Monotone method for fuzzy differential equations

We develop the monotone iterative technique to approximate the extremal solutions for the initial value problem relative to a fuzzy differential equation in a fuzzy functional interval. To our procedure, it is crucial to have at our disposal fuzzy comparison results and several properties of sets of fuzzy functions and fuzzy numbers, including a compactness result in spaces of fuzzy functions, which are useful from the point of view of applications.

[1]  H. Carter.  Fuzzy Sets and Systems — Theory and Applications , 1982 .

[2]  Giovanni Colombo,et al.  A non-stochastic approach for modeling uncertainty in population dynamics , 1998 .

[3]  Rosana Rodríguez-López,et al.  Existence of extremal solutions for quadratic fuzzy equations , 2005 .

[4]  Abraham Kandel,et al.  Numerical solutions of fuzzy differential equations , 1999, Fuzzy Sets Syst..

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

[6]  Dimitris N. Georgiou,et al.  On Cauchy problems for fuzzy differential equations , 2004, Int. J. Math. Math. Sci..

[7]  Juan J. Nieto,et al.  Linear First-Order Fuzzy Differential Equations , 2006, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[8]  V. Lakshmikantham,et al.  Monotone iterative techniques for nonlinear differential equations , 1985 .

[9]  Garyfalos Papaschinopoulos,et al.  Existence, uniqueness and asymptotic behavior of the solutions of a fuzzy differential equation with piecewise constant argument , 2007, Inf. Sci..

[10]  Juan J. Nieto,et al.  The Cauchy problem for continuous fuzzy differential equations , 1999, Fuzzy Sets Syst..

[11]  Congxin Wu,et al.  Existence Theorem to the Cauchy Problem of Fuzzy Differential Equations under Compactness-Type Conditions , 1998, Inf. Sci..

[12]  Abraham Kandel,et al.  A new approach for defuzzification , 2000, Fuzzy Sets Syst..

[13]  Brane Hlebcar.  Forecasting closing overvoltages in high-voltage networks using a fuzzy model , 2001, Fuzzy Sets Syst..

[14]  Ravi P. Agarwal,et al.  A stacking theorem approach for fuzzy differential equations , 2003 .

[15]  Imre J. Rudas,et al.  First order linear fuzzy differential equations under generalized differentiability , 2007, Inf. Sci..

[16]  Saeid Abbasbandy,et al.  Note on "A new approach for defuzzification" , 2002, Fuzzy Sets Syst..

[17]  M. Murty,et al.  INITIAL AND BOUNDARY VALUE PROBLEMS FOR FUZZY DIFFERENTIAL EQUATIONS , 2007 .

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

[19]  V. V. Amelʹkin.  Differential equations in applications , 1990 .

[20]  Eyke Hüllermeier,et al.  Numerical Methods for Fuzzy Initial Value Problems , 1999, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[21]  V. Lakshmikantham,et al.  Initial and boundary value problems for fuzzy differential equations , 2003 .

[22]  Congxin Wu,et al.  Convergence of sequences of fuzzy numbers and fixed point theorems for increasing fuzzy mappings and application , 2002, Fuzzy Sets Syst..

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

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

[25]  Mignon Park,et al.  Design of an adaptive fuzzy model based controller for chaotic dynamics in Lorenz systems with uncertainty , 2002, Inf. Sci..

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

[27]  Dimitris N. Georgiou,et al.  Initial value problems for higher-order fuzzy differential equations , 2005 .

[28]  Kai-Yuan Cai,et al.  Extended neuro-fuzzy models of multilayer perceptrons , 2004, Fuzzy Sets Syst..

[29]  J. Nieto,et al.  Bounded solutions for fuzzy differential and integral equations , 2006 .

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

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

[32]  Ravi P. Agarwal,et al.  Viability theory and fuzzy differential equations , 2005, Fuzzy Sets Syst..

[33]  H Roman Flores,et al.  EMBEDDING OF LEVEL-CONTINUOUS FUZZY SETS ON BANACH SPACES , 2002 .

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

[35]  A. de Korvin,et al.  Analysis by fuzzy difference equations of a model of CO2 level in the blood , 1999 .

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

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

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

[39]  Esmail Babolian,et al.  Numerically solution of fuzzy differential equations by Adomian method , 2004, Appl. Math. Comput..

[40]  Jung-Chan Chang,et al.  Fixed-point theorems in fuzzy real line , 2004 .

[41]  V. Lakshmikantham,et al.  Revisiting fuzzy differential equations , 2004 .

[42]  Abraham Kandel,et al.  Correction to "A new approach for defuzzification" [Fuzzy Sets and Systems 111 (2000) 351-356] , 2002, Fuzzy Sets Syst..

[43]  Osmo Kaleva.  The Cauchy problem for fuzzy differential equations , 1990 .

[44]  Luciano Stefanini,et al.  Parametric representation of fuzzy numbers and application to fuzzy calculus , 2006, Fuzzy Sets Syst..