A couple (σ, τ ) of lower and upper slopes for the resonant second order boundary value problem x ′′ = f(t, x, x′ ), x(0) = 0, x ′ (1) = ∫ 1 0 x ′ (s) dg(s), with g increasing on [0, 1] such that ∫ 1 0 dg = 1, is a couple of functions σ, τ ∈ C 1 ([0, 1]) such that σ(t) ≤ τ (t) for all t ∈ [0, 1], σ ′ (t) ≥ f(t, x, σ(t)), σ(1) ≤ ∫ 1 0 σ(s) dg(s), τ ′ (t) ≤ f(t, x, τ (t)), τ (1) ≥ ∫ 1 0 τ (s) dg(s), in the stripe ∫ t 0 σ(s) ds ≤ x ≤ ∫ t 0 τ (s) ds and t ∈ [0, 1]. It is proved that the existence of such a couple (σ, τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.
We use Brouwer degree to prove existence and multiplicity results for the solutions of some nonlinear second order difference equations with Dirichlet boundary conditions. We obtain in particular upper and lower solutions theorems, Ambrosetti-Prodi type results, and sharp existence conditions for nonlinearities which are bounded from below or from above.
The paper surveys recent results obtained for the existence and multiplicity of radial solutions of Dirichlet problems of the type ∇ · ( ∇v ⁄ √ 1 − |∇v| 2 ) = f(|x|, v) in BR, u = 0 on ∂BR, where BR is the open ball of center 0 and radius R in R n , and f is continuous. Comparison is made with similar results for the Laplacian. Topological and variational methods are used and the case of positive solutions is emphasized. The paper ends with the case of a general domain.