Remarks on semilinear problems with nonlinearities depending on the derivative

Abstract

In this paper, we continue some work by Canada and Drabek [1] and Mawhin [6] on the range of the Neumann and Periodic boundary value problems:egin{gather*} mathbf{u}''(t)+mathbf{g}(t,mathbf{u}'(t))= overline{mathbf{f}}+widetilde{mathbf{f}}(t), quad tin (a,b) mathbf{u}'(a)=mathbf{u}'(b)=0 ext{or}quad mathbf{u}(a)=mathbf{u}(b),quad mathbf{u}'(a)=mathbf{u}'(b) end{gather*} where $mathbf{g}in C([a,b] imes mathbb{R}^n,mathbb{R}^n)$, $overline{mathbf{f}}in mathbb{R}^n$, and $widetilde{mathbf{f}}$ has mean value zero. For the Neumann problem with $n>1$, we prove that for a fixed $widetilde{mathbf{f}}$ the range can contain an infinity continuum. For the one dimensional case, we study the asymptotic behavior of the range in both problems.

Keywords

• Nonlinear boundary-value problem
• Neumann and Periodic problems.