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DCDS

The bifurcation of subharmonics for resonant nonautonomous equations
of the second order is studied. The set of subharmonics is defined by principal
homogeneous parts of the nonlinearities

*provided these parts are not polynomials*. Analogous statements are proved for bifurcations of $p$-periodic orbits of a planar dynamical system. The analysis is based on topological methods and harmonic linearization.
DCDS

This paper is devoted to the computation of the index at
infinity for some asymptotically linear completely
continuous vector fields $x-T(x)$, when the principal
linear part $x-Ax$ is degenerate ($1$ is an eigenvalue
of $A$), and the sublinear part is not asymptotically
homogeneous (in particular do not satisfy Landesman-Lazer
conditions).
In this work we consider only the case of a one-dimensional
degeneration of the linear part, i.e.s $1$ is a simple
eigenvalue of $A$.
For this case we formulate an
abstract theorem and give some general examples for vector
fields of Hammerstein
type and for a two point boundary value problem.

DCDS

We present an approach to study degenerate ODE with periodic nonlinearities;
for resonant higher order nonlinear equations $L(p)x=f(x)+b(t),\;p=d/dt$
with $2\pi$-periodic forcing $b$ and periodic $f$ we give multiplicity results,
in particular, conditions of existence of infinite and unbounded
sets of $2\pi$-periodic solutions.

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