Boundary and periodic value problems for systems of differential equations under Bernstein-Nagumo growth condition. (English) Zbl 0831.34021

Frigon, Marlène

Differ. Integral Equ. 8, No. 7, 1789-1804 (1995).

The author establishes several existence results for systems of second- order differential equations (1) \(x'' = f(t,x,x')\) subjected on the interval \(0,1\) to various boundary conditions (e.g. nonhomogeneous Dirichlet, Neumann, Sturm-Liouville conditions, periodic conditions). It is assumed that \(f : 0,1 \times \mathbb{R} 2n \to \mathbb{R} n\) fulfils the Carathéodory conditions and some (Bernstein-type or Nagumo-type) growth conditions. Proofs are obtained via the theory of topological transversality for continuous compact operators in the case of the Bernstein-type conditions and for upper semi-continuous, compact, multivalued operators in the case of the Nagumo-type conditions. On the contrary to the previously published results concerning the subject, Hartman’s condition is replaced by another condition which is automatically satisfied in the scalar case.

Reviewer: M.Tvrdý (Praha)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems

Keywords:

boundary value problem; periodic solution; Bernstein-type growth conditions; Nagumo-type growth conditions; second-order differential equations; topological transversality