Introduction

In recent years , the topologica l degre e has been reveale d as a very powerfu l an d ver-

satile tool in dealin g with nonlinea r boundary valu e problems fo r ordinar y an d functiona l

differential equations . A glimpse of the earl y history o f degre e theory show s that th e inter -

action betwee n this fundamental topologica l concep t an d th e theor y o f differentia l equa -

tions was present fro m th e ver y beginning . A problem o f electrodynamics le d Gaus s in 1833

to introduc e th e linkin g number o f tw o curve s in

R3,

undoubtedl y a forerunner o f Kronec-

ker's concep t o f characteristi c introduced i n 1869, and th e archetyp e o f th e degre e fo r map -

pings which are continuously differentiabl e o n som e special domain s analytically defined .

Besides Gauss' influence, Kronecker acknowledge d tha t o f th e work of Stur m and Sylveste r

about the number of zeros of a polynomial, a subject which later becam e basic for th e stud y

of the stability of linear ordinary differential equation s with constan t coefficients . A s early

as 1883, Poincard use d Kronecker's characteristic (o r index) in his qualitative stud y o f non -

linear differentia l equations , and discovere d o n this occasion new important consequence s of

Kronecker's theory, in particular th e so-called Miranda' s theorem. I t was some mechanics

problems which led Boh l in 1904 to formulat e an d prove, using Kronecker's ideas, results

equivalent t o th e Brouwer' s and Poincare-Bohl's theorems , at leas t fo r continuousl y differ -

entiable mappings . I f th e motivatio n o f th e tw o basi c papers o f Hadamar d (1910) and

Brouwer (1912) on th e degre e of continuou s mappings in

Rn

i s topological i n nature, it wa s

existence problem s for nonlinea r ordinar y differentia l equation s whic h motivated Birkhof f

and Kellog g in 1922 in provin g the first results in the topologica l fixe d poin t theor y i n in-

finite-dimensional vector spaces . Nonlinea r partia l differentia l equation s are present i n

Schauder's generalizations , given betwee n 1927 and 1930, of th e Birkhoff-Kellog g fixe d

point theorems , as well as in th e generalizatio n o f th e topologica l degre e t o th e clas s of

compact perturbation s o f identity i n Banach spaces, given by Lera y an d Schaude r i n 1934.

On the othe r hand, the basi c idea of relatin g the existenc e o f solution s of nonlinear equa -

tions to tha t o f a priori bounds for th e solutions , which is so clearly expressed i n the Leray -

Schauder paper , can b e traced t o Bernstein' s pioneering wor k o n nonlinea r ellipti c equation s

(1906) and Leray's thesi s (1933) on nonlinear integra l equation s an d hydrodynamical prob -

lems. W e shall not pursu e thi s discussion here an d will refer t o Siegber g [242 ] fo r man y in -

teresting detail s and discussion s concerning th e early histor y o f th e concep t o f topologica l

degree.

The aim o f thi s work i s to illustrat e th e us e o f topologica l degre e in the stud y o f non -

linear ordinar y an d functiona l differentia l equations . Havin g to mak e a choice amon g to-

day's widely availabl e material, we hav e restricte d ourselve s t o existenc e an d bifurcatio n

1

http://dx.doi.org/10.1090/cbms/040/01