Two-parameter nonlinear oscillations: the neumann problem

Abstract

Boundary value problems of the form  are considered, where  In our considerations functions f and g are generally nonlinear. We give a description of a solution set of the problem (i), (ii). It consist of all triples () such that (λ,μ,x(t)) nontrivially ′solves the problem(i),(ii) and |x (z)| = α at zero points z of the function x(t) (iii). We show that this solution set is a union of solution surfaces which are centro-affine equivalent. Each solution surface is associated with nontrivial solutions with definite nodal type. Properties of solution surfaces are studied. It is shown, in particular, that solution surface associated with solutions with exactly i zeroes in the interval (a,b) is centro-affne equivalent to a solution surface of the Dirichlet problem (i), x(a) = 0 = x(b), (iii) corresponding to solutions with odd number of zeros 2j − 1 (i ≠ 2j)in the interval (a,b).

Keyword : Nonlinear oscillations, α-spectrum, α-branch, solution surfaces, solution curves, centro-affine equivalence, Neumann boundary value problem, Dirichlet boundary value problem