Some remarks on semilinear problems at resonance where the nonlinearity depends only on the derivatives. (English) Zbl 0853.34021

Mawhin, Jean

Acta Math. Inform. Univ. Ostrav. 2, No. 1, 61-69 (1994).

The author considers the Neumann and the periodic boundary value problems for the second order nonlinear system \(u'' + g(t,u') = f(t)\), \(t \in [a,b]\), where \(f \in L^1 ([a,b], \mathbb{R}^n)\) and \(g : [a,b] \times \mathbb{R}^n \to \mathbb{R}^n\) is a Carathéodory function such that (1) \(g(t,v)/ |v |\to 0\) uniformly a.e. in \(t \in [a,b]\) whenever \(|v |\to \infty\). In particular, he shows that for each \(\widetilde f \in L^1 ([a,b], \mathbb{R}^n)\) such that \((\int^b_a \widetilde fds)/(b - a) = 0\) there exist \(k \in \mathbb{R}^n\) such that the given problem with \(f = \widetilde f + k\) possesses a family of solutions of the form \(u(t) + c\), \(c \in \mathbb{R}^n\). In the scalar case \(n = 1\) analogous existence results are obtained if \(g\) is continuous ((1) need not be satisfied) and \(f \in L^2 ([a,b], \mathbb{R})\). Some uniqueness results are given, as well.

Reviewer: M.Tvrdý (Praha)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

periodic boundary value problems; second order nonlinear system; uniqueness

References:

[1]	Canada A., Drabek P.: On semilinear problems with nonlinearities depending only on derivatives. Západočeská Univerzita Plzeň, Preprint vědeckých prací 50 (), 1994.
[2]	Fučík S.: Solvability of Nonlinear Equations and Boundary Value Problems. Reidel, Dordrecht, 1980.
[3]	Rouche N., Mawhin J.: Ordinary Differential Equations. Stability and Periodic Solutions. Pitman, Boston, 1980. · Zbl 0433.34001
[4]	Zeidler E.: Nonlinear functional Analysis and its Applications, vol. I. Springer, New York, 1986. · Zbl 0583.47050