On Resonant Differential Equations with Unbounded Non-Linearities

Abstract

We present a method to study asymptotically linear degenerate problems with sublinear unbounded non-linearities. The method is based on the uniform convergence to zero of projections of non-linearity increments onto some finite-dimensional spaces. Such convergence was used for the analysis of resonant equations with bounded non-linearities by many authors. The unboundedness of nonlinear terms complicates essentially the analysis of most problems: existence results, approximate methods, systems with parameters, stability, dissipativity, etc. In this paper we present statements on projection convergence for unbounded non-linearities and apply them to various resonant asymptotically linear problems: existence of forced periodic oscillations and unbounded sequences of such oscillations, existence of unbounded solutions, sharp analysis of integral equations with simple degeneration of the linear part (a scalar twopoint boundary value problem is considered as an example), existence of non-trivial cycles for higher order autonomous ordinary differential equations, and Hopf bifurcations at infinity.

On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point

“…The passage to equations with unbounded sublinear operators F is one possible direction of generalization of our results. In this passage, the main problem in applications is to prove the asymptotic homogeneity; if F is a composition operator, then the methods in [8] can be used. One can consider nonlinearities asymptotically homogeneous in a weaker sense with discontinuous asymptotic limit, say, by using the technique in [9].…”

“…Thus Theorems 1 and 2 supplement each other. Note that in Theorem 1 one additionally assumes the asymptotic homogeneity of the operator F 1 (x, λ 0 ) and the validity of the representations (8).…”

О Числе Неограниченных Ветвей Решений В Окрестности Асимптотической Точки Бифуркации

“…Если F 0 (λ 0 ) ∈ E ⊥ (как в теореме 1), то η ≡ 0 и теорема 2 оказывается неприменимой, т. е. теоремы 1 и 2 дополняют друг друга. Отметим, что в теореме 1 дополнительно предполагается асимптотическая однородность оператора F 1 (x, λ 0 ) и справедливость представлений (8).…”

“…Одно из возможных направлений развития сформулированных результатов -переход к уравнениям с неограниченными сублинейными операторами F . При таком переходе основная проблема в приложениях состоит в доказательстве асимптотической однородности; если F -оператор суперпозиции, то можно использовать методы из [8]. Можно рассматривать асимптотически однородные в более слабом смысле нелинейности с разрывным асимптотическим пределом, например, используя технику из […”