Authors:A. Rahimi; Z. Darvishi; B. Daraby Pages: 335 - 348 Abstract: Abstract Improving and extending the concept of dual for frames, fusion frames and continuous frames, the notion of dual for continuous fusion frames in Hilbert spaces will be studied. It will be shown that generally the dual of c-fusion frames may not be defined. To overcome this problem, the new concept namely Q-dual for c-fusion frames will be defined and some of its properties will be investigated. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0146-4 Issue No:Vol. 7, No. 4 (2017)

Authors:Boris Rubin Pages: 349 - 375 Abstract: Abstract We transfer the results of Part I related to the modified support theorem and the kernel description of the hyperplane Radon transform to totally geodesic transforms on the sphere and the hyperbolic space, the spherical slice transform, and the spherical mean transform for spheres through the origin. The assumptions for functions are formulated in integral terms and close to minimal. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0145-5 Issue No:Vol. 7, No. 4 (2017)

Authors:Alessandro Michelangeli; Alessandro Olgiati Pages: 377 - 416 Abstract: Abstract We study the effective time evolution of a large quantum system consisting of a mixture of different species of identical bosons in interaction. If the system is initially prepared so as to exhibit condensation in each component, we prove that condensation persists at later times and we show quantitatively that the many-body Schrödinger dynamics is effectively described by a system of coupled cubic non-linear Schrödinger equations, one for each component. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0147-3 Issue No:Vol. 7, No. 4 (2017)

Authors:S. Rahman; T. Hayat; B. Ahmad Pages: 417 - 435 Abstract: Abstract The flow of Sisko fluid in an annular pipe is considered. The governing nonlinear equation of an incompressible Sisko fluid is modelled. The purpose of present paper is to obtain the global classical solutions for unsteady flow of magnetohydrodynamic Sisko fluid in terms of the bounded mean oscillations norm. Uniqueness of solution is also verified. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0148-2 Issue No:Vol. 7, No. 4 (2017)

Authors:Farid Messelmi Pages: 437 - 447 Abstract: Abstract We consider in this paper a parabolic partial differential equation involving the infinity Laplace operator and a Leray–Lions operator with no coercitive assumption. We prove the existence and uniqueness of the corresponding approached problem and we show that at the limit the solution solves the parabolic variational inequality arising in the elasto-plastic torsion problem. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0151-7 Issue No:Vol. 7, No. 4 (2017)

Authors:Sergey Tychkov Pages: 449 - 458 Abstract: Abstract A two-dimensional Buckley–Leverett system governing motion of two-phase flow is considered. Travelling-wave solutions for these equations are found. Wavefronts of these solutions may be circles, lines and parabolae. Values of pressure and saturation on the wave fronts are found. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0152-6 Issue No:Vol. 7, No. 4 (2017)

Authors:Der-Chen Chang; Sheng-Ya Feng Pages: 459 - 477 Abstract: Abstract This paper is focused on the approximate procedures for the periodic solutions of the nonlinear Hamilton equation with Gaussian potential. We propose a modified rational harmonic balance method to treat conservative nonlinear equations without the requirements on small perturbation or small parameter. The different approximating orders of this scheme illustrate the excellent agreement of the approximate frequencies with the exact ones. All the numerical results reveal that this effective method can be widely applied to many other truly nonlinear differential equations. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0149-1 Issue No:Vol. 7, No. 4 (2017)

Authors:F. Soleyman; M. Masjed-Jamei; I. Area Pages: 479 - 492 Abstract: Abstract In this paper, a class of finite q-orthogonal polynomials is studied whose weight function corresponds to the inverse gamma distribution as \(q \rightarrow 1\) . Via Sturm–Liouville theory in q-difference spaces, the orthogonality of this class is proved and its norm square value is computed. Also, its general properties such as q-weight function, q-difference equation and the basic hypergeometric representation are recovered in the continuous case. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0150-8 Issue No:Vol. 7, No. 4 (2017)

Authors:Arash Ghaani Farashahi Pages: 493 - 508 Abstract: This paper presents the abstract notion of Poisson summation formulas for homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G, and \(\mu \) be the normalized G-invariant measure over the left coset space G / H associated to the Weil’s formula. We prove that the abstract Fourier transform over G / H satisfies a generalized version of the Poisson summation formula. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0156-2 Issue No:Vol. 7, No. 4 (2017)

Authors:Zaitao Liang; Fanchao Kong Pages: 509 - 524 Abstract: Abstract We study the existence and multiplicity of positive periodic wave solutions for one-dimensional non-Newtonian filtration equations with singular nonlinear sources. We discuss both the attractive singular case and the repulsive singular case. The proof is based on an extension of the continuation theorem of coincidence degree theory. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0153-5 Issue No:Vol. 7, No. 4 (2017)

Authors:Mazhar Hussain Tiwana; Rab Nawaz; Amer Bilal Mann Pages: 525 - 548 Abstract: Abstract This article examines sound radiation from a hard semi-infinite duct placed symmetrically inside an acoustically lined duct. We introduce a wake on right handed region of the duct configuration to analyze sound radiation process for the trailing edge situation. The integral transforms together with Wiener–Hopf techniques render the solution of underlying problem. However expressions for field intensity involve infinite sums/products that enable solution using truncation approach. The sound radiation analysis is then observed graphically while using different choice of some pertinent parameters. It is worth mentioning that results of leading edge situation can be recovered as a limiting case. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0154-4 Issue No:Vol. 7, No. 4 (2017)

Authors:Vitor Balestro; Ákos G. Horváth; Horst Martini Pages: 549 - 575 Abstract: Abstract In this paper a special group of bijective maps of a normed plane (or, more generally, even of a plane with a suitable Jordan curve as unit circle) is introduced which we call the group of general rotations of that plane. It contains the isometry group as a subgroup. The concept of general rotations leads to the notion of flexible motions of the plane, and to the concept of Minkowskian roulettes. As a nice consequence of this new approach to motions the validity of strong analogues to the Euler-Savary equations for Minkowskian roulettes is proved. PubDate: 2017-12-01 DOI: 10.1007/s13324-016-0155-3 Issue No:Vol. 7, No. 4 (2017)

Authors:Nihar Kumar Mahato; Muhammad Aslam Noor; Nabin Kumar Sahu Abstract: Abstract In this paper, we establish the existence and uniqueness solutions of trifunction equilibrium problems using the generalized relaxed \(\alpha \) -monotonicity in Banach spaces. By using the generalized f-projection operator, a hybrid iteration scheme is presented to find a common element of the solutions of a system of trifunction equilibrium problems and the set of fixed points of an infinite family of quasi- \(\phi \) -nonexpansive mappings. Moreover, the strong convergence of our new proposed iterative method under generalized relaxed \(\alpha \) -monotonicity is considered. PubDate: 2017-10-16 DOI: 10.1007/s13324-017-0199-z

Authors:Abderrazek Benhassine Abstract: Abstract This paper concerns the existence of infinitely many solutions for the following fractional Hamiltonian systems: $$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))-L(t).x(t)+\nabla W(t,x(t))=0,\\ x\in H^{\alpha }({\mathbb R}, {\mathbb R}^{N}), \end{array} \right. \end{aligned}$$ where \(\alpha \in \left( {1\over {2}}, 1\right) ,\ t\in {\mathbb R},\ x\in {\mathbb R}^N,\ _{-\infty }D^{\alpha }_{t}\) and \(_{t}D^{\alpha }_{\infty }\) are left and right Liouville–Weyl fractional derivatives of order \(\alpha \) on the whole axis \({\mathbb R}\) respectively, the matrix L(t) is not necessarily positive definite for all \(t\in {\mathbb R}\) nor coercive and the nonlinearity \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) involves a combination of superquadratic and subquadratic terms and is allowed to be sign-changing. Some examples will be given to illustrate our main theoretical results. We also give the proof of new version of Theorem 2.4 in Bartolo et al. (Nonlinear Anal 7(9):981–1012, 1983). PubDate: 2017-10-11 DOI: 10.1007/s13324-017-0197-1

Authors:Sourav Das Abstract: Abstract In this work inequalities for the ratios of q-gamma function are obtained which generalize the results obtained independently by Artin, Wendel, Gautschi and Jameson. Using these inequalities bounds for Gaussian binomial coefficients and q-Wallis ratio are derived and Bohr–Mollerup theorem is also proved as applications. The recent methods developed for gamma functions are used in order to obtain main results. PubDate: 2017-10-09 DOI: 10.1007/s13324-017-0198-0

Authors:Robert Kesler; Darío Mena Arias Abstract: Abstract For each \(\alpha \in \mathbb {T}\) consider the discrete quadratic phase Hilbert transform acting on finitely supported functions \(f : \mathbb {Z} \rightarrow \mathbb {C}\) according to $$\begin{aligned} H^{\alpha }f(n):= \sum _{m \ne 0} \frac{e^{i\alpha m^2} f(n - m)}{m}. \end{aligned}$$ We prove that, uniformly in \(\alpha \in \mathbb {T}\) , there is a sparse bound for the bilinear form \(\left\langle H^{\alpha } f , g \right\rangle \) for every pair of finitely supported functions \(f,g : \mathbb {Z}\rightarrow \mathbb {C}\) . The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes. PubDate: 2017-09-25 DOI: 10.1007/s13324-017-0195-3

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