Nonlinear scalar multipoint boundary value problems at resonance

ABSTRACT

In this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the form $y(t+n)+a_{n-1}\left(t\right)y(t+n-1)+\cdots a_0\left(t\right)y\left(t\right)=g(t,y(t+m-1\left)\right)$ subject to $\sum _{j=1}^n{b}_{ij}\left(0\right)y(j-1)+\sum _{j=1}^n{b}_{ij}\left(1\right)y\left(j\right)+\cdots +\sum _{j=1}^n{b}_{ij}\left(N\right)y(j+N-1)=0$ for $i=1,\dots ,n$. The existence of solutions will be proved under a mild growth condition on the nonlinearity, g, which must hold only on a bounded subset of $\{0,\dots ,N\}\times \mathbb{R}$.

KEYWORDS:

• Multipoint boundary value problems
• resonance
• Lyapunov-Schmidt procedure
• Brouwer's degree

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 34B15

Disclosure statement

No potential conflict of interest was reported by the author.