Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control

In this work we provide an existence and location result for the third-order nonlinear differential equation u (t) = f t,u(t),u (t ), u (t) , where f : [a, b] × R3→R is a continuous function, and two types of boundary conditions: u(a) = A, φ u (b), u (b) = 0, u (a) = B, or u(a) = A, ψ u (a), u (a) = 0, u (b) = C, with φ,ψ :R2→R continuous functions, monotonous in the second variable and A,B,C ∈ R.