Lobachevskii Journal of Mathematics

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ISSN 1818-9962

Vol. 21, 2006, 33–44 \begin{matrix}c \\ ◯ \end{matrix} A. Benbrik, A. Mbarki, S. Lahrech, A. Ouahab

A. Benbrik, A. Mbarki, S. Lahrech, A. Ouahab

EKELAND’S PRINCIPLE FOR VECTOR-VALUED MAPS BASED ON THE CHARACTERIZATION OF UNIFORM

SPACES VIA FAMILIES OF GENERALIZED QUASI-METRICS

(submitted by A. Lapin)

Abstract. Using a new characterization of uniform spaces via Families of generalized quasi-metrics, we present a variant of Ekeland’s variational principle for vector-valued maps being a consequence of minimal point theorem.

1 Introduction

Ekeland’s variational principle [9] is an important tool in nonlinear analysis. In the last decades various theorems had been presented which turned out to be equivalent to Ekeland’s principle. One of them, a lemma due to R. R. Phelps (see [30] and especially the version of ([31] from 1989) can be considered as the first minimal point theorem. Phelp’s lemma yields the existence of minimal point with respect to a partial ordering in a subset of X ×R, where X is a Banach space and R denotes the reals.

Minimal point theorems in a product space X × Y were established by Gopfert and Tammer [13], 1995 and generalized by Gopfert, Tammer and Zalinescu in [15], 2000 and in [14], 1999. In the latest version, X is a complete metric space and Y is a separated locally convex space. These theorems are useful tools in vector optimization. In [15], [14] a variational principle for vector-valued functions f : X → Y was presented to be an easy consequence of the minimal point theorem.

2000 Mathematical Subject Classification. 58E30.

Key words and phrases. Ekeland’s principle for vector-valued maps, minimal point theorem, uniform spaces, family of generalized quasi-metrics.

A generalization of Ekeland’s variational principle with respect to the space X was given by Fang [10], 1996. He introduced the concept of “F-type topological spaces” generating the topology by families of quasimetrics. Andreas Hamel and Andreas Lohne proved in [1] that the class of Fang’s F-type spaces coincides with the class of separated uniform spaces introduced by Weil [34], 1937.

In this paper we present a variant of Ekeland’s variational principle for vector-valued maps as a consequence of minimal point theorem. The proof of the result is based on the Characterization of Uniform Spaces via Families of generalized quasi-metrics.

2 Family of generalized quasi-metrics and Uniform Spaces In this section we present a characterization of uniform spaces via

families of generalized quasi-metrics.

Initially, we shall recall the concept of uniform space. For further details see Kelly [22] or Kothe [23].

Let X be a nonempty set. We consider a system R of subsets N of X × X. For N ⊂ X × X we denote N^{−1} := {(y, x) : (x, y) ∈ N} and N ∘ N := {(x, y) ∈ X × X : ∃z ∈ X (x, z), (z, y) ∈ N}. The set Δ := {(x, x) : x ∈ X} is called the diagonal. The set X is said to be an uniform space if and only if there exists a filter R on X × X satisfying

\begin{matrix}(N1) & ∀N ∈ R Δ ⊂ N; \\ (N2) & ∀N ∈ R N^{−1} ∈ R; \\ (N3) & ∀N ∈ R ∃M ∈ R M ∘ M ⊂ N. \end{matrix}

The system R is called a uniformity on X. By the sets

θ(x) := {U_{N}(x) : N ∈ R},

where U_{N}(x) = {y ∈ X : (x, y) ∈ N}, a topology is given, which is called the uniform topology on X. Of course, an uniform space is already welldefined by a base of its uniformity R, i.e a filter base B of the uniformity R. The topology of an uniform space is separated if and only if

(N4) ⋂_{N∈R} N = Δ.

For a proof see ([23],p. 32).

We recall a well-established result, the characterization of uniform spaces using families of pseudo-metrics see [22].

EKELAND’S PRINCIPLE FOR VECTOR-VALUED MAPS 35

Definition 1. Let X be a nonempty set. A function p : X × X → [0, +∞[ is called pseudo-metric on X if and only if for all x, y, z ∈ X the following conditions are satisfied:

\begin{matrix}(P 1) & p(x, x) = 0; \\ (P 2) & p(x, y) = p(y, x); \\ (P 3) & p(x, z) ≤ p(x, y) + p(y, z). \end{matrix}

Let (ℓ, ≺) be a directed set. A system {p_{λ}}_{λ∈ℓ} of pseudo-metrics p_{λ} : X × X → [0, +∞[ satisfying

(P 4) λ ≺ μ ⇒ ∀x, y ∈ X p_{λ}(x, y) ≤ p_{μ}(x, y)

is called a family of pseudo-metrics. If additionally the condition

p_{λ}(x, y) = 0 ∀λ ∈ ℓ ⇒ x = y

holds, the family of pseudo-metrics is said to be separating.

Proposition 1. A topological space (X, τ) is a separated uniform space if and only if its topology τ can be generated by a separating family of pseudo-metrics.

Proof. See ([22],p. 188, Theorem 15).

Fang [10] introduced so-called F − type topological using families of quasi-metrics.

Definition 2. Let X be a nonempty set and let (ℓ, ≺) be a directed set. A system {q_{λ}}_{λ∈ℓ} of functions q_{λ} : X × X → [0, +∞[ satisfying

(Q1) ∀λ ∈ ℓ ∀x ∈ X q_{λ}(x, x) = 0;
(Q2) ∀λ ∈ ℓ ∀x, y ∈ X q_{λ}(x, y) = q_{λ}(y, x);
(Q3) ∀λ ∈ ℓ ∃μ ∈ ℓ such that λ ≺ μ and
∀x, y, z ∈ Xq_{λ}(x, z) ≤ q_{μ}(x, y) + q_{μ}(y, z);
(Q4) λ ≺ μ ⇒ ∀x, y ∈ Xq_{λ}(x, y) ≤ q_{μ}(x, y)

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

(q_{λ}(x, y) = 0 ∀λ ∈ ℓ) ⇒ x = y

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

Proposition 2. A topological space (X, τ) is a separated uniform space if and only if its topology τ can be generated by a separating family of quasi-metrics.

Proof. See ([1],p. 3, Theorem 4).

Definition 3. Let X be a nonempty set. A system {q_{λ}}_{λ∈ℓ} of functions q_{λ} : X × X → [0, +∞[ satisfying

(Q′1) ∀λ ∈ ℓ ∀x ∈ X q_{λ}(x, x) = 0;
(Q′2) ∀λ ∈ ℓ ∀x, y ∈ X q_{λ}(x, y) = q_{λ}(y, x);
(Q′3) ∃{μ_{1}(λ)}_{λ∈ℓ}, {μ_{2}(λ)}_{λ∈ℓ} ⊂ ℓ such that ∀λ ∈ ℓ ∀x, y, z ∈ X
z ≠ y, y ≠ x q_{λ}(x, z) ≤ q_{μ_{1}(λ)}(x, y) + q_{μ_{2}(λ)}(y, z)

is called a family of generalized quasi-metrics. In this case, (X, {q_{λ}}_{λ∈ℓ}) is called a generalized quasi-metric space. If in addition the condition

(q_{λ}(x, y) = 0 ∀λ ∈ ℓ) ⇒ x = y

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

Our definition is slightly more general because on the one hand ℓ has not to be directed set in our case and on the other hand, the assumption (Q3) in Definition 2 is an optional condition, not automatically satisfied in our Definition.

Our first result clarifies the relation between separated uniform spaces and topological spaces generated by separating families of generalized quasi-metrics.

Proposition 3. A topological space (X, τ) is a separated uniform space if and only if its topology τ can be generated by a separating family of generalized quasi-metrics.

Proof. Let (X, τ) be a topological space where τ is generated by a separating family {q_{λ}}_{λ∈ℓ} of generalized quasi-metrics, i.e. τ is given by

℘(x) = {U(x, n, λ_{1}, .., λ_{n}, t) : t > 0, n ∈ N, λ_{1}, .., λ_{n} ∈ ℓ}

where

U(x, n, λ_{1}, .., λ_{n}, t) = {y ∈ X : q_{λ_{i}}(x, y) < t, 1 ≤ i ≤ n}.

We claim that a base of a uniformity is given by the system

ℵ := {S(n, λ_{1}, .., λ_{n}, t) : t > 0, n ∈ N, λ_{1}, .., λ_{n} ∈ ℓ}

where

S(n, λ_{1}, .., λ_{n}, t) := {(x, y) ∈ X × X : q_{λ_{i}}(x, y) < t, 1 ≤ i ≤ n}.

To show that ℵ is a filter base let t_{1} > 0, n ∈ N, λ_{1}, .., λ_{n} ∈ ℓ and t_{2} > 0, p ∈ N, β_{1}, .., β_{p} ∈ ℓ be arbitrarily given. Set t_{3} := min(t_{1}, t_{2}),r = n + p. Consider the sequence λ_{1}, .., λ_{n}, β_{1}.., β_{p}. Then S(r, λ_{1}, .., λ_{n}, β_{1}.., β_{p}, t_{3}) ⊂ S(n, λ_{1}, .., λ_{n}, t_{1}) ⋂ S(p, β_{1}, .., β_{p}, t_{2}).

Furthermore, ∅ ∉ ℵ since each S(n, λ_{1}, .., λ_{n}, t) contains the diagonal.

Let ℜ be the filter generated by ℵ. We shall show that ℵ is a base of uniformity on X. The axioms (N_{1}) and (N_{2}) are satisfied for ℵ. To verify (N_{3}) let t > 0, n ∈ N, λ_{1}, .., λ_{n} ∈ ℓ be arbitrarily given. Taking (μ_{1} = μ_{1}(λ_{1}), μ_{2} = μ_{2}(λ_{1})), .., (μ_{2n−1} = μ_{1}(λ_{n}), μ_{2n} = μ_{2}(λ_{n})) from (Q’3) we set M := S(3n, λ_{1}, .., λ_{n}, μ_{1}, μ_{2}, μ_{3}, μ_{4}, .., μ_{2n−1}, μ_{2n}, \frac{t}{2}).

Then we have M ∘ M ⊂ S(n, λ_{1}.., λ_{n}, t). Indeed, let (x, y) ∈ M ∘ M, i.e.

∃z ∈ X : (x, z), (z, y) ∈ M.

If z = y or z = x, then (x, y) ∈ M. Therefore,

q_{λ_{i}}(x, y) < \frac{t}{2} < t for all i such that 1 ≤ i ≤ n.

Hence, (x, y) ∈ S(n, λ_{1}, .., λ_{n}, t).

If now z ≠ y and z ≠ x, then q_{μ_{i}}(x, z) < \frac{t}{2}, q_{μ_{i}}(z, y) < \frac{t}{2}, 1 ≤ i ≤ 2n. Hence

q_{λ_{i}}(x, y) ≤ q_{μ_{2i−1}}(x, z) + q_{μ_{2i}}(z, y) < t, 1 ≤ i ≤ n.

Therefore, ℜ is a uniformity generating the topology τ. If additionally the family {q_{λ}}_{λ∈ℓ} of generalized quasi-metrics is separating, then the uniform space (X, τ) is separated.

The opposite assertion follows by Proposition 1 taking into account that a family of pseudo-metrics is in particular a family of generalized quasi-metrics.

An important class of uniform spaces is the class of topological vector spaces. Indeed, For topological vector space the topology can be generated by a family of quasi-norms. This result is due to Hyers [19], 1939 who used the term ”pseudo-norms” instead of ”quasi-norms”.

3. Main Tools

For the convenience of the reader we present the main tools for the proof of our minimal point theorem. The first one is the Brézis-Browder principle.

Theorem 1. Let (W, ⪯) be a quasi-ordered set (i.e. ⪯ is a reflexive and transitive relation) 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).

In particular, if we strengthen (A2) to (A′2) (w_{1} ⪯ w_{2}, w_{1} ≠ w_{2}) ⇒ ϕ(w_{1}) < ϕ(w_{2}) it holds

(ii′) ŵ ⪯ w̄ ⇒ ŵ = w̄, i.e. w̄ is ⪯– minimal in W.

Proof. See [[3], Corollary 1].

Note that A’2 implies the antisymmetry of the relation ⪯.

A second important tool is a scalarization method established by Gerstewitz (Tammer), Iwanow [12] and Gerth (Tammer), Weidner [11].

Theorem 2. Let Y be a topological vector space, K ⊂ Y a convex cone and k_{0} ∈ K \ −cl K. The functional z : Y → R ⋃{+∞}, defined as z(y) := inf{t ∈ R : y ∈ tk_{0} − cl K} has the following properties

(H1) z is sublinear; (H2) y_{1} ⪯_{K} y_{2} ⇒ z(y_{1}) ≤ z(y_{2}); (H3) ∀y ∈ Y ∀α ∈ R z(αk_{0} + y) = z(y) + α; (H4) If Y_{0} ⊂ Y is ⪯_{K} bounded below, then z is bounded below on Y_{0}.

Proof. See [[15], Lemma 7] taking into account that Y has not to be separated for the proof. Moreover, in the definition of the functional the closed cone can be replaced by the closure of a not necessarily closed cone (since y_{1} ⪯_{K} y_{2} implies y_{1} ⪯_{clK} y_{2}). Then, if Y is not separated, we have to choose k_{0} ∈ K \ −cl K to avoid k_{0} ∈ cl{0}. If Y is separated it suffices to suppose k_{0} ∈ K \ −K.

Let Y be a topological vector space and K ⊂ Y a convex cone. We use the following assumption to derive strong (in [15] called ”authentic”) variants of the minimal point theorem.

(C) There exists a proper cone convex C ⊂ Y satisfying K \ {0} ⊂ int C.

Theorem 3. Let Y be a topological vector space, K ⊂ Y a convex cone satisfying assumption (C). Let k_{0} ∈ K \ {0}. The functional z_{C} : Y → R, defined as z_{C}(y) := inf{t ∈ R : y ∈ tk_{0} −cl C} has the following properties

(H′1) z_{C} is sublinear;
(H′2) (y_{1} ⪯_{K} y_{2}, y_{1} ≠ y_{2}) ⇒ z_{C}(y_{1}) < z_{C}(y_{2});
(H′3) ∀y ∈ Y ∀α ∈ R z_{C}(αk_{0} + y) = z_{C}(y) + α;
(H′4) For Y_{0} ⊂ Y, ỹ ∈ Y the condition Y_{0} ⋂(ỹ − int C) = ∅

implies that z_{C} is bounded below on Y_{0}.

Proof. See [[15], Lemma 7] taking into account that z_{C}(y) = ∞ is not possible under our assumptions. Note that we have k_{0} ∈ int C \ −cl C. Therefore, as above, Y has not to be separated.

4. Minimal Point Theorem

Minimal point theorems in product spaces X ×Y presented by Gopfert and Tammer [13], by Gopfert, Tammer and Zalinescu [15], [14], and by Andreas Hamel and Andreas Lohne [1] give useful generalizations of Ekeland’s variational principle. We wish to generalize some of theorems in [1] with respect to the space X. Instead of family of quasi-metrics we consider a family of generalized quasi-metrics.

In what follows let (X, {q_{λ}}_{λ∈ℓ}) be a separated uniform space generated by the separating family {q_{λ}}_{λ∈ℓ} of generalized quasi-metrics and let Y be a topological vector space. We write w = (w_{1}, w_{2}) ∈ W to deal with the two components of an element w of the product space W := X × Y.

It is well-known that a convex cone K ⊂ Y generates a quasi-ordering on Y by

y_{1} ⪯_{K} y_{2} ⇔ y_{2} − y_{1} ∈ K.

If K is pointed, the relation is also antisymmetric, therefore a partial ordering. Using an element k_{0} ∈ K \ −cl K we introduce a relation ⪯_{k_{0}} on W:

(x_{1}, y_{1}) ⪯_{k_{0}} (x_{2}, y_{2}) ⇔ ∀λ ∈ ℓ y_{1} + q_{λ}(x_{1}, x_{2})k_{0} ⪯_{K} y_{2}.

Lemma 1. If K ⊂ Y is a convex cone, the relation ⪯_{k_{0}} is reflexive and transitive on W. If additionally K is pointed, the relation ⪯_{k_{0}} is antisymmetric and thus partial ordering on W.

Proof. Exemplary, we prove the transitivity. Let w_{i} = (x_{i}, y_{i}) ∈ W (i = 1, 2, 3) satisfying (x_{1}, y_{1}) ⪯_{k_{0}} (x_{2}, y_{2}) and (x_{2}, y_{2}) ⪯_{k_{0}} (x_{3}, y_{3}). Therefore,

∀λ ∈ ℓ y_{1} + q_{λ}(x_{1}, x_{2})k_{0} ⪯_{K} y_{2} and y_{2} + q_{λ}(x_{2}, x_{3})k_{0} ⪯_{K} y_{3}.

The transitivity of the relation ⪯_{K} yields

∀λ ∈ ℓ y_{1} + (q_{λ}(x_{1}, x_{2}) + q_{λ}(x_{2}, x_{3}))k_{0} ⪯_{K} y_{3}.

If x_{2} = x_{1} or x_{2} = x_{3}, then the assumption w_{1} ⪯_{k_{0}} w_{3} holds. Otherwise, using Q’3 we deduce that

∀λ ∈ ℓ y_{1} + q_{λ}(x_{1}, x_{3})k_{0} ⪯_{K} y_{3}.

Hence, w_{1} ⪯_{k_{0}} w_{3}.

We continue with our main result, the minimal point theorem in uniform spaces. Just as the Brézis-Browder principle (Theorem 5), the following theorem (as well as its equivalent formulations, Theorems [11], [13], [14]) consists of two parts. The ”weak” assertion (ii) yields the existence of an element w̄ of a certain set A such that some ŵ which is dominated by w̄ with respect to a quasi-ordering necessarily has the same X− component. However, the Y − component may be distinct. The ”strong” (authentic) assertion (ii) yields the minimality of some w̄ ∈ A in A with respect to a partial ordering. Note that assumption (C) of section 3 ensures that we deal in fact with a partial ordering. It plays the key role in establishing the strong assertion and can be traced back to the early work of Bishop and Phelps.

Theorem 4. (Minimal point theorem) Let (X, {q_{λ}}_{λ∈ℓ}) be a separated uniform space generated by the separating family {q_{λ}}_{λ∈ℓ} of generalized quasi-metrics, Y a topological vector space, K ⊂ Y a convex cone and k_{0} ∈ K \ −cl K. Let A ⊂ W be a nonempty subset of the product space W := X × Y and let w_{0} ∈ A be given such that for the set W_{0} := {w ∈ A : w ⪯_{k_{0}} w_{0}} the following assumptions hold true

(M1) The set (W_{0})_{Y} := {y ∈ Y : there exists x ∈ X such that
(x, y) ∈ W_{0}} is ⪯_{K} – bounded below;
(M2) For any ⪯_{k_{0}}- decreasing sequence {w_{n}}_{n∈N} ⊂ W_{0} there exists
some w ∈ W_{0} such thatw ⪯ w_{n} for all n ∈ N.

Then there exists some w̄ ∈ A such that

(i) w̄ ⪯_{k_{0}} w_{0}; (ii) (ŵ ∈ A, ŵ ⪯_{k_{0}} w̄) ⇒ ŵ = w̄.

Under the additional assumption (C) we can relax assumption (M1) to

(M′1) There exists some ỹ ∈ Y such that (W_{0})_{Y} ⋂(ỹ − int C) = ∅ and

(iii) w̄ is ⪯_{k_{0}} minimal point in A

Proof. By Lemma 1, the relation ⪯_{k_{0}} is reflexive and transitive on W_{0}. We apply the Brézis-Browder principle (Theorem 1) on the quasi-ordered set (W_{0}, ⪯_{k_{0}}) using the functional ϕ : W_{0} → R, ϕ(w) = z(w_{Y} − (w_{0})_{Y}), where z : Y → R ⋃{+∞} is the scalarization functional of Theorem 2. First, we must have ϕ(w) ≠ +∞. Indeed, for w ∈ W_{0} it holds w_{Y} ⪯_{K} (w_{0})_{Y}. Hence w_{Y} − (w_{0})_{Y} ∈ −K ⊂ −cl K. By the definition of z we have ϕ(w) ≤ 0.

By (M1) and property (H4) of z (Theorem 2), ϕ is bounded below on W_{0}. Let be w_{1} ⪯_{k_{0}} w_{2}, w_{1}, w_{2} ∈ W_{0}, hence (w_{1})_{Y} ⪯_{K} (w_{2})_{Y}. Property (H2) of z implies assumption (A2) of Theorem 1. Of course, (M2) implies assumption (A3) of Theorem 1.

Theorem 1 yields the existence of some w̄ ∈ W_{0} such that

w̄ ⪯_{k_{0}} w_{0} and ŵ ⪯_{k_{0}} w̄ ⇒ ϕ(ŵ) = ϕ(w̄).

Let us show (ii). Take ŵ ∈ A such that ŵ ⪯_{k_{0}} w̄. The transitivity of ⪯_{k_{0}} yields ŵ ∈ W_{0}. This implies

∀λ ∈ ℓ (ŵ)_{Y} − (w_{0})_{Y} + q_{λ}((ŵ)_{X}, (w̄)_{X})k_{0} ⪯_{K} (w̄)_{Y} − (w_{0})_{Y} .

Applying properties (H2) and (H3) of z we get

∀λ ∈ ℓ q_{λ}((ŵ)_{X}, (w̄)_{X}) ≤ ϕ(w̄) − ϕ(ŵ) = 0.

Consequently,

∀λ ∈ ℓ q_{λ}((ŵ)_{X}, (w̄)_{X}) = 0.

Since (X, {q_{λ}}_{λ∈ℓ}) is separated, then (ŵ)_{X} = (w̄)_{X}.

Now, let assumption (C) be satisfied. We can replace (M1) by (M’1) and proceed analogously, but using the functional z_{C} of Theorem 3 instead of z. In particular, the corresponding functional ϕ_{C} : W_{0} → R, ϕ_{C}(w) = z_{C}(w_{Y}) (the functional can be chosen slightly simpler than before, because z_{C}(y) ≠ ∞ ∀y ∈ Y) is even strict ⪯_{k_{0}}– monotone, i.e. w_{1} ⪯_{k_{0}} w_{2}, w_{1} ≠ w_{2} implies ϕ_{C}(w_{1}) < ϕ_{C}(w_{2}). Indeed, let w_{1} ⪯_{k_{0}} w_{2}, and w_{1} ≠ w_{2}. If (w_{1})_{X} ≠ (w_{2})_{X} then, since (X, {q_{λ}}_{λ∈ℓ}) is separated, there exists some λ ∈ ℓ satisfying q_{λ}((w_{1})_{X}, (w_{2})_{X}) > 0, thus,

(w_{2})_{Y} − (w_{1})_{Y} ∈ {q_{λ}((w_{1})_{X}, (w_{2})_{X})k_{0}} + K ⊂ K \ −cl K + K ⊂ K \ {0}.

Otherwise, if (w_{1})_{X} = (w_{2})_{X}, we have (w_{1})_{Y} ≠ (w_{2})_{Y} and it also holds (w_{2})_{Y} − (w_{1})_{Y} ∈ K \ {0}. Property H’2 of z_{C} yields ϕ(w_{1}) < ϕ(w_{2}). Therefore, assumption A’2 in Theorem 1 is satisfied too. The ⪯_{k_{0}}– minimality of w̄ in A follows from (Theorem 1, (ii’)) taking into account the transitivity of the relation ⪯_{k_{0}}.

5 Ekeland’s principle for vector-valued maps

In this section we present a variant of Ekeland’s variational principle for vector-valued functions. As proposed in [15], [14], we extend the space Y by an element ∞ such that y ⪯_{K} ∞ for all y ∈ Y.

Theorem 5. (Variational Principle) Let (X, {q_{λ}}_{λ∈ℓ}) be a separated uniform space generated by the separating family {q_{λ}}_{λ∈ℓ} of generalized quasi-metrics, Y a topological vector space, K ⊂ Y a convex cone and k_{0} ∈ K \ −cl K. Let f : X → Y ∪ {∞} be a proper function which is ⪯_{K}–bounded below and let for every x ∈ dom f the set

S(x) := {u ∈ X : ∀λ ∈ ℓ f(u) + q_{λ}(u, x)k_{0} ⪯_{K} f(x)}

be sequentially closed.

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

(i) ∀λ ∈ ℓ f(x̄) + q_{λ}(x̄, x_{0})k_{0} ⪯_{K} f(x_{0});

(ii) ∀x ∈ X such that x ≠ x̄ ∃β ∈ ℓ f(x) + q_{β}(x̄, x)k_{0} ⪯̸_{K} f(x̄).

Proof. We consider the set-valued mapping F : X → 2^{Y}, F (x) := {f(x)} if f(x) ≠ ∞ and F (x) := ∅ otherwise. Let dom F denote the domain of F, i.e. dom F := {x ∈ X : F (x) ≠ ∅} and let gr F denote the graph of F, i.e. gr F := {(x, y) ∈ X × Y : y ∈ F (x)}. Setting A := gr F, w_{0} = (x_{0}, f(x_{0})) all assumptions coincide with those of Theorem 4. Indeed, it remains to show that (M2) of Theorem 4 is satisfied. Let {(x_{n}, y_{n})}_{n∈N} ⊂ W_{0} be a ⪯_{k_{0}}– decreasing sequence and let ε > 0, λ ∈ ℓ. Then for any m, n ∈ N with m ≥ n we have

y_{m} − f(x_{0}) + q_{λ}(x_{m}, x_{n})k_{0} ⪯_{K} y_{n} − f(x_{0}).

The properties of z (Theorem 2) yield

ϕ(y_{m}) + q_{λ}(x_{m}, x_{n}) ≤ ϕ(y_{n}),

where ϕ(y) = z(y − f(x_{0})). Thus {ϕ(y_{n})}_{n∈N} is nonincreasing sequence. On the other hand, {ϕ(y_{n})}_{n∈N} is bounded below, hence there exists some n_{0} ∈ N such that for all n, m ≥ n_{0} it holds

q_{λ}(x_{m}, x_{n}) ≤ ϕ(y_{n}) − ϕ(y_{m}) < ε.

Hence, q_{λ}(x_{n}, x_{m}) < ε. This means, that {x_{n}} is a Cauchy sequence in (X, {q_{λ}}_{λ∈ℓ}) and by the sequentially completeness of X convergent to some x ∈ X.

On the other hand, we have x_{m} ∈ S(x_{n}) for all n, m ∈ N with m ≥ n. Since S(x_{n}) is sequentially closed, it follows that x ∈ S(x_{n}) for all n ∈ N. Hence, (x, f(x)) ⪯_{k_{0}} (x_{n}, y_{n}) for all n ∈ N. Theorem 4 implies all assertions.

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44 A. BENBRIK, A. MBARKI, S. LAHRECH, A. OUAHAB

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Department of Mathematics, Faculty of science, Mohamed first university, Oujda, Morocco E-mail address: benbrik@sciences.univ-oujda.ac.ma E-mail address: lahrech@sciences.univ-oujda.ac.ma E-mail address: mbarki@sciences.univ-oujda.ac.ma E-mail address: ouahab@sciences.univ-oujda.ac.ma

Received April 16, 2006