Higher order spectrum and quasi-linear boundary problems of discrete p-Laplacian

Abstract

In the study of problems of the type $-\mathrm{\Delta }\left(\mathrm{\Delta }u\right(k-1\left)\right)=f(k,u(k\left)\right)+h\left(k\right)$, conditions are generally imposed on the asymptotic behaviour of $f(k,u(k\left)\right)$ with respect to the spectrum of $-\mathrm{\Delta }\mathrm{\Delta }$. To study  the quasi-linear problems $-\mathrm{\Delta }\left(\mathrm{\Delta }u\right(k-1\left)\right)=f(k,u(k),\mathrm{\Delta }u(k-1\left)\right)+h\left(k\right)$ type, we introduce a notion of spectrum for $-\mathrm{\Delta }\mathrm{\Delta }$ that takes into account the dependence of the right-hand side with respect to the $\mathrm{\Delta }u(k-1)$ gradient. The object of this work is to define, study and apply this new notion of spectrum.

Keywords:

• Difference equations
• discrete p-Laplacian
• spectrum of order one
• Fredholm alternative
• non-resonance

2010 Mathematics Subject Classifications:

• 35GXX
• 35PXX
• 39AXX

Disclosure statement

No potential conflict of interest was reported by the author(s).