A New Approach to the Hyers–Ulam–Rassias Stability of Differential Equations

Based on the lower and upper solutions method, we propose a new approach to the Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order ordinary differential equations u′=f(t,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u'=f(t,u)$$\end{document}, in the lack of Lipschitz continuity assumption. Apart from extending and improving the literature by dropping some assumptions, our result provides an estimate for the difference between the solutions of the exact and perturbed models better than from that one obtained by fixed point approach which is commonly used method in this topic. Some examples are also given to illustrate the improvement. Particularly, we examine our approach to the Hyers–Ulam stability problem of second-order elliptic differential equations -Δu=g(x,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u =g(x,u)$$\end{document} with homogeneous Dirichlet boundary condition which arise in different applications such as population dynamics and population genetics. This investigation is not only of interest in its own right, but also it supports the usability of our approach to other types of boundary value problems such as p(x)-Laplacian Dirichlet problems, Kirchhoff type problems, fractional differential equations and etc.