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2 2003

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On boundary value problem for a nonlocal elliptic equation

Fijałkowski P., Przeradzki B.

Journal of Applied Analysis

2003

Vol. 9, nr 2

201-209

The existence of a positive radial solution to the Dirichlet boundary value problem for the second order elliptic equation [wzór], where U = B(0, R) \ ‾B(0, ρ), with weak assumptions on the nonlinear term f, is proved. The method based on the Krasnosel'skii Fixed Point Theorem enables to find many solutions to the problem. Solutions for the same problem but with U = B(0, R) and with nonlinear term f depending explicitely on |x| are found as well.

Implicit Mapping Theorem for Singular Points

Fijałkowski P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2003

Vol. 51, nr 4

445-455

This paper deals with an implicit mapping defined by zeros of a differentiable mapping in a neighbourhood of a singular point, in which a number of partial deratives with respect to the range space of an implicit function vanishes. The main theorem is applied to the bifurcation theory, particularly to a certain version of the Lyapunov-Schmidt reduction.