Authors:Ion Chiţescu; Răzvan-Cornel Sfetcu Pages: 1 - 8 Abstract: Abstract We give sufficient conditions for the best approximation of convex, bounded, closed and solid sets in Köthe–Bochner spaces and apply this result to sequence spaces. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0141-2 Issue No:Vol. 155, No. 1 (2018)

Authors:Gongwei Liu; Lin Diao Pages: 9 - 19 Abstract: Abstract In this paper, we consider a weak viscoelastic equation with internal time-varying delay $$\begin{aligned} u_{tt}(x, t)-\triangle u(x, t)+\alpha(t) \int_{0}^{t}g(t-s)\triangle u(x, s)ds+\mu u_{t}\bigl(x, t-\tau(t)\bigr)=0 \end{aligned}$$ in a bounded domain. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general decay result for the energy. This work generalizes and improves earlier results in the literature. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0142-1 Issue No:Vol. 155, No. 1 (2018)

Authors:Yanlin Liu Pages: 21 - 39 Abstract: Abstract In this paper, we consider the axisymmetric MHD system with nearly critical initial data having the special structure: \(u_{0}=u_{0}^{r} e_{r}+u^{\theta}_{0} e_{\theta}+u_{0}^{z} e_{z}\) , \(b_{0}=b_{0}^{\theta}e_{\theta}\) . We prove that, this system is globally well-posed provided the scaling-invariant norms \(\ ru^{\theta}_{0}\ _{L^{\infty}}\) , \(\ r^{-1}b^{\theta}_{0}\ _{L^{\frac{3}{2}}}\) are sufficiently small. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0143-0 Issue No:Vol. 155, No. 1 (2018)

Authors:Palle E. T. Jorgensen; Myung-Sin Song Pages: 41 - 56 Abstract: Abstract We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp distinction between stochastic analysis involving frames in finite vs. infinite dimensions. For the case of infinite-dimensional Hilbert space ℋ, we study three cases of measures. We first show that, for ℋ infinite dimensional, one must resort to infinite dimensional measure spaces which properly contain ℋ. The three cases we consider are: (i) Gaussian frame measures, (ii) Markov path-space measures, and (iii) determinantal measures. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0144-z Issue No:Vol. 155, No. 1 (2018)

Authors:Zuomao Yan; Fangxia Lu Pages: 57 - 84 Abstract: Abstract In this paper, a new class of fractional impulsive stochastic partial integro-differential control systems with state-dependent delay and their optimal controls in a Hilbert space is studied. We firstly prove an existence result of mild solutions for the control systems by using stochastic analysis, analytic \(\alpha \) -resolvent operator, fractional powers of closed operators and suitable fixed point theorems. Then we derive the existence conditions of optimal pairs to the fractional impulsive stochastic control systems. Finally, an example is given to illustrate the effectiveness of our main results. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0145-y Issue No:Vol. 155, No. 1 (2018)

Authors:Séverine Bernard; Catherine Cabuzel; Silvère Paul Nuiro; Alain Pietrus Pages: 85 - 98 Abstract: Abstract This paper deals with variational inclusions of the form \(0 \in K-f(x)\) where \(f : \mathbb{R}^{n} \rightarrow \mathbb{R} ^{m}\) is a semismooth function and \(K\) is a nonempty closed convex cone in \(\mathbb{R}^{m}\) . We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of \(f\) . The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341–347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0146-x Issue No:Vol. 155, No. 1 (2018)

Authors:Ya-Lun Tsai Pages: 99 - 112 Abstract: Abstract In “Counting central configurations at the bifurcation points,” we proposed an algorithm to rigorously count central configurations in some cases that involve one parameter. Here, we improve our algorithm to consider three harder cases: the planar \((3+1)\) -body problem with two equal masses; the planar 4-body problem with two pairs of equal masses which have an axis of symmetry containing one pair of them; the spatial 5-body problem with three equal masses at the vertices of an equilateral triangle and two equal masses on the line passing through the center of the triangle and being perpendicular to the plane containing it. While all three problems have been studied in two parameter cases, numerical observations suggest new results at some points on the bifurcation curves. Applying the improved version of our algorithm, we count at those bifurcation points. As a result, for the \((3+1)\) -body problem, we identify three points on the bifurcation curve where there are 8 central configurations, which adds to the known results of \(8,9,10\) ones. For our 4-body case, at the bifurcation points, there are 3 concave central configurations, which adds to the known results of \(2,4\) ones. For our 5-body case, at the bifurcation point, there is 1 concave central configuration, which adds to the known results of \(0,2\) ones. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0147-9 Issue No:Vol. 155, No. 1 (2018)

Authors:Yuichi Shiozawa Pages: 113 - 150 Abstract: Abstract We find the exponential growth rate of the population outside a ball with time dependent radius for a branching Brownian motion in Euclidean space. We then see that the upper bound of the particle range is determined by the principal eigenvalue of the Schrödinger type operator associated with the branching rate measure and branching mechanism. We assume that the branching rate measure is small enough at infinity, and can be singular with respect to the Lebesgue measure. We finally apply our results to several concrete models. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0148-8 Issue No:Vol. 155, No. 1 (2018)

Authors:Yanyan Zhang; Yu Zhang Pages: 151 - 175 Abstract: Abstract By the vanishing viscosity approach, a class of non-strictly hyperbolic systems of conservation laws that contain the equations of geometrical optics as a prototype are studied. The existence, uniqueness and stability of solutions involving delta shock waves and generalized vacuum states are discussed completely. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0149-7 Issue No:Vol. 155, No. 1 (2018)

Authors:Sabrine Arfaoui; Anouar Ben Mabrouk Pages: 177 - 195 Abstract: Abstract In the present paper, new classes of wavelet functions are developed in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on two-parameters weight functions generalizing the well known Jacobi and Gegenbauer classes when relaxing the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rule have been proved. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0150-1 Issue No:Vol. 155, No. 1 (2018)

Authors:Samuel Nordmann; Benoît Perthame; Cécile Taing Pages: 197 - 225 Abstract: Abstract We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behavior of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamics of such a population. We begin with a simplified model by discarding the effect of mutations, which allows us to introduce the main ideas and state the full result. Then we discuss the general model and its limitations. Our approach uses the eigenelements of a formal limiting operator, that depend on the structuring variables of the model and define an effective fitness. Then we introduce a new method which reduces the convergence proof to entropy estimates rather than estimates on the constrained Hamilton-Jacobi equation. Numerical tests illustrate the theory and show the selection of a fittest trait according to the effective fitness. For the problem with mutations, an unusual Hamiltonian arises with an exponential growth, for which we prove existence of a global viscosity solution, using an uncommon a priori estimate and a new uniqueness result. PubDate: 2018-06-01 DOI: 10.1007/s10440-017-0151-0 Issue No:Vol. 155, No. 1 (2018)

Authors:Sergey Bezuglyi; Palle E. T. Jorgensen Abstract: Abstract In our study of electrical networks we develop two themes: finding explicit formulas for special classes of functions defined on the vertices of a transient network, namely monopoles, dipoles, and harmonic functions. Secondly, our interest is focused on the properties of electrical networks supported on Bratteli diagrams. We show that the structure of Bratteli diagrams allows one to describe algorithmically harmonic functions as well as monopoles and dipoles. We also discuss some special classes of Bratteli diagrams (stationary, Pascal, trees), and we give conditions under which the harmonic functions defined on these diagrams have finite energy. PubDate: 2018-05-31 DOI: 10.1007/s10440-018-0189-7

Authors:Lisette Jager; Jules Maes; Alain Ninet Abstract: Abstract As a first step towards modelling real time-series, we study a class of real-variable, bounded processes \(\{X_{n}, n\in \mathbb{N}\}\) defined by a deterministic \(k\) -term recurrence relation \(X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})\) . These processes are noise-free. We immerse such a dynamical system into \(\mathbb{R}^{k}\) in a slightly distorted way, which allows us to apply the multidimensional techniques introduced by Saussol (Isr. J. Math. 116:223–248, 2000) for deterministic transformations. The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function \(\varphi \) and by products of its first-order partial derivatives. They ensure that the induced transformation \(T\) is dilating. Under these conditions, \(T\) admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for \(X_{n}\) , satisfying integral compatibility conditions. Moreover, if \(T\) is mixing, one obtains the exponential decay of correlations. PubDate: 2018-05-29 DOI: 10.1007/s10440-018-0192-z

Authors:Zhen-Qing Chen; Yan-Xia Ren; Ting Yang Abstract: Abstract The goal of this paper is twofold. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space \(E\) and the branching mechanism takes the form $$ \psi _{\beta }(x,\lambda )=-\beta (x)\lambda +\alpha (x)\lambda ^{2}+ \int _{(0,{\infty })}\bigl(e^{-\lambda y}-1+\lambda y\bigr)\pi (x,dy) $$ with \(\beta \in \mathcal{B}_{b}(E)\) , \(\alpha \in \mathcal{B}^{+}_{b}(E)\) and \(\pi \) being a kernel from \(E\) to \((0,{\infty })\) satisfying \(\sup_{x\in E}\int _{(0,{\infty })} (y\wedge y^{2}) \pi (x,dy)<{\infty }\) . The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap. Our results cover many interesting examples of superprocesses, including super Ornstein-Uhlenbeck process and super stable-like process. The strong law of large numbers for supercritical superprocesses are obtained under the assumption that the strong law of large numbers for an associated supercritical branching Markov process holds along a discrete sequence of times, extending an earlier result of Eckhoff et al. (Ann. Probab. 43(5):2594–2659, 2015) for superdiffusions to a large class of superprocesses. The key for such a result is due to the skeleton decomposition of superprocess, which represents a superprocess as an immigration process along a supercritical branching Markov process. PubDate: 2018-05-29 DOI: 10.1007/s10440-018-0190-1

Authors:Libin Wang; Youren Wang Abstract: Abstract In this paper, we consider an inverse piston problem with small BV initial data for the system of one-dimensional adiabatic flow. Suppose that the original state of the gas on the right side of the piston and the position of the forward shock are known, then we can globally solve the inverse piston problem and estimate the speed of the piston in a unique manner. PubDate: 2018-05-28 DOI: 10.1007/s10440-018-0193-y

Authors:Yaling Zhao; Zuhan Liu; Ling Zhou Abstract: Abstract In this paper we focus on a nonlocal reaction-diffusion population model. Such a model can be used to describe a single species which is diffusing, aggregating, reproducing and competing for space and resources, with the free boundary representing the expanding front. The main objective is to understand the influence of the nonlocal term in the form of an integral convolution on the dynamics of the species. Precisely, when the species successfully spreads into infinity as \(t\rightarrow \infty \) , it is proved that the species stabilizes at a positive equilibrium state under rather mild conditions. Furthermore, we obtain a upper bound for the spreading of the expanding front. PubDate: 2018-05-22 DOI: 10.1007/s10440-018-0188-8

Authors:Feliz Minhós; Robert de Sousa Abstract: Abstract This work considers two types of a second order impulsive coupled system of differential equations with generalized jump conditions in half-line. For both problems it will be presented two localization results with different monotonicity assumptions on the nonlinearities and on the impulsive conditions. The arguments apply lower and upper solutions method combined with Nagumo type condition and truncature techniques. Last section contains an applications of one of this impulsive problems to logging timber by helicopter. PubDate: 2018-05-08 DOI: 10.1007/s10440-018-0187-9

Authors:Kwok-Pun Ho Abstract: Abstract We show that when the infimum of the exponent function equals to 1, the fractional integral operator is a bounded operator from the Morrey space with variable exponent to the weak Morrey space with variable exponent. PubDate: 2018-04-26 DOI: 10.1007/s10440-018-0181-2

Authors:X. Blasco; G. Reynoso-Meza; E. A. Sánchez-Pérez; J. V. Sánchez-Pérez Abstract: Given a finite dimensional asymmetric normed lattice, we provide explicit formulae for the optimization of the associated (non-Hausdorff) asymmetric “distance” among a subset and a point. Our analysis has its roots and finds its applications in the current development of effective algorithms for multi-objective optimization programs. We are interested in providing the fundamental theoretical results for the associated convex analysis, fixing in this way the framework for this new optimization tool. The fact that the associated topology is not Hausdorff forces us to define a new setting and to use a new point of view for this analysis. Existence and uniqueness theorems for this optimization are shown. Our main result is the translation of the original abstract optimal distance problem to a clear optimization scheme. Actually, this justifies the algorithms and shows new aspects of the numerical and computational methods that have been already used in visualization of multi-objective optimization problems. PubDate: 2018-04-26 DOI: 10.1007/s10440-018-0184-z

Authors:Rasul Ganikhodzhaev; Farrukh Mukhamedov; Mansoor Saburov Abstract: Abstract The paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method for stable solutions is also presented. PubDate: 2018-04-26 DOI: 10.1007/s10440-018-0183-0