Research article Special Issues

On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities

  • Received: 08 December 2021 Revised: 07 March 2022 Accepted: 13 March 2022 Published: 23 March 2022
  • MSC : 26D15, 26D10, 26A33

  • In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.

    Citation: Miguel Vivas-Cortez, Pshtiwan O. Mohammed, Y. S. Hamed, Artion Kashuri, Jorge E. Hernández, Jorge E. Macías-Díaz. On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities[J]. AIMS Mathematics, 2022, 7(6): 10256-10275. doi: 10.3934/math.2022571

    Related Papers:

  • In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.



    加载中


    [1] Y. Adjabi, F. Jarad, D. Baleanu, T. Abdeljawad, On Cauchy problems with Caputo Hadamard fractional derivatives, J. Comput. Anal. Appl., 21 (2016), 661–681.
    [2] W. Tan, F. L. Jiang, C. X. Huang, L. Zhou, Synchronization for a class of fractional-order hyperchaotic system and its application, J. Appl. Math., 2012 (2012), 974639. https://doi.org/10.1155/2012/974639 doi: 10.1155/2012/974639
    [3] X. Zhou, C. Huang, H. Hu, L. Liu, Inequality estimates for the boundedness of multilinear singular and fractional integral operators, J. Inequal. Appl., 2013 (2013), 1–15. https://doi.org/10.1186/1029-242X-2013-303 doi: 10.1186/1029-242X-2013-303
    [4] F. Liu, L. Feng, V. Anh, J. Li, Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equations on irregular convex domains, Comput. Math. Appl., 78 (2019), 1637–1650. https://doi.org/10.1016/j.camwa.2019.01.007 doi: 10.1016/j.camwa.2019.01.007
    [5] A. S. Hendy, J. E. Macías-Díaz, A numerically efficient and conservative model for a Riesz space-fractional Klein-Gordon-Zakharov system, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 22–37. https://doi.org/10.1016/j.cnsns.2018.10.025 doi: 10.1016/j.cnsns.2018.10.025
    [6] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
    [7] Z. Cai, J. Huang, L. Huang, Periodic orbit analysis for the delayed Filippov system, Proc. Am. Math. Soc., 146 (2018), 4667–4682. https://doi.org/10.1090/proc/13883 doi: 10.1090/proc/13883
    [8] T. Chen, L. Huang, P. Yu, W. Huang, Bifurcation of limit cycles at infinity in piecewise polynomial systems, Nonlinear Anal., Real World Appl., 41 (2018), 82–106. https://doi.org/10.1016/j.nonrwa.2017.10.003 doi: 10.1016/j.nonrwa.2017.10.003
    [9] J. Wang, X. Chen, L. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl., 469 (2019), 405–427. https://doi.org/10.1016/j.jmaa.2018.09.024 doi: 10.1016/j.jmaa.2018.09.024
    [10] L. Bai, F. Liu, S. Tan, A new efficient variational model for multiplicative noise removal, Int. J. Comput. Math., 97 (2020), 1444–1458. https://doi.org/10.1080/00207160.2019.1622688 doi: 10.1080/00207160.2019.1622688
    [11] H. Yuan, Convergence and stability of exponential integrators for semi-linear stochastic variable delay integro-differential equations, Int. J. Comput. Math., 98 (2020), 903–932. https://doi.org/10.1080/00207160.2020.1792452 doi: 10.1080/00207160.2020.1792452
    [12] M. Aldhaifallah, M. Tomar, K. Nisar, S. D. Purohit, Some new inequalities for $(k, s)$-fractional integrals, J. Nonlinear Sci. Appl., 9 (2016), 5374–5381.
    [13] M. Houas, Certain weighted integral inequalities involving the fractional hypergeometric operators, Sci., Ser. A, Math. Sci. (N.S.), 27 (2017), 87–97.
    [14] M. Houas, On some generalized integral inequalities for Hadamard fractional integrals, Mediterr. J. Model. Simul., 9 (2018), 43–52.
    [15] D. Baleanu, P. O. Mohammed, M. Vivas-Cortez, Y. Rangel-Oliveros, Some modifications in conformable fractional integral inequalities, Adv. Differ. Equ., 2020 (2020), 1–25. https://doi.org/10.1186/s13662-020-02837-0 doi: 10.1186/s13662-020-02837-0
    [16] T. Abdeljawad, P. O. Mohammed, A. Kashuri, New modified conformable fractional integral inequalities of Hermite-Hadamard type with applications, J. Funct. Spaces, 2020 (2020), 1–14. https://doi.org/10.1155/2020/4352357 doi: 10.1155/2020/4352357
    [17] P. O. Mohammed, T. Abdeljawad, Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel, Adv. Differ. Equ., 2020 (2020), 1–19. https://doi.org/10.1186/s13662-020-02825-4 doi: 10.1186/s13662-020-02825-4
    [18] P. O. Mohammed, I. Brevik, A new version of the Hermite-Hadamard inequality for Riemann-Liouville fractional integrals, Symmetry, 12 (2020), 610. https://doi.org/10.3390/sym12040610 doi: 10.3390/sym12040610
    [19] P. L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2 (1882), 93–98.
    [20] F. Qi, G. Rahman, S. M. Hussain, W. S. Du, K. S. Nisar, Some inequalities of Čebyšev type for conformable $k$-fractional integral operators, Symmetry, 10 (2018), 614.
    [21] G. Rahman, S. Nisar, F. Qi, Some new inequalities of the Grüss type for conformable fractional integrals, AIMS Math., 3 (2018), 575–583. https://doi.org/10.3934/Math.2018.4.575 doi: 10.3934/Math.2018.4.575
    [22] S. Rashid, F. Jarad, M. A. Noor, K. I. Noor, D. Baleanu, J. B. Liu, On Grüss inequalities within generalized $k$-fractional integrals, Adv. Differ. Equ., 2020 (2020), 1–18. https://doi.org/10.1186/s13662-020-02644-7 doi: 10.1186/s13662-020-02644-7
    [23] S. Rashid, A. O. Akdemir, F. Jarad, M. A. Noor, K. I. Noor, Simpson's type integral inequalities for $\kappa$-fractional integrals and their applications, AIMS Math., 4 (2019), 1087–1100. https://doi.org/10.3934/math.2019.4.1087 doi: 10.3934/math.2019.4.1087
    [24] K. S. Nisar, G. Rahman, K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 2019 (2019), 1–9. https://doi.org/10.1186/s13660-019-2197-1 doi: 10.1186/s13660-019-2197-1
    [25] R. S. Dubey, P. Goswami, Some fractional integral inequalities for the Katugampola integral operator, AIMS Math., 4 (2019), 193–198. https://doi.org/10.3934/math.2019.2.193 doi: 10.3934/math.2019.2.193
    [26] Z. Dahmani, O. Mechouar, S. Brahami, Certain inequalities related to the Chebyshev's functional involving a Riemann-Liouville operator, Bull. Math. Anal. Appl., 3 (2011), 38–44.
    [27] T. U. Khan, M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389. https://doi.org/10.1016/j.cam.2018.07.018 doi: 10.1016/j.cam.2018.07.018
    [28] F. Usta, H. Budak, M. Z. Sarıkaya, On Chebychev type inequalities for fractional integral operators, In: AIP conference proceedings, Vol. 1833, AIP Publishing LLC, 2017.
    [29] M. E. Özdemir, E. Set, A. O. Akdemir, M. Z. Sarıkaya, Some new Chebyshev type inequalities for functions whose derivatives belongs to $L _ p$ spaces, Afr. Mat., 26 (2015), 1609–1619. https://doi.org/10.1007/s13370-014-0312-5 doi: 10.1007/s13370-014-0312-5
    [30] B. E. Pachpatte, A note on Chebychev-Grüss type inequalities for differentiable functions, Tamsui Oxford J. Math. Sci., 22 (2006), 29–37.
    [31] Z. Liu, A variant of Chebyshev inequality with applications, J. Math. Inequal., 7 (2013), 551–561. https://doi.org/dx.doi.org/10.7153/jmi-07-51 doi: 10.7153/jmi-07-51
    [32] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
    [33] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
    [34] F. Jarad, E. Uǧurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 1–16. https://doi.org/10.1186/s13662-017-1306-z doi: 10.1186/s13662-017-1306-z
    [35] A. O. Akdemir, A. Ekinci, E. Set, Conformable fractional integrals and related new integral inequalities, J. Nonlinear Convex Anal., 18 (2017), 661–674.
    [36] P. O. Mohammed, C. S. Ryoo, A. Kashuri, Y. S. Hamed, K. M. Abualnaja, Some Hermite-Hadamard and Opial dynamic inequalities on time scales, J. Inequal. Appl., 2021 (2021), 1–11. https://doi.org/10.1186/s13660-021-02624-9 doi: 10.1186/s13660-021-02624-9
    [37] Z. Dahmani, L. Tabharit, On weighted Grüss type inequalities via fractional integration, J. Adv. Res. Pure Math., 2 (2010), 31–38.
    [38] E. Set, Z. Dahmani, İ. Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szegö inequality, Int. J. Optim. Control, Theor. Appl., 8 (2018), 137–144. https://doi.org/10.11121/ijocta.01.2018.00541 doi: 10.11121/ijocta.01.2018.00541
    [39] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
    [40] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, theory and applications, CRC Press, 1993.
    [41] R. K. Raina, On generalized Wright's hypergeometric functions and fractional calculus operators, East Asian Math. J., 21 (2005), 191–203.
    [42] R. P. Agarwal, M. J. Luo, R. K. Raina, On Ostrowski type inequalities, Fasc. Math., 56 (2016), 5–27. https://doi.org/10.1515/fascmath-2016-0001 doi: 10.1515/fascmath-2016-0001
    [43] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.
    [44] T. O. Salim, A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl., 3 (2012), 1–13.
    [45] M. Vivas-Cortez, A. Kashuri, J. E. H. Hernández, Trapezium-type inequalities for Raina's fractional integrals operator using generalized convex functions, Symmetry, 12 (2020), 1034. https://doi.org/10.3390/sym12061034 doi: 10.3390/sym12061034
    [46] S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10 (2009), 1–12.
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1330) PDF downloads(82) Cited by(0)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog