Remarks on semilinear problems with nonlinearities depending on the derivative

Almira, Jose María ; Del Toro, Naira

Electronic Journal of Differential Equations, Tome 2003 (2003).

Résumé

Summary: 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: $$\displaylines{ \mathbf{u}''(t)+\mathbf{g}(t,\mathbf{u}'(t))= \overline{\mathbf{f}}+\widetilde{\mathbf{f}}(t), \quad t\in (a,b) \cr \mathbf{u}'(a)=\mathbf{u}'(b)=0 \cr \hbox{or}\quad \mathbf{u}(a)=\mathbf{u}(b),\quad \mathbf{u}'(a)=\mathbf{u}'(b) }$$ where $\mathbf{g}\in C([a,b]\times \mathbb{R}^{n},\mathbb{R}^n), \overline{\mathbf{f}}\in \mathbb{R}^n$, and $\widetilde{\mathbf{f}}$ has mean value zero. For the Neumann problem with $n greater than 1$, 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.

Détail

Classification : 34B15, 34L30
Keywords: nonlinear boundary-value problem, Neumann and periodic problems