Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 18, pp. 1-11.

Title: Remarks on semilinear problems with nonlinearities depending on
       the derivative

Authors: Jose Maria Almira (Univ. de Jaen. E.U.P. Linares, Spain)
 Naira Del Toro (Univ. de Jaen. E.U.P. Linares, Spain)

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:
 \begin{gather*}
 \mathbf{u}''(t)+\mathbf{g}(t,\mathbf{u}'(t))=
 \overline{\mathbf{f}}+\widetilde{\mathbf{f}}(t), \quad t\in (a,b) \\
 \mathbf{u}'(a)=\mathbf{u}'(b)=0 \\
 \text{or}\quad \mathbf{u}(a)=\mathbf{u}(b),\quad \mathbf{u}'(a)=\mathbf{u}'(b)
 \end{gather*}
 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 $n>1$, 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.