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December  2020, 13(12): 3305-3317. doi: 10.3934/dcdss.2020111

Fractional Ostrowski-Sugeno Fuzzy univariate inequalities

George A. Anastassiou

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

Received  August 2018 Revised  December 2018 Published  December 2020 Early access  October 2019

Here we present fractional univariate Ostrowski-Sugeno Fuzzy type inequalities. These are of Ostrowski-like inequalities in the setting of Sugeno fuzzy integral and its special-particular properties. In a fractional environment, they give tight upper bounds to the deviation of a function from its Sugeno-fuzzy averages. The fractional derivatives we use are of Canavati and Caputo types. This work is greatly inspired by [8], [1] and [2].

Keywords: Fractional derivative, Sugeno fuzzy integral, deviation from fuzzy average, fuzzy Ostrowski inequality, fuzzy Polya inequality.
Mathematics Subject Classification: Primary: 26A33, 26D07, 26D10, 26D15, 41A44; Secondary: 26A24, 26D20, 28A25.
Citation: George A. Anastassiou. Fractional Ostrowski-Sugeno Fuzzy univariate inequalities. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3305-3317. doi: 10.3934/dcdss.2020111
References:
[1]

G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-98128-4.  Google Scholar

[2]

G. A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4614-0703-4.  Google Scholar

[3]

G. A. Anastassiou, Intelligent Mathematics: Computational Analysis, Intelligent Systems Reference Library, 5. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-17098-0.  Google Scholar

[4]

G. A. Anastassiou, Intelligent Comparisons: Analytic Inequalities, Studies in Computational Intelligence, 609. Springer, Cham, 2016. doi: 10.1007/978-3-319-21121-3.  Google Scholar

[5]

M. Boczek and M. Kaluszka, On the Minkowaki-Hölder type inequalities for generalized Sugeno integrals with an application, Kybernetika (Prague), 52 (2016), 329-347.  doi: 10.14736/kyb-2016-3-0329.  Google Scholar

[6]

J. A. Canavati, The Riemann-Liouville integral, Nieuw Arch. Wisk., 5 (1987), 53-75.   Google Scholar

[7]

K. Diethelm, The Analysis of Fractional Differential Equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[8]

A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, (German) Comment. Math. Helv., 10 (1938), 226-227.  doi: 10.1007/BF01214290.  Google Scholar

[9]

E. Pap, Null-Additive Set functions, Mathematics and its Applications, 337, Kluwer Academic Publishers Group, Dordrecht; Ister Science, Bratislava, 1995.  Google Scholar

[10]

D. Ralescu and G. Adams, The fuzzy integral, J. Math. Anal. Appl., 75 (1980), 562-570.  doi: 10.1016/0022-247X(80)90101-8.  Google Scholar

[11]

M. Sugeno, Theory of Fuzzy Integrals and Its Applications[J], PhD thesis, Tokyo Institute of Technology, 1974. Google Scholar

[12] Z. Wang and G. J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4757-5303-5.  Google Scholar

show all references

References:
[1]

G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, Dordrecht, 2009. doi: 10.1007/978-0-387-98128-4.  Google Scholar

[2]

G. A. Anastassiou, Advances on Fractional Inequalities, SpringerBriefs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4614-0703-4.  Google Scholar

[3]

G. A. Anastassiou, Intelligent Mathematics: Computational Analysis, Intelligent Systems Reference Library, 5. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-17098-0.  Google Scholar

[4]

G. A. Anastassiou, Intelligent Comparisons: Analytic Inequalities, Studies in Computational Intelligence, 609. Springer, Cham, 2016. doi: 10.1007/978-3-319-21121-3.  Google Scholar

[5]

M. Boczek and M. Kaluszka, On the Minkowaki-Hölder type inequalities for generalized Sugeno integrals with an application, Kybernetika (Prague), 52 (2016), 329-347.  doi: 10.14736/kyb-2016-3-0329.  Google Scholar

[6]

J. A. Canavati, The Riemann-Liouville integral, Nieuw Arch. Wisk., 5 (1987), 53-75.   Google Scholar

[7]

K. Diethelm, The Analysis of Fractional Differential Equations, An application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[8]

A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Integralmittelwert, (German) Comment. Math. Helv., 10 (1938), 226-227.  doi: 10.1007/BF01214290.  Google Scholar

[9]

E. Pap, Null-Additive Set functions, Mathematics and its Applications, 337, Kluwer Academic Publishers Group, Dordrecht; Ister Science, Bratislava, 1995.  Google Scholar

[10]

D. Ralescu and G. Adams, The fuzzy integral, J. Math. Anal. Appl., 75 (1980), 562-570.  doi: 10.1016/0022-247X(80)90101-8.  Google Scholar

[11]

M. Sugeno, Theory of Fuzzy Integrals and Its Applications[J], PhD thesis, Tokyo Institute of Technology, 1974. Google Scholar

[12] Z. Wang and G. J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4757-5303-5.  Google Scholar
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