NONE 20260204072410742 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Alik M. Najafov Higher Mathematics Department Azerbaijan University of Architecture and Construction Ministry of Science and Education Republic of Azerbaijan Institute of Mathematics and Mechanics Ayna Sultanova St., Baku AZ1073 Baku, AZERBAIJAN Ahmet Eroglu Department of Mathematics Nigde Omer Halisdemir University Niğde Merkez, 51200 Niğde Merkez Nigde, TURKIYE In this paper, Sobolev spaces with derivatives of variable order are studied, i.e., first, the norm of this space is determined, after which, using the integral representation, the differential and differential-difference properties of functions from the constructed space are studied.The obtained embedding theorems can be applied to establish of Hòlder regularity of fractional partial differential equations. 12 31 2025 12 31 2025 743 73 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2025.24.73 https://wseas.com/journals/mathematics/2025/b485106-2272.pdf Almeida A., Hasto P., Besov spaces with variables smoothness and integrability, J. Funct. Anal., 258, 2010, 1628-1655. https: //doi.org/10.1016/j.jfa.2009.09.012. (text in Russian) Besov O.V., Embeddings of spaces of differentiable functions of variable smoothness, in: Issled. Po Teor. Differ. Funkts. Mnogikh Perem. I ee Prilozh. 17, Tr. Mat. Inst. Steklova , 214, 1997, 25-58; translation in Proc. Steklov Inst. Math. 214(3), 1996, 19-53. https://www.mathnet.ru/php/archive.pht ml?wshow=paper&jrnid=\tm&paperid=1027& option_lang=eng [Access date: 16/01/20226]. (text in Russian) Besov O.V., Equivalent norming of spaces of functions of variable smoothness Tr. Mat. Inst. Steklova , 243, 2003, Funkts. Prostran., Prilozh. Differ. Uravn., 87-95; translation in Proc. Steklov Inst. Math., 243, 2003, 4, 80-88. https://homepage.mi-ras.r u/~besov/papers/psim080.pdf [Access date: 16/01/20226]. (text in Russian) Besov O.V., Interpolation, embedding, and extension of spaces of functions of variable smoothnessTr. Mat. Inst. Steklova , 248, 2005, Issled. Po Teor. Differ. Funkts. i Differ. Uravn. , 52-63; translation in Proc. Steklov Inst. Math. 248, 2005, 47-58. https://homepage.mi-ras.ru/~besov/pap ers/psim047.pdf [Access date: 16/01/20226]. Bony J.M.,Second microlocalization and propagation of singularities for semi-linear hyperbolic equations Hyperbolic Equations and Related Topics (Katata/Kyoto), Japan, 1986, 11-49, https://doi.org/10.1016/B978-0-1 2-501658-2.50006-3. Diening L., Hasto P., Roudenko S., Function spaces of variable smoothness and integrability, J. Funct. Anal., 256, 2009, 1731-1768. https: //doi.org/10.1016/j.jfa.2009.01.017. Kalyabin G.A., Lizorkin P.I., Spaces of functions of generalized smoothness, Math. Nachr., 133, 1987, 7-32. https: //doi.org/10.1002/mana.19871330102. Leopold H.G., Interpolation of Besov spaces of variable order of differentiation, Arch. Math. (Basel), 53, 1989, 178-187. https://doi.org/10.1007/BF01198569. Leopold H.G., On function spaces of variable order of differentiation, Forum Math., 3, 1991, 1-21.https: //doi.org/10.1515/form.1991.3.1 (text in Russian) Leopold H.G., Schrore E., Trace theorems for Sobolev spaces of variable order of differentiation, Math. Nachr., 179, 1996, 223-245. https://archive.mpim-bonn.mpg.de\/id /eprint/3601/1/preprint_1994_104.pdf [Access date: 16/01/20226]. (text in Russian) Najafov A.M., The differential properties of functions from Sobolev-Morrey type spaces fractional order? J. Math. Research, 73, 2015,149-158.https: //pdfs.semanticscholar.org\/7b8b/d4e87 7704298f0bef627668646a7bd235188.pdf [Access date: 16/01/20226]. Xu J., The relation between variable Bessel potential spaces and Triebel-Lizorkin spacesIntegral Transforms Spec. Funct., 19, 2008, 599-605. https: //doi.org/10.1080/10652460802030631. Yang D. , Zhuo G., Yuan W.,Triebel-Lizorkin type spaces with variable exponent Banach J. Math. Anal. , 9, 2015, 146-202. http: //www.doi.org/10.15352/bjma/09-4-9. Yang D. , Zhno G., Yuan W., Besov type spaces with variable smoothness and integrability, J. Funct. Anal., 269, 2015, 1840-1898. https: //doi.org/10.1016/j.jfa.2015.05.016. Samko S.G., Kilbas A.A., Marichev O.N.,Fractional integrals and derivatives: theory and applications.Switzerland ; Philadelphia, Pa., USA : Gordon and Breach Science Publishers, 1993, 976p, 2881248640, ISBN: 2-88124-864-0.