Abstract
This study presents new interval and fuzzy versions of the Jensen’s integral inequality, which extend the classical Jensen’s integral inequality for real-valued functions, using Aumann and Kaleva integrals. The inequalities for interval-valued functions are interpreted through the preference order relations given by Ishibuchi and Tanaka, which are useful for dealing with interval optimization problems. The order relations adopted in the space of fuzzy intervals are extensions of those considered the interval spaces.
Supported by (PNPD/CAPES/UFPA), FONDECYT 1151154, (CNPq) grant 306546/2017-5, and -CEPID-CEMEAI through São Paulo Research Foundation, FAPESP grant 13/07375-0, respectively.
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References
Aubin, J.P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (1984). https://doi.org/10.1007/978-3-642-69512-4
Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12(1), 1–12 (1965)
Bede, B.: Mathematics of Fuzzy Sets and Fuzzy Logic. Studies in Fuzziness and Soft Computing, vol. 295. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35221-8
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977). https://doi.org/10.1007/BFb0087685
Chalco-Cano, Y., Lodwick, W.A., Rufian-Lizana, A.: Optimality conditions of type KKT for optimization problem with interval-valued objective function via generalized derivative. Fuzzy Optim. Decis. Mak. 12, 305–322 (2013)
Costa, T.M.: Jensen’s inequality type integral for fuzzy-interval-valued functions. Fuzzy Sets Syst. 327, 31–47 (2017)
de Barros, L.C., Bassanezi, R.C., Lodwick, W.A.: A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics: Theory and Applications. Studies in Fuzziness and Soft Computing, vol. 347. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-53324-6
Diamond, P., Kloeden, P.E.: Metric Spaces of Fuzzy Sets: Theory and Applications. World Scientific, Singapore (1994)
Goetschel, R., Voxman, W.: Elementary fuzzy calculus. Fuzzy Sets Syst. 18(1), 31–43 (1986)
Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48, 219–225 (1990)
Kaleva, O.: Fuzzy numbers fuzzy differential equations. Fuzzy Sets Syst. 24(3), 301–317 (1987)
Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)
Negoita, C.V., Ralescu, D.A.: Applications of Fuzzy Sets to Systems Analysis. Wiley, New York (1975)
Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)
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da Costa, T.M., Chalco-Cano, Y., de Barros, L.C., Silva, G.N. (2018). Order Relations, Convexities, and Jensen’s Integral Inequalities in Interval and Fuzzy Spaces. In: Barreto, G., Coelho, R. (eds) Fuzzy Information Processing. NAFIPS 2018. Communications in Computer and Information Science, vol 831. Springer, Cham. https://doi.org/10.1007/978-3-319-95312-0_39
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