Authors:Zhenjie Liu Pages: 219 - 232 Abstract: Abstract This paper is devoted to the construction of solutions for one-dimensional wave equations with Dirichlet or Neumann boundary conditions by means of a Nash-Moser iteration scheme, for a large set of frequencies. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0137-5 Issue No:Vol. 7, No. 3 (2017)

Authors:Maxim Limonov; Roman Nedela; Alexander Mednykh Pages: 233 - 243 Abstract: Abstract In this paper we give a few discrete versions of Robert Accola’s results on Riemann surfaces with automorphism groups admitting partitions. As a consequence, we establish a condition for \(\gamma \) -hyperelliptic involution on a graph to be unique. Also we construct an infinite family of graphs with more than one \(\gamma \) -hyperelliptic involution. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0138-4 Issue No:Vol. 7, No. 3 (2017)

Authors:Ying Wang; Yunxi Guo Pages: 245 - 254 Abstract: Abstract In this paper, we developed, for the first time, the exact expressions of several periodic travelling wave solutions and a solitary wave solution for a shallow water wave model of moderate amplitude. Then, we present the existence theorem of the global weak solutions. Finally, we prove the stability of solution in \(L^{1}(R)\) space for the Cauchy problem of the equation. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0139-3 Issue No:Vol. 7, No. 3 (2017)

Authors:S. A. Bishop; E. O. Ayoola; G. J. Oghonyon Pages: 255 - 265 Abstract: Abstract New results on existence and uniqueness of solution of impulsive quantum stochastic differential equation with nonlocal conditions are established. The nonlocal conditions are completely continuous. The methods applied here are simple extension of the methods applied in the classical case to this noncummutative quantum setting. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0140-x Issue No:Vol. 7, No. 3 (2017)

Authors:Perumal Muthukumar; Saminathan Ponnusamy Pages: 267 - 283 Abstract: Abstract In this article, we define discrete analogue of generalized Hardy spaces and its separable subspace on a homogenous rooted tree and study some of its properties such as completeness, inclusion relations with other spaces, separability, growth estimate for functions in these spaces and their consequences. Equivalent conditions for multiplication operators to be bounded and compact are also obtained. Furthermore, we discuss about point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0141-9 Issue No:Vol. 7, No. 3 (2017)

Authors:Vladimir Ryazanov Pages: 285 - 289 Abstract: Abstract It is proved the existence of nonclassical solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for a special case of the Poincare problem on directional derivatives. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0142-8 Issue No:Vol. 7, No. 3 (2017)

Authors:K. Sayevand; K. Pichaghchi Pages: 291 - 318 Abstract: Abstract In this paper, we were concerned with the description of the singularly perturbed differential equations within the scope of fractional calculus. However, we shall note that one of the main methods used to solve these problems is the so-called \(\mathsf {WKB}\) method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the \(\mathsf {WKB}\) to the scope of fractional derivative, we proposed a relatively new derivative called the local fractional derivative. By use of properties of local fractional derivative, we extend the \(\mathsf {WKB}\) method in the scope of the fractional differential equation. By means of this extension, the \(\mathsf {WKB}\) analysis based on the Borel resummation, for fractional differential operators of \(\mathsf {WKB}\) type are investigated. The convergence and the Mittag-Leffler stability of the proposed approach is proven. The obtained results are in excellent agreement with the existing ones in open literature and it is shown that the present approach is very effective and accurate. Furthermore, we are mainly interested to construct the solution of fractional Schrödinger equation in the Mittag-Leffler form and how it leads naturally to this semi-classical approximation namely modified \(\mathsf {WKB}\) . PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0143-7 Issue No:Vol. 7, No. 3 (2017)

Authors:B. C. Chanyal; S. K. Chanyal Pages: 319 - 334 Abstract: Abstract Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac–Maxwell’s equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons. PubDate: 2017-09-01 DOI: 10.1007/s13324-016-0144-6 Issue No:Vol. 7, No. 3 (2017)

Authors:Yue Hu; Yueshan Wang Abstract: Abstract Let \(\mathcal {L}=-\Delta +V\) be a Schrödinger operator on \(\mathbb {R}^n (n\ge 3),\) where the nonnegative potential V belongs to reverse Hölder class \(RH_{q_1}\) for \(q_1>\frac{n}{2}.\) Let \(H^p_\mathcal {L}(\mathbb {R}^n)\) be the Hardy space related to \(\mathcal {L}.\) In this paper, we consider the Hardy type estimates for the Riesz transform \(T_\alpha =V^\alpha (-\Delta +V)^{-\alpha }\) with \(0<\alpha <n/2.\) We show that \(T_\alpha \) is bounded from \(H^p_\mathcal {L}(\mathbb {R}^n)\) into \(L^p(\mathbb {R}^n)\) for \(\frac{n}{n+\delta '}<p\le 1,\) where \(\delta '=\min \{1, 2-n/q_0\},\) and \(q_0\) is the reverse Hölder index of V. Moreover, we prove that the commutator \([b,T_\alpha ],\) which associated with \(T_\alpha \) and a new BMO function b, maps \(H^{1}_\mathcal {L}(\mathbb {R}^n)\) continuously into weak \(L^1(\mathbb {R}^n)\) . PubDate: 2017-09-23 DOI: 10.1007/s13324-017-0196-2

Authors:Gunter Semmler; Elias Wegert Abstract: Abstract The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to several other topics. Though existence and uniqueness of solutions are established for long, we present new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are equivalent: (i) determining a finite Blaschke product from its critical points, (ii) finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges, and (iii) solving a moment problem for the canonical representation of power moments on the real axis. These equivalences are not only of theoretical interest, but also open up new perspectives for the design of algorithms. For instance, the second problem is closely linked to the determination of certain Stieltjes and Van Vleck polynomials for a second order ODE and characterizes solutions as global minimizers of an energy functional. PubDate: 2017-09-20 DOI: 10.1007/s13324-017-0193-5

Authors:Heng Wang; Shuhua Zheng Abstract: Abstract Wang et al. (Appl Math Comput 249:76–80, 2014) studied the bifurcations and travelling wave solutions of a ( \(2+1\) )-dimensional nonlinear Schrödinger equation. By using the dynamical system method, the authors obtained some exact travelling wave solutions. However, we checked the results and found these solutions were not correct. In this note we present the necessary corrections and give two classes of new travelling wave solutions, namely, the blow-up solutions and the breaking wave solutions. PubDate: 2017-09-18 DOI: 10.1007/s13324-017-0194-4

Abstract: Abstract The aim of this paper is to introduce a new inversion procedure for recovering functions, defined on \(\mathbb R^{2}\) , from the spherical mean transform, which integrates functions on a prescribed family \(\Lambda \) of circles, where \(\Lambda \) consists of circles whose centers belong to a given ellipse E on the plane. The method presented here follows the same procedure which was used by Norton (J Acoust Soc Am 67:1266–1273, 1980) for recovering functions in case where \(\Lambda \) consists of circles with centers on a circle. However, at some point we will have to modify the method in [24] by using expansion in elliptical coordinates, rather than spherical coordinates, in order to solve the more generalized elliptical case. We will rely on a recent result obtained by Cohl and Volkmer (J Phys A Math Theor 45:355204, 2012) for the eigenfunction expansion of the Bessel function in elliptical coordinates. PubDate: 2017-09-09 DOI: 10.1007/s13324-017-0192-6

Authors:Sharief Deshmukh; Nasser Bin Turki Abstract: Abstract Taking clue from the analytic vector fields on a complex manifold, \(\varphi \hbox {-analytic}\) conformal vector fields are defined on a Riemannian manifold (Deshmukh and Al-Solamy in Colloq. Math. 112(1):157–161, 2008). In this paper, we use \(\varphi \hbox {-analytic}\) conformal vector fields to find new characterizations of the n-sphere \( S^{n}(c)\) and the Euclidean space \((R^{n},\left\langle ,\right\rangle )\) . PubDate: 2017-09-06 DOI: 10.1007/s13324-017-0190-8

Authors:Caiyun Fang; Yan Xu Abstract: Abstract Let \(A>1\) be a constant and \(\mathcal {F}\) be a family of meromorphic functions defined in a domain D. For each \(f\in \mathcal {F}\) , f has only zeros of multiplicity at least 3 and satisfies the following conditions: (1) \( f^{\prime \prime \prime }(z) \le A z \) when \(f(z)=0\) ; (2) \(f^{\prime \prime \prime }(z)\ne z\) ; (3) all poles of f are multiple. In this paper, we characterize the non-normal sequences of \(\mathcal {F}\) . PubDate: 2017-09-04 DOI: 10.1007/s13324-017-0191-7

Authors:Yongxia Guo; Guangsheng Wei; Ruoxia Yao Abstract: Abstract The inverse spectral problems for Dirac operator with the potential known on an interior subinterval are considered. We prove that the potential on the entire interval and boundary conditions are uniquely determined in terms of the potential on an interior subinterval including midpoint, the known partial eigenvalues and partial interior spectral data. PubDate: 2017-08-29 DOI: 10.1007/s13324-017-0188-2

Authors:Vagif S. Guliyev; Fatih Deringoz; Sabir G. Hasanov Abstract: Abstract In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator \(M_{\alpha }\) on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator \(M_{b,\alpha }\) and nonlinear commutator of fractional maximal operator \([b,M_{\alpha }]\) on Orlicz spaces, when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given. PubDate: 2017-08-23 DOI: 10.1007/s13324-017-0189-1

Authors:Rahim Kargar; Ali Ebadian; Janusz Sokół Abstract: Abstract Let \(\Delta \) be the open unit disk in the complex plane and \(\mathcal {A}\) be the class of normalized analytic functions in \(\Delta \) . In this paper, we introduce and study the class $$\begin{aligned} \mathcal {BS}(\alpha ):=\left\{ f\in \mathcal {A}: \left( \frac{zf'(z)}{f(z)}-1\right) \prec \frac{z}{1-\alpha z^2}, \, z\in \Delta \right\} , \end{aligned}$$ where \(0\le \alpha \le 1\) and \(\prec \) is the subordination relation. Some properties of this class like differential subordination, coefficient estimates and Fekete–Szegö inequality associated with the k-th root transform are considered. PubDate: 2017-08-22 DOI: 10.1007/s13324-017-0187-3

Authors:Octavian Agratini Abstract: Abstract The paper is focused on general sequences of discrete linear operators, say \((L_n)_{n\ge 1}\) . The special case of positive operators is also to our attention. Concerning the quantity \({\Delta } (L_n,f,g):=L_n(fg)-(L_n f)(L_n g), f\) and g belonging to some certain spaces, we propose different estimates. Firstly, we study its asymptotic behavior in Voronovskaja’s sense. Examples are presented. Secondly, we prove an extension of Chebyshev–Grüss type inequality for the above quantity. Special cases are investigated separately. Finally we establish sufficient conditions that ensure statistical convergence of the sequence \({\Delta }(L_n,f,g)\) . PubDate: 2017-08-22 DOI: 10.1007/s13324-017-0186-4

Authors:Zhonglong Zhao; Bo Han Abstract: Abstract On the basis of bilinear equation of a (3+1)-dimensional B-type KP equation, we construct the lump-type solutions by symbolic computation. The (3+1)-dimensional B-type KP equation can be used to describe the propagation of nonlinear waves in fluid dynamics. The lump solutions of three dimensionally reduced (2+1)-dimensional B-type KP equation are derived. The sufficient and necessary conditions to guarantee the analyticity, positiveness and localization of lump solutions are discussed. Figures are presented to illustrate the energy distribution of these lump wave solutions. PubDate: 2017-08-04 DOI: 10.1007/s13324-017-0185-5

Authors: Gegenhasi Abstract: Abstract In this paper, we derive the Grammian determinant solutions to the modified semi-discrete two-dimensional Toda lattice equation, and then construct the semi-discrete two-dimensional Toda lattice equation with self-consistent sources via source generation procedure. The algebraic structure of the resulting coupled modified differential–difference equation is clarified by presenting its Grammian determinant solutions and Casorati determinant solutions. As an application of the Grammian determinant and Casorati determinant solution, the explicit one-soliton and two-soliton solution of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources are given. We also construct another form of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources which is the Bäcklund transformation for the semi-discrete two-dimensional Toda lattice equation with self-consistent sources. PubDate: 2017-07-17 DOI: 10.1007/s13324-017-0184-6