Abdelhamid Benmezai: Krein-Rutman operators and a variant of Banach contraction principle in ordered Banach spaces, 255-280

Abstract:

Let $E$ be a real Banach space, $K$ be a cone in $E$ and $L$ be a linear positive and compact self mapping defined on $E$. The operator $L$ is said to be a Krein-Rutman operaor if it has a positive characteristic value $% \lambda _{L}$ such that for all $h\in K\smallsetminus \left\{ 0_{E}\right\} ,$ the nonhomogeneous equation $u-\theta Lu=h$ has no positive solution if $% \theta \geq \lambda _{L}$ and a unique positive solution if $\theta \in \left( 0,\lambda _{L}\right)$. M. G. Krein and M. A. Rutman have proved that if $L$ is strongly positive then $L$ is a Krein-Rutman operator with $% \lambda _{L}=1/r(L)$. Here $r(L)$ refers to the spectral radius of $L$.

The main goal of this article is to provide sufficient conditions making of $% L$ a Krein-Rutman operator. The particular case where $E$ is a Hilbert space and $L$ is a self-adjoint operator is examined.

We also present in this article a version of the Banach contraction principle adapted to the case where the cone $K$ is normal and minihedral, making of the Banach space $E$ a Riesz space.

Key Words: Cones, positive operators, Krein-Rutman theory, Banach contraction principle.

2010 Mathematics Subject Classification: Primary 47H07; Secondary 47A10, 34B05.