Existence of solutions for p-Laplacian-like differential equation with multi-point nonlinear Neumann boundary conditions at resonance

Abstract

This work concerns the multi-point nonlinear Neumann boundary-value problem involving a p-Laplacian-like operator $$\displaylines{ (\phi( u'))' = f(t, u, u'),\quad t\in (0,1), \cr u'(0) = u'(\eta), \quad \phi(u'(1)) = \sum_{i=1}^m{\alpha_i \phi(u'(\xi_i))}, }$$ where $\phi:\mathbb{R} \to \mathbb{R}$ is an odd increasing homeomorphism with $\phi(\pm \infty) = \pm \infty$ such that $$ 00. $$ By using an extension of Mawhin's continuation theorem, we establish sufficient conditions for the existence of at least one solution.

Keywords

• Continuation theorem
• p-Laplacian differential equation
• resonance