On transformations \(z(t)=L(t)y\bigl (\alpha (t)\bigr)\) of functional-differential equations (Q2702812)

The author discusses some problems concerning the transformation theory of functional-differential equations. He deals with the delay differential equation NEWLINE\[NEWLINE \begin{aligned} y^{(n+1)}(x)= f\big (x,y(x),\dots ,y^{(n)}(x),y(\xi_1(x)), \dots ,y^{(n)}(\xi_1(x)),\dots & \\ \dots ,y(\xi_m(x)),\dots ,y^{(n)}(\xi_m(x))\big)&, \end{aligned} \tag{1} NEWLINE\]NEWLINE with \(x\in I\subset \mathbb{R}\), \(n,m\in \mathbb{Z}\), \(n,m\geq 1\), \(\xi_1,\dots ,\xi_m\in C^{n}(I)\), \(\xi_j:I\to I\), \(\xi_j\neq \xi_k\) for \(j\neq k\), \(j,k=1,\dots ,m\). As the main result, he derives the general form of (1) which allows stationary transformations NEWLINE\[NEWLINE z(x)=L(x)y\bigl (\varphi (x)\bigr) \tag{2} NEWLINE\]NEWLINE converting equation (1) into itself on the whole definition interval. If \(L\in C^{n+1}(I)\), \(L(x)\neq 0\) on \(I\), \(\varphi \) is a \(C^{n+1}\) diffeomorphism of \(I\) onto \(I\) and (2) is the stationary transformation of (1), then (1) is a linear functional-differential equation. NEWLINENEWLINENEWLINESome related questions concerning the theory of functional equations are discussed. Particularly, a general continuous solution to the auxiliary matrix functional equation is derived.