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.

In this paper, we consider the multiplicity of solutions of the *p*-Laplacian problems involving supercritical Sobolev growth and exponential growth in
via Ricceri principle. By means of the truncation combining with Moser iteration, we can extend the result about the subcritical growth to the supercritical and exponential growth.

In this paper, we give some new properties of the presented asynchronous algorithms of theta scheme combined with finite elements methods (App. Math. Comp., 217 (2011), 6443-6450) for an evolutionary implicit 2-sided obstacle problem to prove the existence and uniqueness of the discrete solution. Furthermore, an error estimate on the uniform norm is given.

]]>In this paper, we study the following Kirchhoff-type equation with critical or supercritical growth

where *a*>0, *b*>0, *λ*>0, *p*≥6 and *f* is a continuous superlinear but subcritical nonlinearity. When *V* and *f* are asymptotically periodic in *x*, we prove that the equation has a ground state solution for small *λ*>0 by Nehari method. Moreover, we regard *b* as a parameter and obtain a convergence property of the ground state solution as *b*↘0. Our main contribution is related to the fact that we are able to deal with the case *p*>6.

In this study, first, a formula for regularized sums of eigenvalues of a Sturm-Liouville problem with retarded argument at 2 points of discontinuity which contains a spectral parameter in the boundary conditions is obtained. After that, oscillation properties of the related problem is investigated. Finally, under the condition that a subset of nodal points is dense in definition set, the potential function is determined.

]]>Let Γ be a simple closed curve that bounds the finite domain *D*, *z*=*z*(*ζ*)=*z*(*r**e*^{iϑ}) be the conformal mapping of the circle {*ζ*:|*ζ*|<1} onto the domain *D*. Furthermore, let the functions *A*(*z*), *B*(*z*) be given on *D* and *U*^{s,2}(*A*;*B*;*D*) be the set of regular solutions of the equation
.

We call the Smirnov class *E*^{p(t)}(*A*;*B*;*D*) the set of those generalized functions *W*in *D* for which

where *p*(*t*) is a positive measurable function on Γ.

We consider the Riemann-Hilbert problem: Define a function *W*(*z*) from the class *E*^{p(t)}(*A*;*B*;*D*) for which the equality,

is fulfilled almost everywhere on Γ.

It is assumed that Γ is a piecewise-smooth curve without external peaks;
, *p* is Log Hölder continuous and

the function
belongs to the class *A*(*p*(*t*);Γ), which is the generalization of the well-known Simonenko class *A*(*p*;Γ), where *p* is constant.

The solvability conditions are established, and solutions are constructed.

This paper mainly focus on the exponential stabilization problem of coupled systems on networks with mixed time-varying delays. Periodically intermittent control is used to control the coupled systems on networks with mixed time-varying delays. Moreover, based on the graph theory and Lyapunov method, two different kinds of stabilization criteria are derived, which are in the form of Lyapunov-type theorem and coefficients-type criterion, respectively. These laws reveal that the stability has a close relationship with the topology structure of the networks. In addition, as a subsequent result, a decision theorem is also presented. It is straightforward to show the stability of original system can be determined by that of modified system with added absolute value into the coupling weighted-value matrix. Finally, the feasibility and validity of the obtained results are demonstrated by several numerical simulation figures.

]]>We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3-space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.

]]>In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed-point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.

]]>In this paper, we consider a nonautonomous impulsive plankton model with mutual help of preys. Sufficient conditions ensuring permanence and global attractivity of the model are established by the relation between solutions of impulsive system and corresponding nonimpulsive system. Also, we propose the conditions for which the species of system are driven to extinction. Numerical simulations are given to verify the main results.

]]>In this short paper, the initial value problem for the Navier-Stokes equations with the Coriolis force is investigated. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. We prove that existence of uniform global large solutions to the Navier-Stokes equations with the Coriolis force for a class of special initial data. The results obtained in this paper are different from the previous 2 types of results.

]]>In this paper, we prove the large-time behavior, as time tends to infinity, of solutions in and for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier-Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.

]]>In this paper, we investigate the existence and multiplicity of harmonic and subharmonic solutions for second-order quasilinear equation

where
, *g* satisfies the superlinear condition at infinity. We prove that the given equation possesses harmonic and subharmonic solutions by using the phase-plane analysis methods and a generalized version of the Poincaré-Birkhoff twist theorem.

In this paper, we provide a detailed convergence analysis for fully discrete second-order (in both time and space) numerical schemes for nonlocal Allen-Cahn and nonlocal Cahn-Hilliard equations. The unconditional unique solvability and energy stability ensures *ℓ*^{4} stability. The convergence analysis for the nonlocal Allen-Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn-Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an *H*^{−1} inner-product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori
bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an *O*(*s*^{3}+*h*^{4}) convergence in
norm, in which *s* and *h* denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint *s*≤*C**h*. Here, we also prove convergence of the scheme in the maximum norm under the same constraint.

In this paper, we consider a class of asymptotically linear second-order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.

]]>We consider 2 transmission problems. The first problem has 2 damping mechanisms acting in the same part of the body, one of frictional type and other of Kelvin-Voigt type. In this case, we show that, even though it has too much dissipation, the semigroup is not exponentially stable. The second problem also has those damping terms but they act in complementary parts of the body. For this case, we show that the semigroup is exponentially stable and it is not analytic.

]]>A two-dimensional sparse-data tomographic problem is studied. The target is assumed to be a homogeneous object bounded by a smooth curve. A nonuniform rational basis splines (NURBS) curve is used as a computational representation of the boundary. This approach conveniently provides the result in a format readily compatible with computer-aided design software. However, the linear tomography task becomes a nonlinear inverse problem because of the NURBS-based parameterization. Therefore, Bayesian inversion with Markov chain Monte Carlo sampling is used for calculating an estimate of the NURBS control points. The reconstruction method is tested with both simulated data and measured X-ray projection data. The proposed method recovers the shape and the attenuation coefficient significantly better than the baseline algorithm (optimally thresholded total variation regularization), but at the cost of heavier computation.

]]>A well-known result on pathwise uniqueness of the solution of stochastic differential equations in is the Yamada-Watanabe theorem. We have extended this result by replacing the Lipschitz assumption on the drift coefficient by much weaker assumption of semi-monotonicity.

We study the shape derivative of the strongly singular volume integral operator that describes time-harmonic electromagnetic scattering from homogeneous medium. We show the existence and a representation of the derivative, and we deduce a characterization of the shape derivative of the solution to the diffraction problem as a solution to a volume integral equation of the second kind.

]]>In this paper, we introduce fractional order into an ecoepidemiological model, where predator consumes disproportionately large number of infected preys following type 2 response function. We prove different mathematical results like existence, uniqueness, nonnegativity, and boundedness of the solutions of fractional order system. We also prove the local and global stability of different equilibrium points of the system. The results are illustrated with several examples.

]]>In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrödinger equations:

where the nonlinearities *f*_{1}(*x*,*s*) and *f*_{2}(*x*,*s*) are superlinear at infinity and have exponential critical growth of the Trudinger-Moser type. The potentials *V*_{1}(*x*) and *V*_{2}(*x*) are nonnegative and satisfy a condition involving the coupling term *λ*(*x*), namely, *λ*(*x*)^{2}<*δ*^{2}*V*_{1}(*x*)*V*_{2}(*x*) for some 0<*δ*<1. For this purpose, we use the minimization technique over the Nehari manifold and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap argument and *L*^{q}-estimates, we get regularity and asymptotic behavior.

In this paper, we introduced geometric algebra to develop a new multisource multisink optimal evacuation route planning method. A dynamically updatable data structure and a matrix-based greedy searching algorithm were developed to support the dynamic evacuation route searching for multiple evacuees. Unlike most existing methods, which iteratively search the optimal path for each evacuee, our method can search all the possible evacuation routes synchronously. The dynamic updating of the network topography, weights, and constraints during the route searching is direct and flexible, thus can support the evacuation in dynamic cases. The method is demonstrated and tested by an evacuation case in the city of Changzhou, China. The simulation experiments suggest that the method can well support the dynamic route searching in large scales with dynamic weight changes.

]]>In this paper, we investigate the dynamics of a time-delay ratio-dependent predator-prey model with stage structure for the predator. This predator-prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.

]]>We consider, in this paper, the following nonlinear equation with variable exponents:

where *a*,*b*>0 are constants and the exponents of nonlinearity *m*,*p*, and *r* are given functions. We prove a finite-time blow-up result for the solutions with negative initial energy and for certain solutions with positive energy.

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configuration manifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group
and the space of translations
, where *n* equals the dimension of the “physical space.” In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua.

We introduce a new type of linear and cyclic codes. These codes are defined over a direct product of 2 finite chain rings. The definition of these codes as certain submodules of the direct product of copies of these rings is given, and the cyclic property is defined. Cyclic codes can be seen as submodules of the direct product of polynomial rings. Generator matrices for linear codes and generator polynomials for cyclic codes are determined. Further, we study the concept of duality.

]]>Using the existence of integrable bi–almost-periodic Green functions of linear homogeneous differential equations and the contraction fixed point, we are able to prove the existence of almost and pseudo–almost-periodic mild solutions under quite general hypotheses for the differential equation with constant delay

in a Banach space *X*, where *τ*>0 is a fixed constant. The results extend the corresponding ones in the case of exponential dichotomy. Some examples illustrate the importance of the concepts.

Inversion of the scalar and vector ray transforms is performed in domain , ie, with the presence of an obstacle or singularity in the origin. Initially, the ray transforms of the basis functions for the scalar and vector fields are evaluated in an analytical form, and next, the inversion procedure is reduced to a linear system of equations by the use of the least squares method.

]]>In this work, we obtain the fundamental solution (FS) of the multidimensional time-fractional telegraph Dirac operator where the 2 time-fractional derivatives of orders *α*∈]0,1] and *β*∈]1,2] are in the Caputo sense. Explicit integral and series representation of the FS are obtained for any dimension. We present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameters *α* and *β*. Finally, using the FS, we study some Poisson and Cauchy problems.

This paper presents 2 new classes of the Bessel functions on a compact domain [0,*T*] as generalized-tempered Bessel functions of the first- and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.

No abstract is available for this article.

]]>The existence of one non-trivial solution for a second-order impulsive differential inclusion is established. More precisely, a recent critical point result is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non-trivial anti-periodic solution. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the variable-coefficient fractional diffusion equations with two-sided fractional derivative. By introducing an intermediate variable, we propose a mixed-type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over
. On the basis of the formulation, we develop a mixed-type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz-like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from *O*(*N*^{2}) to *O*(*N*) and the computing cost from *O*(*N*^{3}) to *O*(*N*log*N*). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.

We consider the initial-boundary value problem for a model of motion of aqueous polymer solutions in a bounded three-dimensional domain subject to the Navier slip boundary condition. We construct a global (in time) weak solution to this problem. Moreover, we establish some uniqueness results, assuming additional regularity for weak solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, by using the fractional calculus, measure of noncompactness, and the Mönch's fixed point theorem, we investigate the controllability results for fractional neutral integrodifferential equations with nonlocal conditions in Banach spaces. In the end, we give an example to illustrate the applications of the abstract conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we prove the existence of ground state sign-changing solutions for the following class of elliptic equation:

where
, and *K*(*x*) are positive continuous functions. Firstly, we obtain one ground state sign-changing solution *u*_{b} by using some new analytical skills and non-Nehari manifold method. Furthermore, the energy of *u*_{b} is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of *u*_{b} as the parameter *b*↘0. Copyright © 2017 John Wiley & Sons, Ltd.

Optical vortices as topological objects exist ubiquitously in nature. In this paper, we use the principle of variational method and mountain pass lemma to develop some existence theorems for the stationary vortex wave solution of a coupled nonlinear Schrödinger equations, which describe the possibility of effective waveguiding of a weak probe beam via the cross-phase modulation-type interaction. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Additionally, as demanded by beam confinement, we prove the exponential decay of the soliton amplitude at infinity. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A new mathematical model included an exposed compartment is established in consideration of incubation period of schistosoma in human body. The basic reproduction number is calculated to illustrate the threshold of disease outbreak. The existence of the disease free equilibrium and the endemic equilibrium are proved. Studies about stability behaviors of the model are exploited. Moreover, control measure assessments are investigated in order to seek out effective control interventions for anti-schistosomiasis. Then, the corresponding optimal control problem according to the model is presented and solved. Theoretical analyses and numerical simulations induce several prevention and control strategies for anti-schistosomiasis. At last, a discussion is provided about our results and further work. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider a class of singular quasilinear Schrödinger equations of the form

where
are given functions, *N*⩾3,*λ* is a positive constant,
. By using variational methods together with concentration–compactness principle, we prove the existence of positive solutions of the aforementioned equations under suitable conditions on *V*(*x*) and *K*(*x*). Copyright © 2017 John Wiley & Sons, Ltd.

A novel second-order two-scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non-periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non-periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we consider an abstract wave equation in the presence of memory. The viscoelastic kernel *g*(*t*) is subject to a general assumption
, where the function *H*(·)∈*C*^{1}(*R*^{+}) is positive, increasing and convex with *H*(0)=0. We give the decay result as a solution to a given nonlinear dissipative ODE governed by the function *H*(*s*). Copyright © 2017 John Wiley & Sons, Ltd.

We analyze some fourth-order partial differential equations that model the ‘propagation of hexagonal patterns’ and the ‘microphase separation of di-block copolymers’. The underlying invariance properties and conservation laws of the models and related partial differential equations are studied. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The oscillation of solutions of the *n*th-order delay differential equation

was studied in [S. R. Grace and A. Zafer, Math. Meth. Appl. Sci. 2016, 39 1150–1158] when *n* is even and the *n* odd case has been referred to as an interesting open problem. In the present work, our primary aim is to address this situation. Our method of the proof that is quite different from the aforementioned study is essentially new. We introduce *V*_{n−1}-type solutions and use comparisons with first-order oscillatory and second-order nonoscillatory equations. Examples are given to illustrate the main results. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the inviscid limit for the periodic solutions to Navier–Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier–Stokes equation is independent of viscosity, and that the solutions of the Navier–Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover, the convergence rate in Gevrey class is presented. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Quasi-periodic piecewise analytic solutions, without poles, are found for the local antiplane-strain problems. Such problems arise from applying the asymptotic homogenization method to an elastic problem in a parallel fiber-reinforced periodic composite that presents an imperfect contact of spring type between the fiber and the matrix. Our methodology consists of rewriting the contact conditions in a complex appropriate form that allow us to use the elliptic integrals of Cauchy type. Several general conditions are assumed including that the fibers are disposed of arbitrary manner in the unit cell, that all fibers present imperfect contact with different constants of imperfection, and that their cross section is smooth closed arbitrary curves. Finally, we obtain a family of piecewise analytic solutions for the local antiplane-strain problems that depend of a real parameter. When we vary this parameter, it is possible to improve classic bounds for the effective coefficients. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The paper deals with the asymptotic formulation and justification of a mechanical model for a dynamic piezoelastic shallow shell in Cartesian coordinates. Starting from the three-dimensional dynamic piezoelastic problem and by an asymptotic approach, the authors study the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. In order to obtain a nontrivial limit problem by asymptotic analysis, we need different scalings on the mass density. The authors show that the transverse mechanical displacement field coupled with the in-plane components solves an problem with new piezoelectric characteristics and also investigate the very popular case of cubic crystals and show that, for two-dimensional shallow shells, the coupling piezoelectric effect disappears. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper is concerned with the Cauchy problem of the two-dimensional Euler–Boussinesq system with stratification effects. We obtain the global existence of a unique solution to this system without assumptions of small initial data in Sobolev spaces. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical simulations to illustrate the analytical results. In addition, as a by-product of our analysis, we obtain an explicit formula for the inverse of a Wronskian matrix that is of independent interest in linear algebra and differential equations theory. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper addresses the analysis of a time noise-driven Allen–Cahn equation modelling the evolution of damage in continuum media in the presence of stochastic dynamics. The nonlinear character of the equation is mainly due to a multivoque maximal monotone operator representing a constraint on the damage variable, which is forced to take physically admissible values. By a Yosida approximation and a time-discretization procedure, we prove a result of global-in-time existence and uniqueness of the solution to the stochastic problem. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we will firstly extend the results about Jiu, Wang, and Xin (JDE, 2015, 259, 2981–3003). We prove that any smooth solution of compressible fluid will blow up without any restriction about the specific heat ratio *γ*. Then we prove the blow-up of smooth solution of compressible Navier–Stokes equations in half space with Navier-slip boundary. The main ideal is constructing the differential inequality. Copyright © 2017 John Wiley & Sons, Ltd.

This paper deals with asymptotic behavior for blow-up solutions to time-weighted reaction–diffusion equations *u*_{t}=Δ*u*+*e*^{αt}*v*^{p} and *v*_{t}=Δ*v*+*e*^{βt}*u*^{q}, subject to homogeneous Dirichlet boundary. The time-weighted blow-up rates are defined and obtained by ways of the scaling or auxiliary-function methods for all *α*,
. Aiding by key inequalities between components of solutions, we give lower pointwise blow-up profiles for single-point blow-up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow-up rates and new blow-up versus global existence criteria are obtained. Time-weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow-up rates, but they do not limit the order of time-weighted blow-up rates and pointwise profile near blow-up time. Copyright © 2017 John Wiley & Sons, Ltd.

The construction of modified two-step hybrid methods for the numerical solution of second-order initial value problems with periodic or oscillatory behavior is considered. The coefficients of the new methods depend on the frequency of each problem so that the harmonic oscillator is integrated exactly. Numerical experiments indicate that the new methods are more efficient than existing methods with constant or variable coefficients. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider second-order ordinary differential equations with discontinuous right-hand side. We analyze the concept of solution of this kind of equations and determine analytical conditions that are satisfied by typical solutions. Moreover, the existence and uniqueness of solutions and sliding solutions are studied. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the dynamics of a Nicholson's blowflies equation with state-dependent delay. For the constant delay, it is known that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and global existence of periodic solutions has been established. Here, we consider the state-dependent delay instead of the constant delay and generalize the results on the existence of slowly oscillating periodic solutions under a set of mild conditions on the parameters and the delay function. In particular, when the positive equilibrium gets unstable, a global unstable manifold connects the positive equilibrium to a slowly oscillating periodic orbit. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to help in designing experiments to develop novel therapeutic strategies against cancer. In this paper, by using theory of functional and ordinary differential equations, we study the existence and stability of nonlinear growth kinetics of breast cancer stem cells. First, we provide a sufficient condition for the existence and uniqueness of the solution for nonlinear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a nontrivial steady-state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions that allow for this criteria to be satisfied. Next, we apply these theorems to a special case of the system of functional differential equations that has been used to model nonlinear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two-dimensional sphere close to its poles. This equation is the counterpart of the well-studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is *C*^{1} smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, by applying nonstandard finite difference scheme, we propose a discrete multigroup Susceptible-Infective-Removed (SIR) model with nonlinear incidence rate. Using Lyapunov functions, it is shown that the global dynamics of this model are completely determined by the basic reproduction number . If , then the disease-free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium and it is globally asymptotically stable. Example and numerical simulations are presented to illustrate the results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We investigate the correctness of the initial boundary value problem of longitudinal impact on a piecewise-homogeneous semi-infinite bar consisting of a semi-infinite elastic part and finite length visco-elastic part whose hereditary properties are described by linear integral relations with an arbitrary difference kernel. Introducing nonstationary regularization in boundary conditions and in the contact conditions, the well-posedness of the considered problem is proved. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We investigate the qualitative behavior of a host-parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole-placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.

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