The properties of ''convolution-type'' operators that are invariant with respect to dilation and to their approximation using a unity in weighted mixed Lebesgue spaces are studied in this paper. Integral representations are obtained for the Marchaud-Hadamard and Marchaud-Hadamard type truncated fractional derivatives (by direction and mixed ones). This paper introduces the concept of a mixed difference of a vector fractional order with a multiplicative step and its properties. Some of these properties are proven using the Mellin transform. In this paper, we give the proof of theorems on coincidence of the definition domains of two different forms of fractional differentiation operators of the Marchaud-Hadamard and Grunwald-Letnikov-Hadamard type (by direction and mixed ones) in weighted mixed Lebesgue spaces. In addition, the necessary and sufficient conditions for the existence of a fractional derivative of the Marchaud-Hadamard type by direction ω are obtained.
1. Benedek A. and Panzone R. The space Lp, with mixed norm. Duke Mathematical Journal, Vol. 28, Issue 3, 301–324 (1961). doi:10.1215/s0012-7094-61-02828-9
2. Berdyshev A.S., Turmetov B.Kh. and Kadirkulov B.Zh. Some properties and applications of integro-differential operators of the Marchaud-Hadamard type in the class of harmonic functions. Sibirsk. Math.J., Vol. 53, No. 4, 752–764 (2012) (in Russian). doi.org/10.1134/S0037446612040039
3. Besov O.V., Il'in V.P. and Nikol'skii S.M. Integral Representations of Functions, and Embedding Theorems, Second edition, Fizmatlit "Nauka", Moscow, (1996) (in Russian).
4. Bugrov Ya.S. Fractional difference operators and function classes. Proc. of Math. Inst. of AS USSR, Vol. 172. 60–70 (1985) (in Russian).
5. Butzer P.L., Kilbas A.A. and Trujillo J.J. Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl., Vol.269, Isssue 1, 1–27, (2002). doi.org/10.1016/S0022-247X0200001-5
6. Butzer P.L., Westphal U. An access to fractional differentiation via fractional difference quotients. Lect. Notes Math., Vol. 457, 116–145 (1975).
7. Ma L. and Li C. On Hadamard fractional calculus. World Scientific, Fractals, Vol. 25, No. 03, (2017). doi.org/10.1142/S0218348X17500335
8. Ostalczyk P.W. A note on the Grnwald-Letnikov fractional-order backward-difference. J. Phys. Ser., Vol. 136, 014–036(2009). doi.org/10.1088/0031-8949/2009/T136/014036
9. Rogosin S. and Dubatovskaya M. Letnikov vs. Marchaud: A Survey on Two Prominent Constructions of Fractional Derivatives. J. Mathematics, Vol. 6, Issue 3, 1–15 (2018). doi:10.33906010003
10. Samko S.G., Kilbas A.A. and Marichev O.I. Fractional Integrals and Derivatives. Theory and Applications. London-New-York: Gordon & Breach. Sci. Publ., (Russian edition - Fractional Integrals and Derivatives and some of their Applications, Minsk: Nauka i Tekhnika, 1987.), (1993).
11. Samko S.G. Hypersingular Integrals and Differences of Fractional Order.Proc. of Math. Inst. of AS USSR, Vol. 192, 164–182 (1990) (in Russian). www.mathnet.ru/rus/agreemen
12. Samko S.G. and Yakhshiboev M.U. A Chen-type Modification of Hadamard Fractional Integro-Dfferentiation. Operator Theory: Advances and Applications, Vol. 242, 325–339 (2014).
13. Samko S.G. and Yakhshiboev M.U. Fractional integrodifferentiation on the half-axis invariant with respect to dilatation. Deponierted in VINITI, N 1604-B90 DEP (March 26, 1990)(in Russian).
14. Tuan Vu Kim and Gorenflo R. The Grunwald-Letnikov Difference operator and Regularization of the Weyl Fractional Differentiation. J. Analysis and its Applications, Vol. 13, Issue 3, 537–545 (1994).
15. Wang Z., Huang X. and Zhou J., A numerical method for delayed fractional-order differential equations: Based on G-L definition. Appl. Math. Inf. Sci., Vol. 7, Issue 2 L, 525–529, (2013). doi:10.12785/amis/072L22
16. Westphal U. An approach to fractional powers of operators via fractional differences. Proc. London Math. Soc., Vol. 29, Issue 3, 557–576 (1974). doi.org/10.1112/plms/s3-29.3.557
17. Wu Y., Yao K. and Zhang X. The Hadamard fractional calculus of a fractal function. World Scientific, Fractals, Vol. 26, Issue 03 (2018). doi.org/10.1142/s0218348x18500251
18. Yakhshiboev M.U. Hadamard-type Fractional Integrals and Marchaud-Hadamard-type Fractional Derivatives in the Spaces with Power Weight. Uzbek Mathematical Journal, No. 3, 155–174 (2019). doi.org/10.29229/uzmj.2019-3-17
"Fractional differentiation of the Grunwald-Letnikov-Hadamard type and the difference of the fractional order with a multiplicative step,"
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