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Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations. (English) Zbl 0946.34054

The following notations and assumptions are used in the paper: (1) Denote by \(E^n\) the set of all mappings \(u: \mathbb{R}^n\to [0,1]\), that are normal, fuzzy convex, upper semicontinuous and for which \([u]^0:= \text{cl}\{x\in \mathbb{R}^n\mid u(x)> 0\}\) is compact; (2) For \(\alpha\in (0,1]\) we define \([u]^\alpha= \{x\in \mathbb{R}^n\mid u(x)> \alpha\}\); (3) For \(p>0\), \(q>0\) and \(x_0\in E^n\), we denote \(\mathbb{R}_0= T\times B(x_0, q)\), where \(T= [t_0,t_0+ p]\), \(B(x_0,q):= \{x\in E^n\mid D(x,x_0)\leq q\}\), here \(D(x, x_0):= \sup_{0\leq \alpha\leq 1} d([x]^\alpha, [x_0]^\alpha)\) and \(d(A,B)\) is the distance between two nonempty bounded sets \(A,B\subset \mathbb{R}^n\) defined by the Hausdorff metric; and (4) let \(g\in C(T\times [0,q])\) be a bounded continuous function, \(g(t, 0)\equiv 0\), and \(g(t,u)\) is nondecreasing in \(u\) for every \(t\) fixed, and such that the next initial value problem has only the solution \(u(t)\equiv 0\) on \(T\): \[ u'= g(t, u),\quad (t,u)\in T\times \mathbb{R},\quad u(t_0)= 0. \] The authors introduce a generalization of the H-differentiation due to M. L. Puri and D. A. Ralescu [J. Math. Anal. Appl. 91, 552-558 (1983; Zbl 0528.54009)] as follows: “A mapping \(F: T\to E^n\) is called differentiable at point \(t_0\in T\), if for every \(\alpha\in [0,1]\) the set-valued mapping \(F_\alpha:= [F(t)]^\alpha\) is Hukuhara differentiable at \(t_0\) and the family \(\{DF_\alpha(t_0)\mid \alpha\in [0,1]\}\) defines a fuzzy number \(F'(t_0)\in E^n\), where \(DF_\alpha\) denotes one of the four Dini derivatives of \(F_\alpha\).”
Applying the new differentiation and the successive approximation method, the authors prove the existence of the local unique \(C^1\)-solution to the following Cauchy problem \[ x'= f(t,x),\quad (t,x)\in \mathbb{R}_0,\quad x(t_0)= x_0\in B(x_0, q), \] with \(f\in C(\mathbb{R}_0, E^n)\), \(D(f(t, x),\widehat 0)\leq M\) \(\forall (t,x)\in \mathbb{R}_0\) and satisfies the generalized Lipschitz condition \[ D(f(t,x),f(t,y))\leq g(t,D(x,y)),\quad \forall(t, x),\;(t,y)\in \mathbb{R}_0, \] herein \(g\in C(T, [0,q])\) is a function satisfying the assumption (4) given above.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26E50 Fuzzy real analysis

Citations:

Zbl 0528.54009
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