Volume 21, Issue 3

We discuss the existence of solutions for a differential equation of fractional order with multi-point nonlocal boundary conditions via standard tools of fixed point theory. The paper concludes with some illustrative examples.

In this paper, we present our recent results on the so-called complex structures in algebras in the context relevant to the following four problems which, at first glance, have nothing in common: (i) solubility of polynomial equations in non-associative algebras, (ii) topological/geometric properties of quadratic maps, (iii) existence of bounded solutions to quadratic differential systems, and (iv) ellipticity of the Dirac equation. The unsolved problems related to (i)–(iv) are formulated as well.

Let ℱ be a family of meromorphic functions in D , M(f)=fn0(f')n1⋯(f(k))nk$M(f)=f^{n_0}(f^{\prime })^{n_1}\cdots (f^{(k)})^{n_k}$ with ΓM≥2$\Gamma _M \ge 2$, let H ( f ) be a differential polynomial of f satisfying Γγ|H<ΓMγM$\frac{\Gamma }{\gamma }|_H&lt; \frac{\Gamma _M}{\gamma _M}$, and let ϕ(z)$\varphi (z)$ (≠0$\ne 0$) be an analytic function in D . If, for each function f∈ℱ$f \in \mathcal {F}$, f ≠ 0, M(f)+H(f)-ϕ(z)$M(f)+H(f)- \varphi (z)$ has at most ΓM-1$\Gamma _M-1$ distinct zeros in D , then ℱ is normal in D .

In [`Robin function and conformal welding – A new proof of the existence', submitted] a new proof of the existence of the Robin function is given using conformal welding. This proof is essentially based on the simply connected case and a conformal mapping which maps the domain onto a complement of a disk and a system of radial spires. The aim of this note is to show that this proof also has a wider meaning, namely that this mapping is the unique solution of extremal problems, which minimizes the length or the diameter of a subset of the boundary curve. As a consequence we obtain sharp inequalities for the Robin capacity and the logarithmic capacity.

We extend the classical Schwarz Lemma to homeomorphisms with integrally bounded p -module in ℝ n .

We provide details for Gromov's proof of Stallings' theorem on groups with infinitely many ends using harmonic functions. The main technical result of the paper is a compactness theorem for a certain family of harmonic functions.

One characterization of an uncountable set E is given in terms of mappings f of E into itself and “large” subsets of E which are invariant with respect to f .

For first order singular functional differential equations, optimal sufficient conditions for the existence of positive solutions of periodic type boundary value problems are established.

We investigate the invariant metrics and complex geodesics in the universal Teichmüller space and the Teichmüller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all non-expanding invariant metrics in either of these spaces coincide with its intrinsic Teichmüller metric. Other applications concern the variational theory for univalent functions with quasiconformal extension. It turns out that geometric features caused by the equality of metrics and connection with complex geodesics provide deep distortion results for various classes of such functions and create new phenomena which do not appear in the classical geometric function theory.

We show by a new example how to obtain a priori estimates for the velocity of the classical potential flow, using distortion theory for conformal mappings.

A function f with bounded variation and integrable Hilbert transform of d f is locally absolutely continuous. We give a direct real-valued proof of this result and connect it with integrability problems for the Fourier transform of a function with bounded variation, in particular, with a Hardy–Littlewood theorem.

The paper considers operators that map the set E onto itself. If A(ϕ)=ψ${A(\varphi )=\psi }$, then ψ preserves some property of the point ϕ (here A is an operator and ϕ,ψ∈E${\varphi ,\psi \in E}$). Such an approach to operators may serve as a method of solving some problems of mathematics, including the theory of orthogonal series.

We study the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood–Paley g -function and gλ*$g^*_\lambda $-function on the generalized Morrey spaces Lp,Φ$L^{p,\Phi }$ for 1≤p<∞$1\le p&lt;\infty $, where Φ is a growth function on (0,∞)$(0,\infty )$ satisfying the doubling condition. The boundedness of commutators generated by BMO(ℝn)$\operatorname{BMO}(\mathbb {R}^n)$ functions and intrinsic square functions is also obtained.

In this paper, we investigate the normal families of meromorphic functions concerning shared analytic functions and prove some normal criteria that generalize or extend some results obtained by J. Ding et al. [Abstr. Appl. Anal. (2012), Art. ID 582854], [Bull. Malays. Math. Sci. Soc. (2) 35 (2012), 449–457].