Band 34, Heft 1

Let p∈[1,∞)$p\in [1,\infty )$, 0≤q∈L1loc(R).$0\le q\in L_1^{\operatorname{loc}}(\mathbb R).$ Consider an equation -y′(x)+q(x)y(x)=f(x),x∈R\begin{equation}\label {1} - y'(x)+q(x)y(x)=f(x),\quad x\in \mathbb R \end{equation} where f  ∈  L p (ℝ), and the weight function space Sp(1)(R,q)={y∈ACloc(R):∥y′-qy∥Lp[(R)+∥q1/py∥Lp(R)<∞}.\begin{align*} S_p^{(1)}(\mathbb R,q)=\{y\in AC^{\operatorname{\loc}}(\mathbb R):\|y'-qy\|_{L_p[(\mathbb R)}+\|q^{1/p}y\|_{L_p(\mathbb R)}\lt\infty\}.\end{align*} The following assertion is the main result of the paper: Theorem. Equation (0.1) is correctly solvable in L p (ℝ), p  ∈ [1,∞ ) if and only if the space S p (1) (ℝ,  q ) is embedded into L p (ℝ).

In this paper we study the concept of almost asymptotically lacunary statistical convergent sequences in probabilistic normed spaces and prove some basic properties.

We introduce a gamma function Γ( x ,  z ) in two complex variables which extends the classical gamma function Γ( z ) in the sense that lim  x →1 Γ( x ,  z ) =Γ ( z ). We will show that many properties which Γ( z ) enjoys extend in a natural way to the function Γ( x ,  z ). Among other things we shall provide functional equations, a multiplication formula, and analogues of the Stirling formula with asymptotic estimates as consequences.

The aim of this paper is to prove a conjecture of Picone concerning a mean value formula for polyharmonic functions. In this formula the mean values are considered on a set of concentric spheres.

In this paper, we define the Euler phi function φ π ( n ) associated to the automorphic L -function L ( s ,  π ), where π is an automorphic irreductible unitary cuspidal representation of GL N (ℚ). Furthermore, we study the asymptotic properties for φ π ( n ) and prove some results on the summation function ∑ n ≤ x φ π ( n ).

With the notion of weighted sharing of values we investigate the uniqueness problem of meromorphic functions when certain non-linear differential polynomials share the same 1-points. We prove two uniqueness results which significantly rectify, improve and generalize the recent result of L. Liu [10].

We obtain the location of the weighted Fermat–Torricelli point A 0 for a geodesic triangle △  A 1 A 2 A 3 on the K -plane (two dimensional sphere S K 2 and two dimensional hyperbolic plane H K 2 ) by deriving a rational algebraic equation which depends on the variable z  ≡  sin  ∠  A 0 A 1 A 3 . Thus, by squaring both parts of this rational equation and by repeating this process until we eliminate the radicals with respect to z , we could derive a polynomial equation P ( z ) = 0, which yields a discrete (finite) set of solutions with respect to the variable z . The uniqueness of the weighted Fermat–Torricelli point leads to the unique optimal solution (global minimum point) which corresponds to the finite set of solutions of the roots of the polynomial P ( z ).

We prove that for sufficiently small δ , the Hausdorff dimension of an m dimensional δ -Reifenberg flat set of ℝ n does not exceed m  +  C ( n ,  m ) δ 2 . An example shows that our estimate is optimal.