Showing papers in "Mathematica Slovaca in 2019"

∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds

Abstract: Abstract In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

Sharp coefficient bounds for starlike functions associated with the Bell numbers

Abstract: Abstract Let SB∗ $\\begin{array}{} \\mathcal{S}^*_B \\end{array}$ be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class SB∗ $\\begin{array}{} \\mathcal{S}^*_B \\end{array}$ several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Bounds on the first three consecutive higher-order Schwarzian derivatives for functions in the class SB∗ $\\begin{array}{} \\mathcal{S}^*_B \\end{array}$ are investigated.

On sequence spaces generated by binomial difference operator of fractional order

Abstract: Abstract In this article we introduce binomial difference sequence spaces of fractional order α, bpr,s $\\begin{array}{} b_p^{r,s} \\end{array}$ (Δ(α)) (1 ≤ p ≤ ∞) by the composition of binomial matrix, Br,s and fractional difference operator Δ(α), defined by (Δ(α)x)k = ∑i=0∞(−1)iΓ(α+1)i!Γ(α−i+1)xk−i $\\begin{array}{} \\displaystyle \\sum\\limits_{i=0}^{\\infty}(-1)^i\\frac{\\Gamma(\\alpha+1)}{i!\\Gamma(\\alpha-i+1)}x_{k-i} \\end{array}$. We give some topological properties, obtain the Schauder basis and determine the α, β and γ-duals of the spaces. We characterize the matrix classes ( bpr,s $\\begin{array}{} b_p^{r,s} \\end{array}$(Δ(α)), Y), where Y ∈ {ℓ∞, c, c0, ℓ1} and certain classes of compact operators on the space bpr,s $\\begin{array}{} b_p^{r,s} \\end{array}$(Δ(α)) using Hausdorff measure of non-compactness. Finally, we give some geometric properties of the space bpr,s $\\begin{array}{} b_p^{r,s} \\end{array}$(Δ(α)) (1 < p < ∞).

On weakly -permutable subgroups of finite groups

Abstract: Abstract Let σ = {σi ∣i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi∣σi ∩ π(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H0 ≤ H1 ≤ … ≤ Ht = G such that either Hi−1 is normal in Hi or Hi/(Hi−1)Hi is σ-primary for all i = 1, …, t. A set 𝓗 of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of 𝓗 is a Hall σi-subgroup of G for some i and 𝓗 contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). Let 𝓗 be a complete Hall σ-set of G. A subgroup H of G is said to be 𝓗-permutable if HA = AH for all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H𝓗, where H𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable. By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.

Common fixed point theorems for a class of (s, q)-contractive mappings in b-metric-like spaces and applications to integral equations

Abstract: Abstract In this paper, we establish fixed point theorems for one and two selfmaps in b-metric-like spaces, using (s, q)-contractive and F-(ψ, φ, s, q)-contractive conditions, defined by means of altering distances and 𝓒-class functions. Our theorems unify, extend and generalize corresponding results in the literature.

The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions

Abstract: Abstract In this paper, with a new approach, a new fractional Hermite-Hadamard type inequalities for convex functions is obtained by using only the left Riemann-Liouville fractional integral. Also, to have new fractional trapezoid and midpoint type inequalities for the differentiable convex functions, two new equalities are proved. Our results generalize earlier studies. We expect that this study will be lead to the new fractional integration studies for Hermite-Hadamard type inequalities.

Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups

Abstract: Abstract The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.

Weakly demicompact linear operators and axiomatic measures of weak noncompactness

Abstract: Abstract In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O’REGAN, D.: On the class of relatively weakly demicompact nonlinear operators, Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators.

Integrals of logarithmic functions and alternating multiple zeta values

Abstract: By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form \[\zeta ( {\bar 1,{\left\{ 1 \right\}_{m - 1}},\bar 1,{\left\{ 1 \right\}_{k - 1}}} ),\ (k,m\in \mathbb{N})\] for $m=1$ or $k=1$, and \[\zeta ( {\bar 1,{\left\{ 1 \right\}_{m - 1}},p,{\left\{ 1 \right\}_{k - 1}}}),\ (k,m\in\mathbb{N})\] for $p=1$ and $2$, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and $\ln 2$. Moreover, we also prove that the multiple zeta values $\zeta ( {\bar 1,{\left\{ 1 \right\}_{m - 1}},3,{\left\{ 1 \right\}_{k - 1}}} )$ can be expressed as a rational linear combination of products of zeta values, multiple polylogarithms and $\ln 2$ when $m=k\in \mathbb{N}$. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at $\frac {1}{2}$.

Some properties associated to a certain class of starlike functions

Abstract: Abstract Let 𝓐 denote the family of analytic functions f with f(0) = f′(0) – 1 = 0, in the open unit disk Δ. We consider a class Scs∗(α):=f∈A:zf′(z)f(z)−1≺z1+α−1z−αz2,z∈Δ, $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γn of f ∈ Scs∗ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$(α): ∑n=1∞γn2≤141+α2π26−2Li2−α+Li2α2, $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li2 denotes the dilogarithm function are investigated.

A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator

Abstract: Abstract Making use of Ruscheweyh q-differential operator, we define a new subclass of uniformly convex functions and corresponding subclass of starlike functions with negative coefficients. The main object of this paper is to obtain, coefficient estimates, closure theorems and extreme point for the functions belonging to this new class. The results are generalized to families with fixed finitely many coefficients.

On a Choquet-Stieltjes type integral on intervals

Abstract: Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

Perfect 1-factorizations

Abstract: Abstract Let G be a graph with vertex-set V = V(G) and edge-set E = E(G). A 1-factor of G (also called perfect matching) is a factor of G of degree 1, that is, a set of pairwise disjoint edges which partitions V. A 1-factorization of G is a partition of its edge-set E into 1-factors. For a graph G to have a 1-factor, |V(G)| must be even, and for a graph G to admit a 1-factorization, G must be regular of degree r, 1 ≤ r ≤ |V| − 1. One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs. A 1-factorization of G is said to be perfect if the union of any two of its distinct 1-factors is a Hamiltonian cycle of G. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs. It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

Stabilization of third order differential equation by delay distributed feedback control with unbounded memory

Abstract: Abstract There are almost no results on the exponential stability of differential equations with unbounded memory in mathematical literature. This article aimes to partially fill this gap. We propose a new approach to the study of stability of integro-differential equations with unbounded memory of the following forms x‴(t)+∑i=1m∫t−τi(t)tbi(t)e−αi(t−s)x(s)ds=0,x‴(t)+∑i=1m∫0t−τi(t)bi(t)e−αi(t−s)x(s)ds=0, $$\\begin{array}{} \\begin{split} \\displaystyle x'''(t)+\\sum_{i=1}^{m}\\int\\limits_{t-\\tau_{i}(t)}^{t}b_{i}(t)\\text{e}^{-\\alpha _{i}(t-s) }x(s)\\text{d} s &=0, \\\\ x'''(t)+\\sum_{i=1}^{m}\\int\\limits_{0}^{t-\\tau _{i}(t)}b_{i}(t)\\text{e}^{-\\alpha _{i}(t-s) }x(s)\\text{d} s &= 0, \\end{split} \\end{array}$$ with measurable essentially bounded bi(t) and τi(t), i = 1, …, m. We demonstrate that, under certain conditions on the coefficients, integro-differential equations of these forms are exponentially stable if the delays τi(t), i = 1, …, m, are small enough. This opens new possibilities for stabilization by distributed input control. According to common belief this sort of stabilization requires first and second derivatives of x. Results obtained in this paper prove that this is not the case.

Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays

Abstract: Abstract Caputo fractional delay differential equations with non-instantaneous impulses are studied. Initially a brief overview of the basic two approaches in the interpretation of solutions is given. A generalization of Mittag-Leffler stability with respect to non-instantaneous impulses is given and sufficient conditions are obtained. Lyapunov functions and the Razumikhin technique will be applied and appropriate derivatives among the studied fractional equations is defined and applied. Examples are given to illustrate our results.

Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables

Abstract: Abstract Let Sψ∗ $\\begin{array}{} \\mathcal {S}^*_\\psi \\end{array}$ be a subclass of starlike functions in the unit disk 𝕌, where ψ is a convex function such that ψ(0) = 1, ψ′(0) > 0, ℜ(ψ(ξ)) > 0 and ψ(𝕌) is symmetric with respect to the real axis. We obtain the sharp solution of Fekete-Szegö problem for the family Sψ∗ $\\begin{array}{} \\mathcal {S}^*_\\psi \\end{array}$, and then extend the result to the case of corresponding subclass defined on the bounded starlike circular domain Ω in several complex variables, which give an unified answer of Fekete-Szegö problem for the kinds of subclasses of starlike mappings defined on Ω. At last, we propose two conjectures related the same problems on the unit ball in a complex Banach space and on the unit polydisk in ℂn.

Global behavior of two third order rational difference equations with quadratic terms

Abstract: Abstract In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations xn+1=axnxn−1bxn−1+cxn−2,n=0,1,… $$\\begin{array}{} \\displaystyle x_{n+1}=\\frac{ax_{n}x_{n-1}}{bx_{n-1}+ cx_{n-2}},\\quad n=0,1,\\ldots \\end{array} $$ where a,b,c are positive real numbers and the initial conditions x−2,x−1,x0 are real numbers.

Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms

Abstract: Abstract This article concerns the oscillatory behavior of solutions to second-order half-linear delay differential equations with mixed neutral terms. The authors present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated with examples.

Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets

Abstract: Abstract Taking two and three shared set problems into background, the uniqueness problem of a meromorphic function together with its shift operator have been studied. Our results will improve a number of recent results in the literature. Some examples have been provided in the last section to show that certain conditions used in the paper, is the best possible.

Fixed point results for F-generalized contractive mappings in partial metric spaces

Abstract: Abstract In this paper, we consider F𝓡-generalized contractivity condition and utilized the same to establish some fixed point results for a self-mapping in partial metric spaces endowed with an amorphous binary relation. Our results generalize several core results of the existing literature. We also furnish some examples to exhibit the utility of our results. Finally, we further deduce fixed point result for cyclic contractions in partial metric spaces.

Some refinements of young type inequality for positive linear map

Abstract: Abstract We obtain a refined Young type inequality in this paper. The conclusion is presented as follows: Let A, B ∈ B(𝓗) be two positive operators and p ∈ [0, 1], then A♯pB+G∗(A♯pB)G≤A∇pB−2r(A∇B−A♯B), $$\\begin{array}{} \\displaystyle A\\sharp_p B+G^*(A\\sharp_p B)G\\le A\ abla_p B-2r(A\ abla B-A\\sharp B), \\end{array}$$ where r = min{p, 1 – p}, G = L(2p)2 $\\begin{array}{} \\displaystyle \\frac{\\sqrt{L(2p)}}{2} \\end{array}$ A–1S(A|B), L(t) is periodic with period one and L(t) = t221−tt2t $\\begin{array}{} \\displaystyle \\frac{t^2}{2}\\left( \\frac{1-t}{t} \\right)^{2t} \\end{array}$ for t ∈ [0, 1]. Moreover, we give the s-th powering of two inequalities related to the above one with s > 0 which refines Lin’s work. In the mean time, we present an inequality involving Hilbert-Schmidt norm.

Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2

Abstract: Abstract In this paper we investigate the asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2 and in particular existence and uniqueness results are established. Two examples are given to illustrate our results.

Functions of bounded variation related to domains bounded by conic sections

Abstract: Abstract The aim of this paper is to bring together two areas of studies in the theory of analytic functions: functions of bounded variation and functions related to domains bounded by conic sections. Some relevant properties are indicated.

Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications

Abstract: Abstract In this paper, we study the limit distributions of upper and lower record values of a stationary Gaussian sequence under an equi-correlated set up. Moreover, the class of limit distribution functions (df’s) of the joint upper (and the lower) record values of a stationary Gaussian sequence is fully characterized. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-mid-range, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit df’s of these statistics are derived.

Regular double p-algebras

Abstract: Abstract In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members.

An abelian subextension of the dyadic division field of a hyperelliptic Jacobian

Abstract: Abstract Given a field k of characteristic different from 2 and an integer d ≥ 3, let J be the Jacobian of the “generic” hyperelliptic curve given by y2=∏i=1d(x−αi) $\\begin{array}{} \\displaystyle y^2 = \\prod_{i = 1}^d (x - \\alpha_i) \\end{array}$ , where the αi’s are transcendental and independent over k; it is defined over the transcendental extension K/k generated by the symmetric functions of the αi’s. We investigate certain subfields of the field K∞ obtained by adjoining all points of 2-power order of J(K̄). In particular, we explicitly describe the maximal abelian subextension of K∞ / K(J[2]) and show that it is contained in K(J[8]) (resp. K(J[16])) if g ≥ 2 (resp. if g = 1). On the way we obtain an explicit description of the abelian subextension K(J[4]), and we describe the action of a particular automorphism in Gal(K∞ / K) on these subfields.

Sufficient conditions for Carathéodory functions and applications to univalent functions

Abstract: Abstract In this paper, the authors derive several sufficient conditions for a function to be the Carathéodory function in the unit disk 𝔻: = {z ∈ ℂ: |z| < 1}. More precisely, for given β ∈ (–π/2, π/2), γ ∈ [0, cosβ) and δ ∈ (0, π/2], we find some sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{e−iβ p(z)} > γ or | arg {p(z)–γ} | < δ for all z ∈ 𝔻 by using the first-order differential subordination. We then apply the results obtained here in order to find some conditions for univalent functions with geometric properties such as spirallikeness and strongly starlikeness.

On the equivalence of various definitions of mixed poisson processes

Abstract: Under mild assumptions the equivalence of the mixed Poisson process with mixing parameter a real-valued random variable to the one with mixing distribution as well as to the mixed Poisson process in the sense of Huang is obtained, and a characterization of each one of the above mixed Poisson processes in terms of disintegrations is provided. Moreover, some examples of "canonical" probability spaces admitting counting processes satisfying the equivalence of all above statements are given. Finally, it is shown that our assumptions are essential for the characterization of mixed Poisson processes in terms of disintegrations.

A geometrical version of Hardy-Rellich type inequalities

Abstract: Abstract We obtained a version of Hardy-Rellich type inequality in a domain Ω ∈ ℝn which involves the distance to the boundary, the diameter and the volume of Ω. Weight functions in the inequalities depend on the “mean-distance” function and on the distance function to the boundary of Ω. The proved inequalities connect function to first and second order derivatives.