BOUNDARY NONLINEARITIES FOR A ONE-DIMENSIONAL p-LAPLACIAN LIKE EQUATION

We study a nonlinear ordinary second order vector equation of pLaplacian type under nonlinear boundary conditions. Applying Leray-Schauder arguments we obtain solutions under appropriate conditions. Moreover, for the scalar case we prove the existence of at least one periodic solution of the problem applying the method of upper and lower solutions. INTRODUCTION We consider a nonlinear one-dimensional problem for a vector function u : [0, T ]→ IR satisfying (1) (φ(u′))′ = f(t, u, u′) in (0, T ) where φ : IR → IR is a homeomorphism. We study problem (1) under the following nonlinear boundary conditions: (NBC) u(0) = h1(u(T ), u′(T )), u′(0) = h2(u(T ), u′(T )) with hi : IR −→ IR continuous. Nonlinear boundary conditions for systems of semilinear ODE’s have been considered by different authors (see e.g. [BL], [E], [KL], [S], [AMP] and [BL2] for more references). Using coincidence degree, Gaines and Mawhin [GM] proved a continuation theorem for a general nonlinear condition which in the scalar second order case reads γ(u) = 0 where γ : C([0, T ], IR) → IR is continuous and takes bounded sets into bounded sets. The particular case γ(u) = h(u(0), u(T ), u′(0), u′(T )), corresponds to the general nonlinear two-point condition. Explicit results are known for the special case h(u) = ( h1(u(0), u′(0)), h2(u(T ), u′(T )) )