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Zeitschrift für Analysis und ihre Anwendungen

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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