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The structural features of Hilbert-type local fractional integral inequalities with abstract homogeneous kernel and its applications. (English) Zbl 1445.26006

Summary: In this paper, by using the theory of local fractional calculus and some techniques of real analysis, the structural characteristics of Hilbert-type local fractional integral inequalities with abstract homogeneous kernel are studied. At the same time, the necessary and sufficient conditions for these inequalities to take the best constant factor are discussed. As an application, some best constant factor inequalities with specific kernels are obtained.

MSC:

26A33 Fractional derivatives and integrals
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[1] Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities, 2nd edition (Cambridge University Press, Cambridge, 1967). · JFM 60.0169.01
[2] Mintrinovic, D. S., Pecaric, J. E. and Kink, A. M., Inequalities Involving Functions and Their Integrals and Derivertives (Kluwer Academic Publishers, Boston, 1991). · Zbl 0744.26011
[3] Yang, B. C., On Hilbert’s integral inequality,J. Math. Anal. Appl.220 (1998) 778-785. · Zbl 0911.26011
[4] Yang, B. C., The Norm of Operator and Hibert-type Inequalities (Science Press, 2009).
[5] Krnić, M., Pečarić, J., Perić, I. and Vuković, P., Recent Advances in Hilbert-Type Inequalities (Element, Zagreb, 2012). · Zbl 1301.26004
[6] Yang, B. C., Hilbert-Type Integral Inequalities (Bentham Science Publishers Ltd, 2009). · Zbl 1469.26004
[7] Kolwankar, K. M. and Gangal, A. D., Fractional differentiability of nowhere differentiable functions and dimensions, Chaos6(4) (1996) 505-513. · Zbl 1055.26504
[8] Yang, X. J., Local Fractional Functional Analysis and Its Applications (Asian Academic publisher Limited, Hong Kong, 2011).
[9] Yang, X. J., Advanced Local Fractional Calculus and its Applications (World Science Publisher, 2012).
[10] Yang, X. J., Machado, J. A. Tenreiro and Nieto, J. J., A new family of the local fractional PDEs, Fundam. Inform.151(1-4) (2017) 63-75. · Zbl 1386.35461
[11] Chen, G. S., Local fractional Mellin transform in fractal space, Adv. Electr. Eng. Syst.1(1) (2012) 89-95.
[12] Yang, X. J., Srivastava, H. M. and Cattani, C., Local fractional homotopy perturbation method for solving fractional partial differential equations arising in mathematical physics, Rom. Rep. Phys.67 (2015) 752-761.
[13] Yang, X. J. and Carlo, C., On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos26(8) (2016) 084312. · Zbl 1378.35329
[14] Yang, X. J., Gao, F. and Srivastava, H. M., Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Comput. Mathe. Appl.73(2) (2017) 203-210. · Zbl 1386.35460
[15] Maitama, S. and Zhao, W. D., Local fractional homotopy analysis method for solving non-differentiable problems on Cantor sets, Adv. Diff. Equ.2019(1) (2019) 127. · Zbl 1459.34033
[16] Jassim, H. K., Analytical approximate solutions for local fractional wave equations, Math. Methods Appl. Sci.43(2) (2020) 939-947. · Zbl 1439.35533
[17] Anastassiou, G., Kashuri, A. and Liko, R., Local fractional integrals involving generalized strongly m-convex mappings, Arab. J. Math.8(2) (2018) 95-107. · Zbl 1414.26014
[18] Maitama, S., Local fractional natural homotopy perturbation method for solving partial differential equations with local fractional derivative, Prog. Fract. Differ. Appl.4(3) (2018) 219-228.
[19] Singh, J., Kumar, D., Baleanu, D. and Rathore, S., On the local fractional wave equation in fractal strings, Math. Methods Appl. Sci.42(5) (2019) 1588-1595. · Zbl 1419.35226
[20] Yang, X. J., Gao, F. and Srivastava, H. M., A new computational approach for solving nonlinear local fractional PDEs, J. Comput. Appl. Math.339 (2018) 285-296. · Zbl 1490.35530
[21] Ziane, D., Baleanu, D., Belghaba, K. and Cherif, M., Local fractional Sumudu decomposition method for linear partial differential equations with local fractional derivative, J. King Saud Univ., Sci.31(1) (2019) 83-88.
[22] G. S. Chen, Generalizations of Hölders and some related integral inequalities on fractal space, arXiv:1109.5567v1.
[23] Sarikaya, M. Z. and Budak, H., New inequalities for local fractional integrals, Iran. J. Sci. Technol. Trans. A, Sci.41(4) (2015) 1039-1046. · Zbl 1391.26049
[24] Sarikaya, M. Z., Tunc, T. and Budak, H., On generalized some integral inequalities for local fractional integrals, Appl. Math. Comput.276 (2016) 316-323. · Zbl 1410.26045
[25] Sarikaya, M. Z. and Budak, H., Generalized Ostrowski type inequalities for local fractional integrals, Proc. Am. Math. Soc.145(4) (2016) 1527-1538. · Zbl 1357.26026
[26] Sun, W. B. and Liu, Q., New inequalities of Hermite-Hadamard type for generalized conves functions on factal sets and its applications, J. Zhejiang Univ.44(1) (2017) 47-52.
[27] Rainier, R. V. C. and Sánchez, J. E., Strongly convexity on fractal sets and some inequalities, Proyecciones (Antofagasta)39(1) (2018) 1-13. · Zbl 1454.26011
[28] Sarikaya, M. Z. and Budak, H., Generalized Ostrowski-type inequalities for local fractional integrals, Proc. Am. Math. Soc.145 (2017) 1527-1538. · Zbl 1357.26026
[29] Batbold, T. S., Krnić, M. and Vuković, P., A unified approach to fractal Hilbert-type inequalities, J. Inequal. Appl.2019 (2019) 117. · Zbl 1499.26095
[30] Almutairi, O. and Kilicman, A., Generalized integral inequalities for hermite-hadamard-type inequalities via s-convexity on fractal sets, Mathematics7(11) (2019) 1065.
[31] Luo, C. Y., Wang, H. and Du, T. S., Fejér-Hermite-Hadamard-type inequalities involving generalized \(h\)-convexity on fractal sets and their applications, Chaos Solitons and Fractals131 (2020) 109547. · Zbl 1495.26031
[32] Liu, Q. and Sun, W. B., A Hilbert-type fractal integral inequality and its applications, J. Inequal. Appl.2017 (2017) 83. · Zbl 1362.26021
[33] Liu, Q. and Chen, D. Z., A Hilbert-type integral inequality on the fractal spaces, Integral Transforms Spec. Funct.28 (2017) 772-780. · Zbl 1379.26024
[34] Liu, Q., A Hilbert-type fractional integral inequality with the kernel of Mittag-Leffler function and its applications, Math. Inequal. Appl.21 (2018) 729-737. · Zbl 1400.26056
[35] M. Krnić and P. Vuković, Multidimensional Hilbert-type inequalities obtained via local fractional calculus, Acta Appl. Math., https://doi.org/10.1007/ s10440-020-00317-x. · Zbl 1357.26037
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