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Axioms 2012, 1(3), 384-394; https://doi.org/10.3390/axioms1030384

Article
On the Equilibria of Generalized Dynamical Systems
Department of Mathematics, Informatics and Training Sciences, Faculty of Sciences, “Vasile Alecsandri” Bacău University, Calea Marasesti 157, 600115, Bacău, Romania
Received: 24 September 2012; in revised form: 26 November 2012 / Accepted: 26 November 2012 / Published: 6 December 2012

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.
Keywords:
Isac’s cone; dynamical system; critical point; efficiency; Choquet’s boundary

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 = {pα: αI} 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 pα ∈ P there exists f ∈ X* such that pα(x) ≤ f(x) for all xK.
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 → 2A is called a generalized dynamical system if Γ(a) is non-empty for every aA.
Let Γ: A → 2A be the generalized dynamical system defined by Γ(a) = A ∩ (aK), aA.
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, a0MINK(A) (a0eff(A, K)) if it satisfies one of the following equivalent conditions;
(i)
A ∩ (a0K) ⊆ a0 + K
(ii)
K ∩ (a0A) ⊆ −K
(iii)
(A + K) ∩ (a0K) ⊆ a0 + K
(iv)
K ∩ (a0AK) ⊆ −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 a0MINK(A), if and only if, A ∩ (a0K) = {a0} or equivalently K ∩ (a0A) = {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 = {pα: αI}) 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 pα ∈ P there exists a lower semicontinuous function φα:AR+ with pα(xy) ≤ φα(x) − φα(y), ∀xA, y ∈ Γ(x), then MINK(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 ABA + K. If K is an Isac’s cone and B ∩ (A0K) is bounded and complete for some non-empty set A0A, then MINK(A) ≠ ϕ.
Corollary 2.2.1 
Let ABA + K. If K is a weak Isac’s cone and B ∩ (A0K) is bounded and weakly complete for some non-empty set A0A, then MINK(A) ≠ ϕ.
In particular, if K is a weak Isac’s cone in X and A∩(aK) or (A + K)∩(aK) is bounded and weakly complete for some aA, then MINK(A) ≠ ϕ When the boundedness and weak completeness properties hold for every aA, then we have the following domination property: AMINK(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 
MINK(A) ≠ ϕ if one of the following conditions holds:
(i)
K is closed, normal, weakly complete and is weakly closed such that A∩(aK) is bounded for some aA We also have AMINK(A) + K if, under the above hypotheses, A∩(aK) is bounded for every aA,
(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)∩(aK) is bounded for some aA.
The domination property AMINK(A) + K holds if, in addition, (A + K)∩(aK) is bounded for any aA.
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 MINK(A) ≠ ϕ and AMINK(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 MINK(A) ≠ ϕ and AMINK(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,yE one defines xk y, if yx + K.
Clearly, this order relation on E is topologically closed, that is, the set GK given by GK = {(x,y) ∈ E × E: xk y} is a closed subset of E × E. Also, the set Gε+K = {(x,y) ∈ E × E: yx + 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 a0A 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 aA such that a0aεK that is, A≤(a0aε) = ϕ.
The ε-critical points set of A with respect to K will be denoted by ε-MINK(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 MINK(A) ⊆ εMINK(A), ∀εK\{θ} and Axioms 01 00384 i001.
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 + Kincreasing if f(x1) ≥ f(x2) whenever x1,x2E and x1x2 + 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,f2S 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,vM+(X) then μsv, if μ(S) ≤ v(S) for all sS.
Following [5] a measure μM+(X) is minimal with respect to the above order relation if for any continuous function f:X → ℜ we have μ(Qsf) = μ(f), where μ(Qsf) = inf{sS: fs}. Particularly, if xX, then the Dirac measure εx is minimal, if εx(Qsf) = εx(f), that is, Qsf(x) = f(x) for every continuous function f:X → ℜ.
The set of all points xX such that εx is minimal measure with respect to ≤s is named the Choquet boundary of X with respect to S and it is denoted by δsX Hence, if C(X) is the usual Banach space of all real continuous functions on X, then
Axioms 01 00384 i002
A closed set AX is called S-absorbent if xA and μs εx implies that μ(X\A) = 0. The trace on δsX 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 {xX : ∃xS with s(x) < θ} is usually named the Choquet topology on δsX.
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] 
MINK(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 MINK(X) endowed with the trace topology τx induced on X by τ is a Baire space. Moreover, if X is metrizable, then MINK(X) is a Gδ—set in (X,τx).
Taking into account Theorem 2.11 and Theorem 2.12 respectively in [5] we obtain
Corollary 3.1.1 
(i) MINK(X) and MINK(X)∩{xX :s(x) ≤ 0} (sS) are compact sets with respect to Choquet’s topology;(ii) MINK(X) is a compact subset of X.
Remark 3.3 
In the conditions of Theorem 3.1, let us consider MINK(X) endowed with the trace topology also denoted by τx, the following game between partners A and B. each partner successively chooses a non-empty set belonging to τx such that the player A makes the first choice and each player must choose a set in τx which should be included in the previously chosen set of the other player.
Let G1, G′1, G2, G′2, …, Gn, G′n, … be the successive options of the two players, (G1, G2, …, Gn, …) 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
Axioms 01 00384 i003
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 MINK(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 xX such that if x1,x2X, 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,τx) is metrizable, then ε-eff(X,K) is a Gδ—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 a0A is a K1-efficient (minimal) point of A, in notation, a0eff(A,K,K1) (or a0MINK+K1 (A)) if it satisfies one of the following equivalent conditions:
(i)
A∩(a0KK1) ⊆ a0 + K + K1
(ii)
(K + K1)∩(a0A) ⊆ −KK1
In a similar manner one defines the maximal efficient points by replacing K + K1 with −(K + K1).
Remark 4.1 
a0eff(A,K,K1), if it is a fixed point for the multifunction F : AA defined by F(t) = {aA: A∩(aKK1) ⊆ t + K + K1}.
Remark 4.2 
In [33] it was shown that whenever K1K\{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 a0eff(A,K,K1) means that A∩(a0KK1) = Ø or, equivalently, (K + K1)∩(a0A) = Ø for θK1 and A∩(a0KK1) = {a0} respectively, if θK1 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 a0eff(A,K) (a0MINK(A)), that is, satisfying the next equivalent properties:
(i)
A∩(a0K) = {a0}
(ii)
A∩(a0K\{θ}) = Ø
(iii)
K∩(a0A) = {θ}
(iv)
(K\{θ})∩(a0A) = Ø
and we notice that
Axioms 01 00384 i004
It is clear that for any εK\{θ} taking K1 = {ε} it follows that a0eff(A,K,K1) if, and only if, A∩(a0εK) = Ø. In all these cases, the set eff(A,K,K1) was denoted by εeff(A,K) (or εMINK(A) as in [11]) and it is obvious that Axioms 01 00384 i005.
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) = {a1A: Aa1 + 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 a0S(A,K,K1) and aA∩(a0KK1) are arbitrary elements, then aa0 + K + K1, that is, a0eff(A,K,K1) by virtue of (i) in Definition 4.1. Suppose now that Axioms 01 00384 i006 and there exists a0eff(A,K,K1)\S(A,K,K1). From Axioms 01 00384 i007 it follows that Axioms 01 00384 i008, that is, Axioms 01 00384 i009 from which, since Axioms 01 00384 i010 and a0eff(A,K,K1) we conclude that Axioms 01 00384 i011. Therefore, Axioms 01 00384 i012 in contradiction with a0S(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 aS(A,K,K1). Then, aa + K + K1 which implies that 0 ∈ K + K1.
Therefore, KK1 + K = K1 + KK. 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(nN, n ≥ 2) endowed with the separated H-locally convex topology generated by the semi-norms Axioms 01 00384 i013 and for each real number c we define Axioms 01 00384 i014, then it is clear that S(Ac,K,K1) is empty and eff(Ac,K,K1) = Ac.
In all our further considerations we suppose that X is a Hausdorff locally convex space having the topology induced by family P = {pα : αI} 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, ∀xK} 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 a0eff(A,K,K1) whenever for each pαp and η ∈ (0, 1) and there exists x* in the polar cone K0 of K such that pα(a0a) + η, ∀aA.
Proof 
Let us suppose that, under the above hypotheses, (K + K1)∩(a0A)⊈−(K + K1), that is, there exists aA so that a0aK + K1\(−KK1) Then, a0a ≠ 0 and, because X is separated in Hausdorff’s sense, there exists pαp such that pα(a0a) > 0. On the other hand, there exists nN* sufficiently large with pα(a0a)/n ∈ (0, 1) and the relation given by the hypothesis of theorem leads to pα(a0a) ≤ x*(a0a) + pα(a0a)/n with x* ∈ K0 and n → ∞ which implies that pα(a0a) ≤ 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 Axioms 01 00384 i015 it follows that for each λ0 ∈ [0, 1] there exists c1, c2 ≤ 0 such that Axioms 01 00384 i016. Taking Axioms 01 00384 i017 one obtains |1 − 4λ| ≤ (c1c2)(1 − 4λ)+1, ∀λ ∈ [0, 1] which for λ = 0 implies that c2c1 and for Axioms 01 00384 i018 leads to c1c2 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: PK*\{0} the full nuclear cone Kφ = {xX : pα(x) ≤ φ(pα)(x), ∀pαp} 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:PK*\{0}, then,
Axioms 01 00384 i019
for any non-empty subset K1 of K.
Proof 
If a0eff(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∩(a0KK1) − a0K + K1KKφ for some φ:PK*\{0}.
Therefore, a0S(A∩(a0KK1), Kφ).
Hence, Axioms 01 00384 i020. Conversely, consider now a1S(A∩(a0KK1), Kφ) for a0A and φ:PK*\{0}. Then a1A∩(a0KK1) and A∩(a0KK1) − a1Kφ, that is,
pα(aa1) ≤ φ(pα)(aa1), ∀aA∩(a0KK1), pαP which implies immediately that
pα(aa1) ≤ −φ(pα)(aa1) + η, ∀aA∩(a0KK1), pαP, η ∈ (0, 1) and, by virtue of Theorem 4.2 one obtains that a1eff(A∩(a0KK1), K, K1).
But eff(A∩(a0KK1), K, K1) ⊆ eff(A, K, K1).
Indeed, for any teff(A∩(a0KK1), K, K1) and hA∩(tKK1) we have h ∈ (a0KK1)∩(tKK1) ⊆ t + K + K1 and by the point (i) of Definition 4.1 it follows that teff(A, K, K1). This completes the proof.
Remark 4.6 
The hypothesis KKφ 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 Axioms 01 00384 i021. When K is any pointed convex cone, A is a non-empty subset of X and a0eff(A,K) then, by virtue of (i) in Remark 4.3, it follows that A∩(a0K) = {a0} that is, A∩(a0K) − a0 = {0} ⊂ Kφ. Hence, a0S(A∩(a0K), Kφ) 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
Axioms 01 00384 i022
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.

Acknowledgements

The author would like to thank the anonymous referees for their valuable suggestions and conclusions concerning this work.

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