Gronwall-Bellman-type inequalities for systems
In this paper the Gronwall-Bellman inequality is generalized as follows. If \[ u(x)\leq g(x)+\int^{x}_{0}K(x,tu(t))dt \] then \[ u(x)\leq g(x)+\int^{x}_{0}\exp \{\int^{x}_{t}K(\xi (s,x),s)ds\}K(\xi (t,x),t)g(t)dt \] (this is how the inequality (2.6) of the paper should have been). Here x, s, t are n vectors; u, g are continuous m vectors; \(K(x,t)\) is an \(m\times m\) matrix whose elements are continuous in x and t and monotone in x; and \(\int^{x}_{0}(\quad)dt=\int^{x_ 1}_{0}...\int^{x_ n}_{0}(\quad)dt_ 1...dt_ n.\) By an unfortunate oversight the condition that K be monotone got omitted in the statement of the hypotheses. Also note that the misprinted equation (2.6) is correct for \(K(x,t)\) non decreasing in x.
scientific article; zbMATH DE number 3987600