Solvability on the axis and the stability of equations of neutral type with decreasing memory
Let \(R^ n\) be the Euclidean space with the norm \(| \cdot |\), \(L_{\infty}=W_{\infty}^{(0)}\) the space of all measurable essential bounded functions \(x: R\to R^ n\) with the norm \(\| x\| =ess \sup | x(t)|\), \(W_{\infty}^{(1)}\) the space of all absolutely continuous bounded functions \(x: R\to R^ n\) which have the derivative in \(L_{\infty}\) with norm \(\| x\| =\| x\|_{L_{\infty}}+\| x'\|_{L_{\infty}}.\) The author considers the Cauchy problem (1) (\({\mathcal L}x)(s)\equiv\) \(x'(s)- A_{\epsilon}x(s)-B_{\epsilon}x'(s)=f(s),\) \(s>t\); (2) \(x(s)=\phi (s)\), \(s<t\), where \(\epsilon\in [0,1]\); \(A_{\epsilon},B_{\epsilon}: L_{\infty}\to L_{\infty}\) the Volterra operators, \(t\in R\), \(f\in L_{\infty}\) and \({\mathcal L}_{\epsilon}: W_{\infty}^{(1)}\to L_{\infty}\). As an example of such an equation he gives the equation \[ x'(t)-[\sum^{\infty}_{i-1}a_ i(t)x(t-\epsilon h_ i)+b_ i(t)x'(t-\epsilon h_ i)]=f(t). \] He states the conditions which guarantee the maintenance of the uniqueness of the solution in the class of bounded functions, of the stability (unstability) by passing from \(\epsilon =0\) to small positive \(\epsilon\).
scientific article; zbMATH DE number 4059905