About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 932696, 8 pages
http://dx.doi.org/10.1155/2014/932696
Research Article

Fuzzy Integral Equations and Strong Fuzzy Henstock Integrals

1College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou 730030, China
2Department of Mathematics, Sichuan University, Chengdu 610065, China

Received 29 April 2013; Revised 6 September 2013; Accepted 18 September 2013; Published 29 January 2014

Academic Editor: Márcia Federson

Copyright © 2014 Yabin Shao and Huanhuan Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–319, 1987. View at Zentralblatt MATH · View at MathSciNet
  2. Z. Gong and Y. Shao, “Global existence and uniqueness of solutions for fuzzy differential equations under dissipative-type conditions,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2716–2723, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. O. Kaleva, “The Cauchy problem for fuzzy differential equations,” Fuzzy Sets and Systems, vol. 35, no. 3, pp. 389–396, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. J. Nieto, “The Cauchy problem for continuous fuzzy differential equations,” Fuzzy Sets and Systems, vol. 102, no. 2, pp. 259–262, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications, vol. 91, no. 2, pp. 552–558, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. Wu and S. Song, “Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions,” Information Sciences, vol. 108, no. 1–4, pp. 123–134, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. Seikkala, “On the fuzzy initial value problem,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 319–330, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. Chalco-Cano and H. Román-Flores, “On new solutions of fuzzy differential equations,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 112–119, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. D. Dubois and H. Prade, “Towards fuzzy differential calculus. I: integration of fuzzy mappings,” Fuzzy Sets and Systems, vol. 8, no. 1, pp. 1–17, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. Allahviranloo, M. Khezerloo, O. Sedaghatfar, and S. Salahshour, “Toward the existence and uniqueness of solutions of second-order fuzzy volterra integro-differential equations with fuzzy kernel,” Neural Computing and Applications, vol. 22, no. 1, supplement, pp. 133–141, 2013. View at Publisher · View at Google Scholar · View at Scopus
  12. J. Y. Park, Y. C. Kwun, and J. U. Jeong, “Existence of solutions of fuzzy integral equations in Banach spaces,” Fuzzy Sets and Systems, vol. 72, no. 3, pp. 373–378, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Y. Park and J. U. Jeong, “On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations,” Fuzzy Sets and Systems, vol. 115, no. 3, pp. 425–431, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. Zhang, W. Feng, and J. Qiu, “Global existence of solutions to fuzzy volterra integral equations,” ICIC Express Letters, vol. 3, no. 3, pp. 707–711, 2009. View at Scopus
  15. X. Xue and Y. Fu, “Carathéodory solutions of fuzzy differential equations,” Fuzzy Sets and Systems, vol. 125, no. 2, pp. 239–243, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. P. Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989. View at MathSciNet
  17. C. Wu and Z. Gong, “On Henstock integrals of interval-valued functions and fuzzy-valued functions,” Fuzzy Sets and Systems, vol. 115, no. 3, pp. 377–391, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. C. Wu and Z. Gong, “On Henstock integral of fuzzy-number-valued functions. I,” Fuzzy Sets and Systems, vol. 120, no. 3, pp. 523–532, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Z. Gong, “On the problem of characterizing derivatives for the fuzzy-valued functions. II. Almost everywhere differentiability and strong Henstock integral,” Fuzzy Sets and Systems, vol. 145, no. 3, pp. 381–393, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Z. Gong and Y. Shao, “The controlled convergence theorems for the strong Henstock integrals of fuzzy-number-valued functions,” Fuzzy Sets and Systems, vol. 160, no. 11, pp. 1528–1546, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. T. S. Chew and F. Flordeliza, “On x=f(t,x) and Henstock-Kurzweil integrals,” Differential and Integral Equations, vol. 4, no. 4, pp. 861–868, 1991. View at Zentralblatt MATH · View at MathSciNet
  22. A. Sikorska-Nowak, “Existence of solutions of nonlinear integral equations and Henstock-Kurzweil integrals,” Annales Societatis Mathematicae Polonae, vol. 47, no. 2, pp. 227–238, 2007. View at Zentralblatt MATH · View at MathSciNet
  23. P. Diamond and P. Kloeden, Metric Space of Fuzzy Fets: Theory and Applications, World Scientific, Singapore, 1994. View at MathSciNet
  24. C. X. Wu and M. Ma, “Embedding problem of fuzzy number space. II,” Fuzzy Sets and Systems, vol. 45, no. 2, pp. 189–202, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. J. Banaś and K. Goebel, Measure of Noncompactness in Banach Space, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. View at MathSciNet
  26. V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London, UK, 2003. View at Publisher · View at Google Scholar · View at MathSciNet