Volume 27, Issue 1

In this paper, we deal with improvements on the constant Mn⁢(γ){M_{n}(\gamma)} in the so-called Videnskiĭ inequality |(Bn⁢f)⁢(x)-f⁢(x)-x⁢(1-x)2⁢n⁢f′′⁢(x)|≤Mn⁢(γ)⁢x⁢(1-x)n⁢ω⁢(f′′;γn)\biggl{|}(B_{n}f)(x)-f(x)-\frac{x(1-x)}{2n}f^{\prime\prime}(x)\biggr{|}\leq M_% {n}(\gamma)\frac{x(1-x)}{n}\omega\bigg{(}f^{\prime\prime};\sqrt{\frac{\gamma}{% n}}\biggr{)} for a fixed constant γ≥1{\gamma\geq 1} and x∈[0,1]{x\in[0,1]}, where Bn⁢f{B_{n}f} is the Bernstein polynomial, f∈C2⁢[0,1]{f\in C^{2}[0,1]} and ω is the first order modulus of continuity. Let M⁢(γ)=supn∈ℕ⁡Mn⁢(γ){M(\gamma)=\sup_{n\in\mathbb{N}}M_{n}(\gamma)}. We prove the Videnskiĭ inequality for arbitrary γ≥1{\gamma\geq 1}. In particular, we improve the constant M⁢(2)=0.9{M(2)=0.9} (Gonska and Ra şa [8], 2008) to M⁢(2)=0.6875{M(2)=0.6875}. Finally, we consider Mn⁢(1){M_{n}(1)} for small values of n .

In this paper, we consider generalized α-ψ{\alpha-\psi}-Geraghty contractive type mappings and investigate the existence and uniqueness of a fixed point for mappings involving such contractions. In particular, we extend, improve and generalize some earlier results in the literature on this topic. An application concerning the existence of an integral equation is also considered to illustrate the novelty of the main result.

In this paper, we implement the optimal homotopy asymptotic method to find the approximate solutions of the Poisson–Boltzmann equation. We also use the results of the conjugate gradient method for comparison with those of the optimal homotopy asymptotic method. Our study reveals that the optimal homotopy asymptotic method gives more effective results than conjugate gradient algorithms for the considered problems.

Let (pn){(p_{n})} and (qn){(q_{n})} be any two non-negative real sequences, with Rn:=∑k=0npk⁢qn-k≠0{R_{n}:=\sum_{k=0}^{n}{p_{k}q_{n-k}}\neq 0} (n∈ℕ{n\in\mathbb{N}}). Let ∑k=0∞ak{\sum_{k=0}^{\infty}a_{k}} be a series of real or complex numbers with partial sums (sn){(s_{n})}, and set tnp,q:=1Rn⁢∑k=0npk⁢qn-k⁢sk{t_{n}^{p,q}:=\frac{1}{R_{n}}\sum_{k=0}^{n}{p_{k}q_{n-k}s_{k}}} for n∈ℕ{n\in\mathbb{N}}. In this paper, we present the necessary and sufficient conditions under which the existence of the limit limn→∞⁡sn=L{\lim_{n\to\infty}{s_{n}}=L} follows from that of limn→∞⁡tnp,q=L{\lim_{n\to\infty}t_{n}^{p,q}=L}. These conditions are one-sided or two-sided if (sn){(s_{n})} is a sequence of real or complex numbers, respectively.

We prove a multilinear version of the reverse Hölder inequality in the theory of Muckenhoupt Ap{A_{p}} weights. We give two applications of this inequality to the study of multilinear weighted norm inequalities. First, we prove a structure theorem for multilinear Ap→{A_{\vec{p}}} weights; second, we give a new sufficient condition for multilinear, two-weighted norm inequalities for a maximal operator.

Green’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function R⁢(z,ζ){R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n -fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

For the nonlinear functional differential equation with several constant delays, the variation formulas for its solution are proved, in which the effects of perturbations of delays and the initial moment are detected.

We establish the well-posedness of a nonlocal multipoint problem for a second-order evolution equation with respect to a time variable with an operator having a discrete spectrum. A nonlocal condition is considered to be satisfied in a weak sense in the space of formal Fourier series that are identified with continuous linear functionals (generalized elements) on some space connected with the operator.

In this paper, a new computational method based on the Chebyshev wavelets (CWs) is proposed for solving nonlinear stochastic Itô–Volterra integral equations. In this way, a new stochastic operational matrix (SOM) for the CWs is obtained. By using these basis functions and their SOM, such problems can be transformed into nonlinear systems of algebraic equations which can be simply solved. Moreover, a new technique for computation of nonlinear terms in such problems is presented. Further error analysis of the proposed method is also investigated and the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient.

In this work, a method for calculating an approximate solution of a singular integral equation of first kind is presented for the Neumann boundary value problems for the Helmholtz equation.

In this paper, we introduce some classes of R -modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set ℍ={M∣M⁢ is an ⁢R⁢-module and ⁢Nil⁢(M)⁢ is a divided prime submodule of ⁢M}.\mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}. For an R -module M∈ℍ{M\in\mathbb{H}}, set T=(R∖Z⁢(R))∩(R∖Z⁢(M)){T=(R\setminus Z(R))\cap(R\setminus Z(M))}, 𝔗⁢(M)=T-1⁢M{\mathfrak{T}(M)=T^{-1}M} and P=(Nil(M):RM){P=(\mathrm{Nil}(M):_{R}M)}. In this case, the mapping Φ:𝔗⁢(M)→MP{\Phi:\mathfrak{T}(M)\to M_{P}} given by Φ⁢(x/s)=x/s{\Phi(x/s)=x/s} is an R -module homomorphism. The restriction of Φ to M is also an R -module homomorphism from M into MP{M_{P}} given by Φ⁢(x)=x/1{\Phi(x)=x/1} for every x∈M{x\in M}. A nonnil submodule N of M is said to be Φ-invertible if Φ⁢(N){\Phi(N)} is an invertible submodule of Φ⁢(M){\Phi(M)}. Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if Φ⁢(N){\Phi(N)} is a principal ideal of Φ⁢(M){\Phi(M)} for every finitely generated submodule N of the R -module M . The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R -modules.

In this paper, we first classify Einstein-like metrics on hypercomplex four-dimensional Lie groups. Then we obtain the exact form of all harmonic maps on these spaces. We also calculate the energy of an arbitrary left-invariant vector field X on these spaces and determine all critical points for their energy functional restricted to vector fields of the same length. Furthermore, we give a complete and explicit description of all totally geodesic hypersurfaces of these spaces. The existence of Einstein hypercomplex four-dimensional Lie groups and the non-existence of non-trivial left-invariant Ricci and Yamabe solitons on these spaces are also proved.

We study the dependence of differential properties of an indefinite integral on a rotation of the coordinate system. Namely, the following problem is studied: For a summable function f , what kind of a set may be the set of rotations θ for which ∫f{\int f} is not differentiable with respect to the θ-rotation of a given basis B ? For translation invariant bases B formed by two-dimensional intervals, some classes of sets of singular rotations are found. In particular, for such bases with symmetric structure, a characterization of at most countable sets of singular rotations is found.

In this article we investigate some commutativity criteria for a ring with involution (R,∗{(R,\ast}) in which generalized derivations satisfy certain algebraic identities. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

We consider generalized (κ,μ){(\kappa,\mu)}-paracontact metric manifolds satisfying certain flatness conditions on the ℳ{\mathcal{M}}-projective curvature tensor. Specifically, we study ξ-ℳ{\mathcal{M}}-projectively flat and ℳ{\mathcal{M}}-projectively flat generalized (κ,μ){(\kappa,\mu)}-paracontact metric manifolds and, further, ϕ-ℳ{\mathcal{M}}-projectively symmetric generalized (κ≠-1,μ){(\kappa\neq-1,\mu)}-paracontact metric manifolds. We prove that they are characterized by certain structures whose properties are discussed in some detail.

First, we correct the mistake in the inversion theorem of the ripplet transform in the literature. Next, we prove a convolution theorem for the ripplet transform and extend the ripplet transform as a continuous, linear, injective mapping from a suitable Boehmian space into another Boehmian space.

We show that the fractional operator Iα⁢(⋅){I^{\alpha(\,\cdot\,)}}, of variable order on a bounded open set in Ω, in a quasimetric measure space (X,d,μ){(X,d,\mu)} in the case α⁢(x)⁢p⁢(x)≡n{\alpha(x)p(x)\equiv n} (where n comes from the growth condition on the measure μ), is bounded from the variable exponent Lebesgue space Lp⁢(⋅)⁢(Ω){L^{p(\,\cdot\,)}(\Omega)} into BMO⁢(Ω){\mathrm{BMO}(\Omega)} under certain assumptions on p⁢(x){p(x)} and α⁢(x){\alpha(x)}.