On the solution sets of differential inclusions and the periodic problem in Banach spaces

Nonlinear Analysis 54 (2003) 707 – 754

www.elsevier.com/locate/na

On the solution sets of di!erential inclusions and the periodic problem in Banach spaces
Ralf Badera , Wojciech Kryszewskib; c;∗ a Mathematisches

Institut, Universitat Munchen, Theresienstr. 39, D-80333 Munchen, Germany of Mathematics and Informatics, Faculty of Mathematics and Computer Sciences, Nicholas Copernicus University, ul. Chopina 12/18, Toru(n 87-100, Poland c Faculty of Mathematics, University of L( od(z, ul. Banacha 22, L(od(z 90-239, Poland

b Department

Received 25 October 2001; accepted 8 January 2003

Abstract In this paper, the topological structure of the solution set of a constrained semilinear di!erential inclusion in a Banach space E is studied. It is shown that the set of all mild solutions, with values in a closed and, in general, thin subset D ⊂ E, is an R -set provided natural boundary conditions and appropriate geometrical assumptions on D (which hold, e.g. when D is convex) are satis4ed. Applications to the periodic problem and to the existence of equilibria are given. ? 2003 Elsevier Science Ltd. All rights reserved. MSC: primary 49K24; 34C30; 34C25; secondary 47D06 Keywords: Set-valued maps; C0 -semigroup; Initial value problem under constraints; R -sets; Periodic solutions; Equilibria

1. Introduction In the 4rst part of the paper (Section 3) we study the topological characterization of the solution set to the initial value problem for the semilinear di!erential


This work was done partially while Bader was visiting the University of Toru@ n supported by Deutsche Forschungsgemeinschaft (DFG). Kryszewski was supported in part by the Faculty of Mathematics and Computer Sciences Research Statute Fund and the KBN Grant 2 P03A 024 16. ∗

Corresponding author. E-mail addresses: [email protected] (R. Bader), [email protected], [email protected] (W. Kryszewski). 0362-546X/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0362-546X(03)00098-1

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inclusion x
(t) ∈ Ax(t) + F(t; x(t));

(1)

x(0) = x0 ∈ D;

(2)

where A is the in4nitesimal generator of a linear C0 -semigroup {U (t)}t¿0 , F : [0; T ] × D ( E is an upper-Carath@eodory set-valued map and D is a closed subset of a Banach space E. Under appropriate assumptions concerning D (satis4ed e.g. if D is convex) and some natural boundary conditions we prove that the set of all mild solutions to (1), (2) is an R -set in the space of continuous maps [0; T ] → E, i.e. the intersection of a decreasing sequence of compact contractible spaces (see [36]). Remark that our results are new even in the case when D is convex, A ≡ 0 and F is a continuous single-valued map, since we do not require any additional assumptions on the set D or the space E (we demand neither that e.g. “D has non-empty interior,” or that “D is a proximate retract” nor that E has any special properties, in this context see [13,34,8] and comp. [32] where the so-called proximate retracts in Hilbert spaces are considered; see also [49]). The proofs of our results rely upon some new approximation-selection techniques for set-valued maps (see e.g., Lemma 17 below, comp. [10] or the proofs to Theorems 16 and 21). Similar results in the 4nite-dimensional setting have been obtained by the present authors in [9] (see the extensive references there); results of this type for the so-called asymptotic problems has been studied by di!erent means in [2,3]. The characterization of the solution sets is useful in the study of the periodic problem for (1) (see Section 4). The set-valued translation operator along the trajectories P associated to (1) can be investigated by topological means such as e.g., 4xed point theory for set-valued maps. Such an approach has been used in 4nite-dimensional space in [33,30]. However, in the case of an in4nite-dimensional Banach space, this method is obstructed by the fact that P has good compactness properties in exceptional cases only (e.g. for the trivial equation x
= 0, P is the identity idE and is compact i! E has 4nite dimension)—see also [38,46]. To overcome this diPculty, one imposes certain compactness properties on the semigroup {U (t)}t¿0 (see [50,8,53]) or on the nonlinearity F. Finally we derive some suPcient conditions for the existence of equilibria in the autonomous case. Our results on periodic trajectories and equilibria improve theorems given in [50,21] and recent results of the 4rst author [8] (see also [10,19,9]).

2. Preliminaries Let (E;  · ) be a Banach space. For any  ¿ 0 and sets Z; D ⊂ E, we put
 BD (Z; ) := x ∈ D: inf z − x ¡  z∈Z

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(the subscript D is omitted if D = E, unless it leads to ambiguity). The interior, the closure, the boundary, the convex and the closed convex envelope of Z will be denoted Q @Z, conv Z and conv Z, respectively. If x is an accumulation point of Z, by int Z, Z, Z then the notation y→x means that y converges to x remaining in Z. The (normed) space of bounded linear endomorphisms of E is denoted by L(E) and E ∗ stands for the normed dual of E; by ·; · we denote the duality pairing: p; x := p(x) for p ∈ E ∗ , x ∈ E. Given U ∈ L(E), U  is the norm of U and %(U ) denotes its resolvent set. A real function  de4ned on the family of bounded subsets of E is called a measure of non-compactness (MNC) if () = (conv ) for any bounded subset  of E. Two examples of MNCs are of importance: given a bounded  ⊂ E, () := inf { ¿ 0:  admits a 4nite covering by sets of diameter 6 }; () := inf { ¿ 0:  admits a 4nite covering by -balls} are the Kuratowski and the Hausdor= MNC, respectively. Recall that both of these measures are regular, i.e. ()=0 i!  is relatively compact; monotone, i.e. if  ⊂ 
then () 6 (
) and non-singular, i.e. ({a}∪)=() for any a ∈ E (for details see [1]). In what follows we always speak of regular, monotone and non-singular MNCs. For a; b ∈ R and D ⊂ E, C([a; b]; D) is the set of continuous maps [a; b] → D; C([a; b]; E) is the Banach space of continuous maps [a; b] → E equipped with the maximum norm. By L1 ([a; b]; E) we mean the Banach space of all (Bochner) integrable maps f : [a; b] → E, i.e. f ∈ L1 ([a; b]; E) i! f is strongly measurable and b fL1 := a f(s) ds ¡ ∞. Recall that strong measurability is equivalent to the usual measurability in case E is separable. The Lebesgue measure of a measurable set # ⊂ R is denoted by |#|. Given metric spaces X and Y , a set-valued map & : X ( Y assigns to any x ∈ X a non-empty subset &(x) ⊂ Y . F is upper semi-continuous (usc), if the (small) inverse image &−1 (V ) := {x ∈ X : &(x) ⊂ V } is open in X whenever V is open in Y and it is lower semi-continuous (lsc) if &−1 (A) is closed in X whenever A is closed in Y . A set-valued map & : D ( E with closed values, where D ⊂ E, is a (-set contraction (( ¿ 0) with respect to an MNC  provided & is usc and, for every bounded  ⊂ D, the set &() is bounded and (&()) 6 ((). In case (=0, we say that & is compact. Let J ⊂ R be an interval. A set-valued map & : J ( E with closed values is strongly (or Bochner) measurable if it has the Lusin property: for each  ¿ 0, there is a closed set J ⊂ J , |J \ J | ¡  such that the restriction &|J is continuous (i.e. usc and lsc simultaneously). A strongly measurable map with compact values  is almost separably valued, i.e. there is a subset N ⊂ J , with |N | = 0, such that {&(t) | t ∈ J \ N } is separable. If E is separable, then & with compact values is strongly measurable i! it is measurable, i.e. &−1 (A) is measurable for any closed (or open) A ⊂ E (see e.g. [22] or [5] for more details concerning measurability and continuity concepts). Let us suppose that: Assumption (A). (A1) A closed, linear, densely de4ned operator A : E ⊃ D(A) → E is the in@nitesimal generator of a strongly continuous linear C0 -semigroup U := {U (t)}t¿0 of type (1; !), ! ∈ R;

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(A2) J := [0; T ] ⊂ R, T ¿ 0; (A3) D is a closed subset of the Banach space E, invariant with respect to U, i.e. for each t ¿ 0, U (t)D ⊂ D. Recall that any C0 -semigroup {U (t)}t¿0 is of type (C; !) (where C ¿ 0; ! ∈ R are constants), i.e. U (t) 6 C exp(!t) for each t ¿ 0; it is well known that one can renorm E in order to make a semigroup of type (C; !) become of type (1; !). Therefore, assumption (A1) does not restrict the generality. We say that U non-expansive if ! = 0; U is compact if U (t) is compact for all t ¿ 0 (for details see e.g [47,54]). Our assumption (A3) is considered e.g. in the monograph [43, Chapter VII]: this condition can be characterized solely in terms of the generator A (see [43, Proposition VII.5.3]); moreover, it holds i! lim inf t→0+ dD (U (t)x)=t = 0 for any x ∈ D, where dD is the distance function dD (x) = inf x − y; y∈D

x∈E

(see [45, Section 4.5.1]). For any triple (x0 ; t0 ; f) ∈ J × E × L1 (J; E), the function  t M (x0 ; t0 ; f)(t) = U (t − t0 )x0 + U (t − s)f(s) ds for every t ∈ J t0

(3)

given by the variation of constants formula is, by de4nition, the mild solution of the inhomogeneous initial value problem 
x (t) = Ax(t) + f(t) (4) x(t0 ) = x0 : Recall that even the continuity of f does not guarantee that (4) has a strong solution (i.e. an almost everywhere (a.e.) di!erentiable function satisfying (4) a.e.). The latter + occurs  t if U is uniformly continuous (i.e. if limt→0 I − U (t) = 0) or if the function t → t0 U (t − s)f(s) ds is a.e. di!erentiable with derivative in L1 (J ) and x0 ∈ D(A); then (3) is a (unique) strong solution (see [47]). Let F : J × D ( E be a set-valued map. Given x0 ∈ D, t0 ∈ J we shall deal with the initial value problem (1), (2), i.e. 
u (t) ∈ Au(t) + F(t; u(t)) (5) u(t0 ) = x0 : A continuous function u : J → D is a mild solution to (5) if there is an integrable w ∈ L1 (J; E) such that w(t) ∈ F(t; u(t)) for almost all (a.a.) t ∈ J and u = M (x0 ; t0 ; w). Therefore, we see that a minimal assumption is that F satis4es the following weak superpositional measurability property: for each u ∈ C(J; D), the set-valued map F(·; u(·)) possesses a strongly measurable selection. We shall also assume the following: Assumption (B). (B1) for each t ∈ J , x ∈ D, the value F(t; x) ⊂ E is non-empty, compact and convex;

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(B2) F is an upper-Carath(eodory map, i.e. for all t ∈ J , the map F(t; ·) is usc and, for all x ∈ D, the map F(·; x) is strongly measurable; (B3) F has linear growth, i.e. there is c ∈ L1 (J; R) such that ∀(t; x) ∈ J × D

sup z 6 c(t)(1 + x);

z∈F(t; x)

(B4) F maps compact subsets of J × D into compact ones. Let us brieSy comment on these assumptions. Assumptions (B1) and (B2) imply the above-mentioned weak superpositional measurability; by (B3), the map F induces the (set-valued) Nemytskij operator NF : C(J; D) ( L1 (J; E) given by NF (x) := {g ∈ L1 (J; E): g(s) ∈ F(s; x(s)) a:e: on J }: Assumption (B4) holds e.g. when either F is (jointly) upper semicontinuous; or dim E ¡ ∞ and c(·) ≡ const in (B3); or when the compactness condition (7) given below is satis4ed (for details see [22, Section 3]). If E is separable, then assumptions (B1), (B2) and (B4) imply the following result (essentially due to Rze˙zuchowski [52], comp. [27, Theorem 5.4], [22, Proposition 5.1], [12, Proposition 3.3] see also [37]) of the Scorza-Dragoni type. Theorem 1. There is a map F0 : J × D → 2E (i.e. having possibly empty values) with compact convex values such that: (i) for all t ∈ J and x ∈ D, F0 (t; x) ⊂ F(t; x); (ii) if A ⊂ J is measurable, u; v : A → D are measurable maps with v(t) ∈ F(t; u(t)) a.e. (almost everywhere) on A, then v(t) ∈ F0 (t; u(t)) a.e. on A; (iii) for any  ¿ 0, there is a closed J ⊂ J with |J \ J | ¡  such that F0 restricted to J × D has non-empty values and is (jointly) upper semicontinuous. The set of all mild solutions to the initial value problem (5), denoted by L(x0 ; t0 ), consists precisely of the elements of the 4xed point set of the set-valued map M (x0 ; t0 ; ·) ◦ NF , i.e. L(x0 ; t0 ) = Fix(M (x0 ; t0 ; ·) ◦ NF ) (see (3)) de4ned on C(J; D). Assumptions (A), (B) are our standing hypotheses for the rest of the paper unless stated otherwise. Concerning the existence of solutions to (5) we have the following result, see [12, Section 7;8]. For some other existence results—see e.g. [16] and rich references therein (comp. [15] for a lower semicontinuous case and e.g. [26] for the so-called one-sided Lipschitz case). Theorem 2. Suppose that F(t; x) ∩ TD (x) = ∅

for every t ∈ J; x ∈ D:

(6)

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Then, for each x0 ∈ D, t0 ∈ J , the initial value problem (5) has a mild solution provided E is separable and one of the following conditions holds: (i) for any bounded  ⊂ D lim (F(J (t; h) × )) 6 k(t)()

h→0+

for each t ∈ J;

(7)

where J (t; h) := (t − h; t + h) ∩ J and k ∈ L1 (J; R); or (ii) the semigroup U is compact. Moreover, in both cases, the set-valued map L : D × J ( C(J; D) is usc with non-empty, compact values. Above TD (x) stands for the Bouligand tangent cone to D at the point x ∈ D, i.e. 
dD (x + hy) = 0 : TD (x) := y ∈ E | lim inf h→0+ h

Remark 3. (i) As was observed above (7) implies (B4). (ii) It is not diPcult to see that L(x0 ; t0 ) (if non-empty) is always compact. Condition (7) is suPcient to get that L(x0 ; t0 ) = ∅. If D = E, then a weaker condition (F({t} × )) 6 k(t)()

(8)

for all bounded  ⊂ D, is suPcient in Theorem 2. We shall see below that if D = E, then (B4) together with (8) is also suPcient provided we know more about D. (iii) It is clear that if a single-valued F is such that, for all x ∈ D, F(·; x) is strongly measurable, F(I × {x}) is compact, for all t ∈ J , F(t; ·) is locally Lipschitz (uniformly with respect to t) and satis4es (6), then (5) possesses a unique mild solution which depends continuously on initial data (x0 ; t0 ). Indeed under these assumptions F satis4es all hypotheses of Theorem 2 locally; hence, a local unique mild solution exists. The existence of the global solution follows by the usual continuation method. (iv) As shown by Bothe [12, Section 7], if F is (jointly) usc (or even almost usc in the sense of Deimling [22]), then the separability of E is not necessary. We shall make use of this important observation in the sequel (see Remark 24). More details and explanations on all the above-introduced concepts may be found in [39].

3. Solution sets of the semilinear system We shall study the topological structure of the set L(x0 ; t0 ) of mild solutions to the semilinear system (5): we shall try to recover a result in the spirit of Aronszajn (valid in RN for ordinary di!erential equation—see [4]) saying that L(x0 ; t0 ) is an R -set

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in C(J; E). Results of this type are known: in a 4nite-dimensional situation (see e.g. [9] for the most general case, [22,34] and the surveys [27,31]) and in the present situation—see [12,35,39] in case D = E. It also holds if D is convex and int D = ∅ or if D is convex proximinal (i.e. each x ∈ E admits a nearest point in D) and U is non-expansive (see [8,13]). However, if the set D is not convex, then the situation changes dramatically. For that reason consider the following examples.  √ Example 4. Let E = R3 , D := {x = (x1 ; x2 ; x3 ) ∈ E | |x| 6 2 and x12 + x22 ¿ x3 }, 2 2 2 2 S := {x ∈ D | x1 + x2 = 1 and x3 = 1} and Z = {x ∈ E | x1 + x2 6 1 and x3 = 1}. Next, for x ∈ D, put  Z for x ∈ D \ S; F(x) = conv{Z ∪ {(−x2 ; x1 ; 0}} for x ∈ S: Clearly F : D ( E is u.s.c. and F(x) ∩ TD (x) = ∅ for all x ∈ D. Hence (A3), (6) hold (with A ≡ 0). But is easy to see that L(x0 ; 0) (with A ≡ 0, x0 = 0) is homeomorphic to the unit sphere S 1 := {x ∈ R2 | x = 1}; hence, it is not an R -set (such sets are always acyclic). Notice that, for all x ∈ D, x = 0, the Bouligand and the Clarke tangent cones TD (x) and CD (x) coincide; however, TD (0) = CD (0) and F(0) ∩ CD (0) = ∅. Above CD (x) denotes the Clarke tangent cone to D at x ∈ D, i.e.   dD (y + hu) CD (x) = u ∈ E | lim =0 : D h h→0+ ; y→x Observe that CD (x) is a closed convex cone and CD (x) ⊂ TD (x). It is not true however that the remedy would be to replace in (6) the Bouligand cone by the Clarke one. Example 5. Let D = S1 ∪ S−1 where Si = {x = (x1 ; x2 ) ∈ E = R2 | (x1 − i)2 + x22 = 1} and, for x ∈ D, let  (x2 ; 1 − x1 ) for x ∈ S1 ; F(x) = (−x2 ; 1 + x1 ) for x ∈ S−1 : Then, for all x ∈ D, F(x) ∈ TD (x) = CD (x), but L(x0 ; 0) (again with A ≡ 0, x0 = 0) is even not connected. Therefore, it seems that in order to state the correct tangency condition which implies the expected topological structure of solutions sets to (5) one should replace the Bouligand cones by the Clarke ones and take care of the geometry of the set D involved. To make it we need to employ some notions of the non-smooth calculus introduced by Clarke (see [17] for the details). Given a locally Lipschitz continuous function f : E → R, by f◦ (x; u) we denote the Clarke generalized directional derivative of f at x ∈ E in the direction u ∈ E. The

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function E  u → f◦ (x; u) ∈ R is 4nite, positive homogeneous, subadditive (hence convex) and Lipschitz continuous (with Lipschitz constant equal to the Lipschitz constant of f around x); the function (x; u) → f◦ (x; u) ∈ R is 4nite, upper semicontinuous and f◦ (x; −u)=(−f)◦ (x; u). Moreover, the Clarke generalized gradient of f at x, @f(x) := {p ∈ E ∗ | p; u 6 f◦ (x; u) for all u ∈ E}, is a non-empty w∗ -compact convex subset of E ∗ ; f◦ (x; u) = supp∈@f(x) p; u and the set-valued x → @f(x) is upper hemicontinuous (i.e. for any u ∈ E, the map x → supp∈@f(x) p; u ∈ R is upper semicontinuous). It is clear that u ∈ @f(x)− (where @f(x)− := {u ∈ E | p; u 6 0 for all p ∈ @f(x)} is the negative polar cone) if and only if f◦ (x; u) 6 0. In particular, CD (x) = @dD (x)− for all x ∈ D. 3.1. Geometry of the domain Below we shall deal with the so-called regular domains. De&nition 6. We say that a closed set D ⊂ E is regular if, for any x ∈ @D, lim inf |@dD (y)| ¿ 0 E\D

y→x

where |@dD (x)| :=

inf

p∈@dD (x)

p:

It is clear that |@dD (x)| = sup

inf

u61 p∈@dD (x)

p; u = − inf d◦D (x; u): u61

(9)

Observe that regularity of D means that the distance function dD has no critical points in a neighborhood of D intersected with the complement of D. If there is a neighborhood  of D such that inf y∈\D |@dD (y)| ¿ 0, then D is regular; if D is regular compact (or @D is compact), then such a neighborhood exists. Regular sets are well designed to study solutions to (5) in case of a non-expansive semigroup U. In order to study a general situation we shall also deal with the so-called strictly regular sets. De&nition 7. We say that a closed set D ⊂ E is strictly regular if there is a fat neighborhood  of D (i.e. such that B(D; r) ⊂  for some r ¿ 0) such that inf |@dD (y)| ¿ 0:

y∈\D

Clearly strictly regular sets are regular and compact regular sets are strictly regular. The class of (strictly) regular sets has been introduced in [19] in a di!erent (and a bit more general) setting and studied in the context of equilibria. This class is rich: for instance the set D in Example 4 is strictly regular and the set D from Example 5 is not regular.

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Example 8. (i) Any convex closed set D ⊂ E is strictly regular: in fact one shows easily that |@dD (y)| ¿ 1 for all y ∈ E \ D (see [19]). (ii) Suppose that a closed D ⊂ E is proximinal, i.e. there is a neighborhood  of D such that, for all y ∈ , the set
Lim inf denotes the lower limit in the sense of Painlev@e–Kuratowski—see e.g. [5, De4nition 1.4.6].

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Proof. Assume (i). In [19] it was proved that 0 ∈ @#D (x) for x ∈ @D (comp. [18] for E = Rn ). Hence @#D (x)− ⊂ CD (x). We shall show that CD (x) ⊂ @#D (x)− . We sustain the notation from De4nition 9. Fix x0 ∈ @D and let A(z0 ; =0 ) = x0 . It is clear that =0 = g(z0 ). In the space Z × R we consider a norm (z; =) := A(z; =) for z ∈ Z, = ∈ R; this norm is equivalent to the original one. Thus A is an isometry and, for y = (z; =) from a suPciently small ball around y0 = (z0 ; =0 ) (contained in U ), we have #D ◦ A(y) = #N (y) := dN (y) − dM (y) where N := Epi g and M := Z × R \ int N = Hyp g where Hyp g := {y = (z; =) | g(z) ¿ =} is the hypograph of g. Hence @#N (y) = A∗ [@#D (A(y))] and A[@#N (y)− ] = @#D (A(y))− . Since clearly A(CN (y0 )) = CD (x0 ), it is enough to show that CN (y0 ) ⊂ @#N (y0 )− . To this end take v0 = (u0 ; >0 ) ∈ CN (y0 ); we have to prove that #◦N (y0 ; v0 ) 6 0. Take a sequence (yn )∞ 1 in Z × R such that yn → y0 and a sequence hn → 0+ . In view of the useful characterization of tangent cones to epigraphs (see [17, Theorem 2.4.9]) we have that >0 ¿ g◦ (z0 ; u0 ). Therefore, we easily see that −v0 = (−u0 ; −>0 ) ∈ CM (y0 ) (recall that M = Hyp g). Suppose that yn ∈ N (for a subsequence). Since v0 ∈ CN (y0 ), there is a sequence vn → v0 such that yn + hn vn ∈ N . Hence, lim sup n→∞

#N (yn + hn v0 ) − #N (yn ) #N (yn + hn vn ) − #N (yn ) = lim sup hn hn n→∞

= lim sup n→∞

−dM (yn + hn v0 ) + dM (yn ) −dM (yn + hn vn ) + dM (yn ) = lim sup hn hn n→∞

6 (−dM )◦ (y0 ; v0 ) = d◦M (y0 ; −v0 ) 6 0 since −v0 ∈ CM (y0 ). If yn ∈ N , then (since −v0 ∈ CM (y0 )) there is a sequence wn → v0 such that yn = yn + hn (−wn ) ∈ M . Hence, if we put yQ n = yn + hn wn , then lim sup n→∞

#N (yn + hn v0 ) − #N (yn ) −#N (yQ n − hn wn ) + #N (yQ n ) = lim sup hn hn n→∞

= lim sup n→∞

−dN (yQ n − hn wn ) + dN (yQ n ) −dN (yQ n − hn v0 ) + dN (yQ n ) = lim sup hn hn n→∞

6 (−dN )◦ (y0 ; −v0 ) = d◦N (y0 ; v0 ) 6 0: This shows that indeed #◦N (y0 ; v0 ) 6 0, i.e. v0 ∈ @#N (y0 )− . Since 0 ∈ @#D (x) for x ∈ @D, there is u ∈ E such that u = 1 and #◦D (x; u) ¡ −  for some  ¿ 0. This together with the upper semicontinuity of #◦D (·; u) shows that, for y ∈ D suPciently close to x, we have d◦D (y; u) = #◦D (y; u) ¡ − . Hence, by (9), lim inf |@dD (y)| ¿ : E\D

y→x

This shows the regularity of D. Remark 12. In a similar manner one shows that a smooth (i.e. C 1 ) Banach submanifold M in E of codimension 1 is regular.

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3.2. The tangency condition Now we shall discuss in detail the proper tangency condition which leads to the R -characterization of the solution set to (5). Let us 4rst study condition (6) with the Bouligand cone replaced by the Clarke one, that is ∀(t; x) ∈ J × D

F(t; x) ∩ CD (x) = ∅:

(10)

It has some important (from our viewpoint) consequences. Since values of F are compact and CD (x) = @dD (x)− (x ∈ D), it is clear that (10) is equivalent to ∀x ∈ D

sup inf

t∈J z∈F(t; x)

d◦D (x; z) 6 0:

(11)

This, in turn, implies that, for all x ∈ D,

◦ lim sup sup inf dD (y; z) 6 0: y→x

(12)

t∈J z∈F(t; x)

To see this 4x x ∈ D; for each t ∈ J , there is zt ∈ F(t; x) such that d◦D (x; zt ) 6 0. Take an arbitrary  ¿ 0. Since d◦D (·; zt ) is upper semicontinuous, lim supy→x d◦D (y; zt ) 6 d◦D (x; zt ) 6 0, i.e. there is ?(t) ¿ 0 such that for all y ∈ B(x; ?(t)), d◦D (y; zt ) ¡ =2. By (B4), the set F(J × {x}) is relatively compact; hence it has a 4nite =4-net. Therefore, one may choose t1 ; : : : ; tm ∈ J such that, for any t ∈ J , there is 1 6 j 6 m with zt − ztj  ¡ =2. Let 0 ¡ ? ¡ min16j6m ?(tj ). Take any y ∈ B(x; ?) and t ∈ J . Then clearly: inf

z∈F(t; x)

d◦D (y; z) 6 d◦D (y; zt ) = d◦D (y; ztj + (zt − ztj )) 6 d◦D (y; ztj ) + d◦D (y; zt − ztj ) ¡

and (12) follows. To simplify the notation let us write BF (y; x) := sup inf

t∈J z∈F(t; x)

d◦D (y; z):

(13)

In order to get the next implication we shall use the notion of the so-called C-convergence. 2 In our setting we have lim sup inf BF (y; x
) 6 lim sup BF (y; x) 6 0 y→x

D

y→x

x →x

(14)

2 Let X; Y be metric spaces, A ⊂ X , B ⊂ Y and let a ∈ X , b ∈ Y be accumulation points of A and B, 0 0 respectively. Finally let f : A × B → R be a function. We de4ne

lim sup inf f(a; b) := sup inf a→a0 b→b0

sup

Observe that lim sup inf f(a; b) 6 lim sup f(a; b0 ): a→a0 b→b0

inf

¿0 ?¿0 a∈BX (a0 ;?) b∈BY (b0 ; )

a→a0

f(a; b):

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R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

for all x ∈ D. Finally observe that in case D is compact, then (14) holds uniformly with respect to x ∈ D, i.e. sup inf

sup

inf

 ¿0 ?¿0 x∈D;y∈B(x;?) x ∈BD (x; )

BF (y; x
) 6 0:

(15)

Let us emphasize that the implication (10) ⇒ (14) holds always; the implication (10) ⇒ (15) is true if D is compact. Assumptions (B2) and (B4) imply that (15) ⇒ (14) ⇒ (10). In order to compensate the lack of compactness (which seems to be an intrinsic problem in an in4nite-dimensional setting), we shall give the following de4nition. De&nition 13. We say that F satis4es the uniform tangency condition on D if (15) holds locally uniformly on D, i.e. any point p ∈ D has a neighborhood V (in D) such that sup inf

sup

inf

 ¿0 ?¿0 x∈V;y∈B(x;?) x ∈BD (x; )

BF (y; x
) 6 0:

(16)

Example 14. Suppose that (10) holds. The uniform tangency (16) is satis4ed in each of the following cases: (i) the set D is proximinal and satis4es assumptions of Example 8(ii); (ii) D is convex and E is reSexive; (iii) D is compact (then (16) holds with V = D, i.e. (15) holds). Situation (iii) has been discussed above (the conclusion may be achieved via standard compactness arguments). Let us show (i) (we sustain the notation from Example 8). Take p ∈ D; there is r ¿ 0 such that B(p; 2r) ⊂ . For  ¿ 0, let ? 6 min{=2; r}, take x ∈ V := B(p; r) ∩ D and y ∈ B(x; ?). Then y − p 6 2r, i.e. y ∈ . Choose xQ ∈
dD (yn + hn zt ) − dD (yn ) dD (yn + hn zn ) − dD (yn ) = lim sup hn hn n→∞ 6 lim sup n→∞

xQn − yn  + dD (xQn + hn zn ) − d(yn ) hn

= 0: Thus zt ∈ @dD (y)− ; hence BF (y; x) Q 6 0 as required. Situation (ii) is similar. The set D being convex in a reSexive space E is proximinal (with  = E); however, it is not clear whether the Lim inf property (from Example 8) of
R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

719

and t ∈ J . There is xQ ∈
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R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

Theorem 16. Suppose that E is separable, condition (16) is satis@ed and: (i) either D is strictly regular, or (ii) D is regular and the semigroup U is non-expansive. Then, for any x0 ∈ D, the set L(x0 ) of all mild solutions to the initial value problem (5) is an R -set in C(J; E) provided the compactness condition (8) is satis@ed or the semigroup U is compact. Proof. The subsequent proof is technical and rather long. For the reader’s convenience we shall proceed in several steps and formulate some claims in its course. Step I: Fix x0 ∈ D. Let & := M (x0 ; 0; ·) ◦ NF : C(J; D) → C(J; E), let the solution of


(t) = Rc(t)(1 + (t)) a:e: on J;

: J → R be

(0) = sup{U (t)x0  | t ∈ J };

where R := sup{e!s | s ∈ J }, and let K0 := {u ∈ C(J; E) | u(t) 6 (t)}: Then the (constant) function J  t → u0 (t) ≡ x0 belongs to K0 and &(K0 ∩C(J; D)) ⊂ K0 . Now, for n ¿ 1, let Kn := conv[&(Kn−1 ∩ C(J; D)) ∪ {u0 }]: It is easy to see that, for all ∞n ¿ 1, u0 ∈ Kn , Kn ⊂ Kn−1 and &(Kn ∩ C(J; D)) ⊂ Kn . Let K := C(J; D) ∩ n=0 Kn . Clearly K is closed and non-empty. Moreover, L(x0 ) = Fix & ⊂ K in view of the Gronwall inequality. Using (8) and exactly as in [14, Proof to Theorem 2], one shows that, for each t ∈ J , K(t) := {u(t) | u ∈ K} is compact. Next it is easy to show (see the appendix) that K is equicontinuous; hence, by the Ascoli theorem, K is compact. Let K := {u(t) | u ∈ K; t ∈ J }. It is clear that K is compact. Without loss of generality we may assume that K ∩ @D = ∅ (if not, then L(x0 ) is an R -set in view of Bothe’s results from [13,14]). Since K is compact, D is regular and the uniform tangency condition (16) holds, there is  ¿ 0 such that: ∃# ¿ 0 ∀y ∈ B(K; 2)\D sup inf

sup

|@dD (y)| ¿ 2#; inf

 ¿0 ?¿0 x∈BD (K;);y∈B(x;?) x ∈B(x; )

BF (y; x
) 6 0:

(17) (18)

If D is strictly regular, then we may assume that ∀y ∈ \D

|@dD (y)| ¿ 2#;

(19)

where  is a fat neighborhood of D (in this case we may assume that B(K; 2) ⊂ ).

R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

721

We modify F in the following manner: consider a continuous function  : E → [0; 1] such that (x) = 1 if dK (x) 6 =3, (x) = 0 if dK (x) ¿ 2=3 and, for x ∈ D, t ∈ J , de4ne ˜ x) = (x)F(t; x): F(t;

(20)

It is clear that ˜ 0 ) = L(x0 ); L(x

(21)

˜ 0 ) stands for the set of all mild solutions to (5) (with F replaced by F). ˜ where L(x De4ne the set-valued map Fn : J × D ( E by the formula ˜ Fn (t; x) = F({t} × BD (x; n−1 )) + B(0; n−1 ):

(22)

Observe that, for all suPciently large n ¿ 1 (without loss of generality for all n ¿ 1), Fn (t; x) = B(0; n−1 )

if x ∈ BD (K; ):

(23)

Claim 1. For any positive integer n, there is ?n ¿ 0 such that if x ∈ BD (K; ) and y ∈ B(x; 4?n )\D, then there is a measurable function z = zx; y : J → E such that z(t) ∈ Fn (t; x)

and

d◦D (y; z(t)) ¡ − #n := −

# n

for t ∈ J:

(24)

Moreover, z takes values in a compact subset of E. To see this, 4x n ¿ 1 and, by (18), choose ?n ¿ 0 such that 4?n ¡  and, for all x ∈ BD (K; ) and y ∈ B(x; 4?n ), there is x
∈ BD (x; n−1 ) such that F(t; x
) ∩ {z ∈ E | d◦D (y; z) 6 #n } = ∅ for all t ∈ J . Take arbitrary x ∈ BD (K; ) and y ∈ B(x; 4?n ) \ D. Since the set {z ∈ E | d◦D (y; z) 6 #n } is closed (in view of the continuity of d◦D (y; ·)), there is a measurable function w : J → E such that, for all t ∈ J , d◦D (y; w(t)) 6 #n and w(t) ∈ F(t; x
) where x
∈ BD (x; n−1 ). Since y ∈ B(K; 2), by (17) and (9), |@dD (y)| = − inf d◦D (y; u) = − inf d◦D (y; u) ¿ 2#; u61

u¡1

i.e. there is uy ∈ E with uy  ¡ n−1 such that d◦D (y; uy ) ¡ − 2#n . For t ∈ J , put z(t) := (x
)w(t) + uy : Obviously z is measurable and, for all t ∈ J , d◦D (y; z(t)) ¡ − #n

and

z(t) ∈ Fn (t; x):

Thus (24) is established. Notice that the values of w (as a selection of F(·; x
)) lie in a compact subset of E in view of assumption (B4). The same observation concerns z = zx; y .

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R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

Changing ?n if necessary we may assume that ?n+1 ¡ ?n ¡ n−1

(25)

!?n ¡ #n =2:

(26)

and

Additionally de4ne Dn := {y ∈ E | dD (y) 6 ?n } and Dn
:= {y ∈ E | dD (y) ¡ 2?n }: If D is strictly regular, then we may assume that Dn
⊂  for all (suPciently large) n. Step II: Let {Us ; as )}s∈S be the so-called Dugundji system for E \ D (see [11, p. 57]), i.e. for all s ∈ S: • Us ⊂ E \ D, as ∈ @D; • if x ∈ Us , then x − as  6 2dD (x); • {Us }s∈S is a locally 4nite covering of E\D. We de4ne now the extension Gn : J × E ( E of Fn by  for x ∈ D; t ∈ J;   Fn (t; x)  Gn (t; x) = =s (x)Fn (t; as ) for x ∈ E\D; t ∈ J;  

(27)

s∈S

where {=s }s∈S is a locally 4nite partition of unity subordinated to the cover {Us }. Note that, for any n ¿ 1, if x ∈ E, then for all t ∈ J ˜ Gn (t; x) ⊂ conv F({t} × BD (x; n−1 + 2dD (x))) + B(0; n−1 ):

(28)

Indeed, D. Therefore let x ∈ E \ D and y ∈ Gn (t; x). Then  by (22), it is obvious for x ∈ y ∈ s∈S =s (x)Fn (t; as ). Hence y = s∈S =s (x)ys where ys ∈ Fn (t; as ). If =s (x) = 0, then x ∈ Us and x − as  6 2dD (x). Thus, ˜ ys ∈ Fn ({t} × BD (x; 2dD (x))) ⊂ F({t} × BD (x; n−1 + 2dD (x))) + B(0; n−1 ): This establishes (28). Finally we modify Gn in the following manner. Let, for t ∈ J , x ∈ E, Hn (t; x) = conv Gn ({t} × B(x; n−1 )) + B(0; n−1 ):

(29)

In view of (28), ˜ Hn (t; x) ⊂ conv F({t} × BD (x; 3n−1 + 2dD (x))) + B(0; 2n−1 ):

(30)

R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

723

m m Indeed, let y ∈ Hn (t; x), then y = i=1 i yi + z where i=1 i = 1 and yi ∈ Gn (t; xi ) ˜ × with xi ∈ B(x; n−1 ) (i = 1; 2; : : : ; m) and z ¡ n−1 . Hence, by (28), yi ∈ conv F({t} ˜ BD (xi ; n−1 + 2dD (xi ))) + B(0; n−1 ). Thus y ∈ conv F({t} × BD (x; 3n−1 + 2dD (x))) + B(0; 2n−1 ). It is clear that, for a continuous function v : J → D, Fn (·; v(·)) has a measurable selector. The maps Gn and Hn enjoy the similar property: let v : J → E be continuous and S ∗ := {s ∈ S | v(J ) ∩ supp =s = ∅} and J ∗ := {t ∈ J | v(t) ∈ D} (both these sets may be empty; however J ∗ = ∅ i! S ∗ = ∅). Clearly J \ J ∗ is closed. Let w0 be a measurable selector of Fn (·; v(·)) on J \ J ∗ and let ws be a selector of Fn (·; as ) for any s ∈ S ∗ . Then we easily see that  for t ∈ J \ J ∗ ;   w0 (t) w(t) =  =s (v(t))ws (t) for t ∈ J ∗   s∈S ∗

is a measurable selector of Gn (·; v(·)). It therefore makes sense to consider the set Ln (x0 ) of all mild solutions to the problem 
x (t) ∈ Ax(t) + Hn (t; x(t)); x(0) = x0 ;

x(t) ∈ Dn

for t ∈ J:

It is easy to see that Ln+1 (x0 ) ⊂ Ln (x0 ) because Fn+1 (t; x) ⊂ Fn (t; x) and, consequently, Gn+1 (t; x) ⊂ Gn (t; x), Hn+1 (t; x) ⊂ Hn (t; x) on J × E. Moreover,  ˜ 0) ⊂ L(x0 ) = L(x Ln (x0 ) n¿1

˜ x) ⊂ Fn (t; x) on J × D. Thus Ln (x0 ) is non-empty for all n ¿ 1. since F(t; Claim 2. Let un ∈ Ln (x0 ) (n ¿ 1). The sequence (un ) possess a subsequence converging ˜ 0 ) = L(x0 ). uniformly to some u0 ∈ L(x The proof of this claim, being long and rather standard, is contained in the appendix. Claim 2 yields also that  L(x0 ) = Ln (x0 ): (31) n¿1

Moreover, in fact we have get that In := supv∈Ln (x0 ) d(v; L(x0 )) → 0; hence Ln (x0 ) ⊂ L(x0 ) + B(0; 2In ), which implies that 0 (Ln (x0 )) 6 2In → 0, where 0 denotes the Hausdor! MNC on C(J; E), since L(x0 ) is compact.

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R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

Step III: To complete the proof of Theorem 16 we shall use the characterization of R -sets provided in the beginning of the subsection. In view of (31) we only need to show that, for any n ¿ 1, Ln (x0 ) is contractible. Claim 3. For any n ¿ 1 and x ∈ Dn
\D, there is a measurable function w=wx : J → E taking values in a compact subset of E and such that w(t) ∈ Gn (t; x)

(32)

and ∀t ∈ J

d◦D (x; w(t)) ¡ − #n

∀t ∈ J

d◦D (x; w(t)) 6 0

if D is strictly regular;

if D is regular:

(33) (34)

Fix n ¿ 1 and x ∈ Dn
\ D (i.e. 0 ¡ dD (x) ¡ 2?n ). In view of (27),  Gn (t; x) = =s (x)Fn (t; as ); s∈S(x)

where S(x) := {s ∈ S | x ∈ Us }; S(x) is 4nite since the cover {Us }s∈S is locally 4nite. Let s ∈ S(x); then, by (25), x − as  6 2dD (x) ¡ 4?n ¡ . If as ∈ B(K; ), then by (23), Fn (t; as ) = B(0; n−1 ) and: (i) in case D is strictly regular we put ws (t) ≡ u where u ¡ n−1 is such that d◦D (x; u) ¡ − 2#n (see (19) and the proof of Claim 1); (ii) if D is regular, then we put ws (t) ≡ 0. If as ∈ B(K; ), then in view of Claim 1 (see (24)) there is a measurable function ws := zas ;x : J → E such that ws (t) ∈ Fn (t; as ) and d◦D (x; ws (t)) ¡ − #n for all t ∈ J . De4ne w : J → E by the formula  w(t) = =s (x)ws (t) for t ∈ J: s∈S(x)

Obviously w is measurable, for all t ∈ J , d◦D (x; w(t)) ¡ − #n if D is strictly regular, and d◦D (x; w(t)) 6 0 if D is regular (since d◦D (x; ·) is a convex function). Moreover w(t) ∈ Gn (t; x). This establishes (32), (33) (resp. (34)). Remark. Observe that if D is regular and (j) x ∈ Dn
\D is such that as ∈ B(K; ) for all s ∈ S(x), then wx ≡ 0 and d◦D (x; wx (t))=0; (jj) however, if there is at least one s ∈ S(x) with as ∈ B(K; ), then wx is, in general, not identically 0 and d◦D (x; wx (t)) ¡ 0. Claim 4. For any n ¿ 1, there is a mapping fn : J × Dn
→ E such that: (i) fn (t; x) ∈ Hn (t; x) on J × Dn
; (ii) fn (·; x) is measurable;

R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

725

(iii) each point p ∈ Dn
has a neighborhood W such that fn (J × W ) lies in a compact subset of E and there is a constant ‘ ¿ 0 such that, for all x; x
∈ W and t ∈ J , fn (t; x) − fn (t; x
) 6 ‘x − x
; i.e. in particular fn (t; ·) is locally Lipschitz (uniformly with respect to t ∈ J ); (iv) for all t ∈ J , x ∈ Dn
, d◦D (x; fn (t; x)) 6 0; if D is strictly regular, then d◦D (x; fn (t; x)) ¡ − #n provided dD (x) ¿ ?n =2. Take any x ∈ Dn
\ D, then recall the measurable function wx : J → E given in Claim 3 (see (33) or (34)). Recall that wx takes values in a compact subset of E. Hence, there is a simple (i.e. measurable and having a 4nite number of values) function vx : J → E such that wx (t) − vx (t) ¡ n−1 and vx (J ) ⊂ wx (J ) (in case (j) we put vx = wx ≡ 0). Hence also ∀t ∈ J

d◦D (x; vx (t)) ¡ − #n

∀t ∈ J

d◦D (x; vx (t)) 6 0

if D is strictly regular;

if D is regular:

Since vx has a 4nite number of values and d◦D is upper semicontinuous, there is rx ∈ (0; n−1 ) such that, for any y ∈ B(x; rx ), ∀t ∈ J

d◦D (y; vx (t)) ¡ − #n

∀t ∈ J

d◦D (y; vx (t)) 6 0

if D is strictly regular;

if D is regular:

(35) (36)

If x ∈ D, then take rx ∈ (0; n−1 ) such that B(x; rx ) ⊂ {y ∈ E | dD (y) ¡ ?n =2} and, moreover, let vx : J → E be an arbitrary measurable selection of Gn (·; x) = Fn (t; x) having values in a compact set and such that d◦D (x; wx (t)) 6 0. We have therefore constructed an open covering {B(x; rx ) ∩ Dn
}x∈Dn . Let {>i : Dn
→ [0; 1]}i∈I be a locally Lipschitz continuous partition of unity inscribed into this cover, i.e. for any i ∈ I , there is xi ∈ Dn
such that supp >i ⊂ B(xi ; ri ) where ri := rxi . For any i ∈ I , let vi := vxi and de4ne  fn (t; x) = >i (x)vi (t); x ∈ Dn
; t ∈ J: (37) i∈I

It is clear that, for any x ∈ Dn
, fn (·; x) is measurable. Since, for all i ∈ I , vi takes values in a compact set, the local 4niteness of the cover {supp >i }i∈I implies condition (iii) of the claim. It is clear that d◦D (x; fn (t; x)) 6 0 on J × Dn
by (35) or (36). Supposing that D is strictly regular and given x ∈ Dn
with ?n =2 6 dD (x) ¡ 2?n , if >i (x) = 0, then the corresponding xi is such that 0 ¡ dD (xi ) ¡ 2?n (for otherwise x ∈ B(xi ; ri ) with xi ∈ D, so dD (x) ¡ ?n =2, a contradiction) and x ∈ B(xi ; ri ). Hence by (35),  >i (x)d◦D (x; vi (t)) ¡ − #n : d◦D (x; fn (t; x)) 6 i∈I

This shows condition (iv).

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R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

Finally take x ∈ Dn
and observe that, if >i (x) = 0, then vi (t) ∈ Gn (t; xi ) + B(0; n−1 ), i.e. vi (t) ∈ Gn ({t} × B(x; n−1 )) + B(0; n−1 ). Hence, fn (t; x) ∈ conv Gn ({t} × B(x; n−1 )) + B(0; n−1 ) = Hn (t; x): This establishes condition (i) and completes the proof of Claim 4. Given t0 ∈ [0; T ) (recall that J = [0; T ]), y0 ∈ Dn , consider an initial value problem v
∈ Av + fn (t; v); v(t0 ) = y0 :

(38)

Claim 5. Problem (38) admits a unique solution on vn = vn (·; t0 ; y0 ) : [t0 ; T ] → Dn . First suppose that D is regular and U is non-expansive. Hence, by (A3), U (t)Dn ⊂ Dn for all t ¿ 0. At the same time, for all x ∈ Dn , t ∈ J , fn (t; x) ∈ CDn (x) since d◦Dn (x; fn (t; x)) 6 d◦D (x; fn (t; x)) 6 0. Hence, Theorem 2 applies and the conclusion follows. Now we shall consider the strictly regular case. The existence theory yields that (38) admits a unique local solution vn : [t0 ; t1 ] → E where t0 ¡ t1 6 T . It is enough to show that, for all t ∈ [t0 ; t1 ], vn (t) ∈ Dn . Having this Claim 5 follows by the use of the usual continuation argument. To simplify the notation 4x n ¿ 1 and write f := fn , v := vn . Suppose to the contrary that the trajectory of v leaves Dn . Without loss of generality we may assume that dD (v(t1 )) ¿ ?n . Let tQ := max{t ∈ [t0 ; t1 ] | dD (v(t)) = ?n }: Hence, tQ ¡ t1 and for all t ∈ (tQ; t1 ], dD (v(t)) ¿ ?n . Let p = v(tQ). By Claim 4(iii), there is a neighborhood W of p having properties enlisted there and such that, for all x ∈ W , 1 2 ?n ¡ dD (x) ¡ 2?n . According to Claim 4(iv), for all t ∈ J , x ∈ W , d◦D (x; f(t; x)) ¡ − #n :

(39)

Now we shall make use of the Scorza–Dragoni property of f (see also Theorem 1 and recall that f|J × W takes values in a compact (and, if we wish, convex) set, say C): for each k ¿ 1, there is a closed set Jk ⊂ J with |J \ Jk | ¡ k −1 such that f|Jk × W is continuous. Since J \ Jk is open, we have  (ik ; ik ); J \ Jk = i¿1

where intervals (ik ; ik ) are disjoint. Let us de4ne gk : J × W → E by  if t ∈ Jk ;   f(t; x) gk (t; x) = (ik − t)f(ik ; x) + (t − ik )f(ik ; x)  if t ∈ (ik ; ik )  ik − ik

(40)

R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

727

and x ∈ W . It is clear that, for each k ¿ 1, gk is jointly continuous, takes values in C and is uniformly Lipschitz (with respect to x) with the Lipschitz rank independent of k. Moreover, by (39), d◦D (x; gk (t; x)) ¡ − #n

(41)

for all k ¿ 1, t ∈ J and x ∈ W . There is B ∈ (tQ; t1 ] such that, for all k ¿ 1, the problem u
= Au + gk (t; u); u(tQ) = p admits the unique solution uk : [tQ; B] → E. Applying the standard arguments (see also the appendix) one shows that uk → v|[tQ; B] in C([tQ; B]; E) as k → ∞. We shall show that, for any k ¿ 1, the trajectory of uk stays in Dn . To this end 4x k ¿ 1 and suppose to the contrary that there is t ∈ [tQ; B] such that dD (uk (t)) ¿ ?n and de4ne z = max{t ∈ [tQ; B] | dD (uk (t)) = ?n }: Then tQ 6 z ¡ B. The semigroup property gives that for small h ¿ 0,  z+h uk (z + h) = U (h)uk (z) + U (z + h − s)gk (s; uk (s)) ds: z

(42)

Since, for any h ¿ 0, U (h)D ⊂ D, we gather that dD (U (h)uk (z)) 6 e!h dD (uk (z)) = e!h ?n : By continuity, there is 1 ¿ 0 such that z + 1 ¡ B and, for h ∈ [0; 1 ), #n gk (z; uk (z)) − U (h)gk (s; uk (s)) ¡ 2 for all s ∈ [z; z + 1 ). Integrating in the interval [z; z + h] we get  z+h #n hgk (z; uk (z)) − U (z + h − s)gk (s; uk (s)) ds ¡ h 2 z for all h ∈ [0; 1 ). In view of (41) d◦D (uk (z); gk (z; uk (z))) ¡ − #n : Hence, by (44), lim sup

dD (U (h)uk (z) +

h→0+

6 lim sup h→0+

 z+h z

U (z + h − s)gk (s; uk (s)) ds) − dD (U (h)uk (z)) h

dD (U (h)uk (z) + hgk (z; uk (z))) − dD (U (h)uk (z)) #n + h 2

6 d◦D (uk (z); gk (z; uk (z))) +

#n ¡ − #n =2: 2

(43)

(44)

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Thus there is 0 ¡  ¡ 1 such that, for all h ∈ (0; ), by (42) and (43), dD (uk (z + h)) ¡ dD (U (h)uk (z)) − h#n =2 ¡ e!h ?n − h#n =2:

(45)

Observe that lim+

h→0

e!h ?n − ?n = !?n ¡ #n =2 h

by (26). Thus diminishing  if necessary we see that, for all h ∈ [0; ), by (45), dD (uk (z + h)) ¡ ?n : This contradicts the de4nition of z and shows that uk (t) ∈ Dn on [tQ; B]. Hence v(t) ∈ Dn for all t ∈ [tQ; B]. This again contradicts the de4nition of tQ and completes the proof of Claim 5. ˜ 0 ) = L(x0 ) Claim 6. For each n ¿ 1, the set Ln (x0 ) is contractible. Consequently L(x is an R -set. Recall Claim 5 and de4ne a homotopy h : [0; 1] × Ln (x0 ) → C(J; E) by the formula  u(s) if s ∈ [0; =T ]; h(=; u)(s) := v(s; =T; u(=T )) if s ∈ [=T; T ] for u ∈ Ln (x0 ) and = ∈ [0; 1]. Since the semigroup {U (t)} is strongly continuous and fn (t; ·) is locally Lipschitz it is easy to see that the solution vn (·; t0 ; y0 ) of Eq. (38) depends continuously on t0 and y0 and therefore h is continuous. Moreover, in view of Claims 4 and 5, we have that h([0; 1] × Ln (x0 )) ⊂ Ln (x0 ). From the continuity of h, we infer that h([0; 1] × Ln (x0 )) ⊂ Ln (x0 ). Finally observe that h(0; u) = v(·; 0; x0 ) and h(1; u) = u for every u ∈ Ln (x0 ), i.e. Ln (x0 ) is contractible. This ends the proof of Theorem 16. 3.4. Convex case If D is convex, then condition (16) is too strong for our purposes. It is well known that, for a convex closed subset X of a normed space Y and x ∈ X , CX (x) = TX (x) = SX (x);

(46)

where SX (x) :=

 X −x : h

(47)

h¿0

Therefore, the cone TX (x) is convex (and closed). Recall (see [5, Theorem 4.2.2]) that the cone-map X  x → TX (x) ⊂ Y is lsc. Below we shall make a comment on this result (see Remark 19).

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Our improvement concerning known results on the structure of the set of solutions living in convex sets in Banach spaces (comp. [14,8]) is based upon the following new result. Lemma 17. Let (; d) be a metric space. Suppose that ’ :  × X ( Y , where X and Y are as above, is an usc map with closed convex values. If, for each ! ∈  and x∈X, ’(!; x) ∩ TX (x) = ∅

(48)

then, for any  ¿ 0, there exists a locally Lipschitz map f :  × X → Y such that ∀ x ∈ X; ! ∈ 

f(!; x) ∈ ’(B (!; ) × BX (x; )) + BY (0; )

(49)

and ∀x ∈ X; ! ∈ 

f(!; x) ∈ TX (x):

(50)

Remark 18. If Y is Banach space, then the existence of a continuous (single-valued) map f :  × X → Y satisfying conditions (49), (50) follows from a general result due to Ben-El-Mechaiekh and Kryszewski (see [10]) stating that given a metric space Z, an arbitrary lsc map & : Z ( Y and an usc map ’ : Z ( Y (both with closed convex values) such that &(z) ∩ ’(z) = ∅ on Z, for any  ¿ 0, there is a continuous mapping f : Z → Y such that f(z) ∈ ’(BZ (z; )) + BY (0; ) and f(z) ∈ &(z), z ∈ Z. However, in our case we need no completeness and we may improve this result obtaining a locally Lipschitzian mapping. Proof. Take  ¿ 0 and (!; x) ∈  × X . There is v(!; x) ∈ Y such that v(!; x) ∈ [’(!; x) + BY (0; =4)] ∩ SX (x)

(51)

in view of (48) and (46). Hence, by (47), there is (!; x) ¿ 0 such that x + (!; x)v(!; x) ∈ X:

(52)

By upper semicontinuity choose a number (!; x), 0 ¡ (!; x) ¡ =4 such that ’(BX (x; 2(!; x))) ⊂ ’(!; x) + BY (0; =2)

(53)

and a number (!; x), 0 ¡ (!; x) ¡ min{(!; x); (!; x)=(!; x)}. Let {=s :  × X → [0; 1]}s∈S be a locally 4nite locally Lipschitzian partition of unity re4ning the open cover {B (!; (!; x)(!; x)) × BX (x; (!; x)(!; x))}(!; x)∈×X . For any s ∈ S, there is !s ∈  and xs ∈ X such that supp =s ⊂ B (!s ; s s ) × BX (xs ; s s ) where we have put s := (!s ; xs ) and s := (!s ; xs ). Additionally let us set vs := v(!s ; xs ) and s := (!s ; xs ). For any s ∈ S, we de4ne a map fs : X → Y by the formula fs (x) :=

1 (xs − x) + vs ; s

x ∈ X:

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Observe, that for s ∈ S, x ∈ X , x + s fs (x) = xs + s vs ∈ X in view of (52); hence fs (x) ∈ SX (x) ⊂ TX (x):

(54)

It is clear that fs , s ∈ S, is Lipschitz continuous (with the Lipschitz constant s−1 ). Now we de4ne f :  × X → Y by the formula  =s (!; x)fs (x); x ∈ X: f(!; x) := s∈S

Observe that f is locally Lipschitz because so are all functions =s , fs for s ∈ S, and the covering {supp =s }s∈S is locally 4nite. Moreover, since, for (!; x) ∈  × X , f(x) is a (4nite) convex combination of vectors fs (x) ∈ TX (x) (see (54)) and since TX (x) is convex, we get condition (50). Take (!; x) ∈  × X and let S(!; x) = {s ∈ S | (!; x) ∈ supp =s }. It is clear that S(!; x) is a 4nite set and  =s (!; x)fs (x): f(!; x) = s∈S(!; x)

For any s ∈ S(!; x), we have (!; x) ∈ supp =s ⊂ B (!s ; s s ) × BX (xs ; s s ), i.e. max{d(!; !s ); x − xs } ¡ s s ¡ s

and

fs (x) − vs  ¡ s ¡ s :

(55)

There is s0 ∈ S(!; x) such that s0 = maxs∈S(!; x) s . If s ∈ S(!; x), then by (55), max{d(!s ; !s0 ); xs − xs0 } 6 max{d(!s ; !); xs − x} + max{d(!s0 ; !); xs0 − x} ¡ s + s0 6 2s0 :

(56)

Therefore, by (55), (51), (56) and (53), for any s ∈ S(x), fs (x) ∈ BY (vs ; s0 )⊂’(!s ; xs )+BY (0; =4 + s0 )⊂’(B (!s0 ; 2s0 )×BX (xs0 ; 2s0 )) + BY (0; =4 + s0 ) ⊂ ’(!s0 ; xs0 ) + BY (0; =4 + =2 + s0 ) ⊂ ’(!s0 ; xs0 ) + BY (0; ): Hence, by convexity of ’(!s0 ; xs0 ) + BY (0; ) and by (55), f(!; x) ∈ ’(!s0 ; xs0 ) + BY (0; ) ⊂ ’(B (!; s0 ) × BX (x; s0 )) + BY (0; ) ⊂ ’(B(!; ) × BX (x; )) + BY (0; ): This establishes property (49) and ends the proof.

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Remark 19. Note that in the course of the proof we have not used the lower semicontinuity of TX (·). Instead we have applied the following, astonishingly simple, observation (already employed in a di!erent situation in [19]). If X is a convex closed set in a normed space Y , then for every x0 ∈ X , v0 ∈ SX (x0 ) and 0 ¿ 0 such that x0 + 0 v0 ∈ X (existing in view of the very de@nition of SX (x0 )), an aDne mapping g(x) = 1=0 (x0 − x) + v0 , x ∈ X , provides a selection of SX (x). This proves the lower semicontinuity of both SX (·) and TX (·). Theorem 20. Suppose that E is separable, D is convex and (6) is satis@ed. The set of mild solutions L(x0 ) of the initial value problem (5) is an R -set in C(J; E) provided that (7) is ful@lled or the semigroup U is compact. Proof. Step 1: First let us assume additionally that F has the Scorza–Dragoni property (i.e. is almost usc in the sense of Deimling [21, De4nition 3.3]), i.e. for any ? ¿ 0, there is a closed J? ⊂ J with |J \ J? | 6 ? such that the restriction F|J? × D is upper semicontinuous. Claim. For any n ¿ 1, there is a continuous map fn : J × D → E such that fn (t; ·) is locally Lipschitz, for all t ∈ J , x ∈ D, fn (t; x) ∈ TD (x)

(57)

fn (t; x) ∈ Fn (t; x) := conv F(B((t; x); n−1 ) ∩ J × D) + B(0; n−1 ):

(58)

and

Indeed, 4x n and let Jn := J(2n)−1 . By Lemma 17, there is a locally Lipschitz map ˜ x) ∈ F(B((t; x); (2n)−1 ) ∩ Jn × D) + f˜ : Jn × D → E such that, for all (t; x) ∈ Jn × D, f(t; ˜ x) ∈ TD (x). B(0; n−1 ) and f(t; Since J \ Jn is open, we have  J \ Jn = (ai ; bi ); i¿1

where intervals (ai ; bi ) are disjoint. Similarly as in (40) we de4ne fn : J × D → E by  ˜ x)  f(t; if t ∈ Jn ; fn (t; x) = ˜ ˜  (bi −t)f(ai ; x)+(t−ai )f(bi ; x) if t ∈ (a ; b ): i i bi −ai and x ∈ D. It is clear that fn is jointly continuous and locally Lipschitz with respect to the second variable. Moreover properties (57), (58) are satis4ed. Clearly F(t; x) ⊂ Fn (t; x) ⊂ Fn+1 (t; x) for all t ∈ J , x ∈ D and n ¿ 1; thus  ∅ = Ln (x0 ) ⊃ Ln+1 (x0 ) and L(x0 ) ⊂ Ln (x0 ); n¿1

where Ln (x0 ) denotes the set of solutions of (5) where F is replaced by Fn .

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Step 2: Let un ∈ Ln (x0 ) for all n ¿ 1. Exactly by the same methods as presented in the appendix, we show that there exists a subsequence (unk )k¿1 such that unk → u0 ∈ L(x0 ) in C(J; E) as k → ∞. If follows that L(x0 ) =

∞ 

Ln (x0 );

n=1

0 (Ln (x0 )) → 0

as n → ∞:

Next, as in Claim 6 (in the proof to Theorem 16) we show that Ln (x0 ) is contractible to the unique solution of the equation analogous to (38) (with t0 = 0). 3 This completes the proof of our result in case F is almost usc. Step 3: Now we return to the general F (satisfying assumption (B)). There exists F0 : J × D → 2E with compact convex values satisfying conditions (i), (ii) of Theorem 1 and, given  ¿ 0, there is a closed J˜  ⊂ J with |J \ J˜  | ¡ =2 such that F0 |J˜  × D is usc (with non-empty values). Next we choose a closed J ⊂ J˜   with |J \ J | ¡  such that each point t ∈ J is a density point of J˜  . We thus have J = m¿0 Jm with |J0 | = 0, Jm closed, |J \Jm | ¡ m−1 and F0 |Jm ×D usc for m ¿ 1. Rede4ne F0 (t; x)={0} for t ∈ J0 , x ∈ D. Now F0 satis4es assumptions (B1), (B3) and (B4); moreover it is almost usc. It satis4es (6): if w is a measurable selection of F(·; x)∩TD (x), then w(t) ∈ F0 (t; x)∩TD (x) a.e. on J , hence F0 (t; x) ∩ TD (x) = ∅ on Jm for all m ¿ 0 because of (B4) and our choice of Jm . Clearly F0 satis4es also (7) (if so does F). Steps 1 and 2 imply that the set L0 (x0 ) of all mild solutions to (5) (with F0 replacing F) is an R -set. But at the same time we see that L(x0 ) = L0 (x0 ). This ends the proof. 3.5. Epi-Lipschitz case We shall now deal with an epi-Lipschitz set D. In this case D is regular and Theorem 16 applies. However, we can improve this result for the uniform tangency may be replaced by (10) (in case F is single valued, by (8)) and we do not require that U is non-expansive. Theorem 21. Suppose that E is separable, D is an epi-Lipschitz set and (10) is satis@ed. For each x0 ∈ D, the set L(x0 ) of all mild solutions to the initial value problem (5) is an R -set provided the compactness condition (8) is satis@ed or the semigroup U is compact. One may replace (10) by (6) in case F is single valued. Proof. Let us start with the last statement. It appears that is this situation (6) ⇒ (10). Indeed, on should deal only with x ∈ @D (for x ∈ int D, CD (x) = TD (x) = E). Take a sequence (yn ) in D such that yn → x. Then, for all t ∈ J , F(t; yn ) ∈ TD (yn ) and, in

3

Due to Theorem 2, for any t0 ∈ [0; T ) and y0 ∈ D, v(·; t0 ; y0 ) exists on [t0 ; T ].

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view of [5, Theorem 4.1.9], F(t; x) = lim F(t; yn ) ∈ Lim inf TD (y) = CD (x) n→∞

D

y→x

and the 4rst part of the theorem applies. Step 1: We shall show that, for any integer n ¿ 1, there is a map fn : J × D → E such that fn (·; x) is measurable, fn (t; ·) is locally Lipschitz, fn (t; x) ∈ CD (x) and fn (t; x) ∈ Fn (t; x) := conv F({t} × BD (x; n−1 )) + B(0; n−1 )

(59)

for all t ∈ J and x ∈ D. The construction follows that from Claim 4 in the proof of Theorem 16. Recall Propositions 10 and 11, 4x n ¿ 1 and take x ∈ @D. Since CD (x) = @#D (x)− , there is a measurable function vx : J → E such that, for all t ∈ J , vx (t) ∈ F(t; x) and #◦D (x; vx (t)) 6 0: Since 0 ∈ @#D (x) and by (9), we have inf u61 #◦D (x; u) ¡ 0; hence, there is uxn ∈ E with uxn  = 1=2n such that #◦D (x; uxn ) ¡ 0: By (B4), the set {vx (t) | t ∈ J } is relatively compact; thus there is a simple (i.e. measurable and having 4nite number of values) function vQnx : J → E such that vQnx (J ) ⊂ vx (J ) and vx (t) − vQnx (t) ¡ 1=2n. Hence #◦D (x; vQnx (t)) 6 0 on J . Let wxn (t) := vQnx (t) + uxn for t ∈ J . Then wxn : J → E is measurable (and has 4nite number of values). By the convexity of #◦D (x; ·), #◦D (x; wxn (t)) ¡ 0: Since wxn admits a 4nite number of values and, for each w ∈ E, the function #◦D (·; w) is upper semicontinuous, there is rxn ∈ (0; n−1 ) such that, for all y ∈ B(x; rxn ), #◦D (y; wxn (t)) ¡ 0

for all

t ∈ J:

If x ∈ int D, then we choose rxn ∈ (0; n−1 ) such that B(x; rxn ) ⊂ int D and let wxn be an arbitrary measurable selection of F(·; x). For any n ¿ 1, we have constructed an open covering {BD (x; rxn )}x∈D of D; let {=s : D → [0; 1]}s∈S be a locally 4nite partition of unity (depending on n) inscribed into this cover. Therefore, for any s ∈ S, there is xs ∈ D such that supp =s ⊂ B(xs ; rxns ). For s ∈ S, let ws = wxns . Finally we de4ne  fn (t; x) := =s (x)ws (t); t ∈ J; x ∈ D s∈S

and easily show that fn satis4es required conditions. 4 4 It is also easy to see that f (t; ·) is locally Lipschitz uniformly with respect to t ∈ J and each point n x ∈ D has a neighborhood W such that fn (J × W ) lies in a compact subset of E.

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Step 2: Now we proceed as in the proof of Theorem 20 (or Theorem 16): we show that ∞  Ln (x0 ); L(x0 ) = n=1

where Ln (x0 ) is the set of all mild solutions to (5) (with F replaced by Fn ) and, then we show that Ln (x0 ) is contractible to a unique solution of the equation analogous to (38)—see also footnote 3. Since 0 (Ln (x0 )) → 0 as n → ∞, we end the proof. The next result corresponds in a sense to Example 15. Corollary 22. Suppose that E is separable and that F : J × ( E satis@es assumption (B) (with D replaced by ) where  is a fat neighborhood of the strictly regular set D (appearing in De@nition 7). If ∀y ∈  \ D ∀t ∈ J

F(t; y) ∩ @dD (y)− = ∅;

(60)

then, for any x0 ∈ D, the set L(x0 ) is an R -set provided condition (7) (with D replaced with ) holds or the semigroup U is compact. It seems that condition (60) has nothing to do with (10). (60) together with (B2) and (B4) imply rather that, for all x ∈ D, t ∈ J , F(t; x) ∩ Lim inf @dD (y)− = ∅ E\D

y→x

but @dD (x)− = CD (x) = Lim inf @dD (y)− E\D

y→x

(the inclusion ⊂ follows from [19, Remark 5.14 (2)], while the inclusion ⊃ may be obtained by the use of arguments similar to those from the proof of [5, Theorem 4.1.9]). Proof. According to De4nition 7, there is # ¿ 0 such that |@dD (y)| ¿ # for all y ∈  \ D. For any n ¿ 1, de4ne ?n ¿ 0 such that ?n+1 ¡ ?n and !?n ¡ #n =4 where #n := #=2n, and let Dn := {y ∈ E | dD (y) 6 ?n } and Dn
:= {y ∈ E | dD (y) 6 2?n }. Without loss of generality we may assume that Dn
+ B(0; n−1 ) ⊂  for all n. Similarly as in Step 1 of the proof of Theorem 21 (see also footnote 4), for any n, we construct a map fn : J ×Dn
→ E such that: for any t ∈ J and x ∈ Dn
, f(·; x) is measurable, f(t; ·) is locally Lipschitz uniformly with respect to t, each point has a neighborhood W with fn (J × W ) relatively compact and fn (t; x) ∈ Fn (t; x) := conv F({t} × B(x; n−1 )) + B(0; n−1 ). Moreover we may demand that d◦D (x; fn (t; x)) ¡ − #n whenever ?n =2 6 dD (x) 6 2?n . To see this: in Step 1 of the proof of Theorem 21, replace @D by the closed set {x | ?n =2 6 dD (x) 6 2?n } and, for any x from this set, take uxn with uxn  ¡ 1=2n such that dD (x; uxn ) ¡−#n ; then take a selection vx of F(·; x)∩@dD (x)− , a 4nite-valued vx and proceed as in the mentioned proof. Next, using arguments similar to those used in Claim 5 of the proof to Theorem 16, we show that, for any t0 ∈ [0; T )

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and y0 ∈ Dn , a unique solution v(·; t0 ; y0 ) to the equation analogous to (38) exists on the whole interval [t0 ; T ]. Hence, for any x0 ∈ D, the set Ln (x0 ) of all mild solutions to (5) (with F replaced  by Fn and D with Dn ) is contractible to v(·; 0; x0 )). It is again clear that L(x0 ) = n¿1 Ln (x0 ) and 0 (Ln (x0 )) → 0 as n → ∞. This completes the proof. Remark 23. The above Theorems 20, 16, 21 and Corollary 22 seem to be the 4rst results proving the R -structure of the set of mild solutions to (5) in Banach spaces satisfying the constraint u(t) ∈ D with D closed and e.g. convex without any additional assumptions (such as e.g.: D has non-empty interior or special Banach spaces are considered) or being merely (strictly) regular. Therefore, our results generalize those given in [12,34] and recent results in [8,9]. Remark 24. We shall discuss shortly the relevance of the separability of E. It appears that we can do without it if one assumes more about F (assumptions (A) and (B) are still sustained): 1. Let us consider 4rst the convex case: suppose that E is arbitrary, D is convex, F is usc 5 and other hypotheses of Theorem 20 are ful4lled. We easily see that the assertion of this theorem holds true: the separability of E (and (B4)) were necessary only to proceed with Step 3 of the proof to Theorem 20. The continuity of the constructed approximation fn allows to use Remark 3 instead of Theorem 2 in order to establish the existence of a solution to Eq. (38). 2. A similar argument helps to deal with the epi-Lipschitz case (or in the context of Corollary 22). For instance if all hypotheses of Theorem 21 are satis4ed but E is arbitrary and F is usc, then the assertion of Theorem 21 is also true. To see this take x from the epi-Lipschitz set D; in view of Remark 18, for any n ¿ 1, there is a continuous vxn : J → E such that vxn (t) ∈ CD (x) and vxn (t) ∈ F((t − n−1 ; t + n−1 × {x}) + B(0; (2n)−1 ). If x ∈ @D; then #◦D (x; vxn (t)) 6 0 on J . De4ne wxn (t) := vxn (t) + uxn . Then #◦D (x; wxn (t)) ¡ 0 on J . Since the function #◦D (·; wxn (·)) is upper semicontinuous on J × D, there is 0 ¡ r(t; x) ¡ n−1 such that #◦D (y; wxn (M)) ¡ 0 for all (M; y) ∈ B((t; x); r(t; x)) ∩ J × D. If x ∈ int D, then we choose 0 ¡ r(t; x) := r(x) ¡ n−1 such that B(x; r(x)) ⊂ int D and put wxn := vxn . Next we take a lointo the cally Lipschitz partition of unity {=s : J × D → [0; 1]}(t; x)∈J ×D inscribed  cover {B((t; x); r(t; x)) ∩ J × D}(t; x)∈J ×D and de4ne fn (t; x) := s∈S =s (t; x)ws (t) where ws = wxs and supp =s ⊂ B((ts ; xs ); r(ts ; xs )). One easily veri4es that fn is continuous, locally Lipschitz with respect to the second variable, fn (t; x) ∈ CD (x) and fn (t; x) ∈ conv F(B((t; x); n−1 )) + B(0; n−1 ). Then we proceed as in the proof of Theorem 9 making again the use of Remark 3. 3. Finally let us discuss Theorem 16. Suppose that in its setting F is usc. In Claim 1 (of the proof to this theorem), using Remark 18, we construct a continuous z = zx; y : ˜ J → E such that, for all t ∈ J , z(t) ∈ Fn
(t; x) := F(B(t; x); n−1 ) + B(0; n−1 ) and d◦D (y; z(t)) ¡ − #n . In Claim 3 we construct a continuous w = wx : J → E such that, 5

In this case we may assume also that F is merely almost usc.

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for all t ∈ J , w(t) ∈ Gn
(t; x) (where Gn
is de4ned as Gn through Fn
). Next, in Claim 4 we construct a continuous (and locally Lipschitz with respect to the second variable) selection fn : J × Dn
→ E of Hn
where Hn
(t; x) = conv Gn
(B((t; x); n−1 )) + B(0; n−1 ) To this end we use arguments similar to those presented above in 2. Then we proceed as in the proof to Theorem 16 by evoking Remark 3 (obviously we do not need any “regularization” of the form (40) for f = fn is already continuous). 4. The periodic problem and equilibria In this section, we will discuss the implications of the results proven above to the periodic problem, the existence of equilibria and 4xed point problems. Recall assumptions (A) and (B). We shall deal with the periodic problem considered as the two point boundary value problem 
u (t) ∈ Au(t) + F(t; u(t)); (61) x(0) = x(T ): By a solution to (61) we mean a mild solution to (1) satisfying u(0) = u(T ) (and u : J → D). In view of Theorem 2, if E is separable (or F is usc), (6) and (7) hold (or the semigroup U is compact), then we associate with (61) the set-valued Poincar(e translation operator along trajectories P = PT : D ( D given by P := eT ◦ L, where L : D ( C(J; D) is the solution operator and eT : C(J; D) → D is the evaluation mapping eT (x) := x(T ). Observe that P maps bounded sets onto bounded ones. Clearly the existence of periodic solutions is equivalent to the existence of 4xed points of P. In the present section, we will be interested in compactness properties of the operator P. In order to do so we assume additionally that (A4) there is N ∈ R such that, for any t ¿ 0, U (t) := (U (t)B(0; 1)) 6 exp(Nt): Observe that since U (t) 6 U (t), the number N always exists and N 6 !. Recall that for a given bounded set  in E the estimate (U (t)) 6 U (t) () holds. We shall work with the MNC ˜0 given by ˜0 () := sup{(C) | C ⊂  countable} for each bounded subset  of E. It can be shown that ˜0 is regular, monotone and non-singular (see [1, Section 1.4]). Using these notions we can formulate the following theorem improving some results given by the 4rst author in [7]. Theorem 25. Assume that P is well de@ned and let  be a bounded subset of E. (i) If the semigroup U is compact, then P() is relatively compact. Thus P is a compact operator.

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(ii) Suppose that (A4) and (8) hold (but we do not assume that the semigroup U is compact). Then • if E is separable, then (P()) 6 exp(NT + kL1 )(); • if E is a weakly compactly generated, then ˜0 (P()) 6 exp(NT + 2kL1 )˜0 (); • if E is arbitrary, then ˜0 (P()) 6 exp(NT + 4kL1 )˜0 (): In the proof we will apply the following Lemma (see [39, Corollary 4.2.5], comp. [22, Proposition 9.3]). Lemma 26. Let (fn ) be a sequence L1 ([a; b]; E). Assume that (i) fn (t) 6 ((t) for all n ¿ 1 and a.e. t ∈ [a; b] where ( ∈ L1 ([a; b]), (ii) ({fn (t)}) 6 c(t) for a.e. t ∈ [a; b] where c ∈ L1 ([a; b]). Then we have the estimate 
 t   t  fn (s) ds 6q c(s) ds a

a

for each t ∈ [a; b] with q = 1 if E is separable and q = 2 in general case. Proof of Theorem 25. (i) Let x ∈  and y ∈ P(x). Then there exist w ∈ NF (u) and u ∈ L(x) such that  T −  T y = U (T )x + U () U (T −  − s)w(s) ds + U (T − s)w(s) ds; (62) 0

T −

where we have chosen an arbitrary  ¿ 0. Since  is bounded and using assumption (B3), there exists R ¿ 0 such that u(t) 6 R on J for each u ∈ P(). T Let  ¿ 0. Choose  ¿ 0 such that e!T (1 + R) T − c(s) ds ¡ . By (62), (P()) 6 (U (T ))   + U () !T

+e

T −

0

 (1 + R)

T

T −

 U (T −  − s)w(s) ds|w ∈ NF (u); u ∈ L() c(s) ds ¡ 

since U () and U (T ) are compact, linear mappings. Thus (P()) = 0. (ii) Let E be an arbitrary Banach space. Let set C ⊂ P() be countable; arrange its elements into a sequence (yn ). Let x n ∈  be such that yn ∈ P(x n ). There exist

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un ∈ L(x n ) and fn ∈ NF (un ) such that  t un (t) = U (t)x n + U (t − s)fn (s) ds;

t ∈J

0

and yn = un (T ); n ¿ 1. De4ne (t) := ({un (t)}n¿1 );

(63)

: J → [0; ∞) by t ∈ J:

By properties of  and (63) we see that, for t ∈ J , 
 t  (t) 6 ({U (t)x n }n¿1 ) +  U (t − s)fn (s) ds 0



n¿1

:

 Let Y be the closed linear subspace spanned by n¿1 un (J ∩ Q); it is clear that Y is separable. We denote the Hausdor! MNC w.r.t. Y by Y . Since  6 Y 6 2 and in view of (A4) and assumption (7) we obtain ({U (t − s)fn (s)}n¿1 ) 6 eN(t−s) ({fn (s)}n¿1 ) 6 eN(t−s) k(s) (s) 6 eN(t−s) k(s)Y ({un (s)}n¿1 ): Using the separability of Y one can show that the function s → Y ({un (s)}n¿1 is measurable (see [39, Corollary 4.2.4], [22, Proposition 9.2]). Since it is also bounded we infer that the function on the right-hand side of the above inequality is integrable. Applying Lemma 26 we obtain 
 t    t  62 U (t − s)fn (s) ds eN(t−s) k(s)Y ({un (s)}n¿1 ) ds (64) 0

and thus (t) 6 eNt ({x n }n¿1 ) + 2

0

n¿1

 0

t

eN(t−s) k(s)Y ({un (s)}n¿1 ) ds:

Now de4ne % : J → [0; ∞) by the right-hand side of the above inequality. Clearly, % is absolutely continuous and we have for a.e. t ∈ J , %
(t) = N%(t) + 2k(t)Y ({un (s)}n¿1 ) 6 N%(t) + 4k(t) (t) 6 (N + 4k(t))%(t): (65) By the Gronwall inequality,

   t (t) 6 %(t) 6 ({x n }n¿1 )exp Nt + k(s) ds : 0

Since (T ) = ({yn }n¿1 ) we have shown that for an arbitrary sequence (yn ) in P() there exists a sequence (x n ) in  such that ({yn }n¿1 ) 6 exp(NT + 4kL1 )({x n }n¿1 ): So we obtain the desired result in the case of a Banach space.

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In case E is a weakly compactly generated it is known that there exists a closed and separable subspace Y0 such that Y ⊂ Y0 ⊂ E and a projection P : E → Y0 with P = 1 (see [24, p. 149]). Applying the arguments in the proof on Y0 instead of Y and using the obvious equality (t) = Y0 ({un (s)}n¿1 ) it is clear that the additional factor 2 in estimate (65) can be avoided. In case that E is separable the passage to Y is not necessary and estimation (64) holds without factor 2. Also in this case it is clear that  = ˜0 . These remarks yield our claim in the remaining cases. This result gives us means to establish conditions necessary for P to be a (-set contraction. For instance if E is an arbitrary Banach space and NT + 4kL1 ¡ 0, then P (if de4ned) is a (-set contraction with ( := exp(NT + 4kL1 ). In the situation of Theorems 16, 20, 21 or Corollary 22 (see also Remark 24), the Poincar@e operator P has a special structure. Namely it is a so-called decomposable map (de4ned in [40]; see also [41] and [30]). Let X be a metric space. A set-valued map & : X ( X is called decomposable if there is a metric ANR (absolute neighborhood retract) M , a usc set-valued map ’ : X ( M such that for every x ∈ X the set ’(x) is an R -subset in M and a continuous single-valued map f : M → X such that &=f ◦’. It is clear that if, for any x0 ∈ D, L(x0 ) is de4ned and is an R -set, then P is a decomposable map. Notice also that then P is homotopic (through a decomposable homotopy) to the identity idD : D → D; indeed the homotopy is provided by the composition D × [0; 1]  (x0 ; =) → P(x0 ; =) = e=T ◦ L(x0 ) where e=T is the evaluation e=T (u) := u(=T ) for u ∈ C(J; D); then P(x0 ; 0) = x0 and P(x0 ; 1) = P. The class of decomposable maps falls into a much broader class of the so-called admissible maps (see [29] and details given therein) particularly well designed for the 4xed-point problems. Roughly speaking an usc map & : X ( X is admissible if it admits a multivalued selection being a 4nite composition of acyclic maps (recall that a map is acyclic if it is usc and has non-empty compact acyclic 6 values). Clearly a decomposable map & = f ◦ ’, where ’ is as above, is admissible (since so is ’ having R -values). We have the following 4xed point result. Proposition 27 (G@orniewicz [29, Comp. Chapter V]). Let X be an ANR 7 and let a decomposable map & : X ( X be homotopic to the identity. (i) If X is compact and Q(X ) = 0, 8 then & has a @xed point. (ii) If X is acyclic (e.g. contractible) and & is compact (here it means that &(X ) is compact), then & has a @xed point. 6 A topological space Z is acyclic if it has the same rational Cech U homology with compact supports—see U [29] (or the rational Cech cohomology)—as a one-point space. 7 If X is closed in E, then it means that X is a neighborhood retract, i.e. there is a retraction r :  → D where  is a neighborhood of D. 8 Q(X ) stands for the Euler characteristic of X de4ned in terms of the rational Cech U homology with compact supports. For any compact ANR X , Q(X ) is a well-de4ned integer. In particular, if X is a compact absolute retract, then Q(X ) = 1.

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4.1. Compactness methods Let us 4rst observe that in Theorems 16, 20, 21 and Corollary 22, assumptions concerning D always imply that D is a regular sets. In [19] it was shown that strictly regular sets are (absolute) neighborhood retracts (one constructs a neighborhood retraction r :  → D via some variational arguments); so are compact regular sets. If D is epi-Lipschitz, then D, by the very de4nition, each point has a neighborhood which is an ANR; hence—by the Hanner theorem (see e.g. [11, Theorem 5.1])—D is an ANR. In particular Q(D) is well-de4ned whenever D is compact. Proposition 27 applies and we get the following results. Theorem 28. Suppose that D ⊂ E is compact and that one of the following conditions hold: (i) D is regular (in particular epi-Lipschitz), Q(D) = 0 and (10) 9 or (60) is satis@ed; (ii) D is convex and (6) holds Then problem (61) admits a solution, i.e. there is a periodic mild solution to (1). Proof. By the results of Section 3, we see that P is a decomposable map (here we do not need separability: since D is compact we may restrict the attention to the separable subspace spanned by D). It is also compact since so is D. Thus Proposition 27(i) completes the proof. Note that if F is single valued, then (10) in (i) may be replaced by (6). In the special case A = 0 the above theorem gives a positive answer to open problem 13.1 in [12]. We now would like to dispense with the assumption on the compactness of D. However, given a closed and bounded set D and a compact map f : J × D → E, the translation operator along trajectories P : D ( D associated with the equation x
=f(t; x) is easily seen to be only a 1-set-contraction and therefore Proposition 27 does not apply. Indeed, Deimling has given an example of such a map without any periodic solutions (see [20]; we recall this example in Remark 40 below). On the other hand a positive result can be obtained in the present case of the semilinear system provided certain compactness conditions on the semigroup are employed. Theorem 29. Assume that the semigroup U is compact and that D is bounded and acyclic. If E is separable and: (i) D is strictly regular and (16) (or (60)) is satis@ed, or; (ii) D is convex (i.e. automatically contractible and, thus, acyclic) and (6) holds, or; 9

Recall that if D is compact, then (10) ⇒ (16).

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(iii) D is epi-Lipschitz and (10) holds, then problem (61) admits a solution. The same assertion holds without separability of E if, instead of (B2), F is ( jointly) usc. Proof. By the results of Section 3, and Theorem 25, P is a decomposable compact map (i.e. P(D) is relatively compact). Hence Proposition 27 applies. The compactness assumption on U may still be relaxed. Theorem 30. Recall assumptions (A4), (7), (6) and assume that D is convex bounded. If E is separable and NT ¡ − kL1 , then (61) has a solution. The same is true if F is almost usc and (i) E is arbitrary and NT ¡ − 4kL1 , or; (ii) E is weakly compactly generated and NT ¡ − 2kL1 . Proof. Theorems 20 (see also Remark 24 1.) and 25 imply that P is a decomposable (-set contraction (with respect to , if E is separable and with respect to ˜0 if E is arbitrary or weakly compactly generated) with ( ¡ 1. In order to complete the proof we shall evoke the following result (see e.g. [6,39]): If D it is convex, closed and bounded,  : D ( D is a decomposable -set contraction with respect to a (non-singular, regular and monotone) MNC  with  ¡ 1, then  has a &xed point. Remark 31. In fact we can do much better. However, the precise proof of this result would involve too much space and, therefore, we shall restrict ourselves to its rough description. Namely following Nussbaum [45], let us suppose that a bounded and closed  set D ∈ F, i.e. assume that D = i∈I Di where {Di }i∈I is a locally 4nite family of closed convex subsets of E. Combining methods of [44] with those from [29] and given and decomposable (-set contraction (with ( ¡ 1) & : D ( D, one is in a position to de4ne an integer valued invariant i(&; D) such that if i(&; D) = 0, then & possesses a 4xed point. Then one shows (see e.g. [45, Section E, Theorem 4] for a hint) that if D is acyclic, then i(&; D) = 1. Having this result we easily see that one may replace the convex bounded D from Theorem 30 by an acyclic closed D ∈ F. Then assuming that D is strictly regular, 10 E is separable and condition (16) is satis4ed, we obtain a generalization of Theorem 30. Observe that in Theorem 30 implicitly N ¡ 0 (this holds e.g. if ! ¡ 0). In the next result we consider also the case ! = 0. 10 It is, for instance, not diPcult to get conditions implying that a @nite union of closed convex sets is strictly regular.

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Theorem 32. Let the assumptions in front of Theorem 30 be ful@lled. Then the periodic problem (61) has a mild solution provided 0 ∈ D, ! 6 0, F is compact (i.e. F(J × D) is compact) and 1 ∈ %(U (T )). Proof. For  ¿ 0 we consider the equation y
∈ Ay−y+F(t; y). Then the C0 -semigroup U := {U (t)}t¿0 , where U (t) = e−t U (t), generated by A − I satis4es U (t)D ⊂ D, for any t ¿ 0, since 0 ∈ D. By Theorem 30 we thus get the existence of a periodic solution u to the perturbed equation. Since the resolvent set %(U (T )) is open we have the invertibility of I − U (T ) for  ¿ 0 suPciently small. Thus the following representation holds: u (t) = U (t)(I − U (T ))

−1

 0

T

 U (T − s)w (s) ds +

0

t

U (t − s)w (s) ds

(66)

for each t ∈ J with some w ∈ NF (u ). Let n  0. Since F is compact and using (66), the Arzela–Ascoli theorem shows that {un : n ¿ 1} is relatively compact in C(J; E). Hence, without loss of generality un → u ∈ C(J; E). Clearly, u is a mild solution to (61). Remark 33. 1. In view of our improvement in the characterization of the solution sets of the initial value problem, the above theorems contain the results on periodic solutions given in [8] as particular cases. Specialized to single-valued maps our Theorems 30, 32 improve Theorems 3 and 4 in [50], where as additional conditions “D has non-empty interior” and “the metric projection on D exists,” respectively, were considered. 2. We may formulate the following periodic existence theorem of the Browder type. Suppose now that E is a separable Hilbert space with the inner product ·; · . If assumptions (A1), (A2), (B), (7) are satis4ed and there is r ¿ 0 such that ∀t ∈ J; x ∈ E; x = r ∃z ∈ F(t; x)

z; x 6 0;

(67)

then the periodic problem (61) has a mild solution provided !T + kL1 ¡ 0. To see this let D := B(0; r). Then assumption (67) means exactly that F(t; x) ∩ TD (x) = ∅ for every t ∈ J , x ∈ D. Since ! ¡ 0, the semigroup U is non-expansive and we also get that U (t)D ⊂ D for each t ¿ 0. Thus, Theorem 30(ii) shows the existence of the periodic solution. 3. We recall Example 24.12 from [21]: Let f : R × D → E and g : R × D → E be continuous, T -periodic in the @rst variable and such that (f({t} × )) 6 k1 (t)() for all bounded  ⊂ D and ∀x; y ∈ D

(g(t; x) − g(t; y); x − y)− 6 k2 (t)x − y2

where (·; ·)− denotes the semi-inner product (·; ·)− : E × E → R given by (x; y)− := y lim+ t→0

y − y − tx ; t

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f0 := f + g is uniformly continuous, bounded and satis@es (6); the functions k1 ; k2 ∈ T L1 (J ) and k := 0 (k1 (s) + k2 (s)) ds ¡ 0. In [21] existence of periodic orbits is shown in case D has non-empty interior. We may dispense with this assumption. As in the proof of Theorem 20 one shows that also in this situation the set of continuously di!erentiable solutions is an R -set. Moreover, the translation operator along the trajectories P : D ( D satis4es (P()) 6 ek () for  ⊂ D (see [21]). Hence, we may again apply the 4xed point result stated in the proof of Theorem 30 to show the existence of a 4xed point of P, i.e. a periodic solution. 4.2. Strongly continuous case In the above subsection, we showed that certain compactness conditions on the semigroup (or the nonlinearity F) are suPcient for the existence of periodic orbits. These conditions implicitly required that the linear part A is non-zero (except for Theorem 28 where D is compact). Now we shall deal with a non-compact domain not excluding the case A ≡ 0 but we shall specialize our discussion as concerns the nonlinearity. We say that a set-valued map & : D ( E is strongly upper semicontinuous (strongly usc) if, for every sequence (x n ) in D converging weakly to some x0 ∈ D (denoted by x n * x0 ) and any sequence (yn ) in E such that yn ∈ &(x n ) for all n ¿ 1, there is a subsequence (ynk ) converging (strongly) to an element y0 ∈ &(x0 ). The notion of a strongly usc set-valued map was 4rst introduced in [28] under the name of completely continuous maps (however, not in connection with di!erential equations or inclusions). Obviously strong upper semicontinuity implies upper semicontinuity. Example 34. Suppose that E
is a Banach space, let j : E → E
be a compact bounded linear map and let ’ : E
( E be usc. Then it is easy to see that & := ’ ◦ j|D : D ( E is strongly usc. For if x n * x0 ∈ D, then j(x n ) → j(x0 ) and the strong upper semicontinuity follows from the upper semicontinuity of ’. In particular, if E ,→,→ E
(i.e. E is compactly embedded into E
) and ’ : E
( E is usc, then & = ’|D is strongly usc. Our interest in strongly usc maps is motivated by the following 4xed point result. Proposition 35. Let E be a Banach space, D be a weakly compact, convex subset of E and let & : D ( D be a decomposable map. Assume there is a linear, bounded operator U : E → E, U  6 1, and a strongly usc map ’ : D ( E such that the following representation holds: &(x) = Ux + ’(x)

for every x ∈ D:

Then & has a @xed point. The proof of this result follows the arguments given in [6, Corollary 11]. Observe that in Proposition 35 one cannot replace the assumption of strong continuity of ’ by the assumption that ’ is usc and compact.

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Example 36. Consider E := ‘2 and D := B(0; 1) ⊂ ‘2 . Let U : E → E be given by U (x) := (0; x1 ; x2 ; : : :) and ’(x) := (1 − x; 0; 0; : : :). Then & := T + ’ : D → D but it has no 4xed points. 11 Theorem 37. In addition to the hypotheses enlisted in assumption (A) assume that the semigroup U is uniformly continuous and U (T ) 6 1. If E is separable, D is convex and weakly compact, F satis@es (B1), (B3), (6), (7) and, instead of (B2), the condition (B2)
for all x ∈ D, F(·; x) is measurable and, for any t ∈ J , F(t; ·) is strongly upper semicontinuous, then problem (61) has an absolutely continuous solution. If F is strongly usc (in both variables), then the separability of E is superFuous. As stated above the case A=0 is not excluded. Thus, in view of the remark following Theorem 28, compactness of the nonlinearity is not suPcient for the existence of periodic solutions. This is why the stronger assumption (B2)
is considered here. Proof of Theorem 37. We will apply Proposition 35. For this purpose let us introduce the map ’ : D ( E. 
 T U (T − s)w(s) ds | w(·) ∈ F(·; u(·)); u ∈ L(x) (68) ’(x) := 0

for x ∈ D, and let U := U (T ). One easily sees that P(x) = Ux + ’(x)

for every x ∈ D:

Our assumptions concerning F imply that all the assumptions of ful4lled (comp. Remark 24); hence P (and ’) is decomposable. representation (69), the theorem will be proved if we can show semicontinuity of ’ given by (68). Let x n ∈ D, suppose that x n * x0 ∈ D and yn ∈ ’(x n ) for all n ¿ 1. un ∈ L(x n ) such that yn = vn (T ) where  t U (t − s)wn (s) ds = un (t) − U (t)x n ; t ∈ J vn (t) :=

(69) Theorem 20 are Thus in view of the strong upper Thus, there exists

0

and wn is an integrable selection of F(·; un (·)). We have to show that there is a subsequence (ynk ) with ynk → y0 ∈ ’(x0 ) as k → ∞. In view of the weak compactness of D, the strong usc of F(t; ·) and Lemma 26, we show that, for each t ∈ J , the set {vn (t)}n¿1 is relatively compact. Next, observe that the family {vn }n¿1 is equicontinuous (since, in general, {un (t)} is not compact we cannot proceed as in the appendix). Indeed: {vn } is uniformly bounded in view (B3) 11

This example is taken from [48]

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and the boundedness of D; for any t ∈ J and small h ¿ 0, we have  t U (t − s)wn (s) ds vn (t) − vn (t − h) 6 I − U (h)vn (t − h) + t−h

and

 vn (t + h) − vn (t) 6 I − U (h)vn (t) +

t

t+h

U (t + h − s)wn (s) ds:

The uniform continuity of U implies that, for any  ¿ 0, there is  ¿ 0 such that vn (t) − vn (s) ¡  for all n provided |t − s| ¡ . On the other hand the family {wn }, being integrably bounded and such that, for all t ∈ J , {wn (t)} is relatively compact, is relatively weakly compact in L1 (J; E) (see [25, Corollary 2.6]). Hence, passing to subsequences if necessary we may assume that vn → v0 ∈ C(J; E) uniformly and wn → w0 ∈ L1 (J; E) weakly. Moreover, U (t)x n * U (t)x0 on J . Thus un (t) * u0 (t) := U (t)x0 + v0 (t) and  t U (t − s)w0 (s) ds on J: v0 (t) = 0

Using the strong upper semicontinuity of F(t; ·) (for all t ∈ J ), and arguing as in the end of the proof presented in the appendix, we infer that w0 (t) ∈ F(t; u0 (t)), i.e. u0 ∈ L(x0 ). Finally yn → y0 := v0 (T ) and y0 ∈ ’(x0 ). Proposition 35 implies that problem (61) admits a mild solution. However, since U is uniformly continuous, A is de4ned everywhere and U (t) = etA , we get that this solution is absolutely continuous. In case F is single valued we can do without the assumption concerning the uniform continuity of U. Theorem 38. Suppose that E is reFexive and separable, U (T ) 6 1, D is convex closed and bounded, and let F : J × D → E be single valued. If F satis@es (6), (B1), (B2)
12 and (B3), then (61) admits a mild solution. Again if F is strongly usc, then one needs no separability. Proof. Again we consider a map ’ given by (68), i.e.
 T  ’(x) := U (T − s)F(s; u(s)) ds | u ∈ L(x) : 0

Clearly representation (69) is also satis4ed and P (and ’) is a decomposable set-valued map (we do not assume the unique solvability of (1)). To complete the proof we again 12 Meaning that F(·; x) is strongly measurable and, for all t ∈ J , F(t; ·) is strongly continuous, i.e. F(t; x ) → n F(t; x0 ) provided x n * x0 in D.

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need to show that ’ is strongly usc (recall that D being closed convex and bounded is weakly compact). First note that the family of functions {J  t → U (t)x: x ∈ D} is weakly equicontinuous. Indeed let t ∈ J and assume to the contrary that there exist  ¿ 0 and p ∈ E ∗ such that, for every n ¿ 1, there are tn ; |t − tn | ¡ n−1 and x n ∈ D such that |p; U (t)x n − U (tn )x n | ¿ . The weak compactness of D implies without loss of generality that x n * x ∈ D. But then |p; U (t)x n − U (tn )x n | 6 |p; U (t)x n − U (t)x | + |p; U (t)x − U (tn )x | +|p; U (tn )(x − x n ) |: First two terms in the right-hand side of this inequality converges to 0 as n → ∞ since the sequence (x n ) converges weakly to x and the semigroup is strongly continuous. To estimate the third term, observe that p; U (tn )(x − x n ) = U (tn )∗ ◦ p; x − x n where U (t)∗ denotes the adjoint to U (t). Hence, |p; U (tn )(x − x n ) | 6 |U (tn )∗ ◦ p; x − x n − U (t)∗ ◦ p; x − x n | +|U (t)∗ ◦ p; x − x n | 6 U (tn )∗ ◦ p − U (t)∗ ◦ px − x n  +|U (t)∗ ◦ p; x − x n | → 0 as n → ∞ since, by the reSexivity of E, the adjoint semigroup {U (t)∗ } is strongly continuous (see [47, p. 41]). Thus we obtain a contradiction. T Again let x n * x0 in D and yn ∈ ’(x n ). There is un ∈ L(x n ) such that yn = 0 U (T − s)F(s; un (s)) ds. Since D is weakly relatively compact it follows that, for every t ∈ J , the set {un (t): n ¿ 1} is weakly relatively compact. Using the established above weak equicontinuity of {U (·)x}x∈D and arguments similar to those from the proof presented in the appendix, we see that {un : n ¿ 1} is also weakly equicontinuous. Thus, the Arzela–Ascoli theorem (in terms of the weak topology—see e.g. [42, p. 5])) implies that, passing to a subsequence if necessary, (un ) converges weakly uniformly on J to a weakly continuous function u. Hence, un (t) * u(t) for every t ∈ J and u(t) ∈ D since D is weakly closed. T De4ne y0 := 0 U (T − s)F(s; u(s)) ds. By (B2)
, F(s; un (s)) → F(s; u(s)) on J and, thus, by the dominated convergence theorem,  t  t U (t − s)F(s; un (s)) ds → U (t − s)F(s; u(s)) ds (70) 0

0

t

for each t ∈ J . On the other hand, 0 U (t − s)F(s; un (s)) ds = un (t) − U (t)x n * u(t) − U (t)x0 and thus  t u(t) = U (t)x0 + U (t − s)F(s; u(s)) ds 0

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for every t ∈ J , i.e. u ∈ L(x0 ). Hence y0 ∈ ’(x0 ) and, setting t = T in (70), we also have yn → y0 . Since A = 0 is the generator of the non-expansive semigroup {U (t) = idE } we get Corollary 39. Suppose D is weakly compact convex, a single-valued F : J × D → E is strongly continuous and such that (6) holds. Then the periodic problem x
= f(t; x);

x(0) = x(T )

(71)

has a strong solution, i.e. a continuously di=erentiable map u such that (71) holds. Proof. Clearly, the mild solution given by the theorem is also a strong solution since A = 0. Observe that in the proof of Theorem 38 the reSexivity of E was used only to establish the weak equicontinuity of {U (·)x}x∈D . Remark 40. In [20] the following example is given: Let f : ‘2 → ‘2 be de4ned by fn (x) := n x n + n (x − 1)2

for n ¿ 1;

where n → 0; n ¡ 0; (n ) ∈ ‘2 and (n =n ) ∈ ‘2 . Then f is compact and satis4es f(x); x ¡ 0 for each x ∈ ‘2 with x = 1. However, the problem u
(t) = f(u(t)) has no solution of any period. It is clear that in view of Corollary 39 the map f is not strongly continuous. Indeed, ‘2

‘2

let en := (in ). Then en *0, f(en ) = (0; : : : 0; n ; 0; : : :) and thus f(en )→0 as n → ∞. But f(0) = (n ) = 0. 4.3. Equilibria Let F : D ( E. We shall study the autonomous inclusion x
(t) ∈ Ax(t) + F(x(t)):

(72)

A stationary solution to (72), i.e. a point u0 ∈ D(A) ∩ D satisfying 0 ∈ Au0 + F(u0 ) is called an equilibrium of (72). The existence of equilibria (for A ≡ 0) has been carefully studied in [19] (comp. also [9] for a 4nite-dimensional setting). Here we assume (A) but, instead of (B), let (C) F be upper semicontinuous and have non-empty, convex and compact values. Moreover, we shall assume that F is subject to one of the following tangency conditions: ∀x ∈ D

F(x) ∩ TD (x) = ∅;

(73)

∀x ∈ D

F(x) ∩ CD (x) = ∅:

(74)

These conditions are “autonomous” analogs of conditions (6) and (10), respectively.

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Theorem 41. Under assumption (C), suppose that D is compact. Then (72) has an equilibrium provided: (i) D regular (in particular epi-Lipschitz), the Euler characteristic Q(D) = 0 and (74) is satis@ed; (ii) D is convex and (73) holds. Proof. Fix T ¿ 0. Theorem 28 implies that, for each n ¿ 1, there is a Tn -periodic mild solution un to (72) where Tn = 2−n T . The compactness of D shows that (passing to a subsequence) un → u ∈ C([0; T ]; E), u is a mild solution to (72) and it is easy to see that u(t) ≡ u0 ∈ D. Thus we have obtained  t u0 = U (t)u0 + U (t − s)w(s) ds; t ∈ [0; T ]; (75) 0

t where w is integrable and w(s) ∈ F(u0 ) for all s ∈ [0; T ]. The function v(t) = 0 w(s) ds is di!erentiable a.e.; suppose that v
(z) = w(z)= : y0 ∈ F(u0 ) exists (z ∈ (0; T )). From (75), we get that, for h ¿ 0,  z+h u0 = U (h)u0 + U (z + h − s)w(s) ds: z

Hence, U (h)u0 − u0 1 = h h

 z

z+h

(w(s) − U (z + h − s)w(s)) ds −

v(z + h) − v(z) : h

Since F(u0 ) is compact, we infer that the 4rst term in the right-hand side tends to 0 as h → 0+ and the second one converges to −y0 . Hence, u0 ∈ D(A) and Au0 = −y0 showing that u0 is a solution to (72). In a similar manner we get the next results. Theorem 42. Suppose that (C) holds and let D be bounded and acyclic. If the semigroup is compact, (i) D is strictly regular and any point p ∈ D has a neighborhood V (in D) such that sup inf

sup

inf

inf d◦D (y; z) 6 0

  ¿0 ?¿0 x∈V;y∈B(x;?) x ∈BD (x; ) z∈F(x )

or (ii) D is epi-Lipschitz and (74) holds or (iii) D is convex and (73) holds, then (72) has an equilibrium.

(76)

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749

Theorem 43. Suppose (C), (73), (A4) and let D be convex bounded. Then (72) has an equilibrium provided one of the following conditions is satis@ed: (i) F is a (-set contraction and N + 4( ¡ 0; (ii) F is a (-set contraction, E is separable and N + ( ¡ 0; (iii) 0 ∈ D, ! 6 0, F is compact and there is t0 ¿ 0 such that 1 ∈ %(U (t)) for each 0 ¡ t ¡ t0 ; (iv) F is strongly upper semicontinuous, D is weakly compact, the semigroup U is uniformly continuous and non-expansive; (v) F is single-valued strongly continuous, U is non-expansive and E is reFexive. For the proof observe only that if F (independent of t) is a (-set contraction, then (8) holds true (on [0; T ]) with k(·) ≡ (. Hence kL1 ([0; T ]) = (T and it follows that NT + 4(T ¡ 0 (or NT + (T ¡ 0 in case E is separable) for each T ¿ 0. Then one can proceed exactly as in the proof of Theorem 41. That (un ) has a convergent subsequence (in C([0; T ]; E)) follows as in the appendix (taking into account that {un (0)}, as the 4xed point set of an appropriate Poincar@e operator, is compact). Finally observe that if F is de4ned on a fat neighborhood  of a strictly regular set D, then condition (76) in Theorem 42(i) may be replaced by: ∀y ∈ \D F(y)∩@d− D = ∅. At the same time in Theorems 41(i) or 42(ii) condition (74) may be replaced by (73) if F is single valued. Remark 44. 1. Problem 0 = f(x) with f : ‘2 → ‘2 given as in Remark 40 has a solution in case n → ! ¡ 0. This follows from the Theorem 43(iii). 2. In [21, p. 211] the following conjecture is formulated: Let D ⊂ E be closed, bounded and convex; f; g : D → E continuous and bounded; (f()) 6 (1 () for all  ⊂ D and (g(x) − g(y); x − y)− 6 (2 x − y2

for each x; y ∈ D:

(77)

Suppose also that f + g is weakly inward (this means exactly that the mapping F := f + g − idE satis4es (73)). Then f + g has a 4xed point provided that (1 + (2 ¡ 1. We may prove this conjecture under the additional assumption that f+g is uniformly continuous improving previous results (see [21]). Indeed let g0 := g − idE . By the properties of (·; ·)− it is clear that g0 satis4es estimate (77) with (2 − 1 instead of (2 . It follows that for each T ¿ 0 the constant k := T ((1 + ((2 − 1)) ¡ 0 and hence, by the result given in Remark 33.3 above, we have the existence of a T -periodic solution to u
(t) = F(u(t)). Now, arguing as in the proof of Theorem 43 we get a constant solution u(t) ≡ u, which means 0 = F(u), i.e. u = f(u) + g(u). Appendix Here we provide the complete proof of Claim 2 from the proof of Theorem 16 (we sustain the notation introduced there). In the course of the proof we do not assume

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separability of E (hence the presented arguments may be used in the context of Remark 24, too). Since un ∈ Ln (x0 ), we have that un ∈ C(J; Dn ), un (0) = x0 and  t U (t − M)wn (M) dM on J; un (t) = U (t)x0 + 0

where wn : J → E is a strongly measurable selection of Hn (·; un (·)). By (30), it is clear that (for large n) sup z 6 [c(t) + 1](1 + x)

z∈Hn (t; x)

on J × Dn :

Therefore, by the Gronwall inequality the sequence (un ) is (uniformly) bounded. Hence, the sequence (wn ) is integrably bounded (i.e. wn (t) 6 >(t) on J for some > ∈ L1 (J; R)). We shall prove that the sequence (un ) is relatively compact. Suppose that condition (8) holds. 13 First we show that the set {un (t)}n¿1 is relatively compact for any t ∈ J . Let Y be  the closed linear space spanned by the set n¿1 [wn (J ) ∪ un (J ) ∪ vn (S)] where S = {(t; s) ∈ J ×J | s 6 t} and vn (t; s)=U (t−s)wn (s) for (t; s) ∈ T . Then Y is, without loss of generality, separable since all involved functions are strongly measurable. According to [22, Proposition 9.2], for (t; s) ∈ S, Y ({vn (t; s)}n¿1 )=limn→∞ lim supk→∞ d(vk (t; s); Yn ) where Y denotes the Hausdor! MNC in Y and{Yn } is an increasing sequence of 4nite-dimensional subspaces of Y such that Y = n¿1 Yn . This shows that map S  (t; s) → Y ({vn (t; s)}n¿1 ) is measurable and integrable (being integrably bounded). Similarly one shows that functions T; I : J → R given for t ∈ J by T(t) := Y ({wn (t)}n¿1 );

I(t) := Y ({un (t)})

are integrable and bounded. By the Fubini theorem and in view of Lemma 26, for every t ∈ J , 
 t    t 6R U (t − M)wn (M) dM T(M) dM; I(t) = Y 0

n¿1

0

(A.1)

where R = sup{e!s | s ∈ J }. Fix t ∈ J and an arbitrary M ∈ [0; t]. We are going to show that T(M) 6 2k(M)I(M):

(A.2)

13 If the semigroup U is compact, then the relative compactness of (u ) follows from a general result in n [39, Theorem 2.2.3].

R. Bader, W. Kryszewski / Nonlinear Analysis 54 (2003) 707 – 754

751

By (30) (see also (25)) and since wn (M) ∈ Hn (M; un (M)) where un (M) ∈ Dn , we have that, for any n, ˜ × BD (un (M); 5n−1 )) + B(0; n−1 ): wn (M) ∈ conv F({M}

(A.3)

Take any m ¿ 1 and let =BD ({un (M)}n¿m ; 5m−1 ). Since  6 Y 6 2, we then obtain by (A.3) and (8), ˜ T(M) = Y ({wn (M)}n¿m ) 6 2({wn (M)}n¿m ) 6 2[F({M} × ) + B(0; m−1 )] 6 2[F({M} × ) + B(0; m−1 )] 6 2k(M)() + m−1 6 2k(M)(({un (M)}n¿m ) + 5m−1 ) + m−1 6 2k(M)(I(M) + 5m−1 ) + m−1 : Thus passing with m → ∞ we establish (A.2). Then, by (A.1), (A.2) and since k(·)I(·) is integrable, for any t ∈ J ,  t I(t) 6 2R k(M)I(M) dM; 0

hence, T(t) = I(t) ≡ 0 on J by the Gronwall inequality. Consequently the set {un (t)} is relatively compact in E for any t ∈ J . To show that so is the sequence (un ) in C(J; E) we need to show its equicontinuity. To this end take t0 ∈ J ,  ¿ 0 and 1 ¿ 0 such that R A >(s) ds ¡ =3 whenever |A| ¡ 21 (where > is an integrable bound for the sequence (wn )). Finally take z := max{0; t0 − 1 }; then t0 − z 6 1 . Since the set {un (z)}n¿1 is compact, the family {U (·)un (z)}n¿1 is equicontinuous, i.e. there is 0 ¡  ¡ 1 such that, for any n, U (t − z)un (z) − U (t0 − z)un (z) ¡ =3 if |t − t0 | ¡  (t ∈ J ). Let t ∈ J with |t − t0 | ¡ . The semigroup property yields that, for any n ¿ 1,   t    un (t) − un (t0 ) = U (t − z)un (z) − U (t0 − z)un (z) + U (t − s)wn (s) ds z

 +

t0

z

 +R

z

t

  U (t0 − s)wn (s) ds 6 U (t − z)un (z) − U (t0 − z)un (z)  >(s) ds + R

z

t0

>(s) ds ¡ :

Hence the equicontinuity of {un }n¿1 . In view of the Ascoli–Arzela theorem, we get that (passing to a subsequence if necessary) un → u0 ∈ C(J; E) uniformly on J . It is clear that, for each t ∈ J , u0 (t) ∈ D (because ?n  0 by (25), (26)). By a result due to Diestel (see [25, Corollary 2.6], comp. [23, Corollary 3]), we also get (again passing to a subsequence) that wn * w0 ∈ L1 (J; E) (weakly). Therefore, we infer that  t u0 (t) = U (t)x0 + U (t − M)w0 (M) dM on J: 0

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We shall now show that ˜ u0 (t)) w0 (t) ∈ F(t;

for a:a: t ∈ J:

(A.4)

First, in view of the Mazur lemma, we 4nd a sequence yn ∈ conv{wk | k ¿ n} such that yn → w0 in L1 . Thus, without loss of generality we may assume that yn (t) → w0 (t) for almost all t. Take such t; then yn (t) → w0 (t) and un (t) → u0 (t). Fix  ¿ 0 and ˜ ·) is usc, there is 0 ¡ ? ¡ =2p take a bounded linear form 0 = p ∈ E ∗ . Since F(t; such that sup p; z 6

˜ z∈F(t;u)

sup

p; z + =2

(A.5)

˜ 0 (t)) z∈F(t;u

mn n n =i zi wherezin ∈ vin + provided u−u0 (t) 6 2?. By (A.3), for each n ¿ 1, wn (t)= i=1 mn −1 n n n −1 n ˜ B(0; n ), vi ∈ F(t; ui ), ui −un (t) ¡ 5n , =i ¿ 0 for all i=1; 2; : : : ; mn and i=1 =in = −1 1. Since un (t) → u0 (t), there is N ¿ 1 such that 5n 6 ? and un (t) − u0 (t) 6 ? for n ¿ N . Therefore, for n ¿ N , uin − u0 (t) 6 2? for all i = 1; 2; : : : ; mn , and, by (A.5), p; wn (t) 6

mn  i=1

=in p; vin + p? 6

sup

p; z + :

˜ 0 (t)) z∈F(t;u

By convexity, for all n ¿ N , p; yn (t) 6 supz∈F(t; ˜ u0 (t)) p; z + . Passing with n → ∞ and  → 0+ , we see that p; w0 (t) 6 supz∈F(t; ˜ u0 (t)) p; z . Since this holds for any ∗ ˜ ˜ u0 (t)). This establishes (A.4) and p ∈ E , we infer that w0 (t) ∈ conv F(t; u0 (t)) = F(t; completes the proof of Claim 2.

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