{"title":"Anisotropic variable exponent Sobolev spaces and applications to elliptic equations with critical growth","authors":"Nabil Chems Eddine,Maria Alessandra Ragusa","doi":"10.1007/s13540-026-00591-w","DOIUrl":"https://doi.org/10.1007/s13540-026-00591-w","url":null,"abstract":"Abstract We establish critical embeddings for anisotropic Sobolev spaces with variable exponents, resolving, under log-Hölder continuity supplemented by finite Sobolev regularity and a quantitative integrability-gap condition, an open problem raised by Fan. We derive a Poincaré-type inequality at the harmonic-mean critical scale, give compactness criteria when the critical exponent is attained on a nontrivial set, and prove a corresponding concentration–compactness principle. These results are then applied to anisotropic Leray–Lions equations with critical growth. Under a global quantitative superlinearity condition, the problem possesses a nontrivial weak solution for a sufficiently large reaction parameter. Under the additional oddness assumptions, for every prescribed integer m , it possesses at least m distinct pairs of nontrivial weak solutions once the parameter exceeds a sufficient threshold depending on m .","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"2 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149210373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fractional calculus operators based on Prabhakar’s Le Roy Functions","authors":"İlkay Onbaşı Elidemir,Mehmet Ali Özarslan","doi":"10.1007/s13540-026-00595-6","DOIUrl":"https://doi.org/10.1007/s13540-026-00595-6","url":null,"abstract":"","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"68 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149208862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francesco Dell’Accio,Filomena Di Tommaso,Ilde Ferrara
{"title":"The univariate multinode Shepard method for the Caputo fractional derivatives: from approximation to the solution of the Bagley–Torvik equation","authors":"Francesco Dell’Accio,Filomena Di Tommaso,Ilde Ferrara","doi":"10.1007/s13540-026-00594-7","DOIUrl":"https://doi.org/10.1007/s13540-026-00594-7","url":null,"abstract":"Abstract In this paper, we propose a numerical approach based on the univariate multinode Shepard operator for the approximation of Caputo fractional derivatives. The fractional integral is evaluated through a Gauss–Jacobi quadrature formula, and the resulting approximation is then employed within a collocation scheme for Bagley–Torvik boundary and initial value problems. Approximation and convergence properties are discussed, together with the influence of the conditioning of the associated collocation systems. Numerical experiments involving different node distributions show high accuracy for both polynomial and non-polynomial solutions. The method is also applied to variable-coefficient and nonsmooth Bagley–Torvik problems, and comparisons with previously published numerical methods confirm its competitive accuracy.","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"8 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149150630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularity of solutions to the fractional Navier–Stokes equations in logarithmic Q spaces covering $$BM O^{-1}$$ and its fractional counterpart","authors":"Meimei Shi,Pengtao Li,Zengjian Lou,Zhichun Zhai","doi":"10.1007/s13540-026-00598-3","DOIUrl":"https://doi.org/10.1007/s13540-026-00598-3","url":null,"abstract":"","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"81 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149150629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral fractional radial Laplacian with Caputo structure on bounded domains","authors":"Margarita Rivero,Juan J. Trujillo","doi":"10.1007/s13540-026-00597-4","DOIUrl":"https://doi.org/10.1007/s13540-026-00597-4","url":null,"abstract":"","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"13 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149150632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximate controllability of linear fractional impulsive evolution equations in Hilbert spaces","authors":"Javad Asadzade,Nazim Mahmudov","doi":"10.1007/s13540-026-00588-5","DOIUrl":"https://doi.org/10.1007/s13540-026-00588-5","url":null,"abstract":"","PeriodicalId":48928,"journal":{"name":"Fractional Calculus and Applied Analysis","volume":"66 3 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149049441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}