Iterated operator inequalities on ordered linear spaces. (English) Zbl 1263.47023

The Gronwall-Bellman inequality asserts that, if the function \(u : \mathbb{R}_+\to \mathbb{B}_+\) is continuous and \[ u(t)\leq b(t)+\int_0^t k(s)u(s)\, ds,\quad t \in\mathbb{R}_+, \] where \(b, k : \mathbb{R}_+\to \mathbb{B}_+\) are continuous functions, then \[ u(t) \leq b(t)+\int_0^t \exp[\int_0^s k(r)dr]b(s)\, ds,\quad t\in \mathbb{R}_+. \] 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.


47A63 Linear operator inequalities
26D10 Inequalities involving derivatives and differential and integral operators
46A40 Ordered topological linear spaces, vector lattices
54E40 Special maps on metric spaces


Zbl 0587.26013
Full Text: Euclid