Solutions of nonlinear equations in cones and positive linear operators
Jeffrey R. L. Webb · Journal of the London Mathematical Society · 2010
We introduce a modification of the concept of a cone invariant linear operator being u0-positive, a concept due to Krasnosel'skiĭ. We show how this definition allows us to prove results closely related to some results for positive operators in a classic text of Krasnosel'skiĭ. We also show how it can be used, in conjunction with the theory of a fixed-point index, to give some short proofs of existence and nonexistence of positive solutions for nonlinear maps. We prove a new fixed-point index result and show that it can be applied in cases where the previous theory is not applicable.