Existence and uniqueness of Cauchy problem for fuzzy differential equations under dissipative conditions. (English) Zbl 1157.34002

Song, Shiji; Wu, Cheng; Xue, Xiaoping

Comput. Math. Appl. 51, No. 9-10, 1483-1492 (2006).

The authors prove the existence and uniqueness of a solution to the Cauchy problem associated with first-order fuzzy differential equations in \(E^n\) in the space \(E^n\) of normal, fuzzy convex, upper semicontinuous and compactly supported fuzzy sets \(u:\mathbb{R}^n \to [0,1] \)
\[ x'(t)=f(t,x(t)),\quad x(t_0)=x_0, \] for \(f\) satisfying Lyapunov dissipative-type conditions.
The procedure is based on the application of the result on the existence of approximate solutions to the Cauchy problem for fuzzy differential equations given in the work by C. X. Wu and S. J. Song [Inf. Sci. 108, 123–134 (1998; Zbl 0931.34041)], the embedding result of the fuzzy number space \((E^n,D)\) in a real Banach space [see H. Rådström, Proc. Am. Math. Soc. 3, 165–169 (1952; Zbl 0046.33304), and P. E. Kloeden, Fuzzy Sets Syst. 44, 161–163 (1991; Zbl 0742.34058)], and comparison results for classical ordinary differential equations [see V. Lakshmikantham and S. Leela, Differential and integral inequalities, New York-London, Academic Press (1969; Zbl 0177.12403)], which are important tools to complete the proof of the main theorems.

Reviewer: Rosana Rodriguez López (Santiago de Compostela)

MSC:

34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26E50	Fuzzy real analysis

Keywords:

existence and uniqueness theorem; fuzzy differential equations; dissipative-type conditions; fuzzy number space \((E^n,D)\)

Citations:

Zbl 0931.34041; Zbl 0046.33304; Zbl 0742.34058; Zbl 0177.12403

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