New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation

Muhammad Bilal Khan; Pshtiwan Othman Mohammed; Muhammad Aslam Noor; Abdullah M. Alsharif; Khalida Inayat Noor; Muhammad Bilal Khan; Pshtiwan Othman Mohammed; Muhammad Aslam Noor; Abdullah M. Alsharif; Khalida Inayat Noor

doi:10.3934/math.2021637

AIMS Mathematics

2021, Volume 6, Issue 10: 10964-10988. doi: 10.3934/math.2021637

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New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation

1.
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
2.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq
3.
Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 26 March 2021 Accepted: 23 July 2021 Published: 29 July 2021
MSC : 26A33, 26A51, 26D10

It is well-known that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis and fuzzy-interval analysis, the inclusion relation (⊆) and fuzzy order relation $\left(\preccurlyeq \right)$ both are two different concepts, respectively. In this article, with the help of fuzzy order relation, we introduce fractional Hermite-Hadamard inequality (HH-inequality) for h-convex fuzzy-interval-valued functions (h-convex-IVFs). Moreover, we also establish a strong relationship between h-convex fuzzy-IVFs and Hermite-Hadamard Fejér inequality (HH-Fejér inequality) via fuzzy Riemann Liouville fractional integral operator. It is also shown that our results include a wide class of new and known inequalities for h-convex fuzz-IVFs and their variant forms as special cases. Nontrivial examples are presented to illustrate the validity of the concept suggested in this review. This paper's techniques and approaches may serve as a springboard for further research in this field.
Citation: Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor, Abdullah M. Alsharif, Khalida Inayat Noor. New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation[J]. AIMS Mathematics, 2021, 6(10): 10964-10988. doi: 10.3934/math.2021637

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Abstract

It is well-known that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis and fuzzy-interval analysis, the inclusion relation (⊆) and fuzzy order relation $\left(\preccurlyeq \right)$ both are two different concepts, respectively. In this article, with the help of fuzzy order relation, we introduce fractional Hermite-Hadamard inequality (HH-inequality) for h-convex fuzzy-interval-valued functions (h-convex-IVFs). Moreover, we also establish a strong relationship between h-convex fuzzy-IVFs and Hermite-Hadamard Fejér inequality (HH-Fejér inequality) via fuzzy Riemann Liouville fractional integral operator. It is also shown that our results include a wide class of new and known inequalities for h-convex fuzz-IVFs and their variant forms as special cases. Nontrivial examples are presented to illustrate the validity of the concept suggested in this review. This paper's techniques and approaches may serve as a springboard for further research in this field.

References

[1]	C. Hermite, Sur deux limites d'une intégrale définie, Mathesis, 3 (1883), 1-82.
[2]	J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann, J. Math. Appl., 7 (1893), 171-216.
[3]	F. Chen, S. Wu, Integral inequalities of Hermite-Hadamard type for products of two h-convex functions, Abst. Appl. Anal., 6 (2014), 1-6.
[4]	Z. B. Fang, R. Shi, On the (p, h)-convex function and some integral inequalities, J. Inequal. Appl., 2014 (2014), 1-16. doi: 10.1186/1029-242X-2014-1
[5]	I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe J. Math. Stat., 43 (2014), 935-942.
[6]	I. Iscan, Hermite-Hadamard type inequalities for p-convex functions, Int. J. Anal. Appl., 11 (2016), 137-145.
[7]	L. Fejér, Uber die fourierreihen Ⅱ, Math. Naturwiss. Anz. Ungar. Akad. Wiss., 24 (1906), 369-390.
[8]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403-2407. doi: 10.1016/j.mcm.2011.12.048
[9]	M. Kunt, I. Iscan, Hermite-Hadamard-Fejér type inequalities for p-convex functions via fractional integrals, Stud. Univ. Babes-Bolyai, Math., 42 (2018), 2079-2089.
[10]	P. O. Mohammed, New generalized Riemann-Liouville fractional integral inequalities for convex functions, J. Math. Inequal., 15 (2021), 511-519.
[11]	I. Iscan, M. Kunt, N. Yazici, Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via fractional integrals, New Trends Math. Sci., 4 (2016), 239-253.
[12]	D. Kaur, P. Agarwal, M. Rakshit, M. Chand, Fractional calculus involving (p, q)-Mathieu type series, Appl. Math. Nonlinear Sci., 5 (2020), 15-34. doi: 10.2478/amns.2020.2.00011
[13]	I. Iscan, Generalization of different type integral inequalities for s-convex functions via fractional integrals, Appl. Anal., 93 (2014), 1846-1862. doi: 10.1080/00036811.2013.851785
[14]	J. Han, P. O. Mohammed, H. Zeng, Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function, Open Math., 18 (2020), 794-806. doi: 10.1515/math-2020-0038
[15]	P. O. Mohammed, Fractional integral inequalities of Hermite-Hadamard type for convex functions with respect to a monotone function, J. Math. Inequal., 15 (2021), 511-519.
[16]	R. E. Moore, Interval analysis, Englewood Cliffs: Prentice Hall, 1966.
[17]	T. M. Costa, Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets Syst., 327 (2017), 31-47. doi: 10.1016/j.fss.2017.02.001
[18]	T. M. Costa, H. Roman-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inform. Sci., 420 (2017), 110-125. doi: 10.1016/j.ins.2017.08.055
[19]	H. Román-Flores, Y. Chalco-Cano, W. A. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2018), 1306-1318. doi: 10.1007/s40314-016-0396-7
[20]	H. Roman-Flores, Y. Chalco-Cano, G. N. Silva, A note on Gronwall type inequality for interval-valued functions, In: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), IEEE, 2013, 1455-1458.
[21]	Y. Chalco-Cano, A. Flores-Franulič, H. Román-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457-472.
[22]	Y. Chalco-Cano, W. A. Lodwick, W. Condori-Equice, Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput., 19 (2015), 3293-3300. doi: 10.1007/s00500-014-1483-6
[23]	K. Nikodem, J. L. Snchez, L. Snchez, Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Math. Aterna, 4 (2014), 979-987.
[24]	J. Matkowski, K. Nikodem, An integral Jensen inequality for convex multifunctions, Results Math., 26 (1994), 348-353. doi: 10.1007/BF03323058
[25]	D. Zhang, C. Guo, D. Chen, G. Wang, Jensen's inequalities for set-valued and fuzzy set-valued functions, Fuzzy Sets Syst., 404 (2020), 178-204.
[26]	H. Budak, T. Tunç, M. Z. Sarikaya, Fractional Hermite-Hadamard type inequalities for interval-valued functions, Proc. Am. Math. Soc., 148 (2019), 705-718. doi: 10.1090/proc/14741
[27]	T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equations with uncertainty, Soft Comput., 16 (2012), 297-302. doi: 10.1007/s00500-011-0743-y
[28]	M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, New Hermite-Hadamard type inequalities for (h1, h2)-convex fuzzy-interval-valued functions, Adv. Differ. Equ., 2021 (2021), 1-20. doi: 10.1186/s13662-020-03162-2
[29]	U. Kulish, W. Miranker, Computer arithmetic in theory and practice, New York: Academic Press, 2014.
[30]	D. F. Zhao, T. Q. An, G. J. Ye, W. Liu, New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 1-14. doi: 10.1186/s13660-017-1594-6
[31]	O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301-317. doi: 10.1016/0165-0114(87)90029-7
[32]	N. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets Syst., 48 (1992), 129-132. doi: 10.1016/0165-0114(92)90256-4
[33]	M. A. Noor, Fuzzy preinvex functions, Fuzzy Sets Syst., 64 (1994), 95-104. doi: 10.1016/0165-0114(94)90011-6
[34]	D. Zhao, T. An, G. Ye, W. Liu, Chebyshev type inequalities for interval-valued functions, Fuzzy Sets Syst., 396 (2020), 82-101. doi: 10.1016/j.fss.2019.10.006
[35]	M. A. Alqudah, A. Kashuri, P. O. Mohammed, M. Raees, T. Abdeljawad, M. Anwar, et al., On modified convex interval valued functions and related inclusions via the interval valued generalized fractional integrals in extended interval space, AIMS Math., 6 (2021), 4638-4663. doi: 10.3934/math.2021273
[36]	Z. Zhang, M. A. Ali, H. Budak, M. Z. Sarikaya, On Hermite-Hadamard type inequalities for interval-valued multiplicative integrals, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., 69 (2020), 1428-1448.
[37]	S. S. Dragomir, On the Hadamard's inequlality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math., 2001 (2001), 775-788.
[38]	H. M. Srivastava, Z. H. Zhang, Y. D. Wu, Some further refinements and extensions of the Hermite-Hadamard and Jensen inequalities in several variables, Math. Comput. Model., 54 (2011), 2709-2717. doi: 10.1016/j.mcm.2011.06.057
[39]	P. Liu, M. B. Khan, M. A. Noor, K. I. Noor, New Hermite-Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense, Complex Intell. Syst., 2021, (2021), 1-15.
[40]	M. B. Khan, M. A. Noor, L. Abdullah, Y. M. Chu, Some new classes of preinvex fuzzy-interval-valued functions and inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1403-1418. doi: 10.2991/ijcis.d.210409.001
[41]	M. B. Khan, P. O. Mohammed, M. A. Noor, Y. S. Hamed, New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities, Symmetry, 13 (2021), 673. doi: 10.3390/sym13040673
[42]	P. Liu, M. B. Khan, M. A. Noor, K. I. Noor, On strongly generalized preinvex fuzzy mappings, J. Math., 2021 (2021), 6657602.
[43]	M. B. Khan, M. A. Noor, K. I. Noor, A. T. Ab Ghani, L. Abdullah, Extended perturbed mixed variational-like inequalities for fuzzy mappings, J. Math., 2021 (2021), 1-16.
[44]	M. B. Khan, M. A. Noor, K. I. Noor, H. Almusawa, K. S. Nisar, Exponentially Preinvex Fuzzy Mappings and Fuzzy Exponentially Mixed Variational-Like Inequalities, Int. J. Anal. Appl., 19 (2021), 518-541.
[45]	M. B. Khan, M. A. Noor, H. M. Al-Bayatti, K. I. Noor, Some new inequalities for LR-Log-h-convex interval-valued functions by means of pseudo order relation, Appl. Math Inf. Sci., 15 (2021), 459-470. doi: 10.18576/amis/150408
[46]	G. Sana, M. B. Khan, M. A. Noor, P. O. Mohammed, Y. M. Chu, Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann-Liouville fractional integral inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1809-1822. doi: 10.2991/ijcis.d.210620.001
[47]	M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, Higher-order strongly preinvex fuzzy mappings and fuzzy mixed variational-like inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1856-1870. doi: 10.2991/ijcis.d.210616.001
[48]	P. O. Mohammed, T. Abdeljawad, Modification of certain fractional integral inequalities for convex functions, Adv. Differ. Equ., 2020 (2020), 1-22. doi: 10.1186/s13662-019-2438-0
[49]	M. A. Alqudah, A. Kshuri, P. O Mohammed, T. Abdeljawad, M. Raees, M. Anwar, et al., Hermite-Hadamard integral inequalities on coordinated convex functions in quantum calculus, Adv. Differ. Equ., 2021 (2021), 1-29. doi: 10.1186/s13662-020-03162-2
[50]	M. B. Khan, M. A. Noor, K. I. Noor, On some characterization of preinvex fuzzy mappings, Earth. J. Mathe. Sci., 5 (2021), 17-42.
[51]	M. B. Khan, M. A. Noor, K. I. Noor, On fuzzy quasi-invex sets, Int. J. Algebra Stat., 9 (2020), 11-26
[52]	P. O. Mohammed, T. Abdeljawad, M. A. Alqudah, F. Jarad, New discrete inequalities of Hermite-Hadamard type for convex functions, Adv. Differ. Equ., 2021 (2021), 122. doi: 10.1186/s13662-021-03290-3
[53]	R. Sahin, O. Yagc, Fractional calculus of the extended hypergeometric function, Appl. Math. Nonlinear Sci., 5 (2020), 369-384. doi: 10.2478/amns.2020.1.00035

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