Geodesic
On the existence of generalized weak solutions to discontinuous fuzzy differential equations
Shao, Ya-Bin 1 ; Gong, Zeng-Tai 2 ; Chen, Zi-Zhong 3
1 School of Science, Chongqing University of Posts and Telecommunications, 400065 Nanan, Chongqing, People's Republic of China
2 College of Mathematics and Statistics, Northwest Normal University, 730070 Lanzhou, Gansu, People's Republic of China
3 College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, 400065 Nanan, Chongqing, People's Republic of China
Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 12, p. 6274-6287.

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

In this paper, by means of replacing the Lebesgue integrability of support functions with its Henstock integrability, the definitions of the Henstock-Pettis integral of $n$-dimensional fuzzy-number-valued functions are defined. In addition, the controlled convergence theorems for such integrals are considered. As the applications of these integrals, we provide some existence theorems of generalized weak solutions to initial value problems for the discontinuous fuzzy differential equations under the strong GH-differentiability.
DOI : 10.22436/jnsa.010.12.12
Classification : 03E72, 26A39, 46G05, 34A07
Keywords: Fuzzy number, fuzzy Henstock-Pettis integral, convergence theorem, discontinuous fuzzy differential equation, generalized weak solution

Shao, Ya-Bin  1 ; Gong, Zeng-Tai  2 ; Chen, Zi-Zhong  3

1 School of Science, Chongqing University of Posts and Telecommunications, 400065 Nanan, Chongqing, People's Republic of China
2 College of Mathematics and Statistics, Northwest Normal University, 730070 Lanzhou, Gansu, People's Republic of China
3 College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, 400065 Nanan, Chongqing, People's Republic of China 
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Shao, Ya-Bin ; Gong, Zeng-Tai ; Chen, Zi-Zhong . On the existence of generalized weak solutions to discontinuous fuzzy differential equations. Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 12, p. 6274-6287. doi : 10.22436/jnsa.010.12.12. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.12.12/

[1] Banaś, J.; Goebel, K. Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1980

[2] Barrosa, L. C.; Gomesa, L. T.; Tonelli, P. A. Fuzzy differential equations: an approach via fuzzification of the derivative operator, Fuzzy Sets and Systems, Volume 230 (2013), pp. 39-52 | Zbl | DOI

[3] Bede, B.; Gal, S. G. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, Volume 151 (2005), pp. 581-599 | Zbl | DOI

[4] Bede, B.; Stefanini, L. Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, Volume 230 (2013), pp. 119-141 | DOI

[5] Bica, A. M. One-sided fuzzy numbers and applications to integral equations from epidemiology, Fuzzy Sets and Systems, Volume 219 (2013), pp. 27-48 | DOI | Zbl

[6] Chalco-Cano, Y.; H. Román-Flores On new solutions of fuzzy differential equations, Chaos Solitons Fractals, Volume 38 (2008), pp. 112-119 | DOI

[7] Chen, M.-H.; Han, C.-S. Some topological properties of solutions to fuzzy differential systems, Inform. Sci., Volume 197 (2012), pp. 207-214 | DOI | Zbl

[8] Chen, M.-H.; Li, D.-H.; Xue, X.-P. Periodic problems of first order uncertain dynamical systems , Fuzzy Sets and Systems, Volume 162 (2011), pp. 67-78 | Zbl | DOI

[9] Chew, T. S.; Flordeliza, F. On \(x' = f(t, x)\) and Henstock-Kurzweil integrals, Differential Integral Equations, Volume 4 (1991), pp. 861-868

[10] Cichoń, M. Convergence theorems for the Henstock-Kurzweil-Pettis integral, Acta Math. Hungar., Volume 92 (2001), pp. 75-82 | Zbl | DOI

[11] Cichoń, M.; Kubiaczyk, I.; Sikorska, A. The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem, Czechoslovak Math. J., Volume 54 (2004), pp. 279-289 | DOI | Zbl

[12] Diamond, P.; Kloeden, P. Metric spaces of fuzzy sets, Theory and applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1994

[13] Piazza, L. Di; K. Musiał Set-valued Kurzweil-Henstock-Pettis integral , Set-Valued Anal., Volume 13 (2005), pp. 167-169 | Zbl | DOI

[14] Amri, K. El; Hess, C. On the Pettis integral of closed valued multifunctions, Set-Valued Anal., Volume 8 (2000), pp. 329-360 | DOI

[15] Gong, Z.-T. 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.-T.; Y.-B. Shao Global existence and uniqueness of solutions for fuzzy differential equations under dissipative-type conditions, Comput. Math. Appl., Volume 56 (2008), pp. 2716-2723 | Zbl | DOI

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

[18] Gong, Z.-T.; Wang, L.-L. The Henstock-Stieltjes integral for fuzzy-number-valued functions, Inform. Sci., Volume 188 (2012), pp. 276-297 | Zbl | DOI

[19] Gong, Z.-T.; Wu, C.-X. Bounded variation, absolute continuity and absolute integrability for fuzzy-number-valued functions, Fuzzy Sets and Systems, Volume 129 (2002), pp. 83-94 | Zbl | DOI

[20] Guo, M.-S.; Peng, X.-Y.; Xu, Y.-Q. Oscillation property for fuzzy delay differential equations, Fuzzy Sets and Systems, Volume 200 (2012), pp. 25-35 | DOI | Zbl

[21] Henstock, R. Theory of integration , Butterworths, London, 1963

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

[23] Kloeden, P. E.; T. Lorenz Fuzzy differential equations without fuzzy convexity, Fuzzy Sets and Systems, Volume 230 (2013), pp. 65-81 | DOI | Zbl

[24] I. Kubiaczyk On a fixed point theorem for weakly sequentially continuous mappings, Discuss. Math. Differential Incl., Volume 15 (1995), pp. 15-20 | EuDML

[25] Kurzweil, J. Generalized ordinary differential equations and continuous dependence on a parameter, (Russian) Czechoslovak Math. J., Volume 7 (1957), pp. 418-446

[26] P. Y. Lee Lanzhou lectures on Henstock integration, Series in Real Analysis, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989

[27] Puri, M. L.; D. A. Ralescu Fuzzy random variables, J. Math. Anal. Appl., Volume 114 (1986), pp. 409-422

[28] S. Schwabik Generalized ordinary differential equations, Series in Real Analysis, World Scientific Publishing Co., Inc., River Edge, NJ, 1992

[29] Schwabik, S.; Ye, G.-J. Topics in Banach space integration, Series in Real Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005 | DOI

[30] Seikkala, S. On the fuzzy initial value problem , Fuzzy Sets and Systems, Volume 24 (1987), pp. 319-330 | DOI

[31] Shao, Y.-B.; Z.-T. Gong Discontinuous fuzzy systems and Henstock integrals of fuzzy number valued functions, Int. Conf. Intell. Comput., Springer, Berlin, Heidelberg, Volume 7389 (2012), pp. 65-72 | DOI

[32] Shao, Y.-B.; Xue, G.-L. Discontinuous fuzzy Fredholm integral equations and strong fuzzy Henstock integrals, Articial Intelligence Res., Volume 2 (2013), pp. 87-95 | DOI

[33] Shao, Y.-B.; Zhang, H.-H. The strong fuzzy Henstock integrals and discontinuous fuzzy differential equations, J. Appl. Math., Volume 2013 (2013), pp. 1-8

[34] Shao, Y.-B.; Zhang, H.-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

[35] Shao, Y.-B.; H.-H. Zhang Fuzzy integral equations and strong fuzzy Henstock integrals, Abstr. Appl. Anal., Volume 2014 (2014), pp. 1-8

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

[37] Wu, C.-X.; Z.-T. Gong On Henstock integral of fuzzy-number-valued functions, I, Fuzzy Sets and Systems, Volume 120 (2001), pp. 523-552 | Zbl | DOI

[38] Wu, C.-X.; Ma, M. Embedding problem of fuzzy number space, II, Fuzzy Sets and Systems, Volume 45 (1992), pp. 189-202 | DOI | Zbl

[39] Xue, X.-P.; Fu, Y.-Q. Carathéodory solutions of fuzzy differential equations, Fuzzy Sets and Systems, Volume 125 (2002), pp. 239-243 | DOI

[40] Zhang, B.-K.; C.-X. Wu On the representation of n-dimensional fuzzy numbers and their informational context, Fuzzy Sets and Systems, Volume 128 (2002), pp. 227-235 | DOI

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