4 J. MAWHI N Among the variou s possible application s o f bifurcation theory , we have chose n in Chapter X the proble m o f periodi c solution s aroun d a n equilibrium i n autonomous ordinar y differential equations . Classica l studie s of thi s problem wer e initiated b y Poincare ' and Lyapunov and th e result s we give, due t o Laloux , are related t o more recen t contribution s of Hiss, Berger, Lazer an d Weistreich . Of course , there ar e man y othe r type s of problem s which ca n be successfull y treate d by topologica l degre e arguments an d ar e not eve n mentioned i n this monograph. Amon g them i s the fixed point theor y i n cones and th e correspondin g stud y o f positiv e solution s of operator equations , with its important application s t o autonomou s functiona l differentia l equations. Th e interested reade r ca n consul t th e important wor k o f Krasnoserski l [128] , Amann [6] , Nussbaum [199] , [200 ] an d Schmit t [231 ] an d th e reference s give n therein . One should als o mention th e importan t are a o f asymptotic fixed point theorem s an d its re- lation t o th e qualitativ e propertie s o f ordinar y an d functiona l differentia l equations . Se e in this respect th e contribution s o f Jone s [112] , Browde r [34] , Nussbaum [196] , [197] , Peitgen [202] , Hal e [99 ] an d others . Th e reade r wil l notice tha t th e asymptoti c fixed point theor y relie s not onl y upo n th e basi c degre e theor y describe d in Chapte r I I but als o upon dee p and difficul t result s o f algebrai c topology . A s already suggeste d b y bifurcatio n theory, th e topologica l degre e i s a powerful too l fo r provin g not onl y existence bu t als o multiplicity result s for th e solution s o f nonlinear equations . Result s of thi s type, du e among others t o Croni n [53] , [54] , Krasnosel'skil [130] , Amann [6] , Schmitt [230] , ca n be rathe r easily deduce d fro m th e basic properties o f degre e given in thi s monograph. On e shoul d finally mentio n that , althoug h outsid e o f th e scop e o f thi s work, many problem s fo r non - linear partia l differentia l equation s o r integral equation s ca n b e successfully treate d b y th e abstract method s describe d here. Th e intereste d reade r ca n consul t th e book s o f Croni n [52], Berge r [23] , Miranda [181] . This monograph originate d i n lectures give n at th e NSF-CBM S Reginal Conference o n Topological Degre e Methods in Nonlinear Boundar y Valu e Problems held i n June 197 7 at the Claremon t Universit y Center , Claremont, California. Th e autho r woul d lik e to acknow - ledge th e Organizin g Committee, an d especiall y S . Busenberg an d K. L . Cooke, for thei r in- vitation an d invaluable wor k i n making the organizatio n superb . Sincer e thank s also go to the participants o f thi s conference whos e pertinent comment s an d remark s helped ver y much in transformin g th e preliminar y versio n o f thos e lecture note s into th e presen t form , and t o J . P. Gossez, B. Laloux an d M . Willem who read importan t part s o f th e manuscrip t and suggeste d numerou s correction s an d improvements . Despit e thi s collective effort , im - perfections an d mistake s are undoubtedly stil l present i n thi s text 5 an d definitel y remai n th e personal propert y o f th e autho r only .

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