Research article

Generalized k-fractional conformable integrals and related inequalities

  • Received: 21 December 2018 Accepted: 25 February 2019 Published: 15 April 2019
  • MSC : Primary 26A33; Secondary 26D10, 35A23, 47A63

  • In the paper, the authors introduce the generalized k-fractional conformable integrals, which are the k-analogues of the recently introduced fractional conformable integrals and can be reduced to other fractional integrals under specific values of the parameters involved. Hereafter, the authors prove the existence of k-fractional conformable integrals. Finally, the authors generalize some integral inequalities to ones for generalized k-fractional conformable integrals.

    Citation: Feng Qi, Siddra Habib, Shahid Mubeen, Muhammad Nawaz Naeem. Generalized k-fractional conformable integrals and related inequalities[J]. AIMS Mathematics, 2019, 4(3): 343-358. doi: 10.3934/math.2019.3.343

    Related Papers:

  • In the paper, the authors introduce the generalized k-fractional conformable integrals, which are the k-analogues of the recently introduced fractional conformable integrals and can be reduced to other fractional integrals under specific values of the parameters involved. Hereafter, the authors prove the existence of k-fractional conformable integrals. Finally, the authors generalize some integral inequalities to ones for generalized k-fractional conformable integrals.


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    [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66. doi: 10.1016/j.cam.2014.10.016
    [2] T. Abdeljawad, R. P. Agarwal, J. Alzabut, et al. Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives, J. Inequal. Appl., 2018 (2018), Paper No. 143: 1-17.
    [3] T. Abdeljawad, J. Alzabut, On Riemann-Liouville fractional q-difference equations and their application to retarded logistic type model, Math. Method. Appl. Sci., 41 (2018), 8953-8962. doi: 10.1002/mma.4743
    [4] T. Abdeljawad, F. Jarad and J. Alzabut, Fractional proportional differences with memory, Eur. Phys. J. Spec. Top., 226 (2017), 3333-3354. doi: 10.1140/epjst/e2018-00053-5
    [5] J. Alzabut, S. Tyagi and S. Abbas, Discrete fractional-order BAM neural networks with leakage delay: Existence and stability results, Asian J. Control, 22 (2020), Paper No. 1: 1-13. Available from:https://doi.org/10.1002/asjc.1918.
    [6] M. Al-Refai and T. Abdeljawad, Fundamental results of conformable Sturm-Liouville eigenvalue problems, Complexity, 2017 (2017), Article ID 3720471: 1-7.
    [7] D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137.
    [8] P. Agarwal, M. Jleli and M. Tomar, Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals, J. Inequal. Appl., 2017 (2017), Paper No. 55: 1-10.
    [9] R. P. Agarwal and A. Özbekler, Lyapunov type inequalities for mixed nonlinear Riemann-Liouville fractional differential equations with a forcing term, J. Comput. Appl. Math., 314 (2017), 69-78. doi: 10.1016/j.cam.2016.10.009
    [10] A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204.
    [11] D. Băleanu, R. P. Agarwal, H. Khan, et al. On the existence of solution for fractional differential equations of order 3<δ1≤4, Adv. Differ. Equ., 2015 (2015), Paper No. 362: 1-9.
    [12] D. Băleanu, R. P. Agarwal, H. Mohammadi, et al. Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), Paper No. 112: 1-8.
    [13] D. Băleanu, H. Khan, H. Jafari, et al. On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Differ. Equ., 2015 (2015), Paper No. 318: 1-14.
    [14] D. Băleanu and O. G. Mustafa, On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl., 59 (2010), 1835-1841. doi: 10.1016/j.camwa.2009.08.028
    [15] D. Băleanu, O. G. Mustafa, and R. P. Agarwal, An existence result for a superlinear fractional differential equation, Appl. Math. Lett., 23 (2010), 1129-1132. doi: 10.1016/j.aml.2010.04.049
    [16] D. Băleanu, O. G. Mustafa, and R. P. Agarwal, On the solution set for a class of sequential fractional differential equations, J. Phys. A: Math. Theor., 43 (2010), Article ID 385209: 1-7.
    [17] A. Bolandtalat, E. Babolian, and H. Jafari, Numerical solutions of multi-order fractional differential equations by Boubaker polynomials, Open Phys., 14 (2016), 226-230.
    [18] A. Debbouche and V. Antonov, Finite-dimensional diffusion models of heat transfer in fractal mediums involving local fractional derivatives, Nonlinear Stud., 24 (2017), 527-535.
    [19] R. Díaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat., 15 (2007), 179-192.
    [20] S. Habib, S. Mubeen, M. N. Naeem, et al. Generalized k-fractional conformable integrals and related inequalities, HAL archives, (2018), hal-01788916.
    [21] P. R. Halmos, Measure Theory, New York: D. Van Nostrand Company, Inc., 1950.
    [22] C. J. Huang, G. Rahman, K. S. Nisar, et al. Some inequalities of the Hermite-Hadamard type for k-fractional conformable integrals, Aust. J. Math. Anal. Appl., 16 (2019), Article ID 7: 1-9. \\Available from: http://ajmaa.org/cgi-bin/paper.pl?string=v16n1/V16I1P7.tex.
    [23] F. Jarad, T. Abdeljawad and J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457-3471. doi: 10.1140/epjst/e2018-00021-7
    [24] F. Jarad, Y. Adjabi, D. Baleanu, et al. On defining the distributions $\delta^r$ and $(\delta')^r$ by conformable derivatives, Adv. Differ. Equ., 2018 (2018), Paper No. 407: 1-20.
    [25] H. Jafari, H. K Jassim, S. P. Moshokoa, et al. Reduced differential transform method for partial differential equations within local fractional derivative operators, Adv. Mech. Eng., 8 (2016), 1-6.
    [26] H. Jafari, H. K. Jassim, F. Tchier, et al. On the approximate solutions of local fractional differential equations with local fractional operators, Entropy, 18 (2016), Paper No. 150: 1-12.
    [27] F. Jarad, E. Uğurlu, T. Abdeljawad, et al. On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), Paper No. 247: 1-16.
    [28] M. Jleli, M. Kirane and B. Samet, Hartman-Wintner-type inequality for a fractional boundary value problem via a fractional derivative with respect to another function, Discrete Dyn. Nat. Soc., 2017 (2017), Article ID 5123240: 1-8.
    [29] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.
    [30] U. N. Katugampola, New fractional integral unifying six existing fractional integrals, arXiv preprint, (2016).
    [31] H. Khan, H. Jafari, D. Băleanu, et al. On iterative solutions and error estimations of a coupled system of fractional order differential-integral equations with initial and boundary conditions, Differ. Equ. Dyn. Syst., (2019), in press. Available from: https://doi.org/10.1007/s12591-017-0365-7.
    [32] H. Khan, Y. J. Li, W. Chen, et al. Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, Bound. Value Probl., 2017, Paper No. 157: 1-16.
    [33] A. Khan, Y. J. Li, K. Shah, et al. On coupled p-Laplacian fractional differential equations with nonlinear boundary conditions, Complexity, 2017 (2017), Article ID 8197610: 1-9.
    [34] H. Khan, Y. J. Li, H. G. Sun, et al. Existence of solution and Hyers-Ulam stability for a coupled system of fractional differential equations with p-Laplacian operator, J. Nonlinear Sci. Appl., 10 (2017), 5219-5229. doi: 10.22436/jnsa.010.10.08
    [35] H. Khan, H. G. Sun, W. Chen, et al. Inequalities for new class of fractional integral operators, J. Nonlinear Sci. Appl., 10 (2017), 6166-6176. doi: 10.22436/jnsa.010.12.04
    [36] Y. J. Li, K. Shah and R. A. Khan, Iterative technique for coupled integral boundary value problem of non-integer order differential equations, Adv. Differ. Equ., 2017 (2017), Paper No. 251: 1-14.
    [37] W. J. Liu, Q. A. Ngô, and V. N. Huy, Several interesting integral inequalities, J. Math. Inequal., 3 (2009), 201-212.
    [38] S. Mubeen and S. Iqbal, Grüss type integral inequalities for generalized Riemann-Liouville k-fractional integrals, J. Inequal. Appl., 2016 (2016), Paper No. 109: 1-13.
    [39] S. Mubeen, S. Iqbal and Z. Iqbal, On Ostrowski type inequalities for generalized k-fractional integrals, J. Inequal. Spec. Funct., 8 (2017), 107-118.
    [40] K. S. Nisar and F. Qi, On solutions of fractional kinetic equations involving the generalized k-Bessel function, Note Mat., 37 (2017), 11-20. Available from: https://doi.org/10.1285/i15900932v37n2p11.
    [41] K. S. Nisar, F. Qi, G. Rahman, et al. Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function, J. Inequal. Appl., 2018 (2018), Paper No. 135: 1-12.
    [42] D. O'Regan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), Paper No. 247: 1-10.
    [43] F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl., 2019 (2019), 1-42. doi: 10.1186/s13660-019-1955-4
    [44] F. Qi, A. Akkurt and H. Yildirim, Catalan numbers, k-gamma and k-beta functions, and parametric integrals, J. Comput. Anal. Appl., 25 (2018), 1036-1042.
    [45] F. Qi and B. N. Guo, Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function, RACSAM Rev. R. Acad. A., 111 (2017), 425-434.
    [46] F. Qi, G. Rahman, S. M. Hussain, et al. Some inequalities of Čebyšev type for conformable k-fractional integral operators, Symmetry, 10 (2018), Article ID 614: 1-8.
    [47] F. Qi, G. Rahman and K. S. Nisar, Convexity and inequalities related to extended beta and confluent hypergeometric functions, HAL archives (2018). Available form: https://hal.archives-ouvertes.fr/hal-01703900.
    [48] G. Rahman, K. S. Nisar and F. Qi, Some new inequalities of the Grüss type for conformable fractional integrals, AIMS Math., 3 (2018), 575-583. doi: 10.3934/Math.2018.4.575
    [49] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
    [50] M. Z. Sarikaya and H. Budak, Generalized Ostrowski type inequalities for local fractional integrals, Proc. Amer. Math. Soc., 145 (2017), 1527-1538.
    [51] M. Z. Sarikaya, Z. Dahmani, M. E. Kiris, et al. (k,s)-Riemann-Liouville fractional integral and applications, Hacet. J. Math. Stat., 45 (2016), 77-89.
    [52] E. Set, M. A. Noor, M. U. Awan, et al. Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., 2017 (2017), Article ID 169: 1-10.
    [53] E. Set, M. Tomar and M. Z. Sarikaya, On generalized Grüss type inequalities for k-fractional integrals, Appl. Math. Comput., 269 (2015), 29-34.
    [54] K. Shah, H. Khalil and R. A. Khan, Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Progr. Frac. Differ. Appl., 2 (2016), 31-39. doi: 10.18576/pfda/020104
    [55] K. Shah and R. A. Khan, Study of solution to a toppled system of fractional differential equations with integral boundary conditions, Int. J. Appl. Comput. Math., 3 (2017), 2369-2388. doi: 10.1007/s40819-016-0243-y
    [56] D. P. Shi, B. Y. Xi and F. Qi, Hermite-Hadamard type inequalities for (m,h1,h2)-convex functions via Riemann-Liouville fractional integrals, Turkish J. Anal. Number Theory, 2 (2014), 23-28.
    [57] D. P. Shi, B. Y. Xi and F. Qi, Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals of (α,m)-convex functions, Fract. Differ. Calculus, 4 (2014), 31-43. Available from: https://doi.org/10.7153/fdc-04-02.
    [58] H. G. Sun, Y. Zhang, W. Chen, et al. Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media, J. Contam. Hydrol., 157 (2014), 47-58. doi: 10.1016/j.jconhyd.2013.11.002
    [59] Y. Z. Tian, M. Fan and Y. G. Sun, Certain nonlinear integral inequalities and their applications, Discrete Dyn. Nat. Soc., 2017 (2017), Article ID 8290906: 1-8.
    [60] M. Tunç, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat, 27 (2013), 559-565. doi: 10.2298/FIL1304559T
    [61] S. H. Wang and F. Qi, Hermite-Hadamard type inequalities for s-convex functions via Riemann-Liouville fractional integrals, J. Comput. Anal. Appl., 22 (2017), 1124-1134.
    [62] X. Yang and K. Lo, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput., 215 (2010), 3884-3890.
    [63] X. J. Yang, J. A. T. Machado and J. J. Nieto, A new family of the local fractional PDEs, Fund. Inform., 151 (2017), 63-75. doi: 10.3233/FI-2017-1479
    [64] X. J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Therm. Sci., 21 (2017), 1161-1171. doi: 10.2298/TSCI161216326Y
    [65] X. J. Yang, J. A. T. Machao and D. Băleanu, Anomalous diffusion models with general fractional derivatives within the kernels of the extended Mittag-Leffler type functions, Rom. Rep. Phys., 69 (2017), Article ID 115: 1-19.
    [66] X. J. Yang, H. M. Srivastava and J. A. T. Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci., 20 (2016), 753-756. doi: 10.2298/TSCI151224222Y
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