Electron. J. Diff. Eqns., Vol. 2003(2003), No. 18, pp. 1-11.

Remarks on semilinear problems with nonlinearities depending on the derivative

Jose Maria Almira & Naira Del Toro

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:
$$\displaylines{ \mathbf{u}''(t)+\mathbf{g}(t,\mathbf{u}'(t))= \overline{\mathbf{f}}+\widetilde{\mathbf{f}}(t), \quad t\in (a,b) \cr \mathbf{u}'(a)=\mathbf{u}'(b)=0 \cr \hbox{or}\quad \mathbf{u}(a)=\mathbf{u}(b),\quad \mathbf{u}'(a)=\mathbf{u}'(b) }$$
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.

Submitted December 5, 2002. Published February 20, 2003.
Math Subject Classifications: 34B15, 34L30.
Key Words: Nonlinear boundary-value problem, Neumann and Periodic problems.

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Jose Maria Almira
Departamento de Matematicas
Universidad de Jaen. E.U.P. Linares
C/ Alfonso X el Sabio, 28.
23700 Linares (Jaen) Spain
e-mail: jmalmira@ujaen.es
Naira Del Toro
Departamento de Matematicas
Universidad de Jaen. E.U.P. Linares
C/ Alfonso X el Sabio, 28.
23700 Linares (Jaen) Spain
e-mail: ndeltoro@ujaen.es

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