On extremal solutions of operator equations in ordered normed spaces

Abstract

In this paper we shall first derive fixed point results for mappings in ordered normed spaces by using a generalized iteration method. Counterexamples are given to illustrate the necessity of the assumptions given for mappings and orderings. The obtained fixed point results are then applied to derive existence results for extremal solutions of the operator equation

, and to consider the dependence of these solutions on y and λ. Under concavity hypotheses we shall prove also uniqueness results for Eq. (1). In the case when y = 0 we obtain results concerning the spectrum of F. No continuity or linearity hypotheses are imposed on F. Finally, we shall present some ordered function spaces suitable for applications of the obtained results.

Keywords:

• Extremal solutions
• nonlinear operators
• ordered normed spaces
• generalized iterations

AMS:

• 47H07
• 47H12
• 47H15