Interntionl Journl of Pure nd Applied Mthemtics Volume 51 No. 2 2009, 189-194 Invited Lecture Delivered t Fifth Interntionl Conference of Applied Mthemtics nd Computing (Plovdiv, Bulgri, August 12 18, 2008) LOWER-SEMICONTINUITY AND OPTIMIZATION OF CONVEX FUNCTIONALS L.A.O. Fernndes 1, R. Arbch 2 1,2 Deprtment of Mthemtics UNESP-Ilh Solteir, Almed Rio de Jneiro 266, Zip Code 15385-000, Ilh Solteir, SP, BRASIL 1 e-mils: lfo@mt.feis.unesp.br 2 e-mil: roseli@mt.feis.unesp.br Abstrct: The result tht we tret in this rticle llows to the utiliztion of clssic tools of convex nlysis in the study of optimlity conditions in the optiml control convex process for Volterr-Stietjes liner integrl eqution in the Bnch spce G([,b],X) of the regulted functions in [,b], tht is, the functions f : [,b] X tht hve only descontinuity of first kind, in Dushnik (or interior) sense, nd with n equlity liner restriction. In this work we introduce convex functionl L β,f (x) of Nemytskii type, nd we present conditions for its lower-semicontinuity. As consequence, Weierstrss Theorem grntees (under compcity conditions) the existence of solution to the problem min{l β,f (x)}. AMS Subject Clssifiction: 45D05 Key Words: Volterr-Stietjes liner integrl equtions, convex optimiztion, regulted functions 1. Introduction The lower-semicontinuity is very importnt notion tht plys n importnt Received: August 14, 2008 c 2009 Acdemic Publictions Correspondence uthor
190 L.A.O. Fernndes, R. Arbch role in optimizing convex functionls, tht ppers in severl clssic pplictions. The result tht we del in this rticle llows to the utiliztion of clssic tools of Convex Anlysis in the study of optimlity conditions in problems with equlity restrictions. We introduce convex functionl of integrl Nemytskii type, nd we present conditions for its lower-semicontinuity. Given Bnch spce X, X stnds for its dul (the spce of ll bounded liner functionls on X). We denote by G([,b],X) the Bnch spce (with the convergent uniform norm) of the regulted functions in [, b], tht is, the functions f : [,b] X tht hve only descontinuity of first kind, nd by SV ([,b],l(x,y )) the spce of functions α of bounded semivrition, tht is, SV ([,b])[α] = sup{sv (d)[α] / d D} < nd SV (d)[α] = sup{ [α(t i ) α(t i 1 )] x i / x i X, x i 1}, such tht α() = 0 (in prticulr SV ([,b],x ) = BV ([,b],x )), see [4]. SV ([,b],l(x)) becomes Bnch spce when endowed with the norm α = SV [α]. The integrls in the Dushnik (or interior) sense tht ppers re defined, when there exists the limit, by d α(t) f(t) = lim d D d [α(t i ) α(t i 1 )] f(ξ i ) (1) i=1 where D is the set of ll divisions d : = t 0 < t 1 <... < t n = b of [,b] (n = d is the order of division d) nd ξ i ]t i 1,t i [. The osciltion of f on [,b] corresponding to division d will be defined by ω d (f) = sup{ω i (f) / i = 1,2,..., d }, where ω i (f) = sup{ f(t) f(s) / s,t ]t i 1,t i [}. If α SV ([,b],l(x)) then the integrl exists. We consider here the liner evolutive process described by the liner Volterr-Stieltjes integrl eqution x(t) x 0 + t d s K(t,s) x(s) = u(t), t [,b], (K) where the kernel K G σ 0 SV u ([,b] [,b],l(x)), tht is, K is simply regulted s function of the first vrible with K(t,t) = 0 nd of uniformly bounded semivrition s function of the second vrible, nd subject to the liner constrint F α [x] = d α(s) x(s) = 0. (F α ) Both sttes x nd prmeter control u re selected in G([,b],X). Mny exmples of systems in mthemticl nlysis re instnces of this system, for exmple, Stieltjes integrl equtions, Volterr integrl equtions, liner dely differentil equtions, functionl equtions, impulsive equtions. We use the nottion x u (s) to the solution of (K) + (F α ) ssocite to u in the sense of
LOWER-SEMICONTINUITY AND OPTIMIZATION... 191 Theorem 3.4, [6]. In other words, x u (t) = u(t) + R(t,) [x 0 + u()] t d s R(t,s) u(s), t [,b], (ρ) where R G σ I SV u (R(t,t) = I X ) is the resolvent (unique) of K. The properties of the notions given here cn be found in [4]. With the support of representtion theorem in [G ([,b],x)], where G ([,b],x) = {f G ([,b],x) / f = f} is closed subspce of G([,b],X), f (t) = f(t ),t ],b], f () = f(t ), we hd study optimiztion of liner functionls defined over set of solutions of system (K) + (F α ). A chrcteriztion of regulted functions is given (see Theorem 3.1 of [4]) by: Theorem 1. Let x : [, b] X be function. The following sttements re equivlents: ) x is the uniform limit of sequence of finite step functions; b) x G([,b],X); c) for every ǫ > 0 there exists d D such tht ω d (x) < ǫ. It follows tht the uniform limit of sequence of regulted functions is regulted function. Lemm 1. Let f : [,b] X X regulted s function of the first vrible nd Lipschitz s function of the second vrible. Let x G([,b],X). Then f 1 : s [,b] f(s,x(s)) X is regulted function. Proof. If t [,b[ we set f(t,x(t + )) X. Then f 1 (s) f(t,x(t + )) = f(s,x(s)) f(t,x(t + )) c x(s) x(t + ). Since x G([,b],X) we hve s t = x(s) x(t + ) = (s,x(s)) (t,x(t + )). So given ǫ/c > 0 we hve f(s,x(s)) f(t,x(t + )) < ǫ, provided s [,b], 0 < s t < δ, tht is, there exists f 1 (t + ) = f(t +,x(t + )). By nlogy we show tht there exists f 1 (t ) = f(t,x(t )). In [3] we introduced, when φ BV ([,b],x ), the notion of φ-convexity. Let g : Y R be convex function. We sy tht function f : [,b] X Y is convex with respect to g s function of the second vrible, or shortly, g- convex in the second vrible, if (g f) s is convex function, s [,b], where (g f) s (x) = (g f)(s,x) s [,b] nd x X. A prticulr cse occurs when Y = X nd g = φ X. For ech φ fixed we will denote the set of ll f : [,b] X X tht re regulted s function of the first vrible nd φ-convex in the second vrible by G Conv φ ([,b] X,X). Note tht the regulrity of f in the first vrible does not mke influence in φ-convexity
192 L.A.O. Fernndes, R. Arbch notion since this is vlued by (g f) s. We introduce now convex functionl of Nemytskii type, L β,f : G([,b],X) R, defined by L β,f [x] = d s β(s) f(s,x(s)), (2) where f G Conv β(s) ([,b] X,X) nd β BV ([,b],x ). Consider the optiml control convex problem min{l β,f (x u ), x u stisties (K + F α )}. (3) 2. Lower-Semicontinuity We introduce now conditions on f nd β such tht L β,f will be lowersemicontinuous function. Moreover, if f is proper function then L β,f is too. As consequence, Weierstrss theorem grntees (under compcity conditions) the existence of solution of (3). We sy tht function f : [,b] X Y is lower semicontinuous with respect to g : Y R s function of the second vrible, or shortly, g- lower semicontinuous in the second vrible, if (g f) s : X R is lower semicontinuous function, s [,b], where (g f) s (x) = (g f)(s,x) s [,b] nd x X. We use this notion when Y = X nd g = φ X. For ech φ fixed we will denote the set of ll f : [,b] X X tht re regulted s function of the first vrible nd φ-lower-semicontinuous in the second vrible by G Lsc φ ([,b] X,X). We hve tht f(s,x n (s)) f(s,x(s)), since tht x n x in G([,b],X). Theorem 2. Suppose f G Conv β(s) ([,b] X,X), s [,b] nd β BV ([,b],x ). Then the functionl L β,f is convex. Proof. Simple clcultion. Theorem 3. Suppose tht β BV ([,b],x ) nd f G Lsc β(s) ([,b] X,X), s [,b]. Then the functionl L β,f is lower-semicontinuous. Proof. Let (y n ) n N be sequence of regulted functions with y n Λ(L β,f,λ), for n N, nd such tht y n y, where for ech λ R, Λ(L β,f,λ) = {x G([,b],X) / L β,f (x) λ}, re the level sets of the functionl L β,f, tht is, for ech n N, L β,f [y n ] = β(s) f(s,y n (s)) λ.
LOWER-SEMICONTINUITY AND OPTIMIZATION... 193 Since β(s) X we hve β(s) f(s,y n (s)) λ, s [,b], for ech λ R. So if β(s) X, nd λ R, it follows tht y n Λ β(s) (f,λ), Λ β(s) (f,λ) = {x G([,b],X)/ β(s) f(s,x(s)) λ}. Since f G Lsc β(s) ([,b] X,X), s [,b], we hve tht Λ β(s) (f,λ) re closed subsets in X. Then y Λ β(s) (f,λ), tht is, β(s) f(s,y(s)) λ nd so L β,f (y) λ, tht is, y Λ(L β,f,λ). Then Λ(L β,f,λ) G([,b],X) is closed set nd is lower-semicontinuous the functionl L β,f. In other words, is closed epigrph of L β,f, epi(l β,f ) = {(x,λ) G([,b],X) R/ L β,f (x) λ} or, for ll x G([,b],X) nd fmily of neighbourhood Υ(x) of x in G([,b],X), L β,f (x) = lim y x infl β,f (y) = sup V Υ(x) As consequence we hve, by Weierstrss Theorem: inf L β,f(y). y V Theorem 4. If Λ G([,b],X) is compct set, then L β,f hs minimun in K, tht is, there exists solution of the optimiztion problem min L β,f(x). x Λ References [1] V. Brbu, T. Precupnu, Convexity nd optimiztion in Bnch spces, Sijthoff nd Noordhoff, Publishing House of Romnin Acdemy, Buchrest, 397 (1978). [2] L.D. Berkovitz, Lower semicontinuity of integrl functionls, Trnsctions of the Americn Mthemticl Society, 192 (1974), 51-57. [3] L.A.O. Fernndes, Convex functionls on the regulted function spce G([, b], X), Brsilin Seminr of Anlysis, 49, Cmpins (1999), 767-772. [4] C.S. Hönig, Volterr-Stieltjes Integrl Equtions, Mth. Studies, 16, North Hollnd Publ. Compny, Amsterdm (1975). [5] C.S. Hönig, The djoint eqution of liner Volterr-Stieltjes integrl eqution with liner constrint, Lecture Notes in Mthemtics, Springer 957 (1982), 110-125.
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