Another approach to the theory of differential inequalities realtive to changes in the initial times
The authors consider an initial value problem for a first-order nonlinear differential equation \(x'(t)=f(t,x(t))\) (1), with \(f\in C(\mathbb{R}^{+}\times \mathbb{R},\mathbb{R})\). First, they prove that the method of upper and lower solutions and the monotone iterative technique can be applied to equation (1) with the initial condition \(x(\tau_{0})=x_{0}\) when the lower solution \(\alpha\) and the upper solution \(\beta\) satisfy the classical inequalities at different points, namely \(\alpha(t_{0})\leq x_{0}\leq\beta(\tau_{0})\), with \(t_{0}<\tau_{0}\) and \(\int_{t_{0}}^{\tau_{0}} f(s,\alpha(s))ds\leq 0\). Afterwards, some other comparison results among solutions to differential equations with different initial data are investigated. In concrete, a kind of variation of parameters formula and stability criteria are obtained.
scientific article; zbMATH DE number 1374884