AN EXTENSION OF HORN ’ S THEOREM FOR A SPACE OF LOCALLY INTEGRABLE FUNCTIONS

In this paper we extend the fixed point theorem of Horn's to the space of locally integrable functions from (- , 0 )into a Banach space X.


INTRODUCTION.
Horn's fixed point theorem 3 has numerous interesting applications, espe- cially to obtain periodic solutions of differential and retarded functional differen_ tial equations.In this subject we refer to the paper of Burton and Dwiggins as well as those references mentioned in it.In these applications, the problem of existence of periodic solutions is reduced to the existence of a fixed point in the phase space associated with the equation.Consequently, in order to apply Horn's theorem it is necessary that the phase space be a Banach space.Nevertheless, this condition is not appropriated for retarded functional differential equations with infinite delay and, in general, for abstract differential equations.
For these reasons Burton and Dwiggins established an extension of Horn's theorem to the Frchet space C((-, 0 IRn) endowed with the compact -open topology.
This extension permits consideration of equations whose initial conditions are continuous functions.However, in many initial values problems the space of continuous func- tions is not the most appropriate, being necessary to consider integrable functions.This occurs, for example, in some models used in control theory 2 ).
We now state Horn's theorem 3 THEOREM I. Let X be a Banach space, S C S o and S 2 compact and S open in $2o Let P:S 2 ---> X be a continuous map.
for some integer m > 0 the following conditions hold PJ(s I) _ $2, for j < m-1 and a) b) PJ(s I) _ So, for m < < 2m-I Then P has a fixed point in S o _C S 2 convex subsets of X with So Suppose that In this note we use the line of reasoning of Burton and Dwiggins to extend the Horn's theorem to a Frchet space of locally integrable functions.
In the sequel we will denote by X a Banach space with norm II" II.We will designate by B the Frchet space consisting of the equivalence classes of func- tions :(-, 0 > X which are locally integrable in the Bochner sense, endo wed with the topology r induced by the family of seminorms o pn(O) [[o(0) -n We will represent by G the set formed by all continuous and increasing func tions g:(-, 0 ]----> 0,I such that g(O) and g(@) O, as O > -.
For each g 6 G we will write B to denote the space of the equivalence classes of g locally integrable functions : The aim of this .paper is to prove that theorem holds if we susbtitute the arbitrary Banach space X by the Frchet space (, ).

RESULTS.
In this section we will prove that Horn's theorem is also true in the space (B, ).
The proof is a consequence of the following four lemmas, that relate the Banach spaces for g 6 G, with the space ( ). g LEMMA I. Let g 6 G.The following conditions hold i) The inclusion mapping i: B --> B is continuous.g ii) If S is a bounded subset of B then the inclusion map i:(S,r ----> B g g.h is continuous for every h 6 . Proof.Since B is a Frchet space whose topology is generated by the semi- norms Pn' n IN, to prove assertion i) it is sufficient to observe that g which yields that pn() g / g(-n) for every e B Next we will prove ii).If S is a bounded subset of B then S C Bgh, for g each h 6 G. Let M 0 be a constant such that q M for all 6 S. For g each e 0 there exists a positive integer n such that h(O) e for 4M every @ -n Let E S be such that pn( ) < From the definition of II-II we obtain the following estimate which shows that the inclusion S B gh is continuous for the relative topo- logy in S.
Concerning this result, it is worth mentioning that, in general, the inclu- sion i: (S, ) ----> B is not continuous.In fact, let us consider the space X= g and let g be the function defined by g(O) 101-2 for-O < -I and g(O) for -I < 0 < 0. From this definition it follows that g .E G. Let us define, for each natural n, the function n ' which equals g in (-, -2n U [-n,O and (0) g(O) + IOl, for O (-2n, -n).It is easy to see that these functions n belong to B and that ---> g, as n ----> in the topology of .We can also n see that the subset S {g U {n: n IN is bounded in B However, does not converge to g in B Hence the which shows that the sequence (n)n g inclusion S ---> B is not continuous.g LEMMA 2. Let S be a compact subset of (B,T).Then for every pair of constants a,b 0 and every positive integer k there exists an increasing and continuous function q: [-k, -k+l ----> O, ) such that q(-k+l) a and -k+l f q(O) ,(o)II dO b, -k for every S.
Proof.For each E B we will denote by R() the restriction of to the interval [-k -k+l It is clear that the mapping R: B> LI( I-k, -k+l ]; X) is continuous, which implies that the set R(S) is compact in the space S LI( [-k, -k+l X).Consequently, there exist functions I' 2' n such that for each S we can find an index i 1,2,..., n for which On the other hand, since the space of continuous functions C( [-k, -k/l X is dense in the space of Bochner integrable functions 4 we may choose conti- nuous functions I''''' n such that -k+l -k 3a We now define the function q(O) ae -a(-k+l-@) a positive constant such that i 1,2,..., n (2.2) for e -k, -k+l where a is -k+l 3) for all i 1,2 n.Therefore, if follows form (2.1), (2.2) and (2.3) that for each q S we can choose a function 9. for which the following estimate hold 1 -k+l -k+l ] q(O)tt 9(0)tt dO f which completes the proof.
If S is a compact subset of B then there exists a function g G such that S C B and S is also compact for the topology of g g Proof.We can assume that S contains a function o such that 90(0)[] I, for < O O. By lemma 2 we may assert that there exists a sequence of cont_i.

and all k IN
We define the function h: (-, 0 ----> IR by h(O) qk(O) for -k 0 -k + 1, and k 1.Then h is a continuous and increasing function such which implies that h(@) --> O, as @ --> It follows from this that h 6 G.
On the other hand, since S is bounded in B, there exists a positive constant M such that @(0)II < M (2.5) for every @ E S In view of (2.4) and (2.5) we conclude that S is a set included and bounded Finally, using lemma I. ii) we obtain that S is compact in g where Our last lemma is the following LEMMA 4. Let S be a compact subset of B and let P'.S ----> 8 be a continuous map.Then there exists afunction g E G such that S C B and P: S --> is g g continuous for the topology induced by in S. g Proof.Proceeding now as in the proof of lemma 3 we obtain that there exist gl' g2 G such that S _c gl P(S) _C g2 and the inclusions i:(S, ) ---> Bgj are continuous.Let us consider the function g gl-g 2. It is clear that g 6 G and that the inclusion mappings --> are continuous.Therefore both S as > (S, z) P > (P(S) z) ------> B we obtain g that P is continuous for the topology induced by the space B g We shall now end this note with statement and demonstration of the following.
THEOREM 2. Horn's theorem holds in the space B, z).
Proof.Having obtained the lemmas 4 we can repeat the proof carried out in [I ]in the context of continuous functions with values in IRn.In fact, the proof only depends on the properties established in the lemmas 4. We include it here for completeness.
By lemma 3 there exists gl G such that Soand S2arecompact subsets of the space B Furthermore, applying lemma 4 we may conclude that there exists gl g2 G such that P: S 2 ----> B is continuous for the B topology.Let us g2 g2 and S 2 are also compact in B and P:S 2 ---> B is still define g gl g2" Then S o g g continuous.On the other hand, from lemma we obtain that S is open in S 2, for the B topology in S 2. Moreover it is clear that conditions a) and b) of theorem g remain unchanged.Thus theorem 1 applied in the space B implies that P has a g fixed point in S o ACKNOWLEDGEMENT.This work was partially supported by FONDECYT Proyect #744/89 and DICYT, Proyect 8933HM.