A functional approach to positive solutions of boundary value problems.

Abstract

We apply a well-known fixed point theorem to guarantee the existence of a positive solution and bounds for solutions for second, third, fourth, and nth order families of boundary value problems. We begin by characterizing second order problems having left and right focal boundary conditions. Via an appropriate substitution, associated third, fourth, and nth order problems are resolved. Our main result centers on the nth order equation y(n) + f(y(t)) = 0, t [is an element of] [0, 1], (1) having boundary conditions, y(ri−1)(0) = 0, 1 < i < k, (2) y(sj−1)(1) = 0, 1 < j < n − k, (3) where {s1, · · · , sn−k} and {r1, · · · , rk} form a partition of {1, · · · , n} such that r1 < · · · < rk, s1 < · · · < sn−k, and {rk−1 · · · rk} [is not equal to] {n − 1, n} and {sn−k−1, sn−k} [is not equal to] {n − 1, n}. Under these assumptions we show that the differential equation (1) with boundary conditions (2) and (3) has a positive solution for all n [is greater than or equal to] 2.