Existence of positive solutions for operator equations and applications to semipositone problems

In this paper we study the existence of positive solutions of the following operator equation in a Banach space E:x = G(x, lambda), lambda is an element of (0, infinity), x is an element of P,where G(x, lambda) = lambda KFx+e(0), K: E bar right arrow E is a linear completely continuous operator, F: P - E is a nonlinear continuous , bounded operator, e(0) is an element of E, lambda is a parameter and P is a cone of Banach space E. Since F is not assumed to be positive and e(0) may be a negative element, the operator equation is a so-called semipositone problem. We prove that under certain super-linear conditions on the operator F the operator equation has at least one positive solution for lambda 0 sufficiently small, and that under certain sub-linear conditions on the operator F the operator equation has at least one positive solution for lambda 0 sufficiently large. In addition, we briefly outline an application of our results which simplify previous theorems in the literature.