On asymptotic behavior of solution of a class of scalar equations with aftereffect
Let \(\mathbb{R}= (-\infty,+\infty)\), \(\mathbb{R}_ += [0,\infty)\), \(\Delta= \{(t,s): t\geq s\geq 0\}\), and let \(L_ 1\) be the space of summable on \(\mathbb{R}_ +\) functions. Let us consider the equation (1) \(\dot x(t)+ ax(t)+ bx(t- r(t))= 0\), \(t\geq\tau\) \((\tau\in \mathbb{R}_ +)\) \((x(\xi)= 0, \xi<\tau)\), where \(a,b\in\mathbb{R}\), the function \(r: \mathbb{R}_ +\to \mathbb{R}_ +\), is measurable and bounded on \(\mathbb{R}_ +\). Further, assume \(r_ 0= \varlimsup_{t\to\infty} r(t)>0\), \(r_ 1= \sup_{t\in\mathbb{R}_ +} r(t)\). By solution of equation (1) we call a function \(x: [\tau,\infty)\to \mathbb{R}\), which is absolutely continuous on every finite segment \([\tau,T]\) and satisfies (1) almost everywhere. If we treat the initial point \(\tau\in\mathbb{R}_ +\) as being arbitrary, then (1) will determine a family of equations; denote their solutions by \(C(t,\tau)\). We shall assume that \(C(\tau,\tau)= 1\). The function \(C: \Delta\to \mathbb{R}\), introduced in such a manner, is usually called the Cauchy function of equation (1). Below, there will be given a number of statements on asymptotic behavior of the Cauchy function of equation (1).
scientific article; zbMATH DE number 640211