×

Some generalizations of Opial type inequalities for interval-valued functions. (English) Zbl 1522.26035

Summary: We establish some new Opial type inequalities for interval-valued functions, and obtain some generalizations and refinements of interval Opial type inequalities derived by T. M. Costa et al. [ibid. 358, 48–63 (2019; Zbl 1423.26039)]. Also, some examples are presented to illustrate our results.

MSC:

26E50 Fuzzy real analysis
26D15 Inequalities for sums, series and integrals

Citations:

Zbl 1423.26039
Full Text: DOI

References:

[1] Zhang, D. L.; Guo, C. M.; Chen, D. G.; Wang, G. J., Jensen’s inequalities for set-valued functions and fuzzy set-valued functions, Fuzzy Sets Syst., 404, 178-204 (2021) · Zbl 1464.26025
[2] Agarwal, R. P.; O’Regan, D.; Saker, S. H., Some Hardy type inequalities with weighted functions via Opial type inequalities, Adv. Dyn. Syst. Appl., 10, 1-9 (2015)
[3] Agarwal, R. P.; Pang, P. Y.H., Opial Inequalities with Applications in Differential and Difference Equations (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0821.26013
[4] Aubin, J. P.; Cellina, A., Differential Inclusions (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0538.34007
[5] Burkill, J. C., Functions of intervals, Proc. Lond. Math. Soc., 22, 275-310 (1924) · JFM 49.0177.02
[6] Chalco-Cano, Y.; Flores-Franulič, A.; Román-Flores, H., Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31, 457-472 (2012) · Zbl 1263.26039
[7] Chalco-Cano, Y.; Lodwick, W. A.; Condori-Equice, W., Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput., 19, 3293-3300 (2015) · Zbl 1362.26028
[8] Chalco-Cano, Y.; Silva, G. N.; Rufián-Lizana, A., On the Newton method for solving fuzzy optimization problems, Fuzzy Sets Syst., 272, 60-69 (2015) · Zbl 1374.90430
[9] Costa, T. M., Jensen’s inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets Syst., 327, 31-47 (2017) · Zbl 1382.28017
[10] Costa, T. M.; Chalco-Cano, Y.; Lodwick, W. A.; Silva, G. N., A new approach to linear interval differential equations as a first step toward solving fuzzy differential, Fuzzy Sets Syst., 347, 129-141 (2018) · Zbl 1397.34015
[11] Costa, T. M.; Román-Flores, H., Some integral inequalities for fuzzy-interval-valued functions, Inf. Sci., 420, 110-125 (2017) · Zbl 1445.28022
[12] Costa, T. M.; Román-Flores, H.; Chalco-Cano, Y., Opial-type inequalities for interval-valued functions, Fuzzy Sets Syst., 358, 48-63 (2019) · Zbl 1423.26039
[13] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049
[14] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE Trans. Autom. Control, 26, 926-936 (1981) · Zbl 1457.68262
[15] Dubois, D.; Prade, H., Gradualness, uncertainty and bipolarity: making sense of fuzzy sets, Fuzzy Sets Syst., 192, 3-24 (2012) · Zbl 1238.03044
[16] Flores-Franulič, A.; Chalco-Cano, Y.; Román-Flores, H., An Ostrowski type inequality for interval-valued functions, (IFSA World Congress and NAFIPS Annual Meeting IEEE, vol. 35 (2013)), 1459-1462
[17] Gasilov, N. A.; Amrahov, Ş. E., Solving a nonhomogeneous linear system of interval differential equations, Soft Comput., 22, 3817-3828 (2018) · Zbl 1398.65161
[18] Hansen, E., Global Optimization Using Interval Analysis (2004), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 1103.90092
[19] Hukuhara, M., Integration des applications mesurables dont la valeur est un compact convexe, Funkc. Ekvacioj, 10, 205-223 (1967) · Zbl 0161.24701
[20] Kolmogorov, A. N., Untersuchungen über Integralbegriff, Math. Ann., 103, 654-696 (1930) · JFM 56.0923.01
[21] Li, Y.; Wang, T. H., Interval analysis of the wing divergence, Aerosp. Sci. Technol., 74, 17-21 (2018)
[22] Lin, C. T., A further generalization of Opial’s integral inequality, Tamkang J. Math., 27, 491-493 (1989) · Zbl 0674.26007
[23] Lodwick, W.; Dubois, D., Interval linear systems as a necessary step in fuzzy linear systems, Fuzzy Sets Syst., 281, 227-251 (2015) · Zbl 1368.15029
[24] Lupulescu, V., Hukuhara differentiability of interval-valued functions and interval differential equations on time scales, Inf. Sci., 248, 50-67 (2013) · Zbl 1339.34098
[25] Lupulescu, V., Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265, 63-85 (2015) · Zbl 1361.26001
[26] Lupulescu, V.; O’Regan, D., A new derivative concept for set-valued and fuzzy-valued functions. Differential and integral calculus in quasilinear metric spaces, Fuzzy Sets Syst., 404, 75-110 (2021) · Zbl 1464.26030
[27] Ma, H. L.; Xu, S. J., Optimization of bounded low-thrust rendezvous with terminal constraints by interval analysis, Aerosp. Sci. Technol., 79, 58-69 (2018)
[28] Moore, R. E., Interval Analysis (1966), Prentice-Hall, Inc.: Prentice-Hall, Inc. Englewood Cliffs, N.J. · Zbl 0176.13301
[29] Nikodem, K.; Sánchez, J. L.; Sánchez, L., Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Math. Æterna, 4, 979-987 (2014)
[30] Opial, Z., Sur une inégalité, Ann. Pol. Math., 8, 29-32 (1960) · Zbl 0089.27403
[31] Qin, Y. M., Analytic Inequalities and Their Applications in PDEs, Operator Theory: Advances and Applications, vol. 241 (2017), Birkhäuser/Springer · Zbl 1362.35002
[32] Rojas-Medar, M. A.; Jiménez-Gamero, M. D.; Chalco-Cano, Y.; Viera-Brandão, A. J., Fuzzy quasilinear spaces and applications, Fuzzy Sets Syst., 152, 173-190 (2005) · Zbl 1086.47063
[33] Román-Flores, H.; Chalco-Cano, Y.; Lodwick, W. A., Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37, 1306-1318 (2018) · Zbl 1393.26029
[34] Stefanini, L.; Arana-Jiménez, M., Karush-Kuhn-Tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability, Fuzzy Sets Syst., 362, 1-34 (2019) · Zbl 1423.90288
[35] Stefanini, L., A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst., 161, 1564-1584 (2010) · Zbl 1188.26019
[36] Stefanini, L.; Bede, B., Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71, 1311-1328 (2009) · Zbl 1188.28002
[37] Štrboja, M.; Grbić, T.; Štajner-Papuga, I.; Grujić, G.; Medić, S., Jensen and Chebyshev inequalities for pseudo-integrals of set-valued functions, Fuzzy Sets Syst., 222, 18-32 (2013) · Zbl 1284.28007
[38] Zhao, D. F.; An, T. Q.; Ye, G. J.; Liu, W., New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 2018, 302 (2018) · Zbl 1498.26078
[39] Zhao, D. F.; Ye, G. J.; Liu, W.; Torres, D. F.M., Some inequalities for interval-valued functions on time scales, Soft Comput., 23, 6005-6015 (2019) · Zbl 1418.26009
[40] Zhao, D. F.; An, T. Q.; Ye, G. J.; Liu, W., Chebyshev type inequalities for interval-valued functions, Fuzzy Sets Syst., 396, 82-101 (2020) · Zbl 1464.26026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.