Almost-normed spaces of functions with given asymptotics, Lagrangian asymptotics, and their application to ordinary differential equations (Q556814)

It is known that any \(m\)-times differentiable function \(x(t)\) has a unique representation of the form \(x(t)=\sum^m_{k=0}a_{k,x}(t)t^k\), where the coefficients \(a_{k,x}(t)\) behave as constants when the function \(x(t)\) is differentiated \(m\)-times. If the coefficients \(a_{k,x}(t)\) have limits \(a_{k,x}\) as \(t\to+\infty\), then the function \(x(t)\) is said to approximate \(L\)-asymptotically to the polynomial \(\sum^m_{k=0}a_{k,x}t^k\) as \(t\to+\infty\). Extending this definition of \(L\)-asymptotic approximation from polynomials to a larger class of functions, the author considers the differential equation \((Lx)(t)=f(t,x(t))\), \(a\leq t<b\), where \(f:[a,b)\times \mathbb R^n\to\mathbb R^n\) is continuous and \(L=\frac d{dt}+A(t)\) with a continuous \(A(t):\mathbb R^n\to\mathbb R^n\). He finds a necessary and sufficient condition on \(f\) and \(A\) under which every solution \(x(t)\) of the above equation \(L\)-asymptotically approximates to a certain solution \(v(t)\) of the homogeneous equation \(Lv=0\), as \(t\to b\).