Research article

On interval-valued $\mathbb{K}$-Riemann integral and Hermite-Hadamard type inequalities

  • Received: 25 July 2020 Accepted: 11 November 2020 Published: 16 November 2020
  • MSC : 26A51, 26D15, 26E25

  • We introduce the concepts of $\mathbb{K}$-Riemann integral and radial $\mathbb{K}$-g$H$-derivative for interval-valued functions. We also give some important properties of interval-valued $\mathbb{K}$-Riemann integral, and extend interval-valued Hermite-Hadamard type inequalities in the case of $\mathbb{K}$-Riemann integral. Several examples are shown to illustrate the results.

    Citation: Zehao Sha, Guoju Ye, Dafang Zhao, Wei Liu. On interval-valued $\mathbb{K}$-Riemann integral and Hermite-Hadamard type inequalities[J]. AIMS Mathematics, 2021, 6(2): 1276-1295. doi: 10.3934/math.2021079

    Related Papers:

  • We introduce the concepts of $\mathbb{K}$-Riemann integral and radial $\mathbb{K}$-g$H$-derivative for interval-valued functions. We also give some important properties of interval-valued $\mathbb{K}$-Riemann integral, and extend interval-valued Hermite-Hadamard type inequalities in the case of $\mathbb{K}$-Riemann integral. Several examples are shown to illustrate the results.


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    [1] T. M. Costa, Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Set. Syst., 327 (2017), 31-47. doi: 10.1016/j.fss.2017.02.001
    [2] R. E. Moore, Interval Analysis, Prentice-Hall, 1966.
    [3] M. Kuczma, An introduction to the theory of functional equations and inequalities, Birkhäuser, Basel, 2009.
    [4] A. Olbryś, On the $\mathbb{K}$-Riemann integral and Hermite-Hadamard inequalities for $\mathbb{K}$-convex functions, Aequationes Math., 91 (2017), 429-444.
    [5] V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Set. Syst., 265 (2015), 63-85. doi: 10.1016/j.fss.2014.04.005
    [6] S. Markov, Calculus for interval functions of a real variables, Computing, 22 (1979), 325-337. doi: 10.1007/BF02265313
    [7] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal-Theor., 71 (2009), 1311-1328. doi: 10.1016/j.na.2008.12.005
    [8] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Set. Syst., 161 (2010), 1564-1584. doi: 10.1016/j.fss.2009.06.009
    [9] Y. Chalco-Cano, Gino G. Maqui-Huamán, G. N. Silva, M. D. Jiménez-Gamero, Algebra of generalized Hukuhara differentiable interval-valued functions: review and new properties, Fuzzy Set. Syst., 375 (2019), 53-69.
    [10] M. Z. Sarikaya, A. Saglam, H. Yildirim, On some Hadamard-type inequalities for h-convex functions, J. Math. Inequal., 2 (2008), 335-341.
    [11] M. Z. Sarikaya, H. Budak, Generalized Hermite-Hadamard type integral inequalities for fractional integrals, Filomat, 30 (2016), 1315-1326. doi: 10.2298/FIL1605315S
    [12] H. Budak, T. Tunç, M. Z. Sarikaya, Fractional Hermite-Hadamard type inequalities for intervalvalued functions, P. Am. Math. Soc., 148 (2019), 705-718. doi: 10.1090/proc/14741
    [13] G. Debreu, Integration of correspondences, Proc. Fifth Berkeley Symp. on Math. Statist. and Prob., 2 (1967), 351-372.
    [14] D. F. Zhao, M. A. Ali, A. Kashuri, H. Budak, Generalized fractional integral inequalities of Hermite-Hadamard type for harmonically convex functions, Adv. Differ. Equ., 2020 (2020), 1-14. doi: 10.1186/s13662-019-2438-0
    [15] D. F. Zhao, T. Q. An, G. J. Ye, D. F. M. Torres, On Hermite-Hadamard type inequalities for harmonical h-convex interval-valued functions, Math. Inequal. Appl., 23 (2020), 95-105.
    [16] Z. H. Sha, G. J. Ye, D. F. Zhao, W. Liu, On some Hermite-Hadamard type inequalities for T -convex interval-valued functions, Adv. Differ. Equ., 2020 (2020), 1-15. doi: 10.1186/s13662-019-2438-0
    [17] D. F. Zhao, M. A. Ali, G. Murtaza, Z. Y. Zhang, On the Hermite-Hadamard inequalities for intervalvalued coordinated convex functions, Adv. Difference Equ. (2020), 570.
    [18] D. Ghosh, R. S. Chauhan, R. Mesiar, A. K. Debnath, Generalized Hukuhara Gateaux and Fréchet derivatives of interval-valued functions and their application in optimization with interval-valued functions, Inform. Sciences, 510 (2020), 317-340. doi: 10.1016/j.ins.2019.09.023
    [19] M. Alomari, M. Darus, On The Hadamard's Inequality for Log-Convex Functions on the Coordinates, J. Inequal. Appl., 2009 (2009), 1-13.
    [20] M. Štrboja, T. Grbić, I. Štajiner-Papuga, G. Grujić, S. Medić, Jensen and Chebyshev inequalities for pseudo-integrals of set-valued functions, Fuzzy Set. Syst., 222 (2013), 18-32.
    [21] D. Zhang, C. Guo, D. Chen, G. Wang, Jensen's inequalities for set-valued and-fuzzy set-valued functions, Fuzzy Set. Syst., https://doi.org/10.1016/j.fss.2020.06.003.
    [22] Z. Boros, Z. Páles, $\mathbb{Q}$-subdifferential of Jensen-convex functions, J. Math. Anal. Appl., 321 (2006), 99-113.
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