2 J. MAWHI N results. I n sketching the basic features of degre e theory, we have chosen to present it di- rectly fo r a class of mappings of the form L 4 - G, where L is a linear Fredholm operator of index zero and G some nonlinear perturbation, between normed spaces, because such map- pings naturally occu r in the abstract formulation o f many problems stemming from ordinary or functional differentia l equations . On e obtains in this way, when enough compactness is present, a systematic topological degre e background for the class of abstract problems usu- ally referred to as alternative problems. A first step in this direction, for local problems, was made by Cronin in the fifties, and Cesari's extension, in 1964 , of the classica l Lyapunov- Schmidt method to th e case where G was not necessarily smal l in some sense, made clear that a corresponding treatment for global problem s should be possible. Thi s was done sys- tematically b y the author in 197 2 in the so-called coincidence degre e theory, which was re- vealed to be an underlying abstract frame for earlier independent existenc e theorem s of Lazer, Nirenberg and the author for ordinary and partial differential equations . W e refer to Gaines and Mawhin [84 ] fo r more details on the development o f the alternativ e method, and briefly describ e now the conten t o f this monograph. Linear Fredholm operators are briefly introduce d in Chapter I and their relation with linear boundary value problems for linear ordinary or functional differentia l equation s is discussed, as well as the concept o f L-compactness o f a mapping G which replaces in the study o f mappings of type L + G the usual compactness condition o f fixed point theory. Our presentation o f degre e theory i n Chapter II is made for the clas s of ^-compact perturbations of a given linear Fredholm mapping L o f index zero between rea l normed spaces X an d Z.W e have chosen to give an axiomatic presentation, but hints are given about the construction of the Brouwer degree (X an d Z have the same finite dimension and L = 0) , the Leray-Schaude r degree (X = Z and L = / ) an d the coincidenc e degree . W e do hope tha t those indications will be sufficient t o give to th e reader a sufficiently intuitiv e feeling of th e concept o f degre e of a mapping in the finite-dimensional cas e and a precise idea of how further generalization s rely upon this notion. Technica l detail s can easily be found in the literature mentioned at the en d of the correspondin g chapter.W e also give re- sults about th e computation o f th e degree and its dependence upo n the choice of L. Finally , our class of mappings being motivated by the one s usually treate d by th e Lyapunov-Schmid t method, we study th e relations between th e classical an d the topological degre e approaches, when both are applicable. Chapter III starts by giving a systematic constructio n o f classe s of fixed point problems in various abstract spaces, which are equivalent t o the question o f existence o f 1-periodi c solutions of first order vector ordinary differentia l equations . W e then use the basic proper- ties of the degree theor y introduced in Chapter II, including its dependence upo n L, t o ob- tain in an almost trivial way dualit y theorem s in the sense o f Krasnoserskii, i.e. relations be- tween the topological degree s of those various equivalent fixed point problems. B y inter- preting the classica l Poincare's method for the stud y of periodi c solutions as a special case of the Lyapunov-Schmidt procedure , we also relate the degrees mentioned abov e to the Brouwer degree of the associate d Poincard's mapping. Chapter IV begins with general Leray-Schauder' s and Borsuk's type existenc e theorem s for equations of the for m Lx 4 - Fx = 0 with L and F as in Chapter II. Thos e result s are
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