{"title":"A Multiparameter Family of Solutions to the Volterra Linear Integral Equation of the First Kind","authors":"I. V. Sapronov","doi":"10.3103/s1066369x24700348","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We study the Volterra integral equation of the first kind with an integral operator of order <span>\\(n\\)</span>, a singularity, and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms in the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows constructing a particular and general solutions to the integro-differential equations in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multiparameter family of solutions is constructed for the original integral equation.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s1066369x24700348","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study the Volterra integral equation of the first kind with an integral operator of order \(n\), a singularity, and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms in the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows constructing a particular and general solutions to the integro-differential equations in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multiparameter family of solutions is constructed for the original integral equation.
Abstract We study the Volterra integral equation of the first kind with an integral operator of order \(n\), a singularity, and a sufficiently smooth kernel in a certain Banach space with weight.它简化为左侧有两个项的微分方程。第一个项对应于一个方程，为其构建了一个明确的多参数解族。对于第二个项，我们得到了一个方程，其算子在任意巴拿赫空间中的规范在零附近是任意小的。对积分算子进行这种拆分，就能在相应的巴拿赫空间中以收敛级数的形式构建特定和一般的微分方程解。因此，在给定积分算子对应的算子铅笔受到一定限制的情况下，可以为原始积分方程构建一个多参数解族。
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