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 $F^t:t\ge 0$ of continuous linear set-valued functions $F^t:K\to cc\left(K\right)$ is a differentiable iteration semigroup with F⁰(x) = x for x ∈ K if and only if the set-valued function $\Phi (t,x)=F^t\left(x\right)$ is a solution of the problem $D_t\Phi (t,x)=\Phi (t,G\left(x\right)):=\bigcup \Phi (t,y):y\in G\left(x\right)$, Φ(0,x) = x, for x ∈ K and t ≥ 0, where $D_t\Phi (t,x)$ denotes the Hukuhara derivative of Φ(t,x) with respect to t and $G\left(x\right):=li{m}_{s\to 0+}(F^s\left(x\right)-x)/s$ for x ∈ K.

We discuss the Hyers-Ulam stability of the nonlinear iterative equation $G(f^{n₁}\left(x\right),...,f^{n_k}\left(x\right))=F\left(x\right)$. By constructing uniformly convergent sequence of functions we prove that this equation has a unique solution near its approximate solution.

We work with a fixed N-tuple of quasi-arithmetic means $M₁,...,M_N$ generated by an N-tuple of continuous monotone functions $f₁,...,f_N:I\to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^N\ni b\mapsto (M₁\left(b\right),...,M_N\left(b\right))$ tend pointwise to a mapping having values on the diagonal of $I^N$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M₁,...,M_N$ taken on b. We effectively...

We prove a new sufficient condition for the asymptotic stability of Markov operators acting on measures. This criterion is applied to iterated function systems.

We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.

In this paper we give some criteria for the existence of compactly supported $C^{k+\alpha }$-solutions ($k$ is an integer and $0\le \alpha <1$) of matrix refinement equations. Several examples are presented to illustrate the general theory.