Applied Mathematics Letters 22 (2009) 501–504

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Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

On generalizing Takahashi’s nonconvex minimization theoremYousuke ArayaGraduate School of Science and Technology, Niigata University, 8050, Ikarashi, 2-no-cho, Niigata 950-2181, Japan

a r t i c l e i n f o

Article history:Received 18 December 2006Received in revised form 9 March 2007Accepted 3 June 2008

Keywords:Takahashi’s nonconvex minimizationtheoremScalarizationNonconvex separation theoremVector-valued functionSolid convex cone

a b s t r a c t

Wepresent a simple proof of vectorial Takahashi’s nonconvexminimization theorembasedon Gopfert, Tammer and Zalinescu [A. Gopfert, C. Tammer, C. Zalinescu, On the vectorialEkeland’s variational principle and minimal points in product spaces, Nonlinear Anal. 39(2000) 909–922; C. Tammer, A variational principle and a fixed point theorem, in: SystemModelling and Optimization (Compiegne, 1993), in: Lecture Notes in Control and Inform.Sci., vol. 197, Springer, London, 1994, pp. 248–257].

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Since Ekeland [1] in 1972, the variational principle and its equivalent formulations have been one of the main subjectsin many fields of nonlinear functional analysis, convex analysis, and optimization. Phelps shows that Ekeland’s variationalprinciple is equivalent to the existence ofmaximal pointswith respect to an appropriate order.Moreover, Takahashi presentsthe following theorem, which is equivalent to the above two.

Theorem 1.1 (Takahashi[5]). Let (X, d) be a complete metric space and f : X → (−∞,∞] a l.s.c. function, 6≡ +∞, boundedfrom below. Suppose that for each u ∈ X with infx∈X f (x) < f (u), there exists v ∈ X such that v 6= u and f (v)+ d(u, v) ≤ f (u).Then there exists x0 ∈ X such that f (x0) = infx∈X f (x).

Weare interested in generalizing Theorem1.1 to an analogy for vector-valued function. Tammer [2,6] presented vectorialversions of Takahashi’s nonconvex minimization theorem as a corollary of vectorial Ekeland’s variational principle.In this paper, replacing the conditions of Tammer’s vectorial Takahashi’s theorem by aweaker one, we present a vectorial

version of Takahashi’s nonconvex minimization theorem whose proof is more simple than Tammer’s one.We give the preliminary terminology and notation used throughout this paper. Let X be a complete metric space and Y a

locally convex space. For a set A ⊂ Y , corA, int A, clA denote the algebraic interior, the topological interior and the topologicalclosure of A, respectively. We assume that a nonempty set C ⊂ Y is a solid closed convex cone, that is,

(a) int C 6= ∅,(b) clC = C ,(c) C + C ⊆ C ,(d) λC ⊆ C for all λ ∈ [0,∞).

A cone C is said to be pointed if C ∩ (−C) = {0}. If a pointed convex cone C ⊆ Y is given, we can define an ordering in Y by‘‘x≤C ywhen y− x ∈ C ’’. This ordering is compatible with the vector structure of Y , that is, for every x ∈ Y and y ∈ Y ,

E-mail address: yousuke@m.sc.niigata-u.ac.jp.

0893-9659/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.aml.2008.06.024

502 Y. Araya / Applied Mathematics Letters 22 (2009) 501–504

(i) x≤C y implies that x+ z≤C y+ z for all z ∈ Y ;(ii) x≤C y implies that αx≤C αy for all α ≥ 0.

We also denote f (X) :=⋃x∈X {f (x)}where f : X → Y and C0 := int C ∪ {0}. We say that a point a ∈ A is an minimal [resp.

weak minimal] point of A if

A ∩ (a− C) = {a} [resp. A ∩ (a− C0) = {a}].

We denote by Min(A; C) [resp. wMin(A; C)] the set of minimal [resp.weak minimal] points of Awith respect to K [resp. C0].We can easily see that

Min(A; C) ⊂ wMin(A; C) ⊂ A.

Tammer andWeidner introduced the following nonlinear scalarizing function, which takes values in R in the setting of thispaper because C is solid.

Lemma 1.2 (Lemma 7 in [3] and Theorem 2.3.1 in [2]). Let C be a closed convex cone. We take k0 ∈ C \ (−C) and definehC,k0 : Y → [−∞,∞] by

hC,k0(y) = inf{t ∈ R|y ∈ tk0 − C}.

Then the function hC,k0 has the following six properties:

(i) hC,k0 is proper;(ii) hC,k0 is lower semicontinuous;(iii) hC,k0 is sublinear;(iv) hC,k0 is C-monotone (i.e., y1≤C y2 implies hC,k0(y1) ≤ hC,k0(y2));(v) {y ∈ Y |hC,k0(y) ≤ t} = tk

0− C;

(vi) hC,k0(y+ λk0) = hC,k0(y)+ λ for every y ∈ Y and λ ∈ R.

Moreover, if k0 ∈ cor C then hC,k0 has the following four properties:

(vii) hC,k0 achieves a real value;(viii) hC,k0 is continuous;(ix) {y ∈ Y |hC,k0(y) ≤ t} = tk

0− cor C;

(x) hC,k0 is strictly int C-monotone (i.e., y2 − y1 ∈ cor C implies hC,k0(y1) < hC,k0(y2)).

As a corollary of the above lemma, Gerth (Tammer) andWeidner presented the following nonconvex separation theorem.

Lemma 1.3 (Theorem 2.3.6 in [2]). Assume that Y is a topological vector space, C a solid closed convex cone and A ⊂ Y anonempty set such that A ∩ (−int C) = ∅. Then hC,k0 is a finite-valued continuous function such that

hC,k0(−y) < 0 ≤ hC,k0(x) ∀x ∈ A, y ∈ int C,

moreover, hC,k0(x) > 0 for all x ∈ int A.

The above two lemmas play important roles in this paper.

2. Main result

Theorem 2.1. Let f : X → Y be a vector-valued function and k0 ∈ int C. Suppose that

(H) {x ∈ X |f (x)≤C rk0} is closed for every r ∈ R,

and there exists y ∈ Y such that f (X) ∩ (y− int C) = ∅. Moreover f satisfies the following condition;

(A) for each u ∈ X with f (X) ∩ (f (u)− C0) 6= {f (u)}, there exists v 6= u such that f (v)+ d(u, v)k0≤C f (u).

Then there exists x ∈ X such that f (x) ∈ wMin(f (X); C0).

Proof. First of all, (hC,k0 ◦ f )(x) is bounded from below on X for all x ∈ X . By Lemma 1.3, we have

hC,k0(−y) < 0 ≤ hC,k0(f (x)− y)

for all x ∈ X , y ∈ int C . Using (iii) of Lemma 1.2, we have

−∞ < hC,k0(−y)− hC,k0(−y) < hC,k0(f (x)).

Second we show that hC,k0 ◦ f is lower semicontinuous. Indeed, from (v) of Lemma 1.2, we get

{x ∈ X |hC,k0(f (x)) ≤ r} = {x ∈ X |f (x) ∈ rk0− C} = {x ∈ X |f (x)≤C rk0},

Y. Araya / Applied Mathematics Letters 22 (2009) 501–504 503

which is closed for all r ∈ R by hypothesis (H). From f (X) ∩ (f (u) − C0) 6= {f (u)} there exists y 6= f (u) such thaty ∈ f (u)− int C and from Lemma 1.2 we have that

infx∈XhC,k0(f (x)) ≤ hC,k0(y) < hC,k0(f (u)).

Moreover, by assumption (A) and conditions (iv) and (vi) in Lemma 1.2, we have that hC,k0(f (v)) + d(u, v) ≤ hC,k0(f (u)).Therefore, we conclude that hC,k0 ◦ f satisfies all assumptions of Theorem 1.1 and there exists x ∈ X such that

hC,k0(f (x)) = infx∈XhC,k0(f (x)).

With the same argument as above, we obtain the conclusion. �

Remark 1. Note that taking Y = R, C = R+ = [0,∞) and k0 = 1 ∈ R+ \ {0} in Theorem 2.1, we obtain Theorem 1.1.

Remark 2. In Theorem 2.1, the solidness of C is used to ensure that the constructed functional hC,k0 takes finite values. Ifwe set

Y = R2 C = {(x, x)|x ∈ R} k0 = (1, 1) a = (3, 3) b = (0, 1).

We have that

hC,k0(a) = 3 but hC,k0(b) = ∞.

This fact guarantees the lower boundedness of the function in Theorem 2.1.

Remark 3. The lower boundedness condition on the vector-valued function f and Condition (A) of Theorem 2.1 are weakerthan Tammer’s result [2,6]. We also note that the pointedness of C is not needed to prove Theorem 2.1.

Example 1. We set

X = R, Y = R2, C = R2+, k0 = (1, 1), f (x) = (2x, 3x).

It is clear that f satisfies condition (H). We have that u = 0 satisfies f (X) ∩ (f (u) − C0) 6= f (u) and there exists v = −1which satisfies

f (u)− f (v)− d(u, v)k0 = (1, 2) ∈ C .

However, there is no point y such that f (X) ∩ (y− int C) = ∅ and f does not have a weak minimal point.

Example 2. We set

X = (−1, 2), Y = R2, C = R2+, k0 = (1, 1), f (x) = (x, x+ 2), y = (−2, 0).

It is clear that f satisfies f (X) ∩ (y− int C) = ∅ and condition (H). An element u = 1 satisfies f (X) ∩ (f (u)− C0) 6= {f (u)}and there exists v = 0 such that

f (u)− f (v)− d(u, v)k0 = (0, 0) ∈ C .

However, X is not complete, f does not have a weak minimal point. If we set X = [−1, 2], f has a weak minimal point(−1, 1) and x = −1 is a solution.

3. Conclusions

Muchwork has focused on the topic of producing conditions which ensure the existence of minimal points. For example,interesting results have been given by Yu, Borwein, Jahn, Luc [4] and their references. We obtain an existence theorem ofweak minimal points without any compactness condition.

References

[1] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974) 324–354.[2] A. Gopfert, H. Riahi, C. Tammer, C. Zalinescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003.[3] A. Gopfert, C. Tammer, C. Zalinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal. 39 (2000)909–922.

504 Y. Araya / Applied Mathematics Letters 22 (2009) 501–504

[4] D.T. Luc, Theory of vector optimization, in: Lecture Notes in Economics and Mathematical Systems, vol. 319, Springer-Verlag, Berlin, 1989.[5] W. Takahashi, Existence theorems generalizing fixed point theorems for multivalued mappings, in: Fixed Point Theory and Applications (Marseille,1989), in: Pitman Res. Notes Math. Ser., vol. 252, Longman Sci. Tech, Harlow, 1991, pp. 397–406.

[6] C. Tammer, A variational principle and a fixed point theorem, in: System Modelling and Optimization (Compiegne, 1993), in: Lecture Notes in Controland Inform. Sci., vol. 197, Springer, London, 1994, pp. 248–257.