{"title":"Initial Value Problem for a Third-Order Nonlinear Integro-Differential Equation of Convolution Type","authors":"S. N. Askhabov","doi":"10.1134/s0012266124040086","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this paper, we obtain two-sided a priori estimates for the solution of a homogeneous\nthird-order Volterra integro-differential equation with a power-law nonlinearity and difference\nkernel. It is shown that the lower a priori estimate, which plays the role of a weight function when\nconstructing a metric in the cone of the space of continuous functions, is sharp. Using these\nestimates, by the weighted metric method (an analog of A. Bielecki’s method), we prove a global\ntheorem on the existence and uniqueness of a nontrivial solution of the initial value problem for\nthis integro-differential equation in the class of nonnegative continuous functions on the positive\nhalf-line and on the method for finding this solution. It is shown that the solution can be found by\nthe successive approximation method, and an estimate of the rate of convergence of the\napproximations to the exact solution is obtained. Examples are given to illustrate the results\nobtained.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124040086","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we obtain two-sided a priori estimates for the solution of a homogeneous
third-order Volterra integro-differential equation with a power-law nonlinearity and difference
kernel. It is shown that the lower a priori estimate, which plays the role of a weight function when
constructing a metric in the cone of the space of continuous functions, is sharp. Using these
estimates, by the weighted metric method (an analog of A. Bielecki’s method), we prove a global
theorem on the existence and uniqueness of a nontrivial solution of the initial value problem for
this integro-differential equation in the class of nonnegative continuous functions on the positive
half-line and on the method for finding this solution. It is shown that the solution can be found by
the successive approximation method, and an estimate of the rate of convergence of the
approximations to the exact solution is obtained. Examples are given to illustrate the results
obtained.
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