In a number of previous works it was found that for binomial functional equations of the form<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7620_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="278" /> </InlineMediaObject> <EquationSource Format="TEX">$\begin{array}{cc}a\left(x\right)u\left(\alpha \left(x\right)\right)-\lambda u(x) = v(x), & x\in X,\end{array}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>a</mi> <mfenced close=")" open="("> <mi>x</mi> </mfenced> <mi>u</mi> <mfenced close=")" open="("> <mi>α</mi> <mfenced close=")" open="("> <mi>x</mi> </mfenced> </mfenced> <mo>-</mo> <mi>λ</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </InlineEquation>where α : X → X is an invertible map of the set X into itself, a situation typical for differential equations is possible: the equation is solvable for any right-hand side and there is no uniqueness of the solution. As in the case of differential equations, the question arises of formulating well-posed boundary-value problems, i.e., of specifying additional conditions under which the solution exists and is unique. In this paper, we discuss the question of what kind of additional conditions lead to well-posed boundary-value problems for the equations under consideration.

On the Formulation of Boundary-Value Problems for Binomial Functional Equations

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发表日期：

2025年1月1日

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A. B. Antonevich,
D. I. Kravtsov

摘要

In a number of previous works it was found that for binomial functional equations of the form

$\begin{array}{cc}a\left(x\right)u\left(\alpha \left(x\right)\right)-\lambda u(x) = v(x), & x\in X,\end{array}$ $\begin{matrix} a (x) u (α, (x)) - λ u (x) = v (x), & x \in X, \end{matrix}$

where α : X → X is an invertible map of the set X into itself, a situation typical for differential equations is possible: the equation is solvable for any right-hand side and there is no uniqueness of the solution. As in the case of differential equations, the question arises of formulating well-posed boundary-value problems, i.e., of specifying additional conditions under which the solution exists and is unique. In this paper, we discuss the question of what kind of additional conditions lead to well-posed boundary-value problems for the equations under consideration.