Let ϕ be an analytic function defined on the unit disk$D$, with ϕ($D$)⊂$D$, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φ_{$n$}/λ^{$n$}converges uniformly on compact subsets of$D$to a function σ analytic in$D$which satisfies$σ$°φ=λ$σ$. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space$H$_{$p$}, the sequence Φ_{$n$}/λ^{$n$}converges to σ in the norm of$H$_{$p$}. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.

We consider the Newton equation(*) $$\ddot x = F\left( x \right), F\left( x \right) = - \nabla \upsilon \left( x \right), x \in R^d ,$$ for |$j$|≤2 and some α>1.

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We study the solvability problem for the multidimensional Riccati equation −∇$u$=|∇$u$|^{q}+ω, where$q$>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation −Δ$u$−ω$u$=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions on$R$^{n}in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type$−Lu=f(x, u, ∇u)$+ω where, and$L$is a uniformly elliptic operator.

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