ON DIFFERENTIABILITY OF ABSOLUTELY MONOTONE SET-VALUED FUNCTIONS

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1 Folia Maemaica Vol. 16, No. 1, pp Aca Universiais Lodziensis c 2009 for Universiy of Lódź Press ON DIFFERENTIABILITY OF ABSOLUTELY MONOTONE SET-VALUED FUNCTIONS ANDRZEJ SMAJDOR Absrac. We prove a e exisence of all Hukuara derivaives H (k) () on [0, b) conaining zero is equivalen o e absoluely monooniciy of a sevalued funcion H on [0, b) wi nonempy convex compac values. 1. Le A and B be wo subses of a real vecor space X. We define e sum of A and B by e formula A + B : {a + b : a A, b B}. Le X be a real normed vecor space and le cc(x) denoe e family of all nonempy compac convex subses of X. A se C cc(x) is e Hukuara difference of A cc(x) and B cc(x) if A B + C (see [2]). If e difference A B exiss, en i is unique. I is a consequence of e following lemma. Lemma 1 (cf. [4]). Le A, B and C be subses of a opological vecor space suc a A + B C + B. If C is convex closed and B is nonempy bounded, en A C. Now, le < a < b + and le H : [a, b) cc(x). We define e p- differences p sh() by e reccurence 0 sh() H(), p+1 s H() p sh( + s) p sh() for every nonnegaive ineger p, [a, b), s > 0 suc a + (p + 1)s < b. Pedagogical Universiy, Podcor ażyc 2, Kraków, Poland. asmajdor@ap.krakow.pl. Key words and prases: absoluely monoone se-valued funcions, Hukuara derivaives. AMS subjec classificaions: 26A48, 26A51, 26E25. 25

2 26 ANDRZEJ SMAJDOR A se-valued funcion is said o be absoluely monoone if ere exis all differences p sh() and eac conains zero. Example 1. Assume a A cc(x), 0 A and : [a, b) [0, ). Ten H() ()A is absoluely monoone se-valued funcion if and only if is an absoluely monoone real funcion. Example 2. Assume a f : [a, b) [0, ) and g : [a, b) [0, ) are suc a f() 0 g() and H() [f(), g()] for [a, b). Ten e se-valued funcion H is absoluely monoone if and only if f and g are absoluely monoone. In [5] e following eorem was proved. Teorem 1. A se-valued funcion H : [0, b) cc(x) is absoluely monoone if and only if ere exis ses A n cc(x), n 0, 1,... conaining zero suc a (1) H() n A n for [0, b). Te convergence of e series in (1) is e convergence in cc(x) wi respec o e Hausdorff meric derived from e norm in X. Assume a H : [a, b) cc(x) is a se-valued funcion suc a e differences H(s) H() exis if, s [a, b) wenever < s. Te Hukuara derivaive of H a is defined by e formula H H(s) H() () lim lim s + s s H() H(s), s wenever bo limis exis wi respec o e Hausdorff meric d in cc(x) derived from e norm in X. Moreover H H(s) H(a) (a) lim. s a+ s a Lemma 2 (cf. Lemma 5 in [6]). Le F, G : [α, β] cc(x) be wo differeniable se-valued funcions suc a F () G () for [α, β] and F (α) G(α), en F () G() for [α, β]. A se-valued funcion F : [α, β] cc(x) is called increasing if F (x) F (y) for all x, y [α, β] suc a x < y. We say a a se-valued funcion F : [α, β] cc(x) is concave if for all x, y [a, b] and λ (0, 1). F ((1 λ)x + λy) (1 λ)f (x) + λf (y)

3 ON DIFFERENTIABILITY OF ABSOLUTELY MONOTONE Te Riemann ype inegral for se-valued funcions was inroduced by A. Dingas in [1] (see also [2]). We ave e following wo lemmas. Lemma 3 (cf. Teorem 2.6 in [3]). If G : [α, β] cc(x) is an increasing se-valued funcion, D cc(x) and F (s) D + s α G(u)du, en F is a concave mulifuncion for wic ere exis all differences F (s) F () wenever α < s β. Lemma 4 (cf. Teorem 1.3 in [3]). If F : [α, β] cc(x) is inegrable, α, β, a, b are real numbers suc a α < β, aα + b α, aβ + b β, en β α F ()d a β α F (au + b)du. Our main resul caracerizes absoluely monoone se-valued funcions. Teorem 2. Le X be a Banac space. A se-valued funcion H : [0, b) cc(x) is absoluely monoone if and only if ere exis e k derivaives H (k) () conaining zero for k 1, 2,... and [0, b). Proof. Le H : [0, b) cc(x) be an absoluely monoone se-valued funcion. I is of e form (1) by Teorem 1. Fix [0, b) and c (, b). Define H(s) H() G(s) s for s (, c]. Teorem 1 yields G(s) (s n 1 + s n n 1 )A n n1 for s (, c]. I is clear a (s n 1 + s n n 1 )A n (s n 1 + s n n 1 ) A n < λ : lim sup n < nc n 1 A n for s (, c] and n 1, 2,... (e norm A of a se A is defined as sup{ a : a A}). Le n An. Fix d (c, b). Te convergence of e series dn A n implies a ere exiss a consan M > 0 suc a d n A n M, n 0, 1, 2,...

4 28 ANDRZEJ SMAJDOR Terefore λd 1. We see a n lim sup nc n 1 A n cλ < dλ 1. n By compleeness of X e series nc n 1 A n is absoluely convergen and e series (s n 1 + s n n 1 )A n n1 is absoluely and uniformly convergen for s (, c]. Te se-valued funcion G(s) is coninuous on (, c] wi respec o Hausdorff meric in cc(x). Since every funcion s (s n 1 + s n n 1 )A n as e limi n n 1 A n a s, e se-valued funcion G may be exended o e coninuous (wi respec o Hausdorff meric in cc(x) derived from e norm in X) funcion on [, c] and H +() H(s) H() lim lim (s n 1 + s n n 1 )A n s + s s + lim G(s) G() n n 1 A n. s + Te proof of e formula H () H(s) H() lim s s n n 1 A n is similar. We see a every absoluely monoone se-valued funcion H : [0, b) cc(x) is differeniable and is derivaive H is absoluely monoone sevalued funcion. I is obvious a 0 H (). I follows by inducion a H (k) () exiss for every k 1, 2,... and [0, b) and 0 H (k) () for e same k and. Conversely, suppose a ere exis all derivaives H (k) () and (2) 0 H (k) () for k 0, 1,..., [0, b). Tey are coninuous. Terefore ese derivaives are inegrable in Riemann sense and e se-valued funcion s

5 s ON DIFFERENTIABILITY OF ABSOLUTELY MONOTONE H(k+1) (u)du is differeniable in [0, b) and ( d s ) H (k+1) (u)du H (k+1) (s) ds 0 (see [2] p. 216). Tis implies a for any [0, b) e se-valued funcion Φ(s) : H (k) () + s H(k+1) (u)du is differeniable in [, b) and d ds Φ(s) H(k+1) (s) d ds H(k) (s). Since Φ() H (k) (), Lemma 2 sows a Φ(s) H (k) (s) for all s [, b). Consequenly ere exis e Hukuara differences (3) H (k) (s) H (k) () s for 0 < s < b. From 0 H (k+1) (u) we ave and ence 0 H (k) (s) H (k) () H (k) () H (k) (s) H (k+1) (u)du for 0 < s < b and k 0, 1,... Tus all H (k) are concave se-valued funcions according o Lemma 3. Tere exiss e difference 1 H() for 0 < b and 0 < < b. By formula (3) + (4) 1 H() H (u)du for e same and. Since 0 H (u) for every u, we ave a 0 1 H() wenever 0 < + < b. We conclude from (4) and Lemma 4 a +2 2 H() 1 H( + ) 1 H() H (u)du H (u + )du + + ( 1 H (u) + H (u))du 1 H (u)du H (u)du + + H (u)du H (u)du

6 30 ANDRZEJ SMAJDOR for, suc a 0 < + 2 < b. By (2) we ave + ( u+ ) + 0 H (v)dv du 1 H (u)du 2 H(). u Now assume a ere exis differences 1 H(),..., k H(), 0 i H(), i 1,..., k and + k H() k 1 H (u)du for and suc a [0, b) and + k < b. Suppose a [0, b) and + (k + 1) < b. On accoun of e assumpion we ave Terefore k H( + ) +2 k+1 + k 1 H (u)du + H() k H( + ) k H () + + k H (u + )du k+1 H (u)du. + k 1 H (u + )du. k H (u)du By inducion we can prove a ( + ( u1+ ( uk 1 k+1 + ) ) ) uk + H()... H (k+1) (v)dv du k... du 1. u 1 u k 1 Tis implies a 0 k+1 H() for k 0, 1,..., [0, b) and 0 < suc a + (k + 1) < b. Tus H is absoluely monoone. u k References [1] A. Dingas, Zum Minkowskiscen Inegralbegriff abgesclossener Mengen, Ma. Z. 66 (1956), pp [2] M. Hukuara, Inégraion des applicaions mesurables don la valeur es un compac convexe, Funcial. Ekvac. 10 (1967), pp [3] M. Piszczek, Inegral represenaions of convex and concave se-valued funcions, Demonsraio Ma. 35 (2002), pp [4] H. Rådsröm, An embedding eorem for spaces of convex ses, Proc. Amer. Ma. Soc. 3 (1952), pp [5] A. Smajdor, On absoluely monoone se-valued funcions, Ann. Polon. Ma. 88, 2 (2006), pp [6] A. Smajdor, On a mulivalued differenial problem, Inerna. J. Bifur. Caos Appl. Appl. Sci. 13 (2003), pp