Existence of solutions for implicit fuzzy differential inclusions

doi:10.1007/s10483-015-1914-6

Abstract

Abstract: A class of implicit fuzzy differential inclusions (IFDIs) are introduced and studied. Some existence theorems under different conditions are proved with the selection theorems for the open situation and the closed situation, respectively. A viable solution for a closed IFDI is proved to exist under the tangential condition. As an application, an implicit fuzzy differential equation, which comes from the drilling dynamics in petroleum engineering, is analyzed numerically. The obtained results can improve and extend some known results for fuzzy differential inclusions (FDIs) and fuzzy differential equations (FDEs), which might be helpful in the analysis of fuzzy dynamic systems.

Key words: fuzzy differential equation (FDE), implicit fuzzy differential inclusion (IFDI), stacking theorem, selection theorem

2010 MSC Number:

Chao MIN;Nanjing HUANG;Zhibin LIU;Liehui ZHANG. Existence of solutions for implicit fuzzy differential inclusions. Applied Mathematics and Mechanics (English Edition), 2015, 36(3): 401-416.

TrendMD

References

[1] Chen, B. and Liu, X. Reliable control design of fuzzy dynamical systems with time-varying delay. Fuzzy Sets and Systems, 146, 349-374 (2000)
[2] Bede, B., Rudas, I. J., and Bencsik, A. L. First order linear fuzzy differential equations under generalized differentiability. Information Sciences, 177, 1648-1662 (2007)
[3] Puri, M. and Ralescu, D. Differential and fuzzy functions. Journal of Mathematical Analysis and Applications, 91, 552-558 (1983)
[4] Kaleva, O. Fuzzy differential equations. Fuzzy Sets and Systems, 24, 301-317 (1987)
[5] Kaleva, O. The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems, 35, 389-396 (1990)
[6] Seikkala, S. On the fuzzy initial value problem. Fuzzy Sets and Systems, 24, 319-330 (1987)
[7] Wu, C. X., Song, S. J., and Lee, E. S. Approximate solutions, existence and uniqueness of the Cauchy problem of fuzzy differential equations. Journal of Mathematical Analysis and Applications, 202, 629-644 (1996)
[8] Song, S., Guo, L., and Feng, C. Global existence of solutions to fuzzy differential equations. Fuzzy Sets and Systems, 115, 371-376 (2000)
[9] O'Regan, D., Lakshmikantham, V., and Nieto, J. J. Initial and boundary value problems for fuzzy differential equations. Nonlinear Analysis: Theory, Methods and Applications, 54, 405-415 (2003)
[10] Choudary, A. and Donchev, T. On Peano theorem for fuzzy differential equations. Fuzzy Sets and Systems, 177, 93-94 (2011)
[11] Nieto, J. The Cauchy problem for continuous fuzzy differential equation. Fuzzy Sets and Systems, 102, 259-262 (1999)
[12] Kaleva, O. Nonlinear iteration semigroup of fuzzy Cauchy problem. Fuzzy Sets and Systems, 209, 104-110 (2012)
[13] Guo, M. S., Peng, X. Y., and Xu, Y. Q. Oscillation property for fuzzy delay differential equations. Fuzzy Sets and Systems, 200, 25-35 (2012)
[14] Ma, M., Friedman, M., and Kandel, A. Numerical solutions of fuzzy differential equations. Fuzzy Sets and Systems, 105, 133-138 (1999)
[15] Abbasbandy, S. and Viranloo, T. Numerical solutions of fuzzy differential equations by Taylor method. Computaional Methods in Applied Mathematics, 2, 113-124 (2002)
[16] Abbasbandy, S. and Viranloo, T. Numerical solutions of fuzzy differential equations by Runge- Kutta method. Nonlinear Studies, 11, 117-129 (2004)
[17] Omar, A. and Hasan, Y. Numerical solution of fuzzy differential equations and the dependency problem. Applied Mathematics and Computation, 219, 1263-1272 (2012)
[18] Yang, L. F. and Li, G. Q. Fuzzy stochastic variable and variational principle. Applied Mathematics and Mechanics (English Edition), 20(7), 795-800 (1999) DOI 10.1007/BF02454902
[19] Lü, E. L. and Zhong, Y. M. Mathematic description about random variable with fuzzy density function (RVFDF). Applied Mathematics and Mechanics (English Edition), 21(8), 957-964 (2000) DOI 10.1007/BF02428366
[20] Ma, J., Chen, J. J., Xu, Y. L., and Jiang, T. Dynamic characteristic analysis of fuzzy-stochastic truss structures based on fuzzy factor method and random factor method. Applied Mathematics and Mechanics (English Edition), 27(6), 823-832 (2006) DOI 10.1007/s10483-006-0613-z
[21] Malinowski, M. T. Some properties of strong solutions to stochastic fuzzy differential equations. Information Sciences, 252, 62-80 (2013)
[22] Malinowski, M. T. Random fuzzy differential equations under generalized Lipschitz condition. Nonlinear Analysis: Real World Applications, 13, 860-881 (2012)
[23] Malinowski, M. T. Itô type stochastic fuzzy differential equations with delay. Systems and Control Letters, 61, 692-701 (2012)
[24] Malinowski, M. T. Existence theorems for solutions to random fuzzy differential equations. Nonlinear Analysis: Theory, Methods and Applications, 73, 1515-1532 (2010)
[25] Malinowski, M. T. Strong solutions to stochastic fuzzy differential equations of Itô type. Mathematical and Computer Modelling, 55, 918-928 (2012)
[26] Malinowski, M. T. Approximation schemes for fuzzy stochastic integral equations. Applied Mathematics and Computation, 219, 11278-11290 (2013)
[27] Diamond, P. Brief note on the variation of constants formula for fuzzy differential equations. Fuzzy Sets and Systems, 129, 65-71 (2002)
[28] Hüllermeier, E. An approach to modeling and simulation of uncertain dynamical systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5, 117-137 (1997)
[29] Lakshmikantham, V., Bhaskar, T. G., and Devi, J. V. Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, Cambridge (2006)
[30] Lakshmikantham, V. and Mohapatra, R. N. Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London/New York (2003)
[31] Aubin, J. P. Mutational and Morphological Analysis: Tools for Shape Evolution and Morphogenesis, Birkhäser, Boston (1999)
[32] Aubin, J. P. Viability Theory, Birkhäser, Boston (1991)
[33] Diamond, P. and Watson, P. Regularity of solution sets for differential inclusions quasi-concave in a parameter. Applied Mathematics Letters, 13, 31-35 (2000)
[34] Bede, B. and Gal, S. G. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems, 151, 581-599 (2005)
[35] Li, J., Zhao, A., and Yan, J. The Cauchy problem of fuzzy differential equations under generalized differentiability. Fuzzy Sets and Systems, 200, 1-24 (2012)
[36] Khastan, A., Nieto, J. J., and Rodríguez-López, R. Variation of constant formula for first order fuzzy differntial equations. Fuzzy Sets and Systems, 177, 20-33 (2011)
[37] Khastan, A. and Nieto, J. J. A boundary value problem for second order fuzzy differential equations. Nonlinear Analysis: Theory, Methods and Applications, 72, 3583-3593 (2010)
[38] Zhang, D. K., Feng, W. L., Zhao, Y. Q., and Qiu, J. Q. Global existence of solutions for fuzzy second-order differential equations under generalized H-differentiability. Computers and Mathematics with Applications, 60, 1548-1556 (2010)
[39] Chen, M. H.,Wu, C. X., Xue, X. P., and Liu, G. Q. On fuzzy boundary value problems. Information Sciences, 178, 1877-1892 (2008)
[40] Chen, M. H., Fu, Y. Q., Xue, X. P., and Wu, C. X. Two-point boundary value problems of undamped uncertain dynamical systems. Fuzzy Sets and Systems, 159, 2077-2089 (2008)
[41] Liu, B. Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2, 3-16 (2008)
[42] Qin, Z. and Li, X. Option pricing formula for fuzzy financial market. Journal of Uncertain Systems, 2, 17-21 (2008)
[43] Zhu, Y. Uncertain optimal control with application to a portfolio selection model. Cybernetics and Systems, 41, 535-547 (2010)
[44] Baidosov, V. A. Fuzzy differential inclusions. Journal of Applied Mathematics and Mechanics, 54, 8-13 (1990)
[45] Dordan, O. Modelling fuzzy control problems with toll sets. Set-Valued Analysis, 8, 85-99 (2000)
[46] Zhu, Y. and Rao, L. Differential inclusions for fuzzy maps. Fuzzy Sets and Systems, 112, 257-261 (2000)
[47] Zadeh, L. A. The concept of a linguistic variable and its applications in approximate reasoning. Information Sciences, 8, 199-251 (1975)
[48] Lojasiewicz, S., Plis, A., and Suarez, R. Necessary conditions for nonlinear control system. Journal of Differential Equations, 59, 257-265 (1985)
[49] Yannelis, N. C. and Prabhakar, N. D. Existence of minimal elements and equilibria in linear topological spaces. Journal of Mathematical Economics, 12, 233-245 (1983)
[50] Michael, E. Continuous selections I. Annals of Mathematics, 63, 361-381 (1956)
[51] Aubin, J. P. and Frankowska, H. Set-Value Analysis, Birkhauser, Boston (1990)
[52] Aubin, J. P. and Cellina, A. Differential Inclusions, Springer-Verlag, Berlin (1984)
[53] Aumann, R. J. Integrals of set-valued functions. Journal of Mathematical Analysis and Applications, 12, 1-12 (1965)
[54] Ho, H. S. General formulation of drillstring under large deformation and its use in BHA analysis. The 61st Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Society of Petroleum Engineers, New Orleans (1986)
[55] Ho, H. S. An improved modeling program for computing the torque and drag in directional and deep wells. The 63rd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Society of Petroleum Engineers, Houston (1988)
[56] Min, C., Liu, Q. Y., Zhang, L. H., and Huang, N. J. Solutions of the stiff-string model with an iterative method. Journal of Computational Analysis and Applications, 15, 424-431 (2013)