American Institute of Mathematical Sciences

Advanced
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  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, 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
[1]

Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118
[2]

Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial & Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581
[3]

Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082
[4]

Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699
[5]

Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619
[6]

Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008
[7]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[8]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
[9]

Wei Wang, Xiao-Long Xin. On fuzzy filters of Heyting-algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1611-1619. doi: 10.3934/dcdss.2011.4.1611
[10]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[11]

Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861
[12]

Guojun Gan, Qiujun Lan, Shiyang Sima. Scalable clustering by truncated fuzzy $c$-means. Big Data & Information Analytics, 2016, 1 (2&3) : 247-259. doi: 10.3934/bdia.2016007
[13]

Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 371-386. doi: 10.3934/jimo.2018157
[14]

Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158
[15]

Jian Luo, Xueqi Yang, Ye Tian, Wenwen Yu. Corporate and personal credit scoring via fuzzy non-kernel SVM with fuzzy within-class scatter. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019078
[16]

Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457
[17]

Pawan Lingras, Farhana Haider, Matt Triff. Fuzzy temporal meta-clustering of financial trading volatility patterns. Big Data & Information Analytics, 2018  doi: 10.3934/bdia.2017018
[18]

Yung Chung Wang, Jenn Shing Wang, Fu Hsiang Tsai. Analysis of discrete-time space priority queue with fuzzy threshold. Journal of Industrial & Management Optimization, 2009, 5 (3) : 467-479. doi: 10.3934/jimo.2009.5.467
[19]

Aliki D. Muradova, Georgios K. Tairidis, Georgios E. Stavroulakis. Adaptive Neuro-Fuzzy vibration control of a smart plate. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 251-271. doi: 10.3934/naco.2017017
[20]

Jian Luo, Shu-Cherng Fang, Yanqin Bai, Zhibin Deng. Fuzzy quadratic surface support vector machine based on fisher discriminant analysis. Journal of Industrial & Management Optimization, 2016, 12 (1) : 357-373. doi: 10.3934/jimo.2016.12.357

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (80)
  • HTML views (267)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences