Iterated operator inequalities on ordered linear spaces (Q421962)

The Gronwall-Bellman inequality asserts that, if the function \(u : \mathbb{R}_+\to \mathbb{B}_+\) is continuous and NEWLINE\[NEWLINEu(t)\leq b(t)+\int_0^t k(s)u(s)\, ds,\quad t \in\mathbb{R}_+,NEWLINE\]NEWLINE where \(b, k : \mathbb{R}_+\to \mathbb{B}_+\) are continuous functions, then NEWLINE\[NEWLINEu(t) \leq b(t)+\int_0^t \exp[\int_0^s k(r)dr]b(s)\, ds,\quad t\in \mathbb{R}_+.NEWLINE\]NEWLINE Some generalizations of this relation have appeared in the literature. In this paper, the author finds an operator version of this inequality for a real linear space \(X\) with a convex cone \(X_+\). He finds an upper bound for \(u\in X_+\) when \(u\leq S(u)\) for some increasing positive map \(S\) on \(X\) with some special conditions, using fixed point theory.