Volume 23, Issue 2

We consider the generalized shift operator associated with the Laplace–Bessel differential operator ΔB=∑i=1n∂2∂xi2+∑i=1kγixi∂∂xi.$ \Delta _{B}=\sum _{i=1}^{n}\frac{\partial ^2 }{\partial x_i^2} +\sum _{i=1}^{k} \frac{\gamma _i }{x_i}\frac{\partial }{\partial x_i}. $ The maximal operator Mγ${M_{\gamma }}$ ( B -maximal operator) and the Riesz potential Iα,γ${I_{\alpha ,\gamma }}$ ( B -Riesz potential), associated with the generalized shift operator are investigated. We prove that the B -maximal operator Mγ${M_{\gamma }}$ and the B -singular integral operator are bounded from the generalized weighted B -Morrey space ℳp,ω1,φ,γ(ℝk,+n)${{\cal M}_{p,\omega _1,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$ to ℳp,ω2,φ,γ(ℝk,+n)${{\cal M}_{p,\omega _2,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$ for all 1<p<∞${1 &lt; p &lt; \infty }$, φ∈Ap,γ(ℝk,+n)${\varphi \in A_{p,\gamma }(\mathbb {R}_{k,+}^{n})}$. Furthermore, we prove that the B -Riesz potential Iα,γ${I_{\alpha ,\gamma }}$, 0<α<n+|γ|${0&lt;\alpha &lt;n+|\gamma |}$, is bounded from the generalized weighted B -Morrey space ℳp,ω1,φ,γ(ℝk,+n)${{\cal M}_{p,\omega _1,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$ to ℳq,ω2,φ,γ(ℝk,+n)${{\cal M}_{q,\omega _2,\varphi ,\gamma }(\mathbb {R}_{k,+}^{n})}$, where α/(n+|γ|)=1/p-1/q${{\alpha }/{(n+|\gamma |)}=1/p-1/q}$, 1<p<(n+|γ|)/α${1&lt;p&lt;(n+|\gamma |)/{\alpha }}$, φ∈A1+q/p',γ(ℝk,+n)${\varphi \in A_{1+{q/p^{\prime }},\gamma }(\mathbb {R}_{k,+}^{n})}$ and 1/p+1/p'=1${{1/p}+{1/p^{\prime }}=1}$.

This paper provides some explicit expressions concerning the formal group laws of the following cohomology theories: BP, the Brown–Peterson cohomology, G ( s ), the cohomology theory obtained from BP by putting vi=0${v_i=0}$ for all i ≥ 1 with i≠s${i \ne s}$, the Morava K -theory and the Abel cohomology. The coefficient rings of Weierstrass universal elliptic curve formal group law are computed in low dimensions.

We consider the Dirichlet problem for an elliptic equation with variable coefficients, the solution of which is obtained by means of a finite-difference scheme of second order accuracy. We establish a two-stage finite-difference method for the posed problem and obtain an estimate of the convergence rate consistent with the smoothness of the solution. It is proved that the solution of the corrected scheme converges at rate O(|h|m)${O(|h|^m)}$ in the discrete L 2 -norm, when the solution of the original problem belongs to the Sobolev space with exponent m∈[2,4]${m\in [2,4]}$.

In [15] we obtained the Hyers–Ulam stability of the functional equation ∫K∫Gf(xtk·y)dμ(t)dk=f(x)g(y),x,y∈G,$ \int _{K}\int _{G} f(xtk\cdot y)\,d\mu (t)\,dk=f(x)g(y), \quad x, y\in G, $ where G is a Hausdorff locally compact topological group, K is a compact subgroup of morphisms of G , μ is a K -invariant complex measure with compact support, provided that the continuous function f satisfies some Kannappan type condition. The purpose of this paper is to remove this restriction.

Let R be a prime ring with center Z ( R ), a ∈ R ( a ≠ 0) and I a nonzero ideal of R . Suppose that F,d:R→R${F,d\colon R\rightarrow R}$ are any two mappings such that F(xy)=F(x)y+xd(y)${F(xy) = F(x)y+xd(y)}$ for all x,y∈R${x, y \in R}$. For all x,y∈I${x,y\in I}$, we investigate the identities a(F(xy)±xy)=0${a(F(xy)\pm xy)=0}$, a(F(xy)±yx)=0${a(F(xy)\pm yx)=0}$, a(F(x)F(y)±xy)=0${a(F(x)F(y)\pm xy)=0}$, a(F(x)F(y)±yx)=0${a(F(x)F(y)\pm yx)=0}$, a(d(x)F(y)±xy)∈Z(R)${a(d(x)F(y)\pm xy)\in Z(R)}$, a(d(x)F(y)±yx)∈Z(R)${a(d(x)F(y)\pm yx)\in Z(R)}$ and a(F(xy)±F(x)F(y))=0${a(F(xy)\pm F(x)F(y))=0}$.

Some trapezoid type inequalities for the Riemann–Stieltjes integral of continuous complex-valued integrands defined on the complex unit circle 𝒞(0,1)${\mathcal {C}(0,1)}$ and various subclasses of integrators of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided.

In this paper, we consider a nonlinear system of higher order three-point boundary value problems on time scales. The Schauder fixed point theorem is used to investigate the existence of solutions of nonlinear dynamic systems on time scales. Furthermore, we establish the criteria for the existence of at least one, two and three positive solutions for higher order nonlinear dynamic systems on time scales by using the four functionals fixed point theorem, the Avery–Henderson fixed point theorem and the five functionals fixed point theorem, respectively.

If α denotes the class of all QTAG-modules M such that M/Hβ(M)${M/H_\beta (M)}$ is totally projective for every ordinal β<α${\beta &lt;\alpha }$, then these modules are called α-modules. Here we study the relation between the structure of fully invariant submodules of certain QTAG-modules and the structure of containing modules. It is found that if F is a fully invariant submodule of the totally projective QTAG-module M , then both F and M / F are totally projective. We show that if for some sequence β=(βk)k<ω${\beta =(\beta _k)_{k&lt;\omega }}$, both M β and M/Mβ${M/M^\beta }$ are totally projective, then M itself is necessarily totally projective.

This paper deals with the existence, uniqueness and maximal regularity of a solution to the multidimensional parabolic equation of order 2 m (with m∈ℕ*${m\in \mathbb {N} ^{\ast }}$) ∂tu+(-1)m∑k=1N∂xk2mu=f,$ \partial _{t}u+(-1)^{m}\sum _{k=1}^{N}\partial _{x_{k}}^{2m}u=f, $ defined on a type of the prism Q=(t,x1)∈ℝ2:0<t<T,φ1(t)<x1<φ2(t)×∏i=1N-1]0,bi[$ Q=\bigl \lbrace ( t,x_{1}) \in \mathbb {R} ^{2}:0&lt;t&lt;T,\, \varphi _{1}( t) &lt;x_{1}&lt;\varphi _{2}( t) \bigr \rbrace \times \prod _{i=1}^{N-1}]0,b_{i}[ $ of ℝ N+1 , which is a nonstandard domain since its base shrinks at t = 0 (φ1(0)=φ2(0)${\varphi _{1}(0)=\varphi _{2}(0)}$). The equation is associated with Cauchy–Dirichlet boundary conditions and the right-hand side term f is taken in a weighted Lebesgue space. More precisely, we look for minimal conditions on the functions φ1${\varphi _{1}}$ and φ2${\varphi _{2}}$ under which the solution is regular. For this purpose, the domain decomposition method is employed. This work is an extension of the one space variable case studied in [19] and of the case m = 1 studied in [11].

In this work, by solving a d -Geronimus problem we introduce new examples of basic hypergeometric d -orthogonal polynomials which are useful to construct a similar table to the q -Askey-scheme in the context of d -orthogonality. For d = 1, this leads to a characterization theorem involving all polynomials belonging to the q -Askey-scheme, except the continuous q -Hermite ones.

In this paper, the T b theorem of the inhomogeneous Besov and Triebel–Lizorkin spaces associated with a para-accretive function on homogeneous type spaces is established. As an application, we obtain that the Riesz potential can be used as the lifting operator of these spaces.

For first order nonlinear delay differential equations, necessary and sufficient conditions are established for the oscillation of all proper solutions as well as for the existence of at least one vanishing at infinity proper Kneser solution.

In the paper, we establish necessary and sufficient conditions for two families of functions involving divided differences of the di- and tri-gamma functions to be completely monotonic. Consequently, we derive necessary and sufficient conditions for two families of functions involving the ratio of two gamma functions to be logarithmically completely monotonic. Furthermore, we deduce some inequalities for bounding the ratio of two gamma functions and divided differences of polygamma functions.