Generalization of an inequality of Gronwall-Reid
The paper provides an extension of Gronwall-Reid inequality in the form: If \(\mu_{\lambda_ j}\) \((j=1,2,...,n)\) is a set of finite measure on \([t_ 0,+\infty)\) and f is non-negative integrable on finite intervals with respect to the measures \(\mu_{\lambda_ j}\) and \[ f(t)\leq \alpha +\beta \sum^{m}_{j=1}\int^{t}_{t_ 0}f^{q_ j}(s)d\mu_{\lambda_ j}+\epsilon \sum^{n}_{j=m+1}\int^{t}_{t_ 0}f(s)d\mu_{\lambda_ j}, \] where \(\alpha\), \(\beta\) are non-negative constants and \(0\leq q_ j\leq\) and \(\mu_{\lambda_ j}[t_ 0,+\infty)\leq M<\infty\) for all \(\lambda_ j\) and \(0<nM\epsilon <1\), then \[ f(t)\leq \frac{\alpha +\beta cmM}{1-nM(\epsilon +\beta c^{-1})},\quad t\geq t_ 0 \] and \(c\geq \max [1,(\beta nm)/(1-nm\epsilon]\). The result finds effective use in the stability analysis of the system of difference equations \(X_{n+1}=X_ n+E_ nF_ n(X_ n,\Phi_ n)\) where \(\Phi_ n\) are perturbations and the \(E_ n\) are diagonal matrices with positive elements on the diagonal.
scientific article; zbMATH DE number 4120551