{"id":6027,"date":"2026-01-21T11:47:27","date_gmt":"2026-01-21T11:47:27","guid":{"rendered":"https:\/\/izvestiyakbncran.ru\/?page_id=6027"},"modified":"2026-06-02T10:22:58","modified_gmt":"2026-06-02T09:22:58","slug":"27-6-4-en","status":"publish","type":"page","link":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/","title":{"rendered":"27.6.4 En"},"content":{"rendered":"\n

On uniqueness of solution to functional-integral equation of fractional order with involution<\/strong><\/h1>\n\n\n\n

L.M. Eneeva<\/strong><\/p>\n\n\n\n

\n

<\/p>\n\n\n\n

\n
\n
PDF<\/a><\/div>\n<\/div>\n\n\n\n
<\/div>\n\n\n\n
\n
JATS XML<\/a><\/div>\n<\/div>\n<\/div>\n\n\n\n

<\/p>\n<\/div>\n\n\n\n

\n\n\n\n

Abstract<\/strong><\/strong>. <\/em>The paper studies a functional integral equation with a fractional integration operator and an involution operator, which arise when solving boundary value problems for differential equations that contain a composition of left- and right-sided fractional derivatives. These equations underlie mathematical models of various physical and geophysical processes, such as describing dissipative oscillatory systems.<\/p>\n\n\n\n

Aim<\/strong>. The study aims to investigate a functional integral equation with an operator of fractional integration involving an involution operator in the critical case. Research methods. To solve the problem, we employ methods of the theory of integral equations of the first kind, operator theory and properties of completely monotone functions.
Results.<\/strong> It has been shown that the equation under study can be reduced to the problem of solving an integral equation of the first kind with a positive kernel, in a class of functions that change sign under the action of an operator, and for this class of functions, a theorem on the uniqueness of the solution has been proven.<\/p>\n\n\n\n

<\/p>\n\n\n\n

Keywords<\/strong><\/em><\/strong>:<\/em> functional integral equation, Riemann\u2013Liouville fractional integral, involution, Mittag-Leffler function, positive operator, completely monotone function<\/p>\n\n\n\n

<\/p>\n\n\n\n

For citation<\/strong>.<\/strong> Eneeva L.M. On uniqueness of solution to functional-integral equation of fractional order with involution. News of the Kabardino-Balkarian Scientific Center of RAS<\/em>. 2025. Vol. 27. No. 6. Pp. 39\u201346. DOI:
10.35330\/1991-6639-2025-27-6-39-46<\/p>\n\n\n\n

<\/p>\n\n\n\n

R<\/strong>eferences<\/summary>\n
    \n
  1. Nakhushev A.M. Drobnoye ischisleniye i yego primeneniye<\/em> [Fractional Calculus and its Application]. Moscow: FIZMATLIT, 2003. 272 p. EDN: UGLEPD. (In Russian)<\/li>\n\n\n\n
  2. Eneeva L.M. On the question of solving a mixed boundary value problem for an equation with fractional derivatives with different origins. Reports of the Adyghe (Circassian) International Academy of Sciences<\/em>. 2023. Vol. 23. No. 4. Pp. 62\u201368. DOI: 10.47928\/1726-9946-2023-23-4-62-68. (In Russian)<\/li>\n\n\n\n
  3. Rekhviashvili S.Sh. Lagrange formalism with fractional derivative in problems of mechanics. Technical Physics Letters. <\/em>2004. Vol. 30. No. 2. Pp. 33\u201337. EDN: RDBIHN. (In Russian)<\/li>\n\n\n\n
  4. Rekhviashvili S.Sh. Fractional derivative physical interpretation. Nonlinear World. <\/em>2007. Vol. 5. No. 4. Pp. 194\u2013197. EDN: IAWYWN. (In Russian)<\/li>\n\n\n\n
  5. Eneeva L.M. Cauchy problem for fractional order equation with involution. Vestnik KRAUNC. Fiz.-mat. nauki<\/em>. 2024. Vol. 48. No. 3. Pp. 43\u201355. DOI: 10.26117\/2079-6641-2024-48-3-43-55. EDN: RHKXQA. (In Russian)<\/li>\n\n\n\n
  6. Eneeva L.M. Initial value problem for a fractional order equation with the Gerasimov\u2013Caputo derivative with involution. News of the Kabardino-Balkarian Scientific Center of RAS. Vol. 26. No. 6. Pp. 19\u201325. DOI: 10.35330\/1991-6639-2024-26-6-19-25. EDN: BOUNKR. (In Russian)<\/li>\n\n\n\n
  7. Eneeva L.M. Fractional integral equation with involution. Vestnik KRAUNC. Fiz.-mat. nauki. Vol. 52. No. 3. Pp. 63\u201374. EDN: EOPADJ. DOI: 10.26117\/2079-6641-2025-52-3-63-74. (In Russian)<\/li>\n\n\n\n
  8. Dzhrbashyan M.M., Bagiyan R.A. On integral representations and measures associated with Mittag-Leffler type functions. Izvestiya Academii nauk Armynskoi SSR. Matematika. 1975. Vol. 10. No. 6. Pp. 483\u2013508. (In Russian)<\/li>\n\n\n\n
  9. Dzhrbashyan M.M., Bagiyan R.A. On integral representations and measures associated with functions of Mittag-Leffler type. Doklady Akademii nauk SSSR. 1975. Vol. 223. No. 6.
    Pp. 1297\u20131300. (In Russian)<\/li>\n\n\n\n
  10. Miller K.S., Samko S.G. Completely monotonic functions. Integral Transforms and Special Functions. 2001. Vol. 12. No 4. Pp. 389\u2013402. DOI: 10.1080\/10652460108819360<\/li>\n\n\n\n
  11. Bitsadze A.V. Integral equations of first kind. Singapore: World Scientific, 1995. 265 p. ISBN10: 9810222637<\/li>\n<\/ol>\n<\/details>\n\n\n\n
    Information about the author<\/strong>s<\/summary>\n
    \n

    Liana M. Eneeva<\/strong>, Candidate of Physics and Mathematics, Senior Researcher, Department of Mathematical Modeling of Geophysical Processes, Institute of Applied Mathematics and Automation \u2013 branch of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences;
    89 A, Shortanov street, Nalchik, 360000, Russia;
    eneeva72@list.ru, ORCID: https:\/\/orcid.org\/0000-0003-2530-5022, SPIN-code: 3403-8412<\/p>\n\n\n\n

    <\/p>\n\n\n\n

    <\/p>\n\n\n\n

    <\/p>\n<\/div>\n<\/details>\n\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    On uniqueness of solution to functional-integral equation of fractional order with involution L.M. Eneeva Abstract. The paper studies a functional integral equation with a fractional integration operator and an involution operator, which arise when solving boundary value problems for differential equations that contain a composition of left- and right-sided fractional derivatives. These equations underlie mathematical […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"wp-custom-template-home","meta":{"footnotes":""},"class_list":["post-6027","page","type-page","status-publish","hentry"],"yoast_head":"\n27.6.4 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/\" \/>\n<meta property=\"og:locale\" content=\"ru_RU\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"27.6.4 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb\" \/>\n<meta property=\"og:description\" content=\"On uniqueness of solution to functional-integral equation of fractional order with involution L.M. Eneeva Abstract. The paper studies a functional integral equation with a fractional integration operator and an involution operator, which arise when solving boundary value problems for differential equations that contain a composition of left- and right-sided fractional derivatives. These equations underlie mathematical […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/\" \/>\n<meta property=\"og:site_name\" content=\"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb\" \/>\n<meta property=\"article:modified_time\" content=\"2026-06-02T09:22:58+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u041f\u0440\u0438\u043c\u0435\u0440\u043d\u043e\u0435 \u0432\u0440\u0435\u043c\u044f \u0434\u043b\u044f \u0447\u0442\u0435\u043d\u0438\u044f\" \/>\n\t<meta name=\"twitter:data1\" content=\"4 \u043c\u0438\u043d\u0443\u0442\u044b\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/index.php\\\/en\\\/27-6-4-en\\\/\",\"url\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/index.php\\\/en\\\/27-6-4-en\\\/\",\"name\":\"27.6.4 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/#website\"},\"datePublished\":\"2026-01-21T11:47:27+00:00\",\"dateModified\":\"2026-06-02T09:22:58+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/index.php\\\/en\\\/27-6-4-en\\\/#breadcrumb\"},\"inLanguage\":\"ru-RU\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/izvestiyakbncran.ru\\\/index.php\\\/en\\\/27-6-4-en\\\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/index.php\\\/en\\\/27-6-4-en\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"\u0413\u043b\u0430\u0432\u043d\u0430\u044f \u0441\u0442\u0440\u0430\u043d\u0438\u0446\u0430\",\"item\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"27.6.4 En\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/#website\",\"url\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/\",\"name\":\"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb\",\"description\":\"\u041d\u0430\u0443\u0447\u043d\u044b\u0439 \u0436\u0443\u0440\u043d\u0430\u043b\",\"publisher\":{\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"ru-RU\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/#organization\",\"name\":\"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb\",\"url\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"ru-RU\",\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/wp-content\\\/uploads\\\/2025\\\/07\\\/oblozhka-zhurnala-na-angl-scaled.jpg\",\"contentUrl\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/wp-content\\\/uploads\\\/2025\\\/07\\\/oblozhka-zhurnala-na-angl-scaled.jpg\",\"width\":1828,\"height\":2560,\"caption\":\"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb\"},\"image\":{\"@id\":\"https:\\\/\\\/izvestiyakbncran.ru\\\/#\\\/schema\\\/logo\\\/image\\\/\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"27.6.4 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/","og_locale":"ru_RU","og_type":"article","og_title":"27.6.4 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb","og_description":"On uniqueness of solution to functional-integral equation of fractional order with involution L.M. Eneeva Abstract. The paper studies a functional integral equation with a fractional integration operator and an involution operator, which arise when solving boundary value problems for differential equations that contain a composition of left- and right-sided fractional derivatives. These equations underlie mathematical […]","og_url":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/","og_site_name":"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb","article_modified_time":"2026-06-02T09:22:58+00:00","twitter_card":"summary_large_image","twitter_misc":{"\u041f\u0440\u0438\u043c\u0435\u0440\u043d\u043e\u0435 \u0432\u0440\u0435\u043c\u044f \u0434\u043b\u044f \u0447\u0442\u0435\u043d\u0438\u044f":"4 \u043c\u0438\u043d\u0443\u0442\u044b"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/","url":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/","name":"27.6.4 En - \u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb","isPartOf":{"@id":"https:\/\/izvestiyakbncran.ru\/#website"},"datePublished":"2026-01-21T11:47:27+00:00","dateModified":"2026-06-02T09:22:58+00:00","breadcrumb":{"@id":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/#breadcrumb"},"inLanguage":"ru-RU","potentialAction":[{"@type":"ReadAction","target":["https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/izvestiyakbncran.ru\/index.php\/en\/27-6-4-en\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"\u0413\u043b\u0430\u0432\u043d\u0430\u044f \u0441\u0442\u0440\u0430\u043d\u0438\u0446\u0430","item":"https:\/\/izvestiyakbncran.ru\/"},{"@type":"ListItem","position":2,"name":"27.6.4 En"}]},{"@type":"WebSite","@id":"https:\/\/izvestiyakbncran.ru\/#website","url":"https:\/\/izvestiyakbncran.ru\/","name":"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb","description":"\u041d\u0430\u0443\u0447\u043d\u044b\u0439 \u0436\u0443\u0440\u043d\u0430\u043b","publisher":{"@id":"https:\/\/izvestiyakbncran.ru\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/izvestiyakbncran.ru\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"ru-RU"},{"@type":"Organization","@id":"https:\/\/izvestiyakbncran.ru\/#organization","name":"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb","url":"https:\/\/izvestiyakbncran.ru\/","logo":{"@type":"ImageObject","inLanguage":"ru-RU","@id":"https:\/\/izvestiyakbncran.ru\/#\/schema\/logo\/image\/","url":"https:\/\/izvestiyakbncran.ru\/wp-content\/uploads\/2025\/07\/oblozhka-zhurnala-na-angl-scaled.jpg","contentUrl":"https:\/\/izvestiyakbncran.ru\/wp-content\/uploads\/2025\/07\/oblozhka-zhurnala-na-angl-scaled.jpg","width":1828,"height":2560,"caption":"\u0418\u0417\u0412\u0415\u0421\u0422\u0418\u042f \u041a\u0410\u0411\u0410\u0420\u0414\u0418\u041d\u041e-\u0411\u0410\u041b\u041a\u0410\u0420\u0421\u041a\u041e\u0413\u041e \u041d\u0410\u0423\u0427\u041d\u041e\u0413\u041e \u0426\u0415\u041d\u0422\u0420\u0410 \u0420\u0410\u041d\u00bb"},"image":{"@id":"https:\/\/izvestiyakbncran.ru\/#\/schema\/logo\/image\/"}}]}},"_links":{"self":[{"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/pages\/6027","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/comments?post=6027"}],"version-history":[{"count":12,"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/pages\/6027\/revisions"}],"predecessor-version":[{"id":12818,"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/pages\/6027\/revisions\/12818"}],"wp:attachment":[{"href":"https:\/\/izvestiyakbncran.ru\/index.php\/wp-json\/wp\/v2\/media?parent=6027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}