Elements of fractional calculus. Fractional integrals


  • Yuliya S. Mishura Taras Shevchenko National University of Kyiv https://orcid.org/0000-0002-6877-1800
  • Olha M. Hopkalo Taras Shevchenko National University of Kyiv
  • Hanna S. Zhelezniak Taras Shevchenko National University of Kyiv




fractional integral, Hölder property, continuity according to the fractional index


The paper is devoted to the basic properties of fractional integrals. It is a survey of the well-known properties of fractional integrals, however, the authors tried to present the known information about fractional integrals as short and transparently as possible. We introduce fractional integrals on the compact interval and on the semi-axes, consider the famous Hardy-Littlewood theorem and other properties of integrability of fractional integrals. Among other basic properties, we consider Holder continuity and establish to what extent fractional integration increases the smoothness of the integrand. Also, we establish continuity of fractional integrals according to the index of fractional integration, both at strictly positive value and at zero. Then we consider properties of restrictions of fractional integrals from semi-axes on the compact interval. Generalized Minkowsky inequality is applied as one of the important tools. Some examples of calculating fractional integrals are provided.

Pages of the article in the issue: 11 - 19

Language of the article: Ukrainian


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How to Cite

Mishura, Y. S., Hopkalo, O. M., & Zhelezniak, H. S. (2022). Elements of fractional calculus. Fractional integrals. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (1), 11–19. https://doi.org/10.17721/1812-5409.2022/1.1



Algebra, Geometry and Probability Theory