Pliska Stud. Math. Bulgar. 12 (1998), STUDIA MATHEMATICA BULGARICA TIME-DEPENDENT DIFFERENTIAL INCLUSIONS AND VIABILITY *

Transcription

1

2 Pliska Sud. Mah. Bulgar. 12 (1998), STUDIA MATHEMATICA BUGARICA TIME-DEPENDENT DIFFERENTIA INCUSIONS AND VIABIITY * uis Marco, José Albero Murillo This paper is devoed o sudy he exisence of viable soluions for nonauonomous higher order differenial inclusions. Two cases are considered, according o he properies of he se-valued maps on he righ-hand side. Firsly, upper semiconinuiy is assumed and boh necessary and sufficien condiions are given by means of higher order angen ses. aer, almos upper semiconinuous case is invesigaed, and similar resuls are saed. Finally, hese resuls are used o solve a mulivalued differenial inequaliy. eywords: viable soluions, differenial inclusions, high order, nonauonomous, higher order angen ses, angenial condiions, mulivalued differenial inequaliies AMS subjec classificaion: Primary 34A60, Secondary 49J52. 1 Inroducion e F : R X 2 X be a nonrivial se-valued map, wih X a finie dimensional vecor space and le X a nonempy se. A funcion x( ) saisfying he differenial inclusion x () F(, x()), a.e., is called viable in, if x() holds. The concep of viabiliy appeared in he framework of differenial inclusions in [26] and [6] under he name of weak invariance and admissibiliy and also in [18]. In he auonomous case, i.e. when an auonomous differenial inclusion is considered, he more general concep of monoone rajecory was sudied in [1] (for convex viabiliy domains and Marchaud mulivalued maps), [10] (for compac viabiliy domains and coninuous maps) and [19] (for locally compac viabiliy domains and upper semiconinuous correspondences). The exisence of viable soluions o a nonauonomous or ime-dependen differenial inclusion was invesigaed in [19], [2] and [12] (for upper semiconinuous se-valued maps) and also in [20], [25] and [13] (for Carahéodory mulivalued maps). The more general * This paper was parially suppored by DGICYT gran PB

3 108 uis Marco, José Albero Murillo case in which he viable se changes along he ime, and herefore he viabiliy condiion becomes x() (), is considered in [19] and [2] (for upper semiconinuous se-valued maps), [25] (for Carahéodory mulivalued maps, by assuming ha () is convex), [7] (for almos upper semiconinuous maps) and also in [16], [21] and [17] (for Carahéodory mulivalued maps), where some applicaions are provided. We refer o [4] for more deailed hisorical noes and references. The viabiliy problem for second order differenial inclusions was firs invesigaed by Corne and Haddad in [11]. They noed ha finding soluions of he auonomous viabiliy problem: (1.1) x () F(x(), x ()) x(0) = x 0, x (0) = v 0 x() is equivalen o he following viabiliy problem of firs order, x () = u(), u () F(x(), u()) (1.2) x(0) = x 0, u(0) = v 0 (x(), u()) G(T ) where T (x) is Bouligand s angen cone o a x, and G(T ) is is graph. However, one of he main assumpions of he viabiliy heorems for firs order differenial inclusions is ha he viabiliy se mus be locally compac, and his is no in general saisfied by G(T ). So if we wan o use his way o solve (1.1), we mus assume ha G(T ) is locally compac. Hence, his approach is very seldomly applicable. In he above menioned paper, Corne and Haddad saed he exisence of viable soluions of a differenial inclusion of second order by imposing resricions on iniial condiions by means of Dubovickii-Miljuin and Clarke s angen cones. In [5], Auslender and Mechler firs esablished a necessary condiion by using he se of second order angens of inroduced by Ben-Tal and laer hey gave a sufficien condiion valid for all iniial saes by inroducing he noion of he second order inerior angen se. Finally, in [22] and [23] he auhors sudied he viabiliy problem of n h order in he auonomous case, and we noed ha he viabiliy condiion on he soluion of a differenial inclusion of n h order involves an underlying viabiliy condiion on he soluion and is derivaives up o order n 1. To describe i we inroduced a class of higher order angen ses. We also saed necessary and sufficien condiions ensuring he exisence of viable soluions o an auonomous differenial inclusion of n h order. In his paper he ime-dependen viabiliy problem of n h order is analysed. Secion 2 presens some preliminaries and ses up noaion and erminology. In Secion 3 we consider he case in which he se-valued map is upper semiconinuous. Firsly, we assume ha G(A (n 1) ) is locally compac, and under his hypohesis boh local and global viabiliy heorems are saed. aer we sudy he general case, and we give necessary and sufficien condiions ensuring he exisence of viable soluions. Secion 4 is devoed o analyse he almos upper semiconinuous case. In his secion we also suppose ha he se-valued map is inegrably bounded, and when G(A (n 1) ) is assumed o be closed, we obain a viabiliy resul similar o ha in he preceding secion. In he general case, we impose regulariy assumpions on he growh of he se-valued map o geing viable

4 Time-dependen differenial inclusions and viabiliy 109 soluions. Finally, in Secion 5 we consider a mulivalued differenial inequaliy of higher order and we solve i by using previous resuls. 2 Preliminaries e us firs recall some noions and noaion, a deailed discussion of hese conceps can be found, e.g. in [4], [3] or [14]. e X, Y be meric spaces. The domain of a sevalued map F : X 2 Y, is he se D(F) = {x X : F(x) Ø}, and i is said o be nonrivial if D(F) Ø. The graph of F is he se G(F) = {(x, y) X Y : y F(x)}. A se-valued map is said o be upper semiconinuous (u.s.c. for shor) on Ω X if F 1 (C) = {x X : F(x) C Ø} is closed in Ω for all closed se C Y. In he case F defined from R X o 2 Y, we call i almos u.s.c. on I Ω, I being a compac inerval, if for every ε > 0, here exiss a closed I ε I wih µ(i \ I ε ) ε such ha F Iε Ω is u.s.c. wih nonempy values; here µ denoes he ebesgue measure. If I is no compac, F is called almos u.s.c. on I Ω if i saisfies his propery on J Ω, for all compac J I. For a family of ses {S σ } σ Σ, he upper limi in he uraowski sense is he se: lim sup S σ = {x X : liminf d(x, S σ) = 0} σ Σ σ Σ The coningen derivaive of a se-valued map F a (x 0, y 0 ) G(F), denoed by DF(x 0, y 0 ), is given by means of is graph: G(F) (x 0, y 0 ) G(DF(x 0, y 0 )) = limsup. h 0 h + Given a nonempy se X and x 0, x 1,..., x n 1 X, he n h order angen se of a (x 0,..., x n 1 ), denoed by A (n) (x 0,...,x n 1 ) is defined as follows: ) A (n) (x n! 0,..., x n 1 ) = limsup ( h 0 + h n x 0 h x 1 hn 1 (n 1)! x n 1. An imporan propery of hese ses is ha A (n) (x 0,..., x n 1 ) Ø, implies x n 1 belongs o A (n 1) (x 0,...,x n 2 ). The n h order inerior angen se of a (x 0,...,x n 1 ), denoed by AI (n) (x 0,..., x n 1 ) is he se of poins y X such ha here are ε > 0 and α > 0 saisfying: n 1 j=0 h j j! x j + hn n! (y + ε U X), 0 h α where U X is he uni ball in X. We refer o [22], [23] or [24] for more informaion on hese higher order angen ses. In he remainder of he paper, X will be a finie dimensional vecor space, X a nonempy closed se and F : R X n 2 X will be a nonrivial se-valued map wih convex compac values. The problem ha we shall consider is he following imedependen viabiliy problem of n h order: (2.1) x (n) () F(, x(), x (),..., x n 1 ())

5 110 uis Marco, José Albero Murillo (2.2) x(0) = x 0, x (0) = v 1,..., x (n 1) (0) = v n 1 (2.3) x(). A soluion of (2) on an inerval [0, T] is a funcion ϕ possessing absoluely coninuous derivaives up o order n 1, i.e. ϕ W n,1 (0, T; X), such ha i is soluion of (2.1) viable in (condiion (2.3)) and saisfying iniial condiions (2.2). 3 Upper semiconinuous case In his secion we will make he assumpion ha F is u.s.c. Under his hypohesis, an analysis similar o ha in he proof of Proposiion 4.1. in [23] shows ha he nex proposiion holds Proposiion 3.1 e ϕ be a soluion of (2) on [0, T], hen for each [0, T[. (ϕ(), ϕ (),..., ϕ (n 1) ()) G(A (n 1) ) From his resul i follows ha he inclusion (x 0, v 1,..., v n 1 ) G(A (n 1) ) mus be saisfied by he iniial condiions in (2.2). Therefore we will assume in he remainder of his secion he nex compaibiliy condiion on F, (3.1) [0, δ[ G(A (n 1) ) D(F) for some 0 < δ. Furhermore, we will suppose ha F is u.s.c. on [0, δ[ G(A (n 1) ). We now disinguish wo siuaions in accordance wih he opological properies of he graph of he n h order angen se of. 3.1 G(A (n 1) ) locally compac Since we are assuming ha F is u.s.c. on [0, δ[ G(A (n 1) ) and G(A (n 1) ) is locally compac, we can use Theorem 4.1. in [23] o ge soluions o he auonomous problem, (3.2) y (n) () G(y(), y (),..., y (n 1) ()) y(0) = (0, x 0 ), y (0) = (1, v 1 ), y (i) (0) = (0, v i ), 2 i n 1 y() [0, δ[ where G(y 1,..., y n ) = {0} F(y 1, π X (y 2 ),..., π X (y n )); π X : R X X being he projecor ono X. Hence (2) has a soluion for each iniial condiion (x 0, v 1,...,v n 1 ) in ) iff he angenial condiion, G(A (n 1) (3.3) G(y, ω 1,..., ω n 1 ) DA (n 1) [0,δ[ (y, ω 1,...,ω n 1 )[ω 1,..., ω n 1 ] Ø holds for all (y, ω 1,..., ω n 1 ) G(A (n 1) [0,δ[ ). Bu he nex echnical lemma allow us o rewrie (3.3) in erms of he se-valued map F and he coningen derivaive of A (n 1), and Theorem 3.1 is esablished.

6 Time-dependen differenial inclusions and viabiliy 111 emma 3.1 e C be a nonempy subse in X and le η > 0, hen A (n) [0,η[ C (y 1,...,y n ) = R + A (n) C (π X(y 1 ),..., π X (y n )) if π R (y i ) = 0 R A (n) C (π X(y 1 ),..., π X (y n )) oherwise Theorem 3.1 (ocal viabiliy) Under he hypoheses of his secion, (2) has a soluion for each iniial condiion in G(A (n 1) ) if and only if, (3.4) F(, x, u 1,..., u n 1 ) DA (n 1) (x, u 1,..., u n 1 )[u 1,...,u n 1 ] Ø holds for all (, x, u 1,...,u n 1 ) [0, δ[ G(A (n 1) ). Moreover, given (x 0, v 1,..., v n 1 ) G(A (n 1) ), here are η > 0 and T 0 > 0, such ha (2) has a soluion on [0, T 0 ] for each iniial condiion (x, u 1,..., u n 1 ) G(A (n 1) ) ((x 0, v 1,..., v n 1 ) + η U X n) If we make he sronger assumpion: G(A (n 1) ) is closed, hen we can sae a global viabiliy heorem. I comes from he nex lemma, which is a generalizaion of Theorem in [4]. emma 3.2 e G(A (n 1) ) be closed. Suppose (3.4) holds. If ϕ is a soluion of (2) on [0, T[ (T < δ) such ha, (3.5) n 1 lim sup ϕ (j) () < + T j=0 hen ϕ and is derivaives up o order n 1 can be exended o he full inerval [0, T]. Theorem 3.2 (Global viabiliy) Under he assumpions of Theorem 3.1, if moreover G(A (n 1) ) is closed and F is bounded on [0, δ[ G(A (n 1) ), hen every soluion of (2) can be exended o he full inerval [0, δ[. Proof. e ϕ be a soluion of (2). By classical argumens (Zorn s emma) here exiss a maximal soluion exending ϕ (again denoed by ϕ for simpliciy) defined on [0, T[. Suppose, conrary o our claim, ha T < δ. Then (3.5) holds by boundedness of F. Hence, ϕ and is derivaives up o order n 1 can be exended o T (emma 3.2). Furhermore, he problem: x (n) () F( + T, x(), x (),..., x (n 1) ()) x (i) (0) = ϕ (i) (T), 0 i n 1 x() has a soluion φ on an inerval [0, γ[, because he assumpions of Theorem 3.1 are saisfied. Finally, he funcion: ϕ : [0, T + γ[ X ϕ() = { ϕ(), [0, T] φ( T), ]T, T + γ[

7 112 uis Marco, José Albero Murillo exends ϕ and is a soluion of (2), a conradicion. Noe 3.1 The preceding heorem remains rue if boundedness of F is replaced by inegrably boundedness, i.e. if we assume he following growh condiion on F: (3.6) F(, y) α()(1 + y ) U X for all (, y) [0, δ[ G(A (n 1) ), wih α 1 (0, δ). Under his hypohesis, if ϕ is a soluion of (2), by using Bellman s Inequaliy we obain, ( 2 ) (3.7) ψ() (1 + y 0 )exp α(s) ds 1 where ψ() = (ϕ(), ϕ (),..., ϕ (n 1) ()) and y 0 = (x 0, v 1,..., v n 1 ). Hence, (3.5) holds for all 0 < T < δ. Theorem 3.2 also remains rue under he weaker hypohesis, (3.8) F(, y) α()(1 + y ) U X Ø for all (, y) [0, δ[ G(A (n 1) ). (See Secion 4.) 3.2 General case In he preceding we have assumed locally compacness on G(A (n 1) ). Unforunaely, his is no usually rue. Furhermore, if his condiion fails angenial condiion (3.4) does no imply he exisence of viable soluions, even in he single-valued case, as he nex example shows. Example 3.1 e [a, b] R (b > 1). Obviously, G(T [a,b] ) is no locally compac, because: [0, + [ if x = a T [a,b] (x) = R if x ]a, b[ ], 0] if x = b e us consider he problem, x = x + x(0) = b x (0) = 0 x [a, b] I is easy o see ha (3.4) is saisfied, because (u, + x) T G(T[a,b] )(x, u) for all (x, u) in G(T [a,b] ). However, he soluion of he iniial value problem, ϕ() = b e + b 1 2 e is no viable in [a, b]. The aim of his secion is o sae boh necessary and sufficien condiions ensuring he exisence of soluions of (2). To ge such condiions we inroduce he following se-valued map, Definiion 3.1 e ϕ be a soluion of (2) on [0, T]. We define he se-valued map: Λ(ϕ, ) : [0, T[ 2 X Λ(ϕ, ) = limsup h 0 + n h n 0 +h ( + h s) n 1 ϕ (n) (s)ds

8 Time-dependen differenial inclusions and viabiliy 113 n Tha is, Λ(ϕ, ) is he se of limi poins of h h h n ( + h s) n 1 ϕ (n) (s)ds, leing The sudy of Λ(ϕ, ) will allow us o give a necessary condiion for (2). Theorem 3.3 e ϕ be a soluion of (2) on [0, T]. Then for each [0, T[, Λ(ϕ, ) is nonempy and compac. Moreover, he inclusion: (3.9) Λ(ϕ, ) F(, ψ()) A (n) (ψ()) holds, here ψ() = (ϕ(), ϕ (),..., ϕ (n 1) ()). Proof. Since F is u.s.c. having convex compac values and ϕ W n,1 (0, T; X), here is β > 0 such ha F(, ψ()) β U X. Hence, for each [0, T[: n! h n +h ( + h s) n 1 ϕ (n) (s)ds β U X which implies ha Λ(ϕ, ) is nonempy (by compacness of U X ) and bounded. I is also closed from properies of upper limis (see e.g. [3]). On he oher hand, by is very definiion, Λ(ϕ, ) A (n) (ψ()). e ε > 0 and [0, T[. Since F is u.s.c., here is γ > 0 such ha F(s, ψ(s)) F(, ψ()) + ε U X if 0 < s < γ. Therefore, for 0 < h small enough we have: n! h n +h ( + h s) n 1 ϕ (n) (s)ds F(, ψ()) + ε U X by using emma 5.1 in [23], and leing h 0 + we complee he proof. Corollary 3.1 (Necessary condiion) If here is a soluion of (2) for he iniial condiion (x 0, v 1,..., v n 1 ) G(A (n 1) ), hen: (3.10) F(0, x 0, v 1,..., v n 1 ) A (n) (x 0, v 1,..., v n 1 ) Ø. Finally, he sufficien condiion is saed in he nex heorem, which proof is similar o ha of Theorem 6.1 in [23]. So we will no give i. Theorem 3.4 (Sufficien condiion) e us suppose ha = M wih G(A (n 1) ) closed and F u.s.c. on [0, δ[ G(A (n 1) ). e us suppose ha he angenial condiion, (3.11) F(, x, u 1,..., u n 1 ) DA (n 1) (x, u 1,..., u n 1 )[u 1,...,u n 1 ] Ø holds for all (, x, u 1,..., u n 1 ) [0, δ[ G(A (n 1) ). Then (2) has a soluion for each iniial condiion (x 0, v 1,..., v n 1 ) G(A (n 1) ) saisfying: (3.12) F(0, x 0, ω 0 ) A (n) (x 0, ω 0 ) AI (n) M (x 0, ω 0 ) where ω 0 = (v 1,...,v n 1 ). In his general case is no possible o obain a global viabiliy resul like Theorem 3.2, bu a procedure similar o ha in he proof of his Theorem can be used o sae he nex resul.

9 114 uis Marco, José Albero Murillo Theorem 3.5 Under he hypoheses of Theorem 3.4, if moreover F is bounded or inegrably bounded on [0, δ[ G(A (n 1) ) and F(, x, ω) A (n) (x, ω) AI(n) M (x, ω) holds, for all (, x, u 1,..., u n 1 ) [0, δ[ G(A (n 1) ), here ω = (u 1,..., u n 1 ), hen given a maximal soluion ϕ of (2) defined on [0, T[ eiher T = δ or (ϕ(t), ϕ (T),...,ϕ (n 1) (T)) G(A (n 1) ). 4 Almos upper semiconinuous case In many problems, he se-valued map F is only measurable in, and no longer upper semiconinuous. This siuaion arises, for insance when we consider variaional inclusions obained by linearizaion of a differenial inclusion, even in he auonomous case (see [3, Chap. 10]) or in he sudy of mulivalued differenial inequaliies (see [15]). So we will devoe his secion o invesigae he exisence of soluions of (2) by assuming ha F is only almos upper semiconinuous. Noice ha in his conex our assumpion on F is equivalen o he Carahéodory propery, as a consequence of Scorzà-Dragoni heorem for se-valued maps (see e.g. [14, Prop. 5.1]). This is no longer rue when changes along he ime (see [7] for a counerexample). We will make anoher assumpion on F: i will be inegrably bounded (or (3.8) will be saisfied). Under hese hypoheses, he saemen of Proposiion 3.1 remains rue, because given ϕ a soluion of (2) on [0, T]: d( n 1 j=0 hj j! ϕ(j) (); ) h n 1 1 /(n 1)! h n 1 1 h n 1 +h (1 + ψ ) +h ( + h s) n 1 ϕ (n) (s) ds ( + h s) n 1 α(s)(1 + ψ(s) )ds +h α(s) ds here ψ as in (3.7) and ψ = sup [0,T] ψ() being. Hence, leing h 0 + we have he desired resul. From now we shall suppose ha here is δ > 0 such ha F is almos u.s.c. and inegrably bounded (or saisfying (3.8)) on [0, δ[ G(A (n 1) ). The firs heorem in his secion saes ha, as in he u.s.c., angenial condiion (3.4) implies ha (2) has a soluion. Theorem 4.1 Under he hypoheses of his secion, if G(A (n 1) ) is closed and (3.4) is saisfied for all (, x, u 1,...,u n 1 ) ([0, δ[ \ N) G(A (n 1) ), N [0, δ[ being a null se, hen (2) has a soluion on [0, δ[. Proof. The exisence of a soluion of (2) comes from [4, Theorem ], because (2)

10 Time-dependen differenial inclusions and viabiliy 115 is equivalen o, y () F(, y()) y(0) = (x 0, v 1,...,v n 1 ) y() G(A (n 1) ) where F(, y) = {(y 2,...,y n 1 )} F(, y), and angenial condiion F(, y) T (n 1) G(A ) (y) Ø can be rewrien as (3.4). Remark 4.1 I is possible o give a differen proof, which does no depend on firs order viabiliy heorem. To ge i we consider he family of u.s.c. problems, x (n) () F h (, x(), x (),..., x (n 1) ()) (4.1) x(0) = x 0, x (0) = v 1,..., x (n 1) (0) = v n 1 x() where F h (, y) = n +h h n (+h s) n 1 F(s, y)ds, and h > 0. Then we apply Theorem 3.2 o have a soluion of (4.1) on [0, δ[, and finally a soluion of (2) on [0, δ[ is obained as a limi of soluions of he problems (4.1), leing h 0 +. In his case, when G(A (n 1) ) is no closed, Λ(ϕ, ) can be empy as he nex example shows. Example 4.1 e us consider he almos u.s.c. se-valued map wih convex compac values, { [ F(, x, y) = a, a ], 0 < < 1 {0}, = 0 being 0 < a < 1. Obviously, F is inegrably bounded, aking: { 2 α() = a, 0 < < 1 0, = 0 I is easy o check ha ϕ() = however, Λ(ϕ, 0) = Ø. 2 a (1 a)(2 a) is a soluion of, x () F(, x(), x ()) x(0) = 0, x (0) = 0 x() [0, 2] Neverheless, i is immediae o show ha Λ(ϕ, ) is no empy and compac if 1 +h is a ebesgue poin of α, i.e. if lim α(s)ds = α(). Moreover, if Λ(ϕ, ) is h 0 + h nonempy, hen i is conained in A (n) (ψ()), here ψ as in Theorem 3.3, and if F(, ψ) is u.s.c. a, hen (3.9) holds. Bu, F(, ψ) is measurable and usin s Theorem (see e.g.

11 116 uis Marco, José Albero Murillo [8]) saes ha i is coninuous a almos every poin in [0, δ[. Therefore, Theorem 3.3 is saisfied almos everywhere, and making assumpions on α we can sae he nex necessary condiion. Theorem 4.2 (Necessary condiion) e us assume ha α is coninuous a zero, hen if (2) has a soluion (3.10) is saisfied. We close his secion wih a sufficien condiion. Theorem 4.3 (Sufficien condiion) e us suppose ha = M wih G(A (n 1) ) closed and F almos u.s.c. and inegrably bounded (or saisfying (3.8)) on [0, δ[ G(A (n 1) ), wih α coninuous a zero. e us suppose ha he angenial condiion, (4.2) F(, x, u 1,..., u n 1 ) DA (n 1) (x, u 1,..., u n 1 )[u 1,...,u n 1 ] Ø holds for all (, x, u 1,...,u n 1 ) ([0, δ[ \ N) G(A (n 1) ), N [0, δ[ being a null se. Then (2) has a soluion for each iniial condiion (x 0, v 1,..., v n 1 ) in G(A (n 1) ) saisfying (3.12). Proof. By Theorem 4.1 here exiss a soluion ϕ of (2.1)-(2.2) on [0, δ[, viable in. Since α is coninuous a zero, (3.10) is saisfied, and in a procedure similar o ha in he proof of Theorem 6.1 in [23], we find 0 < T < δ, such ha ϕ is viable in on [0, T]. 5 Mulivalued differenial inequaliies of higher order In his secion we shall apply he previous resuls o find a soluion of a mulivalued differenial inequaliy of higher order. e F : X n 2 X be an u.s.c. se-valued map having nonempy convex compac values and linear growh, i.e. here exiss a posiive consan β such ha F(y) β (1 + y ) U X, y X n A se-valued map saisfying ha properies is usually called a Marchaud map (see e.g. [4]). We look for a soluion of he iniial value problem: (5.1) such ha, (5.2) x (n) () F(x(), x (),..., x (n 1) ()) x(0) = x 0, x (0) = v 1,..., x (n 1) (0) = v n 1 ω() x() here ω W n,1 (0, δ; X) saisfying ω(0) x 0 and ω (j) (0) v j, 1 j n 1, and refers o he parial ordering given by X + = {x X : π i (x) 0 i}. Obviously, if we define y() = x() ω(), we can rewrie (5.1) (5.2) as he following ime-dependen viabiliy

12 Time-dependen differenial inclusions and viabiliy 117 problem: (5.3) y (n) () F(, y(), y (),..., y (n 1) ()) y(0) = x 0 ω(0), y (j) (0) = v j ω (j) (0), 1 j n 1 y() X + where F(, y 1,..., y n ) = F(y 1 + ω(), y 2 + ω (),..., y n + ω (n 1) ()) ω (n) (). Since F is u.s.c. and ω W n,1, hen F is almos u.s.c. I is also inegrably bounded because, F(, y) β (1 + 2φ())(1 + y ) here φ() = (ω(), ω (),..., ω (n) ()), and so φ 1 (0, δ). From Secion 6 in [23] we can compue he (n 1) h order angen se of X +, and we clearly show ha G(A (n 1) X ) is no locally compac. Therefore, we are under he + hypoheses of Theorem 4.3 aking = X and M = X + and assuming ha ω (n) is coninuous a zero. Then (5.1) (5.2) has a soluion if he iniial condiion saisfies, F(0, y 0, y 1,...,y n 1 ) AI (n) X + (y 0, y 1,..., y n 1 ) where y 0 = x 0 ω(0) and y j = v j ω (j) (0), 1 j n 1. Finally, his expression can be rewrien by using Corollary 6.1 in [23] as, π i (ω (n) (0)) < inf π i(y) y F(x 0,v 1,...,v n 1) for all i such ha π i (x 0 ω(0)) = π i (v j ω (j) (0)) = 0, 1 j n 1. REFERENCES [1] J.-P. Aubin, A. Cellina, J. Nohel. Monoone rajecories of mulivalued dynamical sysems. Ann. Ma. Pura Appl. 115 (1977), [2] J.-P. Aubin, A. Cellina. Differenial Inclusions. Springer-Verlag, Berlin, [3] J.-P. Aubin, H. Frankowska. Se-Valued Analysis. Birkhäuser, Boson, [4] J.-P. Aubin. Viabiliy Theory. Birkhäuser, Boson, [5] A. Auslender, J. Mechler. Second Order Viabiliy Problems for Differenial Inclusions, J. Mah. Anal. Appl. 181 (1994), [6] J. W. Bebernes, J. D. Schuur. The Wazewski opological mehod for coningen equaions. Ann. Ma. Pura Appl. 87 (1970), [7] D. Bohe. Mulivalued differenial equaions on graphs. Nonlinear Anal. 18 (1992), [8] T. F. Bridgland Jr. Trajecory Inegrals of Se Valued Funcions. Pacific J. Mah. 33 (1970), [9] C. Casaing, M. Valadier. Convex Analysis and Measurable Mulifuncions. ecure Noes in Mah., vol. 580, Springer-Verlag, Heidelberg, 1977.

13 118 uis Marco, José Albero Murillo [10] F. H. Clarke, J.-P. Aubin. Monoone invarian soluions o differenial inclusions. J. ondon Mah. Soc. 16 (1977), [11] B. Corne, G. Haddad. Théorèmes de viabilié pour les inclusions différenielles du second ordre. Israel J. Mah. 57 (1987), [12]. Deimling. Mulivalued differenial equaions on closed ses. Differenial Inegral Equaions 1 (1988), [13]. Deimling. Exremal soluions of mulivalued differenial equaions II, Resuls in Mah. 15 (1989), [14]. Deimling. Mulivalued Differenial Equaions. Waler de Gruyer & Co., Berlin, [15]. Deimling, V. akshmikanham. Mulivalued differenial inequaliies. Nonlinear Anal. 14 (1990), [16] H. Frankowska, S. Plaskacz, T. Rzeżuchowski. Théorèmes de viabilié mesurables e l équaion d Hamilon-Jacobi-Bellman. C. R. Acad. Sci. Paris 315 (1992), [17] H. Frankowska, S. Plaskacz. A measurable upper semiconinuous viabiliy heorem for ubes. Nonlinear Anal. 26 (1996), [18] S. Gauier. Equaions différenielles mulivoques sur un fermé. Publicaions Mahémaiques de Pau, [19] G. Haddad. Monoone rajecories of differenial inclusions and funcional differenial inclusions wih memory. Israel J. Mah. 39 (1981), [20] M. arrieu. Exisence des soluions différenielles de Carahéodory sur des ensembles fermés. Rev. Roumaine Mah. Pures e Appl. 32 (1987), [21] Y. S. edyaev. Crieria for viabiliy of rajecories of nonauonomous differenial inclusions and heir applicaions. J. Mah. Anal. Appl. 182 (1994), [22]. Marco, J. A. Murillo. Higher order differenial inclusions and viabiliy. Nonlinear Sudies, (1998), o appear. [23]. Marco, J. A. Murillo. Viabiliy heorems for higher order differenial inclusions. Se-Valued Analysis 6(1998), [24]. Marco, J. A. Murillo. Viabiliy kernels of higher order. Pliska Sud. Mah. Bulgar. 12 (1998), [25] P. Tallos. Viabiliy Problems for Nonauonomous Differenial Inclusions. SIAM J. Conrol Opim. 29 (1991), [26] J. A. Yorke. Differenial inequaliies and non-lipschiz scalar funcions. Mah. Sysems Theory 4 (1970), uis Marco Deparamen de Maemàica Aplicada Universia de València Docor Moliner, Burjasso, Spain uis.marco@uv.es José Albero Murillo Deparameno de Maemáica Aplicada Universidad de Murcia Paseo de Alfonso XIII, 44, Caragena, Spain murilloa@plc.um.es Albero.Murillo@uv.es