Let X be a Banach space, let K(X) := f : X → X Lipschitz: f − id sup < ∞ , and let P : K(X) → K(X), F ∈ K(X). By applying the Banach contraction principle we prove that if P is sufficiently close (in a certain sense) to the identity then the equationhas a unique solution f . As a corollary we obtain results on iterative equations of the types i

Let V be a convex subset of a normed space and let a nondecreasing functionIt is known (Tabor in Control Cybern., 38/3: [656][657][658][659][660][661][662][663][664][665][666][667][668][669] 2009) that if f : V → R is α-midconvex, locally bounded above at every point of V thenwhere Pα(r) := ∞ k=0 1 2 k α(2dist(2 k r, Z)) for r ∈ R. We show that under some additional assumptions the above estimation cannot be improved. Mathematics Subject Classification (2000). 26A51, 39B82.

It is known that the isometry equation is stable in Banach spaces. In this paper we investigate stability of isometries in real p -Banach spaces, that is Frechet spaces with p -homogenous norms, where p ∈ (0, 1] .Let X, Y be p -Banach spaces and let f :ε for all x, y ∈ X . We show that if f is a surjective then there exists an affine surjective isometry U : X → Y and a constant C p such thatWe also show that in general the above estimation cannot be improved.