Open Access Research

Discrete Fučík spectrum - anchoring rather than pasting

Petr Stehlík

Author Affiliations

Department of Mathematics, University of West Bohemia, Univerzitni 22, Pilsen, 30614, Czech Republic

Boundary Value Problems 2013, 2013:67  doi:10.1186/1687-2770-2013-67

Published: 29 March 2013

Abstract

In this short note we study a simple discrete Fučík spectrum. Trying to imitate standard continuous pasting procedures, we derive a more complicated discrete analogue - anchoring. Using this technique, we show that the problem of finding the parametrization of the second discrete Fučík branch is equivalent to solving a transcendent equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M1">View MathML</a>. Based on this equivalence, we state a conjecture that already the second branch has no elementary parametrization, i.e., it cannot be described by a finite number of elementary functions.

MSC: 39A12, 34B15.

Keywords:
Fučík spectrum; nonlinear difference equation; resonance; boundary value problem; elementary functions; second branch