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Abstract and Applied Analysis
Volume 2014, Article ID 381972, 8 pages
http://dx.doi.org/10.1155/2014/381972
Research Article

Properties of Solutions to Stochastic Set Differential Equations under Non-Lipschitzian Coefficients

School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui 241000, China

Received 31 October 2013; Revised 7 January 2014; Accepted 19 January 2014; Published 5 March 2014

Academic Editor: Ilaria Fragala

Copyright © 2014 Weiyin Fei et al. 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. F. S. de Blasi and F. Iervolino, “Equazioni differenziali con soluzioni a valore compatto convesso,” Bollettino della Unione Matematica Italiana, vol. 4, no. 2, pp. 194–501, 1969. View at Google Scholar
  2. F. S. de Blasi, “Banach-Saks-Mazur and Kakutani-Ky Fan theorems in spaces of multifunctions and applications to set differential inclusions,” Dynamic Systems and Applications, vol. 16, no. 1, pp. 73–89, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific Publishing, Singapore, 1994. View at MathSciNet
  4. W. Y. Fei, “Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients,” Information Sciences, vol. 177, no. 20, pp. 4329–4337, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  5. W. Y. Fei, “Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non-Lipschitzian condition,” Nonlinear Analysis A, vol. 76, pp. 202–214, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  6. W. Y. Fei, H. J. Liu, and W. Zhang, “On solutions to fuzzy stochastic eqautions with local martingales,” Systems and Control Letters, vol. 65, pp. 96–105, 2014. View at Google Scholar
  7. M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. View at MathSciNet
  8. V. Lakshmikantham, T. Gnana Bhaskar, and J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge, UK, 2006. View at MathSciNet
  9. V. Laksmikantham, S. Leela, and A. S. Vatsala, “Interconnection between set and fuzzy differential equations,” Nonlinear Analysis A, vol. 54, no. 2, pp. 351–360, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  10. 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
  11. J. J. Nieto and R. Rodríguez-López, “Bounded solutions for fuzzy differential and integral equations,” Chaos, Solitons and Fractals, vol. 27, no. 5, pp. 1376–1386, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at MathSciNet
  13. A. I. Brandão Lopes Pinto, F. S. de Blasi, and F. Iervolino, “Uniqueness and existence theorems for differential equations with convex valued solutions,” Bollettino della Unione Matematica Italiana, vol. 4, no. 3, pp. 1–12, 1970. View at Google Scholar · View at MathSciNet
  14. G. N. Galanis, T. G. Bhaskar, V. Lakshmikantham, and P. K. Palamides, “Set valued functions in Fréchet spaces: continuity, Hukuhara differentiability and applications to set differential equations,” Nonlinear Analysis A, vol. 61, no. 4, pp. 559–575, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  15. V. A. Plotnikov and L. I. Plotnikova, “Averaging of equations of controlled motion in a metric space,” Cybernetics and Systems Analysis, vol. 33, no. 4, pp. 601–606, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  16. B. van Cutsem, “Martingales de multiapplications à valeurs convexes compactes,” Les Comptes Rendus de l'Académie des sciences, vol. 269, pp. 429–432, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. E. P. Avgerinos and N. S. Papageorgiou, “Almost sure convergence and decomposition of multivalued random processes,” The Rocky Mountain Journal of Mathematics, vol. 29, no. 2, pp. 401–435, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Bagchi, “On a.s. convergence of classes of multivalued asymptotic martingales,” Annales de l'Institut Henri Poincaré, vol. 21, no. 4, pp. 313–321, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. F. Hiai and H. Umegaki, “Integrals, conditional expectations, and martingales of multivalued functions,” Journal of Multivariate Analysis, vol. 7, no. 1, pp. 149–182, 1977. View at Google Scholar · View at MathSciNet
  20. E. J. Jung and J. H. Kim, “On set-valued stochastic integrals,” Stochastic Analysis and Applications, vol. 21, no. 2, pp. 401–418, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  21. B. K. Kim and J. H. Kim, “Stochastic integrals of set-valued processes and fuzzy processes,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 480–502, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  22. S. Li and L. Guan, “Fuzzy set-valued Gaussian processes and Brownian motions,” Information Sciences, vol. 177, no. 16, pp. 3251–3259, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  23. S. Li and A. Ren, “Representation theorems, set-valued and fuzzy set-valued Ito integral,” Fuzzy Sets and Systems, vol. 158, no. 9, pp. 949–962, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  24. M. T. Malinowski, “On a new set-valued stochastic integral with respect to semimartingales and its applications,” Journal of Mathematical Analysis and Applications, vol. 408, no. 2, pp. 669–680, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  25. N. S. Papageorgiou, “On the efficiency and optimality of allocations II,” SIAM Journal on Control and Optimization, vol. 24, no. 3, pp. 452–479, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  26. B. van Cutsem, “Martingales de convexes fermés aléatoires en dimension finie,” Annales de l'Institut Henri Poincaré B, vol. 8, pp. 365–385, 1972. View at Google Scholar · View at MathSciNet
  27. M. T. Malinowski and M. Michta, “Stochastic set differential equations,” Nonlinear Analysis A, vol. 72, no. 3-4, pp. 1247–1256, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. Michta, “Set-valued random differential equations in Banach space,” Discussiones Mathematicae Differential Inclusions, vol. 15, no. 2, pp. 124–200, 1995. View at Google Scholar · View at MathSciNet
  29. M. Michta, “Continuity properties of solutions of multivalued equations with white noise perturbation,” Journal of Applied Mathematics and Stochastic Analysis, vol. 10, no. 3, pp. 239–248, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  30. M. Michta, “On set-valued stochastic integrals and fuzzy stochastic equations,” Fuzzy Sets and Systems, vol. 177, pp. 1–19, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  31. I. Mitoma, Y. Okazaki, and J. Zhang, “Set-valued stochastic differential equation in M-type 2 Banach space,” Communications on Stochastic Analysis, vol. 4, no. 2, pp. 215–237, 2010. View at Google Scholar · View at MathSciNet
  32. J. Li, S. Li, and Y. Ogura, “Strong solution of Itô type set-valued stochastic differential equation,” Acta Mathematica Sinica, vol. 26, no. 9, pp. 1739–1748, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  33. N. U. Ahmed, “Nonlinear stochastic differential inclusions on Banach space,” Stochastic Analysis and Applications, vol. 12, no. 1, pp. 1–10, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. J. P. Aubin and G. da Prato, “The viability theorem for stochastic differential inclusions,” Stochastic Analysis and Applications, vol. 16, no. 1, pp. 1–15, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. M. Kisielewicz, “Set-valued stochastic integrals and stochastic inclusions,” Stochastic Analysis and Applications, vol. 15, no. 5, pp. 783–800, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  36. M. Kisielewicz, Stochastic differential inclusions and applications, Springer, New York, NY, USA, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  37. M. Kisielewicz, M. Michta, and J. Motyl, “Set valued approach to stochastic control. Part. I and II,” Dynamic Systems and Applications, vol. 12, no. 3-4, pp. 405–466, 2003. View at Google Scholar · View at MathSciNet
  38. J. Krasińska and M. Michta, “A note on stochastic inclusions approach for fuzzy stochastic differential equations driven by semimartingales,” Dynamic Systems and Applications, vol. 22, no. 4, pp. 503–516, 2013. View at Google Scholar · View at MathSciNet
  39. M. T. Malinowski and M. Michta, “The interrelation between stochastic differential inclusions and set-valued stochastic differential equations,” Journal of Mathematical Analysis and Applications, vol. 408, no. 2, pp. 733–743, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  40. M. Michta and J. Motyl, “Stochastic inclusions with a non-Lipschitz right hand side,” in Stochastic Differential Equations, N. Halidias, Ed., pp. 189–232, Nova Scientic, 2011. View at Google Scholar
  41. W. Y. Fei and Y. Liang, “Stochastic set differential equations driven by local martingales under the non-Lipschitzian condition,” Acta Mathematica Sinica, vol. 56, no. 4, pp. 561–574, 2013. View at Google Scholar · View at MathSciNet
  42. W. Y. Fei and D. F. Xia, “On solutions to stochastic set differential equations to Itô type under the non-Lipschitzian condition,” Dynamic Systems and Applications, vol. 22, no. 1, pp. 137–156, 2013. View at Google Scholar · View at MathSciNet
  43. W. Fei, “Regularity and stopping theorem for fuzzy martingales with continuous parameters,” Information Sciences, vol. 169, no. 1-2, pp. 175–187, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  44. W. Fei, “On the theory of (dual) projection for fuzzy stochastic processes,” Stochastic Analysis and Applications, vol. 23, no. 3, pp. 449–474, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  45. W. Y. Fei, “A generalization of Bihari's inequality and fuzzy random differential equations with non-Lipschitz coefficients,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 15, no. 4, pp. 425–439, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  46. W. Y. Fei and R. Wu, “Doob's decomposition theorem for fuzzy (super) submartingales,” Stochastic Analysis and Applications, vol. 22, no. 3, pp. 627–645, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  47. W. Y. Fei, R. Q. Wu, and S. Shao, “Doob's stopping theorem for fuzzy (super, sub) martingales with discrete time,” Fuzzy Sets and Systems. An International Journal in Information Science and Engineering, vol. 135, no. 3, pp. 377–390, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  48. M. T. Malinowski, “Strong solutions to stochastic fuzzy differential equations of Itô type,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 918–928, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  49. M. T. Malinowski, “Itô type stochastic fuzzy differential equations with delay,” Systems and Control Letters, vol. 61, no. 6, pp. 692–701, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  50. M. T. Malinowski, “Some properties of strong solutions to stochastic fuzzy differential equations,” Information Sciences, vol. 252, pp. 62–80, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  51. M. T. Malinowski, “Approximation schemes for fuzzy stochastic integral equations,” Applied Mathematics and Computation, vol. 219, no. 24, pp. 11278–11290, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  52. X. R. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, NY, USA, 1994. View at MathSciNet
  53. B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, Berlin, Germany, 6th edition, 2005.
  54. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing, Amsterdam, The Netherlands, 1981. View at MathSciNet