Band 22, Heft 1

We consider the differential equation -y'(x)+q(x)y(x)=f(x),x∈ℝ,$ -y^{\prime }(x)+q(x)y(x)=f(x), \quad x \in \mathbb {R},$ where f∈Lp(ℝ)${f \in L_{p}(\mathbb {R})}$, p∈[1,∞)${p\in [1,\infty )}$, and 0≤q∈L1 loc (ℝ)${0\le q \in L_{1}^{\rm loc}(\mathbb {R})}$, ∫-∞0q(t)dt=∫0∞q(t)dt=∞,${\int _{-\infty }^{0}q(t)\,dt=\int _{0}^{\infty }q(t)\,dt=\infty ,}$q0(a)=infx∈ℝ∫x-ax+aq(t)dt=0foranya∈(0,∞).$q_{0}(a)=\inf _{x\in \mathbb {R}}\int _{x-a}^{x+a}q(t)\,dt=0 \quad \text{for any } a\in (0,\infty ).$ Under these conditions, the above equation is not correctly solvable in Lp(ℝ)${L_{p}(\mathbb {R})}$ for any p∈[1,∞)${p \in [1, \infty ) }$. Let q*(x)${q^{*}(x)}$ be the Otelbaev-type average of the function q ( t ), t∈ℝ${t\in \mathbb {R}}$, at the point t=x${t=x}$; let θ(x)${\theta (x)}$ be a continuous positive function for x∈ℝ${x \in \mathbb {R}}$, and Lp,θ(ℝ)=f∈Lp loc (ℝ):∫-∞∞|θ(x)f(x)|pdx<∞,∥f∥p,θ:=∥f∥Lp,θ(ℝ)=∫-∞∞|θ(x)f(x)|pdx1/p.$L_{p,\theta }(\mathbb {R}) = \Biggl \lbrace f\in L_{p}^{\rm loc}(\mathbb {R}): \int _{-\infty }^{\infty }|\theta (x)f(x)|^{p}\,dx&lt;\infty \Biggr \rbrace ,\quad \Vert f\Vert _{p,\theta }:=\Vert f\Vert _{L_{p,\theta }(\mathbb {R})}=\Biggl (\int _{-\infty }^{\infty }|\theta (x)f(x)|^{p}\,dx\Biggr )^{1/p}.$ We show that if there exists a constant c∈[1,∞)${c\in [1, \infty )}$ such that the inequality c-1q*(x)≤θ(x)≤cq*(x)$c^{-1}q^{*}(x)\le \theta (x)\le cq^{*}(x)$ holds for all x∈ℝ${x \in \mathbb {R}}$, then under some additional conditions for q the pair of spaces {Lp,θ(ℝ);Lp(ℝ)}${\lbrace L_{p, \theta }(\mathbb {R}); L_{p}(\mathbb {R})\rbrace }$ is admissible for the considered equation.

Let f be a meromorphic function in the unit disc and (aν)ν=1k${(a_\nu )_{\nu =1}^k}$ a set of distinct meromorphic functions small with respect to f . An analogue of the second main theorem for f and (aν)ν=1k${(a_\nu )_{\nu =1}^k}$ is given. Upper limits for the sum of defects of an admissible meromorphic function and an admissible holomorphic function follow. For meromorphic and holomorphic functions in the unit disc and their small functions the analogues of Ullrich's theorem are presented.

The paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.

Are every two separated nets in the plane bilipschitz equivalent? In the late 1990s, Burago and Kleiner and, independently, McMullen resolved this beautiful question negatively. Both solutions are based on a construction of a density function that is not realizable as the Jacobian determinant of a bilipschitz map. McMullen's construction is simpler than the Burago–Kleiner one, and we provide a full proof of its nonrealizability, which has not been available in the literature.

In the paper, the authors establish an inequality involving the gamma and digamma functions and apply it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.

The aim of this paper is to study the large-time behaviour of mild solutions to the one-dimensional cooperative systems with anomalous diffusion when at least one entry of the initial condition decays slower than a power. We prove that the solution moves at least exponentially fast as time goes to infinity. Moreover, the exponent of propagation depends on the decay of the initial condition and of the reaction term.

In the present paper we study the generalized slow growth of special monogenic functions. The characterizations of generalized order, generalized lower order, generalized type and generalized lower type of special monogenic functions have been obtained in terms of their Taylor series coefficients.

In this paper, we studied the split equality problems (SEP) with a new proposed iterative algorithm and established the strong convergence of the proposed algorithm to solution of the split equality problems (SEP).

We show that, for a special case, equality of the spectra of single layer potentials defined on two segments implies that these segments must have equal length. We also provide an upper bound for the operator norm and exact expression for the Hilbert–Schmidt norm of single layer potentials on segments.