Set-Valued Analysis 9: 289–313, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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On the Solution Sets of Constrained Differential Inclusions with Applications RALF BADER1 and WOJCIECH KRYSZEWSKI2 1 Mathematisches Institut, Universität München, Theresienstr. 39, D-80333 Munich, Germany. e-mail:
[email protected] 2 Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toru´n, Poland. e-mail:
[email protected]
(Received: 9 November 1999; in final form: 5 December 2000) Abstract. We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem for a Carathéodorytype differential inclusion in a closed domain is an Rδ -set provided some mild boundary conditions expressed in terms of functional constraints defining the domain are satisfied. Presented results generalize most of the existing ones. Some applications to the existence of periodic solutions as well as equilibria are given. Mathematics Subject Classifications (2000): 34A60, 34C25, 47H10. Key words: differential inclusion, structure of solutions, periodic solution, equilibrium.
1. Main Results 1.1. THE PROBLEM Assume K is a closed subset of the N-dimensional Euclidean space RN (N 1) and let ϕ: J × K RN , where J := [0, T ] and T > 0, be a set-valued mapping such that: (ϕ1 ) for each t ∈ J , x ∈ K, the value ϕ(t, x) ⊂ RN is nonempty, compact and convex; (ϕ2 ) for almost all (a.a.) t ∈ J , the map K x ϕ(t, x) is upper semicontinuous and, for all x ∈ K, the map ϕ(·, x) has a measurable selection, i.e. there is a measurable function wx : J → RN such that wx (t) ∈ ϕ(t, x) for a.a. t ∈ J ; (ϕ3 ) ϕ is (uniformly) bounded, i.e. there is c > 0 such that, for a.a. t ∈ J and all x ∈ K supy∈ϕ(t,x) |y| c. This work was done partially while the first author visited the University of Toru´n. Support was provided by a scholarship from Deutsche Forschungsgemeinschaft (DFG), Bonn (Germany). The second author was supported by KBN Grant 2 P03A 024 16 and UMK Grant 458-M. We use the following notation: | · | is the standard norm in RN , i.e. |x| = √ x, x, where ·, · is the scalar product; given A ⊂ RN , cl A, int A, bd A and conv A denote the closure, the interior, the boundary of A and the convex envelope of A, respectively. If ε > 0, then B(A, ε) := {x ∈ RN | d(x, A) := infa∈A |x − a| < ε} (resp. D(A, ε) := {x ∈ RN | d(x, A) ε}) is the open (resp. closed) ball around A of radius ε; in particular D(0, 1) = {x ∈ RN | |x| 1}. Moreover DK (A, ε) := D(A, ε) ∩ K and BK (A, ε) := B(A, ε) ∩ K.
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We study the existence and the structure of solutions to the following initialvalue problem u ∈ ϕ(t, u), t ∈ J, u ∈ K, u(0) = x0 ∈ K.
(P)
As usual, by a solution to problem (P) we understand a continuous function u: J → RN such that u(0) = x0 ,u(J ) ⊂ K, for which there is a function t v ∈ L1 (J, RN ) such that u(t) = x0 + 0 v(s) ds and v(t) ∈ ϕ(t, u(t)) on J . Clearly v is the weak derivative of u; u is absolutely continuous and the derivative u exists almost everywhere (a.e.); moreover, u (t) = v(t) for a.a. t ∈ J . Remark 1.1. There is no restriction of generality in considering bounded ϕ instead of ϕ having a sublinear growth, i.e. such that there is c ∈ L1 (J, R) with sup |y| c(t)(1 + |x|) y∈ϕ(t,x)
for a.a. t ∈ J and all x ∈ K. If ϕ in (P) has sublinear growth, then a standard argument involving the Gronwall inequality (comp. [20, p. 52]) implies that we may replace it by a map satisfying (ϕ3 ) with c = 1 retaining the same set of solutions.
1.2. EXISTENCE We recall the following essentially well-known existence (viability) result stating the so-called weak invariance of problem (P). Its simple, constructive (and, in our belief, new) proof will be given in Section 2. THEOREM 1.2. Under assumptions (ϕ1 )–(ϕ3 ), if (ϕ4 ) ϕ satisfies the weak tangency condition with respect to K, i.e. for any x ∈ bd K, the map ϕ(·, x) ∩ conv TK (x) has a measurable selection, then, for any x0 ∈ K, there exists a solution to problem (P). TK (x) stands for the Bouligand tangent cone to K at x ∈ K, i.e. dK (x + hu) N =0 TK (x) := u ∈ R lim inf h→0+ h and dK = d(·, K) is the distance function. Since, for x ∈ int K, TK (x) = RN , (ϕ2 ) together with (ϕ4 ) imply that, for all x ∈ K, ϕ(·, x) ∩ conv TK (x) has a measurable selection. Remark 1.3. As remarked above, Theorem 1.2 is known. In [3], it is called the ‘dual characterization of viability domains’ (comp. also [2] and see, e.g., [20, Cor. 5.1]). However, its usual and perhaps better known formulation (see, e.g., [20, Th. 5.1, 5.2]) relies, instead of (ϕ2 ) and (ϕ4 ), on the following assumptions:
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• ϕ is either upper semicontinuous or ϕ is a Carathéodory map (i.e. ϕ(· , x) measurable for all x ∈ K and ϕ(t, ·) is upper semicontinuous for a.a. t ∈ J ); • ϕ satisfies the tangency condition, i.e. ϕ(t, x) ∩ TK (x) = ∅
(1)
for all (t, x) ∈ J × K. It is clear that these assumptions imply both (ϕ2 ) and (ϕ4 ). It is, however, also clear that in order to check assumption (ϕ2 ) it is reasonable to assume that ϕ is a Carathéodory map and, probably with exception of ad hoc constructed counterexamples, this is the class of maps to be studied. Nevertheless, in the course of the paper, we use only the existence of measurable selections of ϕ(·, x) and not the measurability itself. The proof we give in Section 2 is quite simple and uses Euler approximations of sorts (comp. also [12]). In case ϕ is single-valued, the viability was first observed by Nagumo in 1942 and, then, it has been rediscovered several times by several authors. The set-valued (upper semicontinuous case) is due to Haddad [31]. The possibility to replace the tangency condition (1) by ‘the convexified tangency condition’: ϕ(t, x)∩ conv TK (x) = ∅ on J × K, was noted by Guseinov, Subbotin and Ushakov (see [29, 30]). Let us also note that in the autonomous case (i.e. if ϕ does not depend on t), condition (1) is also necessary for the existence of solutions to (P). Problem (P), in the case of a time-dependent constraint (i.e. the existence of a solution u (t) ∈ ϕ(t, u(u)) such that u(0) = x0 ∈ K(0) and u(t) ∈ K(t) where the set-valued map K: J RN has closed values), has been studied thoroughly in [25] for a closed convex-valued L×B-measurable ϕ having a linear growth with upper semicontinuous ϕ(t, ·) and the absolutely continuous K(·) (resp. in [24] for ϕ(·, x) measurable, ϕ(t, ·) continuous and left absolutely continuous K(·)). The appropriate tangency conditions were stated in terms of of the convex hull of the Bouligand tangent cone to the graph of K(·) (see Theorems 4.2, 4.7 in [24] and Theorem 3.1 in [25]). A related result (under stronger assumptions) in the context of differential inclusions with delays has been established in [38, Th. 2, 3]. Some other issues of the viability theory are studied in detail in [3, 4, 6].
1.3. STRUCTURE OF SOLUTIONS If K = RN , then the set of all solutions to (P) is characterized in the spirit of the celebrated Aronszajn theorem [1]: it is a compact Rδ -set in C(J, RN ) – see, e.g., [18], [33, 34] (autonomous systems), [19] (nonautonomous systems), comp. [4, Cor. 5, p. 109] and the survey [23]. However even if ϕ is defined on J × RN , K RN and the strong tangency condition (i.e. ϕ(t, x) ⊂ TK (x) for all (t, x) ∈ J × K) Recall that a nonempty compact subset L of a metric space X is called an R -set if it is the δ intersection of a decreasing family of compact contractible sets Ln ⊂ X.
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is satisfied, then there are solutions which ‘live’ in K as well as there may exist solutions starting in K but leaving it (see, e.g., [20, Ex. 7.2]). Hence, in our case, the above-mentioned results fail to help in characterization of the set of solutions surviving in K. It is clear that a condition: ∀x ∈ bd K
ϕ(·, x) ∩ CK (x) has a measurable selection,
(2)
where CK (x) stands for the Clarke tangent cone to K at x, i.e. dK (y + hu) N =0 , CK (x) := u ∈ R lim y→x, y∈K, h→0+ h implies (ϕ4 ) (since CK (x) ⊂ TK (x)) and, thus, the existence of solutions to the initial value problem (P). But even this stronger assumption does not prevent solutions to (P) to escape from K and does not give means to characterize the set of solutions to (P). Note here that a stronger form of condition (2), i.e. ϕ(t, x) ⊂ CK (x) implies that all solutions to (P) stay in K provided ϕ(t, ·) is Lipschitz continuous. In order to get a satisfying characterization involving weaker sorts of assumptions, we need to be more specific about the set K and the tangency condition. To see that without additional assumptions nothing in fact can be said about the structure of solutions to (P) consider the following example. EXAMPLE 1.4 (comp. [8, Ex. 1.1]; see also [4, p. 203] and [20, Ex. 7.3]). Let K := S1 ∪ S−1 where Si := z = (x, y) ∈ R2 | (x − i)2 + y 2 = 1 . Let
g(x, y) =
(y, 1 − x) (−y, 1 + x)
for (x, y) ∈ S1 , for (x, y) ∈ S−1 .
For all z ∈ K, g(z) ∈ TK (z) = CK (z) but the set of all solutions to (P) (with ϕ replaced by g and x0 = (0, 0)) is even not connected. In case K is closed convex, a result concerning the structure of solutions to (P) has been proven by Deimling (see, for instance, [20, Th. 7.2 or Cor. 7.3], see also [22, 21] and [37]). This result was generalized by Plaskacz [41] where he studies the class ρ of sets (in RN ) (called proximate retracts in [27]; see also [40]): If K ⊂ RN is a compact proximate retract, then the set of all solutions to (P) is an Rδ -set provided the strong tangency condition is satisfied. Plaskacz’s result was extended to the Hilbert space context (see [28]) while Deimling’s to a It means that K admits a metric retraction, i.e. the projection π from Section 2 is single-valued K
on a neighborhood of K – see also Example 1.10. In this case TK (x) = CK (x) for all x ∈ K.
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Banach space (see [10]). Let us also mention that, for a compact convex K, the Rδ -characterization was obtained in [38] for differential inclusions with delays. In the present paper, we shall further generalize the paper [41] in order to obtain the Rδ -characterization of the set of solutions to (P) under less stringent assumptions on the constraint set K. First we need some notions of nonsmooth analysis (see, e.g., [11]). Assume that f : Dom(f ) → R, where the domain Dom(f ) is open in RN , is a locally Lipschitz function. For x ∈ Dom(f ), the directional derivative of f at x in the direction u ∈ RN in the sense of Clarke is defined by the formula f ◦ (x; u) := lim sup
y→x, h→0+
f (y + hu) − f (y) h
and the generalized gradient of f at x by ∂f (x) := p ∈ RN | p, u f ◦ (x; u) ∀u ∈ RN . For each x ∈ Dom(f ), the function RN u → f ◦ (x; u) is Lipschitz, subadditive and positively homogeneous; f ◦ (x; ·) is the support function of the gradient ∂f (x), i.e. f ◦ (x; u) = sup p, u.
(3)
p∈∂f (x)
Hence, the (negative) polar cone to ∂f (x), ∂f (x)− := u ∈ RN | p, u 0 ∀p ∈ ∂f (x) = u ∈ RN | f ◦ (x; u) 0 . For all x ∈ Dom(f ), the set ∂f (x) is convex and compact. The (real) function Dom(f ) × RN (x, u) → f ◦ (x; u) is upper semicontinuous; consequently, so is the set-valued map Dom(f ) x → ∂f (x) ⊂ RN . It is well known that CK (x) = ∂dK (x)− . Let f be as above and let K = {x ∈ Dom(f ) | f (x) 0}.
(S)
We say that f is a representing function of K and/or K is represented by f . In general sets of the form (S) are not closed. Any closed set K ⊂ RN is represented by its distance function dK . We shall need the following result. PROPOSITION 1.5 (compare [5, Proposition 16, 7.3]). Consider the set K of the form (S) and let x0 ∈ bd K. If 0 ∈ ∂f (x0 ), then ∂f (x0 )− ⊂ CK (x0 ) (in general the inclusion is strict). The first result concerning sets of the form (S) is the following. THEOREM 1.6. Suppose K, represented by f , is closed and, for any x ∈ bd K, 0 ∈ ∂f (x). Let ϕ: J ×K RN satisfy conditions (ϕ1 )–(ϕ3 ) and, instead of (ϕ4 ),
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(ϕ5 ) ∀x ∈ bd K
ϕ(·, x) ∩ ∂f (x)− has a measurable selection
(equivalently: for x ∈ bd K, there is a measurable function wx : J → RN such that wx (t) ∈ ϕ(t, x) and f ◦ (x; wx (t)) 0 a.e. on J ). Let x0 ∈ K and let L := u ∈ C J, RN | u (t) ∈ ϕ(t, u(t)) a.e. on J, u(0) = x0 , u(J ) ⊂ K denote the set of all solutions to (P). Then L is an Rδ -set in C(J, RN ). In the situation of Theorem 1.6, ∂f (x)− ⊂ CK (x) (see Proposition 1.5); thus (ϕ5 ) implies (ϕ4 ). As noted above any closed set K is represented by dK and CK (x) = ∂dK (x)− for x ∈ K. But since 0 ∈ ∂dK (x) for all x ∈ K, Theorem 1.6 does not apply. When analyzing assumptions of Theorem 1.6 one encounters the following class of sets. 1.4. EPI - LIPSCHITZ SETS In many recent papers, a special attention has been paid to the so-called epiLipschitz sets. A nonempty closed subset M of RN is said to be epi-Lipschitz (or wedged in [12]) if, at every x ∈ M, the Clarke normal cone NM (x) := CM (x)− is pointed (i.e. NM (x) ∩ (−NM (x)) = {0}) (equivalently int CK (x) = ∅). This important class of sets has been introduced in optimization by Rockafellar [42] and includes closed convex subsets of Rn with nonempty interiors, C 1 submanifolds of RN with boundaries (or corners) of full dimension and, more generally, sets in RN defined via finite number of C 1 inequality constraints satisfying the so-called constraint qualification assumption. Given an epi-Lipschitz set M ⊂ RN , one considers a Lipschitz continuous function ,M := dM − dM c : RN → R where M c = RN \ M. Clearly M = {x ∈ RN | ,M (x) 0}, i.e. M is represented by ,M . If x ∈ bd M, then HiriartUrruty [35, 36]) shows that NM (x) = cl ( λ0 λ∂,M (x)) and Cornet, Czarnecki (see [15]) show that 0 ∈ ∂,M (x). This follows from the inclusion ∂,M (x) ⊂ conv (NM (x) ∩ S N−1 ),
x ∈ bd M,
where S N−1 stands for the unit sphere in RN , derived from facts due to Rockafellar [42]. Hence CM (x) = ∂,M (x)− and the set-valued map M x CM (x) is lower semicontinuous. On the other hand, consider a set K of the form (S) satisfying assumptions of Theorem 1.6. Then K is epi-Lipschitz since, by Proposition 1.5, NK (x) ⊂ ∂f (x)−− for all x ∈ bd K and NK (x) must be pointed. Hence K is represented by ,K . Putting f = ,K in Theorem 1.6 and recalling that ∂,K (x)− = CK (x), x ∈ bd K, we obtain the following result. Hence, actually, N (x) = M λ0 λ∂,M (x).
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COROLLARY 1.7. Assume that K is an epi-Lipschitz subset of RN. If ϕ: J ×K RN satisfies conditions (ϕ1 )–(ϕ3 ) and (ϕ5 ) ∀x ∈ bd K
ϕ(·, x) ∩ CK (x) has a measurable selection,
then, for any x0 ∈ K, the set of all solutions to (P) is an Rδ -set in C(J, RN ). In particular, if ϕ is a single-valued Carathéodory map, then the same holds if condition (ϕ5 ) is replaced by (ϕ5 ) ϕ(t, x) ∈ TK (x) for any x ∈ bd K and a.a. t ∈ J. Proof. Only the last statement requires a comment. Given x ∈ bd K, take a sequence (yn ) in K such that yn → x as n → ∞. Then, in view of [6, Theorem 4.1.10], ϕ(t, x) = lim ϕ(t, yn ) ∈ Lim inf TK (y) = CK (x) n→∞
y→x,y∈K
for a.a. t ∈ J .
✷
It is obvious that epi-Lipschitz sets have nonempty interiors; thus, e.g., a convex closed set is not epi-Lipschitz in general (it is epi-Lipschitz when considered as a subset of the affine subspace spanned by itself; hence Theorem 1.6 may serve as a generalization of the above-mentioned Deimling result). Therefore Theorem 1.6 (or Corollary 1.7) is not fully satisfying (note that proximate retracts are, in general, ‘thin’, i.e. they have empty interiors) and there is a need to establish a result generalizing Theorem 1.6 and extending the result of Plaskacz. For this reason let us recall after [17] the following class of regular sets. 1.5. REGULAR SETS DEFINITION 1.8. We say that the set K of the form (S), represented by a locally Lipschitz function f : Dom(f ) → R, is regular if: (i) K is closed; (ii) there is an open set U such that K ⊂ U ⊂ Dom(f ) and 0 ∈ ∂f (x) for x ∈ U \ K. The set K is strictly regular if it is closed and (iii) for any x ∈ bd K, lim infy→x, y∈K |||∂f (y)||| > 0. Clearly (ϕ ) is nothing else but (2). Recall the5definition of Lim inf: Given a metric space (X, d), a set A ⊂ X and a set-valued
mapping ψ: A X, let x ∈ X be an accumulation point of A. Then z ∈ Lim inf ψ(y) y→x, y∈A
⇐⇒
∀ (yn ), yn → x, yn ∈ A ∃ zn ∈ ψ(yn ), zn → z.
⇐⇒
∃ (yn ), yn → x, yn ∈ A ∃ zn ∈ ψ(yn ), zn → z.
Recall also that z ∈ Lim sup ψ(y) y→x, y∈A
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Above, for x ∈ Dom(f ), |||∂f (x)||| :=
inf |p| =
p∈∂f (x)
inf
sup p, u.
p∈∂f (x) u∈D(0,1)
By (3), it is easy to see that |||∂f (x)||| =
sup
inf p, u = −
u∈D(0,1) p∈∂f (x)
inf
u∈D(0,1)
f ◦ (x; u)
(4)
and 0 ∈ ∂f (x) if and only if |||∂f (x)||| = 0. Moreover, note that the function Dom(f ) x → |||∂f (x)||| is lower semicontinuous. Below we shall provide examples of (strictly) regular sets but, in order to make them more familiar, let us first discuss the definition and establish some of their basic properties. Remark 1.9. (1) Observe that the (strict) regularity of the set K does not depend only on K itself: it strongly depends on the behaviour of its representing function. It is not difficult to provide an example of a closed set K being represented by two different functions and such that it is (strictly) regular with respect to one of these functions and failing to have this property with respect to another. For an a priori given closed subset K ⊂ RN , there is even a C ∞ -function representing it. Hence condition (ii) (or (iii)) of Definition 1.8 is crucial. However it would be interesting to get a different, more geometric, independent from the representing function f , assumption concerning sets of the form (S) under which our main results (i.e. Theorem 1.11 and Corollary 1.12) hold true. Could it be done, e.g., by the application of hypertangent fields used in [26] (see also Example 1.10)? Note also that our notion of regularity differs from the notion of the so-called tangential regularity (see, e.g., [6]). (2) If K is regular, then {x ∈ U | 0 ∈ ∂f (x)} ⊂ K; this means that f has no critical points in U \K. We emphasize that above we make assumptions concerning the behaviour of ∂f neither on K nor on bd K. (3) If K is regular and C ⊂ U \ K is compact, then infx∈C f (x) > 0 and, since the function |||∂f (·)||| is lower semicontinuous, we gather that also infx∈C |||∂f (x)||| > 0. (4) Assume that the set K of the form (S) is closed and consider the following condition: (∗) there is a neighborhood U of K such that infy∈U \K |||∂f (y)||| > 0. It is clear that (∗) implies (iii). By the lower semicontinuity of the function K x → lim infy→x,y∈K |||∂f (y)|||, if bd K is compact, then condition (iii) implies (∗). In view of these remarks, one may think of strictly regular in terms of the more friendly looking condition (∗). (5) It is clear that a strictly regular set is regular. Actually condition (∗) was used in [17] to define strictly regular sets; in [17] the infinite-
dimensional situation was considered and condition (iii) was too weak for our purposes.
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EXAMPLE 1.10. (1) The set K from Example 1.4 is regular when represented by its distance function dK . There is no way to make it strictly regular. (2) Any set K satisfying assumptions of Theorem 1.6 is strictly regular; in particular any epi-Lipschitz set is strictly regular. This is because |||∂f (·)||| is lower semicontinuous and, hence, for any x ∈ bd K, lim infy→x,y∈K |||∂f (y)||| |||∂f (x)||| > 0. In [26], the authors extend the Fillipov and the Fillipov–Wa˙zewski theorems to the case when the state variable x is constrained to the closure of the open set 0 ⊂ RN assuming the existence of the so-called uniformly hypertangent conical field on bd 0 (see [26, Def. 4.1]). For bounded 0, the existence of such a field is equivalent to the condition: int Ccl 0 (x) = ∅ for all x ∈ bd 0, i.e. to the epi-Lipschitzeanity of cl 0 – see [26, p. 465] and [42, Th. 3]. Therefore our assumption (ϕ5 ) or (ϕ5 ) (with K = cl 0) is similar but weaker then the corresponding assumption H2 from [26] (see also [26, Lemma 4.1]). (3) Let K ⊂ RN be a proximate retract (see [40, 27]). This means that K is closed and there is a neighborhood U of K and a (continuous) retraction r: U → K such that |r(x) − x| = dK (x). Observe that, for instance, proximally smooth sets from [14] are proximate retracts. Take x ∈ U \ K, sequences yn → x, hn → 0+ and let u := r(x) − x. Then dK (yn + hn u) − dK (yn ) hn |(r(yn ) − yn ) − u| + dK (yn + hn (r(yn ) − yn )) − dK (yn ). It is easy to see that dK (yn + hn (r(yn ) − yn )) = (1 − hn )dK (yn ). Hence lim sup n→∞
dK (yn + hn u) − dK (yn ) −|u|, hn
i.e. dK◦ (x; u/|u|) −1 and, by (4), |||∂dK (x)||| 1, so condition (∗) from Remark 1.9(4) is satisfied. Hence K, represented by dK , is strictly regular. (4) In particular any closed convex K ⊂ RN , represented by dK , is strictly regular. (5) Plaskacz shows that any C 2 -manifold (in RN ) is a proximate retract (this is because of the tubular neighborhood theorem). However, he also provides an example of a C 1 -manifold whithout this property. Here let us consider an open set W ⊂ RN and a C 1 -smooth map g: W → Rm (m < N). Let c ∈ Rm be a regular value of g (i.e. M := g −1 (c) = ∅ and the derivative g (x) is surjective for all x ∈ M). Of course M is an (N − m)-dimensional C 1 -manifold. If M is a closed set, then M is strictly regular. To see this put f : W → R by f (y) = |g(y) − c|, y ∈ W . Clearly M is represented by f and f is C 1 on W \M, i.e. ∂f (y) = {∇f (y)} (so |||∂f (y)||| = |∇f (y)|) for all y ∈ W \ M. We easily find that, for all x ∈ M limy→x, y∈M |∇f (y)| = $g (x)$ > 0 ($ · $ is the operator norm). In particular we see that any orientable closed C 1 -manifold is strictly regular. Note here that in order to check whether a closed set K is a proximate retract it is enough to
show that the map x → πK (x) = {y ∈ K | |x − y| = dK (x)} is lower semicontinuous on U ; this is also equivalent to the single-valuedness of πK on U .
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Let us now state the first main result of this paper. THEOREM 1.11. Suppose that K of the form (S) is regular. Let ϕ satisfy conditions (ϕ1 )–(ϕ3 ). Assume that (ϕ6 ) ∀x ∈ bd K ϕ(·, x) ∩ Lim infy→x, y∈K ∂f (y)− has a measurable selection. Then, for any x0 ∈ K, the set of all solutions to (P) is an Rδ -set in C(J, RN ). Observe that in the situation of the above result, assumption (ϕ6 ) cannot be replaced by assumption (ϕ5 ). Example 1.4 provides a suitable counterexample. Condition (ϕ6 ) is by all means restrictive, but is the best we could achieve under that weak assumptions concerning the set K. Theorem 1.11 gives us means to state the second main result of this paper. COROLLARY 1.12. Suppose that K of the form (S) is strictly regular. If ϕ satisfies conditions (ϕ1 )–(ϕ3 ) and (ϕ5 ) then the set of all solutions to (P) is a nonempty compact Rδ -set in C(J, RN ). In particular, if f = dK , then (ϕ5 ) is equivalent to (ϕ5 ); in this situation if ϕ is a single-valued Carathéodory map, then (ϕ5 ) may be replaced by (ϕ5 ). Proof. Let x ∈ bd K, take u ∈ ∂f (x)− and q ∈ Lim supy→x, y∈K ∂f (y)−− . Hence there are sequence yn → x, yn ∈ K and qn → q, qn ∈ ∂f (yn )−− . Since there is n0 such that, for n n0 , |||∂f (yn )||| m for some m > 0 (see Defini−− tion 1.8), we gather that ∂f (yn ) = λ0 λ∂f (yn ), i.e. qn = λn pn where λn 0 and pn ∈ ∂f (yn ). Since ∂f (·) is upper semicontinuous, passing to a subsequence if necessary, we may assume that pn → p ∈ ∂f (x). Since |pn | m for n n0 , p = 0; hence (again for a subsequence) λn → λ 0 and q = λp ∈ ∂f (x)−− . Thus q, u 0 and we see that
− ∂f (x)− ⊂ Lim sup ∂f (y)−− = Lim inf ∂f (y)− y→x, y∈K
y→x, y∈K
(the last equality follows from [44]; see also [6, Th. 1.1.8]). Hence, (ϕ5 ) entails (ϕ6 ) and our result follows from Theorem 1.11. ✷ Corollary 1.12 generalizes results, e.g., from [20, 37] and [41] since if K is a proximate retract (or, in particular, K is closed convex), then, according to Example 1.10(3) and (4), K (being represented by dK ) is strictly regular and condition (ϕ5 ) is equivalent to (ϕ4 ) which is slightly weaker than the usually used condition (1). In Section 3 we shall provide some applications concerning the existence of periodic solutions to (P) as well as the existence of the so-called equilibria of setvalued upper semicontinuous maps defined on compact (strictly) regular sets.
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2. Proofs Let K ⊂ RN be closed. For a point x ∈ RN , put πK (x) := {z ∈ K | |z − x| = dK (x)}. It is clear that πK (x) = ∅. After [6, Prop. 4.1.2], let us recall LEMMA 2.1. If x ∈ RN and z ∈ πK (x), then for any w ∈ conv TK (z), w, x −z 0. Proof of Theorem 1.2. In the proof we follow the idea of proximal aiming, which we learned from [12]. Without loss of generality, we put c = T = 1, i.e. J = [0, 1]. Take x0 ∈ K and fix a positive integer n. We consider a partition {t0n , . . . , tnn2 } of the interval J where tin = n−2 i, i = 0, . . . , n2 . Let x0n = x0 , z0n = x0 and let w0n : J → RN be a measurable function such that w0n (t) ∈ ϕ(t, z0n ) ∩ conv TK (z0n ) for a.a. t ∈ J . Let tn 1 n n w0n (s) ds. x1 := x0 + t0n
n n Assuming that xi−1 ∈ RN , for some 2 i n2 , is constructed, let zi−1 ∈ n n n n N πK (xi−1 ), let wi−1 : J → R be measurable and such that wi−1 (t) ∈ ϕ(t, zi−1 ) ∩ n ) a.e. on J . Put conv TK (zi−1
xin
:=
n xi−1
+
tin
n ti−1
n wi−1 (s) ds.
Thus, for any n, we have defined inductively a measurable function wn : [0, 1] → RN via the following formula n (t) wn (t) := wi−1
n for t ∈ [ti−1 , tin ), i = 1, . . . , n2 ,
and wn (1) := wnn2 −1 (1). Let un : J → RN be given by
t
un (t) := x0 +
wn (s) ds 0
for t ∈ J . Additionally, let us define vn : [0, 1] → K by n vn (t) := zi−1
n for t ∈ [ti−1 , tin ), i = 1, . . . , n2
and vn (1) := znn2 ∈ πK (xnn2 ). We see that un is absolutely continuous, and un = wn , un (0) = x0 , and |wn (t)| 1 a.e. on J.
wn (t) ∈ ϕ(t, vn (t)) (5)
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We claim that for all n and t ∈ J , 1 . n Indeed: if t ∈ [t0n , t1n ), then t |w0n (s)| ds t. |un (t) − vn (t)| |un (t) − vn (t)|
(6)
0
Hence dK2 (un (t)) |un (t) − vn (t)|2 t 2
1 t. n2
By continuity, dK2 (x1n ) = dK2 (un (t1n )) n−2 t1n , as well. Assume that, for some 2 i n2 , we have 1 n t . n2 i−1 n , tin ). Then Let t ∈ [ti−1 n ) dK2 (xi−1
(7)
|un (t) − vn (t)|2
t 2 n n n = |un (t) − = wi−1 (s) ds + xi−1 − zi−1 n ti−1 t n 2 n n 2 n n n (t − ti−1 ) + |xi−1 − zi−1 | + 2
wi−1 (s), xi−1 − zi−1 ds n zi−1 |2
n ti−1
1 1 n 1 1 n n n (t − ti−1 ) + dK2 (xi−1 ) 2 (t − ti−1 ) + 2 ti−1 = 2t 2 n n n n in view of (7) and since, by Lemma 2.1, t n n n
wi−1 (s), xi−1 − zi−1 ds 0
n ti−1
n n (s) ∈ conv TK (zi−1 ) for a.a. s ∈ J . This shows that dK2 (xin ) n−2 tin , because wi−1 so by induction, it proves (6). Obviously (6) implies that
1 on J. (8) n Thus we obtain sequences (un ), (wn ) and (vn ) satisfying (5), (6) and (8). Since |un (t)| |x0 | + 1 on J , |un (t)| = |wn (t)| 1 a.e. on J , by passing to a subsequence if necessary, we may suppose that un → u (uniformly on J ), where u: J → RN is absolutely continuous, and un → u weakly in L1 (J, RN ) (see, e.g., [4, Th. 0.3.4]). In view of (8), u(t) ∈ K for all t ∈ J . By (5) and (6), u(0) = x0 and, for any ε > 0, (un (t), un (t)) belongs to the ε-neighborhood of the graph Gr(ϕ(t, ·)) for a.a. t ∈ J provided n is large enough. Hence, by the Convergence ✷ Theorem (see [5, Th. 3.6]), u (t) ∈ ϕ(t, u(t)) a.e. on J . dK (un (t))
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Remark 2.2. (1) It is well-known and easy to see that the set of all solutions to (P) is compact in C(J, RN ). (2) By inspection of the proof we note that condition (ϕ4 ) may be slightly weakened. Namely, instead of (ϕ4 ), we may assume that, for any x ∈ K, there is a measurable function w(·) such that w(t) ∈ ϕ(t, x) on J and, for any y ∈ πK−1 (x),
w(t), y − x 0 a.e. on J . Observe that the set πK−1 (x) generates the so-called proximal normal cone to K at x. A result similar to the discussed one is therefore called in ‘dual characterization of viability domains’ in the upper semicontinuous case (see [3]). (3) The same proof shows that, given t0 ∈ J and x0 ∈ K, there is a solution on [t0 , T ] to problem (P) with the initial condition replaced by: u(t0 ) = x0 . To proceed with the proof of Theorem 1.6 we need some preliminaries. LEMMA 2.3. Let w: J → RN be a bounded measurable function, say |w(t)| c for all t ∈ J . For any ε > 0 and each set J0 ⊂ J of full measure, there exists a simple function w: J → RN (i.e. measurable and having finite number of values) such that w(J ) ⊂ w(J0 ) and |w(t) − w(t)| < ε a.e. on J ( precisely on J0 ). Proof. For simplicity we set N = 1 (the general case may be treated as below with obvious adjustments). Without loss of generality assume that 2cε = n is a positive integer. Put Ai := [−c + (i − 1)ε, −c + iε) An := [−c + (n − 1)ε, c]
(i = 1, 2, . . . , n − 1),
and Ji := w −1 (Ai ),
i = 1, . . . , n.
For any i, Ji is measurable and these sets are pairwise disjoint. m Suppose that only the sets Ji1 ∩ J0 , . . . , Jim ∩ J0 are nonempty; then J0 ⊂ j =1 Jij . For any j = 1, . . . , m, choose tj ∈ Jij ∩ J0 and put w j = w(tj ) ∈ Aij . Now define m m j =1 w j χJij (t) if t ∈ j =1 Jij , w(t) := w1 if t ∈ J \ m j =1 Jij ⊂ J \ J0 , where χJij stands for the characteristic function of Jij . Clearly w is a simple function and w(J ) ⊂ w(J0 ). Moreover if t ∈ J0 , then there is j = 1, . . . , m such that ✷ t ∈ Jij . Therefore w(t) = wj and w(t) ∈ Aij . Hence |w(t) − w(t)| < ε. Next, recall that due to results of Hyman [39], a compact subset L of a metric space X is an Rδ -set if and only if it is contractible in any of its neighborhoods, i.e. given an open neighborhood U of L, there is a continuous map h: L × [0, 1] → U such that h(x, 0) = x and h(x, 1) = x0 ∈ U ). Therefore if a decreasing family
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{Ln } of Rδ -sets is given, then the intersection L := ∞ n=1 Ln is again an Rδ -set. This follows easily from the above characterization of Rδ -sets. Proof of Theorem 1.6. First observe that for x ∈ bd K, in view Proposition 1.5, ∂f (x)− ⊂ CK (x) ⊂ TK (x). Hence all assumptions of Theorem 1.2 are satisfied (if x ∈ int K, then TK (x) = RN ), i.e. L is a nonempty compact subset of C(J, RN ). Now we are going to use a certain approximation procedure: we shall construct a sequence of locally Lipschitz (in x) maps satisfying the boundary condition (ϕ5 ). In that we follow some ideas of [32] (especially that of a σ -selectionable map) used also by many other authors (see, e.g., [4, Th. 1.13.1], [20, Lemma 2.2]). However since we deal with a nonconvex situation, any version of the Michael selection theorem will not be applied. Take an arbitrary x ∈ bd K. There is a measurable function wx : J → RN such that wx (t) ∈ ϕ(t, x) on J and, for a.a. t ∈ J (say for t ∈ J0 , J0 is of full measure), f ◦ (x; wx (t)) 0.
(9)
Since 0 ∈ ∂f (x), by (4) we have inf
u∈D(0,1)
f ◦ (x; u) < 0.
Choose ux ∈ D(0, 1), ux = 0 such that f ◦ (x; ux ) < 0 and, for any positive integer ux ; hence n, let unx := 2n|u x| |unx | =
1 2n
and
f ◦ (x; unx ) < 0.
(10)
Since wx is bounded (by c), by Lemma 2.3, there is a simple function w nx : J → RN such that w xn (J ) ⊂ wx (J0 ) and |wx (t) − w xn (t)|
r := |x0 | + cT + 1. Replacing ϕ(t, x) by λ(x)ϕ(t, x), we may assume that apart from conditions (ϕ1 )–(ϕ3 ), (ϕ6 ), the map ϕ satisfies an additional condition ϕ(t, x) = {0} for t ∈ J, x ∈ K with |x| r.
(15)
Recall Definition 1.8(ii) and take an open = such that K ⊂ = ⊂ cl = ⊂ U . CLAIM 1. For any ε > 0, there is η > 0 such that, for any x ∈ bd K and y ∈ [B(x, η) ∩ =] \ K, there is a measurable function u = ux,y : J → RN such that u(t) ∈ B(ϕ({t} × BK (x, ε)), ε) ∩ ∂f (y)− for a.a. t ∈ J . To see this take ε > 0 and fix x ∈ bd K. Assume first that |x| r and take a measurable wx : J → RN such that wx (t) ∈ ϕ(t, x) ∩ Lim infy→x,y∈K ∂f (y)− on
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305
J0 where J0 ⊂ J is of full measure. Since wx is bounded, by Lemma 2.3, there is a simple function wx : J → RN such that w x (J ) ⊂ wx (J0 ) and |w x (t) − wx (t)| < ε/2 on J0 .
(16)
It is clear that w x (t) ∈ Lim inf ∂f (y)− y→x,y∈K
for all t ∈ J0 . Since w x (·) has finite number of values, by the definition of Lim inf, there is 0 < η(x) < 2ε such that B(wx (t), ε/2) ∩ ∂f (y)− = ∅
(17)
for all t ∈ J0 provided y ∈ RN , y ∈ K and |y − x| < η(x). Choose x1 , . . . , xn ∈ bd K ∩ D(0, r) such that ni=1 B(xi , η(xi )/2) ⊃ bd K ∩ D(0, r) and define η(xi ) . i=1,...,n 2
η := min
Take any x ∈ bd K ∩ D(0, r). There is 1 i0 n such that x ∈ B(x0 , η(x0 )/2) where x0 := xi0 . Let w0 := w x0 and take y ∈ [B(x, η) ∩ =] \ K. Then y ∈ B(x0 , η(x0 )) and, by (17), for all t ∈ J , B(w0 (t), ε/2) ∩ ∂f (y)− = ∅.
(18)
There exists a measurable u = ux,y : J → RN such that u(t) ∈ B(w0 (t), ε/2) ∩ ∂f (y)− on J . Observe now that, by (16), (18), u(t) ∈ B(w0 (t), ε/2) ⊂ B(ϕ(t, x0 ), ε) ⊂ B(ϕ({t} × BK (x, ε)), ε) on J0 , so u satisfies our requirements. If x ∈ bd K but |x| r, then for any y ∈ [B(x, η) ∩ =] \ K, we put ux,y ≡ 0. Then, by (15), ux,y (t) ∈ ϕ(t, x) ∩ ∂f (y)− for all t ∈ J . II. Let ε > 0 and define ψ = ψε : J × K RN by the formula ψ(t, x) := cl conv D(ϕ({t} × DK (x, ε)), ε) for t ∈ J , x ∈ K. Evidently, ϕ(t, x) ⊂ ψ(t, x) on J ×K and, for a.a. t ∈ J , ψ(t, ·) is upper semicontinuous, ψ is bounded (by c + ε) and has compact convex values. It is clear that, for any x ∈ K, ψ(·, x) has a measurable selector. CLAIM 2. There is an extension ? = ?ε : J × RN RN of ψε . There is a Dugundji system for RN \ K (see [9, Lemma 3.1]), i.e. an indexed (by an index set S) family {(Us , as )}s∈S such that, for all s ∈ S, (1) Us ⊂ RN \ K, as ∈ bd K;
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(2) if x ∈ Us , then |x − as | 2dK (x); (3) U = {Us }s∈S is a locally finite open covering of RN \ K. One defines ?(t, x) = ?ε (t, x) :=
ψ(t, x) for x ∈ K, t ∈ J ,
λ (x)ψ(t, a ) for x ∈ K, t ∈ J , s s∈S s
where {λs }s∈S is a family of continuous functions λs : RN \ K → [0, 1] constituting a locally finite partition of unity subordinated to U, i.e. supp λs ⊂ Us for any s ∈ S. It is routine to check that ?: J × RN RN is well-defined and, for a.a. t ∈ J , the map ?(t, ·) is upper semicontinuous, ? is bounded (again by c + ε) and has compact convex values. Moreover, for any x ∈ RN , ?(·, x) has a measurable selection. III. Now choose 0 < η < ε as in Claim1. CLAIM 3. For all x ∈ [= ∩ B(K, η/2)]\K, there is a measurable w = wx : J → RN such that w(t) ∈ ?(t, x) ∩ ∂f (x)− a.e. on J . Indeed: let x ∈ [= ∩ B(K, η/2)] \ K, i.e. dK (x) < η/2 and x ∈ K. Let S(x) := {s ∈ S | x ∈ Us }. It is clear that the set S(x) is finite and λs (x)ψ(t, as ). ?(t, x) = s∈S(x)
Let s ∈ S(x); then |x − as | 2dK (x) < η. Hence x ∈ [B(as , η) ∩ =] \ K. By Claim 1, there exists a measurable us := uas ,x : J → RN such that us (t) ∈ ψ(t, as ) ∩ ∂f (x)− for a.a. t ∈ J . Define w = wx : J → RN by the formula λs (x)us (t) for t ∈ J. w(t) := s∈S(x)
Obviously w is measurable, w(t) ∈ ∂f (x)− (since the latter set is convex) and w(t) ∈ ?(t, x) by the very definition of ?. For any a ∈ R, let f a := {x ∈ cl = | f (x) a}. Clearly f a is closed and is of the form (S): it is represented by the locally Lipschitz function Dom(f ) x → g(x) := max{,(x), f (x) − a} where ,(x) := d(x, cl =) − d(x, RN \ cl =).
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307
CLAIM 4. It is clear that there is a = a(ε) > 0 such that f a ∩ D(0, r + 2) ⊂ B(K, η/2) ∩ =. Moreover, for any x ∈ bd f a ∩ D(0, r + 2), we have x ∈ =; hence for all y from a (sufficiently small) neighborhood of x, ,(y) < f (y)−a and thus g(y) = f (y)−a. It follows that x ∈ = \ K ⊂ U \ K and 0 ∈ ∂f (x) = ∂g(x). CLAIM 5. For sufficiently small ε > 0, the set f a (a = a(ε)) and ? = ?@ satisfy assumptions of Corollary 2.4. Therefore the set Lε := {u ∈ C(J, RN ) | u (t) ∈ ?ε (t, u(t)) for a.a t ∈ J, u(0) = x0 , u(J ) ⊂ f a(ε)} is an Rδ -subset of C(J, RN ). Indeed, as observed above ? satisfies conditions (ϕ1 )–(ϕ3 ) and, if x ∈ bd f a ∩ D(0, |x0 | + (c + ε)T + 1), then, for sufficiently small ε, x ∈ bd f a ∩ D(0, r + 2) and, in view of Claim 4, 0 ∈ ∂g(x). If x ∈ bd f a and |x| r + 2, then by Claim 4, x ∈ [B(K, η/2) ∩ =] \ K. Hence by Claim 3, there is a measurable w(·) such that w(t) ∈ ?(t, x) ∩ ∂f (x)− = ?(t, x) ∩ ∂g(x)− on J . IV. It follows from the construction that if 0 < ε1 ε2 , then ψε1 ⊂ ψε2 and ?ε1 ⊂ ?ε2 . Moreover, without loss of generality we may assume that also a(ε1 ) a(ε2 ). For a positive integer n, put Ln := L1/n . For all n, Ln is an Rδ -set (in in C(J, RN )) and the family {Ln }∞ n=1 is decreasing, i.e. Ln+1 ⊂ Ln . Let, as above, L denote the set of all solutions to (P), i.e. L = {u ∈ C(J, RN ) | u (t) ∈ ϕ(t, u(t)) for a.a t ∈ J, u(0) = x0 , u(J ) ∈ K}. CLAIM 6. L = ∞ n=1 Ln . Let u ∈ L. Then u(0) = x0 and u(t) ∈ K ⊂ f a(1/n) for all t ∈ J . Since u (t) ∈ ϕ(t, u(t)) ⊂ ψ1/n (t, u(t)) = ?1/n (t, u(t)) for a.a. t ∈ J , we gather that u ∈ Ln . If u ∈ Ln for any n, then clearly u(t) ∈ K on J ; hence u (t) ∈ ?1/n (t, u(t)) = ψ1/n (t, u(t)). This, together with the upper semicontinuity of ϕ(t, ·) for a.a. t ∈ J , implies that u (t) ∈ ϕ(t, u(t)) a.e. on J , i.e. u ∈ L.
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Clearly L, as the intersection of the decreasing family of compact Rδ -sets, is again an Rδ -set. ✷ The idea to use the Hyman result first appeared in [34]. 3. Applications In this section we turn attention to compact (strictly) regular sets. Assume that K, represented by a locally Lipschitz function f : Dom(f ) → R, is a compact set of the form (S). Definition 1.8 and Remark 1.9(4) imply that K is (resp. strictly) regular if there is an open set = with compact closure such that K ⊂ = ⊂ cl = ⊂ Dom(f ) and, for any x ∈ cl =\K, 0 ∈ ∂f (x) (resp. infx∈cl =\K |||∂f (x)||| m > 0). Therefore, M := infx∈bd = f (x) > 0 and, for any a ∈ (0, M), f a := {x ∈ cl = | f (x) a} ⊂ =. Moreover, inf{|||∂f (x)||| | x ∈ cl =, f (x) a} > 0. This shows that f a is a compact strictly regular set (represented by = x → g(x) := f (x) − a). In fact 0 ∈ ∂g(x) = ∂f (x) when x ∈ bd f a . After [17, Theorem 4.2], let us recall PROPOSITION 3.1. (i) Assume that K is regular. Then, for any 0 < a < M, the set f a is a strong deformation retract of =M := {x ∈ cl = | f (x) < M}. In particular, if b ∈ (0, M) and 0 < a b, then f a is a strong deformation retract of f b . (ii) If K is strictly regular, then K is a strong deformation retract of =M and of f a for all a ∈ [0, M). ˇ cohomology with rational coefficients. Denote by Hˇ ∗ (·) = {Hˇ q (·)}q0 the Cech If X is a topological space, then for each nonnegative integer q 0, Hˇ q (X) is a vector space over rationals; it is well-known that if X is a compact absolute neighborhood retract (e.g., a compact neighborhood retract in a metric space), then X is and, for alof finite-type, i.e. for all q 0, the space Hˇ q (X) is finite-dimensional
most all q, Hˇ q (X) = 0. Thus, the Euler characteristic χ(X) := q0 dim Hˇ q (X) is a well-defined integer. Proposition 3.1 implies that a compact strictly regular set K is a neighborhood retract in RN and, hence, is of finite-type and its Euler characteristic χ(K) is defined. The same holds for compact regular sets. Indeed, again after [17, Theorem 4.7], we have PROPOSITION 3.2. If a compact set K ⊂ RN is regular, then the Euler characteristic χ(K) is well-defined. Recall that a closed set A of a (topological) space X is a strong deformation retract of X if there
is a retraction r: X → A such that i ◦ r: X → X, where i: A → X is the inclusion, is homotopic relative to A with the identity idX on X – see, e.g., [43]. If A is a strong deformation retract in X, then, for any homology theory H∗ (resp. cohomology H ∗ ) the inclusion i induces an isomorphism H∗ (A) → H∗ (X) (resp. H ∗ (X) → H ∗ (A)).
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309
Having this we are ready for the main results of this section. THEOREM 3.3. Assume that K is a compact regular set with nontrivial Euler characteristic χ(K) = 0. Suppose that ϕ satisfies conditions (ϕ1 )–(ϕ3 ) and (ϕ6 ). Then there is a solution to the two-point boundary value problem u ∈ ϕ(t, u), t ∈ J, u ∈ K, u(0) = u(T ) ∈ K.
(PP)
The same holds if K is compact strictly regular with the nontrivial Euler characteristic and ϕ satisfies conditions (ϕ1 )–(ϕ3 ) and (ϕ5 ). In both cases, if J = [0, ∞) and for some T > 0, ϕ(t, x) ⊂ ϕ(t + T , x) (t 0, x ∈ K), then the differential inclusion u ∈ ϕ(t, u) admits a periodic solution. Proof. In the course of the proof we shall apply tricks introduced by Haddad and Lasry in [32] and used frequently by other authors (see, e.g., [4, Th. 5.3.4] and [40]). However our argument is more synthetic for we apply some ready fixed point results for set-valued maps with nonnecessarily convex values. It is clear that it is sufficient to prove the first part of Theorem 3.3. Let ε > 0. We shall use the notation employed in the proof of Theorem 1.11. Observe that, in view of the compactness of K, the modification of ϕ in the beginning of this proof is not necessary and there is 0 < a = a(ε) < ε such that f a ⊂ = and ? = ?ε satisfies the assumptions of Theorem 1.6 on f a . In view of Propositions 3.1, 3.2, f a is a neighborhood retract with the nontrivial Euler characteristic. Fix x ∈ f a and let Lε (x) be the set of all solutions to problem (P)ε (that is problem (P) with K, ϕ and x0 replaced by f a , ? and x). Evidently Lε (x) is an Rδ -set in C(J, RN ). For any τ ∈ [0, T ], let eτ : C(J, RN ) → RN be given by the formula eτ (u) := u(τ ). It is clear that a set-valued map D: [0, T ] × f a f a (given by D(τ, x) := eτ (Lε (x)), τ ∈ [0, T ], x ∈ f a ) is well-defined. Obviously D is upper semicontinuous and its values are the continuous images of Rδ -sets. Therefore D falls into the class of the so-called admissible maps considered in many papers (see, e.g., [27] (and the rich bibliography there), [40] and [7] where maps of this particular form were studied). In particular, the following result is true: If ψ: [0, 1] × X → X is an admissible map defined on a compact absolute neighborhood retract X with χ(X) = 0 and ψ(0, x) = {x} on X, then ψ(1, ·) has a fixed point, i.e. there is x ∈ X such that x ∈ ψ(1, x). Since, of course D(0, x) = x on K, this result implies that there is xε ∈ f a such that xε ∈ D(T , xε ), i.e. there is a solution uε to problem (P)ε such that uε (0) = xε = uε (T ). Consider a sequence (εn )∞ n=1 of positive numbers such that limn→∞ εn = 0 and let xn := xεn , un := uεn for any positive integer n. The compactness of cl = implies that (passing to a subsequence if necessary), limn→∞ xn = x ∈ K. Clearly the corresponding solutions un are equicontinuous; hence (again passing to a subsequence if necessary) un → u ∈ C(J, RN ) uniformly as n → ∞, u is absolutely continuous and
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un → u weakly in L1 (J, RN ). By the Convergence Theorem (already used in the proof of Theorem 1.2), u is the required solution to problem (PP). ✷ As usual we obtain immediately the following. COROLLARY 3.4. If K is a compact regular set with the nontrivial Euler characteristic and ψ: K RN is an upper semicontinuous set-valued map with nonempty compact convex values such that ∀x ∈ bd K
ψ(x) ∩ Lim inf ∂f (y)− = ∅, y→x, y∈K
then ψ has and equilibrium, i.e. there is x0 ∈ K such that 0 ∈ ψ(x0 ). The same holds if K is strictly regular and ∀x ∈ bd K
ψ(x) ∩ ∂f (x)− = ∅.
It is enough to consider x0 as the limit of a (sub)sequence (un (0)) where un is a solution to the problem un (t) ∈ ψ(un (t)) a.e. on [0, 1/n] such that un (0) = un (1/n) existing in view of Theorem 3.3. The presented argument follows ideas from [32] and [40]. Corollary 3.4 may be easily generalized to the case of upper hemicontinuous ψ. Recall that ψ is upper hemicontinuous if, for any p ∈ RN , the function K x → sup p, y ∈ R ∪ {+∞} y∈ψ(x)
is upper semicontinuous. THEOREM 3.5. Assume that K ⊂ RN is a compact regular set with the nontrivial Euler characteristic and let A: RN → L(RN , RN ), where L(RN , RN ) stands for the space of linear transformations RN → RN , be continuous. Let ψ: K RN be an upper hemicontinuous map with nonempty closed convex values such that ∀x ∈ bd K
ψ(x) ∩ Lim inf A(y)[∂f (y)− ] = ∅. y→x, y∈K
Then ψ has an equilibrium, i.e. there is x0 ∈ K such that 0 ∈ ψ(x0 ). The same holds if K is strictly regular and the above condition is replaced by ∀x ∈ bd K
ψ(x) ∩ A(x)[∂f (x)− ] = ∅.
(19)
Proof. The second statement follows from the first one since, in case K is strictly regular, for any x ∈ bd K, A(x)[∂f (x)− ] ⊂ Lim inf A(y)[∂f (y)− ]. y→x, y∈K
Observe that any upper semicontinuous set-valued map is upper hemicontinuous, but not
conversely (although an upper hemicontinuous map with compact convex values is upper semicontinuous).
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311
In order to proceed, observe that, for any ε > 0, both the map K x → cl conv D(ψ(DK (x, ε)), ε) and its extension E (defined similarly as ?ε in Claim 2 of the proof of Theorem 1.11) are upper hemicontinuous. Therefore, using arguments similar to those of the proof of Theorem 3.3, without loss of generality we may assume that K satisfies conditions of Theorem 1.6 (i.e. 0 ∈ ∂f (x) on bd K) and that ψ satisfies condition (19) (being an analogue of (ϕ5 ) stated in terms of ψ). Now suppose to the contrary that ψ has no zeros in K. The separation theorem implies that, for each x ∈ K, there is px ∈ RN with |px | = 1 such that infy∈ψ(x) px , y > 0. The upper hemicontinuity of ψ implies that U (x) := {z ∈ K | infy∈ψ(z) px , y > 0} is open (in K) and the collection {U (x)}x∈K forms an open covering of K. Let {λs : K → [0, 1]}s∈S be a locally finite partition of unity inscribed into this cover, i.e. for any
s ∈ S, there is xs ∈ K such that supp λs ⊂ U (xs ). Let, for x ∈ K, p(x) := s∈S λs (x)ps where ps := pxs . Clearly p: K → RN is continuous and ∀x ∈ K
inf p(x), y > 0.
y∈ψ(x)
(20)
We claim that there is x0 ∈ bd K such that sup p(x0 ), A(x0 )q 0.
q∈∂f (x0 )−
(21)
This is clearly a contradiction with (20) and our result is proved. To show (21), suppose that, for any x ∈ bd K, there is qx ∈ ∂f (x)− such that p(x), A(x)qx > 0. Using arguments similar to those used at the beginning of the proof of Theorem 1.6, we easily construct a continuous map q: K → RN such that p(x), A(x)q(x) > 0 on K and q(x) ∈ ∂f (x)− on bd K. But q fulfills assumptions of Corollary 3.4 and, hence, has a zero. This is a contradiction. ✷ Corollary 3.4 and Theorem 3.5 provide generalizations of results from [13] and [16] where epi-Lipschitz sets were studied and those from [5] where convex sets were dealt with (see also the bibliography in, e.g., [17]). Corollary 3.4 is a special case of a result from [17] where a much more general (infinite-dimensional) situation has been considered (however the present proof is independent). Theorem 3.5 is new: it extends [5, Theorem 6.4.12] and (in spite of being finite-dimensional) gives a fresh insight into results of [17]. In the forthcoming paper by present authors some of the results discussed above in Sections 1 and 3 will be extended to infinite dimensional Banach spaces. Acknowledgement We are grateful to the referees for some important suggestions.
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