Showing papers in "Georgian Mathematical Journal in 2016"
TL;DR: In this paper, the authors mainly studied quantitative Voronovskaya-type theorems for q-Szász operators defined in [20] and obtained the quantitative q-VORONOVskaya type theorem in terms of the weighted modulus of continuity of q-derivatives of the approximated function.
Abstract: Abstract In the present paper, we mainly study quantitative Voronovskaya-type theorems for q-Szász operators defined in [20]. We consider weighted spaces of functions and the corresponding weighted modulus of continuity. We obtain the quantitative q-Voronovskaya-type theorem and the q-Grüss–Voronovskaya-type theorem in terms of the weighted modulus of continuity of q-derivatives of the approximated function. In this way, we either obtain the rate of pointwise convergence of q-Szász operators or we present these results for a subspace of continuous functions, although the classical ones are valid for differentiable functions.
66 citations
TL;DR: In this paper, the authors study the relationship between an n-Hom-Lie algebra and its induced (n + 1)-hom-lie algebra and provide an overview of the theory and explore structure properties such as ideals, centers, derived series, solvability, nilpotency, central extensions, and cohomology.
Abstract: The purpose of this paper is to study the relationships between an n-Hom-Lie algebra and its induced (n + 1)-Hom-Lie algebra. We provide an overview of the theory and explore structure properties such as ideals, centers, derived series, solvability, nilpotency, central extensions, and the cohomology.
26 citations
TL;DR: In this paper, Hungar et al. showed that if a ring R with a characteristic different from 2 admits a nonzero derivation d such that d(xx * )=d(x * x)${d(xx^*)=d (x^*x)}$ for all x ∈ R and S(R)∩Z(R),≠(0)${S(R)-cap Z(R,ne (0)}$, then R is commutative.
Abstract: Abstract In [Acta Math. Hungar. 66 (1995), 337–343], Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that d(xy)=d(yx)${d(xy)=d(yx)}$ for all x,y∈R${x,y\\in R}$ , then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that d(xx * )=d(x * x)${d(xx^*)=d(x^*x)}$ for all x ∈ R and S(R)∩Z(R)≠(0)${S(R)\\cap Z(R)\
e (0)}$ , then R is commutative. Moreover, some related results have also been discussed.
20 citations
TL;DR: In this article, the authors studied the asymptotic analysis of a dynamical problem in elasticity with nonlinear friction of Tresca type, where the Lamé coefficients of a thin layer are assumed to vary with respect to the thin layer parameter and to depend on the temperature.
Abstract: Abstract In this paper, we are interested in the study of the asymptotic analysis of a dynamical problem in elasticity with nonlinear friction of Tresca type. The Lamé coefficients of a thin layer are assumed to vary with respect to the thin layer parameter ε and to depend on the temperature. We prove the existence and uniqueness of a weak solution for the limit problem. The proof is carried out by the use of the asymptotic behavior when the dimension of the domain tends to zero.
16 citations
TL;DR: In this paper, a coupled system of nonlinear fractional differential equations involving m nonlinear terms, m ∈ ℕ * ${m\\in\\mathbb{N^{*}}}$, was studied and established new existence and uniqueness results using the Banach contraction principle.
Abstract: Abstract In this paper, we study a coupled system of nonlinear fractional differential equations involving m nonlinear terms, m ∈ ℕ * ${m\\in\\mathbb{N^{*}}}$ . We begin by introducing a new Banach space. Then, we establish new existence and uniqueness results using the Banach contraction principle. We also prove an existence result using the Schaefer fixed point theorem. Finally, we give some illustrative examples.
15 citations
14 citations
TL;DR: In this article, direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the Variable Exponential Grand Lebesge space were established.
Abstract: Abstract In this paper we establish direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the variable exponent grand Lebesgue space.
8 citations
TL;DR: In this article, the Hyers-Ulam stability of the functional equation was obtained for the case where the continuous function f satisfies some Kannappan type condition, provided that f satisfies a continuous function satisfies the Kannappa condition.
Abstract: Abstract In [15] we obtained the Hyers–Ulam stability of the functional equation ∫ K ∫ G f(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.
7 citations
TL;DR: In this paper, the authors established three kinds of endpoint estimates for a class of multilinear singular integral operators and obtained the boundedness of this kind of MISO on the product of BMO spaces, product of LMO spaces and product of λ-central BMO space, respectively.
Abstract: Abstract In this paper, we establish three kinds of endpoint estimates for a class of multilinear singular integral operators and obtain the boundedness of this kind of multilinear singular integral operators on the product of BMO spaces, product of LMO spaces, and product of λ-central BMO spaces, respectively. Moreover, as special cases, the corresponding results of multilinear Calderón–Zygmund operators can be deduced.
6 citations
TL;DR: In this article, the identity of a prime ring with center Z(R), a ∈ R (a ≠ 0) and I a nonzero ideal of R was investigated.
Abstract: Abstract 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${a(F(xy)\\pm F(x)F(y))=0}$ .
6 citations
TL;DR: In this article, the weighted generalized Drazin inverse for elements in rings was introduced and investigated, and the weighted EP elements were introduced and compared to the EP elements in a ring.
Abstract: In this paper we introduce and investigate the weighted generalized Drazin inverse for elements in rings We also introduce and investigate the weighted EP elements
TL;DR: In this paper, the authors give general results which provide asymptotic irrationality measures and estimations for the denominators of the convergents for certain almost periodic simple continued fraction expansions.
Abstract: Abstract We give general results which provide asymptotic irrationality measures and estimations for the denominators of the convergents for certain almost periodic simple continued fraction expansions. As an application we obtain new irrationality measures, for example, to Napier's constant e, certain powers of e, tanh1${\\tanh 1}$ and a ratio of modified Bessel functions.
TL;DR: In this article, the annihilating-ideal graph of a lattice with a least element called zero and denoted by 0 is studied and its properties are studied. And the authors completely determine when the graph is complete bipartite, split and end-regular.
Abstract: Abstract Let L = (L,∧,∨) be a lattice with a least element called zero and denoted by 0. The annihilating-ideal graph of L, denoted by 𝔸𝔾(L), is a graph whose vertex-set is the set of all non-trivial ideals of L and, for every two distinct vertices I and J, I is adjacent to J if and only if I ∧ J = {0}. In this paper, we study some properties of 𝔸𝔾(L). Also, we completely determine when the annihilating-ideal graph is complete bipartite, split and end-regular.
TL;DR: In this article, Bilinear multiplier operator from M (p 1, q 1, ω 1 ) to M (p 2, q 2, ω 2 ) was defined for the unweighted case.
Abstract: Abstract Fix a nonzero window g ∈ 𝒮 ( ℝ n ) ${g\\in\\mathcal{S}(\\mathbb{R}^{n})}$ , a weight function w on ℝ 2 n ${\\mathbb{R}^{2n}}$ and 1 ≤ p , q ≤ ∞ ${1\\leq p,q\\leq\\infty}$ . The weighted Lorentz type modulation space M ( p , q , w ) ( ℝ n ) ${M(p,q,w)(\\mathbb{R}^{n})}$ consists of all tempered distributions f ∈ 𝒮 ′ ( ℝ n ) ${f\\in\\mathcal{S}^{\\prime}(\\mathbb{R}^{n})}$ such that the short time Fourier transform V g f ${V_{g}f}$ is in the weighted Lorentz space L ( p , q , w d μ ) ( ℝ 2 n ) ${L(p,q,w\\,d\\mu)(\\mathbb{R}^{2n})}$ . The norm on M ( p , q , w ) ( ℝ n ) ${M(p,q,w)(\\mathbb{R}^{n})}$ is ∥ f ∥ M ( p , q , w ) = ∥ V g f ∥ p q , w ${\\|f\\/\\|_{M(p,q,w)}=\\|V_{g}f\\/\\|_{pq,w}}$ . This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let 1 < p 1 , p 2 < ∞ ${1
TL;DR: In this paper, the authors derived general solution of new n-dimensional quintic and sextic functional equations and investigated the Hyers-Ulam stability, Hyers Ulam-Rassias stability and generalized Hyers−Ulam−Rassia stability in Felbin type fuzzy normed linear spaces.
Abstract: Abstract In this paper, we derive general solution of new n-dimensional quintic and sextic functional equations and investigate the Hyers–Ulam stability, Hyers–Ulam–Rassias stability and generalized Hyers–Ulam–Rassias stability for quintic and sextic functional equations in Felbin type fuzzy normed linear spaces. Also, we give the counter examples for the Hyers–Ulam–Rassias stability of quintic and sextic functional equations for some cases.
TL;DR: In this article, the relation between the structure of fully invariant submodules of certain QTAG-modules and the structural structure of containing modules was studied, and it was shown that if F is a fully-invariant submodule of the totally projective QTAG module M, then both F and M/F are projective.
Abstract: Abstract If α denotes the class of all QTAG-modules M such that M/H β (M)${M/H_\\beta (M)}$ is totally projective for every ordinal β<α${\\beta <\\alpha }$ , 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=(β k ) k<ω ${\\beta =(\\beta _k)_{k<\\omega }}$ , both Mβ and M/M β ${M/M^\\beta }$ are totally projective, then M itself is necessarily totally projective.
TL;DR: In this paper, the authors studied the properties of bicritical graphs and their relation with critical graphs, and obtained results for BICritical graphs with edge connectivity two or three.
Abstract: Abstract A graph is bicritical if the removal of any pair of vertices decreases the domination number. We study the properties of bicritical graphs and their relation with critical graphs, and we obtain results for bicritical graphs with edge connectivity two or three. We also generalize the notion of the coalescence of two graphs and investigate the bicriticality of such graphs.
TL;DR: In this paper, an asymptotic formula for Bateman's G-function G ( x ) ${G(x)}$ was presented and the double inequality for the error term of the alternating series was derived.
Abstract: Abstract In this paper, we present an asymptotic formula for Bateman’s G-function G ( x ) ${G(x)}$ and deduce the double inequality 1 2 x 2 + 3 / 2 < G ( x ) - 1 x < 1 2 x 2 , x > 0 . $\frac{1}{2x^{2}+3/2}0.$ We apply this result to find estimates for the error term of the alternating series ∑ k = 1 ∞ ( - 1 ) k - 1 k + h $\sum_{k=1}^{\infty}\kern-2.0pt\frac{(-1)^{k-1}}{k+h}$ , h ≠ - 1 , - 2 , - 3 , … $h\kern-1.0pt
eq\kern-1.0pt-1,\kern-0.5pt-2,\kern-0.5pt-3,\ldots\,$ . Also, we study the monotonicity of some functions involving the function G ( x ) ${G(x)}$ . Finally, we propose a sharp double inequality for the function G ( x ) ${G(x)}$ as a conjecture.
TL;DR: In this paper, the authors give one-sided Tauberian conditions of Landau and Hardy type under which (u mns )${(u_{mns})}$ converges in Pringsheim's sense.
Abstract: Abstract Let (u mns )${(u_{mns})}$ be a (C,1,1,1) summable triple sequence of real numbers. We give one-sided Tauberian conditions of Landau and Hardy type under which (u mns )${(u_{mns})}$ converges in Pringsheim's sense. We prove that (u mns )${(u_{mns})}$ converges in Pringsheim's sense if (u mns )${(u_{mns})}$ is slowly oscillating in certain senses. Moreover, we extend a Tauberian theorem given by Móricz [Studia Math. 110 (1994), 83–96] for double sequences to triple sequences.
TL;DR: For higher order nonlinear functional differential equations, sufficient conditions for the solvability and uniqueness of some nonlinear nonlocal boundary value problems are established in this paper, where the authors consider the case of higher order functional differential equation.
Abstract: Abstract For higher order nonlinear functional differential equations, sufficient conditions for the solvability and unique solvability of some nonlinear nonlocal boundary value problems are established.
TL;DR: In this paper, the authors proved the invertibility of boundary integral operators for Dirichlet and Neumann problems in the Besselpotential spaces H s, p ( ∂ D ) ${H^{s,p}(\\partial D)}$, p ∈ ( 1, ∞ ) ${p\\in(1,\\infty)}$.
Abstract: Abstract The paper is devoted to the L p ${L^{p}}$ -theory of boundary integral operators for boundary value problems described by anisotropic Helmholtz operators with variable coefficients in unbounded domains with unbounded smooth boundary. We prove the invertibility of boundary integral operators for Dirichlet and Neumann problems in the Bessel-potential spaces H s , p ( ∂ D ) ${H^{s,p}(\\partial D)}$ , p ∈ ( 1 , ∞ ) ${p\\in(1,\\infty)}$ , and the Besov spaces B p , q s ( ∂ D ) ${B_{p,q}^{s}(\\partial D)}$ , p , q ∈ [ 1 , ∞ ] ${p,q\\in[1,\\infty]}$ . We prove also the Fredholmness of the Robin problem in these spaces and give the index formula.
TL;DR: In this paper, the authors considered three-dimensional Riquier-type and classical mixed boundary value problems for the polymetaharmonic equation ( Δ + k 1 2 ) ( Δ+ k 2 2 ) u = 0, and proved the existence and uniqueness theorems in Sobolev-Slobodetskii spaces.
Abstract: Abstract In the paper we consider three-dimensional Riquier-type and classical mixed boundary value problems for the polymetaharmonic equation ( Δ + k 1 2 ) ( Δ + k 2 2 ) u = 0 ${(\\Delta+k^{2}_{1})(\\Delta+k^{2}_{2})u=0}$ . We investigate these problems by means of the potential method and the theory of pseudodifferential equations. We prove the existence and uniqueness theorems in Sobolev–Slobodetskii spaces, analyse the asymptotic properties of solutions and establish the best Hölder smoothness results for solutions.
TL;DR: In this paper, the authors give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space, especially quasisymmetric homeomorphisms of the circle, which form a group QS (S 1 ) ${\operatorname{QS}(S^{1})}$ with respect to composition.
Abstract: Abstract In this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group QS ( S 1 ) ${\operatorname{QS}(S^{1})}$ with respect to composition. This group acts on the Sobolev space H 0 1 / 2 ( S 1 , ℝ ) ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group QS ( S 1 ) ${\operatorname{QS}(S^{1})}$ and the space H 0 1 / 2 ( S 1 , ℝ ) ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ in terms of noncommutative geometry.
TL;DR: In this article, the Hermite-Hadamard type integral inequalities for n-times differentiable preinvex functions are established by using this identity and Hölder's inequality.
Abstract: Abstract In this paper, we establish a new integral identity for n-times differentiable functions defined on an invex subset of ℝ. Hermite–Hadamard type integral inequalities for n-times differentiable preinvex functions are then established by using this identity and Hölder's inequality.
SIDI1
TL;DR: In this article, the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form ∂ b (x, u ) ∂ ∁� t - div (a(x, t,u, u, ∇ ǫ) + g (x, t, u, ∇ u, gǫ), + H(x t,t, u), ∇ √ u ) + g(x,t,u%, √ nabla u)+H
Abstract: Abstract We study the existence result of a renormalized solution for a class of nonlinear parabolic equations of the form ∂ b ( x , u ) ∂ t - div ( a ( x , t , u , ∇ u ) ) + g ( x , t , u , ∇ u ) + H ( x , t , ∇ u ) = μ in Ω × ( 0 , T ) , ${\\partial b(x,u)\\over\\partial t}-\\operatorname{div}(a(x,t,u,\
abla u))+g(x,t,u% ,\
abla u)+H(x,t,\
abla u)=\\mu\\quad\\text{in }\\Omega\\times(0,T),$ where the right-hand side belongs to L 1 ( Q T ) + L p ′ ( 0 , T ; W - 1 , p ′ ( Ω ) ) ${L^{1}(Q_{T})+L^{p^{\\prime}}(0,T;W^{-1,p^{\\prime}}(\\Omega))}$ and b ( x , u ) ${b(x,u)}$ is unbounded function of u, - div ( a ( x , t , u , ∇ u ) ) ${{-}\\operatorname{div}(a(x,t,u,\
abla u))}$ is a Leray–Lions type operator with growth | ∇ u | p - 1 ${|\
abla u|^{p-1}}$ in ∇ u ${\
abla u}$ . The critical growth condition on g is with respect to ∇ u ${\
abla u}$ and there is no growth condition with respect to u, while the function H ( x , t , ∇ u ) ${H(x,t,\
abla u)}$ grows as | ∇ u | p - 1 ${|\
abla u|^{p-1}}$ .
TL;DR: In this paper, it was shown that there exists a translation invariant measure μ on Ω extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.
Abstract: Abstract The paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure μ on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.
TL;DR: In this paper, screen type mixed boundary value problems for anisotropic pseudo-Maxwell's equations were investigated in tangent Sobolev spaces and the unique solvability results were proven based on the potential method and coercivity result of Costabel on the bilinear form associated with pseudo-maxwells equations.
Abstract: Abstract We investigate screen type mixed boundary value problems for anisotropic pseudo-Maxwell’s equations. We show that the problems with tangent traces are well posed in tangent Sobolev spaces. The unique solvability results are proven based on the potential method and coercivity result of Costabel on the bilinear form associated with pseudo-Maxwell’s equations.
TL;DR: In this paper, the modification of Szász-Durrmeyer operators based on the Jain basis function is considered and Voronovskaya-type estimates of point-wise convergence along with its quantitative version based on weighted modulus of smoothness are given.
Abstract: Abstract In this paper we consider the modification of Szász–Durrmeyer operators based on the Jain basis function. Voronovskaya-type estimates of point-wise convergence along with its quantitative version based on the weighted modulus of smoothness are given. Moreover, a direct approximation theorem for the operators is proved.
TL;DR: In this paper, the convergence of generated iterative sequences in modular spaces is studied and a double sequence iteration is shown to converge strongly to a fixed point of a ρ-quasi contraction mapping.
Abstract: Abstract In this paper, we introduce new nonlinear iterative algorithms. These algorithms are used to study the convergence of generated iterative sequences in modular spaces. Moreover, we introduce a new double sequence iteration and prove that sequences converge strongly to a fixed point of a ρ-quasi contraction mapping in modular spaces. Finally, some illustrative numerical examples (using the Matlab software) are presented.
TL;DR: In this paper, the authors introduced functions of bounded fractional differential variation using the Caputo-type fractional derivative instead of the commonly used first-order derivative, and proved that the space BFDV [ a, b ] ${mathrm{BFDV}[a,b]}$ is a normed algebra under certain type of norms.
Abstract: Abstract The idea of functions of bounded differential variation was introduced by Bhatt, Dabhi and Kachhia in [2]. In the present paper, we introduce functions of bounded fractional differential variation using the Caputo-type fractional derivative instead of the commonly used first-order derivative. Various properties and relation with some known results of classical analysis are also studied. We prove that the space BFDV [ a , b ] ${\\mathrm{BFDV}[a,b]}$ of all functions of bounded fractional differential variation on [ a , b ] ${[a,b]}$ is a normed algebra under certain type of norms.