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

Girg, Petr

Electron. J. Differ. Equ. 2000, Paper No. 63, 28 p. (2000).

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.

Reviewer: Sergiu Aizicovici (Athens/Ohio)

MSC:

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

Keywords:

\(p\)-Laplacian; Leray-Schauder degree; Landesman-Lazer condition