On the existence of generalized weak solutions to discontinuous fuzzy differential equations

Volume 10, Issue 12, pp 6274--6287 http://dx.doi.org/10.22436/jnsa.010.12.12

Publication Date: December 09, 2017 Submission Date: October 01, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2691 Downloads
• 4791 Views

Authors

Ya-Bin Shao - School of Science, Chongqing University of Posts and Telecommunications, 400065 Nanan, Chongqing, People's Republic of China. Zeng-Tai Gong - College of Mathematics and Statistics, Northwest Normal University, 730070 Lanzhou, Gansu, People's Republic of China. Zi-Zhong Chen - College of Computer Science and Technology, Chongqing University of Posts and Telecommunications, 400065 Nanan, Chongqing, People's Republic of China.

Abstract

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.

Share and Cite

Keywords

• Fuzzy number
• fuzzy Henstock-Pettis integral
• convergence theorem
• discontinuous fuzzy differential equation
• generalized weak solution

MSC

•  03E72
•  26A39
•  46G05
•  34A07

References

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


• [2] L. C. Barrosa, L. T. Gomesa, P. A. Tonelli, Fuzzy differential equations: an approach via fuzzification of the derivative operator, Fuzzy Sets and Systems, 230 (2013), 39–52.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581–599.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119–141.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] A. M. Bica, One-sided fuzzy numbers and applications to integral equations from epidemiology, Fuzzy Sets and Systems, 219 (2013), 27–48.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] Y. Chalco-Cano, H. Román-Flores , On new solutions of fuzzy differential equations, Chaos Solitons Fractals, 38 (2008), 112–119.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] M.-H. Chen, C.-S. Han, Some topological properties of solutions to fuzzy differential systems, Inform. Sci., 197 (2012), 207–214.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] M.-H. Chen, D.-H. Li, X.-P. Xue, Periodic problems of first order uncertain dynamical systems , Fuzzy Sets and Systems, 162 (2011), 67–78.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [10] M. Cichoń, Convergence theorems for the Henstock-Kurzweil-Pettis integral, Acta Math. Hungar., 92 (2001), 75–82.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] M. Cichoń, I. Kubiaczyk, A. Sikorska, The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem, Czechoslovak Math. J., 54 (2004), 279–289.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [13] L. Di Piazza, K. Musiał , Set-valued Kurzweil-Henstock-Pettis integral , Set-Valued Anal., 13 (2005), 167–169.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] K. El Amri, C. Hess, On the Pettis integral of closed valued multifunctions, Set-Valued Anal., 8 (2000), 329–360.
  • View Article
  • MathSciNet
  • Google Scholar


• [15] Z.-T. Gong, On the problem of characterizing derivatives for the fuzzy-valued functions, II, Almost everywhere differentiability and strong Henstock integral, Fuzzy Sets and Systems, 145 (2004), 381–393.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] Z.-T. Gong, Y.-B. Shao , Global existence and uniqueness of solutions for fuzzy differential equations under dissipative-type conditions, Comput. Math. Appl., 56 (2008), 2716–2723.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] Z.-T. Gong, Y.-B. Shao , The controlled convergence theorems for the strong Henstock integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems, 160 (2009), 1528–1546.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] Z.-T. Gong, L.-L. Wang, The Henstock-Stieltjes integral for fuzzy-number-valued functions, Inform. Sci., 188 (2012), 276–297.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] Z.-T. Gong, C.-X. Wu, Bounded variation, absolute continuity and absolute integrability for fuzzy-number-valued functions, Fuzzy Sets and Systems, 129 (2002), 83–94.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] M.-S. Guo, X.-Y. Peng, Y.-Q. Xu, Oscillation property for fuzzy delay differential equations, Fuzzy Sets and Systems, 200 (2012), 25–35.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [22] O. Kaleva , Fuzzy differential equations , Fuzzy Sets and Systems, 24 (1987), 301–317.
  • Google Scholar


• [23] P. E. Kloeden, T. Lorenz , Fuzzy differential equations without fuzzy convexity, Fuzzy Sets and Systems, 230 (2013), 65–81.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] I. Kubiaczyk , On a fixed point theorem for weakly sequentially continuous mappings, Discuss. Math. Differential Incl., 15 (1995), 15–20.
  • MathSciNet
  • Google Scholar


• [25] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, (Russian) Czechoslovak Math. J., 7 (1957), 418–446.
  • Google Scholar


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


• [27] M. L. Puri, D. A. Ralescu , Fuzzy random variables, J. Math. Anal. Appl., 114 (1986), 409–422.
  • Google Scholar


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


• [29] S. Schwabik, G.-J. Ye, Topics in Banach space integration, Series in Real Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2005)
  • View Article
  • MathSciNet
  • Google Scholar


• [30] S. Seikkala, On the fuzzy initial value problem , Fuzzy Sets and Systems, 24 (1987), 319–330.
  • View Article
  • MathSciNet
  • Google Scholar


• [31] Y.-B. Shao, Z.-T. Gong , Discontinuous fuzzy systems and Henstock integrals of fuzzy number valued functions, Int. Conf. Intell. Comput., Springer, Berlin, Heidelberg, 7389 (2012), 65–72.
  • View Article
  • Google Scholar


• [32] Y.-B. Shao, G.-L. Xue, Discontinuous fuzzy Fredholm integral equations and strong fuzzy Henstock integrals, Articial Intelligence Res., 2 (2013), 87–95.
  • View Article
  • Google Scholar


• [33] Y.-B. Shao, H.-H. Zhang, The strong fuzzy Henstock integrals and discontinuous fuzzy differential equations, J. Appl. Math., 2013 (2013), 8 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [34] Y.-B. Shao, H.-H. Zhang, Existence of the solution for discontinuous fuzzy integro-differential equations and strong fuzzy Henstock integrals , Nonlinear Dyn. Syst. Theory, 14 (2014), 148–161.
  • MathSciNet
  • Google Scholar
  • MATH


• [35] Y.-B. Shao, H.-H. Zhang , Fuzzy integral equations and strong fuzzy Henstock integrals, Abstr. Appl. Anal., 2014 (2014), 8 pages.
  • Google Scholar


• [36] C.-X. Wu, Z.-T. Gong, On Henstock integrals of interval-valued functions and fuzzy-valued functions, Fuzzy Sets and Systems, 115 (2000), 377–391.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] C.-X. Wu, Z.-T. Gong , On Henstock integral of fuzzy-number-valued functions, I, Fuzzy Sets and Systems, 120 (2001), 523–552.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] C.-X. Wu, M. Ma, Embedding problem of fuzzy number space, II, Fuzzy Sets and Systems, 45 (1992), 189–202.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [39] X.-P. Xue, Y.-Q. Fu, Carathéodory solutions of fuzzy differential equations, Fuzzy Sets and Systems, 125 (2002), 239– 243.
  • View Article
  • MathSciNet
  • Google Scholar


• [40] B.-K. Zhang, C.-X. Wu , On the representation of n-dimensional fuzzy numbers and their informational context, Fuzzy Sets and Systems, 128 (2002), 227–235.
  • View Article
  • MathSciNet
  • Google Scholar