Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions

Muhammad Bilal Khan; Pshtiwan Othman Mohammed; Muhammad Aslam Noor; Khadijah M. Abualnaja; Muhammad Bilal Khan; Pshtiwan Othman Mohammed; Muhammad Aslam Noor; Khadijah M. Abualnaja

doi:10.3934/mbe.2021325

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 6552-6580. doi: 10.3934/mbe.2021325

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Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions

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

Academic Editor: Erkan Oterkus

Received: 23 April 2021 Accepted: 15 July 2021 Published: 30 July 2021

In this study, we introduce and study new fuzzy-interval integral is known as fuzzy-interval double integral, where the integrand is fuzzy-interval-valued functions (FIVFs). Also, some fundamental properties are also investigated. Moreover, we present a new class of convex fuzzy-interval-valued functions is known as coordinated convex fuzzy-interval-valued functions (coordinated convex FIVFs) through fuzzy order relation (FOR). The FOR $\left(\preccurlyeq \right)$ and fuzzy inclusion relation (⊇) are two different concepts. With the help of fuzzy-interval double integral and FOR, we have proved that coordinated convex fuzzy-IVF establish a strong relationship between Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) inequalities. With the support of this relation, we also derive some related HH-inequalities for the product of coordinated convex FIVFs. Some special cases are also discussed. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.
Citation: Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor, Khadijah M. Abualnaja. Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 6552-6580. doi: 10.3934/mbe.2021325

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Abstract

In this study, we introduce and study new fuzzy-interval integral is known as fuzzy-interval double integral, where the integrand is fuzzy-interval-valued functions (FIVFs). Also, some fundamental properties are also investigated. Moreover, we present a new class of convex fuzzy-interval-valued functions is known as coordinated convex fuzzy-interval-valued functions (coordinated convex FIVFs) through fuzzy order relation (FOR). The FOR $\left(\preccurlyeq \right)$ and fuzzy inclusion relation (⊇) are two different concepts. With the help of fuzzy-interval double integral and FOR, we have proved that coordinated convex fuzzy-IVF establish a strong relationship between Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) inequalities. With the support of this relation, we also derive some related HH-inequalities for the product of coordinated convex FIVFs. Some special cases are also discussed. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

References

[1]	C. Hermite, Sur deux limites d'une intégrale définie, Mathesis, 3 (1883), 82-97.
[2]	J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann, J. Math. Pures Appl., 7 (1893), 171-215.
[3]	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
[4]	S. S. Dragomir, R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91-95.
[5]	U.S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004), 137-146.
[6]	L. Fejxer, Uberdie Fourierreihen Ⅱ, Math. Naturwise. Anz, Ungar Akad Wiss., 24 (1906), 369-390.
[7]	S. Erden, M. Z. Sarıkaya, H. Budak, New weighted inequalities for higher order derivatives and applications, Filomat., 32 (2018), 4419-4433. doi: 10.2298/FIL1812419E
[8]	M. Z. Sarikaya, S. Erden, On the Hermite-Hadamard-Fejér type integral inequality for convex function, Turk. J. Anal. Number Theory., 2 (2014), 85-89. doi: 10.12691/tjant-2-3-6
[9]	M. Z. Sarikaya, S. Erden, On the weighted integral inequalities for convex functions, Acta Univ. Sapientiae Math., 6 (2014), 194-208.
[10]	M. A. Latif, M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinates, Int. Math. Forum., 4 (2009), 2327-2338.
[11]	M. E. Ozdemir, M. A. Latif, A. O. Akdemir, On some Hadamard-type inequalities for product of two s-convex functions on the co-ordinates, J. Inequalities Appl., 2012 (2012), 21. doi: 10.1186/1029-242X-2012-21
[12]	M. E. Ozdemir, M. A. Latif, A. O. Akdemir, On some Hadamard-type inequalities for product of two h-convex functions on the co-ordinates, Turk. J. Sci., 1 (2016), 41-58.
[13]	H. Budak, M. Z. Sarikaya, Hermite-Hadamard type inequalities for products of two co-ordinated convex mappings via fractional integrals, Int. J. Appl. Math. Stat., 58 (2019), 11-30.
[14]	R. E. Moore, C. T. Yang, Interval Analysis, Prentice Hall, Englewood Cliffs, 1966.
[15]	E. Sadowska, Hadamard inequality and a refinement of Jensen inequality for set-valued functions, Results Math., 32 (1997), 332-337. doi: 10.1007/BF03322144
[16]	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
[17]	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
[18]	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
[19]	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), 35 (2013), 1455-1458.
[20]	Y. Chalco-Cano, A. Flores-Franuliˇc, H. Román-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457-472.
[21]	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
[22]	K. Nikodem, J. L. Snchez, L. Snchez, Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Math. Aterna., 4 (2014), 979-987.
[23]	J. Matkowski; K. Nikodem, An integral Jensen inequality for convex multifunctions, Results Math., 26 (1994), 348-353. doi: 10.1007/BF03323058
[24]	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
[25]	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
[26]	D. Zhang, C. Guo, D. Chen, G. Wang, Jensen's inequalities for set-valued and fuzzy set-valued functions, Fuzzy Sets Syst. 2020 (2020), 1-27.
[27]	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
[28]	P. O. Mohammed, T. Abdeljawad, M. A. Alqudah, F. Jarad, New discrete inequalities of Hermite-Hadamard type for convex functions, Adv. Differ. Equations, 2021 (2021), 122. doi: 10.1186/s13662-021-03290-3
[29]	D. Zhao, M. A. Ali, G. Murtaza, Z. Zhang, On the Hermite-Hadamard inequalities for interval-valued coordinated convex functions, Adv. Differ. Equations, 2020 (2020), 1-14. doi: 10.1186/s13662-019-2438-0
[30]	H. Kara, M. A. Ali, H. Budak, Hermite-Hadamard-type inequalities for interval-valued coordinated convex functions involving generalized fractional integrals, Math. Methods Appl. Sci., 44 (2021), 104-123. doi: 10.1002/mma.6712
[31]	F. Shi, G. Ye, D. Zhao, W. Liu, Some fractional Hermite-Hadamard-type inequalities for interval-valued coordinated functions, Adv. Differ. Equations, 2021 (2021), 1-17. doi: 10.1186/s13662-020-03162-2
[32]	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.
[33]	P. O. Mohammed, T. Abdeljawad, Modification of certain fractional integral inequalities for convex functions, Adv. Differ. Equations, 2020 (2020), 69. doi: 10.1186/s13662-020-2541-2
[34]	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. Equations, 2021 (2021), 264. doi: 10.1186/s13662-021-03420-x
[35]	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. Equations, 2021 (2021), 6-20. doi: 10.1186/s13662-020-03166-y
[36]	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
[37]	M. B. Khan, P. O. Mohammed, M. A. Noor, A. M. Alsharif and K. I. Noor, New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation. AIMS Math., 6 (2021), 10964-10988. doi: 10.3934/math.2021637
[38]	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
[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, 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
[41]	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., 2021 (2021).
[42]	U. Kulish, W. Miranker, Computer Arithmetic in Theory and Practice, Academic Press, New York, 2014.
[43]	O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301-317. doi: 10.1016/0165-0114(87)90029-7
[44]	N. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets Syst., 48 (1992), 129-132. doi: 10.1016/0165-0114(92)90256-4
[45]	M. A. Noor, Fuzzy preinvex functions, Fuzzy Sets Syst., 64 (1994), 95-104. doi: 10.1016/0165-0114(94)90011-6
[46]	P. Liu, M. B. Khan, M. A. Noor, K. I. Noor, On strongly generalized preinvex fuzzy mappings, J. Math., 2021 (2021).
[47]	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.
[48]	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.
[49]	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., 2021 (2021).
[50]	M. B. Khan, M. A. Noor, K. I. Noor, On some characterization of preinvex fuzzy mappings, Earth. J. Math. Sci., 5 (2021), 17-42.
[51]	M. B. Khan, M. A. Noor, K. I. Noor, On fuzzy quasi-invex sets, Int. J. Algeb. Stat., 9 (2020), 11-26.

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