Journal of the
Korean Mathematical Society JKMS

The aim of this paper is to obtain nonlinear operators in suitable function spaces whose fixed points coincide with the solutions of the nonlinear boundary value problems $$(\phi(u'))' = f(t,u,u'), l(u,u') = 0,$$
where $l(u, u') = 0$ denotes the Dirichlet, Neumann or periodic boundary conditions on $[0, T], \phi:\mathbb R^n\rightarrow\mathbb R^n$ is a suitable monotone homeomorphism and $f : [0, T] \times\mathbb R^n\times\mathbb R^n\rightarrow\mathbb R^n$ is a Caratheodory function. The special case where $\phi(u)$ is the vector p-Laplacian $|u|^{p-2}u|$ with $p > 1$, is considered, and the applications deal with asymptotically positive homogeneous nonlinearities and the Dirichlet problem for generalized Lienard systems.

Keywords: p-Laplacian, Dirichlet problems, Neumann problems, periodic solutions, degree theory, Lienard systems

MSC numbers: 34B15, 47H05, 47HI0, 47Hll