On fuzzy boundary value problems

The statement that a two-point boundary value problem of fuzzy differential equation is equivalent to a fuzzy integral equation was pointed out by Lakshmikantham et al. and O'Regan et al. Recently Bede gave a counterexample to show that this statement does not hold and he also argued that in many cases two-point boundary value problems have no solutions. Under a new structure and certain conditions we show that a two-point boundary value problem is equivalent to a fuzzy integral equation. We also prove the existence of solutions to the two-point boundary value problem. In some sense, this is an amendment to results of Lakshmikantham et al. and O'Regan et al., and it is an answer to one of Bede's problems.

[1]  Weiyin Fei,et al.  Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients , 2007, Inf. Sci..

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

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

[4]  D. Dubois,et al.  Towards fuzzy differential calculus part 3: Differentiation , 1982 .

[5]  Congxin Wu,et al.  On Henstock integral of fuzzy-number-valued functions (I) , 2001, Fuzzy Sets Syst..

[6]  Shiji Song,et al.  Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations , 2000, Fuzzy Sets Syst..

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

[8]  W. Congxin,et al.  Embedding problem of fuzzy number space: part II , 1992 .

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

[10]  D. Dubois,et al.  Towards fuzzy differential calculus part 2: Integration on fuzzy intervals , 1982 .

[11]  D. Dubois,et al.  Operations on fuzzy numbers , 1978 .

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

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

[14]  V. Lakshmikantham,et al.  TWO-POINT BOUNDARY VALUE PROBLEMS ASSOCIATED WITH NON-LINEAR FUZZY DIFFERENTIAL EQUATIONS , 2001 .

[15]  Xiaoping Xue,et al.  Carathéodory solutions of fuzzy differential equations , 2002, Fuzzy Sets Syst..

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

[17]  D. Dubois,et al.  Towards fuzzy differential calculus part 1: Integration of fuzzy mappings , 1982 .

[18]  Barnabás Bede,et al.  A note on "two-point boundary value problems associated with non-linear fuzzy differential equations" , 2006, Fuzzy Sets Syst..

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

[20]  Lotfi A. Zadeh,et al.  Toward a generalized theory of uncertainty (GTU)--an outline , 2005, Inf. Sci..