Two-point boundary value problems associated with non-linear fuzzy differential equations

Abstract

Abstract. This paper presents a criteria for the existence and uniqueness of solutions to two point boundary value problems associated with a second order non-linear fuzzy differential equations. The main tools employed are estimates on Green's function, Ascoli's Lemma and a fixed point theorem of Banach.Mathematics subject classification (2000): 34A10, 26E50, 34B15.

On fuzzy boundary value problems

“…By Theorem 3.1 and continuity of f, the problem (4.1) is the same as those of [13,14], but here the problem (4.1) is confided to F st b , a subset of E 1 . Two-point boundary value problems are proved to have no solutions in many cases (see [2]).…”

“…In some sense, these results are an amendment to those of Refs. [13,14], and also an answer to one of Bede's problems in Ref. [2].…”

“…After we study Lakshmikantham et al [13], O'Regan et al [14] and Bede [2] where the equivalence between a fuzzy two-point boundary value problem and a fuzzy integral equation is proved by means of Green's function, we find some mistakes in the proof of the equivalence because a one-dimensional fuzzy set is only considered as a complete metric space, and only the concepts of H-derivatives and (K) integration [11,12] based on Aumann integration are used, but under these structures usual differential methods and integration by parts are not applicable.…”

Initial and Boundary Value Problems for Fuzzy Differential Equations

“…Here, we use a modified Lipschitz condition that involves all the variables. The results obtained here, include more general class of problems than in ( [3], [5] and [7]) obtained for first and second order fuzzy differential equations.…”

Well-posedness of a class of two-point boundary value problems associated with ordinary differential equations

“…As can be observed in ODEs (1.1), the equations of X t and Y t both have (X t , Y t ) as their components, which makes two equations fully coupled together. It is impossible to solve each equation individually, then many methods adapted to ODEs with one unknown variable are no longer feasible (see, for example, [2,3]). …”