BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios of covex real-valued fuctios, give i terms of quasicovexity ad Jese-type covexity by K. Nikodem [1], F.A. Behriger [2], ad X.M. Yag, K.L. Teo ad X.Q. Yag [3]. 1. Itroductio Throughout the paper we deote by X a liear space ad by Y a topological liear space, partially ordered by a closed covex coe K havig a oempty iterior i Y. Let F : D 2 Y be a set-valued mappig, defied o a oempty covex subset D of X. Recall (see e.g. [4]) that F is said to be K-covex if the iclusio (1) tf (x) (1 t)f (x ) F (tx (1 t)x ) K holds for all x, x D ad for every t [0, 1]. By aalogy to vector-valued fuctios, we say that F is K-quasicovex if for each y Y the level set L F (y) := {x D : y F (x) K} is covex. Sice K is a covex coe, it ca be easily see that F is K-quasicovex wheever it is K-covex. 1991 Mathematics Subject Classificatio. Primary 26B25; Secodary 49J53. Key words ad phrases. Geeralized covexity, set-valued mappigs. This work was supported by a research grat of CNCSU uder Cotract Nr. 46174. 1
2 I order to get sufficiet coditios for a K-quasicovex mappig to be K-covex, we shall cosider the followig cocept of geeralized covexity: F will be called weakly K-covex with respect to a oempty set T ]0, 1[ if for all x, x D there exists some t T for which (1) holds. Note that this cocept exteds several otios of geeralized covexity, which were itesively studied i the literature i the particular case of real-valued fuctios. Ideed, if T ]0, 1[ is a sigleto, we recover the Jese-type covexity (see e.g. [5] ad refereces therei), which is owadays also kow as early covexity (see e.g. [6]). O the other had, for T = ]0, 1[ we recover the otio of weakly covexity, itroduced by A. Alema i [7]. As a itermediate case, if T = [δ, 1 δ] with δ ]0, 1/2[, we recover the otio of uiform covexlikeess, which has bee itroduced by H. Hartwig i [8]. Our aim here is to study the set-valued mappigs, but our mai result also focus o vector-valued fuctios. Actually, if f : D Y is a fuctio defied o a oempty covex subset D of X, the f will be called K-covex (respectively K-quasicovex, or weakly K-covex with respect to a oempty set T ]0, 1[) if ad oly if the set-valued mappig F : D 2 Y, defied by F (x) = {f(x)} for all x D, is K-covex (respectively K-quasicovex, or weakly K-covex with respect to T ). 2. Mai result Theorem 2.1. Let F : D 2 Y be a set-valued mappig defied o a oempty covex subset D of X. If F has K-closed values (i.e. F (x) K is a closed set for every x D), the the followig assertios are equivalet: (i) F is K-covex;
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS 3 (ii) F is both K-quasicovex ad weakly K-covex with respect to a oempty compact set T ]0, 1[. Proof. Obviously (i) implies (ii), the coclusio beig true for ay T ]0, 1[. Coversely suppose that (ii) holds ad let T be a oempty compact subset of ]0, 1[ for which F is weakly K-covex. Let us deote, for all x, x D, T x,x := {t [0, 1] : (1) holds }. I order to prove (i), we just have to show that T x,x = [0, 1] for all x, x D. To this ed, cosider two arbitrary poits x 0, x 1 D ad let us firstly prove that T x0,x 1 is dese i [0, 1]. Suppose o the cotrary that this is ot the case. The there exist some a, b [0, 1], a < b, such that (2) [a, b] T x0,x 1 =. Sice {0, 1} T x0,x 1, we ca defie the real umbers (3) α := sup [0, a] T x0,x 1 ad β := if [b, 1] T x0,x 1. Obviously α a < b β ad, by (2) ad (3), we have (4) ]α, β[ T x0,x 1 =. Let us deote, for all t [0, 1], x t := tx 0 (1 t)x 1 ad Y t := tf (x 0 ) (1 t)f (x 1 ). Recallig (3) ad takig ito accout that T is compact, we ca fid some umbers u [0, α] T x0,x 1 ad v [β, 1] T x0,x 1 such that tu (1 t)v ]α, β[ for all t T. O
4 the other had, sice F is weakly K-covex with respect to T, we ca choose a umber τ T xu,x v T. Hece (5) γ := τu (1 τ)v ]α, β[. Sice u, v T x0,x 1, we have Y u F (x u ) K ad Y v F (x v ) K. Hece τy u (1 τ)y v τf (x u ) (1 τ)f (x v ) K. Recallig that τ T xu,x v, i.e. τf (x u )(1τ)F (x v ) F (τx u (1τ)x v )K, we obtai: Y γ = τy u (1 τ)y v F (τx u (1 τ)x v ) K = F (x γ ) K, which meas that γ T x0,x 1. By (5) it follows that ]α, β[ T x0,x 1, cotradictig (4). So, we have proved that T x0,x 1 is dese i [0, 1]. Now, let us show that T x0,x 1 = [0, 1]. Obviously {0, 1} T x0,x 1 [0, 1]. Cosider a arbitrary t ]0, 1[. We just eed to prove that t T x0,x 1, i.e. Y t F (x t ) K. If Y t = the coclusio is obvious. Otherwise let y Y t. By defiitio of Y t we have y = tz 0 (1t)z 1 for some z 0 F (x 0 ) ad z 1 F (x 1 ). Cosider a poit e itk. By desity of T x0,x 1 i [0, 1], we ifer the existece of two sequeces: (t ) N i T x0,x 1 [0, t] ad (t ) N i T x0,x 1 [t, 1], such that { y, y } y 1 e itk, for all 1, where y = t z 0 (1 t )z 1 ad y = t z 0 (1 t )z 1. The, we have y 1 e t F (x 0 ) (1 t )F (x 1 ) itk F (x t ) K itk F (x t ) K, y 1 e t F (x 0 ) (1 t )F (x 1 ) itk F (x t ) K itk F (x t ) K,
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS 5 implyig that { } x t, x t LF (y 1 e), for all 1. Recallig that F is K-quasicovex ad takig ito accout that x t [x t, x t ] for each N, we ca deduce that x t L F (y 1 e), i.e. y 1 e F (x t) K, for all 1. Fially, by lettig, we ifer that y F (x t ) K = F (x t ) K. Corollary 2.2. Let f : D Y be a fuctio defied o a oempty covex subset D of X. The f is K-covex if ad oly if it is both K-quasicovex ad weakly K-covex with respect to a oempty compact set T ]0, 1[. Proof. It follows by Theorem 2.1, where F : D 2 Y is defied by F (x) = {f(x)} for all x D. I this case F (x) K is closed for every x D, sice the coe K is closed. Remark 2.3. The assumptio o the compactess of T is essetial. Ideed, cosider X = Y = R ad C = R, ad let f : D = [0, 1] R be defied by: f(x) = 1 if x [0, 1/2], ad f(x) = 0 if x ]1/2, 1]. The f is both quasicovex ad weakly covex with respect to T =]0, 1[, but f is ot covex. Remark 2.4. Corollary 2.2 geeralizes some kow characterizatio theorems give for real-valued covex fuctios, such as: (a) Propositio 3 i [1], where X = R, Y = R, K = R, D R is a oempty covex ope set, ad T = {1/2}. (b) Theorem 2 i [2], where X is a liear space, Y = R, C = R, D X is a oempty covex set, ad T = {1/2}.
6 (c) Theorem 3 i [3], where X = R, Y = R, K = R, D R is a oempty covex set, ad T = {α} with α ]0, 1[. Refereces [1] K. Nikodem, O some class of midcovex fuctios, A. Polo. Math. 50 (2) 145 151 (1989). [2] F.A. Behriger, Covexity is equivalet to midpoit covexity combied with strict quasicovexity, Optimizatio 24 (3-4) 219 228 (1992). [3] X.M. Yag, K.L. Teo, ad X.Q. Yag, A characterizatio of covex fuctio, Appl. Math. Lett. 13 (1) 27 30 (2000). [4] D.T. Luc, Theory of vector optimizatio, Spriger-Verlag, Berli (1989). [5] A.W. Roberts ad D.E. Varberg, Covex fuctios, Academic Press, New York-Lodo (1973). [6] W.W. Brecker ad G. Kassay, A systematizatio of covexity cocepts for sets ad fuctios, J. Covex Aal. 4 (1) 109 127 (1997). [7] A. Alema, O some geeralizatios of covex sets ad covex fuctios, Aal. Numér. Théor. Approx. 14 (1) 1 6 (1985). [8] H. Hartwig, Geeralized covexities of lower semicotiuous fuctios, Optimizatio 16 (5) 663 668 (1985). LACO, UMR 6090, Departmet of Mathematics, Uiversity of Limoges, 87060 Limoges, Frace E-mail address: beoist@uilim.fr Faculty of Mathematics ad Computer Sciece, Babeş-Bolyai Uiversity of Cluj, 3400 Cluj-Napoca, Romaia E-mail address: popovici@math.ubbcluj.ro