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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
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