The method of lower and upper solutions for n th-order periodic boundary value problems. (1994) - Cabada, Alberto
Abstract : In this paper we develop the monotone method in the presence of lower and upper solutions for the problem u ( n ) ( t ) = f ( t, u ( t ) ) ; u ( i ) ( a ) − u ( i ) ( b ) = λ i ∈ ℝ, i = 0, …, n − 1 where f is a Carathéodory function. We obtain sufficient conditions for f to guarantee the existence and approximation of solutions between a lower solution α and an upper solution β for n ≥ 3 with either α ≤ β or α ≥ β . For this, we study some maximum principles for the operator L u ≡ u ( n ) + M u . Furthermore, we obtain a generalization of the method of mixed monotonicity considering f and u as vectorial functions.