Geodesic
The fuzzy $C$-delta integral on time scales
You, Xuexiao 1 ; Zhao, Dafang 2 ; Cheng, Jian 3 ; Li, Tongxing 4
1 College of Computer and Information, Hohai University, Nanjing, Jiangsu 210098, P. R. China;School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China
2 School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China;College of Science, Hohai University, Nanjing, Jiangsu 210098, P. R. China
3 School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China
4 LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P. R. China;School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P. R. China
Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 1, p. 161-171.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this paper, we introduce and study the $C$-delta integral of interval-valued functions and fuzzy-valued functions on time scales. Also, some basic properties of the fuzzy $C$-delta integral are proved. Finally, we give two necessary and sufficient conditions of integrability.
DOI : 10.22436/jnsa.011.01.12
Classification : 26A42, 26E50, 26E70
Keywords: \(C\)-Delta integral, fuzzy-valued function, time scale

You, Xuexiao  1 ; Zhao, Dafang  2 ; Cheng, Jian  3 ; Li, Tongxing  4

1 College of Computer and Information, Hohai University, Nanjing, Jiangsu 210098, P. R. China;School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China
2 School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China;College of Science, Hohai University, Nanjing, Jiangsu 210098, P. R. China
3 School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China
4 LinDa Institute of Shandong Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P. R. China;School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, P. R. China 
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You, Xuexiao ; Zhao, Dafang ; Cheng, Jian ; Li, Tongxing . The fuzzy \(C\)-delta integral on time scales. Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 1, p. 161-171. doi : 10.22436/jnsa.011.01.12. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.011.01.12/

[1] Benkhettou, N.; Cruz, A. M. C. Brito da; Torres, D. F. M. A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration, Signal Process., Volume 107 (2015), pp. 230-237 | DOI

[2] Bohner, M.; A. Peterson Dynamic equations on time scales, An introduction with applications, Birkhäuser, Boston, 2001

[3] Bohner, M.; A. Peterson Advances in dynamic equations on time scales, Birkhäuser, , 2003

[4] Bongiorno, B. Un nuovo integrale per il problema delle primitive, Le Matematiche, Volume 51 (1997), pp. 299-313

[5] B. Bongiorno On the minimal solution of the problem of primitives, J. Math. Anal. Appl., Volume 251 (2000), pp. 479-487 | DOI | Zbl

[6] Bongiorno, D. On the problem of nearly derivatives, Sci. Math. Jpn., Volume 61 (2005), pp. 299-311

[7] Bongiorno, B.; Piazza, L. Di; K. Musiał A decomposition theorem for the fuzzy Henstock integral , Fuzzy Sets and Systems, Volume 200 (2012), pp. 36-47 | Zbl | DOI

[8] Bongiorno, B.; Piazza, L. Di; Preiss, D. A constructive minimal integral which includes Lebesgue integrable functions and derivatives, J. London Math. Soc., Volume 62 (2000), pp. 117-126 | Zbl | DOI

[9] Bruckner, A. M.; Fleissner, R. J.; Foran, J. The minimal integral which includes Lebesgue integrable functions and derivatives, Colloq. Math., Volume 2 (1986), pp. 289-293 | DOI | Zbl

[10] Cabada, A.; Vivero, D. R. Criterions for absolute continuity on time scales, J. Difference Equ. Appl., Volume 11 (2005), pp. 1013-1028 | Zbl | DOI

[11] Piazza, L. Di A Riemann-type minimal integral for the classical problem of primitives, Rend. Istit. Mat. Univ. Trieste, Volume 34 (2002), pp. 143-153 | Zbl

[12] K.-F. Duan The Henstock–Stieltjes integral for fuzzy-number-valued functions on a infinite interval , J. Comput. Anal. Appl., Volume 20 (2016), pp. 928-937 | Zbl

[13] Eun, G. S.; Yoon, J. H.; Kim, Y. K.; Kim, B. M. On C-delta integrals on time scales, J. Chungcheong Math. Soc., Volume 25 (2012), pp. 349-357

[14] Goetschel, R.; Voxman, W. Elementary fuzzy calculus, Fuzzy Sets and Systems, Volume 18 (1986), pp. 31-43 | DOI

[15] Gong, Z. On the problem of characterizing derivatives for the fuzzy-valued functions (II): almost everywhere differentiability and strong Henstock integral, Fuzzy Sets and Systems, Volume 145 (2004), pp. 381-393 | DOI | Zbl

[16] Gong, Z.; Shao, Y. The controlled convergence theorems for the strong Henstock integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems, Volume 160 (2009), pp. 1528-1546 | DOI | Zbl

[17] Guseinov, G. S. Integration on time scales, J. Math. Anal. Appl., Volume 285 (2003), pp. 107-127 | DOI

[18] Hilger, S. Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universität Würzburg, Würzburg, Germany, 1988

[19] O. Kaleva Fuzzy differential equations, Fuzzy Sets and Systems, Volume 24 (1987), pp. 301-317 | DOI

[20] Kawasaki, T.; I. Suzuki Criteria for the C-integral , Sci. Math. Jpn., Volume 79 (2016), pp. 155-164 | Zbl

[21] Lee, T.-Y. Multipliers for generalized Riemann integrals in the real line, Math. Bohem., Volume 131 (2006), pp. 161-166 | Zbl

[22] Lee, P. Y.; R. Výborný Integral: An easy approach after Kurzweil and Henstock , Australian Mathematical Society Lecture, Cambridge University Press, Cambridge, 2000

[23] Li, T.; Saker, S. H. A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales, Commun. Nonlinear Sci. Numer. Simul., Volume 19 (2014), pp. 4185-4188 | DOI

[24] K. Musiał A decomposition theorem for Banach space valued fuzzy Henstock integral, Fuzzy Sets and Systems, Volume 259 (2015), pp. 21-28 | DOI | Zbl

[25] Ortigueira, M. D.; Torres, D. F. M.; Trujillo, J. J. Exponentials and Laplace transforms on nonuniform time scales, Commun. Nonlinear Sci. Numer. Simul., Volume 39 (2016), pp. 252-270 | DOI

[26] Park, J. M.; Ryu, H. W.; Lee, H. K.; Lee, D. H. The \(M_\alpha\) and C-integrals, Czechoslovak Math. J., Volume 62 (2012), pp. 869-878 | DOI | Zbl

[27] Peterson, A.; Thompson, B. Henstock–Kurzweil delta and nabla integrals, J. Math. Anal. Appl., Volume 323 (2006), pp. 162-178 | Zbl | DOI

[28] Schwabik, S.; Ye, G. Topics in Banach space integration, World Scientific Publishing Co., Hackensack, 2005 | DOI

[29] Shao, Y.; Zhang, H. Existence of the solution for discontinuous fuzzy integro-differential equations and strong fuzzy Henstock integrals, Nonlinear Dyn. Syst. Theory, Volume 14 (2014), pp. 148-161 | Zbl

[30] E. Talvila The regulated primitive integral , Illinois J. Math., Volume 53 (2009), pp. 1187-1219

[31] C.Wu; Gong, Z. On Henstock integrals of interval-valued functions and fuzzy-valued functions, Fuzzy Sets and Systems, Volume 115 (2000), pp. 377-391 | Zbl | DOI

[32] Wu, C.; Z. Gong On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Sets and Systems, Volume 120 (2001), pp. 523-532 | Zbl | DOI

[33] Ye, G. On Henstock–Kurzweil and McShane integrals of Banach space-valued functions, J. Math. Anal. Appl., Volume 330 (2007), pp. 753-765 | Zbl | DOI

[34] Yoon, J. H. On C-delta integral of Banach space valued functions on time scales, J. Chungcheong Math. Soc., Volume 29 (2016), pp. 157-163 | DOI

[35] You, X.; Zhao, D. On convergence theorems for the McShane integral on time scales, J. Chungcheong Math. Soc., Volume 25 (2012), pp. 393-400

[36] Zhao, D.; Hu, C.-S. The characterization of the primitives of approximately continuous C-integrable functions, J. Math. (Wuhan), Volume 30 (2010), pp. 427-430

[37] Zhao, D.; Li, T. On conformable delta fractional calculus on time scales, J. Math. Computer Sci., Volume 16 (2016), pp. 324-335

[38] Zhao, D.; Ye, G. C-integral and Denjoy-C integral, Commun. Korean Math. Soc., Volume 22 (2007), pp. 27-39 | DOI | Zbl

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