On the asymptotic stability of differential equations with ''maxima''
The scalar differential equation with maxima of the following form is considered: (1) \(\dot x(t)=F(t,x(t),\max\{x(s):s\in[g(t),t]\})\), where \(g(t)<t\), \(t\geq t_ 0\) and \(\dot x\) denotes the right derivative of the function \(x\). Effective sufficient as well as necessary and sufficient conditions for the asymptotic stability of the trivial solution are given. The following formulation gives an idea about the obtained results. Assumptions: \(H_ 1\). The function \(F:(t,x,y)\to R\) is continuous for \(t\in[t_ 0,\infty)=I\), \(x^ 2+y^ 2<Q\leq+\infty\) and satisfies \(xF(t,x,x)<0\) for \(x\neq 0\). \(H_ 2\). The function \(g\) is continuous, nondecreasing for \(t\in I\) and satisfies the inequality \(g(t)<t\). \(H_ 3\). For any \(\sigma\in I\) the initial value problem (1), \(x_ \sigma=0\) has a unique trivial solution. \(H_ 4\). There exists a function \(\ell:I\to I\), such that \(\ell(\sigma)\geq\sigma\) and \(L(x)\subset(\sigma,\ell(\sigma)\) for each solution \(x\) of the initial problem (1), \(x_ \sigma=\varphi\in B\). The definition of \(L(x)\) is clear from the following equalities: \(y(t)=\max\{x(s):s\in[g(t),t]\}\), \(D(x)=\{t\in(\sigma,\infty):x(t)<y(t)\}\), \(L(x)=\{t\in D(x):y(t)\neq x(g(t))\}\). The equation (2) \(\dot x(t)=F(t,x(t),\;x(g(t)))\) is associated to equation (1). Provided that conditions \(H_ 1\), \(H_ 2\), \(H_ 3\) and \(H_ 4\) hold and the trivial solution of equation (2) is stable, then the trivial solution of equation (1) is stable.
scalar differential equation with maxima
scientific article; zbMATH DE number 60498