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
[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. |