{"title":"On the Asymptotic Behavior of Solutions of Third-Order Binomial Differential Equations","authors":"Ya. T. Sultanaev, N. F. Valeev, E. A. Nazirova","doi":"10.1134/s0012266124020095","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper discusses the development of a method for constructing asymptotic formulas as\n<span>\\(x\\to \\infty \\)</span> for the fundamental solution system of two-term\nsingular symmetric differential equations of odd order with coefficients in a broad class of\nfunctions that allow oscillation (with relaxed regularity conditions that do not satisfy the classical\nTitchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation\n<span>\\(({i}/{2})\\bigl [(p(x)y^{\\prime })^{\\prime \\prime }+(p(x)y^{\\prime \\prime })^{\\prime }\\bigr ] +q(x)y =\\lambda y\\)</span>, the asymptotics of solutions in\nthe case of various behavior of the coefficients <span>\\(q(x)\\)</span> and\n<span>\\(h(x)=-1+{1}\\big /{\\sqrt {p(x)}}\\)</span> is studied. New asymptotic\nformulas are obtained for the case in which <span>\\(h(x) \\notin L_1[1,\\infty ) \\)</span>.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"13 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124020095","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The paper discusses the development of a method for constructing asymptotic formulas as
\(x\to \infty \) for the fundamental solution system of two-term
singular symmetric differential equations of odd order with coefficients in a broad class of
functions that allow oscillation (with relaxed regularity conditions that do not satisfy the classical
Titchmarsh–Levitan regularity conditions). Using the example of a third-order binomial equation
\(({i}/{2})\bigl [(p(x)y^{\prime })^{\prime \prime }+(p(x)y^{\prime \prime })^{\prime }\bigr ] +q(x)y =\lambda y\), the asymptotics of solutions in
the case of various behavior of the coefficients \(q(x)\) and
\(h(x)=-1+{1}\big /{\sqrt {p(x)}}\) is studied. New asymptotic
formulas are obtained for the case in which \(h(x) \notin L_1[1,\infty ) \).
Abstract The paper discusses the development of a method for constructing asymptotic formulas as\(x\to \infty \) for the fundamental solution system of two-termsingular symmetric differential equation of odd order with coefficients in a wide class offunctions that allow oscillation (with relaxed regularity conditions that not satisfy the classicalTitchmarsh-Levitan regularity conditions).以三阶二项式方程为例（({i}/{2})\bigl [(p(x)y^{\prime })^{\prime }+(p(x)y^{\prime })^{\prime }\bigr ] +q(x)y =\lambda y\ ）、和（h(x)=-1+{1}\big /\sqrt {p(x)}\）的各种行为情况下的解的渐近性进行了研究。对于 \(h(x) \notin L_1[1,\infty ) 的情况，得到了新的渐近公式。\).
期刊介绍：
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.
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