Singularly perturbed semilinear Neumann problem

In this paper, we investigate the problem of existence and asymptotic behavior of the solutions for the nonlinear boundary value problem ǫy ′′ + ky = f(t, y), t ∈ h a, bi , k > 0, 0 < ǫ << 1


Introduction
We will consider the singularly perturbed Neumann problem ǫy ′′ + ky = f (t, y), t ∈ a, b , k > 0, 0 < ǫ << 1 (1.1) The qualitative behavior of the dynamical systems near a normally hyperbolic manifold of critical points is well known (Theorem on persistence of normally hyperbolic manifold, see [2,3,5,9,12], for reference).However, the framework of the geometric singular perturbation theory is not useful for the non-hyperbolic critical manifolds, i.e. when the characteristic roots of the linearization of (1.1) along a solution u of the reduced problem ku = f (t, u) lie on the imaginary axis.
The main result (Theorem 1) is the existence of a solution y ǫ (t) for ǫ belonging to a non-resonant set and an estimate of the difference between the solution y ǫ (t) and a solution u(t) of the reduced problem.It is accomplished by a construction of a lower and an upper solution for the corresponding boundary value problem.
The conditions (1.5), (1.6) we may write in the form of the system is periodic with period p tendings to 0 for ǫ → 0 + , ǫ ∈ M and using the Floquet theory, then the solution of the linear homogeneous system X ′ = P 1,ǫ (t) P 3,ǫ (t) P 2,ǫ (t) P 4,ǫ (t) X can be written as X hom,ǫ (t) = p ǫ (t)e Θǫt where p ǫ (t) is a periodic function and a matrix Θ ǫ is time independent.This fact is instructive for the numerical description and the computer simulation of the system (1.8), (1.9).
Remark 2. The condition (1.13) is the fundamental assumption for existence of the barrier functions α ǫ , β ǫ for proving Theorem 1.Now let v c,ǫ,i (t) be a solution of Neumann boundary value problem (1.2) for Diff.Eq.
As follows from (1.3), the functions v c,ǫ,i (t) must appear in the region as illustrated in Figure 1 Then the problem (1.1), (1.2) has for ǫ ∈ (0, ǫ 0 ] ∩ M a solution satisfying the inequality on a, b .

Generalization of the assumption (A1)
The assumption of nonnegativity of z ǫ,i (t) in (1.3) and the condition (1.12) may be generalized in the following sense.Denote Let there exist the functions zǫ,i (t) such that where r ǫ,i (t) is from (1.11) and z ǫ,i (t) = r ǫ,i (t) + zǫ,i (t), i = α ǫ , β ǫ .