A nonresonance result for strongly nonlinear second order ODE’s. (English) Zbl 1023.34014

The authors consider the boundary value problem \[ -(\phi(u'))' = f(u) + e(t)\text{ in }(0,T),\quad u(0)=u(T)=0, \] where \(\phi\) is an odd increasing homeomorphism of \(\mathbb{R}\) onto itself, \(f\) is continuous and \(e\) is bounded. They show that, if \(f\) satisfies appropriate nonresonance conditions at infinity, then the problem has a solution for any such \(e\). The nonresonance conditions are formulated in terms of pseudoeigenvalues which have been introduced by M. García-Huidobro, R. Manásevich and F. Zanolin [J. Differ. Equations 114, 132-167 (1994; Zbl 0835.34028)]. The proofs employ a Leray-Schauder-type continuation principle due to A. Capietto, J. Mawhin and F. Zanolin [J. Differ. Equations 88, 347-395 (1990; Zbl 0718.34053)]. A variety of different techniques and delicate estimates is used in order to verify the hypotheses of the continuation principle. At the end of the paper, some examples with \(\phi\) and \(f\) satisfying the nonresonance conditions are discussed.


34B15 Nonlinear boundary value problems for ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces