Recently, Sanhan and Mongkolkeha introduced the concept of Berinde's cyclic contraction, and they established some results. Unfortunately, these results seem to be incorrect. In this paper, some counterexamples are given.

]]>Coevolution can impose density-dependent selection through reciprocal biotic interactions on the fitness of involved species, driving directional and disruptive trait evolution and rich evolutionary possibilities. Coevolution has since Darwin been considered a potential path leading to adaptive diversification that could explain the emergence of ecological networks of biotic interactions that harbour multiple interacting species (eg, pollination networks and food webs). Here, we present adaptive dynamics, a powerful tool of evolutionary invasion analysis that explores how quantitative traits undergo incremental evolution, to exploring the emergence of multi-species networks through coevolution. Specifically, we exemplify the feasibility of using adaptive dynamics to investigate trait evolution in 4 ecological networks, driven, respectively, by resource competition, trophic interactions, as well as bipartite mutualistic and antagonistic interactions. We use a set of ordinary differential equations to describe, at different paces, the population dynamics and trait dynamics of involved species assemblages. Through computing ecological equilibrium, invasion fitness, selection gradient and evolutionary singularity, and testing for evolutionary stability and the coexistence criterion of mutual invasibility, we illustrate the typical evolutionary dynamics and the criteria of evolutionary stability and branching in these ecological networks. Results highlight the importance of the form of trait-mediated interaction kernel (ie, interaction strength as a function of trait difference) to adaptive diversification in these coevolutionary systems. We conclude by advocating that biotic interactions between two species can indeed lead to diffuse and even escape-and-radiate coevolution, making the emerged ecological networks an ideal model for studying complex adaptive systems.

]]>Here, a system of 3 wave equations in
with infinite memories acting in the first 2 equations is considered. Using weighted spaces, we prove the polynomial stability of the system under some conditions on *μ*_{1},*μ*_{2}, and *ϕ* as
.

In this paper, we consider a viscoelastic equation with minimal conditions on the
relaxation function *g*, namely,
, where *H* is an increasing and convex function near the origin and *ξ* is a nonincreasing function. With only these very general assumptions on the behavior of *g*at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when *H*(*s*)=*s*^{p} and *p* covers the full admissible range [1,2). We get the best decay rates expected under this level of generality, and our new results substantially improve several earlier related results in the literature.

The triple conformal geometric algebra (TCGA) for the Euclidean
-plane extends CGA as the product of 3 orthogonal CGAs and thereby the representation of geometric entities to general cubic plane curves and certain cyclidic (or roulette) quartic, quintic, and sextic plane curves. The plane curve entities are 3-vectors that linearize the representation of nonlinear curves, and the entities are inner product null spaces with respect to all points on the represented curves. Each inner product null space entity also has a dual geometric outer product null space form. Orthogonal or conformal (angle preserving) operations (as versors) are valid on all TCGA entities for inversions in circles, reflections in lines, and by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA also provides a method for anisotropic dilations. Intersections of any TCGA entity with a point, point pair, line, or circle are possible. The TCGA defines commutator-based differential operators in the coordinate directions that can be combined to yield a general **n**-directional derivative.

We discuss minimality conditions for the speed of monotone travelling waves in a sample of smectic C^{∗} liquid crystal subject to a constant electric field, dealing with both isotropic and anisotropic cases. Such conditions are important in understanding the properties of domain wall switching across a smectic layer, and our focus here is on examining how the presence of anisotropy can affect the speed of this switching. We obtain an estimate of the influence of anisotropy on the minimal speed, sufficient conditions for linear and non-linear minimal speed selection mechanisms to hold in different parameter regimes, and a characterisation of the boundary separating the linear and non-linear regimes in parameter space.

In this paper, we obtained some useful estimates for convolution corresponding to Kontorovich-Lebedev transform (KL-transform) in Lebesgue space. Some continuity theorems for translation, convolution, and KL-transform in test function space are discussed. Then an integral representation of pseudodifferential operator involving KL-transform is found out, and its estimates in Lebesgue space is obtained. At the end, some applications of KL-transform and its convolution are discussed.

]]>The aim of this paper was to study the junction between a periodic family of beams and two thin plates. This structure depends on 3 small parameters. We use the decompositions of the displacement fields in every beam and plate to obtain a priori estimates. Then in the case for which the displacements of both plates match, we derive the asymptotic behavior of this structure.

]]>We study the Cauchy problem of the 3-dimensional nonhomogeneous heat conducting Navier-Stokes equations with nonnegative density. First of all, we show that for the initial density allowing vacuum, the strong solution to the problem exists globally if the velocity satisfies the Serrin's condition. Then, under some smallness condition, we prove that there is a unique global strong solution to the 3D viscous nonhomogeneous heat conducting Navier-Stokes flows. Our method relies upon the delicate energy estimates.

]]>An efficient method for nonlinear fractional differential equations is proposed in this paper. This method consists of 2 steps. First, we linearize the nonlinear operator equation by quasi-Newton's method, which is based on *Fréchet derivative*. Then we solve the linear fractional differential equations by the simplified reproducing kernel method. The convergence of the quasi-Newton's method is discussed for the general nonlinear case as well. Finally, some numerical examples are presented to illustrate accuracy, efficiency, and simplicity of the method.

In this work, we develop a new integrable equation by combining the KdV equation and the negative-order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.

]]>We consider the undamped Klein-Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions. For any real value of the initial energy, particularly for supercritical values of the energy, we give sufficient conditions to conclude blow-up in finite time of weak solutions. The success of the analysis is based on a detailed analysis of a differential inequality. Our results improve previous ones in the literature.

]]>In this paper, we consider the time-periodic solution to a simplified version of Ericksen-Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals with a time-periodic external force in a periodic domain in . By using an approach of parabolic regularization and combining with the topology degree theory, we establish the existence of the time-periodic solution to the model under some smallness and symmetry assumptions on the external force. Then, we give the uniqueness of the periodic solution of this model.

]]>Quaternion analytic signal is regarded as a generalization of analytic signal from 1D to 4D space. It is defined by an original signal with its quaternion partial and total Hilbert transforms. The quaternion analytic signal provides the signal features representation, such as the local amplitude and local phase angle, the latter includes the structural information of the original signal. The aim of the present study is twofold. Firstly, it attempts to analyze the Plemelj-Sokhotzkis formula associated with quaternion Fourier transform and quaternion linear canonical transform. With these formulae, we show that the quaternion analytic signals are the boundary values of quaternion Hardy functions in the upper half space of 2 complex variables space. Secondly, the quaternion analytic signal can be extended to the quaternion Hardy function in the upper half space of 2 complex variables space. Two novel types of phase-based edge detectors are proposed, namely, quaternion differential phase angle and quaternion differential phase congruency methods. In terms of peak signal-to-noise ratio and structural similarity index measure, comparisons with competing methods on real-world images consistently show the superiority of the proposed methods.

]]>In this paper, we consider the integration of the special second-order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase-lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.

]]>Precomputed radiance transfer (PRT) methods established for handling global illumination (GI) of objects from area lights in real time and many techniques proposed for rotating the light using linear algebra rotation matrices. Rotating area lights efficiently are crucial part for computer graphics since it is one of the main components of real-time rendering. Matrices commonly used for handling such rotations are not quite efficient and require high memory consumption; as a result, the need for proposing new more efficient rotation algorithms has been established. In this work, we use the conformal geometric algebra (CGA) as the mathematical background for “GI in real-time” under distant Image-based lighting (IBL) illumination, for diffuse surfaces with self-shadowing by efficiently rotating the environment light using CGA entities. Our work is based on spherical harmonics (SH), which are used for approximating natural, area-light illumination as irradiance maps. Our main novelty is that we extend the PRT algorithm by representing SH for the first time with CGA.The main intuition is that SH of band index 1 are represented using CGA entities and SH with band index larger than 1 are represented in terms of CGA-SH of band 1. Specifically, we propose a new method for representing SH with CGA entities and rotating SH by rotating CGA entities. In this way, we can visualize the SH rotations, rotate them faster than rotation matrices, and we provide a unique visual representation and intuition regarding their rotation, in stark contrast to usual rotation matrices, and we achieve consistently better visual results from Ivanic rotation matrices during light rotation. Via our CGA expressed SH, we provide a significant boost on the PRT algorithm since we represent SH rotations by CGA rotors (4 numbers) as opposed to 9 × 9 sparse matrices that are usually used. With our algorithm, we pave the way for including scaling (dilation) and translation of light coefficients using CGA motors.

]]>The nonlinear versions of Sturm-Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.

]]>The self-adaptive intelligence gray predictive model (SAIGM) has an alterable-flexible model structure, and it can build a dynamic structure to fit different external environments by adjusting the parameter values of SAIGM. However, the order number of the raw SAIGM model is not optimal, which is an integer. For this, a new SAIGM model with the fractional order accumulating operator (SAIGM_FO) was proposed in this paper. Specifically, the final restored expression of SAIGM_FO was deduced in detail, and the parameter estimation method of SAIGM_FO was studied. After that, the Particle Swarm Optimization algorithm was used to optimize the order number of SAIGM_FO, and some steps were provided. Finally, the SAIGM_FO model was applied to simulate China's electricity consumption from 2001 to 2008 and forecast it during 2009 to 2015, and the mean relative simulation and prediction percentage errors of the new model were only 0.860% and 2.661%, in comparison with the ones obtained from the raw SAIGM model, the GM(1, 1) model with the optimal fractional order accumulating operator and the GM(1, 1) model, which were (1.201%, 5.321%), (1.356%, 3.324%), and (2.013%, 23.944%), respectively. The findings showed both the simulation and the prediction performance of the proposed SAIGM_FO model were the best among the 4 models.

]]>In this paper, a model for the spread of tuberculosis between buffaloes and lions is presented and analyzed. The most important system parameters are identified: vertical and horizontal disease transmission among the buffaloes and the influence of intraspecific competition between healthy and diseased buffaloes on the infected buffaloes population. Removal of diseased prey appears to be the most effective strategy to render the ecosystem disease free.

]]>In this paper, the 2D Navier-Stokes-Voight equations with 3 delays in
is considered. By using the Faedo-Galerkin method, Lions-Aubin lemma, and Arzelà-Ascoli theorem, we establish the global well-posedness of solutions and the existence of pullback attractors in *H*^{1}.

This paper deals with the oscillation of the fourth-order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third-order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained.

]]>We prove that the Volterra-Gause system of predator-prey type exhibits 2 kinds of zero-Hopf bifurcations for convenient values of their parameters. In the first, 1 periodic solution bifurcates from a zero-Hopf equilibrium, and in the second, 4 periodic solutions bifurcate from another zero-Hopf equilibrium. This study is done using the averaging theory of second order.

]]>In living cells, we can observe a variety of complex network systems such as metabolic network. Studying their sensitivity is one of the main approaches for understanding the dynamics of these biological systems. The study of the sensitivity is done by increasing/decreasing, or knocking out separately, each enzyme mediating a reaction in the system and then observing the responses in the concentrations of chemicals or their fluxes. However, because of the complexity of the systems, it has been unclear how the network structures influence/determine the responses of the systems. In this study, we focus on monomolecular networks at steady state and establish a simple criterion for determining regions of influence when any one of the reaction rates is perturbed through sensitivity experiments of enzyme knock-out type. Specifically, we study the network response to perturbations of a reaction rate *j*^{∗} and describe which other reaction rates
respond by non-zero reaction flux, at steady state. Non-zero responses of
to *j*^{∗} are called flux-influence of *j*^{∗} on
. The main and most important aspect of this analysis lies in the reaction graph approach, in which the chemical reaction networks are modelled by a directed graph. Our whole analysis is function-free, ie, in particular, our approach allows a graph theoretical description of sensitivity of chemical reaction networks. We emphasize that the analysis does not require numerical input but is based on the graph structure only. Our specific goal here is to address a topological characterization of the flux-influence relation in the network. In fact we characterize and describe the whole set of reactions influenced by a perturbation of any specific reaction.

We consider a Cahn-Hilliard–type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension *d*=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.

In this note, we study and explore locally symmetric *f*−associated standard static spacetimes *I*_{f}×*M*. Necessary and sufficient conditions on *f*−associated standard static spacetimes to be locally symmetric are derived. Some implications for these conditions are considered.

In this paper we consider a periodic 2-dimensional quasi-geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution for small initial data in the Lei-Lin-Gevrey spaces . Moreover, we establish an exponential type explosion in finite time of this solution.

]]>This paper deals with an adaptation of the Poincaré-Lindstedt method for the determination of periodic orbits in three-dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three-dimensional Lotka-Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.

]]>Human immunodeficiency virus (HIV)/AIDS and cancer coexistence both in vivo and in vitro in a cancer-immune environment leads to specific cytokines being produced by various immune cells and the cancer cells. Most of the studies have suggested that specific cytokines produced by the immune system cells and the tumor play an important role in the dynamics of non-Hodgkin lymphomas (NHLs). In this paper, a mathematical model describing the NHL-immune system interaction in the presence of the HIV, HIV treatment, and chemotherapy is developed. The formulated model, described by nonlinear ODEs, shows existence of multiple equilibria whose stability and bifurcation analysis are presented. From the bifurcation analysis, bistability regions are evident. We observe that with and without HIV treatment, the system results in a nonaggressive tumor size or aggressive tumor (full-blown tumor) depending on the initial conditions. The results further suggest that at a low endemic state, patients can live for longer period with the tumor, which might explain why some patients can live with cancer for many years. However, initiation of HIV treatment in patients with NHL is observed to lower these endemic states of the tumor. Our results explain why late initiation of HIV treatment might not be helpful to NHL patients. We further investigated the effect of chemotherapy on the dynamics of the tumor. Our simulation results might explain why a few of these chemotherapeutic drugs are more effective when given at a slow continuous rate. The model provides a unique opportunity to influence policy on HIV-related cancer treatment and management.

]]>In this article, we consider the Cauchy problem to Keller-Segel equations coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let *u*_{F}:=*e*^{tΔ}*u*_{0}; we prove that there exist 2 positive constants *σ*_{0} and *C*_{0} such that if the gravitational potential
and the initial data (*u*_{0},*n*_{0},*c*_{0}) satisfy

for some *p*,*q* with
and
, then the global solutions can be established in critical Besov spaces.

In this paper, we study constraint minimizers of the following *L*^{2}−critical minimization problem:

where *E*(*u*) is the Schrödinger-Poisson-Slater functional

and *N* denotes the mass of the particles in the Schrödinger-Poisson-Slater system. We prove that *e*(*N*) admits minimizers for
and, however, no minimizers for *N*>*N*^{∗}, where *Q*(*x*) is the unique positive solution of
in
. Some results on the existence and nonexistence of minimizers for *e*(*N*^{∗}) are also established. Further, when *e*(*N*^{∗}) does not admit minimizers, the limit behavior of minimizers as *N*↗*N*^{∗} is also analyzed rigorously.

The determination of a space-dependent source term along with the solution for a 1-dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter *β*>0 is considered. The fractional derivative is generalization of the Riemann-Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over-specified datum at 2 different time is given. The over-specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.

We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis-Navier-Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis-Navier-Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis-Euler system.

]]>Growth models are often used when modelling various processes in life sciences, ecology, demography, social sciences, etc. Dynamical growth models are usually formulated in terms of an ODE (system of ODS's) or by an explicit solution to an ODE, such as the logistic, Gompertz, and Richardson growth models. To choose a suitable growth model it is useful to know the physics-chemical meaning of the model. In many situations this meaning is best expressed by means of a reaction network that possibly induces the dynamical growth model via mass action kinetics. Such reaction networks are well known for a number of growth models, such as the saturation-decay and the logistic Verhulst models. However, no such reaction networks exist for the Gompertz growth model. In this work we propose some reaction networks using mass action kinetics that induce growth models that are (in certain sense) close to the Gompertz model. The discussion of these reaction networks aims to a better understanding of the chemical properties of the Gompertz model and “Gompertzian-type” growth models. Our method can be considered as an extension of the work of previous authors who “recasted” the Gompertz differential equation into a dynamical system of two differential equations that, apart of the basic species variable, involve an additional variable that can be interpreted as a “resource.” Two new growth models based on mass action kinetics are introduced and studied in comparison with the Gompertz model. Numerical computations are presented using some specialized software tools.

]]>In the present paper, we construct a new sequence of Bernstein-Kantorovich operators depending on a parameter *α*. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first- and second-order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein-Kantorovich operators and their approximation behaviors.

Aveiro method is a sparse representation method in reproducing kernel Hilbert spaces, which gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying reproducing kernel Hilbert space. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro method. To avoid those difficulties, we propose an new Aveiro method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so-called pre-orthogonal greedy algorithm involving completion of a given dictionary. The new method is called Aveiro method under complete dictionary. The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition bring available for the classical Hardy and Paley-Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro method and the greedy algorithm.

]]>This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two-point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate.

]]>The swarm behaviour can be controlled by different localizations of attractants (food pieces) and repellents (dangerous places), which, respectively, attract and repel the swarm propagation. If we assume that at each time step, the swarm can find out not more than *p*−1 attractants (
), then the swarm behaviour can be coded by *p*-adic integers, ie, by the numbers of the ring **Z**_{p}. Each swarm propagation has the following 2 stages: (1) the discover of localizations of neighbour attractants and repellents and (2) the logistical optimization of the road system connecting all the reachable attractants and avoiding all the neighbour repellents. In the meanwhile, at the discovering stage, the swarm builds some direct roads and, at the logistical stage, the transporting network of the swarm gets loops (circles) and it permanently changes. So, at the first stage, the behaviour can be expressed by some linear *p*-adic valued strings. At the second stage, it is expressed by non-linear modifications of *p*-adic valued strings. The second stage cannot be described by conventional algebraic tools; therefore, I have introduced the so-called non-linear group theory for describing both stages in the swarm propagation.

Mathematical models of interacting populations are often constructed as systems of differential equations, which describe how populations change with time. Below we study such a model connected to the nonlinear dynamics of a system of populations in presence of time delay. The consequence of the presence of the time delay is that the nonlinear dynamics of the studied system become more rich, eg, new orbits in the phase space of the system arise, which are dependent on the time-delay parameters. In more detail, we introduce a time delay and generalize the model system of differential equations for the interaction of 3 populations based on generalized Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations. Then we solve numerically the system with and without time delay. We use a modification of the method of Adams for the numerical solution of the system of model equations with time delay. By appropriate selection of the parameters and initial conditions, we show the impact of the delay time on the dynamics of the studied population system.

]]>We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials
, by performing double summations involving the products
and
. They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level *m*. The second Mehler's formula generalizes the one appearing as a particular case of the so-called Kibble-Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.

In this paper, we develop a high-order finite difference scheme for the solution of a time fractional partial integro-differential equation with a weakly singular kernel. The fractional derivative is used in the Riemann-Liouville sense. We prove the unconditional stability and convergence of scheme using energy method and show that the convergence order is . We provide some numerical experiments to confirm the efficiency of suggested scheme. The results of numerical experiments are compared with analytical solutions to show the efficiency of proposed scheme. It is illustrated that the numerical results are in good agreement with theoretical ones.

]]>We use a particle method to study a Vlasov-type equation with local alignment, which was proposed by Sebastien Motsch and Eitan Tadmor [*J. Statist. Phys.*, 141(2011), pp. 923-947]. For *N*-particle system, we study the unconditional flocking behavior for a weighted Motsch-Tadmor model and a model with a “tail”. When *N* goes to infinity, global existence and stability (hence uniqueness) of measure valued solutions to the kinetic equation of this model are obtained. We also prove that measure valued solutions converge to a flock. The main tool we use in this paper is Monge-Kantorovich-Rubinstein distance.

The goal in the paper is to advertise Dunkl extension of Szász beta-type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second-order modulus of continuity, the Lipschitz class functions, Peetre's *K*-functional, and modulus of weighted continuity by Dunkl generalization of Szász beta-type operators.

In this paper, we study persistent piecewise linear multidimensional random motions. Their velocities, switching at Poisson times, are uniformly distributed on a sphere. The changes of direction are accompanied with subsequent jumps of random length and of uniformly distributed orientation.

In this paper, we obtain some useful properties and formulae of distributions of these processes. In particular, we get these distributions in the cases of jumps with Gaussian and exponential distributions of jump magnitudes.

We consider the stabilization of the electromagneto-elastic system with Wentzell conditions in a bounded domain of . Using the multiplier method we prove an exponential stability result under some geometric condition. Previous results of this type have recently been obtained for the coupled Maxwell/wave system with Wentzell conditions by H. Kasri and A. Heminna (Evol Equ and Control Theo 5: 235-250, 2016)

]]>In the spectral Petrov-Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the nonorthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein-spectral Petrov-Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.

]]>In this paper, a leader-following consensus of discrete-time multi-agent systems with nonlinear intrinsic dynamics is investigated. We propose and prove conditions ensuring a leader-following consensus. Numerical examples are given to illustrate our results.

]]>In this paper, we study the following fractional Schrödinger equations:

- (1)

where (−△)^{α} is the fractional Laplacian operator with
, 0≤*s*≤2*α*, *λ*>0, *κ* and *β* are real parameter.
is the critical Sobolev exponent. We prove a fractional Sobolev-Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.

We investigate the asymptotic periodicity, *L*^{p}-boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.

In the present article, we study the temperature effects on two-phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2-phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.

]]>In this paper, we introduce and investigate the functions of (*μ*,*ν*)-pseudo *S*-asymptotically *ω*-periodic of class *r*(class infinity). We systematically explore the properties of these functions in Banach space including composition theorems. As applications, we establish some sufficient criteria for (*μ*,*ν*)-pseudo *S*-asymptotic *ω*-periodicity of (nonautonomous) semilinear integro-differential equations with finite or infinite delay. Finally, some interesting examples are presented to illustrate the main findings.

In this study, we consider the stability of tumor model by using the standard differential geometric method that is known as Kosambi-Cartan-Chern (KCC) theory or Jacobi stability analysis. In the KCC theory, we describe the time evolution of tumor model in geometric terms. We obtain nonlinear connection, Berwald connection and KCC invariants. The second KCC invariant gives the Jacobi stability properties of tumor model. We found that the equilibrium points are Jacobi unstable for positive coordinates. We also discussed the time evolution of components of deviation tensor and the behavior of deviation vector near the equilibrium points.

]]>The model of pollution for a system of 3 lakes interconnected by channels is extended using Caputo-Hadamard fractional derivatives of different orders *α*_{i}∈(0,1), *i*=1,2,3. A numerical approach based on ln-shifted Legendre polynomials is proposed to solve the considered fractional model. No discretization is needed in our approach. Some numerical experiments are provided to illustrate the presented method.

Shannon and Zipf-Mandelbrot entropies have many applications in many applied sciences, for example, in information theory, biology and economics, etc. In this paper, we consider two refinements of the well-know Jensen inequality and obtain different bounds for Shannon and Zipf-Mandelbrot entropies. First of all, we use some convex functions and manipulate the weights and domain of the functions and deduce results for Shannon entropy. We also discuss their particular cases. By using Zipf-Mandelbrot laws for different parameters in Shannon entropies results, we obtain bounds for Zipf-Mandelbrot entropy. The idea used in this paper for obtaining the results may stimulate further research in this area, particularly for Zipf-Mandelbrot entropy.

]]>In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on the technique of constructing a proper approximating strong solution system and conducting a limiting type of argument to pass on stability of strong solutions to mild ones. In particular, the stochastic stability under the fixed-index sequence monotonicity condition and under the average dwell-time switching are considered.

]]>In this paper, we introduce a *q*-analog of 1-dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of
. Finally, we give some examples.

We study the well-posedness and dynamic behavior for the KdV-Burgers equation with a force
on **R**. We establish *L*^{p}−*L*^{q} estimates of the evolution
, as an application we obtain the local well-posedness. Then the global well-posedness follows from a uniform estimate for solutions as *t*goes to infinity. Next, we prove the asymptotical regularity of solutions in space
and
by the smoothing effect of
. The regularity and the asymptotical compactness in *L*^{2} yields the asymptotical compactness in
by an interpolation arguement. Finally, we conclude the existence of an
globalattractor.

This paper deals with the attraction-repulsion chemotaxis system with nonlinear diffusion *u*_{t}=∇·(*D*(*u*)∇*u*)−∇·(*u**χ*(*v*)∇*v*)+∇·(*u*^{γ}*ξ*(*w*)∇*w*), *τ*_{1}*v*_{t}=Δ*v*−*α*_{1}*v*+*β*_{1}*u*, *τ*_{2}*w*_{t}=Δ*w*−*α*_{2}*w*+*β*_{2}*u*, subject to the homogenous Neumann boundary conditions, in a smooth bounded domain
, where the coefficients *α*_{i}, *β*_{i}, and *τ*_{i}∈{0,1}(*i*=1,2) are positive. The function *D* fulfills *D*(*u*)⩾*C*_{D}*u*^{m−1} for all *u*>0 with certain *C*_{D}>0 and *m*>1. For the parabolic-elliptic-elliptic case in the sense that *τ*_{1}=*τ*_{2}=0 and *γ*=1, we obtain that for any
and all sufficiently smooth initial data *u*_{0}, the model possesses at least one global weak solution under suitable conditions on the functions *χ* and *ξ*. Under the assumption
, it is also proved that for the parabolic-parabolic-elliptic case in the sense that *τ*_{1}=1, *τ*_{2}=0, and *γ*⩾2, the system possesses at least one global weak solution under different assumptions on the functions *χ* and *ξ*.

In this paper, we present a new algorithm for solving the split common null point and common fixed point problem, to find a point that belongs to the common element of common zero points of an infinite family of maximal monotone operators and common fixed points of an infinite family of demicontractive mappings such that its image under a linear transformation belongs to the common zero points of another infinite family of maximal monotone operators and its image under another linear transformation belongs to the common fixed point of another infinite family of demicontractive mappings in the image space. We establish strong convergence for the algorithm to find a unique solution of the variational inequality, which is the optimality condition for the minimization problem. As special cases, we shall use our results to study the split equilibrium problems and the split optimization problems.

]]>In this paper, our main purpose is to establish the existence results of positive solutions for a *p*−*q*-Laplacian system involving concave-convex nonlinearities:

where Ω is a bounded domain in *R*^{N}, *λ*,*θ*>0 and 1<*r*<*q*<*p*<*N*. We assume 1<*α*,*β* and
is the critical Sobolev exponent and △_{s}·=div(|∇·|^{s−2}∇·) is the s-Laplacian operator. The main results are obtained by variational methods.

We define an abstract setting to treat essential spectra of unbounded coupled operator matrix. We prove a well-posedness result and develop a spectral theory which also allows us to prove an amelioration to many earlier works. We point out that a concrete example from integro-differential equation fit into this abstract framework involving a general class of regular operator in *L*_{1} spaces.

No abstract is available for this article.

]]>In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We prove that the existence of peakon as weak traveling wave solution and as global weak solution for the nonlinear surface wind waves equation, so as to correct the assertion that there exists no peakon solution for such an equation in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We propose a new finite volume scheme for 2D anisotropic diffusion problems on general unstructured meshes. The main feature lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell-centered primary unknowns and cell edge-based auxiliary unknowns. The auxiliary unknowns are interpolated by the multipoint flux approximation technique, which reduces the scheme to a completely cell-centered one. The derivation of the scheme satisfies the linearity-preserving criterion that requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is then called as a cell edge-based linearity-preserving scheme. The optimal convergence rates are numerically obtained on unstructured grids in case that the diffusion tensor is taken to be anisotropic and/or discontinuous. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The purpose of this paper is to introduce a family of *q*-Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well-known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's *K*-functional, and the second-order modulus of continuity. Furthermore, we obtain the approximation results for bivariate *q*-Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.

This paper considers the two-dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non-selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non-selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, we establish sufficient conditions for the regularity of solutions of 3D MHD equations in the framework of the anisotropic Lebesgue spaces. In particular, we obtain the anisotropic regularity criterion via partial derivatives, and it is a generalization of the some previous results. Besides, the anisotropic integrability regularity criteria in terms of the magnetic field and the third component of the velocity field are also investigated. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We discuss bound and anti-bound states for 2×2 matrix Schrödinger operator. We analyze the Fredholm determinant for Hamiltonians that can be represented in a multi-channel framework. Our analysis covers the whole and the half-line problems. We obtain some results on counting anti-bound states between successive bound states. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this work, we study the approximation of traveling wave solutions propagated at minumum speeds *c*_{0}(*h*) of the delayed Nicholson's blowflies equation:

In order to do that, we construct a subsolution and a super solution to (∗). Also, through that construction, an alternative proof of the existence of traveling waves moving at minimum speed is given. Our basic hypothesis is that *p*/*δ*∈(1,*e*] and then, the monostability of the reaction term. Copyright © 2017 John Wiley & Sons, Ltd.

In this article, we discuss the integral representation of quaternionic harmonic functions in the half space with the general boundary condition. Next, we derive a lower bound from an upper one for quaternionic harmonic functions. These results generalize some of the classic results from the case of plane to the case of noncommutative quaterninionic half space. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Global exponential stability for coupled neutral stochastic delayed systems on networks (CNSDSNs) is investigated in this paper. By means of combining the Razumikhin method with graph theory, some sufficient conditions that can be verified easily are derived to ensure the global exponential stability for CNSDSNs. Finally, a specific model of CNSDSNs is discussed, and numerical test manifests the effectiveness of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0,*T*], for a class of higher-order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley & Sons, Ltd.

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three-dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in *B**M**O* norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO-type estimate for the Lam
system and a variant of the Brezis-Waigner's inequality play a critical role in the proof. Copyright © 2017 John Wiley & Sons, Ltd.

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω^{+}) is embedded in an unbounded fluid domain (Ω^{−}). The corresponding physical process is described by the boundary-transmission problem for second-order partial differential equations. In particular, in the bounded domain Ω^{+}, we have a 4×4 dimensional matrix strongly elliptic second-order partial differential equation, while in the unbounded complement domain Ω^{−}, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.

We consider a semi-discrete in time Crank–Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation *u*_{t}−*i*(−Δ)^{α}*u*+*i*|*u*|^{2}*u*+*γ**u*=*f* for
considered in the the whole space
. We prove that such semi-discrete equation provides a discrete infinite-dimensional dynamical system in
that possesses a global attractor in
. We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension. Copyright © 2017 John Wiley & Sons, Ltd.

A five-dimensional ordinary differential equation model describing the transmission of *Toxoplamosis gondii* disease between human and cat populations is studied in this paper. Self-diffusion modeling the spatial dynamics of the *T. gondii* disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans-critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease-free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.

In this work, we integrate both density-dependent diffusion process and Beddington–DeAngelis functional response into virus infection models to consider their combined effects on viral infection and its control. We perform global analysis by constructing Lyapunov functions and prove that the system is well posed. We investigated the viral dynamics for scenarios of single-strain and multi-strain viruses and find that, for the multi-strain model, if the basic reproduction number for all viral strains is greater than 1, then each strain persists in the host. Our investigation indicates that treating a patient using only a single type of therapy may cause competitive exclusion, which is disadvantageous to the patient's health. For patients infected with several viral strains, the combination of several therapies is a better choice. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider a semilinear wave equation with nonlinear damping in the whole space . Local-in-time existence and uniqueness results are obtained in the class of Bessel-potential spaces . Copyright © 2017 John Wiley & Sons, Ltd.

]]>We study the initial boundary value problem for the one-dimensional Kuramoto–Sivashinsky equation posed in a half line
with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line
, the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well-posed in Sobolev space
for any *s*>−2. Copyright © 2017 John Wiley & Sons, Ltd.

The key purpose of the present work is to constitute a numerical scheme based on *q*-homotopy analysis transform method to examine the fractional model of regularized long-wave equation. The regularized long-wave equation explains the shallow water waves and ion acoustic waves in plasma. The proposed technique is a mixture of *q*-homotopy analysis method, Laplace transform, and homotopy polynomials. The convergence analysis of the suggested scheme is verified. The scheme provides
and *n*-curves, which show that the range convergence of series solution is not a local point effects and elucidate that it is superior to homotopy analysis method and other analytical approaches. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, multi-switching combination–combination synchronization scheme has been investigated between a class of four non-identical fractional-order chaotic systems. The fractional-order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional-order Lü and Rössler chaotic systems. In multi-switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional-order chaotic systems, the multi-switching combination–combination synchronization of four fractional-order non-identical systems has been investigated. For the synchronization of four non-identical fractional-order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper presents an operatorial model based on fermionic operators for the description of the dynamics of political parties affected by turncoat-like behaviors. By observing the political landscape in place in Italy over the last years, appropriate macro-groups have been identified on the basis of the behavior of politicians in terms of disloyal attitude as well as openness towards accepting chameleons from other parties. Once introduced, a time-dependent number-like operator for each physical observable relevant for the description of the political environment, the analysis of the party system dynamics is carried out by combining the action of a quadratic Hamiltonian operator with certain rules acting periodically on the system in such a way that the parameters entering the model are repeatedly changed so as to express a sort of dependence of them upon the variations of the mean values of the observables. Copyright © 2017 John Wiley & Sons, Ltd.

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