Minimal Set Theorems

Andreas Hamel  Andreas Löhne

Abstract

We present two existence principles for minimal points of subsets of the product space X × 2^{Y}, where X stands for a separated uniform space and Y a topological vector space. The two principles are distinct with respect to the involved ordering structure in 2^{Y}.

We derive from them new variants of Ekeland’s principle for set-valued maps as well as a minimal point theorem in X × Y and Ekeland’s principle for vector-valued functions. The use of ordering relations in 2^{Y} admits a new approach to set-valued optimization.

Although 2^{Y} is not a vector space, many ideas can be extended to this case.

Keywords and phrases: minimal set theorem, set-valued variational principle, minimal point theorem, set-valued optimization, natural criteria

Mathematical Subject Classification. 58E30, 46N10.

1 Introduction

A set-valued mapping from a set X to a set Y is usually understood to be a relation F ⊂ X × Y not necessarily satisfying the uniqueness property (we have not that (x, y_{1}), (x, y_{2}) ∈ F implies y_{1} = y_{2}).

In contrast to this, we understand a set-valued mapping to be a function from X to 2^{Y}, i.e. a relation F on X ×2^{Y} satisfying the uniqueness property (i.e. (x, V_{1}), (x, V_{2}) ∈ F implies V_{1} = V_{2}). This leads in a natural way to new definitions of concepts like graph, domain and minimal points of set-valued maps as well as new results.

In this paper, we consider subsets of X × 2^{Y} and look for minimal elements of them with respect to appropriate ordering relations based on ideas of Kuroiwa [14], [15]. We present existence results, which we call minimal set theorems, for such elements.

We shall draw several conclusions.

First, we derive new variants of Ekeland’s principle for vector-valued set-valued maps. The special feature of these Ekeland-type theorems consists of the fact that Kuroiwa’s ordering relations admit a direct comparison say of two values F (x_{1}) and F (x_{2}) of a map F : X → 2^{Y} in contrast to known set-valued variational principles [2],

[10] where single elements of the set {(x, y) ∈ X × Y : y ∈ F (x)} are compared. Secondly, we conclude a minimal point theorem in X × Y. Such theorems are well-

established and useful tools in vectorial optimization and related fields, cf. [20], [9], [8],

[10] for example.

Finally, the well-known Ekeland-type principle for vector-valued functions f : X → Y (e.g. [21], [11], [9]) turns out to be a consequence of our new Ekeland-type principles for set-valued maps as well as of the minimal point theorem.

The paper is organized as follows. In the next section we introduce two ordering relations for elements of 2^{Y} where Y is a linear topological space as well as related boundedness concepts in the space 2^{Y}. In Section 3 we present scalarization methods for subsets of 2^{Y}. These methods are essentially used for the proofs of the minimal set theorems and may be of independent interest. In Section 5 our main results, the minimal set theorems, are presented. Section 6 contains the new set-valued variational principles and several conclusions.

Jahn [12] designated the Kuroiwa concept to be ”very promising” for set-valued optimization. We understand our results as some steps to develop this theory.

2 Ordering Relations and Boundedness in 2^{Y}

Let Y be a topological vector space. We denote by 2^{Y} the set of all subsets of Y. As usual, the sum of two sets V_{1}, V_{2} ∈ 2^{Y} is defined by V_{1}+V_{2} := {v_{1}+v_{2} : v_{1} ∈ V_{1}, v_{2} ∈ V_{2}}. We use the convention ∅ + V_{1} = ∅. The product of α ∈ ℝ and V ∈ 2^{Y} is defined by αV := {αv : v ∈ V }.

This section is concerned with ordering relations for sets introduced by Kuroiwa [14], 1998. Moreover, we introduce order boundedness concepts in 2^{Y}.

We define two relations ≼, ⋞ on 2^{Y} as follows.

Definition 2.1 Let K ⊂ Y be a convex cone and V_{1}, V_{2} ∈ 2^{Y}. We define:

V_{1} ≼ V_{2} ⇔ V_{2} ⊂ V_{1} + K;
V_{1} ⋞ V_{2} ⇔ V_{1} ⊂ V_{2} − K.

Both relations can also be expressed by the ordering ≤_{K} in Y which is defined by y_{1} ≤_{K} y_{2} iff y_{2} − y_{1} ∈ K:

V_{1} ≼ V_{2} ⇔ ∀v_{2} ∈ V_{2} ∃v_{1} ∈ V_{1} : v_{1} ≤_{K} v_{2};  (1)
V_{1} ⋞ V_{2} ⇔ ∀v_{1} ∈ V_{1} ∃v_{2} ∈ V_{2} : v_{1} ≤_{K} v_{2}.  (2)

Furthermore, it is an easy task to show the following assertion:

V_{1} ⋞ V_{2} ⇔ −V_{2} ≼ −V_{1}.  (3)

The relations ≼ and ⋞ are reflexive and transitive. We have no antisymmetry but

(V_{1} ≼ V_{2}, V_{2} ≼ V_{1}) ⇔ V_{1} + K = V_{2} + K;  (4)
(V_{1} ⋞ V_{2}, V_{2} ⋞ V_{1}) ⇔ V_{1} − K = V_{2} − K.  (5)

Furthermore, it can be shown that

(V_{1} ≼ V_{2}, V_{3} ≼ V_{4}) ⇒ V_{1} + V_{3} ≼ V_{2} + V_{4};  (6)
(V_{1} ⋞ V_{2}, V_{3} ⋞ V_{4}) ⇒ V_{1} + V_{3} ⋞ V_{2} + V_{4};  (7)

and for α ≥ 0 we have

V_{1} ≼ V_{2} ⇒ αV_{1} ≼ αV_{2};  (8)
V_{1} ⋞ V_{2} ⇒ αV_{1} ⋞ αV_{2}.  (9)

Note that K has not to be pointed (K is pointed iff K ∩ −K = {0}) for proving (1) - (9). Let V ∈ 2^{Y} be a subset of Y. We say that v̄ ∈ V is a ≤_{K}-minimal (≤_{K}-maximal) element of V if v ∈ V, v ≤_{K} v̄ (v̄ ≤_{K} v) implies v = v̄. The set of all ≤_{K}-minimal (≤_{K}-maximal) elements of V is denoted by Min V (Max V). If K is a convex pointed cone then we have

(V_{1} ≼ V_{2}, V_{2} ≼ V_{1}) ⇒ Min (V_{1}) = Min (V_{2});  (10)
(V_{1} ⋞ V_{2}, V_{2} ⋞ V_{1}) ⇒ Max (V_{1}) = Max (V_{2}).  (11)

Definition 2.2 A subset V ⊂ Y is said to be lower externally stable iff:

∀v ∈ V ∃v̄ ∈ Min V : v̄ ≤_{K} v.

A subset V ⊂ Y is said to be upper externally stable iff:

∀v ∈ V ∃v̄ ∈ Max V : v ≤_{K} v̄.

Lemma 2.3 If V_{1}, V_{2} ∈ 2^{Y} are lower externally stable, then it holds

V_{1} ≼ V_{2} ⇔ Min V_{1} ≼ Min V_{2},  (12)

and if V_{1}, V_{2} ∈ 2^{Y} are upper externally stable, we have

V_{1} ⋞ V_{2} ⇔ Max V_{1} ⋞ Max V_{2}.  (13)

Proof. Let V_{1} ≼ V_{2}, i.e. for each v_{2} ∈ V_{2}, in particular for each v̄_{2} ∈ Min V_{2} there exists some v_{1} ∈ V_{1} such that v_{1} ≤_{K} v̄_{2} and by the external stability for each v_{1} there exists v̄_{1} ∈ Min V_{1} such that v̄_{1} ≤_{K} v_{1}. Hence, for each v̄_{2} ∈ Min V_{2} there exists v̄_{1} ∈ Min V_{1} such that v̄_{1} ≤_{K} v̄_{2}, i.e. Min V_{1} ≼ Min V_{2}.

To show the opposite conclusion suppose that Min V_{1} ≼ Min V_{2}. Since V_{2} is externally stable, for each v_{2} ∈ V_{2} there exists v̄_{2} ∈ Min V_{2} and since Min V_{1} ≼ Min V_{2}, we obtain some v_{1} ∈ Min V_{1} ⊂ V_{1} such that v_{1} ≤_{K} v̄_{2} ≤_{K} v_{2}. Hence, V_{1} ≼ V_{2}.

Assertion (13) follows from (12) and (3). ■

It is easy to verify that the following conditions are satisfied:

∀V ∈ 2^{Y} : V ≼ ∅, Y ≼ V, ∅ ⋞ V, V ⋞ Y

\begin{matrix}∅ ≼ V ⇒ V = ∅ & , & V ≼ Y ⇒ Y = V + K \\ V ⋞ ∅ ⇒ V = ∅ & , & Y ⋞ V ⇒ Y = V − K. \end{matrix}

These relations motivate the following boundedness concepts for subsets of 2^{Y}. The crucial point is that we have different definitions of boundedness depending of the ordering relations ≼ and ⋞.

Definition 2.4 A subset V ⊂ 2^{Y} is said to be ≼-bounded below if there exists some topologically bounded subset Ṽ ⊂ Y such that for all V ∈ V it holds Ṽ ≼ V. The set Ṽ is called lower ≼-bound of V.

Definition 2.5 A subset V ⊂ 2^{Y} is said to be ≼-bounded above if there exists some nonempty subset Ṽ ⊂ Y such that for all V ∈ V it holds V ≼ Ṽ. The set Ṽ is called upper ≼-bound of V.

Definition 2.6 A subset V ⊂ 2^{Y} is said to be ⋞-bounded below if there exists some nonempty subset Ṽ ⊂ Y such that for all V ∈ V it holds Ṽ ⋞ V. The set Ṽ is called lower ⋞-bound of V.

Definition 2.7 A subset V ⊂ 2^{Y} is said to be ⋞-bounded above if there exists some topologically bounded subset Ṽ ⊂ Y such that for all V ∈ V it holds V ⋞ Ṽ. The set Ṽ is called upper ⋞-bound of V.

Definition 2.8 A subset V ⊂ 2^{Y} is said to be ≼-bounded if it is ≼-bounded below and ≼-bounded above. A subset V ⊂ 2^{Y} is said to be ⋞-bounded if it is ⋞-bounded below and ⋞-bounded above.

Note that we have a kind of duality between nonempty sets and topologically bounded sets in the definition of the above boundedness concepts. This duality can be observed throughout the paper. Further, we have to take care using the symbol ∅: It denotes an element of 2^{Y} but of course a subset V ⊂ 2^{Y} can also be the empty set.

3 Scalarization Methods on 2^{Y}

We present several nonlinear scalarization functionals defined on subsets of 2^{Y}. These functionals could be considered as generalizations of the scalarization functionals of [5], [6], [9] and [8].

Theorem 3.1 Let Y be a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \ −cl K. Let V ⊂ 2^{Y} be nonempty and ≼-bounded. The functional

z : V → ℝ, z(V) := inf{t ∈ ℝ : tk^{0} + V_{0} ⊂ V + cl K},

where V_{0} is an upper ≼-bound of V, has the following properties

(i) z is bounded;

(ii) ∀V ∈ V ∀α ∈ ℝ : z(V + αk^{0}) = z(V) + α;

(iii) V_{1} ≼ V_{2} ⇒ z(V_{1}) ≤ z(V_{2}).

Proof. Let V ∈ V be given. Define Γ_{V} := {t ∈ ℝ : tk^{0} + V_{0} ⊂ V + cl K}. Since V_{0} is an upper ≼-bound, we have V_{0} ≠ ∅ and V_{0} ⊂ V + K. Hence V ≠ ∅ and z(V) ≤ 0. We prove that Γ_{V} is bounded below with the same constant for all V ∈ V, i.e. z is bounded below. Assuming the contrary, for all n ∈ ℕ we can find some t_{n} ≤ −n and some V_{n} ∈ V such that −nk^{0} + V_{0} = (−n − t_{n})k^{0} + t_{n}k^{0} + V_{0} ⊂ K \ −cl K + V_{n} + cl K ⊂ V_{n} + cl K. Since V is ≼-bounded below, there exists some topologically bounded set Ṽ ⊂ Y such that for all n ∈ ℕ it holds −nk^{0} + V_{0} ⊂ V_{n} + cl K ⊂ Ṽ + K + cl K ⊂ Ṽ + cl K. Hence for an arbitrary v_{0} ∈ V_{0} and for all n ∈ ℕ there exists some v_{n} ∈ Ṽ such that −k^{0} + v_{0}/n − v_{n}/n ∈ cl K. Letting n → ∞ we get k^{0} ∈ −cl K which contradicts the assumption k^{0} ∈ K \ −cl K. Consequently, z is bounded.

Assertion (ii) is obviously. To show (iii) let V_{1} ≼ V_{2}. Choose an arbitrary t ∈ Γ_{V_{2}}. It follows that tk^{0} + V_{0} ⊂ V_{2} + cl K ⊂ V_{1} + K + cl K ⊂ V_{1} + cl K, i.e. t ∈ Γ_{V_{1}}. Hence Γ_{V_{2}} ⊂ Γ_{V_{1}} which implies that z(V_{1}) ≤ z(V_{2}). ■

Theorem 3.2 Let Y be a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \ −cl K. Let V ⊂ 2^{Y} be nonempty and ⋞-bounded. The functional

z̄: V → ℝ, z̄(V) := inf{t ∈ ℝ : V ⊂ V_{0} + tk^{0} − cl K},

where V_{0} is an upper ⋞-bound of V, has the following properties

(i) z̄ is bounded.

(ii) ∀V ∈ V ∀α ∈ ℝ : z̄(V + αk^{0}) = z̄(V) + α;

(iii) V_{1} ⋞ V_{2} ⇒ z̄(V_{1}) ≤ z̄(V_{2}).

Proof. Let V ∈ V be given. Define Γ_{V} := {t ∈ ℝ : V ⊂ V_{0} + tk^{0} − cl K}. Since V_{0} is an upper ⋞-bound we have that V_{0} is topologically bounded and V ⊂ V_{0} − K. The last condition implies that 0 ∈ Γ_{V} for all V ∈ V. Hence z̄ is bounded above. We prove that Γ_{V} is bounded below with the same constant for all V ∈ V, i.e. z̄ is bounded below. Assuming the contrary, for all n ∈ ℕ we can find some t_{n} ≤ −n and some V_{n} ∈ V such that V_{n} ⊂ V_{0} + (t_{n} + n)k^{0} − nk^{0} − cl K ⊂ V_{0} − nk^{0} − cl K. Since V is ⋞-bounded below, there exists some nonempty subset Ṽ ⊂ Y such that Ṽ ⊂ V − K for all V ∈ V. Choosing ṽ ∈ Ṽ we obtain ṽ ∈ V_{n} − K ⊂ V_{0} − nk^{0} − cl K − K ⊂ V_{0} − nk^{0} − cl K for all n ∈ ℕ. Hence for all n ∈ ℕ there exists (v_{0})_{n} ∈ ℕ such that ṽ/n−(v_{0})_{n}/n+k^{0} ∈ −cl K. Letting n → ∞ we get k^{0} ∈ −cl K which contradicts the assumption k^{0} ∈ K \ −cl K. Consequently z̄ is bounded.

Assertion (ii) is obviously. To show (iii) let V_{1} ⋞ V_{2}. Choose t ∈ Γ_{V_{2}} arbitrarily. It follows that V_{1} ⊂ V_{2} − K ⊂ V_{0} + tk^{0} − cl K − K ⊂ V_{0} + tk^{0} − cl K, i.e. t ∈ Γ_{V_{1}}. Hence Γ_{V_{2}} ⊂ Γ_{V_{1}} which implies that z̄(V_{1}) ≤ z̄(V_{2}). ■

Let Y be a topological vector space and K ⊂ Y a convex cone. We use the following assumption for weakening the boundedness condition.

(C)

There exists a proper convex cone C ⊂ Y with nonempty interior satisfying K ⊂ cl C.

Remark 3.3 If Y is a locally convex space and K ⊂ Y a convex cone such that K \ −cl K ≠ ∅ then assumption (C) is satisfied and there exists an element k^{0} ∈ K ∩ int C. Indeed, if k^{0} ∈ K \ −cl K then we have {−k^{0}} ∩ cl K = ∅ and we can apply a classical separation theorem on the convex compact set {−k^{0}} and the closed convex set cl K. We obtain the existence of a linear continuous functional y^{∗} ∈ Y^{∗} such that y^{∗}(−k^{0}) < 0 ≤ y^{∗}(k) for all k ∈ cl K. The appropriated cone C can be defined as C := {y ∈ Y : y^{∗}(y) ≥ 0}.

Theorem 3.4 Let Y be a topological vector space, K ⊂ Y a convex cone satisfying assumption (C). Let k^{0} ∈ K ∩ int C and ∅ ≠ V ⊂ 2^{Y} be such that the following conditions are satisfied:

∀V ∈ V : V ≠ ∅;  (14)
∃ ỹ ∈ Y : ∀V ∈ V : V ∩ (ỹ − int C) = ∅.  (15)

Then, the functional

z_{C} : V → ℝ, z_{C}(V) := inf {t ∈ ℝ : tk^{0} ∈ V + cl C}

has the following properties:

(i) z_{C} is bounded below and attains its values in ℝ;

(ii) ∀V ∈ V ∀α ∈ ℝ : z_{C}(V + αk^{0}) = z_{C}(V) + α;

(iii) V_{1} ≼ V_{2} ⇒ z_{C}(V_{1}) ≤ z_{C}(V_{2});

(iv) If V consists of compact sets V ⊂ Y and K \ {0} ⊂ int C then

(V_{1} ≼ V_{2} , V_{1} ∩ V_{2} = ∅) ⇒ z_{C}(V_{1}) < z_{C}(V_{2}).

Proof. For V ⊂ Y we define Γ_{V} := {t ∈ ℝ : tk^{0} ∈ V + cl C}. We show that Γ_{V} is nonempty for all V ∈ V. We choose v ∈ V which is possible according to (14). Because of k^{0} ∈ int C there exists an absorbing neighborhood U of zero such that U ⊂ −k^{0} + int C. Since U is absorbing, there exists some σ > 0 such that −v ∈ σU. Hence v ∈ −σU ⊂ σ(k^{0} − int C) ⊂ σk^{0} − cl C. We obtain that σk^{0} ∈ V + cl C, i.e. σ ∈ Γ_{V}. Hence z_{C}(V) ≠ ∞ for all V ∈ V.

We prove that Γ_{V} is bounded below with the same constant for all V ∈ V. Assuming the contrary, for all n ∈ ℕ we can find t_{n} ≤ −n and V_{n} ∈ V such that −nk^{0} = (−n − t_{n})k^{0} + t_{n}k^{0} ∈ cl C + V_{n} + cl C ⊂ V_{n} + cl C. Hence, for all n ∈ ℕ there exists some v_{n} ∈ V_{n} such that

−k^{0} − ỹ/n ∈ (v_{n} − ỹ)/n + cl C ⊂^{(15)} (Y \ −int C) + cl C ⊂ Y \ −int C.

Letting n → ∞ we get k^{0} ∉ int C which contradicts the assumption k^{0} ∈ K ∩ int C. Consequently, z_{C} is bounded below.

Assertions (ii) and (iii) are obviously. To show (iv) let {t_{n}}_{n∈ℕ} ⊂ Γ_{V_{2}} with t_{n} → z_{C}(V_{2}). By assumption V_{2} is compact. Without loss of generality we can find a sequence {v_{n}}_{n∈ℕ} ⊂ V_{2} such that v_{n} → v̄ ∈ V_{2} and t_{n}k^{0} ∈ v_{n} + cl C. Hence z_{C}(V_{2})k^{0} ∈ v̄ + cl C. Let V_{1} ≼ V_{2} and V_{1}∩V_{2} = ∅. We have v̄ ∈ V_{2} ⊂ V_{1}+K \{0} ⊂ V_{1}+int C. Hence there is some δ > 0 such that v̄−δk^{0} ∈ V_{1}+int C. It follows z_{C}(V_{2})k^{0} ∈ v̄+cl C ⊂ δk^{0}+V_{1}+cl C. Applying assertion (ii) we obtain z_{C}(V_{1}) + δ = z_{C}(V_{1} + δk^{0}) ≤ z_{C}(V_{2}). ■

Remark 3.5 If V ⊂ 2^{Y} is ≼-bounded below then (15) is satisfied. Indeed, let Ṽ be a lower bound for Y, i.e. Ṽ is topologically bounded and V ⊂ Ṽ + K. If (15) does not hold, then for all n ∈ ℕ there exists v_{n} ∈ V_{n} ∈ V such that v_{n} ∈ −nk^{0} − int C. Since v_{n} ∈ Ṽ + K, there is some ṽ_{n} ∈ Ṽ such that ṽ_{n} ∈ v_{n} − K. We obtain ṽ_{n} ∈ −nk^{0} − int C − K ⊂ −nk^{0} − cl C, hence ṽ_{n}/n + k^{0} ∈ −cl C. Letting n → ∞ yields k^{0} ∈ −cl C. By assumption we have k^{0} ∈ int C. It follows 0 = k^{0} − k^{0} ∈ int C + cl C ⊂ int C. This contradicts the assumption that C is proper.

The preceding remark shows that assumption (15) is not stronger than the assumption that V is ≼-bounded below. Moreover, it can be easily seen that (15) is in fact weaker.

Example 3.6 Let C = K = ℝ_{+}^{2} and V = {Y \ −int C}. Then (15) is satisfied for ỹ = 0. However, V is not ≼-bounded below.

Theorem 3.7 Let Y be a topological vector space, K ⊂ Y a convex cone satisfying assumption (C). Let k^{0} ∈ K ∩ int C and ∅ ≠ V ⊂ 2^{Y} be such that

∀V ∈ V ∃ Ṽ_{V} ∈ Y : Ṽ_{V} is topologically bounded and V ⊂ Ṽ_{V} − cl C;  (16)
∃ ỹ ∈ Y : ∀V ∈ V : (V − ỹ) ∩ (Y \ −int C) ≠ ∅.  (17)

Then, the functional

z̄_{C} : V → ℝ, z̄_{C}(V) := inf {t ∈ ℝ : V ⊂ tk^{0} − cl C}

has the following properties

(i) z̄_{C} is bounded below and attains its values in ℝ;

(ii) ∀V ∈ V ∀α ∈ ℝ : z̄_{C}(V + αk^{0}) = z̄_{C}(V) + α;

(iii) V_{1} ⋞ V_{2} ⇒ z̄_{C}(V_{1}) ≤ z̄_{C}(V_{2});

(iv) If V consists of compact sets V ⊂ Y and K \ {0} ⊂ int C then

(V_{1} ⋞ V_{2} , V_{1} ∩ V_{2} = ∅) ⇒ z̄_{C}(V_{1}) < z̄_{C}(V_{2}).

Proof. For V ∈ 2^{Y} we define Γ_{V} := {t ∈ ℝ : V ⊂ tk^{0} − cl C}. Initially we show that Γ_{V} is nonempty for all V ∈ V. We have k^{0} ∈ int C, hence there exists a neighborhood U of zero such that U ⊂ −k^{0} + int C. Since Ṽ_{V} in (16) is topologically bounded, there exists some σ > 0 such that −Ṽ_{V} ⊂ σU. Hence Ṽ_{V} ⊂ −σU ⊂ σ(k^{0}−int C) ⊂ σk^{0}−cl C. By (16) we obtain that V ⊂ Ṽ_{V} − cl C ⊂ σk^{0} − cl C − cl C ⊂ σk^{0} − cl C, i.e. σ ∈ Γ_{V}.

We prove that Γ_{V} is bounded below with the same constant for all V ∈ V. Assuming the contrary, for all n ∈ ℕ we can find some t_{n} ≤ −n and some V_{n} ∈ V such that V_{n} ⊂ (t_{n} + n)k^{0} − nk^{0} − cl C ⊂ −nk^{0} − cl C. For all n ∈ ℕ we may deduce from (17) the existence of a v_{n} ∈ (V_{n} − ỹ) ∩ (Y \ −int C) and we obtain

−k^{0} − ỹ/n ∈ v_{n}/n + cl C ⊂ (Y \ −int C) + cl C ⊂ Y \ −int C

since int C = Y \(cl (Y \cl C)).

Letting n → ∞ we get k^{0} ∉ int C which contradicts the assumption k^{0} ∈ K ∩ int C. Consequently, z̄_{C} is bounded below.

Assertions (ii) and (iii) are obviously. It remains to prove (iv). By the definition of the infimum there exists a sequence {v_{n}}_{n∈ℕ} ⊂ V_{1} such that v_{n} ∉ (z̄_{C}(V_{1})−1/n)k^{0}−cl C. Without loss of generality, by the compactness of V_{1}, {v_{n}}_{n∈ℕ} ⊂ V_{1} can assumed to be convergent to some v̄ ∈ V_{1}. Hence

v̄ ∉ z̄_{C}(V_{1})k^{0} − int C.  (18)

Let V_{1} ⋞ V_{2} and V_{1} ∩ V_{2} = ∅. We have v̄ ∈ V_{2} − K \ {0} ⊂ V_{2} − int C. Choose v ∈ V_{2} arbitrarily. Then there exists some δ > 0 such that

v̄ ∈ v − δk^{0} − int C.  (19)

Assuming that v ∈ (z̄_{C}(V_{1}) + δ)k^{0} − cl C we obtain

v̄ ∈^{(19)} v − δk^{0} − int C ⊂ z̄_{C}(V_{1})k^{0} − cl C − int C ⊂ z̄_{C}(V_{1})k^{0} − int C.

This contradicts (18). Hence we have v ∉ (z̄_{C}(V_{1}) + δ)k^{0} − cl C. Since v ∈ V_{2} it follows z̄_{C}(V_{1}) + δ ≤ z̄_{C}(V_{2}), i.e. z̄_{C}(V_{1}) < z̄_{C}(V_{2}). ■

Remark 3.8 If V ⊂ 2^{Y} is ⋞-bounded below, then (17) is satisfied. Indeed, let Ṽ be lower ⋞-bound of V, i.e. Ṽ ≠ ∅ and Ṽ ⊂ V − K. Fix some ỹ ∈ Ṽ. Then, for all V ∈ V there exists some v ∈ V such that v − ỹ ∈ K ⊂ cl C ⊂ Y \ −int C, i.e. (V − ỹ) ∩ (Y \ −int C) ≠ ∅.

To see that assumption (17) is in fact weaker than the assumption that V is ⋞-bounded below consider the following example.

Example 3.9 Let C = K = ℝ_{+}^{2} and V = {{y} : y ∈ Y \ −int C}. Then (17) is satisfied for ỹ = 0. However, V is not ⋞-bounded below.

4 Basic Tools

For the convenience of the reader we present two basic tools for the proof of our minimal set theorems.

4.1 The Brézis-Browder Principle

The first tool is a very general existence principle for minimal points in quasi-ordered sets due to Brézis and Browder [1], 1976.

Theorem 4.1 Let (W, ⪯) be a quasi-ordered set (i.e. ⪯ is a reflexive and transitive relation on W) and let ϕ : W → ℝ be a function satisfying

(A1) ϕ is bounded below; (A2) w_{1} ⪯ w_{2} ⇒ ϕ(w_{1}) ≤ ϕ(w_{2});

(A3) For every ⪯-decreasing sequence {w_{n}}_{n∈ℕ} ⊂ W there exists some w ∈ W such that w ⪯ w_{n} for all n ∈ ℕ.

Then, for every w_{0} ∈ W there exists some w̄ ∈ W such that

(i) w̄ ⪯ w_{0};

(ii) ŵ ⪯ w̄ ⇒ ϕ(ŵ) = ϕ(w̄).

Proof. See [1, Corollary 1]. ■

4.2 Uniform Spaces

Our results involve a uniform space X (cf. [13]). Examples for uniform spaces not being necessarily metrizable are locally convex spaces, topological vector spaces and K-metric spaces (see [19] or [17]). If the reader is only interested in results for metric spaces the following considerations can be skipped. Then one has to replace the “families of quasi-metrics”, in the following denoted by {q_{λ}}_{λ∈Λ} or by q_{Λ}, by the metric.

In [10] we presented a characterization of uniform spaces via families of quasi-metrics introduced by Fang [3]. We shall give a short summary of these results.

Definition 4.2 Let X be a nonempty set and let (Λ, ≺) be a directed set. A system {q_{λ}}_{λ∈Λ} of functions q_{λ} : X × X → [0, ∞) satisfying

(Q1) ∀λ ∈ Λ ∀x ∈ X : q_{λ}(x, x) = 0;

(Q2) ∀λ ∈ Λ ∀x, y ∈ X : q_{λ}(x, y) = q_{λ}(y, x);

(Q3) ∀λ ∈ Λ ∃μ ∈ Λ with λ ≺ μ : ∀x, y, z ∈ X : q_{λ}(x, y) ≤ q_{μ}(x, z) + q_{μ}(z, y);

(Q4) λ ≺ μ ⇒ (∀x, y ∈ X : q_{λ}(x, y) ≤ q_{μ}(x, y))

is called a family of quasi-metrics. If, in addition, the condition

(Q5) (∀λ ∈ Λ : q_{λ}(x, y) = 0) ⇒ x = y

is satisfied, the family of quasi-metrics is said to be separating.

Theorem 4.3 A topological space (X, τ) is a (separated) uniform space iff its topology τ can be generated by a (separating) family of quasi-metrics.

Proof. See [10]. ■

Convention 4.4 For an easy dealing with uniform spaces we introduce the following notation. Let {q_{λ}}_{λ∈Λ} the family of quasi-metrics which generates the topology of the uniform space X. We write q_{Λ} iff an assertion holds for all λ ∈ Λ. If X is a metric space, then q_{Λ} is its metric.

5 Minimal Set Theorems

This section contains the main results of this paper. We present two minimal set theorems with respect to the set ordering relations introduced in Section 2. We consider a subset A of X × 2^{Y}, where X is a separated uniform space and Y is a topological vector space. For a subset A ⊂ X × 2^{Y} we introduce the following notation:

V(A) := {V ∈ 2^{Y} : (x, V) ∈ A}.

5.1 Minimal Set Theorem I

Using the relation ≼ we introduce the following ordering relation on X × 2^{Y}

(x_{1}, V_{1}) ≼_{k^{0}} (x_{2}, V_{2}) ⇔ V_{1} + k^{0}q_{Λ}(x_{1}, x_{2}) ≼ V_{2}.

According to Convention 4.4, the last inequality has to be read as

∀λ ∈ Λ : V_{1} + k^{0}q_{λ}(x_{1}, x_{2}) ≼ V_{2}.

It is easy to verify that this relation is reflexive and transitive.

Theorem 5.1 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \ −cl K. Let A be a nonempty subset of X × 2^{Y} such that for some (x_{0}, V_{0}) ∈ A and for A_{0} := {(x, V) ∈ A : (x, V) ≼_{k^{0}} (x_{0}, V_{0})} the following conditions are satisfied:

(M1) V(A_{0}) is ≼-bounded above (i.e. V_{0} is nonempty);

(M2) V(A_{0}) is ≼-bounded below;

(M3) For every ≼_{k^{0}}-decreasing sequence {(x_{n}, V_{n})}_{n∈ℕ} ⊂ A_{0} there exists some (x, V) ∈ A_{0} such that (x, V) ≼_{k^{0}} (x_{n}, V_{n}) for all n ∈ N.

Then, there exists (x̄, V̄) ∈ A such that

(i) (x̄, V̄) ≼_{k^{0}} (x_{0}, V_{0});

(ii) ((x̂, V̂) ∈ A, (x̂, V̂) ≼_{k^{0}} (x̄, V̄)) ⇒ x̂ = x̄.

Under the additional assumption (C) and if k^{0} ∈ K ∩ int C, (M2) can be relaxed to

(M2’) ∃ỹ ∈ Y ∀V ∈ V(A_{0}) : V ∩ (ỹ − int C) = ∅.

If, in addition, k^{0} ∈ K \ {0} ⊂ int C and if for each (x, V) ∈ A_{0}, V is compact, then (ii) can be strengthened to

(ii’) ((x̂, V̂) ∈ A, (x̂, V̂) ≼_{k^{0}} (x̄, V̄)) ⇒ (x̂ = x̄ and V̂ ∩ V̄ ≠ ∅).

Proof. We show that for the quasi-ordered set (A_{0}, ≼_{k^{0}}) and for the functional

ϕ : A_{0} → ℝ, ϕ(x, V) := z(V),

where z : V(A_{0}) → ℝ is the scalarization functional of Theorem 3.1, the assumptions of the Brézis-Browder principle are satisfied. Note that the definition of z involves the upper ≼-bound V_{0}. By (M1), (M2) and Theorem 3.1 (i) it follows that ϕ is bounded on A_{0}, in particular, ϕ is bounded below and ϕ(x, V) ≠ ∞. Theorem 3.1 (ii) and (iii) yield

(x_{1}, V_{1}) ≼_{k^{0}} (x_{2}, V_{2}) ⇒ ϕ(x_{1}, V_{1}) + q_{Λ}(x_{1}, x_{2}) ≤ ϕ(x_{2}, V_{2}).  (20)

Hence, ϕ is ≼_{k^{0}}-monotone on A_{0}, i.e. assumption (A2) of Theorem 4.1 is satisfied. Of course, (M3) implies assumption (A3).

Theorem 4.1 yields the existence of an element (x̄, V̄) ∈ A_{0} (i.e. (i) holds) such that

((x̂, V̂) ∈ A_{0}, (x̂, V̂) ≼_{k^{0}} (x̄, V̄)) ⇒ ϕ(x̂, V̂) = ϕ(x̄, V̄).  (21)

Let (x̂, V̂) ∈ A such that (x̂, V̂) ≼_{k^{0}} (x̄, V̄). The transitivity of ≼_{k^{0}} yields (x̂, V̂) ∈ A_{0}. Applying (21) and (20) we obtain q_{Λ}(x̂, x̄) = 0. Since X is separated, we have x̂ = x̄, i.e. (ii) holds.

To see that (M2) can be replaced by (M2’) we proceed as above but using the functional

ϕ_{C} : A_{0} → ℝ, ϕ_{C}(x, V) := z_{C}(V)

where z_{C} : V(A_{0}) → ℝ is the scalarization functional of Theorem 3.4. Note that we have ∅ ∉ V(A_{0}) because V_{0} is an upper ≼-bound of V(A_{0}). To prove (ii’) assume that V̂ ∩ V̄ = ∅. Then, assertion (iv) of Theorem 3.4 yields ϕ_{C}(x̂, V̂) < ϕ_{C}(x̄, V̄). This contradicts (21). ■

5.2 Minimal Set Theorem II

Using the ⋞-relation we present a second minimal set theorem. We introduce the following ordering relation on X × 2^{Y}:

(x_{1}, V_{1}) ⋞_{k^{0}} (x_{2}, V_{2}) ⇔ V_{1} + k^{0}q_{Λ}(x_{1}, x_{2}) ⋞ V_{2}

which is also a reflexive and transitive relation.

Theorem 5.2 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \ −cl K. Let A be a nonempty subset of X × 2^{Y} such that for some (x_{0}, V_{0}) ∈ A and for A_{0} := {(x, V) ∈ A : (x, V) ⋞_{k^{0}} (x_{0}, V_{0})} the following conditions are satisfied:

(M1) V(A_{0}) is ⋞-bounded above (i.e. V_{0} ⋞ Ṽ_{0} where Ṽ_{0} is topologically bounded);

(M2) V(A_{0}) is ⋞-bounded below;

(M3) For every ⋞_{k^{0}}-decreasing sequence {(x_{n}, V_{n})}_{n∈ℕ} ⊂ A_{0} there exists some (x, V) ∈ A_{0} such that (x, V) ⋞_{k^{0}} (x_{n}, V_{n}) for all n ∈ N.

Then, there exists (x̄, V̄) ∈ A such that

(i) (x̄, V̄) ⋞_{k^{0}} (x_{0}, V_{0});

(ii) ((x̂, V̂) ∈ A, (x̂, V̂) ⋞_{k^{0}} (x̄, V̄)) ⇒ x̂ = x̄.

Under the additional assumption (C) and if k^{0} ∈ K ∩ int C, (M1) and (M2) can be relaxed to

(M1’) There exists a topologically bounded subset Ṽ_{0} ⊂ Y such that V_{0} ⊂ Ṽ_{0} − cl C;

(M2’) There exists ỹ ∈ Y such that

∀V ∈ V(A_{0}) : (V − ỹ) ∩ (Y \ −int C) ≠ ∅.

If, in addition, k^{0} ∈ K \ {0} ⊂ int C and if for each (x, V) ∈ A_{0}, V is compact, then (ii) can be strengthened to

(ii’) ((x̂, V̂) ∈ A, (x̂, V̂) ⋞_{k^{0}} (x̄, V̄)) ⇒ (x̂ = x̄ and V̂ ∩ V̄ ≠ ∅).

Proof. We show that for the quasi-ordered set (A_{0}, ⋞_{k^{0}}) and for the functional

ϕ : A_{0} → ℝ, ϕ(x, V) := z̄(V)

the assumptions of the Brézis-Browder principle are satisfied. Note that the definition of z̄ involves the upper ⋞-bound Ṽ_{0}.

Since (M1), (M2) and Theorem 3.2 (i), ϕ is bounded, in particular, ϕ is bounded below and ϕ(x, V) ≠ ∞ for all (x, V) ∈ A_{0}. Theorem 3.2 (ii) and (iii) yields

(x_{1}, V_{1}) ⋞_{k^{0}} (x_{2}, V_{2}) ⇒ ϕ(x_{1}, V_{1}) + q_{Λ}(x_{1}, x_{2}) ≤ ϕ(x_{2}, V_{2}).  (22)

It follows that ϕ is ⋞_{k^{0}}-monotone on A_{0}, i.e. assumption (A2) of Theorem 4.1 is satisfied. Of course, (M3) implies assumption (A3). Theorem 4.1 yields the existence of an element (x̄, V̄) ∈ A_{0} (i.e. (i) holds) such that

((x̂, V̂) ∈ A_{0}, (x̂, V̂) ⋞_{k^{0}} (x̄, V̄)) ⇒ ϕ(x̂, V̂) = ϕ(x̄, V̄).  (23)

Let (x̂, V̂) ∈ A such that (x̂, V̂) ⋞_{k^{0}} (x̄, V̄). The transitivity of ⋞_{k^{0}} yields (x̂, V̂) ∈ A_{0}. Applying (23) and (22) we obtain q_{Λ}(x̂, x̄) = 0. Since X is separated, we have x̂ = x̄, i.e. (ii) holds.

To see that (M1) and (M2) can be replaced by (M1’) and (M2’) proceed as above but using the functional

ϕ_{C} : A_{0} → ℝ, ϕ_{C}(x, V) := z̄_{C}(V)

where z̄_{C} : V(A_{0}) → ℝ is the scalarization functional of Theorem 3.7. It remains to show that (16) of Theorem 3.7 is satisfied. Indeed, by (M1’) there exists a topologically bounded set Ṽ_{0} ⊂ Y such that V_{0} ⊂ (Ṽ_{0} − cl C). For V ∈ V(A_{0}) it holds V ⋞ V_{0}. This implies V ⊂ V_{0} − K ⊂ Ṽ_{0} − cl C − cl C ⊂ Ṽ_{0} − cl C. Hence, (16) is satisfied.

To prove (ii’) assume that V̂ ∩ V̄ = ∅. Theorem 3.7 (iv) yields ϕ_{C}(x̂, V̂) < ϕ_{C}(x̄, V̄). This contradicts (23). ■

6 Set-Valued Ekeland’s Principles

In this section we present several conclusions of our main results, Theorem 5.1 and 5.2. Recently, Truong [22] proved a variant of Ekeland’s variational principle using one of the ordering relations of Kuroiwa. A similar but more general variant can be obtained from Theorem 5.1. A new variational principle will be derived from Theorem 5.2. Another variant of Ekeland’s principle for set-valued maps (cf. [2], [10]) as well as a minimal point theorem in X × Y are consequences of Theorem 5.1 as well as of Theorem 5.2.

Let X be a set, Y a linear topological space, F : X → 2^{Y} a set-valued mapping and K a convex cone in Y. In contrast to known definitions we call the set

graph F := {(x, V) ∈ X × 2^{Y} : V = F (x)}

the graph of F and the image of a subset M ⊂ X (see also [22]) is defined by

F (M) := {F (x) : x ∈ M}.

Note that graph F is a subset of X × 2^{Y}, not of X × Y and the image F (M) is a subset of 2^{Y}, not of Y!

Moreover, the usual concept of the domain of a set-valued map (dom F := {x ∈ X : F (x) ≠ ∅}) is not appropriated for our considerations.

Definition 6.1 The set of all x ∈ X for which F (x) ∈ 2^{Y} is ≼-bounded is called the domain of F with respect to ≼ and is denoted by ≼-dom F.

The set of all x ∈ X for which F (x) ∈ 2^{Y} is ⋞-bounded is called the domain of F with respect to ⋞ and is denoted by ⋞-dom F.

Definition 6.2 A point x ∈ ≼-dom F is said to be a ≼-minimal point of F iff

F (u) ≼ F (x) ⇒ F (x) ≼ F (u),

and a point x ∈ ⋞-dom F is said to be a ⋞-minimal point of F iff

F (u) ⋞ F (x) ⇒ F (x) ⋞ F (u).

Remark 6.3 If F has lower externally stable values and if K is pointed then by (10) and (12) x ∈ ≼-dom F is a ≼-minimal point of F iff

F (u) ≼ F (x) ⇒ Min F (u) = Min F (x).

If F has upper externally stable values and if K is pointed then by (11) and (13) x ∈ ⋞-dom F is a ⋞-minimal point of F iff

F (u) ⋞ F (x) ⇒ Max F (u) = Max F (x).

Example 6.4 Consider the special case that F : X → Y is a vector-valued function. Then, both kinds of a minimal point, as defined above, coincide with the well-known vector minimum. i.e. F (x) ≤_{K} F (x̄) implies F (x) = F (x̄).

Example 6.5 For the special case Y = ℝ, K = ℝ_{+} and F has lower externally stable values, a minimal point x̄ means that min{y ∈ F (x̄)} ≤ min{y ∈ F (x) : x ∈ X}.

Example 6.6 For the special case Y = ℝ, K = ℝ_{+} and F has upper externally stable values, a minimal point x̄ means that max{y ∈ F (x̄)} ≤ max{y ∈ F (x) : x ∈ X}.

Remark 6.7 An alternative minimality concept for set-valued maps was considered e.g. by Jahn [12]. Let ImF be the union of the sets F (x) over x ∈ X. Then (x̄, ȳ) is said to be a minimizer of F if ȳ is a ≤_{K}-minimal element in ImF.

Note that in the special situation of Example 6.5 this concept coincides with the ≼-minimality concept. However, Example 6.6 shows that the information with respect to the ⋞-minimality concept is lost by looking for minimizers in the union ImF = ∪_{x∈dom F} F (x) in this way.

6.1 Ekeland’s Principle for Set-Valued Maps I

First we give a variational principle involving the ordering relation ≼.

Theorem 6.8 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \ −cl K. Let F : X → 2^{Y} be a set-valued mapping, x_{0} ∈ X, S(x_{0}) := {x ∈ X : F (x) + k^{0}q_{Λ}(x, x_{0}) ≼ F (x_{0})} and A_{0} := {(x, V) ∈ graph F : x ∈ S(x_{0})} such that the following conditions are satisfied:

(E1) x_{0} ∈ ≼-dom F;

(E2) F (S(x_{0})) is ≼-bounded below;

(E3) For every ≼_{k^{0}}-decreasing sequence {(x_{n}, V_{n})}_{n∈ℕ} ⊂ A_{0} there exists some (x, V) ∈ A_{0} such that (x, V) ≼_{k^{0}} (x_{n}, V_{n}) for all n ∈ N.

Then, there exists x̄ ∈ ≼-dom F such that

(i) F (x̄) + k^{0}q_{Λ}(x̄, x_{0}) ≼ F (x_{0});

(ii) ∀x ≠ x̄ : ∃λ ∈ Λ : F (x) + k^{0}q_{λ}(x, x̄) ⋠ F (x̄).

Under the additional assumption (C) and if k^{0} ∈ K ∩ int C, (E2) can be relaxed to

(E2’) There exists some ỹ ∈ Y such that F (S(x_{0})) ∩ (ỹ − int C) = ∅.

Proof. Set A := graph F and apply Theorem 5.1. ■

The assumptions of Theorem 6.8 may look somewhat artificial. The next theorem presents a more familiar situation.

Theorem 6.9 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K\−cl K. Let F : X → 2^{Y} be a set-valued mapping and x_{0} ∈ X such that (E1) and (E2) of Theorem 6.8 are satisfied as well as

(E3’) For every x ∈ X the set S(x) := {x′ ∈ X : F (x′) + k^{0}q_{Λ}(x′, x) ≼ F (x)} is sequentially complete in X.

Then the assertions (i) and (ii) of Theorem 6.8 remain true.

If, additionally, (C) holds and if k^{0} ∈ K ∩ int C then (E2) can be replaced by assumption (E2’) of Theorem 6.8.

Proof. It remains to show that assumption (E3) is in force in the present situation. Indeed, let {(x_{n}, V_{n})}_{n∈ℕ} ⊂ A_{0} be a ≼_{k^{0}}-decreasing sequence, i.e. V_{n} = F (x_{n}) and

F (x_{n+1}) + k^{0}q_{Λ}(x_{n+1}, x_{n}) ≼ F (x_{n})  (24)

for all n ∈ ℕ. By the transitivity of ≼_{k^{0}} we get

F (x_{m}) + k^{0}q_{Λ}(x_{m}, x_{n}) ≼ F (x_{n})  (25)

for all m ≥ n, m ∈ ℕ. Applying the functional ϕ : A_{0} → ℝ from the proof of Theorem 5.1 to relation (24) we obtain

ϕ(x_{n+1}, F (x_{n+1})) + q_{Λ}(x_{n+1}, x_{n}) ≤ ϕ(x_{n}, F (x_{n})).

The sequence {ϕ(x_{n}, F (x_{n}))}_{n∈ℕ} is nonincreasing and bounded below, hence convergent and a fortiori a Cauchy sequence. Applying ϕ to (25) we may conclude that {x_{n}}_{n∈ℕ} is Cauchy as well. Since S(x_{0}) is sequentially complete, {x_{n}}_{n∈ℕ} converges to some x ∈ S(x_{0}). Moreover, (25) implies that x_{m} ∈ S(x_{n}) for all m ≥ n, m ∈ ℕ. Since by (E3’) S(x_{n}) is sequentially complete we must have x ∈ S(x_{n}) for all n ∈ ℕ. This means (x, F (x)) ≼_{k^{0}} (x_{n}, F (x_{n})) as desired.

To see that (M2) can be replaced by (M2’) the same considerations can be done taking ϕ_{C} instead of ϕ, defined in the proof of Theorem 5.1. ■

Remark 6.10 We indicate a special situation where (E3’) is satisfied. Let X be a sequentially complete separated uniform space. Ferro [4] introduced the concept of lower ≼-semicontinuity (D-lower semicontinuity in [4]) and Truong [22] proved an Ekelandtype theorem using this continuity property. A set-valued map F : X → 2^{Y} is said to be sequentially lower ≼-semicontinuous iff for every V ∈ 2^{Y} the set {x′ ∈ X : F (x′) ≼ V } is a sequentially closed subset of X. We claim that (E3’) is satisfied if F is sequentially lower ≼-semicontinuous with ≼-closed values (for each x ∈ X, F (x) + K is a closed set). Indeed, let {x_{n}}_{n∈ℕ} ⊂ S(x) be a sequence such that x_{n} → x′. Fix λ ∈ Λ. Then there exists μ ∈ Λ such that q_{λ}(x′, x) ≤ q_{μ}(x′, x_{n}) + q_{μ}(x_{n}, x). Since x_{n} → x′ we can find for each ε > 0 a number n_{ε} ∈ ℕ such that q_{μ}(x′, x_{n}) ≤ ε for all n ∈ ℕ, n > n_{ε}. Hence q_{μ}(x_{n}, x) ≥ q_{λ}(x′, x) − ε. Since F (x_{n}) + k^{0}q_{μ}(x_{n}, x) ≼ F (x), for n > n_{ε} holds F (x_{n}) + k^{0} (q_{λ}(x′, x) − ε) ≼ F (x). This implies F (x_{n}) ≼ F (x) − k^{0} (q_{λ}(x′, x) − ε) for n > n_{ε}. Since F is lower ≼-semicontinuous we have F (x′) ≼ F (x) − k^{0} (q_{λ}(x′, x) − ε), hence

F (x) ⊂ F (x′) + k^{0} (q_{λ}(x′, x) − ε) + K  (26)

for all n > n_{ε}. Assuming that F (x′) + k^{0}q_{λ}(x′, x) ⋠ F (x), i.e. F (x) ⊄ F (x′) + k^{0}q_{λ}(x′, x)+K we deduce the existence of y ∈ F (x) such that y ∉ F (x′)+k^{0}q_{λ}(x′, x)+K. Since F (x′) + k^{0}q_{λ}(x′, x) + K is closed, its complement Y \ (F (x′) + k^{0}q_{λ}(x′, x) + K) is open. Hence for ε′ > 0 sufficiently small, we have y + ε′k^{0} ∉ F (x′) + k^{0}q_{λ}(x′, x) + K contradicting (26) for ε′ = ε.

Remark 6.11 In [22] the concept of K-boundedness (cf. [18]) is defined as follows: A subset A of a linear topological space Y is said to be K-bounded if there is a topologically bounded set M ⊂ Y such that A ⊂ M + K.

It can easily be seen that the boundedness condition in [22] coincides with condition (E2) of Theorem 6.8.

In [22], Y is supposed to be a locally convex space, K is a closed pointed convex cone and k^{0} ∈ K\{0}. Therefore we have k^{0} ∈ K\−cl K. Remark 3.3 yields that assumption (C) is satisfied and that it holds k^{0} ∈ K ∩ int C. Therefore these assumptions have not to be proposed, additionally. Hence, our boundedness condition (E2’) is in fact weaker than the boundedness condition in [22] (cf. Remark 3.5 and Example 3.6).

Remark 6.12 If x_{0} ∈ X is an εk^{0}-minimal point of F in the sense of [22], i.e. F (x) ⋠ F (x_{0}) − εk^{0} then relation (i) of Theorem of 6.8 can be split into the two relations (i_{1}) F (x̄) ≼ F (x_{0}); (i_{2}) q_{Λ}(x̄, x_{0}) ≤ ε. Indeed, while (i_{1}) is immediately, (i) is equivalent to F (x_{0}) ⊂ F (x̄) + k^{0}q_{Λ}(x̄, x_{0}) + K. If q_{μ}(x̄, x_{0}) > ε for some μ ∈ Λ, we have

F (x_{0}) ⊂ F (x̄) + k^{0} (q_{μ}(x̄, x_{0}) − ε) + εk^{0} + K ⊂ F (x̄) + εk^{0} + K

which contradicts the εk^{0}-minimality of x_{0}.

Remark 6.10 - 6.12 show that Theorem 5.1 of [22] is a special case of Theorem 6.8 and 6.9.

6.2 Ekeland’s Principle for Set-Valued Maps II

Similar to the results in the preceding section we can prove analogous theorems for the relation ⋞.

Theorem 6.13 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \−cl K. Let F : X → 2^{Y} be a set-valued mapping, x_{0} ∈ X, S(x_{0}) := {x ∈ X : F (x) + k^{0}q_{Λ}(x, x_{0}) ⋞ F (x_{0})} and A_{0} := {(x, V) ∈ graph F : x ∈ S(x_{0})}, such that the following conditions are satisfied:

(E1) x_{0} ∈ ⋞-dom F;

(E2) F (S(x_{0})) is ⋞-bounded below;

(E3) For every ⋞_{k^{0}}-decreasing sequence {(x_{n}, V_{n})}_{n∈ℕ} ⊂ A_{0} there exists some (x, V) ∈ A_{0} such that (x, V) ⋞_{k^{0}} (x_{n}, V_{n}) for all n ∈ N.

Then, there exists x̄ ∈ ⋞-dom F such that

(i) F (x̄) + k^{0}q_{Λ}(x̄, x_{0}) ⋞ F (x_{0});

(ii) ∀x ≠ x̄ : ∃λ ∈ Λ : F (x) + k^{0}q_{λ}(x, x̄) ⋞̸ F (x̄).

Under the additional assumption (C) and if k^{0} ∈ K ∩ int C, (E1) and (E2) can be replaced by

(E1’) There exists a topologically bounded subset Ṽ_{0} ⊂ Y such that F (x_{0}) ⊂ Ṽ_{0} − cl C; (E2’) There exists ỹ ∈ Y such that

∀x ∈ S(x_{0}) : (F (x) − ỹ) ∩ (Y \ −int C) ≠ ∅.

Proof. Set A := graph F and apply Theorem 5.2 ■

From Theorem 6.13 one may deduce the following theorem.

Theorem 6.14 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \ −cl K. Let F : X → 2^{Y} be a set-valued mapping and x_{0} ∈ X such that (E1) and (E2) of Theorem 6.13 are satisfied as well as

(E3’) For every x ∈ X the set S(x) := {x′ ∈ X : F (x′) + k^{0}q_{Λ}(x′, x) ⋞ F (x)} is sequentially complete in X.

then the assertions (i) and (ii) of Theorem 6.13 remain true.

If, additionally, (C) holds and k^{0} ∈ K ∩ int C, then (E1) and (E2) can be replaced by (E1’) and (E2’) of Theorem 6.13.

Proof. Follows the lines of the proof of Theorem 6.9. ■

Remark 6.15 A map F : X → 2^{Y} is called sequentially lower ⋞-semicontinuous iff for every V ∈ 2^{Y} the set {x′ ∈ X : F (x′) ⋞ V } is a sequentially closed subset of X. Following arguments similar to those in Remark 6.10, we can prove that (E3’) of Theorem 6.14 is satisfied if F is sequentially lower ⋞-semicontinuous with ⋞-closed values (for each x ∈ X, F (x) − K is a closed set).

Remark 6.16 Considerations similar to Remark 6.12 can be done.

6.3 Ekeland’s Principle with Point Relation and a Minimal Point Theorem

In [10] we proved a minimal point theorem and its equivalence to a variant of Ekeland’s principle for set-valued maps as well as many conclusions of them (e.g. results of [2], [9], [8], [11], [16]). In this subsection we show that these theorems are easy corollaries of Theorem 5.1 as well as of Theorem 5.2. We are concerned with elements w = (w_{X}, w_{Y}) = (x, y) of the product space W = X × Y, X a separated uniform space, Y a topological vector space. We introduce the ordering relation ⪯_{k^{0}} on W using an element k^{0} ∈ K \ −cl K:

(x_{1}, y_{1}) ⪯_{k^{0}} (x_{2}, y_{2}) ⇔ y_{1} + k^{0}q_{Λ} (x_{1}, x_{2}) ≤_{K} y_{2}.  (27)

If K is a convex cone, ⪯_{k^{0}} is a reflexive and transitive relation. If K additionally is pointed, the relation is also antisymmetric. See e.g. [9], [10]. Identifying an element (x, y) ∈ X × Y with (x, {y}) ∈ X × 2^{Y}, it is easy to see that ⪯_{k^{0}} coincides with the above defined ordering ≼_{k^{0}} as well as with ⋞_{k^{0}}.

The following corollary is a variant of Ekeland’s variational principle for set-valued maps involving the ordering relation ⪯_{k^{0}} applied to elements of the set

gr F := {(x, y) ∈ X × Y : y ∈ F (x)}

usually denoting the graph of F. Note the difference to the definition at the beginning of this section. Compare e.g. [2] for a similiar result.

Corollary 6.17 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k_{0} ∈ K \ −cl K. For the set-valued mapping F : X → 2^{Y}, let w_{0} = (x_{0}, y_{0}) ∈ gr F be given such that for the set A_{0} := {w ∈ gr F : w ⪯_{k^{0}} w_{0}} the following assumptions are satisfied: (E2) The set (A_{0})_{Y} := {y ∈ Y : w = (x, y) ∈ A_{0}} is ≤_{K}-bounded below; (E3) For every ⪯_{k^{0}}-decreasing sequence {(x_{n}, y_{n})}_{n∈ℕ} ⊂ A_{0} there exists some point

(x, y) ∈ A_{0} such that (x, y) ⪯_{k^{0}} (x_{n}, y_{n}) for all n ∈ ℕ. Then, there exists some point (x̄, ȳ) ∈ gr F such that

(i) ȳ + k^{0}q_{Λ}(x̄, x_{0}) ≤_{K} y_{0}; (ii) ∀ (x, y) ∈ gr F with x ≠ x̄ ∃λ ∈ Λ : y + k^{0}q_{λ}(x, x̄) ≰_{K} ȳ. Under the additional assumption (C) and if k^{0} ∈ K ∩ int C, (E2) can be relaxed to (E2’) There exists some ỹ ∈ Y such that (A_{0})_{Y} ∩ (ỹ − int C) = ∅; and if k^{0} ∈ K \ {0} ⊂ int C then ȳ is ≤_{K}-minimal in F (x̄). Proof. Choose A = {(x, {y}) : (x, y) ∈ gr F } and apply Theorem 5.1 or 5.2. ■ The following corollary is a minimal point theorem in X × Y. Compare Theorem 9 of [10]. Corollary 6.18 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k^{0} ∈ K \ −cl K. Let A ⊂ W = X × Y such that for some w_{0} ∈ A and for A_{0} := {w ∈ A : w ⪯_{k^{0}} w_{0}} the following conditions are satisfied: (M2) The set (A_{0})_{Y} := {y ∈ Y : (x, y) ∈ A_{0}} is ≤_{K}-bounded below; (M3) For every ⪯_{k^{0}}-decreasing sequence {w_{n}}_{n∈ℕ} ⊂ A_{0} there exists some w ∈ A_{0} such

that w ⪯_{k^{0}} w_{n} for all n ∈ N. Then, there exists w̄ ∈ A such that

(i) w̄ ⪯_{k^{0}} w_{0}; (ii) (ŵ ∈ A, ŵ ⪯_{k^{0}} w̄) ⇒ ŵ_{X} = w̄_{X}; Under the additional assumption (C) and if k^{0} ∈ K ∩ int C, (M2) can be replaced by (M2’) There exists some ỹ ∈ Y such that (A_{0})_{Y} ∩ (ỹ − int C) = ∅. If, in addition, k^{0} ∈ K \ {0} ⊂ int C then (ii) can be strengthened to (ii’) (ŵ ∈ A, ŵ ⪯_{k^{0}} w̄) ⇒ ŵ = w̄, (i.e. w̄ is ⪯_{k^{0}}-minimal in A). Proof. Choose A = {(x, {y}) : (x, y) ∈ A} and apply Theorem 5.1 or 5.2. ■ The equivalence of Corollary 6.17 and Corollary 6.18 as well as a fixed point theorem of Kirk-Caristi type has been established in [10]. Because of this equivalence the minimal point theorem in X × Y can be understood as Ekeland’s principle for set-valued maps with respect to an ordering relation for elements of gr F ⊂ X × Y.

6.4 Ekeland’s Principle for Single-valued Maps

Finally, we present a conclusion for the case of a single-valued map f : X → Y. As above, identifying an element y ∈ Y with {y} ∈ 2^{Y} the ordering relations ≼ and ⋞ coincide (cf. Example 6.4), hence the following corollary may be deduced from Theorem 6.9 as well as from Theorem 6.14. It is also possible to derive it from Corollary 6.17. As proposed in [9], [8], we extend the space Y by an element ∞ such that y ≤_{K} ∞ for all y ∈ Y. As usually, the domain of f is said to be the set domf := {x ∈ X : f(x) ≠ ∞}.

Corollary 6.19 Let X be a separated uniform space, Y a topological vector space, K ⊂ Y a convex cone and k_{0} ∈ K \ −cl K. Let f : X → Y ∪ {∞} be a proper function satisfying the conditions

(E2) f is ≤_{K}-bounded below;

(E3) For every x ∈ dom f the set S(x) := {x′ ∈ X : f(x′) + k^{0}q_{Λ}(x′, x) ≤_{K} f(x)} is sequentially complete.

Then, for each x_{0} ∈ dom f there exists x̄ ∈ X such that

(i) f(x̄) + k^{0}q_{Λ}(x̄, x_{0}) ≤_{K} f(x_{0});

(ii) ∀x ∈ X \ {x̄} : ∃λ ∈ Λ : f(x) + k^{0}q_{λ}(x, x̄) ≰_{K} f(x̄).

Under the additional assumption (C) and if k^{0} ∈ K ∩ int C, (E2) can be replaced by

(E2’) There exists some ỹ ∈ Y such that f(S(x_{0})) ∩ (ỹ − int C) = ∅.

Proof. For a proof with Corollary 6.17 or with Theorem 6.9 set

F : X → 2^{Y} , F (x) := \{ \begin{matrix}\{f(x)\} & if & f(x) ≠ ∞ \\ ∅ & if & f(x) = ∞ \end{matrix}

A third variant of a proof can be given by setting

F : X → 2^{Y} , F (x) := \{ \begin{matrix}\{f(x)\} & if & f(x) ≠ ∞ \\ Y & if & f(x) = ∞ \end{matrix}

and applying Theorem 6.14. ■

Taking into account Remark 6.12 and 6.16, respectively, we see that Corollary 6.19 is a generalization of Theorem 4 of [11] as well as Theorem 4 of [16]. It also covers Corollary 2 of [9] as well as Corollary 2 of [8]. This shows that our minimal set theorems are powerful generalizations of the minimal point theorems of [20], [8], [9], [10].

7 Overview

The following figure visualizes the relationships between the main assertions of this paper:

Single-Valued Ekeland’s Principle

⇓
⇓

⇓

Ekeland’s Principle with Set Relation I

Minimal Point Theorem Ekeland’s Principle with Set Relation II

⇕

Ekeland’s Principle with Point Relation

⇓ ⇓ ⇓ ⇓

Minimal Set Theorem I Minimal Set Theorem II

Figure 1: Overview

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