[handbook of differential equations: ordinary differential equations] volume 3 || chapter 1...

C H A P T E R 1

Topological Principles for Ordinary Differential Equations

Jan Andres* Department of Mathematical Analysis, Faculty of Science, Palaclo] University, Tomkova 40,

779 O00lomouc-Hejgin, Czech Republic E-mail: andres@ infupol.cz

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1. Elements of ANR-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2. Elements of multivalued maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3. Some further preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. Applied fixed point principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1. Lefschetz fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2. Nielsen fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3. Fixed point index theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4. General methods for solvability of boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1. Continuation principles to boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2. Topological structure of solution sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3. Poincarr ' s operator approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5. Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1. Existence of bounded solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2. Solvability of boundary value problems with linear conditions . . . . . . . . . . . . . . . . . . . . 70

5.3. Existence of periodic and anti-periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6. Multiplicity results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.1. Several solutions of initial value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2. Several periodic and bounded solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3. Several anti-periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7. Remarks and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.1. Remarks and comments to general methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.2. Remarks and comments to existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.3. Remarks and comments to multiplicity results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

* Supported by the Council of Czech Government (MSM 6198959214).

H A N D B O O K OF DIFFERENTIAL EQUATIONS

Ordinary Differential Equations, volume 3

Edited by A. Cafiada, P. Dr~ibek and A. Fonda

�9 2006 Elsevier B.V. All rights reserved

Topological principles for ordinary differential equations

1. Introduction

The classical courses of ordinary differential equations (ODEs) start either with the Peano existence theorem (see, e.g., [54]) or with the Picard-Lindeltif existence and uniqueness theorem (see, e.g., [71 ]), both related to the Cauchy (initial value) problems

2 = f (t ,x), (1.1) x (0) = x0,

where f e C([O, r] x R n, Rn), and

I f ( t , x ) - f ( t , y ) l < , L l x - y l , f o r a l l t e [ O , r ] a n d x , y e N n, (1.2)

in the latter case. In fact, if f satisfies the Lipschitz condition (1.2), then "uniqueness implies existence"

even for boundary value problems with linear conditions that are "close" to x(0) = x0, as observed in [53]. Moreover, uniqueness implies in general (i.e. not necessarily, under (1.2)) continuous dependence of solutions on initial values (see, e.g., [54, Theorem 4.1 in Chapter 4.2]), and subsequently the Poincard translation operator Tr : R n ~ R n, at the time r > 0, along the trajectories of 2 -- f (t, x), defined as follows:

Tr (x0) "-- {x(r) Ix(.) is a solution of (1.1)}, (1.3)

is a homeomorphism (cf. [54, Theorem 4.4 in Chapter 4.2]). Hence, besides the existence, uniqueness is also a very important problem. W. Oflicz

[92] showed in 1932 that the set of continuous functions f : U --+ R n, where U is an open subset relative to [0, r] x ]R n, for which problem (1.1) with (0, x0) e U is not uniquely solvable, is meager, i.e. a set of the first Baire category. In other words, the generic continuous Cauchy problems (1.1) are solvable in a unique way. Therefore, no wonder that the first example of nonuniqueness was constructed only in 1925 by M.A. Lavrentev (cf. [71 ] and, for more information, see, e.g., [ 1]). The same is certainly also true for Carathrodory ODEs, because the notion of a classical (C l_) solution can be just replaced by the Carathdodory solution, i.e. absolutely continuous functions satisfying (1.1), almost everywhere (a.e.). The change is related to the application of the Lebesgue integral, instead of the Riemann integral.

On the other hand, H. Kneser [80] proved in 1923 that the sets of solutions to continuous Cauchy problems (1.1) are, at every time, continua (i.e. compact and connected). This result was later improved by M. Hukuhara [75] who proved that the solution set itself is a continuum in C([0, r], Rn). N. Aronszajn [41] specified in 1942 that these continua are R~-sets (see Definition 2.3 below), and as a subsequence, multivalued operators Tr in (1.3) become admissible in the sense of L. Grrniewicz (see Definition 2.5 below).

Obviously if, for f ( t , x) =_ f ( t + r, x), operator Tr admits a fixed point, say ~ ~ ]t~ n ,

i.e. } e Tr (~), then :~ determines a r-periodic solution of 2 = f ( t , x ) , and vice versa. This is one of stimulations why to study the fixed point theory for multivalued mappings in order to obtain periodic solutions of nonuniquely solvable ODEs. Since the regularity of

4 J. Andres

(multivalued) Poincarr 's operator Tr is the same (see Theorem 4.17 below) for differential inclusions k e F (t, x), where F is an upper Carathrodory mapping with nonempty, convex and compact values (see Definition 2.10 below), it is reasonable to study directly such differential inclusions with this respect. Moreover, initial value problems for differential inclusions are, unlike ODEs, typically nonuniquely solvable (cf. [42]) by which Poincarr 's operators are multivalued.

In this context, an interesting phenomenon occurs with respect to the Sharkovskii cycle coexistence theorem [95]. This theorem is based on a new ordering of the positive integers, namely

3 ~,5 ~ 7 ~ . . . t> 2 . 3 ~ 2 . 5 ~ , 2 . 7 ~, . . . ~,22 �9 3 ~,22 �9 5 t> 22 �9 7 ~ . . .

t> 2 n �9 3 ~ 2 n �9 5 t> 2 n �9 7 ~ . . . ~ 2 n+l . 3 ~ 2 n+l . 5 t> 2 n+l . 7 ~, -..

~> 2n+l t> 2 n t>.. . ~> 22t> 2 ~ 1,

saying that i f a continuous function g : R --+ R has a point o f period m with m ~ k (in the above Sharkovskii ordering), then it has also a point o f period k.

By a period, we mean the least period, i.e. a point a e ~ is a periodic point o f period m if gm (a) = a and gJ (a) ~ a, for 0 < j < m.

Now, consider the scalar ODE

:~ = f (t, x), f (t, x) -~ f (t + r, x), (1.4)

where f :[0, r] x ~ --+ II~ is a continuous function. Since

Tm = T~ o . . . o T~ = Tm~ Y

m t i m e s

holds for the Poincar6 translation operator Tr along the trajectories of Eq. (1.4), defined in (1.3), there is (in the case of uniqueness) an apparent one-to-one correspondence between m-periodic points of Tr and (subharmonic) mr-periodic solutions of (1.4). Nevertheless, the analogy of classical Sharkovskii 's theorem does not hold for subharmonics of (1.4). In fact, we only obtain an empty statement, because every bounded solution of (1.4) is, under the uniqueness assumption, either r-periodic or asymptotically r-periodic (see, e.g., [94, pp. 120-122]).

This handicap is due to the assumed uniqueness condition. On the other hand, in the lack of uniqueness, the multivalued operator TT in (1.3) is admissible (see Theorem 4.17 below) which in ~ means (cf. Definition 2.5 below) that Tr is upper semicontinuous (cf. Definition 2.4 below) and the sets of values consist either of single points or of compact intervals. In a series of our papers [16,29,36], we developed a version of the Sharkovskii cycle coexistence theorem which applies to (1.4) as follows:

THEOREM 1.1. I f Eq. (1.4) has an mr-periodic solution, then it also admits a kr-periodic solution, fo r every k ~ m, with at most two exceptions, where k ,~ m means that k is less

Topological principles for ordinary differential equations 5

Fig. 1. Braid ~r.

than m in the above Sharkovskii ordering of positive integers. In particular, if m 7~ 2 ~, for all k E N, then infinitely many (subharmonic) periodic solutions of (1.4) coexist.

REMARK 1.1. As pointed out, Theorem 1.1 holds only in the lack of uniqueness; otherwise, it is empty. On the other hand, the right-hand side of the given (multivalued) ODE can be a (multivalued upper) Carathrodory mapping with nonempty, convex and compact values (see Definition 2.10 below).

REMARK 1.2. Although, e.g., a 3r-periodic solution of (1.4) implies, for every k 6 N, with a possible exception for k = 2 or k = 4, 6, the existence of a kr-periodic solution of (1.4), it is very difficult to prove that such a solution exists. Observe that a 3r-periodic solution of (1.4) implies the existence of at least two more 3r-periodic solutions of (1.4).

The Sharkovskii phenomenon is essentially one-dimensional. On the other hand, it follows from T. Matsuoka's results in [87-89] that three (harmonic) r-periodic solutions of the planar (i.e. in R 2) system (1.4) imply "generically" the coexistence of infinitely many (subharmonic) kr-periodic solutions of (1.4), k 6 N. "Genericity" is this time understood in terms of the Artin braid group theory, i.e. with the exception of certain simplest braids, representing the three given harmonics.

The following theorem was presented in [8], on the basis of T. Matsuoka's results in papers [87-89].

THEOREM 1.2. Assume that a uniqueness condition is satisfied for planar system (1.4). Let three (harmonic) r-periodic solutions of (1.4) exist whose graphs are not conjugated to the braid o "m in B3/Z, for any integer m E N, where ~r is shown in Fig. 1, B3 /Z denotes the factor group of the Artin braid group B3 and Z is its center (for definitions, see, e.g., [22, Chapter 111.9]). Then there exist infinitely many (subharmonic) kr-periodic solutions of(1.4), k E N.

REMARK 1.3. In the absence of uniqueness, there occur serious obstructions, but Theo- rem 1.2 still seems to hold in many situations; for more details see [8].

REMARK 1.4. The application of the Nielsen theory considered in Section 3.2 below might determine the desired three harmonic solutions of (1.4). More precisely, it is more realistic to detect two harmonics by means of the related Nielsen number (see again Sec- tion 3.2 below), and the third one by means of the related fixed point index (see Section 3.3 below).

6 J. Andres

For n > 2, statements like Theorem 1.1 or Theorem 1.2 appear only rarely. Nevertheless, if f = ( f l , f2 . . . . . fn) has a special triangular structure, i.e.

f / ( x ) = f / ( X l . . . . , X n ) = f i ( X l . . . . . X i ) , i = 1 . . . . . n , (1.5)

then Theorem 1.1 can be extended to hold in R n (see [35]).

THEOREM 1.3. Under assumption (1.5), the conclusion of Theorem 1.1 remains valid in 1R n .

REMARK 1.5. Similarly to Theorem 1.1, Theorem 1.3 holds only in the lack of uniqueness. Without the special triangular structure (1.5), there is practically no chance to obtain an analogy to Theorem 1.1, for n ) 2.

There is also another motivation for the investigation of multivalued ODEs, i.e. differential inclusions, because of the strict connection with

(i) optimal control problems for ODEs, (ii) Filippov solutions of discontinuous ODEs,

(iii) implicit ODEs, etc. ad (i): Consider a control problem for

2 = f ( t , x, u), u ~_ U, (1.6)

where f :[0, r] x N n x R n --+ R n and u a U are control parameters such that u(t) ~ N n, for all t ~ [0, r ]. In order to solve a control problem for (1.6), we can define a multivalued map F (t, x) := { f (t, x, u) },eu. The solutions of (1.6) are those of

2 E F(t, x), (1.7)

and the same is true for a given control problem. For more details, see, e.g., [27,79]. ad (ii): If function f is discontinuous in x, then Carath6odory theory cannot be applied

for solving, e.g., (1.1). Making, however, the Filippov regularization of f , namely

:= N N 6>0 rC[O,r]xR n

#(r)--O

conv f ( O 6 ( ( t , x ) \ r)), (1.8)

where /x(r ) denotes the Lebesgue measure of the set r C R n and

06(y) := {z 6 [0, r l x IR n I lY - zl < ~},

multivalued F is well known (see [60]) to be again upper Carath6odory with nonempty, convex and compact values (cf. Definition 2.10 below), provided only f is measurable and satisfies I f ( t , x ) l <<, ~ + r for all ( t ,x) ~ [0, r] x R n, with some nonnegative constants or, ft. Thus, by a Filippov solution of 2 = f (t, x), it is so understood a Carath6odory


solution of (1.7), where F is defined in (1.8). As an example from physics, dry friction problems (see, e.g., [84,91 ]) can be solved in this way.

ad (iii): Let us consider the implicit differential equation

-- f (t, x, ,f), ( 1.9)

where f : [ 0 , r] x IR n • ~n _.._> ]l~n is a compact (continuous) map and the solutions are understood in the sense of Carath6odory. We can associate with (1.9) the following two differential inclusions:

Yc ~ F l ( t , x ) (1.10)

and

Yc E Fz( t ,x) , (1.11)

where Fl(t, x) := F i x ( f ( t , x , .)), i.e. the (nonempty, see [22, p. 560]) fixed point set of f ( t , x, .) w.r.t, the last variable, and F2 C F1 is a (multivalued) lower semicontinuous (see Definition 2.4 below) selection of F1. The sufficient condition for the existence of such a selection F2 reads (see, e.g., [22, Chapter III.11, pp. 558-559]):

dim Fix ( f (t , x, .)) = 0, for all (t, x) E [0, r] x IR n, (1.12)

where dim denotes the topological (covering) dimension. Denoting by S ( f ) , S(F1), S(F2) the sets of all solutions of initial value problems to

(1.9), (1.10), (1.11), respectively, one can prove (see [22, p. 560]) that, under (1.12), S ( f ) -- S(F1) C S(F2) g= 0. For more details, see [19] (cf. [22, Chapter III.11]).

Although there are several monographs devoted to multivalued ODEs (see, e.g., [22,42, 45,58,61,74,79,91,96,97]), topological principles were presented mainly for single-valued ODEs (besides [22,45,58] and [61] for differential inclusions, see, e.g., [62,64,65,82,83, 90]). Hence our main object will be topological principles for (multivalued) ODEs; whence the title. We will consider without special distinguishing differential equations as well as inclusions; both in Euclidean and Banach spaces. All solutions of problems under our consideration (even in Banach spaces) will be understood at least in the sense of Carath6odory. Thus, in view of the indicated relationship with problems (i)-(iii), many obtained results can be also employed for solving optimal control problems, problems for systems with variable structure, implicit boundary value problems, etc.

The reader exclusively interested in single-valued ODEs can simply read "continuous", instead of "upper semicontinuous" or "lower semicontinuous", and replace the inclusion symbol 6 by the equality - , in the given differential inclusions. If, in the single-valued case, the situation simplifies dramatically or if the obtained results can be significantly improved, then the appropriate remarks are still supplied.

We wished to prepare an as much as possible self-contained text. Nevertheless, the reader should be at least familiar with the elements of nonlinear analysis, in particular of fixed point theory, in order to understand the degree arguments, or so. Otherwise, we recommend the monographs [69] (in the single-valued case) and [22] (in the multivalued case).

8 J. Andres

Furthermore, one is also expected to know several classical results and notions from the standard courses of ODEs, functional analysis and the theory of integration like the Gron- wall inequality, the Arzel?a-Ascoli lemma, the Mazur Theorem, the Bochner integral, etc.

We will study mainly existence and multiplicity of bounded, periodic and anti-periodic solutions of (multivalued) ODEs. Since our approach consists in the application of the fixed point principles, these solutions will be either determined by, (e.g., r-periodic solutions x( t ) by the initial values x(0) via (1.3)) or directly identified (e.g., solutions of initial value problems (1.1)) with fixed points of the associated (Cauchy, Hammerstein, etc.) operators.

Although the usage of the relative degree (i.e. the fixed point index) arguments is rather traditional in this framework, it might not be so when the maps, representing, e.g., problems on noncompact intervals, operate in nonnormable Fr6chet spaces. This is due to the unpleasant locally convex topology possessing bounded subsets with an empty interior. We had therefore to develop with my colleagues our own fixed point index theory. The application of the Nielsen theory, for obtaining multiplicity criteria, is very delicate and quite rare, and the related problem is named after Jean Leray who posed it in 1950, at the first Interna- tional Congress of Mathematics held after World War II in Cambridge, Mass. We had also to develop a new multivalued Nielsen theory suitable for applications in this field. Before presenting general methods for solvability of boundary value problems in Section 4, we therefore make a sketch of the applied fixed point principles in Section 3. Hence besides Section 4, the main results are contained in Section 5 (Existence results) and Section 6 (Multiplicity results). The reference sources to our results and their comparison with those of other authors are finally commented in Section 7 (Remarks and comments).

2. Preliminaries

2.1. Elements o f ANR-spaces

In the entire text, all topological spaces will be metric and, in particular, all topological vector spaces will be at least Fr6chet. Let us recall that by a Fr~chet space, we understand a complete (metrizable) locally convex space. Its topology can be generated by a countable family of seminorms. If it is normable, then it becomes Banach.

DEFINITION 2.1. A (metrizable) space X is an absolute neighbourhood retract (ANR) if, for each (metrizable) Y and every closed A C Y, each continuous mapping f : A --+ X is extendable over some neighbourhood of A.

PROPOSITION 2.1. (i) I f X is an ANR, then any open subset o f X is an ANR and any neighbourhood

retract o f X is an ANR. (ii) X is an ANR if and only if it is a neighbourhood retract o f every (metrizable) space

in which it is embedded as a closed subset. (iii) X is an ANR if and only if it is a neighbourhood retract o f some normed linear

space, i.e. if and only if it is a retract o f some open subset o f a normed space.


(iv) I f X is a retract o f an open subset o f a convex set in a Fr~chet space, then it is an

ANR .

(v) I f X1, X2 are closed ANRs such that X1 fq X2 is an ANR, then X1 U X2 is an ANR.

(vi) Any finite union o f closed convex sets in a Fr~chet space is an ANR.

(vii) I f each x ~ X admits a neighbourhood that is an ANR, then X is an ANR.

DEFINITION 2.2. A (metrizable) space X is an absolute retract (AR) if, for each (metrizable) Y and every closed A C Y, each continuous mapping f : A --+ X is extendable over Y.

PROPOSITION 2.2. (i) X is an AR i f and only i f it is a contractible (i.e. homotopically equivalent to a one

point space) ANR.

(ii) X is an AR if and only i f it is a retract o f every (metrizable) space in which it is

embedded as a closed subset.

(iii) I f X is an AR and A is a retract o f X , then A is an AR.

(iv) I f X is homeomorphic to Y and X is an AR, then so is Y.

(v) X is an AR i f and only i f it is a retract o f some normed space.

(vi) I f X is a retract o f a convex subset o f a Fr~chet space, then it is an AR.

(vii) I f X1, X2 are closed ARs such that X1 N X2 is an AR, then X1 U X2 is an AR.

Furthermore, it is well known that every ANR X is locally contractible (i.e. for each x 6 X and a neighbourhood U of x, there exists a neighbourhood V of x that is contractible in U) and, as follows from Proposition 2.2(i) that every AR X is contractible (i.e. if idx : X --+ X is homotopic to a constant map).

DEFINITION 2.3. X is called an R~-set if, there exists a decreasing sequence {Xn} of

compact, contractible sets Xn such that X = ~{Xn I n = 1, 2 . . . . }.

Although contractible spaces need not be ARs, X is an R~-set if and only if it is an intersection of a decreasing sequence of compacts ARs. Moreover, every R~-set is acyclic

w.r.t, any continuous theory of homology (e.g., the (~ech homology), i.e. homologically equivalent to a one point space, and so it is in particular nonempty, compact and connected.

The following hierarchies hold for metric spaces:

contractible C acyclic

t3

convex C A R C ANR,

compact + convex C compact A R C compact + contractible C R~ C compact + acyclic, and all the above inclusions are proper.

For more details, see [47] (cf. also [22,67,69]).

10 J. Andres

2.2. Elements of multivalued maps

In what follows, by a multivalued map ~o'X ---0 y, i.e. qg"X --+ 2Y\{0}, we mean the one with at least nonempty, closed values.

DEFINITION 2.4. A map qg" X ----o y is said to be upper semicontinuous (u.s.c.) if, for every open U C Y, the set {x 6 X I qg(x) C U} is open in X. It is said to be lower semicontinuous (1.s.c.) if, for every open U C Y, the set {x 6 X I ~0(x) A U ~ 0} is open in X. If it is both u.s.c, and 1.s.c., then it is called continuous.

Obviously, in the single-valued case, if f : X --+ Y is u.s.c, or 1.s.c., then it is continuous. Moreover, the compact-valued map qg:X ---o Y is continuous if and only if it is Hausdorff-continuous, i.e. continuous w.r.t, the metric d in X and the Hausdorff- metric dn in {B C Y I B is nonempty and bounded}, where dH(A, B) := inf{e > 0 1 A C Oe(B) and B C Oe(A)} and Oe(B) := {x 6 X I3y E B: d(x, y) < e}. Every u.s.c, map qg:X --o y has a closed graph 1-'~0, but not vice versa. Nevertheless, if the graph 17~o of a compact map qg:X ---o Y is closed, then q9 is u.s.c.

The important role will be played by the following class of admissible maps in the sense of L. G6rniewicz.

DEFINITION 2.5. Assume that we have a diagram X ,~ p F q) Y (F is a metric space), where p : F =~ X is a continuous Vietoris map, namely

(i) p is onto, i.e. p (F ) = X, (ii) p is proper, i.e. p -1 (K) is compact, for every compact K C X,

(iii) p -1 (x) is acyclic, for every x 6 X, where acyclicity is understood in the sense of the (~ech homology functor with compact carriers and coefficients in the field Q of rationals,

and q : F --+ Y is a continuous map. The map q9 : X ---o y is called admissible if it is induced by qg(x) = q(p-1 (x)), for every x E X. We, therefore, identify the admissible map q9 with the pair (p, q) called an admissible (selected) pair.

Pl> DEFINITION 2.6. Let X ~ F0 q0> y and X ~ - - 171 Y be two admissible maps, i.e.

- 1 - 1 qg0 -- q0 o P0 and ~Pl = q l o pl " We say that tp0 is admissibly homotopic to q91 (written

tp0 ~ qgl or (P0, q0) "~ (Pl, ql)) if there exists an admissible map X x [0, 1] ~ 170 q > Y such that the following diagram is commutative:

Pi qi

X ~. I"i > Y

1 I ki fi / / / q

X x [0, 1] < F

for ki (x) -- (x, i), i = 0, 1, and f/" I" i ~ 17 is a homeomorphism onto p

i.e. koPo = Pfo, qo = qfo, kl Pl -- Pfl and ql = q f l .

- l ( x • 1,


Thus, admissible maps are always u.s.c, with nonempty, compact and connected values. Moreover, their class is closed w.r.t, finite compositions, i.e. a finite composition of admissible maps is also admissible. In fact, a map is admissible if and only if it is a finite composition of acyclic maps with compact values, i.e.u.s.c, maps with acyclic and compact values.

The class of admissible maps so contains u.s.c, maps with convex and compact values, u.s.c, maps with contractible and compact values, R6-maps (i.e.u.s.c. maps with R~-values), acyclic maps with compact values and their compositions.

The class of compact admissible maps qg:X ~ Y, i.e. ~0(X) is compact, will be denoted by K(X, Y), or simply by K(X), provided q9 is a self-map (an endomorphism). If the admissible homotopy in Definition 2.6 is still compact, then we say that 99o E K(X, Y) and q91 E K(X, Y) are compactly admissibly homotopic.

Another important class of admissible maps are condensing admissible maps denoted by C(X, Y). For this, we need to recall the notion of a measure of noncompactness (MNC).

Let E be a Fr6chet space endowed with a countable family of seminorms II.lls, s E S (S is the index set), generating the locally convex topology. Denoting by/3 = /3 (E) the set of nonempty, bounded subsets of E, we can give

DEFINITION 2.7. The family of functions ot -- {~s}sES" 13 --+ [0, CX~) s, where us(B) "= inf{3 > 0l B 6 /3 admits a finite covering by the sets of diams ~< 6}, s 6 S, for B E/3, is called the Kuratowski measure of noncompactness and the family of functions y = {Ys}s~s'13 --+ [0, cx~) s, where ys(B) := inf{6 > 0l B E /3 has a finite es-net}, s E S, for B E/3, is called the Hausdorffmeasure ofnoncompactness.

These MNC are related as follows"

v(B) <~ c~(B) <~ 2v(B), i.e. Vs(B) <~ ~s(B) <~ 2ys(B), for each s E S.

Moreover, they satisfy the following properties:

PROPOSITION 2.3. Assume that B, B1, B2 E 13. Then we have (component-wise)" (/z 1) (regularity) lZ (B) -- 0 r B is compact, (/z2) (nonsingularity) {b} E/3 :=~ {b} U B E 13 and/z({b} U B) = / z (B) , (#3) (monotonicity) B1 C B2 ::~/z(B1) ~</z(B2), (#4) (closed convex hull) #(c--6-~ B) -- /z(B), (#5) (closure) #(B) = #(B) , (#6) (Kuratowski condition) decreasing sequence of closed sets Bn ~ 13 with

lim # (Bn) = 0 n---+ ~

~, n { B n l n - l , 2 . . . . } r

(/z7) (semiadditivity)/z(B1 -Jr- B2) ~</z(B1) 4-/z(B2), (#8) (union) #(B1 U B2) = max{#(B1),/z(B2)}, (#9) (intersection)/z(B1 N B2) = min{#(B1),/z(B2)},

(/ZlO) (seminorm) #()~B) = I)~l/z(B), for every )~ ~ ~, and/Z(Bl U B2) ~</z(Ba) + //~(B2),

12 J. Andres

where/z denotes either ot or F.

DEFINITION 2.8. A bounded mapping ~0: E D U ---o E, i.e. qg(B) E 13, for 13 9 B C U, is said to be ~z-condensing (shortly, condensing) if/z(qg(B)) < / z ( B ) , whenever 13 9 B C U and/z (B) > 0, or equivalently, if #(~0(B)) ~>/z(B) implies/z(B) = 0, whenever 13 9 B C U, where/z = {/zs }sEs :13 ~ [0, cx~) s is a family of functions satisfying at least conditions (/zl)-(/zs). Analogously, a bounded mapping ~0 : E D U ---o E is said to be a k-set contraction w.r.t./z = {/zs}s~S :13 --+ [0, cx~) s satisfying at least conditions (/zl)-(/zs) (shortly, a k-contraction or a set-contraction) if/z(~0(B)) ~< k/z(B), for some k 6 [0, 1), whenever B ~ B C U .

Obviously, any set-contraction is condensing and both a-condensing and F-condensing maps are/z-condensing. Furthermore, compact maps or contractions with compact values (in vector spaces, also their sum) are well known to be (or, F)-set-contractions, and so (c~, F)-condensing.

Besides semicontinuous maps, measurable and semi-Carath6odory maps will be also of importance. Hence, assume that Y is a separable metric space and (f2, L/, v) is a measurable space, i.e. a set f2 equipped with a-algebra/.4 of subsets and a countably additive measure v on L/. A typical example is when f2 is a bounded domain in R n, equipped with the Lebesgue measure.

DEFINITION 2.9. A map ~0:f2 ---o y is called strongly measurable if there exists a sequence of step multivalued maps ~0n :g2 ---o y such that dI-I(~On(co), ~0(co)) --+ 0, for a.a. co 6 S2, as n ~ ~ . In the single-valued case, one can simply replace multivalued step maps by single-valued step maps and dI4(~On(co), ~o(co)) by II~0n (co) - ~o(co) II.

A map ~p : f2 ~ Y is called measurable if {co 6 f2 I qg(co) C V} 6 L/, for each open V c Y. A map qg:f2 --o y is called weakly measurable if {co E S2 I ~o(co) C V} E L/, for each

closed V C Y.

Obviously, if ~o is strongly measurable, then it is measurable and if ~p is measurable, then it is also weakly measurable. If ~o has compact values, then the notions of measurability and weak measurability coincide. In separable Banach spaces Y, the notions of strong measurability and measurability coincide for multivalued maps with compact values as well as for single-valued maps (see [78, Theorem 1.3.1 on pp. 45-49]). If Y is a not necessarily separable Banach space, then a strongly measurable map ~p:f2 --~ Y with compact values has a single-valued strongly measurable selection (see, e.g., [58, Proposition 3.4(b) on pp. 25- 26]). Furthermore, if Y is a separable complete space, then every measurable qg:f2 ---o y has, according to the Kuratowski-Ryll-Nardzewski theorem (see, e.g., [22, Theorem 3.49 in Chapter 1.3]), a single-valued measurable selection.

Now, let f2 = [0, a] be equipped with the Lebesgue measure and X, Y be Banach.

DEFINITION 2.10. A map q9 : [0, a] • X ---o Y with nonempty, compact and convex values is called u-Carath~odory (resp. 1-Carath~odory, resp. Carath~odory) if it satisfies

(i) t ---o qg(t, x) is strongly measurable, for every x 6 X, (ii) x ~ qg(t, x) is u.s.c. (resp. 1.s.c., resp. continuous), for almost all t 6 [0, a],


(iii) IlYlIv ~ r(t)(1 + Ilxllg), for every (t, x) 6 [0, a] x X, y E ~0(t, x), where r ' [ 0 , a] --+ [0, ec) is an integrable function.

For X = ~ m and Y - - ~n, one can state

PROPOSITION 2.4. (i) Carathdodory maps are product-measurable (i.e. measurable as the whole (t, x) ---o

qo(t, x)), and (ii) they possess a single-valued Carathdodory selection.

It need not be so for u-Carath6odory or 1-Carath6odory maps. Nevertheless, for u-Carath6odory maps, we have at least (again X -- ]1~ m and Y = IRn).

PROPOSITION 2.5. u-Carathdodory maps (in the sense of Definition 2.10) are weakly superpositionally measurable, i.e. the composition qg(t,q(t)) admits, for every q C ([0, a], Rm), a single-valued measurable selection. If they are still product-measurable, then they are also superpositionally measurable, i.e. the composition q)(t, q(t)) is measurable, for every q E C ([0, a], ] ~ m ) .

REMARK 2.1. If X, Y are separable Banach spaces and ~0:X ---o Y is a Carath6odory mapping, then ~0 is also superpositionally measurable, i.e. qg(t, q(t)) is measurable, for every q 6 C([0, a], X) (see [78, Theorem 1.3.4 on p. 56]). Under the same assumptions, Proposition 2.4 can be appropriately generalized (see [73, Proposition 7.9 on p. 229 and Proposition 7.23 on pp. 234-235]).

If q):X ---o Y is only u-Carath6odory and X, Y are (not necessarily separable) Banach spaces, then q9 is weakly superpositionally measurable, i.e. qg(t,q(t)) admits a single- valued measurable selection, for every q 6 C([0, a], X) (see, e.g., [58, Proposition 3.5 on pp. 26-27] or [78, Theorem 1.3.5 on pp. 57-58]).

For more details, see [22,40,58,67,73,78].

2.3. Some further preliminaries

Assume we have again a diagram (see Definition 2.5)

X~, p [' q > y

where p" 1-" ------5 X is a Vietoris map and q ' F > Y is continuous. Taking ~0(x) = q(p - l ( x ) ) , for every x E X, and denoting as

Fix(p, q) = Fix(q)) "-- {x E X Ix E qg(x)},

C(p, q):-- {z E r I p ( z ) - q(z)}

14 J. Andres

the sets off ixed points and coincidence points of the admissible pair (p, q), it is clear that p ( C ( p , q)) - Fix(p, q), and so

F ix (p ,q ) ~:0 ~, F C ( p , q ) ~ 0 .

The following Aronszajn-Browder-Gupta-type result (see [21, Theorem 3.15]; cf. [22, Theorem 1.4]) is very important in order to say something about the topological structure

of Fix(qg).

PROPOSITION 2.6. Let X be a metric space, E a Fr~chet space, {Uk} a base of open convex symmetric neighbourhoods of the origin in E, and let qg : X --o E be a u.s.c proper map with compact values. Assume that there is a convex symmetric subset C of E and a sequence of compact, convex-valued u.s.c, proper maps qgk : X --o E such that

(i) qgk(x) C q g ( O l / k ( X ) ) -Jr- Uk,for every x E X, where

(ii) for every k ~ 1, there is a convex, symmetric set Vk C Uk N C such that Vk is closed in E and 0 E qg(x) implies qgk(X) n Vk ~ 0,

(iii) for every k ~ 1 and every u E Vk, the inclusion u E qgk(x) has an acyclic set of

solutions. Then the set S -- {x 6 X I qg(x) n {0} ~ 0} is compact and acyclic.

Now, let us assume that E is a Fr6chet space, C is a convex subset of E, U is an open subset of C, /z : B --+ [0, cx~) s is a measure of noncompactness satisfying at least conditions ( # l ) - ( t t s ) in Proposition 2.3 (see Definitions 2.7 and 2.8).

If q9 6 C(U, C), then Fix(qg) can be proved relatively compact. We can say more about

Fix(~).

DEFINITION 2.1 1. Let (p, q) E C(U, C). A nonempty, compact, convex set S C C is called a fundamental set if:

(i) q ( p - l ( u n S)) C S,

(ii) if x 6 conv(qg(x) U S), then x E S. For a homotopy X E C(U x [0, 1 ], C), S C C is called fundamental if it is fundamental to

X (.,)~), for each ;~ 6 [0, 1].

PROPOSITION 2.7. Assume (p, q) E C(U, C). (i) I f S is a fundamental set for (p, q), then Fix(p, q) C S.

(ii) Intersection of fundamental sets, for (p, q), is also fundamental, for (p, q). (iii) The family of all fundamental sets for (p, q) is nonempty. (iv) I f S is a fundamental set for X ~ C(U x [0, 1], C) and P C S, then the set

conv(x((U N S) x [0, 1]) U P) is also fundamental.

For more details, see [22,67], and the references therein.


3. Applied fixed point principles

3.1. Lefschetz fixed point theorems

We start with the Lefschetz theory, because it is a base for our further investigation. More precisely, the generalized Lefschetz number can be used for the definition of essential classes in the Nielsen theory as well as the possible normalization property of the fixed point index. We restrict ourselves only to the presentation of necessary facts.

Consider a multivalued map ~o'X --o X and assume that (i) X is a (metric) ANR-space, e.g., a retract of an open subset of a convex set in a

Fr6chet space, (ii) ~0 is a compact (i.e. ~0(X) is compact) composition of an R~-map p -1 "X ---o F and

a continuous (single-valued) map q ' F --~ X, namely ~0 = q o p - l , where F is a metric space.

Then an integer A((p) -- A(p , q), called the generalized Lefschetz number for ~p 6 IE(X), is well-defined (see, e.g., [12; 22, Chapter 1.6; 67]) and A(~0) # 0 implies that

Fix(~0) "-- {x 6 X Ix 6 ~0(x)} # 0.

Moreover, A is a homotopy invariant, namely if ~0 is compactly homotopic (in the same class of maps) with ~ ' X ---0 X, then A (~p) -- A (~').

In order to define the generalized Lefschetz number, one should be familiar with the elements of algebraic topology, in particular, of homology theory. Therefore, we only briefly sketch this definition without proofs. For more details, we recommend [51,68] (in the single-valued case) and [12,22,67] (in the multivalued case).

At first, we recall some algebraic preliminaries. In what follows, all vector spaces are taken over Q. Let f : E --+ E be an endomorphism of a finite-dimensional vector space E. If Vl, . . . , Vn is a basis for E, then we can write

f (vi) -- ~f~ a i jv j , j -1

for all i -- 1, . . . , n.

The matrix [aij ] is called the matrix of f (with respect to the basis Vl . . . . . Vn). Let A = [aij] be an (n z n)-matrix; then the trace of A is defined as ~in __1 aii. If f " E --+ E is an endomorphism of a finite-dimensional vector space E, then the trace of f , written t r ( f ) , is the trace of the matrix of f with respect to some basis for E. If E is a trivial vector space then, by definition, t r ( f ) - -0 . It is a standard result that the definition of the trace of an endomorphism is independent of the choice of the basis for E.

Hence, let E = { Eq } be a graded vector space of a finite type. If f = { fq } is an endomorphism of degree zero of such a graded vector space, then the

(ordinary) Lefschetz number )~(f) of f is defined by

X ( f ) - Z ( - 1 ) q tr(fq). q

16 J. Andres

Let f " E --+ E be an endomorphism of an arbitrary vector space E. Denote by f n . E --+ E the nth iterate of f and observe that the kernels

ker f C ker f2 C . . . C ker f n C "'"

form an increasing sequence of subspaces of E. Let us now put

N ( f ) = U k e r f n and / ~ - E / N ( f ) . n

Clearly, f maps N ( f ) into itself and, therefore, induces the endomorphism f " E --+ E on the factor space E = E / N ( f ) .

Let f ' E --~ E be an endomorphism of a vector space E. Assume that dim E < e~. In this case, we define the generalized trace Tr(f ) of f by putting Tr(f ) = t r ( f ) .

LEMMA 3.1. Let f " E --+ E be an endomorphism. I f dim E < oo, then Tr(f) -- t r ( f ) .

For the proof, see [22]. Let f = {fq } be an endomorphism of degree zero of a graded vector space E -- {Eq}.

We say that f is a Leray endomorphism if the graded vector space E = {Eq } is of finite type. For such an f , we define the (generalized) Lefschetz number A ( f ) of f by putting

A ( f ) = E ( - 1)q Tr(fq) . q

It is immediate from Lemma 3.1 that

LEMMA 3.2. Let f ' E --+ E be an endomorphism of degree zero, i.e., f = {fq} and fq " Eq --+ Eq is a linear map. I f E is a graded vector space o f finite type, then A ( f ) --

)~(f).

Now, the Lefschetz number will be defined for admissible compact mappings. For our needs in the sequel, it is enough to consider only the compact compositions of Rs-maps and continuous single-valued maps as above (by which Lefschetz sets simplify into Lefschetz numbers). Let q)" E ---o E be an admissible compact map and (p, q) C ~0 be a selected pair of ~0. Then the induced homomorphism q, o p , l ' H , ( E ) --+ H , ( E ) is an endomorphism of the graded vector space H , ( E ) into itself. So, we can define the Lefschetz number A(p, q) of the pair (p, q) by putting A(p, q) = A(q, o p ,1 ) , provided the Lefschetz number A (q, o p , 1) is well-defined.

It allows us to define the Lefschetz set A of q) as follows"

A(~0) = {A(p , q)I (P, q) C ~0}.

In what follows, we say that the Lefschetz set A(~0) of ~0 is well-defined if, for every (p, q) C q), the Lefschetz number A(p, q) of (p, q) is defined.

Moreover, from the homotopy property of A, we get:


LEMMA 3.3. (i) I f ~o, O " E ---o E are compactly homotopic (q) ~ gr), then: A(q)) n A(~p) 5/: 0.

(ii) I f q ) E --o E is admissible and E is acyclic, then the Lefschetz set A(q)) is well- defined and A (99) -- { 1 }.

It is useful to formulate

THEOREM 3.1 (Coincidence theorem). Let U be an open subset of a finite dimensional normed space E. Consider the following diagram:

U,( p I-" q> u

in which q is a compact map. Then the Lefschetz number A(p, q) of the pair (p, q), given by the formula

A ( p , q ) - - A ( q . o p . 1 ) ,

is well-defined, and A(p, q) :/: 0 implies that p(y) - q (y ) , f o r some y E F.

Theorem 3.1 can be reformulated in terms of multivalued mappings as follows. Let U C E be the same as in Theorem 3.1 and let q) 'U --o U be a compact, admissible

map, i.e., q) 6 IN(U). We let A(q)) -- {A(p, q) I (P,q) C q)}, where A(p , q) -- A(q , o p , 1 ) . Then we have:

THEOREM 3.2.

(i) The set A(q)) is well-defined, i.e. for every (p, q) C q), the generalized Lefschetz number A (p, q) of the pair (p, q) is well-defined, and

(ii) A (q)) --/: {0} implies that the set Fix(q)) "-- {x 6 U Ix 6 q)(x)} is nonempty.

Theorem 3.2 can be generalized, by means of the Schauder-like approximation technique (for more details, see [22]), for compact admissible maps 99 6 IK on ANR-spaces, e.g., on retracts of open subsets of convex sets in Fr6chet spaces, as follows.

THEOREM 3.3 (The Lefschetz fixed point theorem). Let X be an ANR-space, e.g., a retract of an open subset U of a convex set in a Frdchet space. Assume, furthermore, that 99 6 IN(X). Then:

(i) the Lefschetz set A (q)) of q) is well-defined, (ii) if A (q)) 7~ {0}, then Fix(q)) 7~ 0.

REMARK 3.1. If admissible map 99 c K(X) is a composition of an R~-map and a continuous single-valued map, then A (q)) is an integer. If, in particular, X is an AR-space, then a ( q ) ) - 1.

REMARK 3.2. The definition of a generalized Lefschetz number for condensing maps is far from to be obvious, and so it can not be used as a normalization property for the related

18 J. Andres

fixed points index. Roughly speaking, it requires to assume additionally the existence of a compact attractor or to impose some additional restrictions on the set X like to be a special neighbourhood retract of a Fr6chet space (cf. [68]).

3.2. Nielsen fixed point theorems

The standard Nielsen theory allows us to obtain the lower estimate of the number of fixed points. More precisely, if f ' X --+ X is a compact (continuous) map on a (metric) ANR- space X, then a nonnegative integer N ( f ) , called the Nielsen number of f , is defined such that

�9 N ( f ) <, # F i x ( f ) : = card{)? ~ X Lf(~) = ~}, �9 N ( f ) = N ( f ) , for any compact f " X ~ X which is compactly homotopic to f , i.e. if

there is a compact map h ' X x [0, 1] --+ X such that h0 = f , h l = f , where h t (x) :-- h(x, t), for t ~ [0, 1].

Given a compact f " X --~ X on X a ANR, we say that x, y a F i x ( f ) are Nielsen related if there exists a path u ' [ 0 , 1] ~ X such that u(0) = x, u(1) = y, and u, f ( u ) are homotopic keeping the endpoints fixed. Since the Nielsen relation is an equivalence, F i x ( f ) splits into fixed point classes. Since the classes are open and f is compact, we have a finite number of fixed point classes.

If, for a Nielsen class N" c F ix ( f ) , we have ind(N', f ) 7~ 0, i.e. if the associated fixed point index is nontrivial, then N" is called essential. The Nielsen number N ( f ) is then defined to be the number of essential Nielsen classes. For more details, see, e.g., [77].

To compute N ( f ) can be a difficult task. In the multivalued case, the situation is even more delicate, because the above definition can not be directly generalized. Thus, we only indicate this subtle definition again. Nevertheless, in the single-valued case, these definitions are equivalent.

Consider a multivalued map ~0"X ---o X and assume that (i) X is a connected ANR-space, e.g., a connected retract of an open subset of a convex

set in a Fr6chet space, (ii) X has a finitely generated abelian fundamental group,

(iii) ~0 is a compact (i.e. ~0(X) is compact) composition of an R~-map p-1 "X ---o F and a continuous (single-valued) map q ' F --+ X, namely ~0 = q o p - I , where F is a metric space.

Then a nonnegative integer N(q)) - N(p , q),l called the Nielsen number for q) 6 K, exists (see [24] and [22, Chapter 1.10] or [12]) such that N(~0) ~< #C(~0), where

#C(q)) = #C(p, q) "= card{z e F [ p(z) = q(z) }

and N(qg0) = N(qgl) , for compactly homotopic maps q)0 "~ qgl.

REMARK 3.3. Condition (ii) is satisfied, provided X is the toms qr n (7rl (~n ) = z n ) and it can be avoided if X is compact and q = id is the identity (cf. [5]).

1We should write more correctly NH(qg) = NH( p, q), because it is in fact (mod H)-Nielsen number, as can be seen below. For the sake of simplicity, we omit the index H in the following sections.


REMARK 3.4 (Important). We have a counter-example in [24] (cf. [22, Example 10.1 in Chapter 1.10]) that, under the above assumptions (i)-(iii), the Nielsen number N(qg) is rather the topological invariant for the number of essential classes of coincidences than of fixed points. On the other hand, for a compact X and q = id, N (q9) gives even without (ii) a lower estimate of the number of fixed points of q9 (see [5]), i.e. N (q9) <~ # Fix(qg), where Fix(qg) := card{x ~ X Ix ~ qg(x)}. We have conjectured in [381 that if q9 = q o p-1 assumes only simply connected values, then also N(qg) ~< # Fix(qg).

The following sketch demonstrates how subtle is the definition of the Nielsen number for multivalued maps. Let

X ,(po F qo ,(pl ql > Y and X F > Y

be two admissible maps. If (p0, q0) "~ (Pl, ql), i.e. if (p0, q0) is admissibly homotopic to (Pl, ql) (see Defin-

ition 2.6), and h : Y --+ Z is a continuous map, then we write (P0, hqo) ~ (p, hq) . We

say that a multivalued map X < p F q> Y represents a single-valued map p" X --+ Y if q = pp. Now, we assume that X -- Y and we are going to estimate the cardinality of the coincidence set

C(p , q ) " - {z ~ r I p (z )= q(z)}.

We begin by defining a Nielsen-type relation on C (p, q). This definition requires the fol-

lowing conditions on X < p F q> Y" (i') X, Y are connected, locally contractible metric spaces (observe that then they

admit universal coverings), (iii') p : F ------> X is a Vietoris map, (iii") for any x ~ X, the restriction ql -- q l p - l ( x ) ' p - l ( x ) --~ Y admits a lift ql to the

universal covering space ( p r " Y --~ Y)"

Y

ql/ / /

/

-1 (X) > Y ql

pr

Let us note that the following implications hold: (i) =r (i'), (iii) =~ (iii'), (iii"). Consider a single-valued map p ' X ~ Y between two spaces admitting universal cov-

erings p x ' X =:~ X and p r ' X =r Y. Let Ox = { a ' X --+ X I pxt~ = p x } be the group of natural transformations of the covering p x. Then the map p admits a lift ~ ' X --+ Y. We can define a homomorphism ~., "Ox --+ 0r by the equality

= Ox,

20 J. Andres

It is well known that there is an isomorphism between the fundamental group 7rl ( X )

and Ox which may be described as follows. We fix points x0 6 X, Y 6 X and a loop co ' l --+ X based at x0. Let ~ denote the unique lift of o) starting from Y0. We subordi- nate to [co] 6 Zrl(X, x0) the unique transformation from Ox sending ~(0) to ~(1). Then the homomorphism "fi!'Ox ~ Or, corresponds to the induced homomorphism between the fundamental groups p#" Jr I (X, x0) --+ Jr 1 ( Y,/9 (x0) ).

It can be shown that, under the assumptions (i'), (iii'), (iii"), a multivalued map (p, q) admits a lift to a multivalued map between the universal coverings. These lifts will split the coincidence set C(p, q) into Nielsen classes. Besides that the pair (p, q) induces a homomorphism Ox --+ OF giving the Reidemeister set.

We start with the following lemma.

LEMMA 3.4. Suppose we are given Y, a locally contractible metric space, F a metric space, F0 C F a compact subspace, q" F --+ Y, "qo" F0 ~ Y continuous maps for which

the diagram

, . . . ,

qo Fo >

F ~ Y

commutes (here, p r " Y --+ Y denotes the universal covering). In other words, "qo is a partial lift o f q. Then "qo admits an extension to a lift onto an open neighbourhood of No in F.

p q Consider again a multivalued map X < F > Y satisfying (i'). Define (a pullback)

? - {(~, z) e ~ x F I p x ( ~ ) = p(z)}.

Now, we can apply Lemma 3.4 to the multivalued map X ,~ p F qP~ Y, and so we get a

lift ~"F ~ Y such that the diagram

p q

X < F ~ Y

X < F > Y p q

is commutative, where if(Y, z) - Y and pr ('~, z) = z. Let us note that the lift ~ is given by the above formula, but ~ is not precised. We fix such a ~.


Observe that p ' F ---> X and the lift ~ induce a homomorphism "~!'Ox ~ Or by the formula if! (ot)(~, z) - (ot~, z). It is e a ~ to check that the homomorphism if! is an isomorphism (any natural transformation of F is of the form ot �9 (~, z) - (ot~, z)) and that p is inverse to if!. Recall that the lift ~ defines a homomorphism ~!'Or --~ Or by the equality

In the sequel, we will consider the composition ~'! "p!'Ox -->. Og.

LEMMA 3.5. Let a multivalued map (p, q) satisfying (i ~) represent a single-valued map p, i.e. q pp Let "fi be the lift o f p which satisfies "~ - "fi'fi. Then "fi! "~' - - , p - - p~.

Now, we are in a position to define the Nielsen classes. Consider a multivalued self-

map X ,~ p F q> X satisfying (i~). By the above consideration, we have a commutative diagram

F

F

> X

l Px P,q

> X

Following the single-valued case (see, e.g., [77]), we can prove (see [24] and cf. [22]) the following lemma.

L E M M A 3 . 6 .

�9 C ( p , q ) - g ~ ~Ox p ~ c ('~, ot'~), �9 i f prC( 'p , a'q) A prC( 'p , ~'q) is not empty, then there exists a F ~ Ox such that ~ -

• o ~ o (~F~• -~,

�9 the sets Pr C ('~, ~'~) are either disjoint or equal.

) Define an action of Ox on itself by the formula F o ot - Fot (~'! P F ) . The quotient set will

be called the set o f Reidemeister classes and will be denoted by R(p , q). The above lemma defines an injection:

set of Nielsen classes --+ R (p, q),

given by A --+ [or] E R(p , q), where ot E Ox satisfies A = p r (C(p, ot~)). One can prove that our definition does not depend on ~. Let us recall that the homomorphism ~'!" Or --+ Or is defined by the relation ~ot - ~'! (ot)~,

for ot 6 Or. If ~ ' - v q is another lift of q (~ ~ Or), then the induced homomorphism ~ : 0 r ~ 0r is defined by the relation ~'~c~ = ~'((a)~ '~.

One can also show that the Reidemeister sets obtained by different lifts of q are canoni- cally isomorphic. That is why we write R (p, q) omitting tildes.

P q PROPOSITION 3.1. I f X x [0, 1] < F > Y is a homotopy satisfying (i'), (iii'), (iii"),

N) then the homomorphism "qt! Pi " Ox ~ Or" does not depend on t ~ [0, 1 ], where the lifts used

22 J. Andres

in the definitions of these homomorphisms are restrictions of some fixed lifts p, q of the given homotopy.

REMARK 3.5. If (p, q) represents a single-valued map p ' X --+ Y (q = pp), then ~!ff! equals ~., (here the chosen lifts satisfy ~ = ~'ff).

Let us point out that the above theory can be modified into the relative case (i.e. modulo a normal subgroup H C Ox). The index H will denote the relative modification.

Assuming X = Y, we can give

LEMMA 3.7.

(i) C(p, q) = U~eox, Pr'HC('ffH, OtQH), (ii) if Pr n C ( ~ , , etCH) A P r , C ('fill, ~ , ) is not empty, then there exists a y ~ Ox ,

such that fl - ~, o ~ o ('qH ! PI-l Y ) "~' -1, (iii) the sets prHC('pH, ~qH) are either disjoint or equal.

Hence, we get the splitting of C(p, q) into the H-Nielsen classes and the natural injection from the set of H-Nielsen classes into the set of Reidemeister classes modulo H, namely, RH (P, q).

Now, we would like to exhibit the classes which do not disappear under any compact (admissible) h omotopy. For this, we need however (besides (it), (iiit), (iii")) the following

two assumptions on the pair X ,~ p 1-' q> Y" (i ') Let X be a connected ANR-space, e.g., a connected retract of an open subset of

(a convex set in) a Fr6chet space, p is a Vietoris map and cl (q (F)) C X is compact, i.e. q is a compact map.

(ii t) There exists a normal subgroup H C Ox of a finite index satisfying ~'!~'!(H) C H. Let us note that the following implications hold: (i), (ii) =~ (iit), (i), (iii) =:~ (i").

DEFINITION 3.1. We call a pair (p, q) N-admissible if it satisfies (it), (i"), (iit), (iii'), (iii") (r (i)-(iii)).

Let us recall that, under the assumption (iit), the Lefschetz number A (p, q) 6 Q is defined (see Section 3.1). This is a homotopy invariant (with respect to the homotopies satisfying (iit)) and A(p , q) ~ 0 implies C(p, q) ~ 0 (cf. Section 3.1).

Let A = pvHC ('p, ~'q) be a Nielsen class of an N-admissible pair (p, q). We say that (the N-Nielsen class) A is essential if A (~', eta') ~ 0. This definition it correct, i.e. does not depend on the choice of c~.

DEFINITION 3.2. Let (p ,q) be an N-admissible multivalued map (for a subgroup H C Ox). We define the Nielsen number modulo H as the number of essential classes in OxH. We denote this number by NH(p, q).

The following theorem is an easy consequence of the homotopy invariance of the Lef- schetz number.


THEOREM 3.4. NI4(p, q) is a homotopy invariant (with respect to N-admissible homotopies

X • [0,1]xx~--I -' q> X).

Moreover, (p, q) has at least N14(P, q) coincidences.

The following theorem shows that the above definition is consistent with the classical Nielsen number for single-valued maps.

THEOREM 3.5. If an N-admissible map ( p , q ) is N-admissibly homotopic to a pair (p~, ql), representing a single-valued map p (i.e. q~= pp~), then (p, q) has at least NH (p) coincidences (here H denotes also the subgroup of rrl X corresponding to the given H COx in (ii')).

Although in the general case the theory requires special assumptions on the considered pair (p, q), in the case of multivalued self-maps on a toms it is enough to assume that this pair satisfies only ( i ' ) , i.e. it is admissible. This is due to the fact that any pair satisfying (i") is homotopic to a pair representing a single-valued map.

THEOREM 3.6. Any multivalued self-map (p, q) on the torus satisfying (i") is admissibly homotopic to a pair representing a single-valued map.

THEOREM 3.7. Let 7s n e, p F q ,~n > be such that p is a Vietoris map. Let p" ~n ~ ~n

be a single-valued map representing a multivalued map homotopic to (p, q) (according to Theorem 3.6, such a map always exists). Then (p, q) has at least N(p) coincidences.

REMARK 3.6. Let us also recall that, on the toms ~,n, N(p) = IA(p)I = [ det(I - A)I, where A is an integer (n x n)-matrix representing the induced homotopy homomorphism t9# :Trl ,~n .__+ Jrl ~n. Moreover, if det(I - A) ~- 0, then

card(zr, (72n)/Im(p#))= Idet(I - A) I.

In particular, for p = id, we have

N(id) = I A (id) l - IX (q,n) ] _ I det O] = 0,

while for p = - id, we have N ( - id) = I A ( - id)[ = [ det 211 = 2 n . For more details, see [22,12].

3.3. Fixed point index theorems

Consider a multivalued map ~0:X ---0 X and assume, similarly as in Section 3.1, that (i) X is ANR-space, e.g., a retract of an open subset of a convex set in a Fr6chet space,

24 J. Andres

(ii) ~0 is a compact composition of an R~-map ~ : X ---o Y and a continuous single- valued map f : Y --+ X, namely ~0 = f o ~, where Y is an ANR-space.

Let D C X be an open subset of X with no fixed points of ~o on its boundary 0 D. Then and integer ind(qg, X, D), called the f ixed point index over X w.r.t. D exists such that the following proposition holds (see, e.g., [ 12,22,44]).

PROPOSITION 3.2. Let ~o : X --o X be a map satisfying (i), (ii). Then ind(~o, X, D) ~ Z is well-defined satisfying the following properties:

�9 (Existence) Ifind(~o, X, D) ~ 0, then Fix(~o) ~ 0. �9 (Localization) I f D1 C D are open subsets o f X such that Fix(~o) C D1 C D, then

ind(~0, X, D) = ind(~p, X, D1).

�9 (Additivity) I f D j, j = 1 . . . . . n, are open disjoint subsets o f D and all f ixed points o f ~P[D are located in n Uj=I Dj, then ind(~o, X, Dj) , j -- 1 . . . . , n, are well-defined satisfying

n

ind(~p, X, D) -- ~ ind(qg, X, Dj) .

j = l

�9 (Homotopy) I f there is a compact homotopy X :X x [0, 1] ---o X (in the same class o f maps under consideration) with X (', O) = qg, X (', 1) = ~ , and ~ D is fixed point free w.r.t. X, then

ind(~o, X, D) = ind(~, X, D).

�9 (Multiplicity) I f ap" X ---o X satisfies (i), (ii) and an open D C X is fixed point free w.r.t. ~ , then

ind(~o x ap, X x ~', D x D) --ind(~o, X, D) . ind(ap, ~' , /9).

�9 (Contraction) I f X ' C X are ANR-spaces such that ~p(X) C X r and ~olx, satisfies (ii) with Fix(~01x,) n O(D N X') = 0, then

ind(~o, X, D) = ind(~plx,, X', D N X').

�9 (Normalization) I f X = D, then

ind(~o, X, D) = ind(~o, X, X) = A (~o).

Because of possible applications, it is very useful to formulate sufficiently general continuation principles.

For compact admissible maps from open subsets of a neighbourhood retract of a Fr6chet space E into E, the fixed point index was just indicated.


Now, we will apply Proposition 3.2 to formulating the appropriate continuation principles. We restrict ourselves only to a particular class of admissible maps, namely to R~-maps O : D ---o E (written here �9 6 J (D, E)), i.e.u.s.c, maps with R~-values.

We often need to study fixed points for maps defined on sufficiently fine sets (possibly with an empty interior), but with values out of them. Making use of the previous results, we are in position to make the following construction.

Assume that X is a retract of a Fr6chet space E and D is an open subset of X. Let �9 E J(D, E) be locally compact, Fix(O) be compact and let the following condition hold:

Vx E Fix(O) 3Ux ~ x, Ux is open in D such that O(Ux) C X. (A)

The class of locally compact J-maps from D to E with the compact fixed point set and satisfying (A) will be denoted by the symbol Ja (D, E). We say that O, qJ E Ja (D, E) are homotopic in JA(D, E) if there exists a homotopy H E J(D • [0, 1], E) such that H (., 0) = O, H (., 1) = qJ, for every x E D, there is an open neighbourhood Vx of x in D such that HI v• • [0,1] is compact, and

Yx E D Vt E [0, 1]

[x E H(x, t) :=> 3Ux ~ x, Ux is open in D, H(Ux • [0, 1]) C X]. (AH)

Note that the condition (AH) is equivalent to the following one: �9 If {xj}j>>a C D converges to x E H(x, t), for some t E [0, 1], then H({xj} • [0, 1]) C

X, for j sufficiently large. Let �9 E Ja (D, E). Then Fix(O) C U{Ux Ix ~ Fix(O)} M V =: D' C D and O(D') C

X, where V is a neighbourhood of the set Fix(O) such that OIv is compact (by the compactness of Fix(O) and local compactness of O) and Ux is a neighbourhood of x as in (A). Define

IndA (O, X, D) -- ind(OlD,, X, D'),

where ind(OI D', X, D') is defined as in Proposition 3.2. This definition is independent of the choice of D'.

In the following theorem, we give some properties of IndA which will be used in the proof of the continuation Theorem 3.8. The simple proof is omitted.

PROPOSITION 3.3. (i) (Existence) IfInda (O, X, D) ~ O, then Fix(O) r 0.

(ii) (Localization) If D1 C D are open subsets of a retract X of a space E, �9 E Ja (D, E) is compact, and Fix(O) is a compact subset of D1, then

IndA (O, X, D) = IndA (O, X, D1).

(iii) (Homotopy) If H is a homotopy in Ja (D, E), then

IndA (H (., 0), X, D) -- IndA (H (., 1), X, D).

26 J. Andres

(iv) (Normalization) I f ~ E J (X) is a compact map, then IndA ((I), X, X) = 1.

THEOREM 3.8 (Continuation principle). Let X be a retract of a Fr~chet space E, D be an open subset of X and H be a homotopy in JA (D, E) such that

(i) H(., O)(D) C X, (ii) there exists H I E J (X) such that HI[D = H(., 0), H t is compact andFix(H~) n ( x \

D) = 0 . Then there exists x E D such that x E H (x, 1).

PROOF. Applying the localization property (ii), we obtain

IndA (H (., 0), X, D) = IndA (H (., 0), X, X).

By the normalization property (iv), IndA (H(., 0), X, X) = 1. Thus, by the homotopy property (iii), IndA (H(., 0), X, D) = IndA (H(., 1), X, D) = 1, which implies by (i) that H(., 1) has a fixed point. Fq

COROLLARY 3.1. Let X be a retract of a Fr~chet space E and H be a homotopy in JA (X, E) such that H (x, O) C X, for every x E X, and H (., O) is compact. Then H (., 1) has a fixed point.

COROLLARY 3.2. Let X be a retract of a Frdchet space E, D be an open subset of X and H be a homotopy in Ja (D, E). Assume that H(x , O) = xo, for every x E D. Then there

exists x E D such that x E H (x, 1).

COROLLARY 3.3. Let X be a retract of a Frdchet space E and �9 E J ( X ) be compact.

Then �9 has a fixed point.

REMARK 3.7. If E -- X is a Banach space, then it follows from Proposition 3.2 that the "pushing" condition (AH), related to JA (D, E), can be reduced to Fix(~o) n 0D ~ 0.

Some applications motivate us to consider weaker than (AH) condition on H. Unfor- tunately, we cannot use the fixed point index technique described above. The proof of the following theorem is based on a Schauder-type approximation technique (for more details,

see [ 19,22]).

THEOREM 3.9 (Continuation principle). Let X be a closed, convex subset of a Frdchet

space E and let H E J (X • [0, 1], E) be compact. Assume that

(i) H (x, O) C X, for every x E X, (ii) for any (x, t) E 0X x [0, 1) with x E H(x , t), there exist open neighbourhoods Ux

of x in X and It o f t in [0, 1) such that H((Ux n OX) • It) C X.

Then there exists a fixed point of H (., 1).

REMARK 3.8. Note that the convexity of X in Theorem 3.9 is essential only in the infinite- dimensional case. For the proof, we have namely to intersect X with a finite-dimensional subspace L.


Now, we would like to consider condensing maps. Hence, let this time go:X ---o X be a multivalued map such that

(I) X is a closed, convex subset of a Fr6chet space E, (II) go is a condensing composition of an R~-map ~ : X --o y and a continuous single-

valued map f :Y ~ X, namely go = f o ~ , where Y is an ANR-space. Assume that Fix(go) n O D 7/: 0, for some open subset D C X. Since go is condensing, it

has a nonempty compact fundamental set T (see Definition 2.11 and Proposition 2.7(iii)). Let ind(go, X, D) -- 0, whenever Fix(go) -- 0. Since T is an AR-space, we may choose a retraction r : X ~ T in order to define the fixed point index, for the composition

,.~ _ _ ap ro f go. D G T --o Y --o T,

by putting

ind(go, X, D)" - - ind(~, ~', D n T),

where ind on the right-hand side is defined as in Proposition 3.2. This correct definition is independent of the chosen fundamental set, and so the index has all the appropriate properties as in Proposition 3.2, but (see Remark 3.2 and cf. [37]) the normalization property. Instead of it, a weak normalization property can be formulated as follows:

�9 (Weak normalization) If f in go = f o ~ is a constant map, i.e. f (y) = a ~ 0 D, for each y 6 Y, then

1, ind(go, X, D) -- 0,

f o r a 6 D , f o r a ~ D.

As already pointed out above, in the applications of the fixed point theory, we often need to consider maps with values in a Fr6chet space and not in a closed convex set. We will also extend our theory to this case.

Again, let E be a Fr6chet space and X be a closed and convex subset of E. Let U C X be open and consider the map go ~ JA (U, E), where the symbol JA (U, E) is again reserved for J -maps from U to E satisfying condition (A). The notion of homotopy in JA

will be understood analogously. Thus, Fix(go) is compact and go has a compact fundamental set T (see Section 2.3). Set IndA (go, X, U) = 0, whenever Fix(go) -- 0. Otherwise, let

n Xl . . . . . Xn ~ Fix(go) such that Fix(go) C Ui=l Uxi =" V, where Uxi are neighbourhoods of xi such that Uxi C U and satisfy condition (A). Then golv 'V --o X is a J -map with compact fundamental set T and satisfies Fix(go) N 0 V -- 0. Thus, we can define

IndA (go, X, U) " - ind(golV, X, V).

The independence of this definition of the chosen set V follows from the additivity property. Furthermore, if go" U --o X has a compact fundamental set and Fix(go) n O U - 0, then IndA (go, X, U) is defined and

IndA (go, X, U) -- ind(go, X, U).

28 J. Andres

The following proposition easily follows from the above argumentation.

PROPOSITION 3.4. (i) (Existence) I f lndA (go, X, U) # O, then Fix(go) -fi 0.

(ii) (Additivity) Let Fix(go) C U1 U U2, where U1, U2 are open disjoint subsets o f U. Then

IndA (go, X, U) -- IndA (golu~, X, U1) -+- IndA (golu2, X, U2).

(iii) (Homotopy) Let ~ : U ---o E be homotopic in Ja to the map go. Assume that the

homotopy X : U x [0, 1 ] ---o E has a compact fundamental set and the set

:= {(x, t) ~ u • [o, 1] Ix ~ x(x , t )}

is compact. Then

IndA (go, X, U) -- IndA (~, X, U).

(iv) (Weak normalization) Assume that go : U --> E is a constant map go(x) = a ~ E, for all x ~ U. Then

1, IndA (go, X, U) = O,

for a ~ U, for a r U.

Using Proposition 3.4, we can easily formulate a continuation principle which is convenient for various applications.

THEOREM 3.1 0 (Continuation principle). Let X be a closed, convex subset of a Fr~chet space E, let U C X be open and let X : U • [0, 1] --o E be a homotopy in JA such that E (see (iii) above) is compact. Let X be condensing and assume that there is a condensing

go ~ J ( X ) such that golF = X(', O) andFix (go )N(X \ U) = 0. Then X(', 1) has a fixedpoint.

PROOF. The proof follows, in view of the existence property (i) in Proposition 3.4, from the following equations:

IndA (X (', 1), X, U) -- IndA (X (', 0), X, U),

by the homotopy property (iii),

IndA (X (', 0), X, U) = IndA (golu, X, U) = IndA (go, X, X),

by the additivity property (ii). Finally, we see that

IndA (go, X, X) = ind(go, X, X) = 1. {-1


COROLLARY 3.4. Let x : X x [0, 1] --o E be a condensing homotopy in JA such that

X (x, 0) C X, f o r every x ~ X. Then X (', 1) has a fixed point.

REMARK 3.9. If E = X is a Banach space, then it follows from the properties of the above fixed point index ind(x, X, D) that the "pushing" condition (A/4), related to JA (D, E) , can be reduced to Fix(x) A 0 D ~ 0. If, in particular, U C E is an open convex subset such that X (', 0) = q9lu : U --o U, then the same is true even without requiring Fix(qg) A (X \ U) = 0.

4. General methods for solvability of boundary value problems

4.1. Continuation principles to boundary value problems

In this part, fixed point principles in Section 3 will be applied to differential equations and inclusions.

At first, we are interested in the existence problems for ordinary differential equations and inclusions in Euclidean spaces on not necessarily compact intervals. Let us start with some definitions.

Let J be an interval in R. We say that a map x : J ~ ]1~ n is locally absolutely continuous if x is absolutely continuous on every compact subset of J . The set of all locally absolutely continuous maps from J to ~n will be denoted by ACloc (J, Rn).

Consider the inclusion

~ F(t , x) , (4.1)

where F is a set-valued u-Carathdodory map, i.e. it has i.a. the following properties: �9 the set of values of F is nonempty, compact and convex, for all (t, x) 6 J x R n, �9 the map F(t , .) is u.s.c., for almost all t 6 J , �9 the map F (., x) is measurable, for all x E R n. By a solution of the inclusion (4.1), we mean a locally absolutely continuous function x

such that (4.1) holds, for almost all t 6 J . We recall two known results which are needed in the sequel.

PROPOSITION 4.1 (Cf. [42, Theorem 0.3.4] and Lemma 4.4 below). Assume that the sequence o f absolutely continuous functions xk : K -+ IR n ( K is a compact interval) satisfies the following conditions:

�9 the set {xk(t) [k ~ N} is bounded, fo r every t ~ K, �9 there is an integrablefunction (in the sense ofLebesgue) o t :K ~ 1R such that

12k(t)l ~< oe(t), f o r a.a. t ~ K and for all k ~ N.

Then there exists a subsequence (denoted just the same) {xk } convergent to an absolutely continuous function x : K --+ ~n in the following sense:

(i) {x/~} uniformly converges to x, and (ii) {2~ } weakly converges in L 1 (K, IR n) to 2.

30 J. Andres

The second one is the well-known (see, e.g., [22, Theorem 1.33 in Chapter 1.1]) Mazur theorem.

The following result is crucial.

PROPOSITION 4.2. Let G : J x ]1~ n • ]I{ m ---o ]1{ n be a u-Carathdodory map and let S be a nonempty subset of ACloc (J, IR n). Assume that

(i) there exists a subset Q of C(J, IR n) such that, for any q E Q, the set T(q) of all solutions of the boundary value problem

{ J(t) ~ G(t, x(t) , q(t)), for a.a. t E J, x E S

(4.2)

is nonempty, (ii) T(Q) is bounded in C(J, Rn),

(iii) there exists a locally integrable function or: J ~ R such that

IG(t,x(t~,q(t~)[ =sup{lyl [y 6 G ( t , x ( t ) , q ( t ) ) } ~ot(t) , a.e. in J,

for any pair (q, x) 6 I-'T, where 1`T denotes the graph of T. Then T(Q) is a relatively compact subset of C(J, IRn). Moreover, under the assumptions (i)-(iii), the multivalued operator T : Q --o S is u.s.c, with compact values if and only if the following condition is satisfied:

(iv) given a sequence {(qk,Xk)} C 1`T, if {(qk, Xk)} converges to (q ,x) with q ~ Q, then x ~ S .

PROOF. For the relative compactness of T (Q), it is sufficient to show that all elements of T (Q) are equicontinuous.

By (iii), for every x ~ T(Q), we have I~(t)l ~ co(t), for a.a. t 6 J, and

IX(tl) -- x ( t2) I ~< fl t2 o t ( s ) d s

This implies an equicontinuity of all x 6 T (Q). We show that the set 1117- is closed (cf. Section 2.1). Let 1-'T 3 {(qk, xk)} --+ (q,x) . Let K be an arbitrary compact interval such that ot is

integrable on K. By conditions (ii) and (iii), the sequence {x~ } satisfies the assumptions of Proposition 4.1.

Thus, there exists a subsequence (denoted just the same) {xk }, uniformly convergent to x on K (because the limit is unique) and such that {2k} weakly converges to 2 in L 1. Therefore, :~ belongs to the weak closure of the set conv{km [m ~> k}, for every k >~ 1. By the mentioned Mazur theorem, k also belongs to the strong closure of this set. Hence, for every k ~> 1, there is Zk ~ conv{km I m/> k} such that IIz~ - kllL~ ~< 1/k. This implies that there exists a subsequence Zk~ -+ Yc a.e. in K.

Let s E K be such that

G(s, ., .) is u.s.c., lim Zkl (s) -- ~c(s), YCk(S) ~ G(s, xk(s), qk(s)). l--+ cxz


Let e > 0. There is ~ > 0 such that G(s, z, p) C Ne(G(s ,x(s ) , q (s))), whenever I x ( s ) - zl < 6 and I q ( s ) - p l < 6. But we know that there exists N ) 1 such that ]x (s ) -xm(s ) l < and Iq(s) - qm(s)l < 3, for every m ~> N. Hence,

2k(s) ~ G(s, xk(s), qk(s)) C Ne(G(s ,x ( s ) , q(s))).

By the convexity of G(s, x(s) , q(s)) , for kl ~> N, we have

z~(s) E Ne (G(s , x ( s ) , q ( s ) ) ) .

Thus, :~(s) 6 Ne(G(s, x(s) , q(s) ) ) , for every e > 0. This implies

/c(s) ~ G(s, x(s), q(s)).

Since K was arbitrary, 2(t) ~ G(t, x(t) , q(t)), a.e. in J . E3

We can now state one of the main results of this subsection.

THEOREM 4.1. Consider the boundary value problem

:f(t) 6 F( t , x ( t ) ) , for a.a. t ~ J, (4.3) x E S ,

where J is a given real interval, F : J x I~ n ---o ]~n is a u-Carath~odory map and S is a subset of AC]oc (J, R n).

Let G : J x ~n x R n x [0, 1] --o R n be a u-Carath~odory map (cf Definition 2.10) such that

G(t, c, c, 1) C F(t, c), for all (t, c) ~ J x ~n.

Assume that (i) there exist a retract Q of C(J, IR n) and a closed bounded subset $1 of S such that

the associated problem

{ 2(t ) ~ G(t, x(t) , q(t), )~), for a.a. t E J, x ~ S 1

(4.4)

is solvable with an R~-set of solutions, for each (q,)~) E Q x [0, 1], (ii) there exists a locally integrable function or: J ~ ]R such that

IG(t ,x( t ) ,q( t ) , )~) l <~ or(t), a.e. in J,

for any (q,)~, x) E 1-'T, where T denotes the set-valued map which assigns to any (q, ~) E Q • [0, 1] the set ofsolutions of (4.4),

(iii) T(Q x {0}) C Q,

32 J. Andres

(iv) if Q 9 qj ~ q ~ Q, q ~ T (q, X), then there exists jo ~ N such that, for every j j0, 0 E [0, 1] and x E T(qj , 0), we have x E Q.

Then problem (4.3) has a solution.

PROOF. Consider the set

Q ' - {y e C(J, R~+I)I y(t)= (q(t), x), q e Q, x ~ [0, 1]}.

By Proposition 4.2, we obtain that the set-valued map T: Q x [0, 1] --o S1 is u.s.c., and so it belongs to the class J ( Q x [0, 1], C(J, I~n)). Moreover, it has a relatively compact image. Assumption (iv) implies that T is a homotopy in Ja (Q, C (J, IR n)). Corollary 3.1 in Section 3 now gives the existence of a fixed point of T (., 1). However, by the hypothesis, it is a solution of (4.3). D

Note that the conditions (iii) and (iv) in the above theorem hold if S1 C Q.

COROLLARY 4.1. Consider the boundary value problem (4.3). Let G : J x R n x IR n --o R n be a u-Carath~odory map such that

G(t, c, c) C F(t, c), for all (t, c) E J x ]t~ n.

Assume that (i) there exists a retract Q of C (J, R n) such that the associated problem

k(t) e G(t, x( t) , q(t)) , for a.a. t e J, x E S A Q

has an R~-set of solutions, for each q ~ Q, (ii) there exists a locally integrable function ot : J --+ R such that

Ia(t, x(t), q(t))l ~ c~(t), a.e. in J,

(4.5)

for any (q, x) E FT, (iii) T (Q) is bounded in C (J, R n) and T (Q) c S.


Making use of the special case of Theorem 3.3 and modifying appropriately the proof of Theorem 4.1, we can easily obtain the following:

COROLLARY 4.2. Consider problem (4.3) and assume that all the assumptions of Corol- lary 4.1 hold with the convex closed set Q and nonempty acyclic sets of solutions (4.5). Then the problem (4.3) has a solution.

Let us note that in applications solution sets are, in fact, R~-sets. If, in particular, J = [a, b] (i.e. compact), then Theorem 4.1 can be easily reformulated

(see Remark 3.7) as follows.


COROLLARY 4.3. Consider the boundary value problem (4.3), where J --" [a, b] is a compact interval, F : J x It~ n --o IR n is a u-Carath~odory map and S C AC(J , Rn).

Let G : J x R n x IR n x [0, 1] --o R n be a u-Carath~odory map such that G(t , c, c, 1) C

F (t, c), f o r all (t, c) ~ J • IR n . Assume that (i) there exist a (bounded) retract Q o f C(J , IR n) such that Q \ OQ is nonempty (open)

and a closed bounded subset $1 o f S such that the associated problem (4.4) is solvable with an R~-set o f solutions, for each (q, k) 6 Q x [0, 1], and conditions (ii) and (iii) in Theorem 4.1 hold true,

(ii) the solution map T (defined in condition (ii) of Theorem 4.1) has no f ixed points on

the boundary OQ o f Q , f o r every (q, ~,) c Q x [0, 1]. Then problem (4.3) has a solution.

REMARK 4.1. In the (single-valued) case of Carath6odory ODEs, we can only assume

in Theorem 4.1(i), Corollary 4.1 (i), Corollary 4.2 and Corollary 4.3(i) that the related lin-

earized problems are uniquely solvable.

Since c ( n - 1 ) ( J ) c a n be considered as a subspace of C(J , IRn), we can also apply the

previous results to nth-order scalar differential equations and inclusions. To solve an existence problem, one should check suitable a priori bounds for all the derivatives up to the order n - 1. Our technique simplifies a work. Let us describe it below.

We need the following lemma [52, Lemma 2.1] related to the Banach space H n'l ( i) :2

LEMMA 4.1. Let I be a compact real interval and let a0, a l . . . . . an-1 "I x ]1~ n ~ ]~ be u-Carath~odory functions. Given any q E c ( n - 1 ) ( I ) , consider the fol lowing linear nth- order differential operator Lq " H n'l ( I ) --+ L 1 (I)"

n-1

L q ( x ) ( t ) -- x(n) (t) 4- Z ai( t , q( t ) . . . . . q

i=o

(n-1)( t ) )x( i ) ( t ) .

Assume there exists a subset Q o f C (n-l) ( I ) and an L 1 -function ~" I --+ R such that, for any q E Q and any i - O, 1 . . . . . n - 1, we have

l a i ( t , q ( t ) . . . . . q ( ' - l ) ( t ) ) l <~ ~( t ) , a.e. in I.

Then the fol lowing two norms are equivalent in H n'! (I)"

n-1

I l x l l - suplx(')(t)l + f d,, i=0 tel

IlxllQ - suplx( t ) l + sup f Itq(x)(t)l dt. tel qEQ

2By H n, 1(i), we denote the Banach space of all c(n-l)-functions x ' l --+ ItS, where I is a compact interval, with absolutely continuous (n - 1)th derivatives.

34 J. Andres

COROLLARY 4.4. Consider the scalar problem

n - 1

X (n) (t) E Z ai (t, x( t ) . . . . . x ( n - l ) ( t))x (i) (t) i=0

+ F( t , x ( t ) . . . . . x (n-1)(t)), for a.a. t ~ J, x E S ,

(4.6)

where J C R, S C C ( J) and a i , F are u-Carathgodory maps on J x ~n. Suppose that there exists a u-Carathgodory map G : J x R n • R n • [0, 1] ---o R n such

that, for every c ~ R n and )~ ~ [0, 1], G(t, c, c, 1) C F(t , c), a.e. in J. Then problem (4.6) has a solution, provided the following conditions are satisfied:

(i) there is a retract Q o f the space c ( n - 1 ) ( J ) such that, for every (q , )0 6 Q x [0, 1], the following problem,

n-1 x (n) (t) E Z ai (t, q( t ) . . . . . q(n-1) ( t))x(i)( t)

i=0

+ G(t , x ( t ) . . . . . X ( n - l ) (t), q(t) . . . . . q

x ~ S O Q ,

(n-1)(t), ~), for a.a. t ~ J,

(4.7)

has an R~-set o f solutions, (ii) there is a locally integrable function ~ " J --+ R such that, for every i - 0 . . . . . n - 1"

la i ( t ,q ( t ) . . . . . q('-l)(t))l ~<ot(t), a.e. in J,

and

I G ( t , x ( t ) . . . . . x ( n - 1 ) ( t ) , q ( t ) . . . . . q(n-1)(t),)~)l <<,~(t), f o r a . e . t ~ J,

for each (q,)~, x) 6 Q x [0, 1] x c ( n - 1 ) ( J ) satisfying (4.7), (iii) T (Q x {0}) c Q, where T denotes the set-valued map which assigns to any (q, X)

Q x [0, 1] the set o f solutions o f (4.7), (iv) the set T ( Q x [0, 1]) is bounded in C ( J ) and its c(n-1)-closure is contained in S

(in particular, this holds i f S A C (n-a) ( J ) is closed in C (n-l) (J)) , (v) i f {qj} C Q converges to q ~ Q, q ~ T(q , )O in c (n -1 ) ( J ) , then there exists jo ~ N

such that, for every j >~ jo, 0 ~ [0, 1] and x ~ T (q j , 0), we have x ~ Q.

PROOF. We construct a new problem in the following way: Define F" J x IR n ---o IR n,

. . . . . F(,, x(,) . . . . . x n-l (t))

n--1

-+- Z ai ( t , x(t) . . . . . x ( n - l ) (t))x (i) (t). i=0


Denote Y(t) = (x(t) . . . . . X ( n - l ) (t)) E R n and define F'" J x N n --o N n,

F'(t,-Y(t)) - { (2(t) . . . . . x(n-1)(t), Y) I Y E 17(t,x(t) . . . . . x(n-1)(t))}.

So, we have a problem

x(t) E F ' ( t , g( t)) , for a.a. t E J, (4.8)

-YES,

where S is an image of S A c ( n - 1 ) ( J ) via the inclusion i" c(n-1)(J) ~ C(J, Rn). Analogously, we find the associated problem

x(t) e o'(t, ~(t), #(t), z), ~ E S NQ.

for a.a. t E J, (4.9)

Notice that (1) a'(t ,Y(t) ,-~(t) , 1) C f ' ( t ,-~(t)), (2) the set Q = i ( Q ) is a retract of C(J , Rn),

m

(3) S c ACloc(J, Rn), (4) for every (q,)~) E Q x [0, 1], the sets of solutions of the problems (4.7) and (4.9)

are the same,

(5) r (Q x [0, 1]) c S, where r is a suitable map corresponding to T and

Ia'(t,-~(t),~(t),x)l <<. Ia(t ,x(t) , . . . ,x(n-l~,q(t) . . . . . q(n-1)(t),)~)[

n-1

-I- ~ j la i (t, q(t) . . . . . q(~-l~ (t)) l lx(i~ (t) I i=0

n - 1

c~(t) +ol(t) ~_~lxei)(t)l. i=0

Since T(Q x [0, 1]) is bounded in C(J), there exists a positive continuous function m ' J --+ IR such that Ix(t)l ~< m(t), for all t E J and any x E r ( e x [0, 1]). We will show that T(Q x [0, 1]) is also bounded in c (n - l l ( j ) . It is sufficient to prove that, for any compact subinterval I in J , there is a constant M > 0 such that

n - 1

p,(x) - ~ sup lx (i) (t) I ~< M,

i=0

for all x E T(Q • [0, 1]).

Let I C J be an arbitrary compact interval. Using the notation in Lemma 4.1, we see that pI (x) ~< [Ix II and, by the equivalence of norms,

Ilxll ~ cllxllQ ~ c ( m a x m ( f ) + f ~(t)dt) <. M. \ t~t

36 J. Andres

We conclude that T(Q x [0, 1]) is bounded in c(n-1)(J) which implies that T(Q x [0, 1]) is bounded in C (J, R n). Moreover, there exists a continuous function go" J ~ R such that

IG'(t, ~-(,), ~(t), &)l ~ c~(t)(1 +g0(t)).

Obviously, the right-hand side of the above inequality is a locally integrable function. Finally, an easy computation shows that the condition (iv) in Theorem 4.1 holds for Q

and T. By Theorem 4.1, there exists a solution of (4.8) as well as the one of (4.6). D

The same argument as in Corollary 4.1 shows how to modify Corollary 4.4 for the following scalar problem, namely

n-1

X (n) (t) E ~ ai (t, x ( t ) . . . . . x ( n - l ) ( t ) )x (i) (t)

i=0

+ F(t , x( t) . . . . . X ( n - l ) (t)), for a.a. t 6 J, x E S ,

(4.10)

where J C R, S C C (J) and ai, F are u-Carath6odory maps on J • ~ n , by means of the following linearized problem

n -1

x (n) (t) E Z ai (t, q( t ) . . . . . q ( n - 1 ) ( t ) )x( i ) ( t )

i=0

+ G(t, x( t) . . . . . X (n- l ) (t), q(t) . . . . . q x ~ S N Q ,

(~-1) (t)), for a.a. t 6 J,

(4.11)

where Q is a retract of the space C (n- 1) ( j ) . Theorem 3.9 in Section 3.3 gives similar consequences as those of Theorem 3.8 in Sec-

tion 3.3. Unfortunately, the weakness of the assumption on solutions causes that we have to assume the convexity of the set Q. In spite of it, the results given below are important because of the applications.

THEOREM 4.2. Consider the boundary value problem (4.3), where J is a given real interval, F : J x R n --o R n is a u-Carath~odory map and S is a subset of ACloc (J, R n).

Let G : J x R n x R n • [0, 1 ] --o R n be as in Theorem 4.1. Assume that the assumptions (i)-(iii) of Theorem 4.1 hold, with the convexity of the set Q, and

(iv) if 0Q x [0, 1] D {(qj ,~j )} converges to (q,)0 ~ 0Q x [0, 1], q ~ T(q,)O, then there exists jo E N such that, for every j >>, jo, and xj ~ T(qj , ~j), we have xj ~ Q.


The proof can be obtained immediately by using our continuation principle presented in Theorem 3.9 in Section 3.3.


REMARK 4.2. In the (single-valued) case of Carath6odory ODEs, we can again only assume in Corollary 4.4(i) and Theorem 4.2 that the linearized problems are uniquely solvable. If the associated problem (4.4) for G is so uniquely solvable, for every (q,)~) �9 Q x [0, 1], then, by continuity of T, we can reformulate the above condition (iv) as follows:

(iv ~) if {(Xj , )~j)} is a sequence in $1 x [0, 1], with)~j ~ )~ �9 [0, 1) andxj is converging to a solution x �9 Q of (4.4) (corresponding to (x,)0), then xj belongs to Q, for j sufficiently large.

Now, we are interested in the existence of several solutions of problem (4.3). For this, the Nielsen theory developed in Section 3.2 will be applied. It will be convenient to use the following definition.

DEFINITION 4.1. We say that the mapping T : Q ---o U is retractible onto Q, where U is an open subset of C(J, R n) containing Q, if there is a (continuous) retraction r : U --+ Q and p �9 U \ Q with r(p) = q implies that p ~ T (q).

Its advantage consists in the fact that, for a retractible mapping T : Q ---o U onto Q with a retraction r in the sense of Definition 4.1, its composition with r, rIT(Q) o T : Q --o Q, has a fixed point ~" �9 Q if and only if ~" is a fixed point of T.

The following principal statement characterizes the matter.

THEOREM 4.3. Let G : J • ]R n • R n ----o ~n be u-Carathdodory map ( c f Defini t ion 2.10) and assume that

(i) there exists a closed, connected subset Q of C(J, It~ n) with a finitely generated abelian fundamental group such that, for any q �9 Q, the set T (q) of all solutions of the linearized problem (4.2) is R~,

(ii) T(Q) is bounded in C(J, R n) and T(Q) c S, (iii) there exists a locally integrable function ~ : J -+ R such that

[G( t , x ( t ) ,q ( t ) ) l " -sup{ ly l ] y �9 G ( t , x ( t ) , q ( t ) ) J ~ ot(t), a.e. in J,

for any pair (q, x) e 1-';r, where Fr denotes the graph of T. Assume, furthermore, that

(iv) the operator T: Q ---o U, related to (4.2), is retractible onto Q with a retraction r in the sense of Definition 4.1.

At last, let

G(t, c, c) C F(t, c) (4.12)

fora.a, t e J andany c e N n. Then the originalproblem (4.3) admits at least N(r]T(Q) o T) solutions belonging to Q, where N stands for the Nielsen number defined in Definition 3.2 in Section 3.2.

PROOF. By the hypothesis, Q is a connected (metric) ANR-space with a finitely generated abelian fundamental group and T (q) is an R6-mapping. Since T is also, according to

38 J. Andres

Proposition 4.2, u.s.c, and such that T (Q) is compact, r o T is compact and admissible. This follows from the commutativity of the following diagram:

T F ~ U ~ - Q

l-'r

where (pr , qr ) is a pair of natural projections of the graph F r and p r is Vietoris. Therefore, according to Theorem 3.4 in Section 3.2, (pr , r[r(Q) o qr) admits at least N(rIr (Q) o T( . ) ) coincidence points. Because of Definition 4.1, they represent the solutions of problem (4.2) and, in view of (4.12), they also satisfy the original problem (4.3). D

REMARK 4.3. In the (single-valued) case of Carath6odory ODEs, we can only assume in Theorem 4.3(i) that the linearized problem (4.2) is uniquely solvable. Moreover, the requirement that the fundamental group re(Q) of Q to be finitely generated and abelian can be then omitted (see Section 3.2).

Furthermore, we will consider boundary value problems on arbitrary (possibly infinite) intervals for differential inclusions in Banach spaces. We start with some definitions.

Let E be a Banach space with the norm I1" II. Denote by C (J, E) the space of all continuous functions x : J --+ E with the locally convex topology generated by the uniform convergence on compact subintervals of J (possibly, the whole N). This topology is completely metrizable, and thus C (J, E) is a Fr6chet space.

Recall that a mapping x : J --+ E is locally absolutely continuous if x is absolutely continuous on every compact subinterval of J. Unfortunately, in general, on each interval [a, b] C J, there need not exist 2(t) (in the sense of Fr6chet), for almost all (a.a.) t E [a, b] with 2 E L 1 ([a, b], E) (the set of all Bochner integrable functions [a, b] --+ E) and so need not be

fa t x ( t ) = xo + 2(s) ds.

It is so if E satisfies the Radon-Nikodym property, in particular, if E is reflexive. Moreover, we have the following result (cf. [48]).

LEMMA 4.2. Suppose x " [a, b] --+ E is absolutely continuous, 2 exists a.e., and

~< y( t ) , a . e . , f o r some y E L l ( [a ,b] , lR ) .

Then 2 E L1 ([a, b], E) and

f t 2 (s ) ds -- x ( t ) - x ( r ) (t, r ~ [a, b]). (4.13)


The set of all locally absolutely continuous functions from J to E, satisfying all the above properties, will be denoted by ACloc(J, E).

Consider now the differential inclusion

Jr E F(t , x), (4.14)

where F : J x E --o E is a u-Carath6odory map, i.e. (C1) F ( t , x ) is nonempty, compact and convex, for every ( t , x ) E J x E, (C2) F(t , .) is u.s.c., for a.a. t 6 J, (C3) F ( . , x ) is strongly measurable (cf. Definition 2.9), on every compact interval

[a, b], for each x ~ E. By a solution of this differential inclusion we mean again a map x E ACloc (J, E) satis-

fying (4.14), for a.a. t E J. To a u-Carath6odory map F, we associate the Nemytskii (or superposition) operator

NF" C (J, E) ---o L~o c (J, E) given by

NF(X) "-- {f E L~oc(J, E) If(t) E F(t,x(t)), a.e. on J},

for each x 6 C (J, E). In the sequel, we will need the following lemma (see [100, p. 88] and cf. Remark 2.1).

LEMMA 4.3. Let [a, b] be a compact interval. Let F : [a, b] x E --o E be a u-Carathdodory mapping and assume in addition that, for every nonempty bounded set ~2 C E, there exists v = v(f2) 6 L l([a, b]) such that

IIF(t,x)ll "-- sup{llzll I z E F ( t , x ) } <, v(t),

for a.e. t E [a, b] and every x E ~2. Then the Nemytskff operator

NF "C ([a, b], E) --o L1 ([a, b], E)

has nonempty, convex values. Moreover, given sequences {Xn } C C ([a, b], E) and {fn } C Ll([a,b], E), fn E NF(Xn), n ~ 1, such that Xn --+ x in C([a,b], E) and fn --+ f weakly in L1 ([a, b], E), then f E NF(X).

The following lemma extends Proposition 4.1 to infinite-dimensional spaces (see again [42, Theorem 0.3.4]).

LEMMA 4.4. Assume that a sequence {x~ I [a, b] --+ E} of AC-maps satisfies the following conditions:

(i) {x~ (t)} is relatively compact, for each t ~ [a, b], (ii) there exists ~ E Ll([a,b]) such that 112k(t)l[ ~< ~( t ) , fora .a . t E [a,b].

Then there exists a subsequence (again denoted by {xk}) that converges to an absolutely continuous map x : [a, b] --+ E in the following sense:

(iii) x~ ~ x in C([a, b], E),

40 J. Andres

(iv) ,f~ --+ ,~ weakly in L 1 ([a, b], E).

PROPOSITION 4.3. Let G : J x E • E ---o E be a u-Carath~odory map and let S be a nonempty subset of ACloc(J, E). Assume that:

(i) there exists a closed Q c C(J, E) such that, for any q E Q, the boundary value problem

. f(t) E G(t ,x( t ) ,q ( t ) ) , fora.a, t ~ J, x E S ,

has a solution. Denote by T : Q ---o S the solution mapping. (ii) There exist or,/3, y E L~oc(J) such that

IlG( t , x , Y)II ~ or(t) +/3(t)llxll + y(t)llyll,

for a.a. t ~ J and every (x, y) E E 2. (iii) I f {(qn,Xn)} is a sequence in the graph o f T and (qn,Xn) --+ (q, x), then x E S.

Then T : Q ---o S has a closed graph ( S is endowed with the topology of C ( J, E) ).

PROOF. Let {(qn, Xn)} be an arbitrary sequence in the graph of T, i.e. Xn E T(qn), for every n 6 N, and assume that (qn, Xn) --+ (qo, xo). Thus, we see that

YCn(t) 6 G(t, Xn(t), qn(t)), for a.a. t 6 J,

and Xn ~ S. Then qo 6 Q and, by assumption (iii), xo 6 S. Now, let [a, b] be an interval in J . Using assumption (ii), we see that the sequence {Xn }

satisfies the assumptions of Lemma 4.4. Thus, {Xn} converges uniformly on [a, b] to xo (because this limit is unique) and {Xn} converges to xo, weakly in L 1 ([a, b], E). Using Lemma 4.3, it follows that ko(t) 6 G(t, xo(t), qo(t)), for a.a. t 6 [a, b]. Since [a, b] was arbitrary, we see that indeed k0(t) 6 G(t, xo(t), qo(t)), for a.a. t 6 J and x0 6 T (qo). []

As another of the main results of this subsection, we can formulate the following continuation principle.

THEOREM 4.4. Consider the boundary value problem (e.g., in a reflexive Banach space E; c f Lemma 4.2):

k(t) E F(t , x( t ) ) , for a.a. t ~ J, (4.15) x E S ,

where F : J x E ---o E is a u-Carath~odory map and S is a subset of ACloc(J, E). Let G : J x E x E x [0, 1] ---o E be a u-Carath~odory map (cf Definition 2.10) such that

G(t, c, c, 1) C F(t, c), for all (t, c) E J x E. (4.16)

Assume that:


(i) there exists a closed, convex Q c C ( J, E) and a closed subset $1 of S such that the problem

{ .~(t) E G(t, x( t) , q(t) , )~), for a.a. t ~ J, x E S 1

is solvable with an Ra-set T (q , )~), for each (q,)~) C Q x [0, 1 ]. (ii) There exist or, fl, }, E L~o c (I) such that

[I G ( t , x , Y,)~) II <~ or(t) + r Y(t)IlYlI,

for a.a. t E J, every (x, y) E E 2 and every )~ E [0, 1]. (iii) T is quasi-compact, i.e. T maps compact subsets onto compact subsets, and there

exists a measure of noncompactness /z in the sense of Definitions 2.7 and 2.8 in Section 2.2 such that, for each S2 C Q, if

• [o,

then f2 is relatively compact. (iv) T ( Q x {0}) C Q. (v) For each )~0 E [0, 1] and q E T (qo, s if qn --+ qo in Q, then there is no E N such

that, for each n ~> no, s E [0, 1] and x E T (qn, )~), we have x ~ Q. Then problem (4.15) has a solution.

PROOF. Using Proposition 4.3, we see that the map T : Q x [0, 1] --o $1 has a closed graph. Since T is also quasi-compact (assumption (iii)), we can easily derive that T is indeed an u.s.c, set-valued map (see, e.g., [78, Theorem 1.1.12]). From assumption (i), we get therefore that T E J (Q x [0, 1], C (~, E)) and assumption (iii) implies that T is also /z-condensing. By (v), we finally see that T is a homotopy in Ja, and thus Corollary 3.4 in Section 3.3 implies the existence of a fixed point of T (., 1). However, by the inclusion (4.16), it is a solution of (4.15). E3

REMARK 4.4. As we can see, Theorem 4.4 extends Theorem 4.1 into the infinite- dimensional setting, when replacing ~n by a real Banach space. On the other hand, this is possible with some loss, namely Q is only convex and the solution operator T is assumed to be quasi-compact, additionally. Because of those restrictions, we are unfortunately un- able to establish a full infinite dimensional analogy of Theorem 4.1.

If, in particular, J = [a, b] (i.e. compact), then Theorem 4.4 can be simplified, in view of Remark 3.9, similarly as Corollary 4.3 w.r.t. Theorem 4.1, as follows.

COROLLARY 4.5. Consider the problem

{ /c(t) E F(t ,x( t ) ) , fora .a . t E[a,b], x E S ,

(4.17)

42 J. Andres

where F : [a, b] • E ---r E is a u-Carathdodory map and S is a subset o f absolutely continuous functions x : [a, b] --+ E, all in a reflex&e Banach space E. Let G : [a, b] • E • E • [0, 1 ] ---r E be a u-Carathgodory map such that

G(t , c, c, 1) C F(t , c), for all (t, c) ~ [a, b] • E. (4.18)

Furthermore, assume that

(i) there exists a convex, bounded subset Q c C ([a, b], E) such that Q \ 0 Q is nonempty and a closed subset S1 o f S such that the problem

{ k( t ) ~ G(t ,x( t ) ,q( t ) ,X) , fora.a, t ~ [a,b], x E S 1

is solvable with R~-set T (q , X), for each (q,)0 ~ Q • [0, 1 ]. (ii) There exists ~, ~, y ~ L 1 ([a, b]) such that

[[G(t,x, Y,)0 [[ ~< o~(t) +/3(t)llxll + Y(t)llyll,

for a.a. t ~ [a, b], every (x, y) E E 2 and every )~ ~ [0, 1]. (iii) T is quasi-compact, i.e. T maps compact subsets onto compact subsets, and there

exists a measure o f noncompactness It (see Definitions 2.7 and 2.8) such that, for each f2 C Q, if

then f2 is relatively compact. (iv) T ( Q • {0})C Q. (v) The map T has no fixed points on the boundary 0 Q o f Q, for every (q, x) 6 Q •

[0, ~]. Then problem (4.17) has a solution.

PROOF. We can proceed quite analogously as in the proof of the foregoing Theorem 4.4. The only difference consists of modifying Corollary 3.4 in the sense of Remark 3.9, both in Section 3.3. D

REMARK 4.5. In the (single-valued) case of Carath6odory ODEs, we can again only assume in Theorem 4.4(i) and Corollary 4.5(i) that the linearized problems are uniquely solvable.

Sometimes it is convenient to consider the asymptotic problems sequentially. For this purpose, it can be useful to employ

PROPOSITION 4.4. Let J1 C J2 C . . . be compact intervals such that J - - ~Jm=l~ Jm and to ~ J1. Let F : J • E ---o E be a u-Carathdodory mapping with nonempty, compact and convex values. Assume, furthermore, that


(i) There are c~, fl �9 L1o c (J, JR) with

]]F(t,x)]] <<. ~(t) + fl(t)IlxII.

(ii) There is some Carath~odory mapping g" J x [0, o~) ~ [0, oo) (called a Kamke function) such that the only nonnegative measurable solution o f

x( t ) <<, fo ~ g ( s , x ( s ) ) d s

is 0 (a.e.), and such that, for a.a. t �9 J, F(F({t} x C)) ~< g(t, F (C) ) , f o r countable, bounded subsets C C E, where F denotes the Hausdorff measure of noncompact- heSS.

(iii) E has the so called retraction property in the sense o f [99], e.g., E is separable or

reflexive. I f Xm �9 AC(Jm, E) satisfies

5Cm(t) �9 F(t , Xm(t)), for a.a. t �9 Jm, m �9 N,

and {Xm(to) I m �9 N} is a relatively compact set, then there is a solution x �9 ACloc(J, E) of the inclusion Yc(t) �9 F(t , x ( t ) ) , f o r a.a. t �9 J, such that, for some subsequence,

Xmk ~ X, uniformly on each Jm,

and

)Cmk ~ X, weakly in L 1 ( j , E).

I f still (iv) sup{llXm(t)ll lm �9 l~, t �9 Jm} <

and the values o f Xm, m �9 N, are located in a closed subdomain 7) of E, then there exists an entirely bounded solution x on J with x( t ) �9 7) , for all t �9 JR.

PROOF. By (i) and the well-known Gronwall inequality (see, e.g., [72]), we get the a priori estimates

IlXm<t> II and II m (t)II ~(t),

for some Y 6 L~o c (J, JR). We claim that {Xm(t) I m ~> mt } is a relatively compact set, for a.a. t 6 J , where mt =

min{m i t �9 Jm}. To show it, put h(t) " - y({Xm(t) I m ~ mr}). Then h is measurable (for more details, see Proposition 11.12 in [99]). Moreover, by means of Proposition 11.12 in [99] and (ii), we obtain

({s I /)

44 J. Andres

~y({ X m ( S ) l m > / m t } ) d s fo~ g(s, h(s)) ds

Applying (ii) again, we arrive at h(t) = 0, for a.a. t 6 J , as claimed. Since F(t, .) maps compact sets into compact sets, {-~m (t) I m ~> mt} becomes relatively

compact as well, for a.a. t 6 J . An application of the standard diagonalization argument implies, jointly with Lemma 4.4, the existence of a subsequence such that Xmk --+ x, uniformly on each Jm, and Xmk ~ )C, weakly in L l( J, E), where x ~ ACloc(J, E).

It follows from Lemma 4.3 that 2(t) E F(t, x(t)). Since the remaining part of the assertion is implied by the foregoing one (just proved) and (iv), the proof is completed. []

REMARK 4.6. Let E be a Banach space and assume that F :R x E --o E is a u-Carath6odory mapping (cf. Definition 2.10) such that

/x (F (t, B)) ~< k (t)/z (B), for bounded subsets B C E, t 6 N,

where k e L~o c (R) and/z denotes either the Kuratowski MNC ot or the Hausdorff MNC y. Then it is well known (see, e.g., [58, Theorem 9.2 and Remark 9.5.4 in Chapter 4.9.3]) that the initial value problem

[ 2 ( t ) 6 F(t,x(t)), x (0) -- xo,

for a.a. t e [ - m , m], m e N,

admits a solution Xm E A C ( [ - m , m]), for each m 6 N, i.e. x 6 ACloc(N, E). If, in particular, the values of Xm, m 6 N, are located in a given bounded, closed subdomain 79 of E, then there exists an entirely bounded solution x 6 ACloc(IR, E) on IR, x(0) = x0, with values in 7), provided E is separable or reflexive. It is namely enough to apply Proposi- tion 4.4, for to = 0 and g(t, x) := 2k(t)x. Such special g is a Kamke function by means of the Gronwall inequality.

If in particular, E = R n, then (ii) and (iii) hold automatically. Hence, Proposition 4.4 can be then simplified as follows.

PROPOSITION 4.5. Let F : IR x ]I{ n ----o I[{ n be a u-Carathdodory mapping with nonempty, compact and convex values, satisfying (i) in Proposition 4.4, for J = ( - c~ , e~). Then, for every xo ~ R n, there exists a solution x ~ ACloc(R, R n) of the Cauchy problem

2(t) 6 F(t, x(t)) , x (0) = xo.

for a.a. t ~_ ( - cx~ , cx:~ ) ,

Let {Xm(t) } be a sequence of absolutely continuous functions such that (i) For every m ~ N, Xm ~ A C ( [ - m , m], N n) is a solution of

2(t) ~ F(t , x(t)) , for a.a. t ~ [ -m , m],


(ii) sup{lxm(t)l l m �9 N, t �9 [ - m , m]} := M < oo and Xm(t) �9 79 C R n, for every t �9 [ - m , m ] .

Then there exists an entirely bounded solution x �9 ACloc(N, R n) of the inclusion

2(t) �9 F( t , x ( t ) ) , for a.a. t �9 ( - c ~ , oo),

such that

s u P l x ( t ) l < ~ M ( < o o ) and x(t) �9 7), for all t �9 N. tEN

4.2. Topological structure o f solution sets

In this part, various methods for investigating the topological structure of solution sets, required in statements of the foregoing subsection, will be presented. Both initial and boundary value problems will be considered.

The classical result, due to F.S. De Blasi and J. Myjak in [57], deals with Cauchy problems for the u-Carath6odory differential inclusions in Euclidean spaces:

2 �9 F( t , x), (4.19) x (0) = x0,

where F : J • ]t~ n ---o ]1~ n is a u-Carath6odory mapping, i.e. a multivalued mapping, satisfying conditions from the beginning of Section 4.1 (cf. Definition 2.10), and such that

IF(t, x)l ~< c~ + ~lxl, for all t �9 J, x �9 N n,

where or,/3 are nonnegative constants.

THEOREM 4.5. Problem (4.19), where J is a compact interval, has under the above assumptions an R~-set o f solutions.

We omit the proof of this theorem, because below we will prove its generalized version (cf. Theorem 4.9).

We recall that a multivalued mapping F : J x R ---o R n is said to be integrably bounded (resp. locally integrably bounded) if there exists an integrable (resp. locally integrable) function # : J --+ [0, ec) such that lyl ~< #( t ) , for every x �9 ]1~ n, t e J and y e F(t , x). We say that F has at most a linear growth (resp. a local linear growth) if there exist integrable (resp. locally integrable) functions #, v: J --+ [0, oo) such that

lyl ~< tx(t)lxl + v(t) ,

for every x 6 R n, t 6 J and y e F( t , x). It is obvious that F has at most a linear growth if there exists an integrable function

# : J ~ [0, oo) such that lYl ~< lz(t)(Ixl + 1), for every x e R n, t �9 J and y e F( t , x).

46 J. Andres

Let us also recall that a single-valued map f : J x ~n ~ ]1~n is said to be measurable- locally Lipschitz (mLL) if, for every x 6 R n, there exists a neighbourhood Vx of x in R n and an integrable function Lx :J --+ [0, c~) such that

I f( t , X l ) - f ( t , x2 ) l< ,Lx( t ) lX l -X2l , f o r e v e r y t ~ J a n d x l , X 2 ~ Vx,

where f (., x) is measurable, for every x 6 R n. Now, for the considerations below, fix J as the halfline [0, c~) and assume that

F : J x ~n -----o ~n is again a multivalued u-Carath6odory map. Consider the Cauchy problem (4.19). By S(F, O, xo), we denote the set of solutions of (4.19). For the characterization of the topological structure of S(F, O, xo), it will be useful to recall the following well-known uniqueness criterion (see, e.g., [60, Theorem 1.1.2]).

THEOREM 4.6. If f is a single-valued, integrably bounded, measurable-locally Lipschitz map, then the set S( f , 0, xo) is a singleton, for every xo E R n.

The following result will be employed as well.

THEOREM 4.7. If F is locally integrably bounded, mLL-selectionable (i.e. if there exists a measurable-locally Lipschitz single-valued selection), then S(F, 0, xo) is contractible, for every xo E R n .

PROOF. Let f C F be measurable-locally Lipschitz. By Theorem 4.6, the following Cauchy problem

[~c = f(t, x), (4.20) x(to) =x0,

has exactly one solution, for every to 6 J and x0 6 ]~n. For the proof, it is sufficient to define a homotopy h:S(F , 0, x0) x [0, 1] --+ S(F, O, xo) such that

h ( x , s ) - I x, for s = 1 and x ~ S(F, 0, x0), ~, fors = 0 , /

where Y = S(f , 0, x0) is exactly one solution of the problem (4.20). Define y :[0, 1) --+ [0, ~ ) , y(s) = tan(zrs/2) and put

x(t), = s ( f ,

x(t),

for0 ~< t <~ y(s ) , s < 1, for y(s) ~<t < cxz, s < 1, for0 ~< t < cx~, s = 1.

Then h is a continuous homotopy, contracting S(F, O, xo) to the point S(f , O, xo). D

Analogously, we can get the following result.


THEOREM 4.8. I f F is locally integrably bounded, Ca-selectionable (i.e. i f there exists a

Carathdodory single-valued selection), or in particular c-selectionable (i.e. i f there exists a

continuous single-valued selection), then S(F, O, xo) is R~-contractible,for every xo E R n.

Observe that, if F : J • I~ n ---o ]t~ n is an intersection of the decreasing sequence of Fk " J • R n --o R n, F(t , x) = (']~=1Fk(t, x) and Fk+l (t, x) C Fk(t, x) , for almost all t E J and for all x E IR n, then

oo

S(F, O, xo) -- A S(Fk, O, xo). k = l

(4.21)

From Theorems 4.7 and 4.8, we obtain

PROPOSITION 4.6. Let F : J • Nn --o Nn be a multivalued map with nonempty, closed values.

(i) I f F is a-mLL-selectionable (i.e. it is an intersection of a decreasing sequence of mLL-selectionable mappings), then the set S(F, O, xo) is an intersection of a decreasing sequence of contractible sets,

(ii) i f F is ~-Ca-selectionable, i.e., it is an intersection of a decreasing sequence of Ca- selectionable mappings, then the set S(F, O, xo) is an intersection of a decreasing sequence of R~-contractible sets.

Before formulating the following important theorem, recall that, for two metric spaces X, Y and the interval J, the multivalued map F : J x X ---o Y is almost upper semicontinuous (a.u.s.c.), if for every e > 0 there exists a measurable set A~ C J such that m ( J \ Ae) < e and the restriction F]A~ xX is u.s.c., where m stands for the Lebesgue measure.

It is clear that every a.u.s.c, map is u-Carath~odory. In general, the reverse is not true. The following Scorza-Dragoni type result describing possible regularizations of Carath~odory maps (see, e.g., [76]) will be employed.

PROPOSITION 4.7. Let X be a separable metric space and J be an interval. Suppose that F : J x X ---o ]t~ n is a nonempty, compact, convex valued u-Carathdodory map. Then there exists an a.u.s.c, map ~ : J • X ---o ]~n with nonempty compact convex values and such that:

(i) f r ( t , x ) C F ( t , x ) , f o r every ( t , x ) ~ J • X,

(ii) i f A C J is measurable, u : A -+ IK n and v : A -+ X are measurable maps and

u(t) E F(t , v ( t ) ) , f o r almost all t E A, then u(t) E ~(t , v ( t ) ) , f o r almost all t E A.

The proof of the following statement can be found in [67].

PROPOSITION 4.8. Let E, E1 be two separable Banach spaces, J be an interval and

F : J x E ---o E be an a.u.s.c, map with compact convex values. Then F is a-Ca-

selectionable (i.e. it is an intersection of a decreasing sequence o f Ca-selectionable map-

pings Fk : J • E --o El). The maps Fk : J • E ---o E1 are a.u.s.c., and we have Fk(t, e) C

48 J. Andres

conv(~xe E F(t , x)) , for all (t, e) ~ J x E. Moreover, i f F is integrably bounded, then F is cr-mLL-selectionable, i.e., it is an intersection of a decreasing sequence of mLL- selectionable mappings.

Now, we are ready to give

THEOREM 4.9. I f F" J x R n ---o R n is a u-Carath~odory map with compact convex values having at most the linear growth, then S(F, O, xo) is an R~-set, for every xo ~ R n .

PROOF. By the hypothesis, there exists an integrable f u n c t i o n / z ' J --+ [0, c~) such that sup{lYl [ y E F ( t , x ) } <<,/z(t)(lx[ + 1), for every ( t , x ) ~ J x R n. By means of the well- known Gronwall inequality (see [71]), we obtain that Ix(t)l ~< (Ix01 + y)exp(y) -- M,

where x E S(F, O, xo) and y = f o / x ( s ) ds. Take r > M and define F ' J x R n ~ R n as follows"

"~ [ F (i ' x ) ' ) i f l x l <<" r

F ( t , x ) = F t , r -~ l , iflxl > r .

One can see that F is an integrably bounded u-Carath6odory map and

S(F , O, xo) = S(F, O, xo).

By Proposition 4.7, there exists an a.u.s.c, map G ' J x R n ---o R n with nonempty, convex, compact values such that S(G, O, xo) - S (F , O, xo). Applying Proposition 4.8 to the map G, we obtain the sequence of maps G~. As in Proposition 4.6, we see that S(G, O, xo) is an intersection of the decreasing sequence S(G~, O, xo) of contractible sets. By the well- known Arzel~-Ascoli lemma and Theorem 4.6, we obtain that, for every k 6 N, the set S(G~, O, xo) is compact and nonempty, which completes the proof. IS]

Using the above results and the unified approach to the u.s.c, and 1.s.c. case due to A. Bressan (cf. [49,50]), we can obtain the following result.

PROPOSITION 4.9. Let G" J x ~ n ---o ~n be a l.s.c, bounded map with nonempty closed values. Then there exists a u.s.c, map F" J x ~ n ----o ]1~ n with compact convex values such

that, for any xo ~ R n, the set S(G, O, xo) contains an R~-set S(F, O, xo) as a subset.

REMARK 4.7. In [22], topological structure of solution sets is also treated, provided F is not necessarily convex-valued. However, the absence of convexity seems to be a big handicap, because to prove the connectedness and compactness of the related solution set can be a difficult task (see, e.g., [22, Example 2.18 on p. 258]).

REMARK 4.8. It follows from the result in [98] (cf. also [55] or [78, Corollary 5.3.1], where mild solutions were considered for semilinear differential inclusions) that, in a

Topological principles for ordinary differential equations 4 9

real separable Banach space E, the solution sets to initial value problems are R~, provided conditions in Remark 4.6 hold. In general Banach spaces E, it is at least so when F(t , . ) : E --o E is completely continuous (cf. [58, Corollary 9.1(b) on pp. 118-119]).

We can also say something about the covering (topological) dimension of solution sets to the Cauchy problem (4.19).

Let ~ be an open set in R n+l such that [to, to + h] x B(xo, r) C f2, where B denotes the closed ball centered at x0 and with the radius r. Assume that F :f2 --o R n satisfies the following conditions:

(C1) the set of values of F is nonempty, compact and convex, for all (t, x) 6 ~ , (C2) F(t , .): B(x0, r) --o R n is continuous, for a.a. t 6 [to, to + h], (C3) F(. , x) : [to, to + h] ---> IR ~ is measurable, for all x ~ B(xo, r), (C4) there exist Lebesgue-integrable nonnegative functions or,/3 :[to, to § h] ---> [0, co)

such that, for any x E B(xo, r), IF(t, x)l ~< or(t) § for a.a. t 6 [to, to § h], where IF(t, x)l ~< sup{lYl I y E F(t , x)}.

Denote by S([t0, to + d], x0) the set of solutions x 6 AC([t0, to + d], R n) of (4.19) on the interval [to, to + d], 0 < d ~< h.

The following two theorems are due to B.D. Gel 'man [66] (cf. [22, Theorems 2.60 and 2.61 in Chapter III.2]).

THEOREM 4.10. Let the assumptions (C1)-(C4) be satisfied. Assume that the set

A - - {t ~ [to, t o + h ] I d i m ( F ( t , x ) ) ~ 1, f o r a n y x ~ B(xo, r)}

is measurable and

lim # (A A [to, to + h]) h-->O h

> 0 ,

where dim(.) denotes the covering dimension and lz(.) stands for the Lebesgue measure. Then there exists a number do such that, for any 0 < d <, do, we have S = S([t0, to + d], x0) =fi {0} and dim(S) = ec.

THEOREM 4.1 1. Let the assumptions of Theorem 4.10 be satisfied jointly with (C2 I) F(t , .): B(xo, r) ---o IR n is Lipschitz-continuous.

Then there exists a number do such that, for any 0 < d <. do, any E > 0 and any solution x E S([to, to + d], xo) (:/: {0}), we have dim(Sx,e) = oc, where Sx,e = {y ~ S I IIx - Yll e}.

Now, we shall study the reverse Cauchy problem when, instead of the origin, the value of solutions is prescribed at infinity, namely

for a.a. t 6 [0, ec), 2(t) ~ F ( t , x ( t ) ) , R n lim x( t ) - x~c ~ ,

t----~ o o

(4.22)

where F :[0, cxz) x R n ---o R n is a u-Carath6odory map, i.e.

50 J. Andres

(i) values of F are nonempty, compact and convex, for all (t, x) e [0, oo) x I~ n, (ii) F (t, .) is upper semicontinuous, for a.a. t ~ [0, c~),

(iii) F (., x) is measurable, for all x 6 I~ n. We will prove acyclicity of the solution set of problem (4.22). Recalling that any contractible set is acyclic, we can give

THEOREM 4.12. Consider the target (terminal) problem (4.22), where F : [0, oo) x R n --o I[~ n is a u-Carathdodory map and x ~ e ]I~ n is arbitrary. Assume that there exists a globally integrable function v ' [0 , oo) --+ [0, oo), where f o v(t) dt - E < 1, such that

d H ( F ( t , x ) , F ( t , y ) ) <<, v ( t ) l x - y[, f o r a l l t ~ [ O , c ~ ) a n d x , y ~ I R n. (4.23)

Moreover, assume that dH(F(. , 0), 0) can be absolutely estimated by some globally integrable function. I f E is a sufficiently small constant, then the set o f solutions to problem (4.22) is compact and acyclic, for every xoo ~ I~ n .

PROOF. Observe that condition (4.23) implies the existence a globally integrable function ot : [0, oo) --+ [0, c~) and a positive constant B such that

[F(t,x)] <<, a ( t ) (B + Ixl), for every x 6 •n and a.a. t ~ [0, cx~), (4.24)

where IF(t, x)l = sup{lyl I y ~ F(t, x)}. Thus, problem (4.22) can be equivalently replaced by the problem

{ ~c(t) ~ G(t, x( t )) , for a.a. t ~ [0, c~), lim x(t) - x ~ ~ R n,

I - -~ o o

(4.25)

where G is a suitable Carath6odory map which can be estimated by a sufficiently large positive constant M, i.e.

[G(t,x)[ <<. M, for every x 6 IR n and a.a. t 6 [0, oe),

and which satisfies condition (4.23) as well. In other words, the solution set S for problem (4.22) is the same as for problem (4.25), where

8 - {x ~ C([O, oo), Rn) [ A(t) ~ F ( t , x ( t ) ) ,

for a.a. t ~ [0, cx~) and x(oo) - xc~ }.

For the structure of S, we will modify the approach from above. Observe that, under the above assumptions, F as well G are well known to be product-measurable (see Proposi- tion 2.4), and subsequently having a Carath6odory selection g C G which is Lipschitzian with a not necessarily same, but again sufficiently small constant (see, e.g., [73, pp. 101- 103]). By the sufficiency we mean that, besides others,

[g(t ,x) - g(t, y)[ ~< y( t ) lx - y[


holds, for all x, y E IR n and a.a. t 6 [0, oo), with a Lebesgue integrable function V'[0, oc) --+ [0, oc) such that f o V ( t ) d t < 1.

Considering the single-valued problem (g C G)

{ 2 ( t ) - g ( t , x ( t ) ) , lim x ( t ) = xoc, t-----~ oo

for a.a. t �9 [0, oc), (4.26)

we can easily prove the existence of a unique solution 2(t) of problem (4.26). The uniqueness can be verified in a standard manner by the contradiction, when assuming the existence of another solution y(t) of that problem, because so we would arrive at the false inequality

sup I~-(t)-y(t)l= sup tE[0,oc) tE[0,oc) f2 f2 g(s , -Y(s)) ds - g(s , y ( s ) ) ds

f0 ~ <~ Ig(s, 2 ( s ) ) - g(s , y(s)) I dt

f0 ~ ~< t '(t) sup 12(t) - y ( t ) ldt

tE[0,oc)

f0 ~ sup 12-(t) - y(t) I • (t) dt

tE[0,oc)

sup I~( t ) - y ( t ) l . tE[0,oc)

Hence, according to the definition of contractibility in Section 2.1, it is sufficient to show that the solution set S of problem (4.25) is homotopic to a unique solution 2(t) of problem (4.26), which is at the same time a solution of problem (4.25) as well. The desired homotopy reads (,k E [0, 1])

x(t), h(x , )~)(t) -- -~(t),

-Y(t),

for t /> 1/)~ -)~, ~. ~ 0, for0 < t ~< 1/)~ -)~, )~ 7~ 0, for)~ = O,

where ~ is a unique solution to the reverse Cauchy problem

2(t) - g( t , z ( t ) ) , for a.a. t E [0, 1/)~ - )~], z ( 1 / L - F~) -- x ( 1 / ~ - )Q,

for each )~ E [0, 1 ]. Then h is a continuous homotopy such that h (x, 0) = 2, h (x, 1) = x, as required, and subsequently, the set S is acyclic. Using the convexity assumption on values of F, we can prove by the standard manner [22, Mazur's Theorem 1.33 in Chapter 1.1] that S is closed in C([0, oc), Rn). By Arzelh-Ascoli's lemma, this set is compact, and the proof is complete. D

52 J. Andres

Now, we shall consider the boundary value problem

2(t) + A(t)x(t) e F( t , x ( t ) ) , for a.a. t e [0, T], (4.27) Lx -- r,

where (i) A :[0, T] ~ E(R n, R n) is a measurable linear operator such that IA(t)l ~< y(t) , for

all t e [0, T] and some integrable function 9/:[0, T] -+ [0, oc), (ii) the associated homogeneous problem

2(t)+A(t)x(t)--O, Lx = 0

for a.a. t e [0, T],

has only the trivial solution, (iii) F :[0, T] x R n --o R n has nonempty, compact, convex values, (iv) F (., x) is measurable, for every x e R n, (v) there is a constant M ~> 0 such that

di4(F(t ,x) , F(t, y)) <, Mix - Yl, for all x, y e R n and a.a. t 6 [0, T],

where d/4 stands for the Hausdorff metric, (vi) there are two nonnegative Lebesgue-integrable functions 61,62:[0, T] ~ [0, cx~)

such that,

IF( t ,x ) l<,61( t )+62( t ) lx l , fora.a, t e [ 0 , T l a n d a l l x e R n,

where IF(t ,x)l = sup{lyl I y e F(t ,x)} . In [43], the authors have proved the functional generalization of following theorem.

THEOREM 4.13. Under the assumptions (i)-(vi), a certain "critical" value )~ exists such that if M < )~, then the set of solutions of (4.27) is a (nonempty) compact AR-space. More- over, if the Lebesgue measure of the set {t I d i m F ( t , x ) < 1,for some x 6 1R} is still zero, then the set of solutions of (4.27) is an infinite dimensional compact AR-space, where dim X denotes the covering (topological) dimension of a space X.

REMARK 4.9. Observe that for A --- 0 and Lx = x (0), the related Cauchy problem can have, under the assumptions of Theorem 4.13, infinitely many linearly independent solutions on the whole interval [0, T].

REMARK 4.10. The first assertion of Theorem 4.13 can be still improved (see [22, Theo- rem 3.13 in Chapter III.3]), namely that, for the problem

2(t) + A(t)x(t) 6 ~F( t , x ( t ) ) , for a.a. t E [0, T], (4.28) Lx =0 ,

where ot ~< )~, when )~ is the critical value in Theorem 4.13, the set of solutions of (4.28) is nonempty, compact and acyclic.


REMARK 4.1 1. In the case of ODEs, the solution set in Theorem 4.13 consists, unlike in the critical case for ot = k in Remark 4.10, of a unique solution. The same is true for Theorem 4.12.

In view of Remark 4.11, a nontrivial structure of solution sets to single-valued boundary problems can be seen as a delicate problem. The following result in this field in [46] is rather rare.

THEOREM 4.14. Consider the Floquet problem

Yc(t) = f (t, x( t)) , for a.a. t �9 [a, b], x ( a ) + X x ( b ) - ~ (X > 0,~ 6IRn),

(4.29)

where f :[a, b] x I~ n --+ R n is a bounded Carath~odory function. Assume, furthermore, that f satisfies

I f ( t , x ) - f ( t , y ) l < , p ( t ) l x - yl, fora.a, t e [ a , b ] a n d x , y e R n, (4.30)

where p : [ a , b] --> [0, cx~) is a Lebesgue-integrable function such that

fa b p(t) dt ~< V/Tr 2 -ff In 2 X.

Then the set of solution to (4.29) is an R~-set.

(4.31)

REMARK 4.12. As pointed out in [46], if the sharp inequalities take place in (4.31), then problem (4.29) has a unique solution. On the other hand, for equalities (4.31), problem (4.29) can possess more solutions, respectively.

Unlike in the above theorems, the following problems can be regarded as those with "limiting" boundary conditions. In [53], the following result has been proved (as Theo- rem 3.1) for the boundary value problem

Yc - f (t, x), (4.32) Lx -- r,

where f : I x ]~n ~ ]t~n is a continuous function, L : C 1 (I, R n) --+ ]1~ n is a linear operator

and I -- [a, b] is a compact interval.

PROPOSITION 4.10. Let f : I x •n __+ ]t~n be a fixed continuous function such that, for every to e I and xo �9 R n, there exists a unique (smooth) solution x(t) of the equation ~c = f ( t ,x) , satisfying x(to) = xo.

Let bt be an open (in the norm topology) subset of the Banach space of all continuous linear operators L" C~ IR n) --+ I~ n, where C o denotes the set CI (IR n) C C~ Rn),

topologized by the induced topology of C~ I~ n) ( := C(I, ~)).

54 J. Andres

If, for every L e lg and r e ]1~ n, the boundary value problem (4.32) has at most one solution, then, for every L e bl and r e R n , problem (4.32) has exactly one solution.

Our aim is to prove, by means of Proposition 2.6 in Section 2.3, the following theorem.

THEOREM 4.15. Let f " I x ~n __+ iRn be a fixed continuous function and p" I --+ R n be

a continuous function such that, for every to e I, xo e R n and p e C (I, ~n) with II p II ~< 1, there exists a unique solution x ( t ) o f

2 -- f ( t , x) + p( t ) , (4.33)

satisfying x(to) = xo. Let bl be an open (in the norm topology) subset o f the Banach space o f all continuous lin-

ear operators L " C~ R n) --+ ~n, where C O has the same meaning as in Proposition 4.10.

Assume that, for every L e bl, r e R n and p e C 1 (~n) with IIP[I ~< 1, the boundary value

problem

Yc = f (t, x) + p( t ) , (4.34) Zx = r,

has at most one solution and that, for every L e ld, all solutions o f problem (4.32) are uniformly (i.e. independently o f L e ld) a priori bounded, where bl denotes the closure o f bt in the cl- topology.

Then, for every L e Old and r e Ii~ n, where Obt denotes the boundary o f 1.4 in the C~ problem (4.32) has an R~-set o f solutions.

PROOF. Since all assumptions of Proposition 4.10 are satisfied, problem (4.34) is solvable, for every L e b/, r e I~ n and p e C (I, IR n) with II P II ~< 1.

Furthermore, since b / i s a closed subset of the Banach space of all continuous linear operators, each element L e 0b/can be regarded as a uniform limit of a suitable sequence {L~} such that L = l i m k ~ L~, where Lk e b / ( = intb/), for every k e I~.

Fix such an [, e 0b/and consider the compact operators ~k, ~ ' / 3 --+ C~ It~n):

fa t �9 k ( x ) ( t ) = x ( a ) + L k x - - r + f ( s , x ( s ) ) d s

and

fa' ~ ( x ) ( t ) -- x(O) + Lx - r + f (s, x ( s ) ) ds,

where 13 C C O (I, ]~n ) is a suitable closed ball centered at the origin, which is implied by the assumption of a uniform a priori boundedness of solutions. The compactness of operators follows directly by means of the well-known Arzel?~-Ascoli lemma.


One can readily check the one-to-one correspondence between the fixed points of �9 and the solutions of problem (4.32) as well as those of ~k and the solutions of the equation 2 - f ( t , x ( t ) ) , satisfying

L k x = r.

Thus, one can associate to ~k and �9 the proper maps r -- i d - -~k and r -- i d - ~ , respectively, where id denotes the identity, namely

fa t r = x ( t ) -- x ( a ) -- L k x + r -- f (s, x ( s ) ) ds

and

fa t r -- x ( t ) - x ( a ) - L x + r - f (s, x ( s ) ) ds.

So, the nonempty kernel q9 -1 (0) of r corresponds to the fixed points of ~ , and subsequently to solutions of (4.32), i.e.

x 6 ~ ( x ) ~, ,~ 0 E x - ~ ( x ) -- (id-(I:,)(x) - r

We can assume without any loss of generality that, for a sufficiently large k 6 N, we have

1 I~o~(x)(,)- ~o(x)(,)l = IL~x- Zxl = I(L~- Z)xl ~ ; , (4.35)

because, otherwise, we can obviously select a subsequence with this property. Since [Iqgk(x)(t) -- ~o(x)(t)ll ~< 1 / k holds, for every x 6 /3 , condition (i) of Proposi-

tion 2.6 in Section 2.3 is satisfied. In order to prove (ii) in Proposition 2.6, it is sufficient to verify the following inequalities

1 1 I~<x~<t~l ~ ~ and I(~<x~)<t~l ~ ~, k ~ N,

for every x with qg(x) = 0.

However, since (~pk(x))( t ) = 2 ( t ) - f ( t , x ( t ) ) = 0, k E N, and the first inequality follows from (4.35), we are done.

In order to verify (iii) in Proposition 2.6, we should realize that, for any u E V~ -- {u

C I ( I , R n ) " Ilullc0 ~< 1 / k and II~llc0 ~< 1 / k } , for some k ~ N, x ( t ) is a solution of the equation u ( t ) = qgk(x)( t ) , i.e.

fa t u ( t ) -- x ( t ) - x ( a ) -- L k x + r - f (s, x ( s ) ) ds ,

56 J. Andres

if and only if it satisfies

{ ~c -- f ig) + f (t, x) , (4.36) L~x = r - u ( a ) .

By the hypothesis, problem (4.36) has a unique solution, for every L~ 6/.4, r 6 ]1~ n and u C I ( I , R n) with Ilull ~< 1, as required. Therefore, applying Proposition 2.6 in Section 2.3, the set {qg(0)} is R~. In other words, the solution set of the original problem (4.32) is R~ as well. D

REMARK 4.13. One can observe that the sole existence can be easily proved by means of the well-known Schauder fixed point theorem.

EXAMPLE 4.1. According to Example 2 in [85], problem

[ -~i = fi (t, Xl, X2) + Pi (t), i -- 1, 2, axl (0) -+- X2(0) = rl , bxl (1) -+- x2(1) -- r2,

(4.37)

is uniquely solvable, for every a 2 < 1, b 2 > 1, ri E R (i = 1, 2) and p = (Pl, P2) E C ([0, 1 ], ~) , provided f / 6 C 1 ([0, 1], R), i = 1, 2, and

~fl ~A ~f2 ~f2 U 2-~- U U2-- U2-- U 2 ~ 0 OqXl ~X2 1 ~Xl ul ~X2

(i = 1, 2),

for each triple ( t , X l , X 2 ) E [0, 1] x IF[ 2 and each double (Ul , U2) E ]~2.

Therefore, according to Theorem 4.15 (more precisely, according to its modified version, where the set of all continuous linear operators can be restricted (see [85]) to the set of all real (n x n)-matrices), problem

[ -~i = ft'(t, Xl, X2), i = 1, 2, axl (0) -k- X2(0) - - rl, bxl (1) + x2(1) -- r2,

(4.38)

has an R~-set of solutions, for certain (a, b) 6 ]~2 in a closed subset of ]I~ 2 with a 2 = 1, b 2/> 1 or a 2 ~< 1, b 2 -- 1 (rl, r2 can be arbitrary), whenever all solutions of problem (4.38)

are uniformly a priori bounded, for such a 2 ~< 1, b 2/> 1. This can be achieved for a ~ b, i.e. particularly with the exception of a = b -- 1 or

a - b - - 1 , and b 2 ~< b 2, for some b. > 1, when, e.g.,

I f ( t , x ) ] <~ ~lxl + ~, for all ( t , x ) ~ [0, 1] • ]I~ 2, (4.39)

where or,/3 are suitable nonnegative constants (ct must be sufficiently small as below) and

x -- (Xl, x2), f = ( f i , f2). Indeed. Since the linear homogeneous problem

Xi =0, i - 1,2, axl (0) + x2(0) -- 0, b x l ( 1 ) + x 2 ( 1 ) = O ,


Fig. 2.

has for a ~ b obviously only a trivial solution, every solution x ( t ) - (Xl (t), x2 (t)) of (4.38) takes the form (see, e.g., [22, Lemma 5.136 in Chapter III.5])

f0 1

xi( t ) -- G i ( t , s , a , b ) f i ( s , x l ( s ) , x 2 ( s ) ) d s -Jr- "xi, i - 1,2,

where G - (G1, G2) is the related Green function of the linearized problem (4.38), namely

{ )r -- f i ( t , Xl ,X2), ax 1 (0) -Jr- X2 (0) - - O,

i - - 1,2, bxl (1) + x2(1) - -0 ,

i .e.

1) G ( t , s a , b ) - - b - a - a b - a ' l(a 1)

b - a - a b - b '

f o r 0 ~< t ~<s ~< 1,

f o r 0 ~<s ~< t ~< 1,

and Y = (~1, ~2) is a unique solution of the problem

{ xi - - 0 , i -- 1,2,

a x l (0) + X2(0) - - r l , b x l (1) + x2(1) - r2,

i.e. ~1 -- (r2 - r l ) / ( b - a ) , "22 - r l - a ( r 2 - r l ) / ( b - a ) .

Let us fix (a, b) at the boundary 0Lt - { (a, b) e R 2 I a2 - 1, b 2/> 1 or a 2 ~< 1, b 2 - 1 }

with a 7~ b, for which we intend to get the result and cut off appropriately the comers with

a - b, jointly with those (a, b) with b 2 > b 2 for some b, > 1, as in Fig. 2. The bold curve in Fig. 2 so indicates the part of the boundary of our interest.

58 J. Andres

Denoting g = maX(a,b)~, ga,b, ga,b -= maxt,sz[0,1] IG(t, s, a, b)l, where b/, is indicated

in Fig. 2 by the shaded region (observe that since b/, is compact, g certainly exists), we obtain by means of (4.39) that

IIx,:,)ll g( llx(t)l1-4- m a x _ I ; I , (a,b)eLt ,

i.e.

IIx<t)ll fig + maX(a,b)~, I~1

1 - o t g

whenever ot < g-1, as claimed. This completes the example.

REMARK 4.14. One can easily check that, for fixed values of (a, b) 6 Ob/with a # b, the condition c~ < g-1 can take, e.g., the form

Ol < I b - a l

Ibl 4" max(l, labl)

A continuous function f " I x ~n _..+ I~n can be approximated with an arbitrary accuracy by locally Lipschitzian (in the second variable) functions (see, e.g., [22, Theorem 3.37 in Chapter 1.3] or [71]), say ( f + e~), k E N, such that limk__.~ Ile~ II = 0. Therefore, applying at first, for fixed k 6 N, Theorem 4.15 to the system

Yc = f (t, x) 4" ek(t, x) , (4.40)

we can still avoid (for more details, see [22, Chapter 111.3]) the uniqueness assumption in Theorem 4.15 as follows.

THEOREM 4.16. Let f : I x I[~ n -"+ ~pn be a fixed continuous function and ek :I x I[~ n •n, k E N, be continuous functions with Ile~ II <~ E (~ - a sufficiently small constant) such that ( f + ek)(t, .) :R n --+ IR n, k ~ N, are locally Lipschitzian, for every t ~ I, and p : I --+

IR n be a continuous function with II P II ~< 1. Let bl be an open (in the norm topology) subset o f the Banach space o f all continuous

linear operators L " C~ IR n) --+ R n .

Assume that, for every L ~ bt, r ~ IR n, k ~ N and p ~ C( I , R n) with IlPll ~< 1, the bound-

ary value problem

Yc = f (t, x) 4" ek(t, x) 4" p( t ) , Z x = r

has at most one solution and that, f o r every L ~ bl and k ~ N, all solutions o f the problem

Yc -- f (t, x) 4" e~(t, x) , L x - - r


m m

are uniformly (i.e. independently of L E Lt) a priori bounded, where lg denotes the closure oflg in the C~

Then, for every L ~ OLt and r ~ ]R n, where OLt denotes the boundary of Lt in the C~ the problem (4.32), i.e.

/c-- f ( t , x ) , L x - - r

has a (nonempty) compact acyclic set of solutions.

4.3. Poincard's operator approach

By the Poincar6 operators, we mean the translation operators along the trajectories of the associated differential systems. The translation operator is sometimes also called as Poincard-Andronov or Levinson or, simply, T-operator.

In the classical theory, these operators are defined to be single-valued, when assuming among other things, the uniqueness of the initial value problems. At the absence of uniqueness, one usually approximates the fight-hand sides of the given systems by the locally Lipschitzian ones (implying already uniqueness), and then applies the standard limiting argument (for more details, see, e.g., [71,81 ]).

On the other hand, set-valued analysis allows us to handle directly with multivalued Poincar6 operators which become, under suitable natural restrictions imposed on the fight- hand sides of given differential systems, admissible in the sense of Definition 2.5 in Sec- tion 2.2.

Hence, consider the u-Carath6odory system

k ~ F( t ,x ) , x 6 Ii~ n, (4.41)

where F : [0 , r ] x ]1~ n - - - o ]~n satisfies all conditions in Definition 2.10. By a solution x(t) of (4.41), we mean an absolutely continuous function x(t)

AC([0, r], R n) satisfying (4.41), for a.a. t 6 [0, r], i.e. the one in the sense of Carath6- odory, such solutions of (4.41) exist on [0, r].

Hence, if x(t, xo) :-- x(t, 0, x0) is a solution of (4.41) with x(0, x0) = x0 6 R n, then the translation operator TT :JR n --o IR n at the time r > 0 along the trajectories of (4.41) is defined as follows:

Tr (x0)"-- {x(v, x0) I x(., x0) is a solution of (4.41) with x(O, xo) -- x0}. (4.42)

More precisely, Tr can be considered as the composition of two maps, namely Tr = 7z o ~0,

~n ~ AC([O, r], ]K n) 7, > ]Rn,

60 J. Andres

where qg(xo)'xo ---o {x(t, x0) Ix(t , xo) is a solution of (4.41) with x(0, x0) = x0} is well known to be an R~-mapping (see Theorem 4.5) and ~p(y)'y --+ y( r ) is obviously a continuous (single-valued) evaluation mapping.

In other words, we have the following commutative diagram:

IR n ~o o AC([O, r], ]R n)

�9

r~ ~ 0 ]t~ n

The following characterization of Tr has been proved on various levels of abstraction in several papers (see, e.g., [22, Theorem 4.3 in Chapter 111.4] and the references therein).

THEOREM 4.17. Tr defined by (4.42) is admissible and admissibly homotopic (see Defi- nitions 2.5 and 2.6) to identity. More precisely, Tr is a composition of an Ra-mapping and a continuous (single-valued) evaluation mapping.

PROOF. According to Theorem 4.5, the mapping ~0 has an R~-set of values. We will show that it is u.s.c, by proving the closedness of the graph 1-'~0 of ~0 (cf. Section 2.2).

Let (Xn, Yn) e 1-'~0, i.e. Yn e qg(Xn), and (Xn, Yn) --+ (x, y) as n --+ oe. Since the functions Yn are absolutely continuous on [0, r], the application of the well-known Gronwall inequality (see, e.g., [71]) leads to the estimates (cf. (iii) in Definition 2.10)

IlYnll ~ M := sup(Ixnl + y r ) e x p ( y r ) and IlYnll <~ y(1 + M), n E N

where y = max{a,/3 }. It follows that {Yn } are equibounded. Proposition 4.1 guarantees the existence of a sequence {Yn } such that Yn --+ Y, uniformly,

and Yn --+ Y, weakly in L 1 ([0, r ], IR n). According to Mazur's theorem (see, e.g., [22, The- orem 1.33 in Chapter 1.1]), ~ belongs to the strong closure ~ e c--6-~{~n In >~ 1}, for all 1 >~ 1. Thus, there also exists a subsequence {zt } such that zl ~ Y, in the Ll-topology, where zl e conv{~n In >/1}. Moreover, there exists a subsequence (for the simplicity, de-

noted again by {zt }) satisfying zl ~ y, a.e. on [0, r]. Let I C [0, r] be a set of a full measure on [0, r], i.e. # ( I ) = r , where /z denotes the

Lebesgue measure, such that zl --+ y as I --+ cx~, for all t e I. It follows from the definition

of zl that zl(t) e Z i )~i F(t, Yni (t)), where Z i ~,i = 1. Since F(t, .) is u.s.c., for a.a. t e [0, r], and Yni (t) is sufficiently close to x(t) as well as

zt(t) to 2(t), we obtain

x(t) E E )~i F(t, x(t)) + eB i

for an arbitrary e > O, where B is an open unit ball. This already means that Yc(t) e F(t, x(t)), and subsequently the graph 1-'~0 of q) is closed. Since the arbitrary closed set


{(x, y), (xl, yl) . . . . , (Xn, Yn) . . . . } is, according to the well-known Arzelh-Ascoli lemma, compact, q9 is u.s.c.

For the remaining part of the proof, it is sufficient to consider the admissible homotopy T;~r, ,k E [0, 1]. D

REMARK 4.15. Since a composition of admissible maps is admissible as well (see Sec- tion 2.2), Tr can be still composed with further admissible maps 4~ such that 4~ o Tr becomes an (admissible) self-map on a compact ENR-space (i.e. homeomorphic to ANR in I~ n), for computation of the well-defined (cf. Section 3.1) generalized Lefschetz number:

A ( r o T~) = A ( ~ ) .

Tr considered on ENRs can be even composed, e.g., with suitable homeomorphisms 7-/ (again considered on ENRs), namely 7-/o Tr, for computation of the well-defined (cf. Sec- tion 3.3) fixed point index:

ind(7-/o Tr) = ind 7-/,

provided the fixed point set of 7-/o Tzr is compact, for )~ ~ [0, 1 ].

REMARK 4.16. In [ 10] (cf. also [22, Chapter III.4]), translation operators are also studied, e.g., for systems with constraints, systems in Banach spaces, for directionally semicontinuous systems, etc. In particular, in real separable Banach spaces, one can check that, under the conditions in Remark 4.6 (cf. also Remark 4.8 and [58, Corollary 9.1 in Chapter 9.4]), the related translation operator Tr is like in Theorem 4.17. In order Tr to be also condensing, one should however impose some further restrictions. Since these restrictions are rather technical (cf. [78, Theorem 6.3.1] or [22, Theorem 4.16 in Chapter III.4]), and so this Poincar6's translation operator will not be more employed, we omit them here.

5. Existence results

5.1. Existence of bounded solutions

We start with the application of Theorem 4.4. Hence, let (E, II" II) be a reflexive Banach space and let E(E) be the space of all linear continuous transformations in E. The Haus- dorff measure of noncompactness (MNC) will be denoted by y.

We are interested in the existence of a bounded solution to the semilinear differential inclusion

k(t) + A(t)x(t) ~ F(t, x(t)), for a.a. t E R (5.1)

with A(t) ~ s and a set-valued transformation F. Our assumptions concerning the inclusion (5.1) will be the following:

62 J. Andres

(A1) A : R --+ E(E) is strongly measurable (cf. Definition 2.9) and Bochner integrable, on every compact interval [a, b].

(A2) Assume that

~c + A (t)x -- 0 (5.2)

admits a regular exponential dichotomy (cf. Remark 5.3 below; for more details see, e.g., [56]). Denote by G the principal Green's function for (5.2).

(F1) Let F :R • E ---o E be a u-Carath6odory set-valued map (cf. Definition 2.10) such that

IIF(t,x)[[ <~ m(t), for a.a. t 6 R, x 6 E.

Here m 6 L~o c (•) is such that, for a constant M,

sup{f '+1 m ( s ) d s l t 6 R } <M.

(F2) Assume that

y(F(t , f2)) <<, g(t)h(y(f2)), for a.a. t 6 R

and each bounded f2 C E, where g, h, are positive functions, g is measurable, h is nondecreasing such that

and qh(t)L < t, for each t > 0, with a constant q = 1, if E is separable, and q = 2, in the general case.

THEOREM 5.1. Under the assumptions (A1), (A2), (F1), (F2), the semilinear differential inclusion (5.1) admits a bounded solution on R.

The main obstruction in the application of Theorem 4.4 will be the estimation of a suit- ably chosen MNC. For this purpose, we recall the following rule of taking the MNC under the sign of the integral (see [78, Corollary 4.2.5]).

LEMMA 5.1. Let {fn } C L 1 ([a, b], E) be a sequence of functions such that (i) Ilfn(t)ll <<. v(t) , forall n ~ N anda.a, t E [a,b], where v ~ Ll([a,b]),

(ii) y({fn(t)}) <~ c(t),fora.a, t ~ [a,b], where c ~ Ll([a,b]). Then we have the estimate

Y ( { f a t f n ( s ) ds} ) <,qfatC(S) ds,

for each t ~ [a, b], with q = 1, if E is separable, and q = 2, in general case.


PROOF OF THEOREM 5.1. We carry out the proof in several steps. (i) Let

a - - {x �9 c (IR, E) ] [I x (,)ll ~< K, for each, �9 IR, II x (t l ) - x ( ,2) ll

S, } g II A(s) ll z;(E) as 4- m (s) as, for all t l, t2 �9 IR, t l ~< '2

with a constant K to be specified below. Clearly Q is a closed convex subset of C (IR, E). For a given q e Q, we are interested in bounded solutions to the differential inclusion

~c(t) 4- A(t)x(t) e F(t, q(t)), for a.a. t �9 IR. (5.3)

Take f �9 NF(q) (recall that such f exists, in view of Lemma 4.3), where NF denotes the Nemytskii operator. Since A admits an exponential dichotomy, we know that the problem

Yc(t) 4- A(t)x(t) = f (t), for a.a. t �9 IR,

has a unique, entirely bounded solution given by

x ( f ) -- fI~ G(t, s) f (s) ds

(cf. [56,86]). Thus, problem (5.3) has a nonempty set of solutions T (q). Using Lemmas 4.3 and 4.4, it is also clear that this set is closed convex, and since its compactness will become clear in the subsequent steps of the proof, it is in fact an R~-set.

(ii) We will show that, for each q e Q, we actually have T(q) C Q. Let x e T(q). Then, for suitable f �9 NF(q), we have

f IIx(t)ll j IIc(t,s)llc( )llfu)ll ds

ff <~ k e-U(t-S)m(s) ds + k e-U(s-t)m(s) ds (x)

o0 fj+l ~~jj+l = k ~ e-n~m(t - a) da +k e-n'~m(t + o-) da

j=0 aj j=0

oo [,+1 f i f,+l <~ k ~ e -~j m(, - a) da + k e -~j m(t + a) da

j -O dJ j -O JJ

oo tf_--j ~ /t+j+i <~ k ~ e -#j m(s) ds + k e -#j m(s) ds

j =0 j - 1 j =0 d t + j

(x)

<~ 2kM Z e-#j -- 2kM(1 - e - " ) -1 --" K. j=0

(5.4)

64 J. A n d r e s

In this estimation, we have used the fact that, by assumption (A2), there exist positive constants k,/z such that

II a(t, s ) I I ke-Ult-sl"

Now, let tl, t2 E ~. Then

ftl t2 ]Ix(q) - x(t2) ll ~ II~(s) ll ds

II e (s) ll II x (s) ll ds + [I f (s)[I ds

~< g I[A(s) lie(E)as + m(s)ds.

Consequently, T (q) C Q. (iii) Let A//be the power set of Q and define, for each ~2 6 .M, the real-valued MNC 7t

by

7t (~2):= max(~2) (sup z(D(t ) ) ) , DED \tER

where D(Q) denotes the collection of all denumerable subsets of f2 and D(t) = {d(t) I d 6 D} C E. Then 7t is well-defined and from the corresponding properties of y it is clear that 7r has monotone and nonsingular properties of measure of noncompactness (see Proposition 2.3 in Section 2.2). Finally, observe that 7r is regular in view of the Arzel~t-

Ascoli lemma. We wish to show that the mapping T given in step (i) is condensing w.r.t, the MNC 7r. Take S2 6 A//. Considering T(f2), we see that by the definition of 7r there exists a se-

quence {Xn} C T(f2) such that

~p ( T ( ~ ) ) = sup y({Xn(t)}). tER

Thus, for each n E N, there is Zn ~ ~2 and fn ~ NF(Zn) such that

Xn(t) -- f• G(t,s)fn(s)ds. (5.5)

Let e > 0 be fixed. Choose a number a > 0 such that Ke -ua < e. Analogously to the estimation (5.4), one shows that

f_-a G(t, s) fn (s) ds + G(t, s) fn (s) ds oo a

<<, e,

Topological principles for ordinary differential equations 65 for every n 6 N. Using (5.5), we thus infer for an arbitrary t 6 IR that

({[t+a }) y({xn(t) }) ~ s + g G(t, s) fn(s) ds .

dt--a

Now, from assumption (F1), we get that

IIG(,,s)f.(s)ll <<. IIG(t,s)llc( )m(s),

for a.a. s 6 R and each n 6 N. Furthermore, using assumption (F2) and properties of 9/(see Proposition 2.3 in Section 2.2), we see that the following estimate holds, namely

y({G(t,s)fn(s)}) IIG(t,s)ll (e)y(lf (s)}) IIG(t,s)l

<. [Ia(t,s)llc(e)g(s)h(O(S2)),

for a.a. s E R. Hence, an application of Lemma 5.1 gives us

t+a y({Xn(t)}) ~ e +qh(g/(a)) IlG(t,s)llc(E)g(s)d,.

dt--a

It follows that

O(T(f2)) ~< e + qh(O(a))L,

and subsequently, since e > 0 was arbitrary,

O(T(a)) <<, qh(g/(a))L. (5.6)

Let us now assume that ~ is not relatively compact. Then ~(f2) > 0 and so, by assumption (F2) and (5.6), we obtain

Finally, observe that the estimate (5.6) also implies the quasi-compactness of the mapping T which subsequently justifies the compactness of the solution set to (5.3), as claimed.

Hence, we have verified all the assumptions of Theorem 4.4 (cf. also Definition 2.8) and we can establish the existence of a bounded solution to problem (5.1). D

REMARK 5.1. One can easily check that the assumption (F1) can be replaced by a weaker one, namely

[F(t,x) I<.m(t)+KIIxll, fora.a, t E R , x 6 E , (5.7)

where K/> 0 is a sufficiently small constant and m is the same as above.

66 J. Andres

REMARK 5.2. If A" E --+ E is a linear, bounded operator whose spectrum does not intersect the imaginary axis, then the constant K in (5.7) can be easily taken as K < 1/C(A), where

sup tER

[ II a (t,s) lt as ,) --CxZ

e A(t-s)P_, for t > s, (5.8) <~ C(A), G(t, s) -- e -A(t-s) P+, for t < s

and P_, P+ stand for the corresponding spectral projections to the invariant subspaces of A.

REMARK 5.3. For E = IR n (=, (F2) holds automatically), condition (A2) is satisfied, provided there exists a projection matrix P (P = p2) and constants k > 0, )~ > 0 such that

{Is(t)ex-l(s)[~ k e x p ( - ) ~ ( t - s)), [X(t)(l - P)X-I(s)I <<. kexp(-~.(s - t)),

for s ~< t, (5.9) for t ~<s,

where X (t) is the fundamental matrix of (5.2), satisfying X (0) - I, i.e., the unit matrix.

If A in (A1) is a piece-wise continuous and periodic, then it is well known that (5.9) takes place, whenever all the associated Floquet multiplies lie off the unit cycle. If A in (A1) is (continuous and) almost-periodic, then it is enough (see [93, p. 70]) that (5.9) holds only on a half-line [to, cx~) or even on a sufficiently long finite interval.

Now, the information concerning the topological structure of solution sets in Section 4.2 will be employed for obtaining existence criteria, on the basis of general methods estab- lished in Section 4.1.

EXAMPLE 5.1. Consider the system

{ -~1 E F1 (t, Xl, x2)xl q- F2(t , Xl, x2)x2 Jr- E1 (t, Xl, x2),

J;2 E - F 2 ( t , Xl, x2)xl d- F1 (t, Xl, x2)x2 q- E2(t , Xl, x2), (5.10)

where E l , E2, F1, F2 " [0, 0~:~) x R 2 --o R 2 are product-measurable u-Carathtodory maps. Assume, furthermore, the existence of positive constants El, E2, F1, F2, X such that

esssup[ sup Fl(t, Xl,X2)] <<.-)~, (5.11) t E[0,oe) "[xi[<. D,i=1,2

esssup[ sup IFl(t, xl,x2)l] <<, F1, (5.12) t ~[0,cx~)Ixil<.D,i=l,2

esssup[ sup ]F2(t, Xl,X2)[] ~</~2, (5.13) t~[0,~) [xil<.D,i=l,2

esssup[ sup [El(t, Xl,X2)l] <<, El, t E[0,~x~)Ixil<~D,i=l,2

(5.14)


esssup[ sup tE[0,o~) Ixil<~D,i--1,2

IE2(t,x,,x2)l] ~ ~2, (5.15)

where D - 1/)~(E1, E2). Observe that, under the assumptions (5.12)-(5.15), we have

esssup[Ai(t)l ~< D', i - - 1,2, (5.16) tE[0,o~)

where D ' = (F1 + F2)D + max(E1, E2), so long as the solution (Xl( t) ,x2(t)) of (5.10) satisfies

sup Ix/(t)l ~ D, i - - 1,2. (5.17) te[0,o~)

Our aim is to prove, under the assumptions (5.11)-(5.15), the existence of a solution x(t) = (Xl (t), xz(t)) satisfying

x(0) = 0 and sup Ixi(t)l ~ D, i - - 1,2. (5.18) t~[0,o~)

In order to apply Corollary 4.1 for this goal, define two sets

Q - - { r ( t ) - (rl (t), r2(t)) E C([0, ~ ) • [0, 00), IR2) I sup [ri(t)[ <~ D, tr

i - - 1,2},

S "-- { s ( t ) = (s l (t),s2(t)) E C([0, co) x [0, 00), ]1~ 2) O e llsi(t) I <. O',,

i - -1 ,2}

(observe that s(0) = 0), where Q is a closed convex subset of C([0, 00) • [0, ~ ) , ]~2) and S is a bounded closed subset of Q.

For q(t) = (ql (t), qz(t)) ~ Q, consider still the family of systems

Y Cl -- Pl (t)Xl + p2(t)x2 -+- rl (t), Jc2 -- -p2 ( t )X l + Pl (t)x2 q-- r2(t),

(5.19)

where pl (t) C F1 (t, q(t)), p2(t) C F2(t, q(t)), rl (t) C E1 (t, q(t)), r2(t) C E2(t, q(t)) are measurable selections (see Proposition 2.5).

To show the solvability of (5.10) and (5.18) by means of Corollary 4.1, we need to verify that, for each q E Q, the linearized system

21 E F1 (t, q(t))Xl -+" F2(t, q(t))x2 + E1 (t, q(t)), x2 E -F2(t, q(t))Xl -+- F1 (t, q(t))x2 -+- E2(t, q(t)) (5.20)

has an R~-set of solutions in S.

68 J. Andres

It is well known that the general solution x(t, O, ~) of (5.19), where ~ = (~l, ~2) ~ IR 2, reads as follows:

[ (/0 ) (£ )] l x~ ( t , 0 , ~ )= ~]cos p2(s)ds +~2sin p2(s)ds exp pl(s)ds

I'[ l' ( I ' ) ~ + rl (s) exp pj (w) dw cos p2 (w) dw ds .1'¢

~'I r' ( ~ ' ) ] + r2 (s) exp P l (w) dw sin pz(w) dw ds, d s

[ (r' ) (~' )] 1' x 2 ( t , 0 , ~ ) = - - ~ l s i n p z ( s ) d s + ~ 2 c o s p z ( s ) d s exp p l ( s ) d s \ J 0

- ~t[rl(s)exp f t pl(w)dwsin({t p2(w)dw)]ds

1'[ ; ( I ' ) ] + rz(s ) exp Pl (w) dw cos p2(w) dw ds. k J S

Because of (5.11 ), (5.14) and (5.15), we get

l[ r' ((' )]~, sup ri (s) ds exp p l (w) dw cos P2 (w) dw te[O,~) ) as ",,,,, s

~ ' [ f ' ] ~< Ei sup e x p - [p~(w)[dw as <~ Ei t ~[0,o,z) (1 ~ - '

1'[ ¢ (l' )] sup ri (s) ds exp Pl (w) dw sin P2 (w) dw ds t~ O,c~) 0 s s \ , . , s

~<Ei sup ~texp[--ft[pl(W)[dw]ds<~, tE[0,~) 0

for i = 1,2, and subsequently we arrive at

sup Ixi(t,O,~)l<~l~ll+l~21+D, i = 1 , 2 , (5.21) tE[0,~)

and x (0, 0, ~) = ~. According to Theorem 4.9 (see also Proposition 2.5 and estimate (5.21)), problem

(5.17) A (5.20) has the R~-set of solutions x(t, 0, 0). Moreover, in view of the indicated implication ((5.17) =~ (5.16)), these solutions x(t, 0, 0) belong obviously to S, for every q 6 Q, as required.

Thus, it follows from Corollary 4.1 that problem (5.10) A (5.18) has, under the assumptions (5.11) (5.15), at least one solution.


If the inequality (5.11) or both inequalities (5.14) and (5.15) are sharp, then the same conclusion is true for x(0) = 0 in (5.18) replaced by x(0) = ~, where I~1 is sufficiently small. For bigger values of I~ [, the above assumptions can be appropriately modified as well.

THEOREM 5.2. Consider the target problem

for a.a. t �9 [0, cx~), )c(t) e F( t , x ( t ) ) , iR n lim x ( t ) - x ~ �9 (5.22)

and assume that F : [ 0 , oo) x IR n ---o ~n is a product-measurable u-Carath~odory map.

Let, furthermore, there exist a globally integrable function ot : [0, oo) --+ [0, oo) and a positive constant ~ such that, f o r every x �9 IR n and for a.a. t �9 [0, oo), we have IF(t, x)] ~< c~(t)(/3 + ]x]), where ] F ( t , x ) ] - sup{ly] ] y �9 F ( t , x ) } , and f o O t ( t ) d t < oo. Then problem (5.22) admits a (bounded) solution, for every xc~ �9 IR n .

PROOF. It is convenient to consider, instead of problem (5.22), the equivalent problem

for a.a. t e [0, oo), x( t ) e G ( t , x ( t ) ) , i R n lim x( t ) -- x ~ e ,

t----> o~ (5.23)

where

G(t, x) - F t, D-~l ,

for Ix l ~ D and t e [0, cx~),

for Ixl ~ D and t �9 [0, oo),

O ~ (Ix~l + A B ) exp A, f0 ~

A = c~(t) d t < cx~.

Moreover, there certainly exists a positive constant y such that

Ix01-+-]G(t,x)l ~ Ix01-t- A ( B + D) <~ y,

for all x �9 IR n and a.a. t �9 [0, oo). (5.24)

Besides problem (5.23), consider still a one-parameter family of linear problems

)c(t) e G ( t , q ( t ) ) , x e Q A S ,

for a.a. t e [0, oo), q e Q, (5.25)

where

s-{x ct[0, n)I Q - {q �9 C([0, oo), R~)] [q(t)l ~ I~1 + A ( B + D), for t ~> 0}.

70 J. Andres

Consider the set

{ i } S1 ~--- x E Q I x ( t ) - x~[ ~< (B + D) ot(s)ds, for t ~> 0 C S.

It is evident that S1 is a closed subset of S and all solutions to problem (5.25) belong to $1. At first, we assume that G = g is single-valued. Then we have a single-valued continu-

ous operator

L T(q) -- xoo + g(s, q(s)) ds, for every q E Q.

Thus, to apply Corollary 4.2, only the condition T (Q) c Q should be verified. But this follows immediately from (5.24), because

sup t6[0,c~) L xc~ + G(s, q(s)) ds

f0 ~176 Ix l + Ia(t, q(t))l dt

~< Ix~l + (n + D) or(t) dt

= Ix~l + A(B + D) < cx~. (5.26)

By Corollary 4.2, we obtain a solution to the problem with g as a right-hand side. This existence result can be used jointly with Theorem 4.12 which is needed to prove our statement in a general case. In fact, in view of the Oust proved) existence result and Theorem 4.12 (cf. also Proposition 2.5), the map T which assigns to every q E Q the set of solutions to the linear problem (5.25), has nonempty, acyclic sets of values. Once more, we use Corollary 4.2, obtaining a solution to problem (5.22), and the proof is complete. []

5.2. Solvability of boundary value problems with linear conditions

Now, we shall deal with boundary value problems of the type

k(t) + A(t)x(t) ~ F( t ,x( t ) ) , for a.a. t 6 [0, r], (5.27) L x = |

where (i) A ' [0 , r] --+ s n, Nn) is a measurable linear operator such that IA(t)l ~ y(t) , for

all t 6 [0, r] and some integrable function y ' [ 0 , r] --+ [0, c~), (ii) the associated homogeneous problem

{ . ~ ( t ) + A ( t ) x ( t ) = O, L x = O

for a.a. t ~ [0, r],

has only the trivial solution,


(iii) F :[0, r] x R n --o IR n is a u-Carath6odory mapping with nonempty, compact and convex values (cf. Definition 2.10),

(iv) there are two nonnegative Lebesgue-integrable functions 61,62:[0, r] -+ [0, oo) such that

IF(t, x) I ~< S1 (t) + 62(t)lxl, for a.a. t �9 [0, r] and all x �9 R n,

where IF(t,x)[ = sup{ly[ [ y �9 F(t ,x)} . Applying Theorem 4.13 (cf. also Proposition 2.5) to replace condition (i) in Corol-

lary 4.3 for (5.27), we can immediately give

PROPOSITION 5.1. Consider problem (5.27) with (i)-(iv) above and let G:[0 , r] • IR n x R n • [0, 1] --+ N n be a product-measurable u-Carathdodory map (cf Definition 2.1 O) such that

G(t, c, c, 1) C F(t, c), for all (t, c) �9 [0, r] • R n.

Assume, furthermore, that (v) there exists a (bounded)retract Q of C([O, r ] , N n) such that Q \ OQ is nonempty

(open) and such that G(t, x, q(t), )~) is Lipschitzian in x with a sufficiently small Lipschitz constant (see Theorem 4.13), for a.a. t �9 [0, r] and each (q,)~) �9 Q x [0, 1],

(vi) there exists a Lebesgue integrable function ~:[0 , r] -~ [0, oo) such that

IG(t ,x( t ) ,q( t ) , )O[ <<. oe(t), a.e. in [0, r],

for any (x, q, )~) �9 FT (i.e. from the graph of T), where T denotes the set-valued map which assigns, to any (q , )0 �9 Q x [0, 1 ], the set of solutions of

2(t) + A(t)x(t) �9 G(t, x(t), q(t), )~), Lx - - |

for a.a. t �9 [0, 1],

(vii) T(Q x {0}) c Q holds and OQ is fixed point free w.r.t. T , fo r every (q,)~) �9 Q x [o, 11.


REMARK 5.4. Rescaling t in (5.27), the interval [0, r] can be obviously replaced in Propo- sition 5.1 by any compact interval J , e.g., J = [--m, m], m e N. Therefore, the second part of Proposition 4.5 can be still applied for obtaining an entirely bounded solution.

EXAMPLE 5.2. Consider problem (5.27). Assume that conditions (i)-(iv) are satisfied. Taking (for a product-measurable F :[0, r] x R n --o R n)

G(t, q(t)) - F(t, q(t)), for q e Q,

72 J. Andres

where Q = {/z 6 C([0, r], ~n) I maxt~[0,r] I/z(t)l ~ D} and D > 0 is a sufficiently big constant which will be specified below, we can see that (v) holds trivially. Furthermore, according to (iv), we get

IG(t, q(t))[ ~< 61(t) -q- 62(t)D, for a.a. t E [0, r], (5.28)

i.e. (vi) holds as well with ct(t) = 61 (t) + 62(t)D. At last, the associated linear problem

:f(t) + A(t)x(t) ~ F(t, q(t)), Lx = |

for a.a. t 6 [0, r],

has, according to Theorem 4.13, for every q 6 Q, an R~-set of solutions of the form

f0 T T(q) -- H(t ,s)f(s ,q(s))ds,

where H is the related Green function and f C F is a measurable selection (see again Proposition 2.5).

Therefore, in order to apply Proposition 5.1 for the solvability of (5.27), we only need to show (cf. (vii)) that T (Q) c Q (and that 0 Q is fixed point free w.r.t. T, for every q 6 Q, which is, however, not necessary here). Hence, in view of (5.28), we have that

f0 z" max I T ( q ) [ - max H(t,s) f(s ,q(s))ds t~[0,r] tE[0,r]

~< max IH(t, s)l(61 (s) + 6z(s)D)ds te[0,rl

[fo fo ] = max IH(t,s l 61(t)dt + D 6z(t)dt , t ,se[0,rl

and subsequently the above requirement holds for

D~> maxt,s~[O,r] IH(t, s)l fo 61 (t) dt

1 - maxt,s~[O, rl IH(t, s)l fo 62(t) dt

provided

fo r 1 62(t) dt < maxt,s~[O,r] IH(t, s)l

(Observe that for D strictly bigger than the above quantity, 8 Q becomes fixed point flee.)


5.3. Existence of periodic and anti-periodic solutions

Now, consider the following special cases of Floquet boundary value problems in a reflex-

ive Banach space E:

2(t) + A(t)x(t) E F(t, x(t)), x ( r ) - - M x ( O ) ,

for a.a. t 6 [0, r], where M = id or M = - i d .

(5.29)

Besides (A1), (A2), assume still

A ( t ) = _ A ( t + r )

for some r > 0,

and F ( t , x ) =-- F(t + r , x ) or F ( t , x ) = - F ( t + r , - x )

(5.30)

and, instead of (F1), that only

IlF(t,x ll + (t llxll (5.31)

holds for a u-Carath6odory map F : [ 0 , r] x E ---o E, for a.a. t E [0, r] and every x E E, where c~0, ell E L 1([0, r]). Then one can check, as in the proof of Theorem 5.1, that the solution operator T : Q x [0, 1] ---o E, where Q = {q E C(R, E) I q(t) =-- q(t + r) or q(t) =-- - q ( t + r)}, associated with the fully linearized problem

2(t) + A(t)x(t) E )~F(t, q(t)), x ( r ) = M x ( O ) ,

for a.a. t e [0, r], )~ e [0, 1], where M = id or M = - id,

(5.32)

is condensing and that the set of (bounded, after r-periodic or 2r-periodic prolongation) solutions is convex and compact, provided an analogy of (F2) holds.

Therefore, taking G(t, c, c, O) = - A (t)c, Corollary 4.5 can be simplified as follows.

COROLLARY 5.1. Assume that conditions (5.30) and (5.31) hold, jointly with (i) A:[0 , r] --+ /2(E) is strongly measurable (cf Definition 2.9) and Bochner inte-

grable on the interval [0, r]. (ii) The linear equation 2(t) + A( t )x ( t ) = 0 admits a regular exponential dichotomy.

Denote by G the related principal Green's function. (iii) F : [ 0 , r] x E ---o E is a u-Carathdodory map with nonempty, compact and convex

values. Assume, furthermore, that there exists a nonempty, bounded, closed, convex subset Q of

{q E C(R, E) I q(t) -- q(t + r) or q(t) -- - q ( t + r)} with nonempty interior int Q such

that (iv) g(F( t , ~2)) <<. g( t )h(y( f2)) , for a.a. t E [0, r], and each ~2 C {q(t) 6 E I t E

[0, r], q E Q}, where g, h are positive functions, g is measurable, h is nonde-

creasing such that

{f0 i } L " - sup II G(t, s)IIc~g<s~ as t <

74 J. Andres

and gh( t )L < t, for each t ~ [0, r], with a constant q = 1, if E is still separable, and q -- 2, in general case.

(v) {0} C int Q and the boundary O Q of Q is fixed point free w.r.t. T, for every (q , )O Q x (0, 1], where T is the map assigning, to any (q, Jk) 6 Q x [0, 1], the set of solutions of the fully linearized problem (5.32).

Then problem (5.29) admits a solution.

REMARK 5.5. Obviously, under the assumptions (i), (ii), (iii), (5.7), (5.30) and (F2) in Section 5.1, problem (5.29) admits a solution, provided K in (5.7) is sufficiently small.

For periodic and anti-periodic problems, Proposition 5.1 can be easily simplified as follows (cf. Remark 5.3).

COROLLARY 5.2. Consider problem

{ :~(t) + A(t )x( t ) ~ F(t , x( t ) ) , x ( 0 ) - x ( r ) ,

for a.a. t ~ [0, r],

where F ( t , x ) -- F(t + r , x ) satisfies conditions (iii) and (iv) in Proposition 5.1. Let G :[0, r] x ~n x R n x [0, 1] ---o ~n be a product-measurable u-Carath~odory map such

that

G(t, c, c, 1) C F(t, c), for all (t, c) ~ [0, r] x R n.

Assume that A is a piece-wise continuous (single-valued) bounded r-periodic (n x n)- matrix whose Floquet multipliers lie off the unit cycle, jointly with (v)-(vii) in Proposi- tion 5.1, where Lx = x(0) - x ( r ) and tO = O. Then the inclusion k + A( t )x ~ F ( t , x ) admits a r-periodic solution.


{ k(t) E F(t , x( t ) ) , for a.a. t ~ [0, r], x(0) = - x ( r ) ,

where F ( t , x ) =-- - F ( t + r , - x ) satisfies conditions (iii) and (iv) in Proposition 5.1. Let G :[0, r] • ]R n x R n x [0, 1] ---o R n be a product-measurable u-Carath~odory map such

that

G( t , c , c , 1) C F(t ,c ) , fora l l ( t ,c) ~ [0, r] x R n.

Assume that (v)-(vii) in Proposition 5.1 hold, where Lx = x(0) + x( r ) and tO = O. Then the inclusion ~c ~ F(t , x) admits a 2r-periodic solution.

In the case of ODEs, Corollary 5.3 can be still improved, in view of Theorem 4.14, where )~ = 1 and ~ = 0, as follows.



{ 2(t) = f ( t , x ( t ) ) , fora.a, t ~ [0, r], x ( 0 ) - - x ( r ) ,

where f ( t ,x ) =_ - f (t + r , - x ) is a Carathdodory function. Let g:[0 , r] x N n x IR n x [0, 1] ~ IK n be a Carathdodory function such that

g(t, c, c, 1) = f ( t , c), for all (t, c) E [0, r] x N n.

Assume that (i) there exists a bounded retract Q of C([0, r], N n) such that Q \ OQ is nonempty

(open) and such that g(t, x, q(t), )~) satisfies

Ig ( t , x ,q ( t ) , )~) -g( t , y ,q ( t ) , )~) l <~p( t ) lx - yl, x, y E R n

for a.a. t E [0, r] and each (q, ;~) E Q x [0, 1], where p: [0 , r] --+ [0, oc) is a Lebesgue integrable function with (see (4.31))

fo r p(t) dt <. re,

(ii) there exists a Lebesgue integrablefunction ot : [0, r] --+ [0, oc) such that

g( t ,x ( t ) ,q( t ) , )~) l <~ ~(t), a.e. in [0, rl ,

for any (x, q, )~) E ['T, where T denotes the set-valued map which assigns, to any (q, ,k) E Q x [0, 1 ], the set of solutions of

2(t) = g(t, x(t) , q(t), )~), for a.a. t E [0, r], x ( 0 ) = - x ( r ) ,

(iii) T(Q x {0}) c Q holds and OQ is f ixedpointfree w.r.t. T , f o r every (q,)O E Q x [0, 11.

Then the equation Yc = f (t, x) admits a 2r-periodic solution.

REMARK 5.6. Since in Corollaries 5.2 and 5.3 the associated homogeneous problems (cf. (ii) at the beginning of Section 5.2) have obviously only the trivial solution, the requirement T(Q x {0}) C Q reduces to {0} c Q, provided G(t ,x ,q , )~) = )~G(t,x,)~),)~ E [0, 1].

REMARK 5.7. The requirement concerning a fixed point free boundary O Q of Q in Propo- sition 5.1, Corollaries 5.1, 5.2, 5.3 and 5.4 can be verified by means of bounding (Liapunov- like) functions (see [11,32-34] and cf. [22, Chapter III.8]).

76 J. Andres

EXAMPLE 5.3. Consider the anti-periodic problem

2 ~ Fl (t, x) + F2(t, x) , (5.33) x(a) = - x ( b ) ,

where x = (x1 . . . . . Xn), F = F1 + F2 = ( f l l , . . . , f l n ) Jr- ( f 2 1 . . . . . f2n), F1, F2 : [a, b] x R n ---o R n are globally upper semicontinuous multivalued functions with nonempty, convex, compact values which are bounded in t E [a, b], for every x ~ R n, and linearly bounded in x ~ R n, for all t ~ [a, b].

Assume, furthermore, that there exist positive constants Ri, i -- 1 . . . . . n such that

I f l i ( t , x ( + R i ) ) l > max I f2 ir l, t~_[a,b] xEK

i - - 1 . . . . . n, t ~ (a ,b ) ,

where x(-d-Ri) - - (Xl . . . . . xi-1,-nt-Ri,xid-1 . . . . . Xn), Ixjl ~ Rj and K -- {x 6 R n I lxi[ < Ri, i = 1 . . . . . n},

[ f l i (a , x ( - f -g i ) ) n t- f2i(a, y ) ] " [ f l i (b , -x ( -+-gi ) ) -4- f2 i (b ,z )] < O,

i - - 1 . . . . . n,

m

where x, y, z 6 K,

F1 (t, .) is Lipschitzian with a sufficiently small constant L,

for every t 6 [a, b]; in the single-valued case, when F ~ C([a, b] x R n, Rn), it is enough to take L <~ rc/(b - a) (see condition (i) in Corollary 5.4).

It can be checked (see [32]; cf. [22, Example 8.40 in Chapter III.8]) that all assumptions of Corollary 5.3 (in the single-valued case, of Corollary 5.4) are satisfied, and so problem (5.33) admits a solution.

6. Multiplicity results

6.1. Several solutions o f initial value problems

Let us recall that in order to apply Theorem 4.3, the following main steps have to be taken: (i) the R~-structure of the solution set to (4.2) must be verified,

(ii) the inclusion T (Q) c S or, most preferably, T (Q) c Q f3 S must be guaranteed, together with the retractibility of T in the sense of Definition 4.1,

(iii) N(r[T(Q) o T) must be computed. For initial value problems, condition (i) can be easily verified, provided G is still

product-measurable. In fact, since u-Carath6odory inclusions (cf. Definition 2.10) with product-measurable right-hand sides G possess (according to Theorem 4.9; cf. also Propo- sition 2.5), for each q 6 Q c C (I, R n), an R~-set of solutions x (., x0) with x (0, x0) = x0, for every x0 6 R n, such a requirement can be, in Theorem 4.3 with S := {x E ACloc (I, R n) I

Topological principles f o r ordinary differential equations 77

x(0, x0) = x0}, simply avoided. Moreover, if Q is still compact and such that T(Q) C Q A S, then (see Remark 4.3) 7rl (Q) need not be abelian and finitely generated.

Thus, Theorem 4.3 simplifies, for initial value problems, as follows:

COROLLARY 6.1. Let G :I x N n • ][{n - -o 11~ n be a product-measurable u-Carathdodory mapping, where I = [0, oo) or I = [0, r], r e (0, oo). Assume, furthermore, that there exists a (nonempty) compact, connected subset Q c C(I, R n) which is a neighbourhood retract of C(I , N n) such that IG(t, x(t) , q(t))l <<, Iz(t)(Ixl + 1) holds, for every (t, x, q) e I x ]R n x Q. Let the initial value problem

{ 2(t) e G(t, x(t), q(t)) , x (0) - xo

for a.a. t e l ,

have, for each q E Q, a nonempty set of solutions T (q) such that T (Q) c Q M S, where S := {x e ACloc(I, R n) I x(0) = x0}. Then the original initial value problem

2(t) E F ( t , x ( t ) ) , for a.a. t E I, x (0) -- xo

admits at least N ( T ) solutions, provided G(t, c, c) C F(t, c) holds a.e. on I, for any C E ]I~ n .

EXAMPLE 6.1. Consider the scalar (n = 1) initial value problem with xo = 0 and I = [0, r], r > 0. Letting

Q "- {q 6 AC([0, rl, R) I q(0) - 0 and 82 ~< q(t) <. 81

or - 81 ~< 0 (t) <~ -82, for a.a. t 6 [0, r] },

where 0 < 82 < 61 are suitable constants, Q can be easily verified to be a disjoint (!) union of two convex, compact sets, and consequently Q is a compact ANR, i.e. also a neighbourhood retract of C ([0, r], N). Unfortunately, Q is disconnected which excludes the direct application of Corollary 6.1.

Nevertheless, e.g., the inclusion

2(t) E 8 Sgn(x(t)) , for a.a. t 6 [0, r], 8 > 0, (6.1)

where

Sgn(x) - - [ ,1], 1,

for x c ( - ~ , 0), fo rx = 0 , for x e (0, ~ ) ,

admits obviously two classical solutions Xl ( t ) = St with Xl ( 0 ) --" 0 and x2(t) = - S t with x2 (0) = O, satisfying the given inclusion everywhere.

78 J. Andres

The linearized inclusion

Jr(t) ~ 8 Sgn(q(t)), for a.a. t 6 [0, r], 8 > 0,

possesses, for each q 6 Q, either the solution Xl (t) = 8t with Xl (0) = 0 or x 2 ( t ) -~ -S t with x2(0) = 0, depending on sgn(q(t)), provided 82 ~ 8 ~ 81. Observe that there are no more solutions, for each q E Q. Thus, we also have T(Q) c Q M S (i.e. condition (ii)), where S := {x 6 ACloc(I, Rn) I x(0) = 0}.

The only handicap is related to the mentioned disconnectedness of Q. However, since T: Q --+ Q, where

[ St, for q ~> O, T(q)

I -6 t , forq ~<0,

is obviously single-valued, the application of the multivalued Nielsen theory, in the proof of Theorem 4.3 (and subsequently of Corollary 6.1), can be replaced by the single-valued one, where Q 6 ANR can be disconnected (see the definition of the Nielsen number at the beginning of Section 3.2). We can, therefore, conclude, on the basis of the appropriately modified Corollary 6.1, that the original inclusion (6.1) admits at least N(T) = 2 solutions x(t) with x(0) = 0, as observed by the direct calculations. In fact, it must therefore have, according to Theorem 4.5, a nontrivial R~-set of infinitely many piece-wise linear solutions x(t) with x(0) = 0. The computation of N(T) = 2 (i.e. condition (iii)) is trivial, because Q = Q+ u Q- , where

(AR 9) Q+ "= {q 6 AC([0, r], R) ]q(0) = 0 and

82 ~ O(t) <<. 81, for a.a. t 6 [0, r]},

(AR 9) Q- "= {q E AC([0, r], IR) [q(0) = 0 and

- 61 <. el(t) ~< -82, for a.a. t 6 [0, r]},

and so for the computation of the generalized Lefschetz numbers we have A (TI Q+) - -

A(TIQ-) = 1, where TIQ+" Q +--+ Q+ and TIQ- " Q- --+ Q-. The same is obviously true for the inclusion

2(t) ~ [8 + f ( t , x(t))] Sgn(x(,)), for a.a.t 6 [0, r], 8 > 0,

where f :[0, r] • IR --+ R is a Carathrodory and locally Lipschitz function in x, for a.a. t E [0, r], such that 62 ~< 8 + f (t, x) ~< 81, for some 0 < 62 < 61, because again T : Q --+ Q.

Of course, we could arrive at the same conclusion even without an explicit usage of the Nielsen theory arguments, just through double application (separately on Q+ and Q - ) of Theorem 3.3 (i.e. of the Lefschetz theory arguments).

REMARK 6.1. In view of Example 6.1, it is more realistic to suppose in Corollary 6.1 that (at least, for n = 1) the solution operator T is single-valued and that Q can be disconnected and not necessarily compact. Naturally, the first requirement seems to be rather associated with differential equations than inclusions.


6.2. Several periodic and bounded solutions

Now, Theorem 4.3 will be applied to boundary value problems associated with semilinear differential inclusions.

Hence, consider the problem

Yc + A( t )x �9 F(t, x), Lx - - |

(6.2)

Since the composed multivalued function F(t, q (t)), where F : J • ~t~ n - - - o ~n is a product- measurable u-Carath6odory mapping with nonempty, compact and convex values and q �9 C (J, Rn), is, according to Proposition 2.5, measurable, we can also employ Theorem 4.13 to the associated linearized system

{ ~ + A( t )x �9 F(t, q(t)), (6.3) Lx - - |

provided

IF(t,x)l ~ zz(t)(Ixl + 1), (6.4)

where # : J ~ [0, oc) is a suitable (locally) Lebesgue integrable bounded function. We can immediately give

THEOREM 6.1. Consider boundary value problem (6.2) on a compact interval J. Assume

that A" J --+ IR n2 is a single-valued continuous (n x n)-matrix and F" J x IR n ---o IR n is a product-measurable u-Carathdodory mapping with nonempty, compact and convex values satisfying (6.4). Furthermore, let L : C(J, IR n) --+ IR n be a linear operator such that the homogeneous problem

k + A( t )x --O, Lx - -0

has only the trivial solution on J. Then the original problem (6.2) has at least N(rlT(Q) o T(.)) solutions ~or the definition of the Nielsen number N, see Definition 3.2 in Sec- tion 3.2), provided there exists a closed connected subset Q of C(J, R n) with a finitely generated abelian fundamental group such that

(i) T (Q) is bounded, (ii) T(q) is retractible onto Q with a retraction r in the sense of Definition 4.1,

(iii) T(Q) C {x �9 AC(J, ]Rn) I L x - | where T (q) denotes the set of (existing) solutions to (6.3).

REMARK 6.2. In the single-valued case, we can obviously assume the unique solvability of the associated linearized problem. Moreover, Q need not then have a finitely generated abelian fundamental group (see Remark 4.3). In the multivalued case, the latter is true, provided Q is compact and T (Q) c Q (see again Remark 4.3).

80 J. Andres

Before presenting a nontrivial example, it will be convenient to have the following reduction property (see [6] and cf. [22, Lemma 6.6 in Chapter III.6]).

LEMMA 6.1 (Reduction). Let X and its closed subset Y be ANR-spaces. Assume that f : X --+ X is a compact map, i.e. f (X) is compact, such that f (X) C Y. Denoting by f t : y __+ y the restriction of f , we have

(i) Fix(f ' ) -- Fix(f) , (if) the Nielsen relations coincide,

(iii) ind(C, f ' ) -- ind(C, f ) , for any Nielsen class C C Fix(f) . Thus, U ( f ' ) -- U ( f ) .

Consider the u-Carath6odory system (the functions e, f , g, h have the same regularity as in Theorem 6.1)

{ 2 + ax �9 e(t, x, y)y(1/m) _+_ g(t, x, y), y + by �9 f ( t , x , y)x (1/n) + h( t ,x , y),

(6.5)

where a, b are positive numbers and m, n are odd integers with min(m, n) ~> 3. Let suitable positive constants E0, F0, G, H exist such that

le(t, x, y)[ ~< E0,

[g( t ,x ,y) l<~G,

[ f ( t , x , y)[ ~< F0,

Ih( t ,x, Y)I ~ H,

hold, for a.a. t �9 ( -oo , oo) and all (x, y) �9 ]R 2. Furthermore, assume the existence of positive constants e0, f0, 61, 62 such that

0 < eo <<. e(t, x, y), (6.6)

for a.a. t, all x and lyl/> ~2, jointly with

0 < fo <~ f ( t , x, y), (6.7)

for a.a. t, Ix l ~ •1 and all y. As a constraint S, consider at first the periodic boundary condition

(x(0), y ( 0 ) ) - (x(o~), y(co)). (6.8)

More precisely, we take S = Q -- Q1 n Q2 n Q3, where

Q1 = [q( t )E c([0, o)], ]~2) [

Ilq(t)[I " -max{ max Iql(t) l, max Iq2(t)[} ~< D] , t6[0,co] t6[0,co]


Fig. 3.

- { min ]ql<t)l~ 81 > 0 Q2 q(t) E C([0, co], •2) ] t~[o,co]

or min Iq2~t) I >/~2 > o], te[0,co] /

Q3 - { q ~ t ) ~ c([o, ~o], R2)I q~0)- q(co)},

the constants 81, 82, D will be specified below. For (Q1 N Q2)N R 2, the situation is schematically sketched in Fig. 3.

Important properties of the set Q can be expressed as follows.

LEMMA 6.2. The set Q defined above satisfies: (i) Q is a closed connected subset of C ([0, co], R2),

(ii) Q E ANR, (iii) Jr1 (Q) - Z.

PROOF. Since Q is an intersection of closed sets Q1, Q2, Q3, we conclude that Q is a closed subset of C ([0, co],/I~ 2) as well. The connectedness follows from the proof of (iii) below.

For (ii), it is enough to realize (see [69, Corollary 4.4 on p. 284]) that Q is the union of four closed, convex sets in the Banach space C([0, co], ~2), namely Q = (Q1 n Q21 N Q3) u (Q1 n Q22 n Q3) u (Q1 n Q23 n Q3) u (Q1 n Q24 N Q3), where

Q 2 1 - { q ( t ) 6C([O, co],R2)] min ql( t )~>31>O}, t~[o,co]

0 2 2 - {q(t) 6 C([0, co],11~2) I rain q2(t)~> 82 > 0}, t~[0,co]

82 J. Andres

023 -- {q(t) ~ C([0, co], N2) l min ql (') ~< -31 < 0}, tE[0,o~]

Q 2 4 - {q(t) ~ C([0, co], R2)] min qe(t)~< -62 < 0}. tE[0,co]

At last, we will show (iii). It is obvious that ~1 (A) = Z, where

A-- {(x, y) E ]R2lmax(lx[, [y[) ~ D and [[x[ ~> 61 or [y[ ~> 62]}.

At the same time, A = Q n I[~ 2, when regarding ]R 2 as a subspace of constant functions of Q3. For (iii), it is sufficient to show that A is a deformation retract of Q.

We define p: Q x [0, 1] --+ A by the formula

p(q, )~) = (Xql -+- (1 - )~)~]-, )~q2 + (1 - )~)~),

where q = (ql, q2) E Q and qi-= ql (0), ~ = q2(0). One can readily check that p is a deformation retraction, which completes the proof of our lemma. D

Besides (6.5) consider still its embedding into

2 + a x ~ [(1 - /z )eo + lze(t ,x , y)]yl/m + lzg(t ,x , y),

+ by ~ [(1 - #)f0 + l z f ( t , x , y)]x 1/n + lzh( t ,x , y), (6.9)

where/z E [0, 1] and observe that (6.9) reduces to (6.5), for/z = 1. The associated linearized system to (6.9) takes, for # 6 [0, 1], the form

2 + ax E [(1 --/z)eo + lze(t, ql (t), q2(t))]q2(t) 1/m -+- lzg(t, ql (t), q2(t)),

+ b y E [(1 - / z ) f o + lz f (t, ql (t), q2(t))]ql (t)l/n _+_ lzh(t, ql (t), q2(t)), (6.10)

or, equivalently,

2 + ax = [(1 - - /z )eo + lzet]q2(t) 1/m + lzgt,

+ by = [(1 --/z)f0 -k- lzft]ql (t) 1/n + lzht, (6.11)

where et C e(t, ql (t), q2(t)), ft C f (t, ql (t), q2(t)), gt C g(t, ql (t), q2(t)), ht C h(t, ql (t), qz(t)) are measurable selections. These exist, because the u-Carath6odory functions e, f , g, h are weakly superpositionally measurable (see Proposition 2.5).

It is well known that problem (6.11) N (6.8) has, for each q(t) ~ Q and every fixed quadruple of selections et, f t , gt, ht, a unique solution X (t) = (x(t), y(t)), namely

fo 9 x(t) = Gl( t , s )[ ( (1 - lz)eo q- lZes)q2(s) 1/m -k- lZgs]ds,

X( t ) - ,,~o y(t) - [ Gz(t ,s)[((1 - lz)fo + I~fs)qa(s) 1/" + #h~]ds,

do


where

e-a(t-s+co)

Gl (t, s) -- 1 - e -a~ e-a(t-s)

1 - e - a ~

for 0 ~< t ~<s ~co,

f o r 0 ~ < s ~<t ~<co,

e-b(t-s+co)

Gz(t ,s) -- 1 - e -b~ ' e-b(t-s)

1 -- e -be~ '

for 0 ~ t ~<s ~<co,

for 0 ~< s ~<t ~<co.

In order to verify that Tu (Q) c S = Q, where Tu (-) is the solution operator to (6.10) A (6.8), it is just sufficient to prove that Tu(Q) C Q, Iz ~ [0, 1], because S = Q is closed. Hence, the Nielsen number N(Tu) is well-defined, for every # 6 [0, 1], provided only product-measurabili ty of e, f , g, h and T# (Q) c Q.

Since x ( 0 ) = X(co), i.e. Tu(Q) c Q3, it remains to prove that Tu(Q) c Qa as well as T# (Q) c Q2. Let us consider the first inclusion. In view of

-boo e -ac~ e min G l ( t , s ) ~ > > 0 and min G e ( t , s ) ) >0,

t,se[O,o)] 1 - e -ac~ t,sc[O,co] 1 - e -be~

we obtain, for the above solution X (t), that

and

fo ) max Ix(t)l ~ max Ial( t ,s) l[[(1 - lz)eo 4- lZes]q2(s) 1/m 4- lZgs]ds t~[0,o)] te[0,co]

fo ~< [(eo + Eo)D 1/m + G] a l ( t , s ) d s

- l[(eo + Eo)D 1/m + G] a

f0 ) max ly(t) I ~< max Iae(t,s)l[[(1 - # ) fo 4- lZfs]ql(s) 1/n 4- #hs]ds te[O,co] tr

fo ~< [(fo + Fo)D 1/n + H] Ge(t, s) ds

---- - l [ ( fo 4- Fo)D 1/n 4- H]. b

Because of

IIx r max{ max Ix(r max ly(r te[O,~o] te[O,co]

84 J. Andres

{1 1 } ~< max - [ (eo + E o ) D 1/m n t- G], ~ [ ( f o n t- Fo) D1/n q- H] ,

a

a sufficiently large constant D certainly exists such that IlX(t)ll ~ R, i.e. Tu(Q) c Q1, independently of/x E [0, 1] and et, ft, gt, ht.

For the inclusion T~ (Q) c Q2, we proceed quite analogously. Assuming that q(t) ~ Q2, we have

min [ql( t)] ~ 31 > 0 or min Iq2(t) I ~ ~2 > 0. either t~[0,co] t~[0,co]

Therefore, we obtain for the above solution X (t) that (see (6.6))

f0 ) min Ix(t)I- min IGl(t,s)l[[(1- lz)eo + #es]q2(s) 1/m + tzgs]ds t6[0,o9] tE[0,co]

/o [ ~ ~l/m (t s) ds 1 _ ~l/m e0o 2 -G[ G1 , - - - I~oo2 - G I > o , a

provided G < e03~/m, for Iq2l ~ 32, or (see (6.7))

f0 ) min ly(t)l = min IG2(t, s)l[[(1 - / z ) f 0 + lZfs]ql(s) 1In + lZhs]ds t~[0,co] tE[0,co]

~ l_e ~l/n fo w 1 1/n JOO 1 -- HI a 2 ( t , s) ds -- ~ l f 0 6 1 - Hi > 0,

1/n for ]ql] ~ 31 provided H < f031 , So, in order to prove that X (t) ~ Q2, we need to fulfill simultaneously the following

inequalities:

{ (1/a)leo6~/m - G I ~ 31 > ( H / f o ) n 1/m (1/b)]fo6l - HI >/32 > (G /eo ) m.

(6.12)

Let us observe that the "amplitudes" of the multivalued functions g, h must be sufficiently small. On the other hand, if e0 and f0 are sufficiently large (for fixed quantities a, b, G, H), then we can easily find 61, $2 satisfying (6.12).

After all, if there exist constants 31, 32 obeying (6.12), then we arrive at X(t) ~ Q2,

i.e., Tu(Q) C Q2, independently of/z E [0, 1] and et, ft, gt, hr. This already means that Tu (Q) c Q,/z E [0, 1 ], as required.

Now, since all the assumptions of Theorem 6.1 are satisfied, problem (6.9) N (6.8) possesses at least N(Tu (.)) solutions belonging to Q, for every # 6 [0, 1]. In particular, problem (6.5) n (6.8) has N(T1 (.)) solutions, but according to the invariance under homotopy, N(T1 (.)) = N(To(.)). So, it remains to compute the Nielsen number N(To(.)) for the operator To : Q --+ Q, where

(fo fo ) To(q) = eo Gl(t, s)q2(s) 1/m ds, fo G2(t, s)ql (S) 1/n ds . (6.13)


Hence, besides (6.13), consider still its embedding into the one-parameter family of operators

TV(q) -- vTo(q) + (1 - v)r o To(q), v �9 [0, 1],

where r ( q ) " - (r(ql) , r(q2)) and

r(qi ) -- qi(O), for i -- 1, 2.

One can readily check that r : Q --+ Q n ]~2 is a retraction and T0(~-) : Q n ]1{ 2 ---+ Q is retractible onto Q n ]~2 with the retraction r in the sense of Definition 4.1. Thus, r o T0(~)" Q n ] ~ 2 ---+ Q N]R 2 has a fixed point ~" �9 Q n ] ~ 2 if and only if ~"-- T0 (~). Moreover, r o To(q)" Q --+ Q n ]~2 has evidently a fixed point ~ ' - Q n I[{2 if and only if ~ ' - T0(qA). So, the investigation of fixed points for T ~ = r o To(q) turns out to be equivalent with the one for T o (~) : Q n ]R 2 ---+ Q n ]1{2.

Since, in view of invariance under homotopy, we have

N(T1 (.))- N(To(.)) - N ( T 1 (.))- N ( T O ( . ) ) ,

where

eoe_aC o fo c~ T ~ -- 1 - e -a~~ eaSq2(s)l /m ds, f oe-bc~ fo c~ 1/n ) 1 - e -b~ e bs ql (s) ds

and

) T O (~-) _ q--~(1 / m), T ~-]-(1 / n) ,

for ~ -- (qi-, q-z) -- (ql (0), q2(0)) �9 Q N ]1{ 2,

it remains to estimate N ( T ~ It will be useful to do it by passing to a simpler finite- dimensional analogy, namely by the direct computation of fixed points of the operator

T O (~-)" Q N R2 ___+ Q n ]t{2,

belonging to different Nielsen classes. There are two fixed points ~'+ -- (~'], q'2) and ~'_ - (-~ '] , -q'2) in O n R 2, where

~'1_ (e~Oa)mn/(mn-1)(~) 1/(mn-1)

"~2-- (e~Oa )m/(mn--1)(_~) mn/(mn-1)

These fixed points belong to different Nielsen classes, because any path u connecting them in Q n • 2 and its image T ~ are not homotopic in the space Q AIR 2, as it is schematically

86

Fig. 4.

sketched in Fig. 4. Thus, according to the equivalent definition of the Nielsen number from the beginning of Section 3.2, N(T~ -- 2. By means of the reduction property which is true here (see Lemma 6.1), we have, moreover,

N(T1 (.))= N ( T ~ = N(T~ = 2

and so, according to Theorem 6.1, system (6.5) admits at least two solutions belonging to Q, provided suitable positive constants 31, 32 exist satisfying (6.12) and e, f , g, h are product-measurable.

In fact, system (6.5) possesses at least three solutions satisfying (6.8), when the sharp inequalities appear in (6.12), by which the lower boundary of Q becomes fixed point free. Indeed. Since

e ) = A(r~ e ) = e n R :)

holds for the generalized and ordinary Lefschetz numbers (see Section 3.1 and cf. [22, Chapter 1.6]) and one can easily check that

I~.(T~ Q n R 2 ) l - 2,

we obtain

[A(T~(.), Q)I - 2.

Furthermore, since for the self-map T1 (.) on the convex set Q1 n Q3 such that T1 (Ol N Q3) is compact we have (see Remark 3.1)

A(TI('), Q1N Q3)--1,


it follows from the additivity, contraction and existence properties of the fixed point index (see Proposition 3.2 in Section 3.3 and cf. [22, Chapter 1.8]) that the mapping T1 (-) has the third coincidence point in Q1 n Q3 \ Q representing a solution of problem (6.5) N (6.8) and belonging to Q1 \ Q.

As we could see, problem (6.5) n (6.8) admits at least two solutions in Q1 n Q2, for an arbitrary co > 0. Furthermore, because of rescaling (6.5), when replacing t by t + (co/2), there are also two solutions of (6.5) satisfying X (-co/2) = X (co/2), for an arbitrary co > 0, and belonging to Q1 n Q2.

Therefore, according to Proposition 4.5 and by obvious geometrical reasons, related to the appropriate subdomains of Q1 n Q2, system (6.5) possesses at least two entirely bounded solutions in Q1 n Q2.

Of course, because of replacing t by ( - t ) , the same result holds for (6.5) with negative constants a, b as well.

Finally, let us consider again system (6.5), where a, b, m, n are the same, but e, f , g, h are this time 1.s.c. in (x, y), for a.a. t E (--cx~, c~), multivalued functions with nonempty, convex, compact values and with the same estimates as above. Since each such mapping e, f , g, h has, under our regularity assumptions including the product-measurability, a Carath6odory selection (see, e.g., [22,73]), the same assertion must be also true in this new situation.

So, after summing up the above conclusions, we can give finally:

THEOREM 6.2. Let suitable positive constants 61, 62 exist such that the inequalities

e0o 2

1 "r~l/njo~ ( G ) m-~O -HI > 6 2 >

(6.14)

are satisfied for constants e0, f0, G, H estimating the product-measurable u-Carath~odory or l-Carath~odory multivalued functions (with nonempty, convex and compact values) e, f , g, h as above,for constants a, b with ab > 0 and for odd integers m, n with min(m, n) ~> 3. Then system (6.5) admits at least two entirely bounded solutions. In particular, if multivalued functions e, f , g, h are still co-periodic in t, then system (6.5) admits at least tree co-periodic solutions, provided the sharp inequalities appear in (6.14).

REMARK 6.3. Unfortunately, because of the invariance (w.r.t. the solution operator 7'1) of the subdomains

q(t) E C - 2 ' -2- , ]~2 0 < 61 ~< ql (t) ~< R/X 0 < 62 ~ qz(t) ~< R

and

q(t) E C - -~ , -~ , - R ~ ql(t) ~ -61 < 0 A - R ~ q2(t) ~< -62 < 0 ,

88 J. Andres

for each co ~ (-cx~, c~), the same result can also be obtained, for example, by means of the fixed point index.

In order to avoid the handicap in Remark 6.3, let us still consider the planar system of integro-differential inclusions. For the sake of transparency, our presentation will be as simple as possible. Thus, the right-hand sides can apparently take a more general form and the multiplicity criteria in terms of inequalities can be improved.

Hence, let Xi" [0, co] --+ ]1~, for i = 1, 2, x = (Xl, x2), q9 E [0, %], a > 0 and consider the following system of integro-differential inclusions

21 + a x l E x~/p2(x) cos q9 -- ~ /p l (x) sinq) +~0e, (6.15)

22 -+- ax2 E ~/pl (x) cos q9 + ~/p2(x) sing) -+- q)e, (6.16)

where e : [0 , co] x ]t~ 2 --o I~ is a product-measurable u-Carathfodory map with nonempty, convex and compact values with le ( t , x)l ~< E, for a.a. t ~ [0, col and all x ~ R 2, and

1/0 (,f0 ) Pi ( x ) - - - - xi (s) ds - B -- xi (s) ds - xi , co co

jr the system takes the form with B > 0. For q9 = -~,

) Xl -a t- a x l e - - ~ 2 ( x ) - ~ P l (x) + ~-e, (6.17)

22 -~ ax2 E - -~ Pl (x) -+- ~/pz(X) -Jr- ~ e , (6.18)

while, for ~o = 0, it reduces to

21 + a x l = ~p2(x ) , x2 -Jr" ax2 -= , J P l (X).

We shall be again looking for the lower estimate of the number of co-periodic solutions

to (6.17), (6.18). Let us define sets S = Q c C ([0, co], R 2) as follows. Function q = (ql, q2) belongs to

Q if the following conditions are satisfied: (i) q (0) = q (co) (w-periodicity),

(ii) Iq(t)l ~< R, for all t e [0, co] (boundedness), (iii) Iql (t)l >/8 or Iq2(t)i >/8, for all t ~ [0, co] (uniform boundedness of one component

from below), 1 (iv) q ( t ) = ~ + ( ( t ) , where ~ := -y f o q ( s ) ds is the integral average of q on [0, co]

1 (thus, ~ fo ~(s) ds = 0) and I~'(t)l ~< e, for all t 6 [0, co] (function q differs from its integral average by less than e).

Values of a and co are given, we shall specify the values of B, 8, R, E and e in the subse-

quent parts.


Set Q is again a union of four closed, convex sets in the Banach space C ([0, co], R2),

namely Q - - Q+ U O2 + U Q~- U Q z, where

Q~ - {q(t) E C([O, co],R2) Iq satisfies (i), (ii), ( iv)and

i q i ( t ) / > ~ > 0 / , min tE[0,~o] J

i = 1,2.

Thus, as in the proof of Lemma 6.2, it is a closed connected ANR-space such that

7rl (Q) - - Z . For homotopic parameter ~0 = 0, system (6.15), (6.16) reduces to a simpler case, which

can be easily handled (in fact, we can explicitly compute two constant fixed points). For ~ 0 - ~-, the situation becomes non-trivial. We shall show that set Q is invariant under the solution operator, which takes a parameter q E Q to the solution x of the linearized inclusion. In this case, no obvious or easily detectable subset of Q can be recognized to be separately invariant.

In order to apply a slightly modified special case of Theorem 6.1 (cf. [18]), we use again the method of Schauder linearization. Let us take an arbitrary q E Q. The system of fully linearized inclusions takes the form:

YCl + ax l E ~/pz(q) cos q9 -- ~ P l (q) sinq9 + ~pe(t, q), (6.19)

x2 -+-ax2 E ~ p l (q) cosq9 + ~p2 (q ) sinq9 + q g e ( t , q ) , (6.20)

where

lf0 (lf0 Pi (q) = -- qi (s) ds - B -- qi (s) ds - qi , co o)

(6.21)

for i = 1,2. - - 1 Denoting q :-- -g f o q ( s ) d s the integral average of q on [0, co], we can write p ' Q c

C ([0, co], R 2) ~ C ([0, co], IR 2) in the form

p ( q ) = -~ - B(-~ - q). (6.22)

For B = 1, operator p reduces to identity. For B < 1, the operator "shrinks" function q closer to its integral average. Indeed, if q - ~ + ~, where Iq(t) I ~< e, for all t E [0, co], then operator p takes q to p ( q ) = - ~ - B'~.

The fully linearized system (6.19), (6.20) possesses, for any q E Q, and any Lebesgue integrable (single-valued) selection e0 C {e(t, q)}, t 6 [0, co], a unique solution x ( t ) which is given by the known convolution with the Green operator

f0 9 xi (t) = G( t , s) f i (s) ds, (6.23)

90 J. Andres

where

f l (s) "-- ~p2(q) cos q) - ~ P l (q) sing) 4- q)e0, (6.24)

fz(s) "= ~/Pl (q) cos q) 4- ~/p2(q) sinq) 4- qge0. (6.25)

Let us denote by T~o the solution operator which takes q e Q to the solutions x, given by (6.23), of the linearized system (6.19), (6.20). We shall prove that Q is invariant under T~0, namely that T~0 (q) C Q, for each q e Q.

Let us take q 6 Q arbitrary. Operator p defined by (6.22) takes q to p ( q ) such that p ( q ) = ~- + ~', where I~(t)l ~< Be. Substituting this p ( q ) into (6.24) and (6.25), we obtain f i (t) -- Fi (t) 4- f (t), where

F1 "-- ~ 2 cos q) - ~ 1 sin q), F2 :-- ~ 1 cos q9 4- ~/q-2 sin q), (6.26)

and Ij~(t)l <~ 3~FB--~ + - ~ for all t ~ [0, col. This estimate can be shown as follows. For [q-il >~ 1, one can get by direct calculation

that I~/~-i 4- pi - ~ / I <~ ~ n e , provided that I~'il ~ BE ~< 1. For [q/I <~ 1, a careful exam-

ination of function I~qi 4- pi - ~ / I reveals that it is bounded from above by the value

22/3 ~B--s, provided again I~il ~< B~ ~< 1. Altogether, f/differs from Fi not more than by

yrE yrE q/2 2 2/3"~/Be + ~< 3~/Be 4- 2 2 - - ~ 4 '

as claimed. The Green's function G in (6.23) takes the form

1 -at eas ~ e for0 ~<s ~< t, G(t s) -- 1 - e -a~

' 1 - a t e -aCoeaS ~ e for t ~< s ~ co. 1 - - e - a ~

(6.27)

Substituting (6.26) and (6.27) into (6.23), we obtain Xi ( t ) - - ~ a + Yi (t), where ~'i satisfies

the inequality

(t) l 7r 3 ~ / B e + -g E

for all t 6 [0, co]. We can now take B and E small enough to fulfill

E) s (6.28) <~2'

for example

)3 82 a and E ~<

a 8

45


which implies This means that function x differs from a constant function by less than ~, that it differs from its integral average by less than e. The above calculations ensure that the solution x satisfies condition (iv) of the definition of the parameter set Q, independently on q 6 Q and q9 E [0, % ]. Condition (i) is trivially satisfied by the form of the Green

function G. We must further ensure that, for each q 6 Q, function x satisfies conditions (ii) and (iii).

Since both q and x differ from their integral averages by less than e, let us first deal with constant functions. This is easy, because the solution mapping T~0 takes constant functions

to functions that differ from a constant function by less than ~r E and it is a composition of -4-a-- �9 reflection (ql, q2) --+ (q2, q l ) ,

1 �9 re-scaling (ql, q-2) --+ a ( ~ ' x~2) , �9 rotation (ql, q2) ~ (q-1 cos q9 - q-2 sin qg, q-2 COS 99 -Jr- q-1 sin ~o) by angle ~0 in the anti-

clockwise direction. The re-scaling part of the composition ensures that constants R and 3 can be specified

so that the solution operator T~o takes constant functions satisfying (ii) and (iii) to functions that again satisfy (ii) and (iii). Since functions in Q differ from their integral averages by less than e, we need to find R, 3 and e such that the following conditions are satisfied:

1 ~ /~ ~< R - e - ~ and ~ >~ 3 + x/2e + ~ . (6.29) a 4a 2a 4a

Inequalities (6.29) guarantee that x satisfies conditions (ii) and (iii) of the definition of the parameter set Q. Taking further e ~< ~ ensures that Q is a non-trivial ANR-space (leaving the "hole" inside).

Starting from a > 0 and co > 0, we have specified constants R, 6, B, E and e such that set Q becomes invariant under the solution operator T~0 which takes any q 6 Q to the solutions x of the linearized problem (6.19) (6.20), for ~0 6 [0, Jr , 7]" Moreover, observe

Jr there are no easily detectable subdomains of Q separately invariant under that, for ~0 = 7, operator T~r/4. Figure 5 shows how operator T~r/4 treats constant functions in Q, for a particular choice of R and 6 and helps understanding why we can not easily detect any subinvariant domains of Q.

Hence, a slightly modified special case of Theorem 6.1 (cf. [18]) ensures that system (6.15), (6.16) admits at least N(Tjr/4) solutions. Since Tjr/4 is compactly admissibly ho-

motopic to To, we have N (Trr/4) -- N (T0). Let us further consider the retraction r ' Q --+ Q which sends a function q 6 Q to its

integral average ~. Let us define the homotopy TU �9 [0, 1 ] • Q --+ Q by

TU(q) "-- lzTo(q) + (1 - lz)r(To(q)).

This compact homotopy guarantees that N (T 1) = N (To) equals to N ( T ~ -- N (r o To). We can thus restrict ourselves to the computation of N(r o To). Let us denote by Q the subset of Q consisting of constant functions. Since r o To" Q --+ Q, all the fixed points of r o To have to belong to Q. Let us therefore deal with the restriction L "-- r o To I~" Q -+ Q, which can be explicitly written in the form

92 J. Andres

Fig. 5. Behaviour of TTr/4 on constant functions on Q for R = 10, 8 -- 1 and a = 4 " Rectangular grids of points represent constant functions q 6 Q, the irregular grid represents their images under Tjr/4. For simplicity, we take here E = 0, so that the images of constant functions become constant again. No easily detectable regions of the

domain are subinvariant.

1 ~ / ~ 2 , �9 L(ql , q2)" a

One can easily check by an explicit computation that L has two fixed-points in Q which belong to different Nielsen classes. Therefore, according to (reduction) Lemma 6.1, N(L) = N(r o To) = 2. This finally shows that system (6.17), (6.18) admits at least two co-periodic solutions.

We are in the position to formulate the multiplicity criterion for co-periodic solutions to system (6.17), (6.18).

THEOREM 6.3. Let the following inequalities be satisfied:

0.247 ~ r - R < ~ R 5 8 a8 ( a ) 3 8 2 8 < a3/2 , a - --~-, E ~< 2~r r , B ~< ~ 4x/2' (6.30)

and take e - ~. Then system (6.17), (6.18) admits at least three co-periodic solutions.


PROOF. The assumptions guarantee that inequalities (6.28) and (6.29) are satisfied. Thus, two o~-periodic solutions have been already obtained by means of the Nielsen number, as above. The third can be proved quite analogously as in the proof of Theorem 6.2, by the additivity property of the fixed point index (see Proposition 3.2). [--1

REMARK 6.4. The inequalities in Theorem 6.3 are satisfied, e.g., for a = -~ , 6 = 1, R =

10, B ~< , and E ~< a--Y, as in Fig. 5, where E = 0.

REMARK 6.5. If the equality appears for 8 in (6.30), then in the lack of periodicity, at least two entirely bounded solutions can be proved as in Theorem 6.2.

6.3. Several anti-periodic solutions

The following approach is via the Poincar6 translation operator treated in Section 4.3. Hence, consider the system of differential inclusions

~c E F(t, x), (6.31)

where F:~:{ n+l ---o ]1{ n is a u-Carath6odory mapping with nonempty, compact and convex values, satisfying (6.4). Then all solutions of (6.31) exist in the sense of Carath6odory, namely they are locally absolutely continuous and satisfy (6.31) a.e.

If x(t , xo) := x(t , O, xo) is a solution of (6.31) with x(O, xo) = xo, then we can define the Poincar6 map (translation operator at the time T > 0) along the trajectories of (6.31) as follows:

~T "IR n ---o IR n, c~T(XO) "= {x(T, xo) I x(t , xo) satisfies (6.31)}. (6.32)

The goal is to represent the admissible (see Theorem 4.17) map ~T in terms of an admissible pair (see Definition 2.5 in Section 2.2). We let ~0 :JR n --o C ([0, T], IRn), where

q)(X) := {x ~ C([0, T], R n) Ix(0) = X and x satisfies (6.31)}

and C([0, T], IR n) is the Banach space of continuous maps. According to Theorem 4.17, ~o is an Ra-mapping.

Now, we let

eT " C([0, T), R n) --> R n, eT(x) -- x (T) ,

where eT is evidently continuous. One can readily check that ~T = eT o ~0. Moreover, 7Y o ~T = 7-/o ~p o eT is admissible

for any homeomorphism 7Y:IR n --, ~n (see Remark 4.15). In fact, we have the diagram

Rn <p~o lp~0 q~0> C([0, a], ]~n) eT> ]R n ~> Rn '

where p~0, q~0 are natural projections.

94 J. A n d r e s

In what follows, we identify the Poincar6 map (I)T o r its composition with 7-/, i.e. ~ o ~T, with the admissible pair (p~, eT o q~) or (p~, 7-[ o eT o q~), i.e. we let

~ T -- (P~o, eT o q~o) or ~ o ~T = (P~0, ~ o eT o q~), (6.33)

respectively. Let C ( ~ o ~T) denote the set of coincidence points of the pair (p~0, ~ o eT o q~0), while

F ix (~ o ~ T ) -- {X ~ R n I X ~ 7-[(eT(q~o(p-~l(x))))}. A pair for ~T can be easily shown to be homotopic to the identity map, so that the

pair (p~, ~ o eT o q~) is homotopic to ~ (see Theorem 4.17). We have a one-to-one correspondence between coincidence points and solutions. Since a coincidence point of (p~, 7-[ o eT o q~) gives us in this way a solution x ( t ) of (6.31) such that x(0) = ~ ( x ( T ) ) ,

the following proposition is self-evident.

PROPOSITION 6.1. I f #C(p~o, 7-[ oeT oq~) = cardC(7-/o ~ T ) >~ k, then system (6.31) has

at least k solutions Xl (t) . . . . . xk( t ) such that xi(O) = 7-~(xi(T)), i = 1 . . . . . k.

The following lemma immediately follows from Theorem 4.17.

LEMMA 6.3. Assume that Y C R n is a compact connected ANR-space such that Cbs(Y) C

Y, f o r every s ~ [0, T] and 7-[(Y) C Y. Then the pair (p~, 7-[ o eT o q~o) restricted to Y is

admissibly homotopic to ~ l r .

As a consequence of Theorems 3.4 and 3.7 in Section 3.2, Proposition 6.1 and Lemma 6.3, we have the following

PROPOSITION 6.2. Assume that ~s , f o r every s ~ [0, T] in (6.32) is a self-map on the torus qF n = R n / z n. Then system (6.31) has at least N ( ~ ) solutions x ( t ) such that x(O) =

7-[(x(T)) on 72 n, where N ( ~ ) denotes the Nielsen number o f a homeomorphism 7-[ : qr n --+ 7 f n .

PROOF. The proof follows directly from Theorems 3.4 and 3.7 in Section 3.2, Proposi- tion 6.1, Lemma 6.3 and the properties of the Nielsen number mentioned in Section 3.2.73

If in particular 7-/= id, then according to Remark 3.6,

N(id) = [)~(id)]- I x ( . > l

holds on tori, and consequently the problem considered in Proposition 6.2 should have at least Ix(')l solutions, where )~ is the Lefschetz number and X denotes the Euler-Poincar6 characteristic of T n . Since, unfortunately, X (') - 0 for toil, this is not a suitable case for applications. In other words, we are not able to establish several T-periodic solutions in

this way. On the other hand, as the simplest application of Proposition 6.2, we can give immedi-

ately


THEOREM 6.4. Assume that

F( t . . . . , x j + 1 , . . . )=- F( t . . . . . x j , . . . ) , for j - 1 . . . . . n, (6.34)

where x = ( X l . . . . . Xn) and consider system (6.31) on the set [0, cxz) x qF n, where qF n = R n / Z n . Then system (6.31) admits, for every positive constant T, at least N (7-[) solutions x (t) such that

x(O) -- ~ ( x ( T ) ) (mod 1),

where 7-[ is a continuous self-map on ~n and N (7-[) denotes the associated Nielsen number

As a consequence, we obtain for 7-[ = - id easily:

COROLLARY 6.2. If, in addition to the assumptions o f Theorem 6.4,

F(t + T , - x ) - - - - - F ( t , x ) ,

then system (6.31) admits at least 2 n anti-T-periodic (or 2T-periodic) solutions x( t ) on 7s n, namely x ( t + T) =_ - x ( t ) (mod 1).

PROOF. According to Theorem 6.4, system (6.31) admits at least N ( - id) anti-T-periodic solutions on qF n. On q[,n, the following formula holds (see Remark 3.6), N ( - i d ) - 1)~(-id)l = ] d e t 2 I I - 2 n, which completes the proof. 5

7. Remarks and comments

7.1. Remarks and comments to general methods

Theorems 4.1, 4.2 and Corollaries 4.1, 4.2, 4.3 are taken from [ 19]. Corollaries 4.1 and 4.4 generalize many single-valued situations, e.g., in [52] and [63], where the parameter set Q was only convex. Corollary 4.3 was employed for the first time in [32,33]. Corollary 4.4 generalizes the single-valued case in [39].

Theorem 4.3, for the lower estimate of the number of solutions, comes from [23]. Its slightly modified version (cf. Remark 4.3) was presented in [5]. As far as we know, there are no other general methods, using the Nielsen number, as our Theorem 4.3.

Theorem 4.4, extending Theorem 4.1 to Banach spaces, was published in [14]. Corol- lary 4.5 is formally new. Intuitively clear Propositions 4.4 and 4.5 (see also Remark 4.6) are contained in [22, Chapter III.1 ].

In view of Remark 4.6, one can check by means of the Gronwall inequality (cf. [71]) that differential inclusion ~ e F(t , x) admits an entirely bounded solution x e ACloc(R, E) with x (0) = x0 such that

)f" I x ( t ) l[ ~< Ilxoll + r(t) at exp r(t) at, o o o o

t 6 ~ ,

96 J. Andres

provided E is a separable or reflexive Banach space and F :R x E ---o E is a u- Carath6odory mapping (cf. Definition 2.10) such that:

(i) # ( F ( t , B)) <<, k ( t ) t x (B) , for bounded subsets B C E, t 6 R, where/x = ot or/x = y,

and k e L~oc(N), (ii) IlYll ~< r(t)(1 + Ilxll), for every ( t , x ) ~_ IR x E, y E F ( t , x ) , where r :IR ~ [0, oc)

is an integrable function such that f _ ~ r( t ) dt < ee. Many alternative continuation principles for ODEs can be found, e.g., in [45,58,61,62,

64,65,82,83,90], and the references therein. Theorems 4.6, 4.8, 4.9 and Proposition 4.9, dealing with the topological structure of

solution sets are taken from [19]. Theorem 4.9 extends Theorem 4.5 in [57] for arbitrary (possibly infinite) intervals. In [20,21 ], we have developed and applied a powerful inverse limit method for the investigation of the topological structure of the solution sets. In these papers, the structure was also systematically studied for less regular right-hand sides of multivalued ODEs than those in Section 4.2.

Theorems 4.10 and 4.11 concerning the topological dimensions of solution sets come from [66]. An extension to multivalued ODEs in Banach spaces was recently published in [45]. For further results concerning the solution sets to initial value problems, see, e.g.,

[42,55,58,59,67,74,78,97]. Unlike for initial value problems, there are only several results concerning the topolog-

ical structure of solution sets to boundary value problems. One of the most important is Theorem 4.13 in [43] which was improved by us (cf. Remark 4.10) in [21]. Theorem 4.12 comes from [20], Theorem 4.14 from [46] and Theorems 4.15, 4.16 from [7]. In [22, Chap- ter III.3], we collected for the first time practically all results in this field (cf. also [59, Chapter 6]). Paper [43] contains also the information about the topological dimension of

the solution sets. Theorem 4.17 about admissibility of Poincar6's operators appeared on various levels

of abstraction in many papers (cf. [22, Chapter III.4 and the related comments on pp. 592-593]). For some generalizations and extensions, see [2,10,28], and the references

therein.

7.2. Remarks and comments to existence results

Theorem 5.1 from [ 14] can be regarded as an infinite-dimensional extension of our earlier results in [3,4], where under suitable additional restrictions also almost-periodic solutions were detected. In [15], the methods applied to Theorem 5.1 were modified for obtaining almost-periodic solutions with values in separable Banach spaces. Theorem 5.2 is con-

tained in [20]. Proposition 5.1, for solvability of a rather general class of boundary value problems with

linear conditions, was presented for the first time in [3]. There is an enormous amount of results about the existence of periodic and anti-periodic

solutions (see, e.g., [22,62,64,65,70,74,78,81,82,90]). Our Corollary 5.1 is formally new. Corollaries 5.2, 5.3, 5.4 can be found in [22, Chapter 111.5]. As pointed out in Remark 5.7, the requirement concerning a fixed point free boundary of a parameter set Q were satisfied


in [ 11,32-34] by means of bound sets. An alternative approach for satisfying this requirement can be found, e.g., in [30,31 ], where canonical domains or upper and lower solutions technique were applied, in the single-valued case (cf. also [65,70,81,82,90]).

7.3. Remarks and comments to multiplicity results

Our multiplicity results are based on the application of the Nielsen theory in Section 3.2, eventually combined with the additivity property of the fixed point index in Section 3.3. For further Nielsen theories which can be also used here, see [12,22-26,38], and the references therein. Practically all results obtained in this way were collected in [ 13].

Corollary 6.1 is from [5]. Variants of Theorems 6.1 and 6.2 can be found in [5,6,9], and [23], while Theorem 6.3 is a multivalued generalization of a single-valued version in [17]. Theorem 6.4 and Corollary 6.2 were presented for the first time in [24].

As pointed out in Section 1, the delicate problem of application of the Nielsen theory to differential equations is associated with the name of J. Leray.

For further multiplicity results obtained by different methods, see, e.g., [22, Chap- ter III.6; 62; 82, Chapter 6], and the references therein.

Acknowledgements

The collaboration and fruitful discussions with the coauthors of the quoted papers, in particular with Prof. Lech G6rniewicz (Torufi), are highly appreciated. The author is also indebted to Dr. T o m ~ Ftirst (Olomouc) for his critical reading of the manuscript.

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