On the application of linear methods to polynomial approximation of solutions of ordinary differential equations and Hammerstein integral equations

Starting from known linear polynomial operators $U_n(\psi;x)$ which generate good approximations to continuous functions $\psi(x)$, the author proposes a method which for a given right-hand side of the equation
\begin{equation} y'=f(x,y) \tag{1} \end{equation}
and given initial conditions enables us to construct polynomials $y_n(x)=y_n(U_n;f;x)$ approximating to the unknown solution of the equation (1) with essentially the same precision as these operators $U_n$ would yield if the solution were given. More precisely, it is shown in this paper that $|y(x)-y_n(U_n;f;x)|\leqslant(1+\alpha_n)\cdot C\|y(x)-U_n(y;x)\|$, $C=\operatorname{const}$, $\alpha_n\downarrow0$, and effective upper bounds are placed on the quantities $C$ and $\alpha_n$. The same procedure is used also for the polynomial approximation of the solutions of $k$-th order equations with $k\geqslant2$, systems of equations, Hamrnerstein integral equations and other integral equations.