NONLINEAR BOUNDAR Y VALU E PROBLEM S 3 then specialized in important particular cases which extend in various ways previous fixed point theorems like those associated with the names of Schaefer , Altman, Krasnosel'skii, Rothe, Schauder and others. I t is then shown how the assumption s of the general existence theorems can be verified when the solutions of an associated famil y o f equations are a priori bounded. Chapter V is devoted to boundary value problems for second order systems of non- linear differential equations . Althoug h the approach also holds for other boundary condi- tions, we have put the emphasis upon the Picard (or Dirichlet) condition.W e first recall some elementary results for the linear case and then prove a rather general existence theorem containing as a special cas e previous results of Picard, Tippett, Lees and others. Thi s result can be considered a s a nonlinear generalization o f the nonresonance situatio n for the linear case. A second result is then given which allows resonance for some values of f, bu t re- quires a somewhat more complicated treatment . Throughou t th e chapter, the required point- wise a priori bounds are obtained via the use of intermediate L 2 -bounds, an approach which also works in many other situations. In Chapter VI we consider the existence o f periodic solution s for first order systems of ordinar y differentia l equation s whose nonlinear terms satisfy som e one-side d growth re- striction. Th e main results are due to Walter, Gossez and the author and contain a s special cases improvements of a variety of well-known results in the line of Krasnosel'skii's method of guiding functions o r Landesman-Lazer's type shar p sufficient conditions . Th e a priori bounds in this chapter are deduced from simple differential inequalities . Chapter VII shows how the geometric properties of th e vector field defined by sys- tems of firs t orde r retarded functional differentia l equation s can be used to obtain a priori estimates on the solutions o f corresponding boundary value problems. Thos e geometric properties are described in terms of the so-called bound set s introduced by Gaine s for or- dinary differential equations . Thos e bound sets also have links with the guiding functions mentioned above, and applications are given to th e stud y o f periodic solutions of retarded functional differentia l equations , with generalizations of previous results of Krasnosel'skii and Gustafson an d Schmitt. In Chapter VIII we localize th e theory o f degre e of Chapte r II by introducing the concept o f index of a n isolated zero of a mapping. I n many cases , this index can be related to the spectral properties of the linearized mapping, and this requires the study o f the L- characteristic values of Z-compact linear mappings, which reduce t o the classical character- istic values of a linear compact operator when X = Z and L = /. I n particular, the Leray- Schauder formula relating the index of a linear compact perturbation of identity t o its spec- tral properties is extended t o the general class of mappings considered in Chapter II. Mos t of the results of this chapter ar e due to Lalou x and the author. Chapter IX is devoted to an analysis of bifurcation theor y usin g degree arguments and again, following th e approac h due to Lalou x and the author, one develops the genera l theory for I-compact perturbation s of a given linear Fredholm mapping. Th e important specia l case of X = Z and L = I corresponds to the fundamental theor y o f Krasnosel'skii (for th e local situation) and Rabinowitz (for th e global case).W e restrict ourselves for simplicit y t o the case in which the parameter occurs linearly in the equation linearized with respect to the unknown variable.

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