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

Abstract

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 ∈ W 1, 1([ a, b]). We consider the nonlinear boundary conditions L( u( a), u( b)) = 0, with L ∈ C( R 2, R ) nondecreasing in x or nonincreasing in y, and the linear boundary conditions a 0 u(0) − b 0 u( T) = λ 0, with a 0, b 0 and λ 0 ∈ 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.