Volume 37, Issue 1

In this paper, we study a periodic boundary value problem for a nonautonomous second-order differential equation with a p -Laplacian. Using the dual least-action principle and a perturbation technique, we obtain existence results under a sub- p th-polynomial condition. A generalized Wirtinger’s inequality is employed to extend some recent results.

Some double integral inequalities of Hermite–Hadamard type for h -convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.

Let 𝒮H${\mathcal{S}_{H}}$ denote the usual class of all normalized functions f=h+g¯${f=h+\overline{g}}$ harmonic and sense-preserving univalent on the unit disk |z|<1${|z|<1}$. In this article we show that the set, consisting of those mappings f from 𝒮H${\mathcal{S}_{H}}$ for which all Taylor coefficients of the analytic and co-analytic parts of f are integers, consists of only nine functions. The second aim is to discuss the set 𝒮H${\mathcal{S}_{H}}$ of those functions which have half-integer coefficients. More precisely, we determine the set of univalent harmonic mappings with half-integer coefficients which are convex in real direction or convex in imaginary direction. This work generalizes the recent paper of Hiranuma and Sugawa. One of the examples generated in this way helps to disprove a conjecture of Bharanedhar and Ponnusamy.

We consider weighted composition operators Wψ,φ:f↦ψ⁢(f∘φ)${W_{\psi,\varphi}\colon f\mapsto\psi(f\circ\varphi)}$ on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of Wψ,φ${W_{\psi,\varphi}}$ on general function spaces, and in particular on weak vector-valued spaces. As an application, we characterize the weak compactness of Wψ,φ${W_{\psi,\varphi}}$ between two different vector-valued Bloch-type spaces. This characterization appears to be new also in the scalar-valued case.

Let u∈C⁢(ℝ,H2)${u\in C(\mathbb{R},H^{2})}$ be the solution to the initial value problem for a fourth-order semi-linear Schrödinger equation with pure power nonlinearity. We prove that some Lr${L^{r}}$-norms of u decay as t→±∞${t\to\pm\infty}$.