The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (324 KB) | Metadata | Table of Contents | ZAA summary
Volume 21, Issue 3, 2002, pp. 639–668
DOI: 10.4171/ZAA/1100

Published online: 2002-09-30

On Resonant Differential Equations with Unbounded Non-Linearities

A. M. Krasnosel'skii[1], N. A. Kuznetsov[2] and D. Rachinskii[3]

(1) Russian Academy of Sciences, Moscow, Russian Federation
(2) Russian Academy of Sciences, Moscow, Russian Federation
(3) Russian Academy of Sciences, Moscow, Russian Federation

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 two-point 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.

Keywords: Non-linearity sublinear at infinity, degenerate linear parts, periodic solutions, cycles, integral equations, two-point problems, Hopf bifurcation, existence results

Krasnosel'skii A., Kuznetsov N., Rachinskii D.: On Resonant Differential Equations with Unbounded Non-Linearities. Z. Anal. Anwend. 21 (2002), 639-668. doi: 10.4171/ZAA/1100