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Hukuhara's differentiable iteration semigroups of linear set-valued functions

Andrzej Smajdor (2004)

Annales Polonici Mathematici

Let K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. A family F t : t ≥ 0 of continuous linear set-valued functions F t : K → c c ( K ) is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function Φ ( t , x ) = F t ( x ) is a solution of the problem D t Φ ( t , x ) = Φ ( t , G ( x ) ) : = ⋃ Φ ( t , y ) : y ∈ G ( x ) , Φ(0,x) = x, for x ∈ K and t ≥ 0, where D t Φ ( t , x ) denotes the Hukuhara derivative of Φ(t,x) with respect to t and G ( x ) : = l i m s → 0 + ( F s ( x ) - x ) / s for x ∈ K.

Hyers-Ulam stability for a nonlinear iterative equation

Bing Xu, Weinian Zhang (2002)

Colloquium Mathematicae

We discuss the Hyers-Ulam stability of the nonlinear iterative equation G ( f n ₁ ( x ) , . . . , f n k ( x ) ) = F ( x ) . By constructing uniformly convergent sequence of functions we prove that this equation has a unique solution near its approximate solution.

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