The Australian Journal of Mathematical Analysis and Applications

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1 The Australia Joural of Mathematical Aalysis ad Applicatios Volume 6, Issue 1, Article 10, pp. 1-10, 2009 DIFFERENTIABILITY OF DISTANCE FUNCTIONS IN p-normed SPACES M.S. MOSLEHIAN, A. NIKNAM AND S. SHADKAM TORBATI Special Issue i Hoor of the 100th Aiversary of S.M. Ulam Received 20 Jue, 2008; accepted 20 July, 2008; published 4 September, DEPARTMENT OF PURE MATHEMATICS CENTRE OF EXCELLENCE IN ANALYSIS ON ALGEBRAIC STRUCTURES CEAAS) FERDOWSI UNIVERSITY OF MASHHAD P. O. BOX 1159, MASHHAD, IRAN moslehia@ferdowsi.um.ac.ir ikam@math.um.ac.ir shadkam.s@wali.um.ac.ir ABSTRACT. The farthest poit mappig i a p-ormed space X is studied i virtue of the Gateaux derivative ad the Frechet derivative. Let M be a closed bouded subset of X havig the uiformly p-gateaux differetiable orm. Uder certai coditios, it is show that every maximizig sequece is coverget, moreover, if M is a uiquely remotal set the the farthest poit mappig is cotiuous ad so M is sigleto. I additio, a Hah Baach type theorem i p-ormed spaces is proved. Key words ad phrases: Frechet derivative, quasi-orm, p-ormed space, farthest mappig, Hah-Baach theorem, remotal set Mathematics Subject Classificatio. Primary 41A52. secodary 41A65, 58C20. ISSN electroic): c 2009 Austral Iteret Publishig. All rights reserved.

2 2 M.S. MOSLEHIAN, A. NIKNAM, S. SHADKAM TORBATI 1. INTRODUCTION Let X be a real liear space. A quasi-orm is a real-valued fuctio o X satisfyig the followig coditios: i) x 0 for all x X, ad x = 0 if ad oly if x = 0. ii) λx = λ x for all λ R ad all x X. iii) There is a costat K 1 such that x + y K x + y ) for all x, y X. The pair X, ) is called a quasi-ormed space if is a quasi-orm o X. The smallest possible K is called the module of cocavity of. By a quasi-baach space we mea a complete quasi-ormed space, i.e., a quasi-ormed space i which every Cauchy sequece coverges i X. This class icludes Baach spaces. The most sigificat class of quasi-baach spaces, which are ot Baach spaces are L p -spaces for 0 < p < 1 equipped with the L p -orms p. A quasi-orm is called a p-orm 0 < p < 1) if x + y p x p + y p for all x, y X. I this case a quasi-ormed quasi Baach) space is called a p-ormed p-baach) space. By the Aoki Rolewicz theorem [7] each quasi-orm is equivalet to some p-orm. Sice it is much easier to work with p-orms tha quasi-orms, heceforth we restrict our attetio maily to p-orms. See [1, 4, 8] ad the refereces therei for more iformatio. Throughout the paper, SX) ad BX) deote the uit sphere ad the uit ball of X, respectivelty. If x is i X, the dual of X, ad x X we write x x) as x, x. We also cosider quasi-orms with K > 1. The case where K = 1 turs out to be the classical ormed spaces, so we will ot discuss it ad refer the iterested reader to [2]. Let k > 0. A real valued fuctio f o X is said to be k-gateaux differetiable at a poit x of X if there is a elemet dfx) of X such that for each y i X, 1.1) lim t 0 sgt) t k fx + ty) fx)) = dfx), y, ad we call dfx) the k-gateaux derivative of f at x. We say that f is k-frechet differetiable at a poit x of X if there is a elemet f x) of X such that 1.2) lim y 0 y k fx + y) fx) f x), y ) = 0, ad we call f x) the k -Frechet derivative of f at x. See [2]. The orm of X is called uiformly k-gateaux differetiable if for fx) = x k equality 1.1) holds uiformly for x, y X, ad the operator orm dfx) is less tha or equal 1. The dfx) is deoted by D k x). We say that a o-zero elemet x of X strogly exposes BX) at x SX), provided x, x = x, ad a sequece {y } i BX) coverges to x wheever { x, y } coverges to x, x. The reader is referred to [2, 5, 6] for aalogue results cocerig ormed spaces ad to the book of D.H. Hyers, G. Isac ad Th.M. Rassias [3] for extesive theory ad applicatios of oliear aalysis methods. I this paper we use some ideas of [2] to study the farthest poit mappig i a p-ormed space X by virtue of the Gateaux derivative ad the Frechet derivative. Let M be a closed bouded subset of X which has a uiformly p-gateaux differetiable orm. Uder certai coditios, we show that every maximizig sequece is coverget, moreover, if M is a uiquely remotal set the the farthest poit mappig is cotiuous ad so M is a sigleto. I additio, we prove a Hah Baach type theorem for p-ormed spaces ad give a applicatio., Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

3 DIFFERENTIABILITY OF DISTANCE FUNCTIONS 3 2. MAIN RESULTS We start this sectio with followig defiitio. Defiitio 2.1. Let X be a p-ormed space ad M be a bouded closed subset of X. Settig Ψx) := sup{ x y p : y M} x X), F x) := {y M : x y p = Ψx)} is called the set of farthest poits i M form x CHECK). We say that M is a uiquely remotal set if F is a sigleto for each x X ad the we deote the sigle elemet of F x) by F x). The map F is called the atiprojectio or the farthest poit mappig for M. A ceter of a bouded set M i a p-ormed space X is a elemet c i X such that rm) := sup c y p = if sup x y. y M x X y M I fact rm), the so called Chebyshev radius of M, is the smallest ball i X cotaiig M. We call a sequece {y } i M a maximizig sequece for x provided x y p Ψx) as. Lemma 2.1. If M is a oempty bouded subset of X ad x, y X the 2.1) Ψx) Ψy) x y p. Proof. Ψx) Ψy) = sup{ x m p : m M} sup{ y m p : m M} sup{ x m y + m p : m M} = x y p. Lemma 2.2. If x X is a poit of k-gateaux differetiability of Ψ ad y F x) the { 0, k < 1; dψx), x y 1 x y) p x y p k, k = 1. Proof. Sice y F x) we have x y p = Ψx). Hece for 0 < t < 1, we get 2.2) Ψx + tx y)) Ψx) x + tx y) y p x y p = 1 + t) p x y p x y p = 1 + t) p 1) x y p. Let s [0, ). The 2.3) dψx), sz = lim t 0 + t k Ψx + tsz) Ψx)) = s k s k lim t k Ψx + tsz) Ψx)) t 0 + = s k lim t k )Ψx + tsz) Ψx)) t 0 +s k = s k dψx), z,, Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

4 4 M.S. MOSLEHIAN, A. NIKNAM, S. SHADKAM TORBATI whece The dψx), x y 1 x y) = lim sgt) t k x y k )Ψx + tx y)) Ψx)) t 0 lim t t p 1) x y p k, t > 0; t = k lim t 0 1+t p 1) x y p k, t < 0. t) k dψx), x y 1 x y) = { 0, k < 1; p x y p k, k = 1. Theorem 2.3. Suppose that M is a closed bouded subset of X with a uiformly p-gateaux differetiable orm, Ψ is p-gateaux differetiable at a poit of x X\M with y F x) ad dψx) strogly exposes BX) at x y 1 x y). The every maximizig sequece for x coverges to y. Moreover, if M is a uiquely remotal set the F is cotiuous at x. Proof. Suppose that {y } is a maximizig sequece for x. For each t, we have 2.4) Fix z X. We have 2.5) ad 2.6) Ψx + tz) Ψx) sup x + tz y p lim x y P y M lim sup x + tz y p x y p). dψx), z = lim sgt) t p Ψx + tz) Ψx)) t 0 lim t) p lim sup x + tz y p x y p)) t 0 = lim lim if t) p x + tz y p x y p ) )) t 0 lim if D px y ), z dψx), z = lim sgt) t p Ψx + tz) Ψx)) t 0 + lim t p lim sup x + tz y p x y p)) t 0 + = lim lim sup t p x + tz y p x y p ) )) t 0 + lim sup D p x y ), z. Usig 2.5) ad 2.6) we get 2.7) lim D p x y ), z = dψx), z. We also have 2.8) D p x y ), x y = lim t 0 + t p x y + tx y ) p x y p ) ) = lim t 0 + t p x y ) p 1 + t) p 1) )., Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

5 DIFFERENTIABILITY OF DISTANCE FUNCTIONS 5 Usig equalities 2.7) for z = x y, 2.3) ad 2.8) we obtai dψx), x y 1 x y ) = lim D p x y ), x y 1 x y ) = lim lim t 0 + t p 1 + t) p 1) ) = 0 = dψx), x y 1 x y). Sice dψx) strogly exposes BX) at x y 1 x y), we get lim x y 1 x y ) = x y 1 x y). Sice {y } is a maximizig sequece for x, we therefore easily get 2.9) lim y = y. Next, assume that {x } coverges to x. We shall show that {F x )} coverges to F x). Sice M is a uiquely remotal set we have x F x ) p = sup{ x y p ; y M}. Hece 2.10) x F x) p x F x ) p. By 2.10), 2.11) x F x) p x F x ) p x F x) p + x x p x F x ) p x F x) p + x x p x F x) p x x p + x F x) x + F x) p 2 x x p. It follows from 2.11) ad the covergece of {x } to x that lim x F x ) p = x F x) p. Hece {F x )} is a maximizig sequece for x ad so, by the first part of the theorem lim F x ) = F x). Theorem 2.4. Suppose M is a closed bouded subset of a p-ormed space X ad x X\M is a poit of p-frechet differetiability of Ψ ad if y F x) ad Ψ x) strogly exposes BX) at x y 1 x y), the every maximizig sequece for x coverges to y. Proof. Let {y } be a maximizig sequece for x ad let k = if x y 2p. Sice M is closed, k > 0. Hece there is N such that for N the Ψx) x y p < k x y 2p, ad so we ca choose a sequece {α } of positive umbers such that α 0 ad Hece x y 2p > α 2 > Ψx) x y p N). 2.12) 1 + t) p x y p > 1 + t) p Ψx) α 2 ) N, 1 < t < 1). Let 0 < t < 1. Sice y M we have 2.13) Ψx ty x)) x ty x) y p = 1 + t) p x y p. It follows from 2.12) ad 2.13) that 2.14) Ψx ty x) Ψx) 1 + t) p x y p Ψx) > Ψx)1 + t) p 1) 1 + t) p α 2., Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

6 6 M.S. MOSLEHIAN, A. NIKNAM, S. SHADKAM TORBATI Fix ε > 0. By the defiitio of Ψ x), there is δ > 0 such that if y p < δ the 2.15) Ψx + y) Ψx) Ψ x), y ε y p. Let t p = α x y ) p < 1 ad α < δ for large. Replacig y by t x y ) i 2.15) ad otig 2.14), we get ε t x y ) p + Ψ x), t x y ) Ψx + t x y )) Ψx) Ψx)1 + t ) p 1) 1 + t ) p α 2, whece 2.16) 1 α Ψ x), t x y ) 1 α Ψx)1 + t ) p 1) 1 + t ) p α 2 ε t y x) p) 1 α Ψx)1 + t ) p 1) 1 + t ) p α ε Usig 2.16), we get 1 t p y x) p Ψx)1 + t ) p 1) 1 + t ) p α ε. 2.17) Ψ x), x y 1 x y ) = Ψ x), x y 1 t x y ) t = x y p t p Ψ x), t x y ) = 1 α Ψ x), t x y ) 1 Ψx) t p y x) 1 + t ) p 1) 1 + t p ) p α ε. Let i 2.17). The α 0, y x Ψx) ad t 0 + such that 2.18) lim Ψ x), x y 1 x y ) 0 Chagig t to t i 2.13) ad 2.14) ad utilizig a similar strategy as above we get whece lim Ψ x), y x 1 y x) 0, 2.19) lim Ψ x), x y 1 x y ) 0. By 2.18), 2.19) we have, By Lemma 2.2, lim Ψ x), x y 1 x y ) = 0. dψx), x y 1 x y) = 0, Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

7 DIFFERENTIABILITY OF DISTANCE FUNCTIONS 7 ad the 0 = lim tx y) 0 tx y) p Ψx + tx y))) Ψx) Ψ x), tx y) ) = lim tx y) p Ψx + tx y))) Ψx)) tx y) p Ψ x), tx y) ) ) t 0 + ) t p Ψx + tx y)) Ψx)) = lim Ψ x), x y 1 x y) t 0 + x y) p = lim t 0 + t p Ψx + tx y)) Ψx)) x y) p Ψ x), x y 1 x y) = x y p dψx), x y) Ψ x), x y 1 x y) = dψx), x y 1 x y) Ψ x), x y 1 x y) = Ψ x), x y 1 x y), therefore lim Ψ x), x y 1 x y ) = 0 = Ψ x), x y 1 x y). Sice Ψ x) strogly exposes BX) at x y 1 x y), we deduce that x y 1 x y ) x y 1 x y) which yields y y. 3. HAHN BANACH THEOREM AND ITS APPLICATION This sectio is devoted to oe versio of the celebrated Hah Baach theorem. We follow the strategy ad termiology of [9]. We begi with a lemma. Lemma 3.1. Let 0 < p < 1, let M be a liear subspace of a real vector space X ad let ρ : X R be a mappig such that i) ρx + y) ρx) + ρy) x, y X); ii) ρtx) = t p ρx) x X, 0 t R). If ϕ 0 : M R is a liear mappig such that ϕ 0 x) ρ 1 p x) for all x M, the there is a liear mappig ϕ : X R such that ϕ M = φ 0 ad ϕx) ρ 1 p x) x X). Proof. We use Zor s lemma. Cosider the partially ordered set P, whose typical member is a pair Y, ψ), where i) Y is a liear subspace of X which cotais X 0 ; ad ii) ψ : Y R is a liear mappig which is a extesio of ϕ 0 ad satisfies ψx) ρ 1 p x) x Y ; the partial order o P is defied by settig Y 1, ψ 1 ) Y 2, ψ 2 ) precisely whe a) Y 1 Y 2, ad b) ψ 2 Y1 = ψ 1. Furthermore, if Γ = {Y i, ψ i ) : i I} is ay totally ordered set i P, a easy verificatio shows that a upper boud for the family Γ is give by Y, ψ), where Y = i I Y i ad ψ : Y R is the uique ecessarily liear mappig satisfyig ψ Yi = ψ i for all i. Hece, by Zor s lemma, the partially ordered set P has a maximal elemet, call it Y, ψ). The proof of the lemma will be completed oce we have show that Y = X. Suppose that Y X; fix x 0 X Y, ad let Y 1 = Y +Rx 0 = {y+tx 0 : y Y, t R}. The defiitios esure that Y 1 is a subspace of X which properly cotais Y. Also, otice that ay liear mappig ψ 1 : Y 1 R which exteds ψ is prescribed uiquely by the umber t 0 = ψx 0 ) ad the equatio ψ 1 y + tx 0 ) = ψx) + tt 0 ). We assert that it is possible to fid a umber t 0 R such that the associated mappig ψ 1 would - i additio to extedig ψ - also satisfy ψ 1 ρ 1 p. This would the establish the iequality Y, ψ) Y 1, ψ 1 ), cotradictig the maximality of Y, ψ); which would i tur imply that we must have had Y = X i the first place, ad the proof would be complete., Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

8 8 M.S. MOSLEHIAN, A. NIKNAM, S. SHADKAM TORBATI First, observe that if y 1, y 2 Y are arbitrary, the ad cosequetly, 3.1) sup y 1 Y ψy 1 ) + ψy 2 ) = ψy 1 + y 2 ) = ψy 1 x 0 + y 2 + x 0 ) = ψy 1 x 0 ) + ψy 2 + x 0 ) ρ 1 p y1 x 0 ) + ρ 1 p y2 + x 0 ) [ ] [ ] ψy 1 ) ρ 1 p y1 x 0 ) if ρ 1 p y2 + x 0 ) ψy 2 ). y 2 Y Let t 0 be ay real umber which lies betwee the supremum ad the ifimum appearig i equatio 3.1). We ow verify that this t 0 does the job. Ideed, if t > 0, ad if y Y, the, sice the defiitio of t 0 esures that we fid that ψy 2 ) + t 0 ρ 1 p y2 + x 0 ) y 2 Y, ψ 1 y + tx 0 ) = ψy) + tt 0 [ y ) ] = t ψ + t 0 t tρ 1 p y 0) t + x = ρ 1 p y + tx0 ). Similarly, if t < 0, the, sice the defiitio of t 0 also esures that we fid that Thus, ψy + tx 0 ) ρ 1 p y + tx0 ) ψy 1 ) t 0 ρ 1 p y1 x 0 ) y 1 Y, ψ 1 y + tx 0 ) = ψy) + tt 0 [ ) ] y = t ψ t 0 t ) tρ 1 p y t x 0 = ρ 1 p y + tx0 ). y Y, t R, ad the proof of the lemma is complete. Theorem 3.2. Hah Baach theorem) Let V be a p-ormed space ad let V 0 be a subspace of V. Suppose ϕ 0 V0 ; the there exist a ϕ V such that i)ϕ V0 = ϕ 0 ii) ϕ = ϕ 0. Proof. Let V 0 be a real) liear subspace of V. Now, apply the precedig lemma with X = V, M = V 0 ad ρ 1 p x) = ϕ0 x, to fid that the desired coclusio follows immediately. Remark 3.1. Oe ca follow the method of [9] to prove that the Hah Baach theorem holds for complex quasi-ormed spaces. Corollary 3.3. Let X be a p-ormed space ad 0 x 0 X. The there exists ϕ X such that ϕ = 1 ad ϕx 0 ) = x 0., Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

9 DIFFERENTIABILITY OF DISTANCE FUNCTIONS 9 Proof. Set X 0 = Cx 0 = {αx 0 : α C}, cosider the liear fuctioal ϕ 0 X defied by ϕ 0 λx) = λ x 0, ad appeal to the Hah Baach theorem. Theorem 3.4. Let X be a p-ormed space ad let M be a uiquely remotal subset of X admittig a ceter. Suppose that for every x i M + rm) 1 p BX) the farthest poit mappig F : X M restricted to the lie segmet [x, F x)] is cotiued at x. The M is a sigleto. Proof. We may assume that ) 0 is a ceter of M. Sice rm) = sup{ y ) 0 p : y M}, we have M B 0, rm) 1 p. For each x M, 0 B x, rm) 1 p M + rm) 1 p BX). Set x 0 = F 0) ad g = F x 0 ). By the Hah Baach theorem for each g x 0 fuctioal Φ such that Φ = 1, ad Φ, g x 0 = g x 0. The whece Sice Φ, x 0 = Φ, g Φ, g x 0 Φ g g x 0 = g g x 0, g x 0. 1 Φ x 0 ) g g p sup 0 g p 0 y p sup y M sup x o z M z p = x 0 F xo ) p x 0 = g p, we get Φ x 0 ) 0 for all. The fuctio F is cotiuous at x 0, so F x 0 ) F 0) hece g x 0 Φ g x 0 ) Φ x 0 ), there exists a x 0 0 = x 0. Therefore lim Φ x 0 ) = lim Φ g x ) 0 g = lim x 0 = x 0. If x 0 > 0, the Φ x 0 ) > 0 for some which is a cotradictio. Hece x 0 = 0 ad so for y M we have y 0 p F 0) 0 p = x 0 P = 0, whece M = {0}. REFERENCES [1] Y. BENYAMINI ad J. LINDENSTRAUSS, Geometric Noliear Fuctioal Aalysis, Vol. 1, Colloq. Publ. 48, Amer. Math. Soc., Providece, [2] S. FITZPATRICK, Metric projectio ad the differetiability of distace fuctios, Bull. Austral. Math. Soc., ), [3] D. H. HYERS, G. ISAC ad Th. M. RASSIAS, Topics i Noliear Aalysis ad Applicatios, World Scietific Publishgig Compay, Sigapore, New Jersey, Lodo, [4] M. S. MOSLEHIAN ad Gh. SADEGHI, Stability of liear mappigs i quasi-baach modules, Math. Iequal. Appl., ), o. 3, , Vol. 6, No. 1, Art. 10, pp. 1-10, 2009

10 10 M.S. MOSLEHIAN, A. NIKNAM, S. SHADKAM TORBATI [5] A. NIKNAM, Cotiuity of the farthest poit map, Idia J. Pure Appl. Math., ), No. 7, [6] B. B. PANDA ad O. P. KAPOOR, O farthest poits of set, J. Math. Aal., ), [7] S. ROLEWICZ, Metric Liear Spaces, PWN-Polish Sci. Publ., Reidel ad Dordrecht, [8] L. SCHWARTZ, Geometry ad Probability i Baach Space, Lecture Notes i Mathematics 852, Spriger Verlag, New York, [9] V. S. SUNDER, Fuctioal Aalysis Spectral Theory, Birkhäuser Advaced Texts, Basel, 1997., Vol. 6, No. 1, Art. 10, pp. 1-10, 2009