Proceedings of Symposia in Pure Mathematics

Volume 45 (1986), Part 1

A Note on the Regularity of Solutions Obtained

by Global Topological Methods

J. C. ALEXANDER1

Over the past decade or so, there have been a number of results establishing

"global" families of solutions of functional-analytic operators. The paper that set

the ideas and terminology is Rabinowitz [6], although the classic work of

Leray-Schauder [5] also fits in the framework. More recently, this author and

other workers have found it useful to consider global families of dimension

greater than one. One way of expressing and proving the results is via cohomol-

ogy theory. The solution set of the operator, considered as a topological space, is

shown to carry some kind of nonzero Cech cohomology class. The dimension of

the class is a lower bound on the dimension of the solution set. If the solution set

is to be connected, as in Rabinowitz's paper, the class is one dimensional. In the

classic existence results, the class is zero dimensional; i.e., the solution set is not

empty. In virtually all cases the underlying space on which the operator acts is a

Banach space (possibly finite dimensional). This is more than a technical point; in

general the results are false in Frechet spaces.

The purpose of this note is to consider what happens when "one" operator acts

on an ordered family of Banach spaces. The result is motivated by some work on

differential operators; however, it was felt it would be worthwhile to isolate a

general functional analytic result.

Consider the following motivating and prototypical example. Let c€r^a be the

Holder space of # r + a functions on some domain with the ^r+a topology.

Suppose P is a parameter space. Suppose J^ is an integral operator which inverts

a differential operator and which defines a compact operator J^: P X ^ r + a -

c€r+a

for each r. Under suitable standard assumptions, the solution set 5^ of J^ in

? X ^

r + a

carries a nonzero cohomology class, and of course ^ c S^r_v Thus

1980 Mathematics Subject Classification. Primary 47H99, 35B65, 35B32; Secondary 58C30, 34A34.

1

Partially supported by NSF.

© 1986 American Mathematical Society

0082-0717/86 $1.00 + $.25 per page

1

http://dx.doi.org/10.1090/pspum/045.1/843543