Authors:Valentin Lychagin; Valeriy Yumaguzhin Pages: 107 - 115 Abstract: In this paper (cf. Lychagin and Yumaguzhin, in Anal Math Phys, 2016) a class of totally geodesics solutions for the vacuum Einstein equations is introduced. It consists of Einstein metrics of signature (1,3) such that 2-dimensional distributions, defined by the Weyl tensor, are completely integrable and totally geodesic. The complete and explicit description of metrics from these class is given. It is shown that these metrics depend on two functions in one variable and one harmonic function. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0130-z Issue No:Vol. 7, No. 2 (2017)

Authors:Boris Rubin Pages: 117 - 150 Abstract: We suggest new modifications of the Helgason’s support theorem and description of the kernel for the hyperplane Radon transform and its dual. The assumptions for functions are formulated in integral terms and close to minimal. The proofs rely on the properties of the Gegenbauer–Chebyshev integrals which generalize Abel type fractional integrals on the positive half-line. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0133-9 Issue No:Vol. 7, No. 2 (2017)

Authors:Min Wang Pages: 151 - 163 Abstract: This paper aims to establish the Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces. For this purpose, we firstly prove a very general existence result for generalized mixed variational inequalities, provided that the mapping involved has the so-called mixed variational inequality property and satisfies a rather weak coercivity condition. Finally, we establish the Tikhonov regularization method for generalized mixed variational inequalities. Our findings extended the results for the generalized variational inequality problem (for short, GVIP(F, K)) in \(R^n\) spaces (He in Abstr Appl Anal, 2012) to the generalized mixed variational inequality problem (for short, GMVIP \((F,\phi , K)\) ) in reflexive Banach spaces. On the other hand, we generalized the corresponding results for the generalized mixed variational inequality problem (for short, GMVIP \((F,\phi ,K)\) ) in \(R^n\) spaces (Fu and He in J Sichuan Norm Univ (Nat Sci) 37:12–17, 2014) to reflexive Banach spaces. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0134-8 Issue No:Vol. 7, No. 2 (2017)

Authors:Yehonatan Salman Pages: 165 - 185 Abstract: The aim of this article is to derive a reconstruction formula for the recovery of \(C^{1}\) functions, defined on the unit sphere \({{\mathbb {S}}}^{n - 1}\) , given their integrals on a special family of \(n - 2\) dimensional sub-spheres. For a fixed point \(\overline{a}\) strictly inside \({{\mathbb {S}}}^{n - 1}\) , each sub-sphere in this special family is obtained by intersection of \({{\mathbb {S}}}^{n - 1}\) with a hyperplane passing through \(\overline{a}\) . The case \(\overline{a} = 0\) results in an inversion formula for the special case of integration on great spheres (i.e., Funk transform). The limiting case where \(p\in {{\mathbb {S}}}^{n - 1}\) and \(\overline{a}\rightarrow p\) results in an inversion formula for the special case of integration on spheres passing through a common point in \({{\mathbb {S}}}^{n - 1}\) . PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0135-7 Issue No:Vol. 7, No. 2 (2017)

Authors:Haibo Lin; Suqing Wu; Dachun Yang Pages: 187 - 218 Abstract: Let \((\mathcal {X},d,\mu )\) be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let T be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and \(b\in \widetilde{\mathrm{RBMO}}(\mu )\) (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator \(T_b:=bT-Tb\) generated by T and b from the atomic Hardy space \(\widetilde{H}^1(\mu )\) with the discrete coefficient into the weak Lebesgue space \(L^{1,\,\infty }(\mu )\) . From this and an interpolation theorem for sublinear operators which is also proved in this paper, the authors further show that the commutator \(T_b\) is bounded on \(L^p(\mu )\) for all \(p\in (1,\infty )\) . Moreover, the boundedness of the commutator generated by the generalized fractional integral \(T_\alpha \,(\alpha \in (0,1))\) and the \(\widetilde{\mathrm{RBMO}}(\mu )\) function from \(\widetilde{H}^1(\mu )\) into \(L^{1/{(1-\alpha )},\,\infty }(\mu )\) is also presented. PubDate: 2017-06-01 DOI: 10.1007/s13324-016-0136-6 Issue No:Vol. 7, No. 2 (2017)

Authors:Ovidiu Calin; Der-Chen Chang; Jishan Hu Pages: 9 - 18 Abstract: This paper deals with integrability conditions for a sub-Riemannian system of equations for a step 2 distribution on the sphere \(\mathbb {S}^3\) . We prove that a certain sub-Riemannian system \(Xf =a\) , \(Yf =b\) on \(\mathbb {S}^3\) has a solution if and only if the following integrability conditions hold: \(X^2 b + 4b = (XY + [X, Y]) a \) , \(Y^2 a + 4a = (YX-[X, Y]) b\) . We also provide an explicit construction of the solution f in terms of the vector fields X, Y and functions a and b. PubDate: 2017-03-01 DOI: 10.1007/s13324-016-0126-8 Issue No:Vol. 7, No. 1 (2017)

Authors:Hui Nie; Junyi Zhu; Xianguo Geng Abstract: The Gerdjikov–Ivanov equation is investigated by the Riemann–Hilbert approach and the technique of regularization. The trace formula and new form of N-soliton solution are given. The dynamics of the stationary solitons and non-stationary solitons are discussed. PubDate: 2017-06-19 DOI: 10.1007/s13324-017-0179-3

Authors:Jin-Yun Yang; Wen-Xiu Ma; Zhenyun Qin Abstract: Based on the Hirota bilinear form of the \((2+1)\) -dimensional Ito equation, one class of lump solutions and two classes of interaction solutions between lumps and line solitons are generated through analysis and symbolic computations with Maple. Analyticity is naturally guaranteed for the presented lump and interaction solutions, and the interaction solutions reduce to lumps (or line solitons) while the hyperbolic-cosine (or the quadratic function) disappears. Three-dimensional plots and contour plots are made for two specific examples of the resulting interaction solutions. PubDate: 2017-06-17 DOI: 10.1007/s13324-017-0181-9

Authors:Jun-ichi Inoguchi; Marian Ioan Munteanu; Ana Irina Nistor Abstract: We study magnetic trajectories corresponding to contact magnetic fields in 3-dimensional quasi-Sasakian manifolds. We show that they are slant curves, that is their contact angles are constant. We prove that such magnetic curves are geodesics for a certain linear connection for which all four structure tensor fields are parallel. PubDate: 2017-06-13 DOI: 10.1007/s13324-017-0180-x

Authors:Heng Wang; Shuhua Zheng Abstract: By using the dynamical system approach, the exact travelling wave solutions for a system of coupled nonlinear electrical transmission lines are studied. Based on this method, the bifurcations of phase portraits of a dynamical system are given. The two-dimensional solitary wave solutions and periodic wave solutions on coupled nonlinear transmission lines are obtained. With the aid of Maple, the numerical simulations are conducted for solitary wave solutions and periodic wave solutions to the model equation. The results presented in this paper improve upon previous studies. PubDate: 2017-06-01 DOI: 10.1007/s13324-017-0178-4

Authors:Aymen Ammar; Aref Jeribi; Bilel Saadaoui Abstract: The main goal of this paper is to give a characterization of the essential pseudospectra of \(2\times 2\) matrix of linear relations on a Banach space. We start by giving the definition and we investigate the characterization and some properties of the essential pseudospectra. Furthermore, we apply the obtained result to determine the essential pseudospectra of two-group transport equation with general boundary conditions in the Banach space. PubDate: 2017-06-01 DOI: 10.1007/s13324-017-0170-z

Authors:Diganta Borah; Pranav Haridas; Kaushal Verma Abstract: We study several quantities associated to the Green’s function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and \(L^2\) -cohomology of the capacity metric and critical points of the Green’s function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green’s function and related objects. PubDate: 2017-05-25 DOI: 10.1007/s13324-017-0177-5

Authors:S. A. Buterin; V. A. Yurko Abstract: We consider the Sturm–Liouville differential equation with a constant delay, which is not less than the half length of the interval. An inverse spectral problem is studied of recovering the potential from subspectra of two boundary value problems with one common boundary condition. The conditions on arbitrary subspectra are obtained that are necessary and sufficient for the unique determination of the potential by specifying these subspectra, and a constructive procedure for solving the inverse problem is provided along with necessary and sufficient conditions of its solvability. PubDate: 2017-05-23 DOI: 10.1007/s13324-017-0176-6

Authors:Nguyen Xuan Hong; Tran Van Thuy Abstract: In this paper, we prove the Hölder continuity for solutions to the complex Monge–Ampère equations on non-smooth pseudoconvex domains of plurisubharmonic type m. PubDate: 2017-05-17 DOI: 10.1007/s13324-017-0175-7

Authors:Palle Jorgensen; Erin Pearse; Feng Tian Abstract: Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces whose intersection contains a fixed vector space \(\mathscr {D}\) . In the case when \(\mathscr {D}\) is dense in one of the Hilbert spaces (but not necessarily in the other), we make precise an operator-theoretic linking between the two Hilbert spaces. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and the operator theory of reflection positivity. PubDate: 2017-05-17 DOI: 10.1007/s13324-017-0173-9

Authors:Fuliang Wang; Mingqi Xiang Abstract: The aim of this paper is to investigate the multiplicity of solutions to the following nonlocal fractional Choquard–Kirchhoff type equation involving critical exponent, $$\begin{aligned}&\left( a+b[u]_{s,p}^p\right) (-\Delta )_p^su=\int _{\mathbb {R}^N}\frac{ u(y) ^{p_{\mu ,s}^*}}{ x-y ^{\mu }}dy u ^{p_{\mu ,s}^*-2}u +\lambda h(x) u ^{q-2}u\quad&\text{ in } \,\,\mathbb {R}^N,\\&[u]_{s,p}=\left( \int _{\mathbb {R}^{N}}\int _{\mathbb {R}^N}\frac{ u(x)- u(y) ^p}{ x-y ^{N+sp}}dxdy\right) ^{1/p} \end{aligned}$$ where \(a\ge 0, b>0\) , \(0<s<\min \{1,N/2p\}\) , \(2sp\le \mu <N\) , \((-\Delta )_p^s\) is the fractional p-Laplace operator, \(\lambda >0\) is a parameter, \(p_{\mu ,s}^*=\frac{(N-\frac{\mu }{2})p}{N-sp}\) is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, \(1<q<p_s^*=\frac{Np}{N-sp}\) and \(h\in L^{\frac{p_s^*}{p_s^*-q}}(\mathbb {R}^N)\) . Under some suitable assumptions, we obtain the multiplicity of nontrivial solutions by using variational methods. In particular, we get the existence of infinitely many nontrivial solutions for the degenerate Kirchhoff case by using Krasnoselskii’s genus theory. PubDate: 2017-05-06 DOI: 10.1007/s13324-017-0174-8

Authors:Natalia P. Bondarenko Abstract: The Sturm–Liouville operator on a star-shaped graph is considered. We assume that the potential is known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential on the remaining edge from the part of the spectrum. A constructive method is developed for the solution of this problem, based on the Riesz-basicity of some sequence of vector functions. The local solvability of the inverse problem and the stability of its solution are proved. PubDate: 2017-04-24 DOI: 10.1007/s13324-017-0172-x

Authors:Yehonatan Salman Abstract: The aim of the article is to recover a certain type of finite parametric distributions and functions using their spherical mean transform which is given on a certain family of spheres whose centers belong to a finite set \(\Gamma \) . For this, we show how the problem of reconstruction can be converted to a Prony’s type system of equations whose regularity is guaranteed by the assumption that the points in the set \(\Gamma \) are in general position. By solving the corresponding Prony’s system we can extract the set of parameters which define the corresponding function or distribution. PubDate: 2017-04-18 DOI: 10.1007/s13324-017-0171-y

Authors:Anna Duyunova; Valentin Lychagin; Sergey Tychkov Abstract: Algebras of symmetries and the corresponding algebras of differential invariants for plane flows of inviscid fluids are given. Their dependence on thermodynamical states of media are studied and a classification of thermodynamical states is given. PubDate: 2017-03-04 DOI: 10.1007/s13324-017-0169-5

Authors:Guangyue Huang; Zhi Li Abstract: In this paper, we consider Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\begin{aligned} \Delta _V u+au\log u=0, \end{aligned}$$ where a is a nonzero real constant. By using gradient estimates, we obtain upper bounds of \( \nabla u \) with respect to \(\sup u\) and the lower bound of Bakry-Emery Ricci curvature. In particular, for complete noncompact manifolds with \(a<0\) , we prove that any positive solution must be \(u\equiv 1\) under a suitable condition for a with respect to the lower bound of Bakry-Emery Ricci curvature. It generalizes a classical result of Yau. PubDate: 2017-03-01 DOI: 10.1007/s13324-017-0168-6