Combined effects in some initial value problems involving Riemann–Liouville fractional derivatives in bounded domains
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We consider the following semilinear fractional initial value problem$$D^{\alpha }u(x)=a_{1}(x)u^{\sigma _{1}}(x)+a_{2}(x)u^{\sigma _{2}},\quad x\in (0,1) \quad and \quad \lim_{x\longrightarrow0^{+}}x^{1-\alpha }u(x)=0,$$where \({0 < \alpha < 1}\), \({\sigma _{1},\sigma _{2}\in (-1,1)}\) and \({a_{1},a_{2}}\) are positive measurable functions on \({(0,1]}\) satisfying appropriate assumptions related to Karamata regular variation theory. We establish the existence and the uniqueness of a positive solution in the space of weighted continuous functions. We also give the boundary behavior of such solution, where appear the combined effects of singular and sublinear terms in the nonlinearity.KeywordsFractional differential equationKaramata classpositive solutionasymptotic behaviorschauder fixed point TheoremMathematics Subject Classification26A3331B2534A1234B18References1.Bliedtner J., Hansen W.: Potential Theory. An Analytic and Probabilistic Approach to Balayage. 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Anal. 71, 2087–2093 (2009)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsSonia Ben Makhlouf1Majda Chaieb1Malek Zribi1Email author1.Université Tunis elManar, Faculté des sciences de Tunis, Département de MathémathiquesTunisTunisia About this article CrossMark Print ISSN 1660-5446 Online ISSN 1660-5454 Publisher Name Springer International Publishing About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s00009-016-0797-2_Combined effects in some initial v", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s00009-016-0797-2_Combined effects in some initial v", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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