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A Generalization of Hahn–Banach Extension Theorem ✩ J. Math. Anal. Appl. 302 (2005) 441–449 www.elsevier.com/locate/jmaa A generalization of Hahn–Banach extension theorem ✩ Jianwen Peng a,b,∗, Heung Wing Joseph Lee c, Weidong Rong b, Xinmin Yang a a Department of Mathematics, Chongqing Normal University, Chongqing 400047, PR China b Department of Mathematics, Inner Mongolia University, Hohhot 010021, Inner Mongolia, PR China c Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Received 3 April 2003 Submitted by C.E. Chidume Abstract In this paper, the notion of affinelike set-valued maps is introduced and some properties of these maps are presented. Then a new Hahn–Banach extension theorem with a K-convex set-valued map dominated by an affinelike set-valued map is obtained. 2004 Published by Elsevier Inc. Keywords: Hahn–Banach extension theorem; Affinelike; Set-valued map 1. Introduction The Hahn–Banach extension theorem plays a central role in optimization theory. Gener- alizations and variants of the Hahn–Banach theorem were developed in different directions in the past. Early in the sixties, Day [1] generalized the Hahn–Banach extension theorem to the partially ordered linear space. Later, the equivalence of the Hahn–Banach extension ✩ This research was partially supported by the National Natural Science Foundation of China (Grant No. 10261005) and Education Committee project Research Foundation of Chongqing (Grant No. 030801), and the research Committee of the Hong Kong Polytechnic University. * Corresponding author. E-mail address: [email protected] (J. Peng). 0022-247X/$ – see front matter 2004 Published by Elsevier Inc. doi:10.1016/j.jmaa.2004.03.038 442 J. Peng et al. / J. Math. Anal. Appl. 302 (2005) 441–449 theorem and the least upper bound property was proved by Bonnice and Silverman [3]. In recent years, Hirano et al. [4] proved the Hahn–Banach extension theorem in which a concave functional is dominated by a sublinear functional in a nonempty convex set by us- ing a theorem of Fan on a system of convex inequalities. Chen and Craven [5] proved two Hahn–Banach extension theorems for vector-valued functions and used a Hahn–Banach extension theorem to prove the existence results of the strong subgradient of vector-valued functions. Yang [6] obtained a Hahn–Banach theorem for a convex relation (i.e., a set- valued map with a convex graph) and used it to prove the existence of a weak subgradient of set-valued maps which was different from the weak subgradient in [7]. Borwein [8] in- troduced a strong subgradient of set-valued maps and proved its existence under convex relation. Chen and Wang [9] proved a Hahn–Banach theorem by using Lemma 1 in [2] and existence of a linear subgradient of set-valued maps. There are also many other forms of Hahn–Banach theorems (see [11–15] etc.). In this paper, the notion of affinelikeness for set-valued map is introduced and some properties of the affinelike set-valued map are pre- sented. Then a new Hahn–Banach extension theorems which used a set-valued map instead of a single-valued map as a dominating map for the K-convex set-valued maps is obtained. This Hahn–Banach extension theorem and its Corollaries contain the Hahn–Banach exten- sion theorems in [5,9,11] as special cases and are different from the results in [8]. Throughout this paper, let X, Y and Z be three real linear topological spaces, let K ⊂ Y be a pointed closed convex cone and (Y, K) be a partially ordered linear space. The partial order on a partially ordered linear space (Y, K) is defined by y1,y2 ∈ Y , y1 y2 ⇔ y2 − y1 ∈ K.LetC be a nonempty subset of X. The algebraic interior of C is defined by Cor C = x ∈ C |∀x1 ∈ X, ∃δ>0, s.t. ∀λ ∈ (0,δ), x+ λx1 ∈ C . By X∗ and Y ∗ we denote the topological dual space of X and Y , respectively. Let L(X, Y ) be the space of continuous linear operators from X into Y .LetF : X → 2Y be a set-valued map. Definition 1.1 [10]. A topological vector space Y , partially ordered by a convex cone K, is order-complete if every subset M which has an upper bound b in terms of the ordering (that is (∀y ∈ M), b − y ∈ K) then has a supremum bˆ (that is, there exists bˆ ∈ Y such that bˆ is an upper bound to M, and each upper bound b to M satisfies b − bˆ ∈ K). It is easy to prove the following result. Proposition 1.1. Let (Y, K) be an order-complete topological vector space. Then, every subset M which has a lower bound d in terms of the ordering (that is (∀y ∈ M), y −d ∈ K) has a infimum dˆ (that is, there exists dˆ ∈ Y such that dˆ is a lower bound to M, and each lower bound d to M satisfies dˆ − d ∈ K). Definition 1.2 [7]. Let C ⊂ X be a convex set. A set-valued map F : C → 2Y is called K-convex on C,ifforeveryx,y ∈ C and λ ∈ (0, 1), λF (x) + (1 − λ)F (y) ⊂ F λx + (1 − λ)y + K. J. Peng et al. / J. Math. Anal. Appl. 302 (2005) 441–449 443 Remark 1.1. By Proposition 2.2.3 in [16], a set-valued map F : C → 2Y is K-convex on C if and only if EpiF ={(x, y) | x ∈ C, y ∈ Y, y ∈ F(x)+ K} is a convex set. From [6] and [8], we know that F : C → 2Y is called a convex relation if Gr F ={(x, y) | x ∈ C, y ∈ Y, y ∈ F(x)} is a convex set. As a generalization of the vector-valued affine maps, we introduce the definition of the affinelike set-valued map. Definition 1.3. The set-valued map H : X → 2Y is called an affinelike map if there exists a continuous linear map f : X → Y and a nonempty convex set M ⊂ Y such that H(x)= f(x)+ M, ∀x ∈ X. Proposition 1.2. If a vector-valued map f : X → Y is continuous at x0 ∈ X and ∅ = Y M ⊂ Y , then the set-valued map F : X → 2 is lower semicontinuous at x0,where F(x)= f(x)+ M, ∀x ∈ X. Proof. Let y0 ∈ F(x0) = f(x0) + M. Then there exists m = m(y0) ∈ M such that y0 = f(x0) + m, for arbitrary neighborhood N(y0) of y0, N(y0) − m is a neighborhood of y0 − m = f(x0). From the continuity of f at x0, there exists a neighborhood of U(x0) of x0 such that for all x ∈ U(x0), f(x)∈ N(y0) − m. Hence f(x)+ m ∈ N(y0),soF(x0) ∩ N(y0) =∅. Therefore, F is lower semicontinuous at x0. 2 Remark 1.2. From Proposition 1.2, we know that an affinelike map must be lower semi- continuous. Proposition 1.3. Let the set-valued map H : X → 2Y be an affinelike map. Then for all α>0, β>0 with α + β = 1 and for all x,y ∈ X, αH(x) + βH(y)= H(αx+ βy). (1) Proof. Suppose M is a convex set, it is obvious that αM + βM = M. Then the formula (1) is satisfied from Definition 1.3. 2 2. Hahn–Banach extension theorems of set-valued maps In this section, a new generalized Hahn–Banach extension theorems in which a K- convex set-valued map is dominated by an affinelike set-valued map in a nonempty convex set is first proven. By this general result, we get some useful conclusions. Theorem 2.1. Let X be a real linear topological space, and let (Y, K) be a real order- complete linear topological space, and C ⊂ X be a convex set, let set-valued map F : C → Y 2 be K-convex, and for all x ∈ C, F(x) =∅.LetX0 be a proper subspace of X, with Y X0 ∩ Cor C =∅.LetH : X0 → 2 be an affinelike map such that F(x)− H(x)⊂ K for Y each x ∈ X0 ∩ C. Then there exists an affinelike map L : X → 2 such that L(x) = H(x) for all x ∈ X0 ∩ C, and F(x)− L(x) ⊂ K for all x ∈ C. 444 J. Peng et al. / J. Math. Anal. Appl. 302 (2005) 441–449 Proof. The theorem holds trivially if C = X0. Assume that C = X0.SinceX0 is a proper subspace of X, there exists x0 ∈ X \ X0.LetX1 ={x + rx0: x ∈ X0,r∈ R}. It is easy to verify that X1 be a subspace of X, X0 ⊂ X1 and X1 ∩ Cor C =∅. Note that for a fixed point xˆ ∈ X0 ∩ Cor C,if|k| is sufficiently small, then xˆ ± kx0 ∈ X1 ∩ C. For arbitrary x1,x2 ∈ X0 ∩ Cor C ⊂ X1 ∩ Cor C,ifx1 + λx0,x2 − µx0 ∈ X1 ∩ C (λ,µ > 0), set α = λ/(λ + µ), β = 1 − α = µ/(λ + µ),thenα(x2 − µx0) + β(x1 + λx0) ∈ X1 ∩ C since X1 ∩ C is a convex set. That is, αx2 + βx1 + (λβ − µα)x0 ∈ X1 ∩ C. By the definition of X0 and λβ − µα = 0, we get αx2 + βx1 ∈ X0.Soαx2 + βx1 ∈ X0 ∩C. By Proposition 1.3, we have αH(x2) + βH(x1) = H(αx2 + βx1). (2) By Definition 1.2, there exists a continuous linear map f : X → Y and a nonempty convex set M ⊂ Y such that H(x)= f(x)+ M. From (2), we obtain αf (x2) + βf (x1) + M = f(αx2 + βx1) + M = H(αx2 + βx1). (3) And by the assumption, we get F(αx2 + βx1) − H(αx2 + βx1) ⊂ K.
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