Index at infinity and bifurcations of twice degenerate vector fields

Alexander Krasnosel'skiĭ

Abstract


We present a method to study twice degenerate at infinity
asymptotically linear vector fields, i.e the fields with
degenerate principal linear parts and next order bounded terms.
The main features of the method
are sharp asymptotic expansions
for projections of nonlinearities onto the kernel of the linear part.
The method includes theorems in abstract Banach spaces,
the expansions which are the main assumptions of these abstract theorems,
and lemmas on the exact form of the expansions for generic
functional nonlinearities with saturation.
The method leads to several new results on solvability and bifurcations
for various classic BVPs.

If the leading terms in the expansions
are of order $0$, then solvability conditions (and conditions for
the index at infinity to be non-zero) coincide with Landesman-Lazer
conditions, traditional for the BVP theory.
If the terms of
order $0$ vanish (the Landesman-Lazer conditions
fail), then it is necessary to determine and to take into account
nonlinearities that are smaller at infinity.
The presented method uses such nonlinearities and makes it possible
to obtain the expansions with the leading terms of arbitrary
possible orders.

The method is applicable if the linear part has simple degeneration,
if the corresponding eigenfunction vanishes,
and if the small nonlinearities decrease at infinity sufficiently fast.
The Dirichlet BVP for a second order ODE is the main model example,
scalar and vector cases being considered separately. Other applications
(the Dirichlet problem for the Laplace PDE and the Neumann
problem for the second order ODE) are given rather schematically.

Keywords


Operator equations; index at infinity; bifurcations at infinity; degenerate vector fields; degree theory; asymptotic expansions; Dirichlet problem

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