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Mathematical Biosciences and Engineering

2022, Volume 19, Issue 1: 812-835. doi: 10.3934/mbe.2022037
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Fuzzy-interval inequalities for generalized preinvex fuzzy interval valued functions

  • 1.
    Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
  • 2.
    Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
  • 3.
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
  • 4.
    Department of Mathematics and Informatics, Azerbaijan University, Baku AZ1007, Azerbaijan
  • 5.
    Section of Mathematics, International Telematic University Uninettuno, Rome I-00186, Italy
  • 6.
    Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq
  • 7.
    Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, Cartagena 30203, Spain
  • 8.
    Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
  • 9.
    Department of Mathematics and Statistics, College of Sciences, Taif University, Taif 21944, Saudi Arabia
  • Received: 26 August 2021 Accepted: 26 October 2021 Published: 22 November 2021

  • In this paper, firstly we define the concept of h-preinvex fuzzy-interval-valued functions (h-preinvex FIVF). Secondly, some new Hermite-Hadamard type inequalities (H-H type inequalities) for h-preinvex FIVFs via fuzzy integrals are established by means of fuzzy order relation. Finally, we obtain Hermite-Hadamard Fejér type inequalities (H-H Fejér type inequalities) for h-preinvex FIVFs by using above relationship. To strengthen our result, we provide some examples to illustrate the validation of our results, and several new and previously known results are obtained.

    Citation: Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Taghreed M. Jawa. Fuzzy-interval inequalities for generalized preinvex fuzzy interval valued functions[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 812-835. doi: 10.3934/mbe.2022037

    Related Papers:

  • In this paper, firstly we define the concept of h-preinvex fuzzy-interval-valued functions (h-preinvex FIVF). Secondly, some new Hermite-Hadamard type inequalities (H-H type inequalities) for h-preinvex FIVFs via fuzzy integrals are established by means of fuzzy order relation. Finally, we obtain Hermite-Hadamard Fejér type inequalities (H-H Fejér type inequalities) for h-preinvex FIVFs by using above relationship. To strengthen our result, we provide some examples to illustrate the validation of our results, and several new and previously known results are obtained.


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    [1] J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann, J. Math. Pures Appl., (1893), 171–215.
    [2] C. Hermite, Sur deux limites d'une intégrale définie, Mathesis, 3 (1883), 1–82.
    [3] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory, 5 (2007), 126–131.
    [4] M. Matłoka, Inequalities for h-preinvex functions, Appl. Math. Comput., 234 (2014), 52–57. doi: 10.1016/j.amc.2014.02.030. doi: 10.1016/j.amc.2014.02.030
    [5] M. Avci, H. Kavurmaci, M.E. Ozdemir, New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications, Appl. Math. Comput., 217 (2011), 5171–5176. doi: 10.1016/j.amc.2010.11.047. doi: 10.1016/j.amc.2010.11.047
    [6] E. Ata, I. O. Kıymaz, A study on certain properties of generalized special functions defined by Fox-Wright function, Appl. Math. Nonlinear Sci., 5 (2020), 147–162. doi: 10.2478/amns.2020.1.00014. doi: 10.2478/amns.2020.1.00014
    [7] E. ˙Ilhan, ˙I. O. Kıymaz, A generalization of truncated M-fractional derivative and applications to fractional differential equations, Appl. Math. Nonlinear Sci., 5 (2020), 171–188. doi: 10.2478/amns.2020.1.00016. doi: 10.2478/amns.2020.1.00016
    [8] H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100–111. doi: 10.1007/BF01837981. doi: 10.1007/BF01837981
    [9] I. Iscan, A new generalization of some integral inequalities for (α, m)-convex functions, Math. Sci., 7 (2013), 1–8. doi: 10.1186/2251-7456-7-22. doi: 10.1186/2251-7456-7-22
    [10] R. E. Moore, C. T. Yang, Interval Analysis, Prentice Hall, Englewood Cliffs, 1966.
    [11] U. Kulish, W. Miranker, Computer Arithmetic in Theory and Practice, Academic Press, New York, 2014. doi: 10.1137/1025138.
    [12] H. Rezazadeh, A. Korkmaz, A. E. Achab, W. Adel, A. Bekir, New travelling wave solution-based new Riccati Equation for solving KdV and modified KdV Equations, Appl. Math. Nonlinear Sci., 5 (2021), 447–458. doi: 10.2478/amns.2020.2.00034. doi: 10.2478/amns.2020.2.00034
    [13] J. M. Snyder, Interval analysis for computer graphics, SIGGRAPH Comput. Graph., 26 (1992), 121–130. doi: 10.1145/142920.134024. doi: 10.1145/142920.134024
    [14] E. de Weerdt, Q. P. Chu, J. A. Mulder, Neural network output optimization using interval analysis, IEEE Trans. Neural Networks, 20 (2009), 638–653. doi: 10.1109/TNN.2008.2011267. doi: 10.1109/TNN.2008.2011267
    [15] 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., 2 (2018), 1–14. doi: 10.1186/s13660-018-1896-3. doi: 10.1186/s13660-018-1896-3
    [16] K. Touchent, Z. Hammouch, T. Mekkaoui, A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives, Appl. Math. Nonlinear Sci., 5 (2020), 35–48. doi: 10.2478/amns.2020.2.00012. doi: 10.2478/amns.2020.2.00012
    [17] B. Bede, Studies in Fuzziness and Soft Computing, in Mathematics of Fuzzy Sets and Fuzzy Logic, Springer, 295 (2013). doi: 10.1007/978-3-642-35221-8.
    [18] R. S¸ahin, O. Yagci, Fractional calculus of the extended hypergeometric function, Appl. Math. Nonlinear Sci., 5 (2020), 369–384. doi: 10.2478/amns.2020.1.00035. doi: 10.2478/amns.2020.1.00035
    [19] 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. doi: 10.2478/amns.2020.2.00011
    [20] S. Kabra, H. Nagar, K. S. Nisar, D. L. Suthar, The Marichev-Saigo-Maeda fractional calculus operators pertaining to the generalized k-Struve function, Appl. Math. Nonlinear Sci., 5 (2020), 593–602. doi: 10.2478/amns.2020.2.00064. doi: 10.2478/amns.2020.2.00064
    [21] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets Syst., 35 (1990), 241–249. doi: 10.1016/0165-0114(90)90197-E. doi: 10.1016/0165-0114(90)90197-E
    [22] M. Gürbüz, Ç. Yıldız, Some new inequalities for convex functions via Riemann-Liouville fractional integrals, Appl. Math. Nonlinear Sci., 6 (2021), 537–544. doi: 10.2478/amns.2020.2.00015. doi: 10.2478/amns.2020.2.00015
    [23] J. R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18 (1986), 31–43. doi: 10.1016/0165-0114(86)90026-6. doi: 10.1016/0165-0114(86)90026-6
    [24] O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301–317. doi: 10.1016/0165-0114(87)90029-7. doi: 10.1016/0165-0114(87)90029-7
    [25] M. L. Puri, D. A. Ralescu, Fuzzy random variables, in Readings in fuzzy sets for intelligent systems, Morgan Kaufmann, (1993), 265–271. doi: 10.1016/B978-1-4832-1450-4.50029-8.
    [26] E. J. Rothwell, M. J. Cloud, Automatic error analysis using intervals, IEEE Trans. Ed., 55 (2012), 9–15. doi: 10.1016/B978-1-4832-1450-4.50029-8. doi: 10.1016/B978-1-4832-1450-4.50029-8
    [27] A. O. Akdemir, E. Deniz, E. Yüksel, On some integral inequalities via conformable fractional integrals, Appl. Math. Nonlinear Sci., 6 (2021), 489–498. doi: 10.2478/amns.2020.2.00071. doi: 10.2478/amns.2020.2.00071
    [28] L. A. Zadeh, Fuzzy sets, in Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh, (1996), 394–432.
    [29] I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe J. Math. Stat., 43 (2014), 935–942.
    [30] 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. doi: 10.1590/S1807-03022012000300002. doi: 10.1590/S1807-03022012000300002
    [31] M. A. Noor, Fuzzy preinvex functions, Fuzzy Sets Syst., 64 (1994), 95–104. doi: 10.1016/0165-0114(94)90011-6. doi: 10.1016/0165-0114(94)90011-6
    [32] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets Syst., 151 (2005), 581–599. doi: 10.1016/j.fss.2004.08.001. doi: 10.1016/j.fss.2004.08.001
    [33] A. Ben-Isreal, B. Mond, What is invexity? The Anziam J., 28 (1986), 1–9.
    [34] J. Cervelati, M. D. Jiménez-Gamero, F. Vilca-Labra, M. A. Rojas-Medar, Continuity for s-convex fuzzy processes, in Soft methodology and random information systems, Springer, Berlin, Heidelberg, 7 (2004), 653–660. doi: 10.1007/978-3-540-44465-7_81.
    [35] Y. Chalco-Cano, M. A. Rojas-Medar, H. Román-Flores, M-convex fuzzy mappings and fuzzy integral mean, Comput. Math. Appl., 40 (2000), 1117–1126. doi: 10.1016/S0898-1221(00)00226-1. doi: 10.1016/S0898-1221(00)00226-1
    [36] M. S. Mohan, S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901–908. doi: 10.1006/jmaa.1995.1057. doi: 10.1006/jmaa.1995.1057
    [37] A. Vanli, I. Ünal, D. Özdemir, Normal complex contact metric manifolds admitting a semi symmetric metric connection, Appl. Math. Nonlinear Sci., 5 (2020), 49–66. doi: 10.2478/amns.2020.2.00013. doi: 10.2478/amns.2020.2.00013
    [38] M. Sharifi, B. Raesi, Vortex theory for two dimensional boussinesq equations, Appl. Math. Nonlinear Sci., 5 (2020), 67–84. doi: 10.2478/AMNS.2020.2.00014. doi: 10.2478/AMNS.2020.2.00014
    [39] M. R. Kanna, R. P. Kumar, S. Nandappa, I. Cangul, On solutions of fractional order telegraph partial differential equation by Crank-Nicholson finite difference method, Appl. Math. Nonlinear Sci., 5 (2020), 85–98. doi: 10.2478/AMNS.2020.2.00017. doi: 10.2478/AMNS.2020.2.00017
    [40] P. Harisha, Ranjini, V. Lokesha, S. Kumar, Degree sequence of graph operator for some standard graphs, Appl. Math. Nonlinear Sci., 5 (2020), 99–108. doi.: 10.2478/AMNS.2020.2.00018.
    [41] R. Osuna-G´omez, M. D. Jim´enez-Gamero, Y. Chalco-Cano, M. A. Rojas-Medar, Hadamard and Jensen inequalities for s-convex fuzzy processes, in Soft Methodology and Random Information Systems. Advances in Soft Computing, Springer, Berlin, Heidelberg, 26 (2004), 1–15. doi: 10.1007/978-3-540-44465-7_80.
    [42] 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. doi: 10.1016/j.fss.2017.02.001
    [43] 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. doi: 10.1016/j.ins.2017.08.055
    [44] I. Iscan, Hermite-Hadamard type inequalities for p-convex functions, Int. J. Anal. Appl., 11 (2016), 137–145.
    [45] S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Sets Syst., 48 (1992), 129–132. doi: 10.1016/0165-0114(92)90256-4. doi: 10.1016/0165-0114(92)90256-4
    [46] 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/s13662-021-03245-8. doi: 10.1186/s13662-021-03245-8
    [47] 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. doi: 10.1007/s40747-021-00379-w. doi: 10.1007/s40747-021-00379-w
    [48] M. B. Khan, M. A. Noor, L. Abdullah, K. I. Noor, New Hermite-Hadamard and Jensen inequalities for Log-h-Convex Fuzzy-interval-valued functions, Int. J. Comput. Intell. Syst., 14 (2021), 155. doi: 10.1007/s44196-021-00004-1. doi: 10.1007/s44196-021-00004-1
    [49] 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. doi: 10.2991/ijcis.d.210409.001
    [50] 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. doi: 10.3390/sym13040673
    [51] M. B. Khan, P. O. Mohammed, M. A. Noor, K. Abuahalnaja, Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions, Math. Biosci. Eng., 18 (2021), 6552–6580. doi: 10.3934/mbe.2021325. doi: 10.3934/mbe.2021325
    [52] 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.
    [53] M. B. Khan, P. O. Mohammed, M. A. Noor, D. Baleanu, J. L. G. Guirao, Some new fractional estimates of inequalities for LR-p-convex interval-valued functions by means of pseudo order relation, Axioms, 10 (2021), 1–18. doi: 10.3390/axioms10030175. doi: 10.3390/axioms10030175
    [54] 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. doi: 10.1155/2021/6652930. doi: 10.1155/2021/6652930
    [55] M. B. Khan, M. A. Noor, T. Abdeljawad, B. Abdalla, A. Althobaiti, Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions, AIMS Math., 7 (2022), 349–370. doi: 10.3934/math.2022024. doi: 10.3934/math.2022024
    [56] R. Şahin, O. Yağcı, Fractional calculus of the extended Hypergeometric function, Appl. Math. Nonlinear Sci., 5 (2020), 369–384. doi: 10.2478/AMNS.2020.1.00035. doi: 10.2478/AMNS.2020.1.00035
    [57] M. B. Khan, M. A. Noor, K. I. Noor, K. S. Nisar, K. A. Ismail, A. Elfasakhany, Some Inequalities for LR-(h1, h2)(h1, h2)-Convex Interval-Valued Functions by Means of Pseudo Order Relation, Int. J. Comput. Intell. Syst., 14 (2021), 1–15. doi: 10.1007/s44196-021-00032-x. doi: 10.1007/s44196-021-00032-x
    [58] M. B. Khan, M. A. Noor, T. Abdeljawad, B. Abdalla, A. Althobaiti, Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions, AIMS Math., 7 (2022), 349–370. doi: 10.3934/math.2022024. doi: 10.3934/math.2022024
    [59] M. B. Khan, M. A. Noor, P. O. Mohammed, J. L. Guirao, K. I. Noor, Some integral inequalities for generalized convex fuzzy-interval-valued functions via fuzzy Riemann integrals, Int. J. Comput. Intell. Syst., 4 (2021), 1–15. doi: 10.1007/s44196-021-00009-w. doi: 10.1007/s44196-021-00009-w
    [60] M. B. Khan, P. O. Mohammed, M. A. Noor, Y. S. Hameed. 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. doi: 10.3934/math.2021637
    [61] 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). doi: 10.2991/ijcis.d.210616.001. doi: 10.2991/ijcis.d.210616.001
    [62] 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). doi:10.2991/ijcis.d.210620.001. doi: 10.2991/ijcis.d.210620.001
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