Application de la théorie de la transversalité topologique à des problèmes non linéaires pour des équations différentielles ordinaires. (Application of topological transversality theory to nonlinear problems for ordinary differential equations). (French) Zbl 0728.34017

This monograph is in large part taken from the PhD thesis that the author wrote at the University of Montreal under the supervision of A. Granas. This thesis deals with the application of the so-called topological transversality method in the study of second order boundary value problems, both for functions and multifunctions (second order differential inclusions) under Carathéodory type conditions. This work extends previous results by A. Granas, R. B. Guenther and J. W. Lee [Nonlinear boundary value problems for ordinary differential equations. Diss. Math. 244, 128 p. (1985; Zbl 0615.34010)]. The topological tool is the notion of an essential map which can be formulated both in the function and the multivalued compact, convex function cases and is preserved under boundary free zero homotopies. The author refers to a book by J. Dugundji and A. Granas [Fixed point theory. Vol. I. Warszawa (1982; Zbl 0483.47038)] for details about this notion. Here the method is used both in the study of initial value problems (in \({\mathbb{R}}^ n)\) and of second order boundary value problems (for real functions). The author points out the fact that the use of a homotopy type argument in the solution of initial value problems is quite recent [J. W. Lee and D. O’Regan, Ann. Pol. Math. 48, No.3, 247-252 (1988; Zbl 0674.34006)]. It is done here under additional complication of Carathéodory conditions. She considers quite a variety of situations: nonlinearities of the Bernstein type (at most quadratic) as well as of the Bernstein-Nagumo type (which can be higher), sign conditions as well as conditions in terms of sub- and super-solutions. She considers both the problem on a bounded interval, with all possible non homogeneous linear boundary conditions, and the problem on the semi- infinite axis, with zero limit at \(+\infty\). It should be noted however that in the latter case, the issue is rather (in the author’s dissertation) to prove that if a solution which goes to zero at \(+\infty\) exists, then a solution which goes to zero as well as its first and second derivatives at \(+\infty\) exists too. Another remark is that the paper lacks some examples, such as one which would illustrate the advantage of weakening the Bernstein-Nagumo condition. In conclusion, the work covers an appreciable range of problems: all the proofs are detailed as well as the technical results such as, for example, an extended version of the theorem of change of variable under the integral. It should be considered as an updated addition to the above quoted book by Granas, Guenther and Lee.
Reviewer: O.Arino (Pau)


34B15 Nonlinear boundary value problems for ordinary differential equations
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)