)

A∩(a₀ − K\{θ}) = Ø

(iii)

K∩(a₀ − A) = {θ}

(iv)

(K\{θ})∩(a₀ − A) = Ø

and we notice that

It is clear that for any ε ∈ K\{θ} taking K₁ = {ε} it follows that a₀ ∈ eff(A,K,K₁) if, and only if, A∩(a₀ − ε − K) = Ø. In all these cases, the set eff(A,K,K₁) was denoted by ε − eff(A,K) (or ε − MIN_K(A) as in [11]) and it is obvious that .

Remark 4.3 

The following theorem offers the first important connection between the strong optimization and the approximate efficiency in the environment of the ordered vector spaces, described initially in the previous Definition 4.1.

Theorem 4.1 

If we denote by S(A,K,K₁) = {a₁ ∈ A: A ⊆ a₁ + K + K₁} and S(A,K,K₁) ≠ Ø, then S(A,K,K₁) = eff(A,K,K₁).

Proof 

Clearly, S(A,K,K₁) ⊆ eff(A,K,K₁). Indeed, if a₀ ∈ S(A,K,K₁) and a ∈ A∩(a₀ − K − K₁) are arbitrary elements, then a ∈ a₀ + K + K₁, that is, a₀ ∈ eff(A,K,K₁) by virtue of (i) in Definition 4.1. Suppose now that and there exists a₀ ∈ eff(A,K,K₁)\S(A,K,K₁). From it follows that , that is, from which, since and a₀ ∈ eff(A,K,K₁) we conclude that . Therefore, in contradiction with a₀ ∉ S(A,K,K₁) as claimed.

Remark 4.4 

If S(A,K,K₁) ≠ Ø then K + K₁ = K hence eff(A,K,K₁) = eff(A,K). Indeed, let a ∈ S(A,K,K₁). Then, a ∈ a + K + K₁ which implies that 0 ∈ K + K₁.

Therefore, K ⊆ K₁ + K = K₁ + K ⊆ K. The above theorem shows that, for any non-empty subset of an arbitrary vector space, the set of all strong minimal elements with respect to any convex cone, through the agency of every non-void subset of it, coincides with the corresponding set of the efficient (minimal) points whenever there is at least a strong minimal element, the result remaining obviously valid for the strong maximal elements and the maximal efficient points, respectively. Using this result and our abstract construction given in [36] for the H-locally convex spaces introduced by Th. Precupanu in [37], as separated locally convex spaces with any seminorms satisfying the parallelogram law, we established in [16] that the only simultaneous and vectorial approximation for each element, in the direct sum of a (closed) linear subspace and its orthogonal with respect to a linear (continuous) operator between two H-locally convex spaces, is its spline function. We also note that it is possible to have S(A,K,K₁) = Ø and eff(A,K,K₁) = A. Thus, for example, if one considers X = Rⁿ(n ∈ N, n ≥ 2) endowed with the separated H-locally convex topology generated by the semi-norms and for each real number c we define , then it is clear that S(A_c,K,K₁) is empty and eff(A_c,K,K₁) = A_c.

In all our further considerations we suppose that X is a Hausdorff locally convex space having the topology induced by family P = {p_α : α ∈ I} of seminorms, ordered by a convex cone K and its topological dual space X^*. In this framework, the next theorem contains a significant criterion for the existence of the approximate Pareto (minimal) efficient points, in particular, for the usual Pareto (minimal) efficient points, taking into account that the dual cone of K is defined by K^* = {x^* ∈ X^* : x^*(x) ≥ 0, ∀x ∈ K} and its attached polar cone is K⁰ = −K^* The version for the approximate maximal efficient points is straightforward.

Theorem 4.2 

If A is any non-empty subset of X, and K₁ is every non-void subset of K, then a₀ ∈ eff(A,K,K₁) whenever for each p_α ∈ p and η ∈ (0, 1) and there exists x* in the polar cone K⁰ of K such that p_α(a₀ − a) + η, ∀a ∈ A.

Proof 

Let us suppose that, under the above hypotheses, (K + K₁)∩(a₀ − A)⊈−(K + K₁), that is, there exists a ∈ A so that a₀ − a ∈ K + K₁\(−K − K₁) Then, a₀ − a ≠ 0 and, because X is separated in Hausdorff’s sense, there exists p_α ∈ p such that p_α(a₀ − a) > 0. On the other hand, there exists n ∈ N* sufficiently large with p_α(a₀ − a)/n ∈ (0, 1) and the relation given by the hypothesis of theorem leads to p_α(a₀ − a) ≤ x*(a₀ − a) + p_α(a₀ − a)/n with x* ∈ K⁰ and n → ∞ which implies that p_α(a₀ − a) ≤ 0 is a contradiction and the proof is complete.

Remark 4.5 

The above theorem represents an immediate extension of Precupanu’s result given in Proposition 1.2 of [38]. In general, the converse of this theorem is not valid at least in (partially) ordered separated locally convex spaces as we can see from the example considered in Remark 4.4. Indeed, if one assumes the contrary in the corresponding, mathematical calculation’s background, then, taking it follows that for each λ₀ ∈ [0, 1] there exists c₁, c₂ ≤ 0 such that . Taking one obtains |1 − 4λ| ≤ (c₁ − c₂)(1 − 4λ)+1, ∀λ ∈ [0, 1] which for λ = 0 implies that c₂ ≤ c₁ and for leads to c₁ ≤ c₂ that is, |1 − 4λ| ≤ 1, ∀λ ∈ [0, 1], a contradiction.

The content of Section 4 in [17] motivated us to consider for each function φ: P → K*\{0} the full nuclear cone K_φ = {x ∈ X : p_α(x) ≤ φ(p_α)(x), ∀p_α ∈ p} and to give the next generalization of Theorem 7 [17] in a more general context, which represents also a new important link between strong optimization and the approximate vector optimization together with its usual particular variant, respectively.

Theorem 4.3 

If there exists φ:P → K*\{0}, then,

for any non-empty subset K₁ of K.

Proof 

If a₀ ∈ eff(A,K,K₁) is an arbitrary element, then, in accordance with point (i) of the Definition 4.1 and the hypothesis of the above theorem, we have A∩(a₀ − K − K₁) − a₀ ⊆ K + K₁ ⊆ K ⊆ K_φ for some φ:P → K*\{0}.

Therefore, a₀ ∈ S(A∩(a₀ − K − K₁), K_φ).

Hence, . Conversely, consider now a₁ ∈ S(A∩(a₀ − K − K₁), K_φ) for a₀ ∈ A and φ:P → K*\{0}. Then a₁ ∈ A∩(a₀ − K − K₁) and A∩(a₀ − K − K₁) − a₁ ⊆ K_φ, that is,

p_α(a − a₁) ≤ φ(p_α)(a − a₁), ∀a ∈ A∩(a₀ − K − K₁), p_α ∈ P which implies immediately that

p_α(a − a₁) ≤ −φ(p_α)(a − a₁) + η, ∀a ∈ A∩(a₀ − K − K₁), p_α ∈ P, η ∈ (0, 1) and, by virtue of Theorem 4.2 one obtains that a₁ ∈ eff(A∩(a₀ − K − K₁), K, K₁).

But eff(A∩(a₀ − K − K₁), K, K₁) ⊆ eff(A, K, K₁).

Indeed, for any t ∈ eff(A∩(a₀ − K − K₁), K, K₁) and h ∈ A∩(t − K − K₁) we have h ∈ (a₀ − K − K₁)∩(t − K − K₁) ⊆ t + K + K₁ and by the point (i) of Definition 4.1 it follows that t ∈ eff(A, K, K₁). This completes the proof.

Remark 4.6 

The hypothesis K ⊆ K_φ imposed upon the convex cone K is automatically satisfied whenever K is an Isac’s cone and it was used only to prove the inclusion . When K is any pointed convex cone, A is a non-empty subset of X and a₀ ∈ eff(A,K) then, by virtue of (i) in Remark 4.3, it follows that A∩(a₀ − K) = {a₀} that is, A∩(a₀ − K) − a₀ = {0} ⊂ K_φ. Hence, a₀ ∈ S(A∩(a₀ − K), K_φ) for every mapping φ: P: K*\{0} and the next corollary is valid.

Corollary 4.3.1 

For every non-empty subset A of any Hausdorff locally convex space ordered by an arbitrary, pointed convex cone K with its dual cone K* we have

Remark 4.7 

Clearly, the announced theorem represents a significant result concerning the possibilities of scalarization for the study of the above efficiency in separated locally convex spaces, so for the particular cases of Hausdorff locally convex spaces ordered by closed, pointed and normal cones. The coincidence between the equilibrium points sets and Choquet’s boundaries together with its immediate corollaries remains valid if one replaces ε by K₁.

5. Conclusions

The main goal of this paper was to exhibit some pertinent results on the equilibrium point sets for the generalized dynamical systems with respect to Isac’s cones. The relations between this kind of equilibria, the efficiency in the infinite dimensional ordered vector spaces and Choquet’s boundaries were also examined.