Authors:Miao Ouyang; Huashui Zhan Pages: 523 - 544 Abstract: The paper studies the posedness of the convection diffusion equation $$\begin{aligned} u_{t}=\text {div}\left( \left \nabla u^{m}\right ^{p-2}\nabla u^{m}\right) +\sum _{i=1}^{N}\frac{\partial b_{i}\left( u^{m}\right) }{\partial x_{i}}-u^{mr}. \end{aligned}$$ with homogeneous boundary condition and with the initial value \(u_0(x)\in L^{q-1+\frac{1}{m}}(\Omega )\) . By considering its regularized problem, using Moser iteration technique, the local bounded properties of the \(L^{\infty }\) -norm of \(u_{k}\) and that of the \(L^{p}\) -norm of the gradient \(\nabla u_{k}\) are got, where \(u_{k}\) is the solution of the regularized problem of the equation. By the compactness theorem, the existence of the solution of the equation itself is obtained. By using some techniques in Zhao and Yuan (Chin Ann Math A 16(2):179–194, 1995), the stability of the solutions is obtained too. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0459-y Issue No:Vol. 40, No. 2 (2017)

Authors:Junfeng Liu; Guangjun Shen; Yang Yang Pages: 565 - 582 Abstract: In this paper, we study a fourth-order stochastic heat equation with homogeneous Neumann boundary conditions and double-parameter fractional noises. We formally replace the random perturbation by a family of noisy inputs depending on a parameter \(n\in {\mathbb {N}}\) which approximates the noises. Then we provided sufficient conditions ensuring that the real-valued mild solution of the fourth-order stochastic heat equation driven by this family of noises converges in law, in the space of \(C([0,T]\times [0,\pi ])\) of continuous functions, to the solution of a class of fourth-order stochastic heat equation driven by fractional noises. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0457-0 Issue No:Vol. 40, No. 2 (2017)

Authors:L. M. Camacho; S. Gómez-Vidal; B. A. Omirov; I. A. Karimjanov Pages: 599 - 615 Abstract: In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras \(sl_2^1\oplus sl_2^2\oplus \dots \oplus sl_2^s\oplus R,\) where R is a solvable radical. The classifications of such Leibniz algebras in the cases \(\mathrm{dim} R=2, 3\) and \(\mathrm{dim} I\ne 3\) have been obtained. Moreover, we classify Leibniz algebras with \(L/I\cong sl_2^1\oplus sl_2^2\) and some conditions on ideal \(I=\mathrm{id}<[x,x] \ \ x\in L>\) . PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0458-z Issue No:Vol. 40, No. 2 (2017)

Authors:Dijana Mosić Pages: 655 - 664 Abstract: We define and characterize reflexive–EP elements in rings, that is elements which commute with their image-kernel (p, q)-inverse. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0445-4 Issue No:Vol. 40, No. 2 (2017)

Authors:Antonio J. Calderón Martín Pages: 679 - 695 Abstract: In the literature, many of the descriptions of different classes of Leibniz algebras \((L, [\cdot , \cdot ])\) have been made by giving the multiplication table on the elements of a basis \({{\mathcal {B}}}=\{v_{k}\}_{k \in K}\) of L, in such a way that for any \(i,j \in K\) we have that \([v_i,v_j] = \lambda _{i,j} [v_j,v_i]\in {{\mathbb {F}}} v_k\) for some \(k \in K\) , where \({{\mathbb {F}}}\) denotes the base field and \(\lambda _{i,j} \in {{\mathbb {F}}}\) . In order to give an unifying viewpoint of all these classes of algebras, we introduce the more general category of Leibniz algebras admitting a multiplicative basis and study its structure. We show that if a Leibniz algebra L admits a multiplicative basis, then it is the direct sum \(L=\bigoplus \nolimits _{\alpha }{{{\mathcal {I}}}}_{\alpha }\) with any \({{\mathcal {I}}}_{\alpha }\) a well-described ideal of L admitting a multiplicative basis inherited from \({\mathcal {B}}\) . Also the \({{\mathcal {B}}}\) -simplicity of L is characterized in terms of the multiplicative basis. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0446-3 Issue No:Vol. 40, No. 2 (2017)

Authors:Richard Foote; Matthew Welz Pages: 697 - 706 Abstract: For G a finite group and p a prime this paper proves two theorems under hypotheses that restrict the index of the subgroup generated by every p-element x in certain subgroups generated by pairs of its conjugates. Under one set of hypotheses G is shown to be supersolvable. Simple groups satisfying a complementary fusion-theoretic hypothesis are classified. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0448-1 Issue No:Vol. 40, No. 2 (2017)

Authors:Wladimir G. Boskoff; Bogdan D. Suceavă Pages: 707 - 733 Abstract: We explore the connection between the geometries generated by logarithmic oscillations and the class of metric spaces satisfying the Gromov hyperbolicity condition. We investigate the most fundamental examples, inspired from classical geometries, e.g. the Euclidean distance on the infinite strip or Hilbert’s distance on the unit disk. We continue our study with the Barbilian’s distance, which historically appeared as a natural extension of a model of hyperbolic geometry. We introduce and investigate a new metric, called the stabilizing metric. In a natural development, we explore a class of extensions of this distance which, under some analytic conditions, produces infinitely many new examples of Gromov hyperbolic metric spaces. Using similar procedures, we construct Vuorinen’s stabilizing metric and its extensions, and we discuss their Gromov hyperbolicity. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0479-7 Issue No:Vol. 40, No. 2 (2017)

Authors:Nadeem Ur Rehman; Mohd Arif Raza Pages: 747 - 764 Abstract: Let R be a prime ring of characteristic different from 2, L be a non-central Lie ideal of R and m, n be the fixed positive integers. If R admits a generalized derivation F associated with a deviation d such that \(F(u^2)^m -d(u)^{2n}\in Z(R)\) for all \(u \in L\) , then R satisfies \(s_4\) , the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0453-4 Issue No:Vol. 40, No. 2 (2017)

Authors:Kah Wai Cheah; Noor Atinah Ahmad Pages: 783 - 794 Abstract: In this paper, Ruspini partitioning is used to represent the reduced form of the fuzzy basis function. Among all the predefined membership functions, Triangular-shaped membership function is used to represent the whole partitioned space of a given fuzzy system. According to the formulation, it shows that the reduced fuzzy basis function using Ruspini partitioning is indeed a strong fuzzy partition and a complete rule base method. By using the Stone–Weierstrass Theorem, for any real continuous function on a compact set, the reduced fuzzy basis function is proven to be a universal approximator at arbitrary accuracy. PubDate: 2017-04-01 DOI: 10.1007/s40840-015-0269-z Issue No:Vol. 40, No. 2 (2017)

Authors:Chaoliang Luo; Shangjiang Guo Pages: 795 - 817 Abstract: In this paper, the stability and bifurcation of a class of two-dimensional stochastic differential equations with multiplicative excitations are investigated. Firstly, we employ Taylor expansions, polar coordinate transformation and stochastic averaging method to transform the original system into an Itô averaging diffusion system. Secondly, we apply the maximum Lyapunov exponent and the singular boundary theory to analyze the local and global stability of the fixed point. Thirdly, we explore the stochastic dynamical bifurcation of the Itô averaging amplitude equation by studying the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker–Planck equation. Finally, an example is given to illustrate the effectiveness of our analyzing procedure. PubDate: 2017-04-01 DOI: 10.1007/s40840-016-0313-7 Issue No:Vol. 40, No. 2 (2017)

Authors:Shelly Verma; V. Ravichandran Pages: 819 - 840 Abstract: Let \(-1 \le B<A \le 1\) and \(F= f/f'\) for normalized locally univalent functions f. For analytic function p on the open unit disc \(\mathbb {D}\) with \(p(0)=1\) and satisfying the subordination \(p(z) \prec (1+A z)/(1+B z)\) , we determine bounds for \( p(z)- z p'(z)-1 \) and \( z p'(z) /{{\mathrm{Re\,}}}p(z)\) . As an application, we investigate the radius problem for F to satisfy \( F'(z)(z/F(z))^2-1 <1\) , where f is a Janowski starlike function and the radius of univalence of F when f is a starlike function of order \(\alpha \) . Also, we have discussed the radius of starlikeness of F when f is a Janowski starlike function with fixed second coefficient. Apart from the radius problems, we give the sharp coefficient bounds of F when f is a Janowski starlike function and a sufficient condition for starlikeness of f when F is a Janowski starlike function. Our results generalize some of the earlier known results. PubDate: 2017-04-01 DOI: 10.1007/s40840-016-0363-x Issue No:Vol. 40, No. 2 (2017)

Authors:Young Jae Sim; Oh Sang Kwon Pages: 841 - 855 Abstract: For real \(\alpha \) and \(\beta \) such that \(0 \le \alpha {<} 1 {<} \beta \) , we denote by \(\varSigma _{s}(\alpha ,\beta )\) and \(\varSigma _{c}^{0}(\alpha ,\beta )\) the class of meromorphic univalent functions f such that \(\alpha < Re \left\{ zf'(z)/f(z) \right\} {<} \beta \) in \(\varDelta \) and \(\alpha {<} \mathfrak {R}\left\{ f'(z) \right\} {<} \beta \) in \(\varDelta \) , where \(\varDelta {=} \left\{ z \in \mathbb {C} : 1< z <\infty \right\} \) , respectively. In the present investigation, estimates on the coefficients of functions of the above classes are investigated. Furthermore, we consider the Fekete-Szegö problem for functions in the above classes. Also, the estimates of the initial coefficients of functions from some subclasses of meromorphic bi-univalent functions are given. PubDate: 2017-04-01 DOI: 10.1007/s40840-016-0335-1 Issue No:Vol. 40, No. 2 (2017)

Authors:Stacey Muir Pages: 857 - 880 Abstract: Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk \({\mathbb {D}}\) have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on \({\mathbb {D}}\) , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, \(\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}\) , \(c>0\) , and consider the convolution \(\Psi _c * f\) for a harmonic mapping \(f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}\) . We prove it is sufficient for \(h \pm g\) to be starlike for \(\Psi _c *f\) to be locally univalent and sense-preserving. Moreover, \(\Psi _c * f\) decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of \(\Psi _c\) , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator. PubDate: 2017-04-01 DOI: 10.1007/s40840-016-0336-0 Issue No:Vol. 40, No. 2 (2017)

Authors:Asif R. Khan; Josip Pečarić; Marjan Praljak Pages: 881 - 889 Abstract: We give an integral version and a refinement of M. Niezgoda’s extension of the variant of Jensen’s inequality given by A. McD. Mercer. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0449-0 Issue No:Vol. 40, No. 2 (2017)

Authors:Ricardo Estrada; Jasson Vindas Pages: 907 - 918 Abstract: It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence \(\left\{ y_{n}\right\} _{n=1}^{\infty }\) of linear continuous functionals in a Fréchet space converges pointwise to a linear functional Y, \(Y\left( x\right) =\lim _{n\rightarrow \infty }\left\langle y_{n} ,x\right\rangle \) for all x, then Y is actually continuous. In this article, we prove that in a Fréchet space the continuity of Y still holds if Y is the finite part of the limit of \(\left\langle y_{n},x\right\rangle \) as \(n\rightarrow \infty .\) We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as LF-spaces, DFS-spaces, and DFS \(^{*}\) -spaces and give examples where it does not hold. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0450-7 Issue No:Vol. 40, No. 2 (2017)

Authors:Xun-Hua Gong; Fang Liu Pages: 919 - 929 Abstract: In this paper, we consider of finding efficient solution and weakly efficient solution for nonconvex vector optimization problems. When X and Y are normed spaces, F is an anti-Lipschitz mapping from X to Y, and the ordering cone is regular, we present an algorithm to guarantee that the generated sequence converges to an efficient solution with respect to normed topology. If the domain of the mapping is compact, we prove that the generated sequence converges to an efficient solution with respect to normed topology without requiring that mapping is anti-Lipschitz. We also give an algorithm to guarantee that the generated sequence converges to a weakly efficient solution with respect to normed topology. PubDate: 2017-04-01 DOI: 10.1007/s40840-017-0455-2 Issue No:Vol. 40, No. 2 (2017)

Authors:K. Teerapabolarn Pages: 931 - 939 Abstract: The Stein–Chen method is used to determine uniform and non-uniform bounds on the ratio between the distribution function of a sum of independent negative binomial random variables and a Poisson distribution function with mean \(\lambda = \sum _{i=1}^nr_iq_i\) , where \(r_i\) and \(p_i=1-q_i\) are parameters of each negative binomial distribution. With these bounds, it indicates that the Poisson distribution function with this mean can be used as an estimate of the independent summands when all \(q_i\) are small or \(\lambda \) is small. Finally, some numerical examples for each result are given. PubDate: 2017-04-01 DOI: 10.1007/s40840-016-0328-0 Issue No:Vol. 40, No. 2 (2017)

Authors:Yaojun Ye Abstract: The initial-boundary value problem for a system of higher-order nonlinear Kirchhoff-type equations with damping and source terms in bounded domain is studied. We prove a global nonexistence result for certain solutions under positive initial energy. PubDate: 2017-02-20 DOI: 10.1007/s40840-017-0452-5

Authors:Tiwei Zhao; Yunge Xu Abstract: In this paper, we investigate the properties of right orthogonal modules of \({{\mathscr {C}}}\) , where \({{\mathscr {C}}}\) is a class of left R-modules. As an application, we investigate the properties of right orthogonal modules of Ding injective left R-modules, and present various characterizations of semisimple and von Neumann regular rings and so on. Moreover, we also consider another cohomology, strong Tate cohomology, which connects the usual cohomology with the Ding cohomology. PubDate: 2017-02-13 DOI: 10.1007/s40840-017-0461-4

Authors:Xiangkui Zhao; Xingjuan Zhang; Weigao Ge Abstract: In this paper, we study the Hyers–Ulam–Rassias stability and Hyers–Ulam stability for a class second differential equation $$\begin{aligned} y''(x)+p(x)y'(x)+q(x)y(x)=F(y(x)) \end{aligned}$$ by a generalized fixed point theorem. Moreover, we show that the differential equation $$\begin{aligned} y'(x)=F(x,y(x)) \end{aligned}$$ has not the Hyers–Ulam stability on an infinite interval. PubDate: 2017-02-10 DOI: 10.1007/s40840-017-0454-3