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
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