## Neumann and periodic boundary-value problems for quasilinear ordinary differential equations with a nonlinearity in the derivative.(English)Zbl 0974.34018

The authors discuss the existence and nonexistence of solutions to the quasilinear differential equation $\biggl( \varphi\bigl( u'(t)\bigr) \biggr)'+ g\bigl(u'(t)\bigr)+ h\bigl(u(t)\bigr)= f(t),\;0<t<T,\tag{E}$ with either periodic or Neumann boundary conditions. In (E), $$\varphi$$ is an increasing homeomorphism of $$I_1$$ onto $$I_2$$, where $$I_1$$ and $$I_2$$ are real open intervals containig zero, with $$\varphi(0)=0$$, $$g\in C(\mathbb{R})$$, $$f\in C[0, T]$$ with $$\int^T_0f(s)ds=0$$, and $$h$$ is a real continuous, bounded function with limits at $$\pm\infty$$, satisfying $$h(-\infty) <h(+\infty)$$. Some uniqueness results are given.

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 47H14 Perturbations of nonlinear operators
Full Text: