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42 6 2013 12









ADVANCES IN MATHEMATICS

Vol.42, No.6 Dec., 2013

A Vectorial Ekeland’s Variational Principle on Bornological Vector Spaces HE Fei1, 2, ∗ , QIU Jinghui2 (1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, Inner Mongolia, 010021, P. R. China; 2. School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, 215006, P. R. China) Abstract: We establish a vectorial Ekeland’s variational principle where the objective function is from bornological vector spaces into real vector spaces, and the ordering cone in real vector spaces is not necessarily solid. Meanwhile, a vectorial Caristi’s fixed point theorem and a vectorial Takahashi’s nonconvex minimization theorem are obtained and the equivalences between the three theorems are shown. Key words: Ekeland’s variational principle; Caristi’s fixed point theorem; Takahashi’s nonconvex minimization theorem; bornological vector space; real vector space MR(2000) Subject Classification: 58E30; 46A17 / CLC number: O177.3 Document code: A Article ID: 1000-0917(2013)06-0889-07

0 Introduction Ekeland’s variational principle is an important tool for nonlinear analysis and optimization theory. Since it was established by Ekeland[1] in 1972, the principle and its equivalent theorems have been applied to numerous problems in various fields of nonlinear analysis. A number of generalizations and equivalent results have been investigated by several authors; see [2–9]. In particular, Wong[9] considered the drop theorem and Ekeland’s variational principle on bornological vector spaces. Since every locally convex space becomes a convex bornological vector space when equipped with the canonical Von Neumann bornology, using his method Wong recovered Qiu’s results[7] . Recently, vectorial Ekeland’s variational principle is considered in the literature, see [2, 10]. In general, the objective function is from complete metric (or quasi-metric) spaces or topological vector spaces into a partial ordered locally convex spaces, where the ordering cone is solid. In this paper, we first give a generalized vectorial Ekeland’s variational principle, where the objective function is defined on a bornological vector space and takes values in a partical ordered real vector space. Besides, the perturbation has a more general form, which involves subadditive, strictly increasing continuous functions of generalized Minkowski functionals. By using the generalized vectorial Ekeland’s variational principle, we deduce a vectorial Caristi’s fixed point theorem and a vectorial Takahashi’s nonconvex minimization theorem in the framework of Received date: 2011-11-14. Revised date: 2012-01-12. Foundation item: Supported by NSFC (No. 10871141) and Scientific Studies of Higher Education Institution of Inner Mongolia (No. NJZZ13019). E-mail: ∗ [email protected]



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bornological vector spaces. Finally we show that the above three theorems are indeed equivalent to each other. It is well known that bornological vector spaces are more general frameworks than topological vector spaces[9, 11] . In our results, the objective function is from bornological vector spaces into real vector spaces. The ordering cone in valued field spaces is only assumed to be vectorially closed convex and pointed (not necessary solid).

1 Preliminaries Throughout this paper, X and Y are vector spaces over R. First we recall some notions on bornological vector spaces, refer to [9, 11]. A vector bornology on X is a collection B of subsets of X that satisfies the following conditions: (B1) x ∈ X implies that {x} ∈ B; (B2) B1 ⊂ B2 and B2 ∈ B implies that B1 ∈ B; (B3) B1 , B2 ∈ B implies that B1 ∪ B2 ∈ B; (B4) B1 , B2 ∈ B implies that B1 + B2 = {x1 + x2 : x1 ∈ B1 , x2 ∈ B2 } ∈ B; (B5) For any bounded interval I ⊂ R, B ∈ B implies that I · B = {αx : α ∈ I, x ∈ B} ∈ B. The ordered pair (X, B) is called a bornological vector space (BVS) and every candidate of B is called a bounded subset (with respect to B). We simply denote (X, B) by X if there is no ambiguity. A sequence {xn } in X is said to be Mackey-convergent (briefly, M-convergent) to a point x, denoted by limn→∞ xn = x, if there exists a balanced set B ∈ B and a sequence of positive real numbers {λn } such that limn→∞ λn = 0 and xn − x ∈ λn · B, ∀n ∈ N. Similarly, a sequence {xn } in X is said to be Mackey-Cauchy (briefly, M-Cauchy) if there exists a balanced set B ∈ B and a double sequence of positive real numbers {λmn } such that limm,n→∞ λmn = 0 and xm − xn ∈ λmn · B, ∀m, n ∈ N. A set A ⊂ X is said to be Mackey-closed (briefly, M-closed) if the conditions {xn } ⊂ A and limn→∞

xn = x imply x ∈ A. A set A ⊂ X is said to be Mackey-complete (briefly, M-complete) if every M-Cauchy sequence in A is M-convergent to some element in A. A BVS (X, B) is said to be separated if every M-convergent sequence is M-convergent to exactly one bornological limit. From now on, we only consider separated BVS. In the following, we recall some concepts and results in real vector spaces.



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Let C ⊂ Y be a pointed convex cone (i.e., C ∩ (−C) = {0}, C + C ⊂ C and λC ⊂ C for all λ ∈ [0, +∞)). Then C specifies a partial order in Y as follows: For x, y ∈ Y , x C y, if and only if y − x ∈ C. Let A be a nonempty subset of Y . A point a ∈ A is called a minimal point of A if A∩(a−C) = {a}. Let Min(A, C) denote the set all minimal points of A with respect to C. We define the vector closure of A as follows: vcl(A) = {y ∈ Y : ∃v ∈ Y, ∀t > 0, ∃α ∈ [0, t], y + αv ∈ A}. The set A is called vectorially closed if A = vcl(A), refer to [12, Definition 2.1]. Similar to [5, Theorem 2.3.1], we can show the following lemma. Lemma 1.1 Let C ⊂ Y be a proper (i.e., Y = C = ∅) vectorially closed convex cone and k 0 ∈ C\(−C). For any y ∈ Y , define ξk0 (y) := inf{t ∈ R : y ∈ tk 0 − C}. Then ξk0 : Y → R ∪ {+∞} is a positively homogeneous and subadditive proper function. Moreover, for every r ∈ R, ξk0 has the following properties: (i) ξk0 (y) < r ⇔ y ∈ rk 0 − [(0, +∞)k 0 + C]; (ii) ξk0 (y)  r ⇔ y ∈ rk 0 − C; (iii) ξk0 (y) = r ⇔ y ∈ rk 0 − {C\[(0, +∞)k 0 + C]}; / rk 0 − [(0, +∞)k 0 + C]; (iv) ξk0 (y)  r ⇔ y ∈ (v) ξk0 (y) > r ⇔ y ∈ / rk 0 − C; (vi) ξk0 is C-monotone (i.e., y1 C y2 implies ξk0 (y1 )  ξk0 (y2 )); (vii) ξk0 (y + rk 0 ) = ξk0 (y) + r for every y ∈ Y . Finally, we introduce some useful notations related to vector-valued functions. Y • := Y ∪ {∞}, where ∞ satisfies ∞ ∈ / Y , y + ∞ = ∞, λ · ∞ = ∞, and +∞ · k 0 = ∞, y C ∞ for all y ∈ Y • , k 0 ∈ C\{0} and λ > 0. If f : X → Y • is a vector-valued function, the domain of f is the set domf := {x ∈ X : f (x) ∈ Y }. A function f is said to be proper if domf = ∅. And f (X) = {y ∈ Y • : ∃x ∈ X, y = f (x)}.

2 Main Results Let ϕ : [0, +∞) → [0, +∞) be a subadditive, strictly increasing, continuous function with ϕ(0) = 0 and let the class Θ consist of all such functions ϕ. Clearly, the converse of ϕ, ϕ−1 : ϕ([0, +∞)) → [0, +∞), exists and ϕ−1 is a supadditive, strictly increasing, continuous function with ϕ−1 (0) = 0. Here ϕ is said to be subadditive if ϕ(t + s) ≤ ϕ(t) + ϕ(s), ∀t, s ∈ [0, +∞), and ϕ−1 is said to be supadditive if ϕ−1 (x + y) ≥ ϕ−1 (x) + ϕ−1 (y), ∀x, y ∈ ϕ([0, +∞)).



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√ n t belong

It is easy to see that some functions like ϕ(t) = εt (ε > 0), ϕ(t) = ln (1 + t), ϕ(t) =

to Θ. More examples are introduced in [8]. In this section, we always assume that (X, B) is a BVS, P : X → [0, +∞] is a subadditive and positively homogenous function such that B := {x ∈ X : P (x)  1} is a bounded set, Y is a real vector space, C ⊂ Y is a proper vectorially closed convex pointed cone, k 0 ∈ C\{0}, and ϕ ∈ Θ. We define ϕ(+∞) = lims→+∞ ϕ(s) and ξk0 (∞) = +∞. Theorem 2.1 Let f : X → Y • be a proper vector-valued function. If there exists r ∈ R and x0 ∈ dom f such that f (X) ∩ (f (x0 ) + rk 0 − C) = ∅ and f satisfies (H1) S(x0 ) := {x ∈ X : f (x ) + ϕ(P (x − x0 ))k 0 C f (x0 )} is M-complete; (H2) S(x) := {x ∈ X : f (x ) + ϕ(P (x − x))k 0 C f (x)} is M-closed for every x ∈ S(x0 ), then there exists x ∈ X such that (i) f (x) + ϕ(P (x − x0 ))k 0 C f (x0 ) (i.e., x ∈ S(x0 )); (ii) f (x) + ϕ(P (x − x))k 0 C f (x) for all x = x (i.e., S(x) = {x}). First, we give a lemma as follows. Lemma 2.1 For every x ∈ dom f , x ∈ S(x) and y ∈ S(x) imply S(y) ⊂ S(x). Proof Let x ∈ dom f . It is clear that x ∈ S(x). Assume now that y ∈ S(x), then f (y) + ϕ(P (y − x))k 0 C f (x). For any z ∈ S(y), f (z) + ϕ(P (z − y))k 0 C f (y). And therefore f (z) + ϕ(P (z − x))k 0 C f (z) + ϕ(P (z − y) + P (y − x))k 0 C f (z) + ϕ(P (z − y))k 0 + ϕ(P (y − x))k 0 C f (y) + ϕ(P (y − x))k 0 C f (x). It follows that z ∈ S(x), and so S(y) ⊂ S(x). / f (x0 ) + rk 0 − C for Proof of Theorem 2.1 Since f (X) ∩ (f (x0 ) + rk 0 − C) = ∅, f (x) ∈ all x ∈ X. By Lemma 1.1, ξk0 (f (x) − f (x0 )) > r, ∀x ∈ X. We denote η(x) := ξk0 (f (x) − f (x0 )) = inf{t ∈ R : f (x) ∈ f (x0 ) + tk 0 − C}, ∀x ∈ X. It follows that r  inf{η(x) : x ∈ S(x0 )}  sup{η(x) : x ∈ S(x0 )}  0. Hence, there exists x1 ∈ S(x0 ) such that η(x1 )  inf{η(x) : x ∈ S(x0 )} + 2−1 . By induction, we can obtain a sequence {xn } ⊂ X such that for every n  0, xn+1 ∈ S(xn ) and η(xn+1 )  inf{η(x) : x ∈ S(xn )} +

1 2n+1

.

(2.1)

By Lemma 2.1, for every n  0, S(xn+1 ) ⊂ S(xn ), and so η(xn+1 )  inf{η(x) : x ∈ S(xn )} +

1 2n+1

 inf{η(x) : x ∈ S(xn+1 )} +

It follows that η(xn ) − inf{η(x) : x ∈ S(xn )} 

1 , ∀n ∈ N. 2n

1 2n+1

.

(2.2)



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We claim that {xn } is M-Cauchy in (X, B). Indeed, for every n ∈ N, since xn+1 ∈ S(xn ), f (xn+1 ) + ϕ(P (xn+1 − xn ))k 0 C f (xn ), and so f (xn+1 ) − f (x0 ) + ϕ(P (xn+1 − xn ))k 0 C f (xn ) − f (x0 ). By Lemma 1.1, ξk0 (f (xn+1 ) − f (x0 )) + ϕ(P (xn+1 − xn ))  ξk0 (f (xn ) − f (x0 )), i.e., η(xn+1 ) + ϕ(P (xn+1 − xn ))  η(xn ), ∀n ∈ N. Taking into account the formula (2.2), for every n ∈ N, P (xn+1 − xn )  ϕ−1 (η(xn ) − η(xn+1 ))  ϕ−1 (η(xn ) − inf{η(x) : x ∈ S(xn )})   1 −1 ϕ . 2n Therefore, for any n, k ∈ N, P (xn+k − xn )  P (xn+k − xn+k−1 ) + P (xn+k−1 − xn+k−2 ) + · · · + P (xn+1 − xn )       1 1 1  ϕ−1 n+k−1 + ϕ−1 n+k−2 + · · · + ϕ−1 n 2 2 2   1 1 1  ϕ−1 n+k−1 + n+k−2 + · · · + n 2 2 2   1  ϕ−1 n−1 , 2 and so xn+k − xn ∈ ϕ−1 (2−n+1 ) · B ⊂ ϕ−1 (2−n+1 ) · Bb , where Bb := [−1, 1] · B is the balanced hull of B. Since B is bounded, Bb is bounded, and so {xn } is M-Cauchy. Since {xn } ⊂ S(x0 ) and S(x0 ) is M-complete, there exists x ∈ S(x0 ) such that limn→+∞ xn = x. By the condition (H2), it is easy to see that x ∈ S(xn ) for all n ∈ N. To prove S(x) = {x}, we take any x ∈ S(x). For every n ∈ N, by S(x) ⊂ S(xn ), we have that x ∈ S(xn ), and so f (x) + ϕ(P (x − xn ))k 0 C f (xn ). Similar to the preceding argument, we can obtain that for every n ∈ N,

P (x − xn )  ϕ−1 (ξk0 (f (xn ) − f (x0 )) − ξk0 (f (x) − f (x0 ))) = ϕ−1 (η(xn ) − η(x))  ϕ−1 (η(xn ) − inf{η(y) : y ∈ S(xn )})   1 −1 ϕ . 2n

Taking into account the boundedness of B, we have limn→+∞ xn = x. Since (X, B) is separated, x = x, and so S(x) = {x}. Theorem 2.2 Let f : X → Y • be a proper extended vector-valued function. If there exists r ∈ R and x0 ∈ dom f such that f (X) ∩ (f (x0 ) + rk 0 − C) = ∅ and f satisfies (H1), (H2) and the following condition (C), (C) if T : X → 2X is a multivalued mapping such that for every x ∈ X, there exists y ∈ T x such that f (y) + ϕ(P (y − x))k 0 C f (x),

(2.3)



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then there exists x ∈ S(x0 ) such that x ∈ T x. Furthermore, if for every x ∈ X we have T x = ∅ and f satisfies the formula (2.3) for all y ∈ T x, then there exists x ∈ S(x0 ) such that T x = {x}. Proof By Theorem 2.1, there exists x ∈ S(x0 ) such that f (x) + ϕ(P (x − x))k 0 C f (x), ∀x = x.

(2.4)

By the condition (C), there exists y ∈ T x such that f (y) + ϕ(P (y − x))k 0 C f (x).

(2.5)

By the formulae (2.4) and (2.5), x = y ∈ T x. Moreover, if for any y ∈ T x the formula (2.5) holds, then y = x, and so T x ⊂ {x}. Since T x = ∅, T x = {x}. Theorem 2.3 Let f : X → Y • be a proper extended vector-valued function. If there exists r ∈ R and x0 ∈ dom f such that f (X) ∩ (f (x0 ) + rk 0 − C) = ∅ and f satisfies (H1), (H2) and the following condition (T), (T) for each u ∈ S(x0 ) with f (X) ∩ (f (u) − C) = {f (u)}, there exists v = u such that f (v) + ϕ(P (v − u))k 0 C f (u),

(2.6)

then there exists x ∈ S(x0 ) such that f (x) ∈ Min(f (X), C). Proof We define T x := S(x) = {x ∈ X : f (x ) + ϕ(P (x − x))k 0 C f (x)}, ∀x ∈ X. For every x ∈ X, x ∈ T x = ∅ and for every y ∈ T x the formula (2.3) holds. By Theorem 2.2, there exists x ∈ S(x0 ) such that T x = {x}. / Min(f (X), C), i.e., f (X) ∩ (f (x) − C) = {f (x)}. By the condition (T), Suppose that f (x) ∈ there exists v = x such that f (v) + ϕ(P (v − x))k 0 C f (x), i.e., v ∈ S(x), and so v ∈ T x = {x}. Therefore, v = x, a contradiction. It follows that f (x) ∈ Min(f (X), C). Theorem 2.4 Theorems 2.1–2.3 are equivalent to each other. Proof By the proofs of Theorems 2.2 and 2.3, we have Theorem 2.1 ⇒ Theorem 2.2 ⇒ Theorem 2.3. We only need to prove that Theorem 2.3 ⇒ Theorem 2.1. Assume now that the conditions of Theorem 2.1 hold, we prove that there exists x ∈ S(x0 ) such that S(x) = {x}. Suppose that for every u ∈ S(x0 ), S(u) = {u}, then there exists v = u such that v ∈ S(u), i.e., the formula (2.6) holds, and therefore the condition (T) in Theorem 2.3 is satisfied. By Theorem 2.3, there exists x ∈ S(x0 ) such that f (X) ∩ (f (x) − C) = {f (x)}. Suppose that there exists v = x such that v ∈ S(x), i.e., f (v) + ϕ(P (v − x))k 0 C f (x),

(2.7)

and so f (v) C f (x). It follows that f (v) ∈ f (X) ∩ (f (x) − C) = {f (x)}, and therefore f (v) = f (x). By the formula (2.7), P (v−x) = 0. However, by [9, Lemma 4.4] and v−x = 0, P (v−x) > 0. This is a contradiction. Therefore, there exists x ∈ S(x0 ) such that S(x) = {x}.



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3 Remarks Remark 3.1 If X is a topological vector space, by [9, Lemma 4.11], we can obtain the corollaries of Theorems 2.1–2.3 only replacing M-closedness and M-completeness in assumptions by sequential closedness and sequential completeness. Furthermore, the equivalences of those corollaries can also be established. Remark 3.2 If Y is a topological vector space and C ⊂ Y is a proper sequentially closed pointed convex cone, then the results related to Theorems 2.1–2.4 can be obtained. Remark 3.3 The results of [3, Theorem] and [6, Theorem 2] are the immediately applications of the scalar form of Theorem 2.1.

References [1] Ekeland, I., Sur les probl`emes variationnels, C. R. Acad. Sci. Paris, S´ er. A-B, 1972, 275: 1057-1059 (in French). [2] Araya, Y., Ekeland’s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl., 2008, 346(1): 9-16. [3] Bosch, C., Garc´ıa, A. and Garc´ıa, C.L., An extension of Ekeland’s variational principle to locally complete spaces, J. Math. Anal. Appl., 2007, 328(1): 106-108. [4] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 1974, 47(2): 324-353. [5] G¨ opfert, A., Riahi, H., Tammer, C. and Zalinescu, C., Variational Methods in Partially Ordered Spaces, New York: Springer-Verlag, 2003. [6] Hamel, A.H., Phelps’ lemma, Danes’ drop theorem and Ekeland’s principle in locally convex spaces, Proc. Amer. Math. Soc., 2003, 131(10): 3025-3038. [7] Qiu J.H., Local completeness, drop theorem and Ekeland’s variational principle, J. Math. Anal. Appl., 2005, 311(1): 23-39. [8] Qiu J.H., Ekeland’s variational principle in Fr´ echet spaces and the density of extremal points, Studia Math., 2005, 168(1): 81-94. [9] Wong, C.-W., A drop theorem without vector topology, J. Math. Anal. Appl., 2007, 329(1): 452-471. [10] Lin, L.-J. and Du, W.-S., Some equivalent formulations of the generalized Ekeland’s variational principle and their applications, Nonlinear Anal., 2007, 67: 187-199. [11] Hogbe-Nlend, H., Bornologies and Functional Analysis, Amsterdam: North-Holland, 1977. [12] Hern´ andez, E., Jim´enez, B. and Novo, V., Benson proper efficiency in set-valued optimization on real linear spaces, In: Recent Advances in Optimization, Lecture Notes in Econom. and Math. Systems, Vol. 563, Berlin: Springer-Verlag, 2006, 45-59.

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