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 183 3 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 190 4 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 192 2 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 192 7 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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