In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An isogeometric discontinuous Galerkin method for Euler equations is proposed. It integrates the idea of isogeometric analysis with the discontinuous Galerkin framework by constructing each element through the knots insertion and degree elevation techniques in non-uniform rational B-splines. This leads to the solution inherently shares the same function space as the non-uniform rational B-splines representation, and results in that the curved boundaries as well as the interfaces between neighboring elements are naturally and exactly resolved. Additionally, the computational cost is reduced in contrast to that of structured grid generation. Numerical tests demonstrate that the presented method can be high order of accuracy and flexible in handling curved geometry. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study existence of invariant curves of an iterative equation

which is from dissipative standard map. By constructing an invertible analytic solution *g*(*x*) of an auxiliary equation of the form

invertible analytic solutions of the form *g*(*λ**g*^{ − 1}(*x*)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |*λ*| < 1, we focus on those *λ* on the unit circle *S*^{1}, that is, |*λ*| = 1. We discuss not only those *λ* at resonance, that is, at a root of the unity, but also those *λ* near resonance under the Brjuno condition. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a generalized Kirchhoff equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem, we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with the following chemotaxis system:

under homogeneous Neumann boundary conditions in a bounded domain
with smooth boundary. Here, *δ* and *χ* are some positive constants and *f* is a smooth function that satisfies

with some constants *a*⩾0,*b* > 0, and *γ* > 1. We prove that the classical solutions to the preceding system are global and bounded provided that

Copyright © 2016 John Wiley & Sons, Ltd.

We consider a fourth-order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low-dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high-order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ-convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed-finite elements with well-established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the initial boundary value problem for generalized logarithmic improved Boussinesq equation. By using the Galerkin method, logarithmic Sobolev inequality, logarithmic Gronwall inequality, and compactness theorem, we show the existence of global weak solution to the problem. By potential well theory, we show the norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions. Furthermore, for the generalized logarithmic improved Boussinesq equation with damped term, we obtain the decay estimate of the energy. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of – except for arbitrary one eigenvalue and the Borg type theorem of – except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time-harmonic current source. We perform the two-scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one-dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross-validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Ductal carcinoma *in situ* – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma *in situ* model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytönen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the *T*1 type theorem for the boundedness of Calderón–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff-type wave equations with a memory condition on one part of the boundary. Without the condition *u*_{0} = 0 on Γ_{0}, we establish a general decay of energy depending on the behavior of relaxation function by introducing suitable energy and Lyapunov functionals. This result allows a wider class of relaxation functions. Copyright © 2016 John Wiley & Sons, Ltd.

We analyze a highly nonlinear system of partial differential equations related to a model solidification and/or melting of thermoviscoelastic isochoric materials with the possibility of motion of the material during the process. This system consists of an internal energy balance equation governing the evolution of temperature, coupled with an evolution equation for a phase field whose values describe the state of material and a balance equation for the linear moments governing the material displacements. For this system, under suitable dissipation conditions, we prove global existence and uniqueness of weak solutions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations

where *w* is a constant,
and
. By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that *a*(*x*) and *f*(*x*,*u*) are nonperiodic in *x*,*a* does not fulfil the coercive condition, and *f* does not satisfy the well-known (*A**R*)-condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth-order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.

We consider the Lamé system for an elastic medium consisting of an inclusion embedded in a homogeneous background medium. Based on the field expansion method and layer potential techniques, we rigorously derived the asymptotic expansion of the perturbed displacement field because of small perturbations in the interface of the inclusion. We extend these techniques to determine a relationship between traction-displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moment tensors because of the presence of small changes in the interface of the inclusion. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a class of fourth-order Sturm-Liouville problems with transmission conditions is considered. The eigenvalues depend not only continuously but also smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a transmission condition, a coefficient, or the weight function, is found. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The process of transporting nanoparticles at the blood vessels level stumbles upon various physical and physiological obstacles; therefore, a Mathematical modeling will provide a valuable means through which to understand better this complexity. In this paper, we consider the motion of nanoparticles in capillaries having cylindrical shapes (i.e., tubes of finite size). Under the assumption that these particles have spherical shapes, the motion of these particles reduces to the motion of their centers. Under these conditions, we derive the mathematical model, to describe the motion of these centers, from the equilibrium of the gravitational force, the hemodynamic force and the van der Waals interaction forces. We distinguish between the interaction between the particles and the interaction between each particle and the walls of the tube. Assuming that the minimum distance between the particles is large compared with the maximum radius *R* of the particles and hence neglecting the interactions between the particles, we derive simpler models for each particle taking into account the particles-to-wall interactions. At an error of order *O*(*R*) or *O*(*R*^{3})(depending if the particles are 'near' or 'very near' to the walls), we show that the horizontal component of each particle's displacement is solution of a nonlinear integral equation that we can solve via the fixed point theory. The vertical components of the displacement are computable in a straightforward manner as soon as the horizontal components are estimated. Finally, we support this theory with several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term. Based on a low-frequency and high-frequency decomposition, Green's function method and the classical energy method, we not only obtain *L*^{2} time-decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data *u*_{0}(*x*) satisfies the smallness condition on
, but not on
. Furthermore, by taking a time-frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm integral equations of the second kind. Using the properties of three-dimensional modification of hat functions, these are types of equations to a nonlinear system of algebraic equations. Also, convergence results and error analysis are discussed. The efficiency and accuracy of the proposed method is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, new Lyapunov-type inequalities are obtained for the case when one is dealing with a class of fractional two-point boundary value problems. As an application of this result, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the long-time dynamics of solutions to a nonlinear nonautonomous extensible plate equation with a strong damping. Under some suitable assumptions on the initial data, the nonlinear term and external force, we establish the existence of global solutions that generate a family of processes for the problem and obtain uniform attractors corresponding to strong and weak symbol spaces in a bounded domain . Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the one-dimensional compressible Navier–Stokes equations with periodic boundary conditions, with initial conditions in a small neighborhood of a state of uniform density and uniform nonzero velocity. We prove that, with a control given only by a body force localized in a subinterval, we can steer the system to uniform density and velocity. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Recently, numerous inventory models were developed for ameliorating items (say, fish, ducklings, chicken, etc.) considering the constant demand rate. However, such types of problems are not useful in the real market. The demand rate of ameliorating items is fluctuates in their life-period. The consumption and demand of ameliorating items are not generally steady. In a few seasons, the demand rate increases; ordinarily, it is static, and sometimes, it declines. With the outcome that their demand rate can be properly portrayed by a trapezoidal-type. In the proposed model, an inventory model for ameliorating/deteriorating items are considered with inflationary condition and time discounting rate. Additionally, having shortages that is completely backlogged. The demand rate is taken as the continuous trapezoidal-type function of time. The amelioration and deterioration rate are considered as Weibull distribution. To obtain the minimum cost, mathematical formulation of the proposed model with solution procedure is talked about. Numerical cases are given to be checked the optimal solution. Additionally, we have talked about the convexity of the proposed model through graphically. Conclusion with future worked are clarified appropriately. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The Dirac-type time-frequency distribution (TFD), regarded as *ideal* TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac-type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac-type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd.

We present a new Lyapunov function for laminar flow, in the *x*-direction, between two parallel planes in the presence of a coplanar magnetic field for three-dimensional perturbations with stress-free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(*R*_{e}) and magnetic Reynolds numbers(*R*_{m}) below *π*^{2}/2*M*. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero-zero-Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper describes the procedure of extracting information about the dynamics of highway traffic speed. The wavelet shrinkage is used to diminish the effect of the noise. Afterwards, the dynamical properties of the system are estimated through the 0–1 test for chaos, Lyapunov exponents and the notion of Shannon entropy. The results indicate the strong chaotic dynamics in the traffic speed data. In addition to that, the predictability of the system is related to the values of the maximal Lyapunov exponent and Shannon entropy. The higher those values are, the worse the predictability of the system is. Furthermore, it is shown that Shannon entropy can be used to detect changes in dynamics on different time scales. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the following second-order dynamical system:

where *c*⩾0 is a constant,
and
. When *g* admits a singularity at zero of repulsive type without the restriction of strong force condition, we apply the coincidence degree theory to prove that the system admits nonplanar collisionless rotating periodic solutions taking the form *u*(*t* + *T*) = *Q**u*(*t*),
with *T* > 0 and *Q* an orthogonal matrix under the assumption of Landesman–Lazer type. Copyright © 2016 John Wiley & Sons, Ltd.

The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore-scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore-scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm-liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE–ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are introducing pertinent Euler–Lagrange–Jensen type *k*-quintic functional equations and investigate the ‘Ulam stability’ of these new *k*-quintic functional mappings *f*:*X**Y*, where *X* is a real normed linear space and *Y* a real complete normed linear space. We also solve the Ulam stability problem for Euler–Lagrange–Jensen alternative *k*-quintic mappings. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, Sturmian comparison theory is developed for the pair of second-order differential equations; first of which is the nonlinear differential equations of the form

- (1)

and the second is the half-linear differential equations

- (2)

where Φ_{α}(*s*) = |*s*|^{α − 1}*s* and *α*_{1} > ⋯ > *α*_{m} > *β* > *α*_{m + 1} > ⋯ > *α*_{n} > 0. Under the assumption that the solution of (2) has two consecutive zeros, we obtain Sturm–Picone type and Leighton type comparison theorems for (1) by employing the new nonlinear version of Picone formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for (1). Examples are given to illustrate the relevance of the results. Copyright © 2016 John Wiley & Sons, Ltd.

Let
be a metric measure space of homogeneous type and *L* be a one-to-one operator of type *ω* on
for *ω*∈[0, *π*/2). In this article, under the assumptions that *L* has a bounded *H*_{∞}-functional calculus on
and satisfies (*p*_{L}, *q*_{L}) off-diagonal estimates on balls, where *p*_{L}∈[1, 2) and *q*_{L}∈(2, *∞*], the authors establish a characterization of the Sobolev space
, defined via *L*^{α/2}, of order *α*∈(0, 2] for *p*∈(*p*_{L}, *q*_{L}) by means of a quadratic function *S*_{α, L}. As an application, the authors show that for the degenerate elliptic operator *L*_{w}: =− *w*^{ − 1}div(*A*∇) and the Schrödinger type operator
with *a*∈(0, *∞*) on the weighted Euclidean space
with *A* being real symmetric, if *n*⩾3,
with *q*∈[1, 2],
, *p*∈(1, *∞*) and
with
, then, for all
,
, where the implicit equivalent positive constants are independent of *f*,
denotes the class of Muckenhoupt weights,
the reverse Hölder class, and *D*(*L*_{w}) and
the domains of *L*_{w} and
, respectively. Copyright © 2016 John Wiley & Sons, Ltd.

This paper studies the chemotaxis-haptotaxis system with nonlinear diffusion

subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where *χ*, *ξ* and *μ* are positive constants, and
(*n*⩾2) is a bounded and smooth domain. Here, we assume that *D*(*u*)⩾*c*_{D}*u*^{m − 1} for all *u* > 0 with some *c*_{D} > 0 and *m*⩾1. For the case of non-degenerate diffusion, if *μ* > *μ*^{∗}, where

it is proved that the system possesses global classical solutions which are uniformly-in-time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Based on the weak form quadrature element method, a perturbation approach is developed. Waves propagating in periodic beams on a nonlinear elastic foundation are studied by using the new proposed method. The feasibility and accuracy of the proposed method are verified by comparing the present results with those available in literatures in linear cases. Detailed modal analysis of the linear cases is conducted in order to obtain the dispersion relations of the nonlinear cases. The theoretical results show that the dispersion relations of the nonlinear cases are amplitude dependent. Furthermore, the effects of geometric parameters and degree of nonlinearity on the amplitude-dependent dispersion relations are discussed in detail. This work provides a new method for analyzing the dispersion relations of nonlinear periodic structures and gives some useful guidelines for designing periodic beams or pipelines with nonlinear structure–foundation interaction. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The evaluation of the diagonal of matrix functions arises in many applications and an efficient approximation of it, without estimating the whole matrix *f*(*A*), would be useful. In the present paper, we compare and analyze the performance of three numerical methods adjusted to attain the estimation of the diagonal of matrix functions *f*(*A*), where
is a symmetric matrix and *f* a suitable function. The applied numerical methods are based on extrapolation and Gaussian quadrature rules. Various numerical results illustrating the effectiveness of these methods and insightful remarks about their complexity and accuracy are demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.

The equations describing the steady flow of Cosserat–Bingham fluids are considered, and existence of weak solution is proved for the three-dimensional boundary-value problem with the use of the Lipschitz truncation argument. In contrast to the classical Bingham fluid, the micropolar Bingham fluid supports local micro-rotations and two types of plug zones. Our approach is based on an approximation of the constitutive relation by a generalized Newtonian constitutive relation and a subsequent limiting process. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Evolution of human language and learning processes have their foundation built on grammar that sets rules for construction of sentences and words. These forms of replicator–mutator (game dynamical with learning) dynamics remain however complex and sometime unpredictable because they involve children with some predispositions. In this paper, a system modeling evolutionary language and learning dynamics is investigated using the Crank–Nicholson numerical method together with the new differentiation with non-singular kernel. Stability and convergence are comprehensively proven for the system. In order to seize the effects of the non-singular kernel, an application to game dynamical with learning dynamics for a population with five languages is given together with numerical simulations. It happens that such dynamics, as functions of the learning accuracy *μ*, can exhibit unusual bifurcations and limit cycles followed by chaotic behaviors. This points out the existence of fickle and unpredictable variations of languages as time goes on, certainly due to the presence of learning errors. More interestingly, this chaos is shown to be dependent on the order of the non-singular kernel derivative and speeds up as this derivative order decreases. Hence, can people use that order to control chaotic behaviors observed in game dynamical systems with learning! Copyright © 2016 John Wiley & Sons, Ltd.

This article investigates the solvability and optimal controls of systems monitored by fractional delay evolution inclusions with Clarke subdifferential type. By applying a fixed-point theorem of condensing multivalued maps and some properties of Clarke subdifferential, an existence theorem concerned with the mild solution for the system is proved under suitable assumptions. Moreover, an existence result of optimal control pair that governed by the presented system is also obtained under some mild conditions. Finally, an example is given to illustrate our main results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this study, the generation of smooth trajectories of the end effector of a rotating extensible manipulator arm is considered. Possible trajectories are modelled using Cartesian and polar piecewise cubic interpolants expressed as polynomial Hermite-type functions. The use of polar piecewise cubic interpolants devises continuous first-order and – in some cases – second-order derivatives and allows easy calculation of kinematics variables such as velocity and acceleration. Moreover, the manipulator equations of motion can be easily handled, and the constrained trajectory of the non-active end of the manipulator derived directly from the position of the end-effector. To verify the proposed approach, numerical simulations are conducted for two different configurations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose an eco-epidemiological predator–prey model, modeling the spread of infectious keratoconjunctivitis among domestic and wild ungulates, during the summer season, when they intermigle in high mountain pastures. The disease can be treated in the domestic animals, but for the wild herbivores, it leads to blindness, with consequent death. The model shows that the disease can lead infected herbivores or their predators to extinction, even if it does not affect the latter. Boundedness of solutions and equilibria feasibility are obtained. Stability around the different equilibrium points is analyzed through eigenvalues and the Routh–Hurwitz criterion. Simulations are carried out to support the theoretical results. Sensitivity with respect of some parameters is investigated. The prey vaccination as control measure is introduced and simulated, although at present, the vaccine is not yet available, but just being developed. It would then possibly eradicate the infection in the domestic animals, which are considered a disease reservoir. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose a space-time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space-time spectral-Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this work is to study *μ*-pseudo almost automorphic solutions of abstract fractional integro-differential neutral equations with an infinite delay. Thanks to some restricted hypothesis on the delayed data in the phase space, we ensure the existence of the ergodic component of the desired solution. Copyright © 2016 John Wiley & Sons, Ltd.

We propose and investigate a delayed model that studies the relationship between HIV and the immune system during the natural course of infection and in the context of antiviral treatment regimes. Sufficient criteria for local asymptotic stability of the infected and viral free equilibria are given. An optimal control problem with time delays both in state variables (incubation delay) and control (pharmacological delay) is then formulated and analyzed, where the objective consists to find the optimal treatment strategy that maximizes the number of uninfected *C**D*4^{ + } T cells as well as cytotoxic T lymphocyte immune response cells, keeping the drug therapy as low as possible. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we investigate a boundary problem with non-local conditions for mixed parabolic–hyperbolic-type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a non-stationary Stokes system in a thin porous medium Ω_{ϵ} of thickness *ϵ* which is perforated by periodically solid cylinders of size *a*_{ϵ}. We are interested here to give the limit behavior when *ϵ* goes to zero. To do so, we apply an adaptation of the unfolding method. Time-dependent Darcy's laws are rigorously derived from this model depending on the comparison between *a*_{ϵ} and *ϵ*. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the solvability of a fully nonlinear third-order *m*-point boundary value problem at resonance posed on the half line. The nonlinearity which depends on the first and the second derivatives satisfies a sublinear-like growth condition. Our main existence result is based on Mawhin's coincidence degree theory. An illustrative example of application is included. Copyright © 2016 John Wiley & Sons, Ltd.

Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy-conserving, non-quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Graph theory is a fundamental tool in the study of economic issues, and input–output tables are one of the main examples. We use the interpretation of the labour market through networks to obtain a better understanding on its overall functioning. One benefit of the network perspective is that a large body of mathematics exists to help analyze many forms of networks models. If an economic system has obtained a suitable model, then it becomes possible to utilize relevant mathematical tools, such as graph theory, to better understand the way the labour market works. This interpretation allows us to employ the concepts of coverage, invariance, orbit and the structural functions supply–demand and competition and interpret them from the point of view of circular flow. In this paper, we aim to interpret the labour market through networks that are represented by graphs and where characteristic concepts of chaos theory such as cover, invariance and orbits interact with the concept circular flow. Finally, an example of this approach to labour markets is described, and some conclusions are drawn. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the reachable set estimation problem of singular systems with time-varying delay and bounded disturbance inputs. Based on a novel Lyapunov–Krasovskii functional that contains four triple integral terms, reciprocally convex approach and free-weighting matrix method, two sufficient conditions are derived in terms of linear matrix inequalities to guarantee that the reachable set of singular systems with time-varying delay is bounded by the intersection of ellipsoid. Finally, two numerical examples are given to demonstrate the effectiveness and superiority of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Recent studies have shown that the initiation of human cancer is due to the malfunction of some genes (such as E2F, CycE, CycD, Cdc25A, P27^{Kip1}, and Rb) at the R-checkpoint during the G_{1}-to-S transition of the cell cycle. Identifying and modeling the dynamics of these genes provide new insight into the initiation and progression of many types of cancers. In this study, a cancer subnetwork that has a mutual activation between phosphatase Cdc25A and the CycE/Cdk2 complex and a mutual inhibition between the Cdk inhibitor P27^{Kip1} and the CycE/Cdk2 complex is identified. A new mathematical model for the dynamics of this cancer subnetwork is developed. Positive steady states are determined and rigorously analyzed. We have found a condition for the existence of positive steady states from the activation, inhibition, and degradation parameter values of the dynamical system. We also found a robust condition that needs to be satisfied for the steady states to be asymptotically stable. We determine the parameter value(s) under which the system exhibits a saddle–node bifurcation. We also identify the condition for which the system exhibits damped oscillation solutions. We further explore the possibility of Hopf and homoclinic bifurcations from the saddle–focus steady state of the system. Our analytic and numerical results confirm experimental results in the literature, thus validating our model. Copyright © 2016 John Wiley & Sons, Ltd.

We first prove the uniqueness of weak solutions (*ψ*,*A*) to the 3-D Ginzburg–Landau model in superconductivity with zero magnetic diffusivity and the Coulomb gauge if
, which is a critical space for some positive constant *T*. We also prove the global existence of solutions when
and *A*_{0}∈*L*^{3}. Copyright © 2016 John Wiley & Sons, Ltd.

Our work is devoted to an inverse problem for three-dimensional parabolic partial differential equations. When the surface temperature data are given, the problem of reconstructing the heat flux and the source term is investigated. There are two main contributions of this paper. First, an adjoint problem approach is used for analysis of the Fréchet gradient of the cost functional. Second, an improved conjugate gradient method is proposed to solve this problem. Based on Lipschitz continuity of the gradient, the convergence analysis of the conjugate gradient algorithm is studied. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the problem of the periodic solutions bifurcated from a homoclinic orbit for a pair of coupled ordinary differential equations in
. Assume that the autonomous system has a degenerate homoclinic solution *γ* in
. A functional analytic approach is used to consider the existence of periodic solution for the autonomous system with periodic perturbations. By exponential dichotomies and the method of Lyapunov–Schmidt, the bifurcation function defined between two finite dimensional subspaces is obtained, where the zeros correspond to the existence of periodic solutions for the coupled ordinary differential equations near
. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non-isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial-boundary value problem of the non-isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero-order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The global stability of equilibria is investigated for a nonlinear multi-group epidemic model with latency and relapses described by two distributed delays. The results show that the global dynamics are completely determined by the basic reproduction number under certain reasonable conditions on the nonlinear incidence rate. Moreover, compared with the results in Michael Y. Li and Zhisheng Shuai, Journal Differential Equations 248 (2010) 1–20, it is found that the two distributed delays have no impact on the global behaviour of the model. Our study improves and extends some known results in recent literature. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Our interest is to quantify the spread of an infective process with latency period and generic incidence rate that takes place in a finite and homogeneous population.

Within a stochastic framework, two random variables are defined to describe the variations of the number of secondary cases produced by an index case inside of a closed population. Computational algorithms are presented in order to characterize both random variables. Finally, theoretical and algorithmic results are illustrated by several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we prove a global well posedness of the three-dimensional incompressible Navier–Stokes equation under an initial data, which belong to the non-homogeneous Fourier–Lei–Lin space
for *σ*⩾ − 1 and if the norm of the initial data in the Lei–Lin space
is controlled by the viscosity. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents all possible exact explicit peakon, pseudo-peakon, cuspon and smooth solitary wave solutions for a nonlocal Kerr-like media. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. We present peakon, pseudo-peakon, cuspon soliton solution in an explicit form. We also have obtained smooth soliton. Mathematical analysis and numeric graphs are provided for those soliton solutions of the nonlocal Kerr-like media. Copyright © 2016 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu-type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of -hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Objective: The study aimed to analyze sexual networks and sex role preference as factors of HIV transmission among men who have sex with men (MSM) in China.

Methods: We have developed a new scale-free network model with a sex role preference framework to study HIV transmission among MSM. We have studied the influence of different sexual networks and the effect of different proportion of sex role preference upon HIV transmission. The results are that the average ones drawn from the scenarios have been simulated for more than 30 times.

Results: Compared with the traditional mathematical model, the sexual networks provide a different prediction of the HIV transmission in the next 30 years. Without any intervention, the proportion of HIV carriers will descend after some time.

Conclusions: There is significant associations among network characteristics, sex role preference, and HIV infection. Although network-based intervention is efficient in reducing HIV transmission among MSM, there are only few studies of the characteristics of sexual network, and such gaps deserve more attention and exploration. Copyright © 2016 John Wiley & Sons, Ltd.

The present paper is concerned with the approximation properties of discrete version of Picard operators. We first give exact equalities for the moments of the operators. In calculations of these moments, Eulerian numbers play a crucial role. We discuss convergence of these operators in weighted spaces and give Voronovskaya-type asymptotic formula. The weighted approximation of the operators in quantitative mean in terms of different modulus of continuities is also considered. We emphasize that the rate of convergence of the operators is better than the one obtained in . Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg–de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd.

]]>For 1-D quasilinear wave equations with different types of boundary conditions, based on the theory of the local exact boundary controllability, using an extension method, the author establishes the exact controllability in a shorter time by means of internal controls acting on suitable domains. In particular, the exact controllability can be realized only by internal controls, and the control time can be arbitrarily small. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, first, we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first-order partial integro-differential equation, and it has been studied extensively. In particular, it is well posed on the biologically relevant state space of Lebesgue integrable functions. We also formulate a delayed integral equation (renewal equation) for the distributed birth rate of the population. We aim to illustrate the connection between the partial integro-differential and the delayed integral equation formulation of the model utilising a recent spectral theoretic result. In particular, we consider the equivalence of the steady state problems in the two different formulations, which then lead us to characterise irreducibility of the semigroup governing the linear partial integro-differential equation. Furthermore, using the method of characteristics, we investigate the connection between the time-dependent problems. In particular, we prove that any (non-negative) solution of the delayed integral equation determines a (non-negative) solution of the partial differential equation and vice versa. The results obtained for the particular distributed states at birth model then lead us to present some very general results, which establish the equivalence between a general class of partial differential and delay equation, modelling physiologically structured populations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We are interested in a mixed reaction diffusion system describing the organic pollution in stream-waters. In this work, we propose a mixed-variational formulation and recall its well-posedness. Next, we consider a spectral discretization of this problem and establish nearly optimal error estimates. Numerical experiments confirm the interest of this approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We give some integrability conditions for the coefficients of a sequence of elliptic systems with varying coefficients in order to obtain the stability for homogenization. In the case of equations, it is well known that equi-integrability and bound in *L*^{1} are enough for this purpose; however, this is based on the maximum principle, and then, it does not work for systems. Here, we use an extension of the Murat–Tartar div-curl lemma because of M. Briane, J. Casado-Díaz, and F. Murat in order to obtain the stability by homogenization for systems uniformly elliptic, with bounded coefficients in
, with *N* the dimension of the space. We also show that a weaker ellipticity condition can be assumed, but then, we need a stronger integrability for the coefficients. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the stability of the zero equilibria of the following systems of difference equations:

and

where *a*, *b*, *c* and *d* are positive constants and the initial conditions *x*_{0} and *y*_{0} are positive numbers. We study the stability of those systems in the special case when one of the eigenvalues has absolute value equal to 1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the relative approximate controllability of functional systems with infinite delay and delayed control in Hilbert spaces. In particular, we begin with studying some criteria of the controllability of linear systems. Based on those results, sufficient conditions are derived for the relative approximate controllability of nonlinear functional systems. Finally, an example is included to illustrate the effectiveness of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider a variable coefficient Calogero–Degasperis equation, a variable coefficient potential Kadomstev–Petviashvili equation and the generalized (3+1)-dimensional variable coefficient Kadomtsev–Petviashvili equation with time-dependent coefficients. Shock wave solutions for the considered models are obtained by using ansatz method in the form of tanh*p* function. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Copyright © 2016 John Wiley & Sons, Ltd.

Macro-hybrid mixed variational models of two-phase flow, through fractured porous media, are analyzed at the mesoscopic and macroscopic levels. The mesoscopic models are treated in terms of nonoverlapping domain decompositions, in such a manner that the porous rock matrix system and the fracture network interact across rock–rock, rock–fracture, and fracture–fracture interfaces, with flux transmission conditions dualized. Alternatively, the models are scaled to a macroscopic level via an asymptotic process, where the width of the fractures tends to zero, and the fracture network turns out to be an interface system of one less spatial dimension, with variable high permeability. The two-phase flow is characterized by a fractional flow dual mixed variational model. Augmented two-field and three-field variational reformulations are presented for regularization, internal approximations, and macro-hybrid mixed finite element implementation. Also abstract proximal-point penalty-duality algorithms are derived and analyzed for parallel computing. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely, counter-migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider a coupled system of mixed hyperbolic–parabolic type, which describes the Biot consolidation model in poro-elasticity. We establish a local Carleman estimate for Biot consolidation system. Using this estimate, we prove the uniqueness and a Hölder stability in determining on the one hand a physical parameter arising in connection with secondary consolidation effects *λ*^{∗} and on the other hand the two spatially varying densities by a single measurement of solution over *ω* × (0,*T*), where *T* > 0 is a sufficiently large time and a suitable subdomain *ω* satisfying *∂**ω*⊃*∂*Ω. Copyright © 2016 John Wiley & Sons, Ltd.

We analyze a bounded confidence model, introduced by Krause, on isolated time scales. In this model, each agent takes into account only the assessments of the agents whose opinions are not too far away from its own opinion. We show that the behavior of the model depends strongly on the graininess function *μ*: If *μ* takes values in the interval ]0,1], then our discrete time scale model behaves similarly to the classical one, but if *μ* takes values in ]1,+*∞*[, then the model has different properties. Simulations are performed to validate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper studies the local-in-time existence of classical solutions to a hyperbolic system with differential boundary conditions modelling a flow in an elastic tube. The well-known Lax transformations used for obtaining a priori estimates for conservation laws are difficult to apply here because of the inhomogeneity of the partial differential equations (PDE). Rather, our method relies on a suitable splitting of the original system into the PDE part and the ODE part, the characteristics for the PDE part, appropriate modulus of continuity estimates and a compactness argument. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we consider the interior transmission eigenvalue problem for a spherically stratified medium, which can be formulated as *y*^{′′}(*r*) + *k*^{2}*η*(*r*)*y*(*r*) = 0 endowed with boundary conditions
, where the refractive index *η*(*r*) is positive and real. We obtain the distribution of transmission eigenvalues under assumptions that
and one of these conditions that (i) *η*(1) ≠ 1, (ii) *η*(1) = 1,*η*′(1) ≠ 0, and (iii) *η*(1) = 1,*η*′(1) = 0,*η*^{″}(1) ≠ 0, respectively. Moreover, in the case *a* = 1, we prove that if partial information on *η*(*r*) is known on subdomain, then only a part of eigenvalues can uniquely determine *η*(*r*) on the whole interval, and the relationship between the proportion of the missing eigenvalues and the subinterval of the known information on *η*(*r*) is revealed. Copyright © 2016 John Wiley & Sons, Ltd.

This paper discusses the problem of stability and periodic behaviour of linear planar difference systems appearing in modelling of discrete problems of population biology and Hopfield neural networks. We concentrate especially on procedures, which enable to formulate optimal (i.e. necessary and sufficient) conditions guaranteing such a behaviour. As a main tool, we analyse in detail location of zeros of characteristic polynomials with respect to the unit circle. From this viewpoint, derived results contribute also to the polynomial theory. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Consider the following *ν*-th order Caputo delta fractional equation

- (0.1)

The following asymptotic results are obtained.

A. **Assume**0 < *ν* < 1**and there exists a constant***b*_{2}**such that***c*(*t*)≥*b*_{2}>0. **Then the solutions of the equation (0.1) with***x*(*a*) > 0**satisfy**

B. **Assume**0 < *ν* < 1**and there exists a constant***b*_{1}**such that**−*ν* < *c*(*t*)≤*b*_{1}<0. **Then the solutions of the equation (0.1) with***x*(*a*) > 0**satisfy**

This shows that the solutions of the Caputo delta fractional equation
, with *x*(*a*) > 0 have similar asymptotic behavior with the solutions of the first-order delta difference equation Δ*x*(*t*) = *c**x*(*t*),*c* >− 1. Copyright © 2016 John Wiley & Sons, Ltd.

We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in
. The system in question is *u*_{t}=*a*Δ*u* − *f*(*x*,*t*,*u*,*v*), *v*_{t}=*c*Δ*u* + *d*Δ*v* + *ρ**f*(*x*,*t*,*u*,*v*),
, *t* > 0 with *f*(*x*,*t*,0,*η*) = 0 and *f*(*x*,*t*,*ξ*,*η*)≤*K**φ*(*ξ*)*e*^{ση}, for all *x*∈Ω, *t* > 0, *ξ*≥0, *η*≥0; where *ρ*, *K* and *σ* are real positive constants. Copyright © 2016 John Wiley & Sons, Ltd.

In a singular limit, the Klein–Gordon (KG) equation can be derived from the Klein–Gordon–Zakharov (KGZ) system. We point out that for the original system posed on a *d*-dimensional torus, the solutions of the KG equation do not approximate the solutions of the KGZ system. The KG system has to be modified to make correct predictions about the dynamics of the KGZ system. We explain that this modification is not necessary for the approximation result for the whole space
with *d*≥3. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2-D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub-domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time-stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we consider a model with one basal resource and two species of predators feeding by the same resource. There are three non-trivial boundary equilibria. One is the saturated state *E*_{K} of the prey without any predator. Other two equilibria, *E*_{1} and *E*_{2}, are the coexistence states of the prey with only one species of predators. Using a high-dimensional shooting method, the Wazewski' principle, we establish the conditions for the existence of traveling wave solutions from *E*_{K} to *E*_{2} and from *E*_{1} to *E*_{2}. These results show that the advantageous species *v*_{2} always win in the competition and exclude species *v*_{1} eventually. Finally, some numerical simulations are presented, and biological interpretations are given. Copyright © 2016 John Wiley & Sons, Ltd.

The persistence of subsonic phase boundaries in a multidimensional Van der Waals fluid is analyzed. The phase boundary is considered as a sharp free boundary that connects liquid and vapor bulk phase dynamics given by the isothermal Euler equations. The evolution of the boundary is driven by effects of configurational forces as well as surface tension.

To analyze this problem, the equations and trace conditions are linearized such that one obtains a general hyperbolic initial boundary value problem with higher-order boundary conditions. A global existence theorem for the linearized system with constant coefficients is shown. The proof relies on the normal mode analysis and a linear form in suitable spaces that is defined using an associated adjoint problem. Especially, the associated adjoint problem satisfies the uniform backward in time Kreiss–Lopatinskiĭ condition. A new energy-like estimate that also includes surface energy terms leads finally to the uniqueness and regularity for the found solutions of the problem in weighted spaces. Copyright © 2016 John Wiley & Sons, Ltd.

A virus infection model with time delays and humoral immunity has been investigated. Mathematical analysis shows that the global dynamics of the model is fully determined by the basic reproduction numbers of the virus and the immune response, *R*_{0} and *R*_{1}. The infection-free equilibrium *P*_{0} is globally asymptotically stable when *R*_{0}≤1. The infection equilibrium without immunity *P*_{1} is globally asymptotically stable when *R*_{1}≤1 < *R*_{0}. The infection equilibrium with immunity *P*_{2} is globally asymptotically stable when *R*_{1}>1. The expression of the basic reproduction number of the immune response *R*_{1} implies that the immune response reduces the concentration of free virus as *R*_{1}>1. Copyright © 2016 John Wiley & Sons, Ltd.

We point out some mistakes in a known paper. Some existence results for solutions of two classes of boundary value problems for nonlinear impulsive fractional differential equations are established. Our analysis relies on the well-known Schauder fixed point theorem. Examples are given to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the control systems of semilinear fractional evolution hemivariational inequalities and their optimal controls in Banach space. Firstly, the existence of mild solutions is obtained and proved mainly by using a well-known fixed point theorem of multivalued maps and the properties of generalized Clarke subdifferential. Then, by applying generally mild conditions of cost functionals, we investigate the existence results of the optimal controls for fractional differential evolution hemivariational inequalities. Finally, an example is given to demonstrate the applicability of the main results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the convergence to equilibria, as time tends to infinity, of trajectories of dissipative wave systems with time-dependent velocity feedbacks and subject to nonlinear potential energies. Estimates for the speed of convergence are obtained in terms of the damping coefficient and the Łojasiewicz–Simon exponent. We allow for both restoring and amplifying effects of exterior forces, which makes our results possess wide applicability. As an example of application, we show that the trajectories of a sine-Gordon system, with nonautonomous damping, approach equilibria at least polynomially. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision-maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide-treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost-effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A rate of *rational decay* is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time-evolving displacements along the flexible portion Γ_{0} of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter *η*∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ_{0}. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth-order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of
. By way of deriving these stability results, necessary *a priori* inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the zero viscosity and capillarity limit problem for the one-dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We provide sufficient conditions for the nonexistence of nontrivial nonnegative solutions for some nonlinear elliptic inequalities involving the fractional Laplace operator and variable exponents. The used techniques are based on the test function method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Hepatitis B virus (HBV) and its vaccination strategy may affect human immunodeficiency virus (HIV) transmission dynamics because both viruses have synergistic effects. To quantitatively assess the potential impact of HBV and its vaccination strategy on HIV transmission dynamics at the population level, in this paper, we formulate a deterministic compartmental model that describes the joint dynamics of HBV and HIV. We first derive the explicit expressions for the basic reproduction numbers of HIV and HBV and analyze the dynamics of HIV and HBV subsystems, respectively. Then a systematic qualitative analysis of the full system is also provided, which includes the local and global behavior. By using a set of reasonable parameter values, the full system is numerically investigated to assess the potential impact of HBV and its vaccination strategy on HIV transmission. The direct and indirect population level impact of HBV on HIV is demonstrated by calculating the fraction of HIV infections attributable to HBV and the difference between HIV prevalence in the presence and absence of HBV, respectively. The findings imply that the increase of HBV vaccination rate may unusually accelerate HIV epidemics indirectly, although the direct effect of HBV on HIV transmission decreases as HBV vaccination rate increases. Finally, the potential impact of HIV on HBV transmission dynamics is investigated by way of parenthesis. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with the existence of positive solutions of the discrete counterpart of nonlinear elliptic problems. We apply our methods to the study of positive solutions under different hypotheses about the nonlinearities. For example, we consider the cases that are superlinear and sublinear at infinity. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish some new sufficient conditions on the existence of homoclinic solution for a class of second-order impulsive Hamiltonian systems. By using the mountain pass theorem, we demonstrate that the limit of a 2*k**T*-periodic approximation solution is a homoclinic solution of our problem. We also present some examples to illustrate the applications of our main results. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the *S*-spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic *S*-functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.

The present paper first introduces the notion of quaternion infinite series of positive term and establishes its several tests. Next, we give the definitions of the positive-definite quaternion sequence and the positive semi-definite quaternion function, and we extend the classical Herglotz's theorem to the quaternion linear canonical transform setting. Then we investigate the properties of the two-sided quaternion linear canonical transform, such as time shift characteristics and differential characteristics. Finally, we derive its several basic properties of the quaternion linear canonical transform of a probability measure, in particular, and establish the Bochner–Minlos theorem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Dynamical behavior of many nonlinear systems can be described by fractional-order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)-expansion method coupled with the so-called fractional complex transform. The solution procedure is elucidated through two generalized time-fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We prove an existence theorem for an abstract operator equation associated with a quasi-subdifferential operator and then apply it to concrete elliptic variational and quasi-variational inequalities. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we will propose a Durrmeyer variant of *q*-Bernstein–Schurer operators. A Bohman–Korovkin-type approximation theorem of these operators is considered. The rate of convergence by using the first modulus of smoothness is computed. The statistical approximation of these operators is also studied. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a new impulsive Lasota–Wazewska model with patch structure and forced perturbed terms is proposed and analyzed on almost periodic time scales. For this, we introduce the concept of matrix measure on time scales and obtain some of its properties. Then, sufficient conditions are established which ensure the existence and exponential stability of positive almost periodic solutions of the proposed biological model. Our results are new even when the time scale is almost periodic, in particular, for periodic time scales on or . An example is given to illustrate the theory. Finally, we present some phenomena which are triggered by almost periodic time scales. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Drawing on viral dynamics theory, this paper presents a differential equations model with time delay to investigate the stock investor behavior driven by new product announcement (NPA) signal. Visually, we look upon investors in stock market as cells *in vivo* and the NPA signals as free virus. The potential investors will be ‘infected’ by the dissociative NPA signal and then make investment decisions. In order to better understand the ‘infection’ process, we extract and establish a multi-stage process during which NPA signal is delivered and ‘infects’ the potential investors. A time-delay effect is employed to reflect the evaluation stage at which potential investors comprehensively evaluate and decide whether to invest or not. In addition, we introduce a set of external and internal factors into the model, including information sensitivity and investor sentiment, and so on, which are pivotal for examining investor behavior. Equilibrium analysis and numerical simulations are employed to check out the properties of the model and highlight the practical application values of the model. Copyright © 2016 John Wiley & Sons, Ltd.

We obtain in this paper the expression of the solutions of the following recursive sequences:

where the initial conditions are arbitrary real numbers. Also, we study the behavior of the solution of these equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We will prove that for piecewise C^{2}-concave domains in
Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant
. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a nonperiodic discrete nonlinear equation with Jacobi operators is considered. By using the critical point theory, we establish some new sufficient conditions on the existence and multiplicity of homoclinic solutions. Recent results are generalized and significantly improved. Furthermore, our results greatly improve some existing ones even for some special cases. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with interval general bidirectional associative memory (BAM) neural networks with proportional delays. Using appropriate nonlinear variable transformations, the interval general BAM neural networks with proportional delays can be equivalently transformed into the interval general BAM neural networks with constant delays. The sufficient condition for the existence and uniqueness of equilibrium point of the model is established by applying Brouwer's fixed point theorem. By constructing suitable delay differential inequalities, some sufficient conditions for the global exponential stability of the model are obtained. Two examples are given to illustrate the effectiveness of the obtained results. This paper ends with a brief conclusion. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the global existence and asymptotic behavior of the Boussinesq-Burgers system subject to the Dirichlet boundary conditions. Based on the *L*^{p}(*p* > 2) estimates of the solution, which are different from the standard *L*^{2}-based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the issue of stability of multi-group coupled systems on networks with multi-diffusion (MCSNMs) is mainly analyzed. Utilizing graph theory, a novel and practical method of constructing a proper Lyapunov function for the MCSNMs is presented. Furthermore, based on the graph-theoretic approach and the proposed Lyapunov function, sufficient criteria, in the term of Lyapunov function and coefficients of the system, respectively, are derived to ensure the stability of the MCSNMs. Apart from accessibility to checking, the proposed results can generalize the corresponding results published in a previous time. Finally, the effectiveness and feasibility of the obtained results are demonstrated by a numerical example. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish finite-region stability (FRS) and finite-region boundedness analysis methods to investigate the transient behavior of discrete two-dimensional Roesser models. First, by building special recursive formulas, a sufficient FRS condition is built via solvable linear matrix inequalities constraints. Next, by designing state feedback controllers, the finite-region stabilization issue is analyzed for the corresponding two-dimensional closed-loop system. Similar to FRS analysis, the finite-region boundedness problem is addressed for Roesser models with exogenous disturbances and corresponding criteria, and linear matrix inequalities conditions are reported. To conclude the paper, we provide numerical examples to confirm the validity of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the fourth-order nonlinear Schrödinger equation with parameterized nonlinearity that is generalized from regular cubic-quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth-order generalized cubic-quintic nonlinear Schrödinger equation through modified *F*-expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.