Three positive solutions for higher order m-point boundary value problems ✩

In this paper, we consider the existence of at least three positive solutions for the 2nth order mpoint boundary value problem y(2n)(t ) = f t,y(t),y (t ), . . . , y(2(n−1))(t ) , 0 t 1, y(2i)(0) = 0, y(2i)(1) = m−2 j=1 kij y(2i)(ξj ), 0 i n− 1, where kij > 0 (i = 0, 1, . . . ,n− 1; j = 1, 2, . . .,m − 2), 0 < ξ1 < ξ2 < ··· < ξm−2 < 1. The associated Green’s function for the higher order m-point boundary value problem is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett–Williams fixed point theorem.  2003 Elsevier Inc. All rights reserved.