Continuation method for $\alpha$-sublinear mappings

Abstract:

Let $B$ be a real Banach space partially ordered by a closed convex cone $P$ with nonempty interior $\mathring {P}$. We study the continuation method for the monotone operator $A: \mathring {P} \rightarrow \mathring {P}$ which satisfies \begin{eqnarray*} A(tx) \geq t^{\alpha (a,b)} A(x), \end{eqnarray*} for all $x \in \mathring {P}$, $t \in [a, b] \subset (0, 1)$, where $\alpha (a,b) \in (0, 1)$. Thompson’s metric is among the main tools we are using.