Some general existence principles in the Carathéodory theory of nonlinear differential systems. (English) Zbl 0687.34009

Granas, A.; Guenther, R. B.; Lee, J. W.

J. Math. Pures Appl., IX. Sér. 70, No. 2, 153-196 (1991).

Some general existence principles for a system of nonlinear differential equations of the form \(y^{(k)}=f(x,y,...,y^{(k-1)}\) subject to certain affine or more general nonlinear boundary conditions are established. When f is continuous, solutions are classical solutions, while when f is Carathéodory solutions are in appropriate Sobolev spaces. The principal features of the three part paper are: Existence principles of fixed point type are developed in Part I. Methods based on topological transversality are used. In particular, a convenient nonlinear alternative is employed. Applications to the Cauchy problem, Dirichlet problem, and periodic problem for systems are given in Part II. Certain singular systems are also considered. Part III develops existence principles of coincidence type. The theoretical development, which relies on topological transversality and avoids degree and coincidence degree considerations, is simpler than a degree theoretic approach. Applications to Neumann and periodic problems are given.
When existence principles are established, both the classical and Carathéodory problems are treated simultaneously in a classical setting. This is accomplished by replacing the boundary value problem by an equivalent integral-differential equation and applying the topological methods in the integral equations setting. Despite the fact that Sobolev spaces are not used explicitly, given existence, solutions automatically lie in the expected Sobolev space.

Reviewer: R.B.Guenther

MSC:

34A34

Nonlinear ordinary differential equations and systems

Keywords:

topological transversality; avoids degree; coincidence degree; integral- differential equation