The monotone method for first-order problems with linear and nonlinear boundary conditions

摘要

In this paper, we develop the monotone method for the first-order problem u′(t) = f(t, u(t)) for a.e.t ∈ [a, b] when f is a Carathéodory function and u ∈ W1, 1([a, b]). We consider the nonlinear boundary conditions L(u(a), u(b)) = 0, with L ∈ C(R2, R) nondecreasing in x or nonincreasing in y, and the linear boundary conditions a0u(0) − b0u(T) = λ0, with a0, b0 and λ0 ∈ R. We prove the existence of solutions and the validity of the monotone method if there exists a lower solution α and an upper solution β, with either α ⩽ β or α ⩾ β. For the linear conditions, we obtain eight new concepts of lower and upper solutions which generalize previous known cases.