Generalized differentiability of fuzzy-valued functions

In the present paper, using novel generalizations of the Hukuhara difference for fuzzy sets, we introduce and study new generalized differentiability concepts for fuzzy valued functions. Several properties of the new concepts are investigated and they are compared to similar fuzzy differentiabilities finding connections between them. Characterization and relatively simple expressions are provided for the new derivatives.

[1]  Barnabás Bede,et al.  Note on "Numerical solutions of fuzzy differential equations by predictor-corrector method" , 2008, Inf. Sci..

[2]  V. Lakshmikantham,et al.  Set Differential Equations and Flow Invariance , 2003 .

[3]  Luciano Stefanini,et al.  On Fuzzy Arithmetic Operations: Some Properties and Distributive Approximations , 2007 .

[4]  Svetoslav Markov,et al.  Calculus for interval functions of a real variable , 1979, Computing.

[5]  Siegfried Gottwald,et al.  Applications of Fuzzy Sets to Systems Analysis , 1975 .

[6]  Svetoslav Markov.  On the Algebra of Intervals and Convex Bodies , 1998, J. Univers. Comput. Sci..

[7]  M. Hukuhara.  INTEGRATION DES APPLICAITONS MESURABLES DONT LA VALEUR EST UN COMPACT CONVEXE , 1967 .

[8]  S. Seikkala.  On the fuzzy initial value problem , 1987 .

[9]  Eyke Hüllermeier,et al.  An Approach to Modelling and Simulation of Uncertain Dynamical Systems , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[10]  Phil Diamond,et al.  Stability and periodicity in fuzzy differential equations , 2000, IEEE Trans. Fuzzy Syst..

[11]  P. Kloeden,et al.  Metric Spaces Of Fuzzy Sets Theory And Applications , 1975 .

[12]  V. Lakshmikantham,et al.  Theory of Fuzzy Differential Equations and Inclusions , 2003 .

[13]  E.Stanley Lee,et al.  Approximate Solutions, Existence, and Uniqueness of the Cauchy Problem of Fuzzy Differential Equations☆ , 1996 .

[14]  Siegfried Gottwald,et al.  Applications of fuzzy sets to systems analysis , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[15]  Phil Diamond,et al.  Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations , 1999, IEEE Trans. Fuzzy Syst..

[16]  Luciano Stefanini,et al.  A generalization of Hukuhara difference and division for interval and fuzzy arithmetic , 2010, Fuzzy Sets Syst..

[17]  Didier Dubois,et al.  Soft Methods for Handling Variability and Imprecision , 2008 .

[18]  N V Plotnikova,et al.  Systems of linear differential equations with -derivative and linear differential inclusions , 2005 .

[19]  H. Román-Flores,et al.  On new solutions of fuzzy differential equations , 2008 .

[20]  Thomas Feuring,et al.  Fuzzy di erential equations , 1999 .

[21]  Yurilev Chalco-Cano,et al.  Fuzzy differential equations and the extension principle , 2007, Inf. Sci..

[22]  Luciano Stefanini,et al.  A generalization of Hukuhara difference for interval and fuzzy arithmetic , 2008 .

[23]  Y. Chalco-Cano,et al.  Comparation between some approaches to solve fuzzy differential equations , 2009, Fuzzy Sets Syst..

[24]  Osmo Kaleva,et al.  A note on fuzzy differential equations , 2006 .

[25]  F. Chorlton,et al.  Differential Equations and Applications , 1966 .

[26]  Jacek Tabor.  DIFFERENTIAL EQUATIONS IN METRIC SPACES , 2002 .

[27]  R. Goetschel,et al.  Elementary fuzzy calculus , 1986 .

[28]  M. Puri,et al.  Differentials of fuzzy functions , 1983 .

[29]  Rosana Rodríguez-López,et al.  Comparison results for fuzzy differential equations , 2008, Inf. Sci..

[30]  Luciano Stefanini.  On the generalized LU-fuzzy derivative and fuzzy differential equations , 2007, 2007 IEEE International Fuzzy Systems Conference.

[31]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[32]  Alexandru Mihai Bica,et al.  Error estimation in the approximation of the solution of nonlinear fuzzy Fredholm integral equations , 2008, Inf. Sci..

[33]  Yurilev Chalco-Cano,et al.  Generalized derivative and π-derivative for set-valued functions , 2011, Inf. Sci..

[34]  Barnabás Bede,et al.  Solutions of fuzzy differential equations based on generalized differentiability , 2010 .

[35]  Maria Letizia Guerra,et al.  Fuzzy Numbers and Fuzzy Arithmetic , 2008 .

[36]  Luciano Stefanini,et al.  Some notes on generalized Hukuhara differentiability of interval-valued functions and interval differential equations , 2012 .

[37]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[38]  Juan J. Nieto,et al.  Euler polygonal method for metric dynamical systems , 2007, Inf. Sci..

[39]  Juan J. Nieto,et al.  Hybrid metric dynamical systems with impulses , 2006 .

[40]  Kevin D. Reilly,et al.  Simulating continuous fuzzy systems , 2005, Inf. Sci..

[41]  Phil Diamond,et al.  Brief note on the variation of constants formula for fuzzy differential equations , 2002, Fuzzy Sets Syst..

[42]  Luciano Stefanini,et al.  A Generalization of Hukuhara Difference , 2008, SMPS.

[43]  P. Kloeden,et al.  Metric spaces of fuzzy sets , 1990 .

[44]  Tofigh Allahviranloo,et al.  Numerical solution of fuzzy differential equations by predictor-corrector method , 2007, Inf. Sci..

[45]  Couette Viscometry Equation.  APPROXIMATE SOLUTIONS FOR THE , 2005 .

[46]  Luis J. Rodríguez-Muñiz,et al.  The s-differentiability of a fuzzy-valued mapping , 2003, Inf. Sci..