Existence of Monotone Positive Solutions for semipositone right focal boundary value problems with dependence on the derivatives

Abstract

We study the existence of monotone positive solutions for the semipositone right focal boundary value problems ⎧⎪⎪⎨ ⎪⎪⎩ (−1)(n−k)u(n)(t) = λf(t, u(t), u (t), . . . , u(k−1)(t)), t∈ (0, 1), u(i)(0) = 0, 0 ≤ i ≤ k − 1, u(j)(1) = 0, k≤ j ≤ n − 1, where λ > 0 is a parameter, n ≥ 3, 1 < k ≤ n − 1 is fixed, f may change sign for 0 < t < 1 and we allow f is both semipositone and lower unbounded. Without making any monotone type assumption, the existence results of at least one and two monotone positive solutions are obtained by means of the fixed point theorems in cones.