Study of a generalized boundary value problem for an infinite order

A. L. Dzhabrailov

Taurida Journal of Computer Science Theory and Mathematics, no. 3 (2024), pp. 44-60

Résumé

The article studies various aspects of the existence of solutions to a linear differential equation of infinite order with constant coefficients. It is noted that earlier researchers did not obtain any significant results even for a linear differential equation of infinite order with constant coefficients devoted to the study of boundary value problems. This is primarily due to the fact that there is no more or less universal method for reducing a differential equation of infinite order to an infinite system of differential equations, the theory of which was well developed in [3, 5]. The work is devoted to the consideration of a general differential equation of infinite order \begin{equation*} \sum_{j=0}^{\infty }a_{j} (x)y^{\left(j\right)} =f\left(x,y,y',\ldots,y^{\left(\nu -1\right)} ,\ldots\right) \end{equation*} boundary value problem with multipoint functional conditions \begin{equation*} y^{\left(k_{i} -1\right)} \left(x_{i,k_{i} } \right)=\Phi _{i,k_{i} } \left(y,y',\ldots,y^{\left(\nu -1\right)} ,\ldots\right), \end{equation*} where ${x_{i,k_{i}} \in [a,b],}$ $k_{i} =\overline{1,n_{i} },$ $i=\overline{1,m},$ $m\in \{ 1,2,\ldots,n\},$ $n$ — finite natural number, $n_{i} \in \{ 0,1,\ldots,n\} $, and $n_{1} +n_{2} +\ldots+n_{m} =n$; \begin{equation*} y^{\left(i-1\right)} \left(x_{i} \right)=\Phi _{i} \left(y,y',\ldots,y^{\left(\nu -1\right)},\ldots\right),\, \, \, i=n+1,\, \, n+2,\ldots \end{equation*} That is, the problem of reducing one generalized multipoint-functional boundary value problem for a nonlinear differential equation of infinite order to a boundary value problem for an infinite system of differential equations is considered by using theories of infinite definitions and solvability of infinite systems of algebraic equations, substantiated in the works of Koch and Poincaré [4]. The author uses the method of linear mappings, which establishes a connection between the space of infinitely differentiable functions $C^{\left(\infty\right)}\left(a, b\right)$ and the space of infinite-dimensional continuous vector functions $C_{\infty}\left(a, b\right)$ with continuous derivatives on the segment $[a, b]$ with using some given non-singular matrix $A(x)$ with continuously differentiable elements. The solution to the problem posed and its derivatives up to infinite order are sought in the form of some functional series compiled from the matrix $A(x)$ and elements of the space $C_{\infty}\left(a, b\right)$. Theorems on the existence and uniqueness of solutions are proved using the results of A. Poincaré related to the convergence of certain series to ensure the solvability of the corresponding infinite algebraic systems in the space $l_1$ [8]. When using Koch conditions for this purpose [3, 9], which provide solutions to such systems in $l_2$ order to construct a system of integral-functional equations, after substitution, the proof of the corresponding theorem is given in space $C_{\infty}\left(a, b\right)$ with the norm $||\bar{y}||=\mathop{\max }\limits_{a\le x\le b}||\bar{y}||_{l_2}=\mathop{\max }\limits_{a\le x\le b} \left(\sum _{i=1}^{\infty }|y_{i} (x)|^2\right)^{1/2}$ and the necessary metric.

Détail

Keywords: differential equation of infinite order, infinite determinants, systems of differential and algebraic equations, multipoint functional conditions, linear mappings, theorems on the existence and uniqueness of a solution