AIMS Mathematics, 2019, 4(3): 343-358. doi: 10.3934/math.2019.3.343

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Generalized k-fractional conformable integrals and related inequalities

1 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
2 School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
3 Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, China
4 Department of Mathematics, Government College University Faisalabad, Pakistan
5 Department of Mathematics, University of Sargodha, Sargodha, Pakistan
6 Department of Mathematics, Government College University Faisalabad, Pakistan, Siddra Habib, Shahid Mubeen, Muhammad Nawaz Naeem

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.    

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.    

4.T. Abdeljawad, F. Jarad and J. Alzabut, Fractional proportional differences with memory, Eur. Phys. J. Spec. Top., 226 (2017), 3333-3354.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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.    

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