We deal with the generalized Emden–Fowler equation *f*^{″}(*x*) + *g*(*x*)*f*^{−θ}(*x*) = 0, where belongs to *L*^{p}((*a*,*b*)). We obtain a priori estimates for the solutions, as well as information about their asymptotic behavior near boundary points. As a tool, we derive new nonlinear variants of first-order and second-order Poincaré inequalities, which are based on strongly nonlinear multiplicative inequalities obtained recently by first author and Peszek. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we employed the bifurcation method and qualitative theory of dynamical systems to study the peakons and periodic cusp waves of the generalization of the Camassa-Holm equation, which may be viewed as an extension of peaked waves of the same equation. Through the bifurcation phase portraits of traveling wave system, we obtained the explicit peakons and periodic cusp wave solutions. Further, we exploited the numerical simulation to confirmthe qualitative analysis, and indeed, the simulation results are in accord with the qualitative analysis. Compared with the previous works, several new nonlinear wave solutions are obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the existence and multiplicity of solutions to the following second-order impulsive Hamiltonian systems:

where is a continuousmap form the interval [0, *T*] to the set of N-order symmetric matrices. Our methods are based on critical point theory for nondifferentiable functionals. Copyright © 2014 John Wiley & Sons, Ltd.

The nonstationary Stokes system with slip boundary conditions is considered in a bounded domain . We prove the existence and uniqueness of solutions to the problem in anisotropic Sobolev spaces . Thanks to the slip boundary conditions, the Stokes problem is transformed to the Poisson and the heat equation. In this way, difficult calculations that must be performed in considerations of boundary value problems for the Stokes system are avoided. This approach does not work for the Dirichlet and the Neumann boundary conditions. Because solvability of the Poisson and the heat equation is carried out by the regularizer technique, we have that *σ* > 3,*α* > 0. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of *n*-fold Darboux transformation. From known solution *Q*, the determinant representation of *n*-th new solutions of *Q*^{[n]} are obtained by the *n*-fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third-order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is devoted to the investigation of the global dynamics of a SEIR model with information dependent vaccination. The basic reproduction number is derived for the model, and it is shown that gives the threshold dynamics in the sense that the disease-free equilibrium is globally asymptotically stable and the disease dies out if , while there exists at least one positive periodic solution and the disease is uniformly persistent when . Further, we give the approximation formula of . This answers the concerns presented in [B. Buonomo, A. d'Onofrio, D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl. 404 (2013) 385–398]. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Dedicated to Professor Dr. Martin Costabel on the occasion of his 65th birthday.Martin Costabel showed in his celebrated paper on boundary integral operators on Lipschitz domains also that the trace theorem for functions in *H*^{s}(Ω) holds for and not only for . Here, we show that his approach can be extended to *C*^{k − 1,1}-domains with . Copyright © 2014 John Wiley & Sons, Ltd.

The weighted *L*^{r}-asymptotic behavior of the strong solution and its first-order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half-space. Further, the *L*^{∞}-decay rates of the second-order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.

We consider a weakly dissipative modified two-component Dullin–Gottwald–Holm system. The existence of global weak solutions to the system is established. We first give the well-posedness result of viscous approximate problem and obtain the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the system. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In the present study, we propose and analyze a predator–prey system with disease in the predator population. To understand the role of cannibalism, we modify the model considering predator population is of cannibalistic type. Local and global stability around the biologically feasible equilibria are studied. The conditions for the persistence of the system are worked out. We also analyze and compare the community structure of the model systems with the help of ecological and disease basic reproduction numbers. Finally, through numerical simulation, we observe that inclusion of cannibalism in predator population may control the disease transmission in the susceptible predator population. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper revisits and complement in different directions the classical work by W. T. Reid on symmetrizable completely continuous transformations in Hilbert spaces and a more recent paper by one of the authors. More precisely, we deal with spectral properties of *% non-compact* operators *G* on a complex Hilbert space *H* such that *SG* is self-adjoint where *S* is a (*not* necessarily injective) nonnegative operator. We study the isolated eigenvalues of *G* outside its essential spectral interval and provide variational characterization of them as well as stability estimates. We compare them also to spectral objects of *S**G*. Finally, we characterize the Schechter essential spectrum of strongly symmetrizable operators in terms singular Weyl sequences; in particular, we complement J. I. Nieto's paper on the essential spectrum of symmetrizable. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we present a collection of *a priori* estimates of the electromagnetic field scattered by a general bounded domain. The constitutive relations of the scatterer are in general anisotropic. Surface averages are investigated, and several results on the decay of these averages are presented. The norm of the exterior Calderón operator for a sphere is investigated and depicted as a function of the frequency. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we investigate the computability of the solution operator of the generalized KdV-Burgers equation with initial-boundary value problem. Here, the solution operator is a nonlinear map *H*^{3m − 1}(*R*^{+}) × *H*^{m}(0,*T*)*C*([0,*T*];*H*^{3m − 1}(*R*^{+})) from the initial-boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integer *m* ≥ 2, and computable real number *T* > 0. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the controllability and stabilizability problem for control systems described by a time-varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we constructed the split-step *θ *(SS*θ*)-method for stochastic age-dependent population equations. The main aim of this paper is to investigate the convergence of the SS *θ*-method for stochastic age-dependent population equations. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from the theory, and comparative analysis with Euler method is given, the results show the higher accuracy of the SS *θ*-method. Copyright © 2014 John Wiley & Sons, Ltd.

We systematically studied the optical properties (narrows peaks position) of the transmission spectra for microspheres coated by a multilayered stack. Three different sequences of spherical stack—periodic, quasiperiodic, and disordered—are studied by the transfer matrix approach. Dependence of the number of resonances in the transmission spectrum as function of number of layers in the stack is numerically investigated with details. It is shown that characteristic shape of the recursive return map forms well-defined ordered spectrum in the state space. The shape of such structures is different for different frequency ranges and various spherical quantum numbers. For quasiperiodic case, the latter leads to a specific signature of studied sequences and generates the self-similarity in the transmittance spectra. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non-oscillatory high-order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz-Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This work deals with a mathematical model of an age-cycle length structured cell population. Each cell is distinguished by its age and its cycle length. The cellular mitosis is mathematically described by non-compact boundary conditions. We prove then that this mathematical model is governed by a positive *C*_{0}-semigroup. Copyright © 2014 John Wiley & Sons, Ltd.

In the present review, we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm–Liouville equations with variable coefficients. We consider Sturm–Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet–Bloch solutions; quantum-mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov–Shabat system. In all these cases, we obtain a characteristic equation of the problem, which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples, which show that at present, the SPPS method is the easiest in the implementation, the most accurate, and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and leads to a powerful numerical method, but most important, it offers a different insight into the spectral and transmission problems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this study, we investigate the existence of mild solutions for a class of impulsive neutral stochastic integro-differential equations with infinite delays, using the Krasnoselskii–Schaefer type fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems:

- (
**FHS**)

where *α* ∈ (1 ∕ 2,1), , , and are symmetric and positive definite matrices for all , , and ∇ *W* is the gradient of *W* at *u*. The novelty of this paper is that, assuming *L* is coercive at infinity, and *W* is of subquadratic growth as | *u* | + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.

We give a detailed study of the infinite-energy solutions of the Cahn–Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this article, we study an explicit scheme for the solution of sine-Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo-spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth-order Runge–Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo-spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix–vector multiplications. Therefore, we propose a multidomain pseudo-spectral method for the numerical simulation of the sine-Gordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long-time numerical behavior for the sine-Gordon equation in one and two dimensions. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner, and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific nonstandard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials, and further terms are genuine nonsmooth functions generated by the piecewise constant zeroth order term of the operator. Copyright © 2013 John Wiley & Sons, Ltd.

In France, 90*%* of surface water suffer from antibiotic pollution that increases the number of antibiotic resistant bacteria. The first indicator of water quality is revealed by the fish quality. According to the Le Conseil Supérieur de la Pêche, only 15*%* of rivers in France are considered in good condition, whereas 22*%* are in very bad condition. The bacterial resistance to antibiotics is a public health problem as it affects humans through drinking water; the treatment of water is costly. Mathematical modeling may estimate and predict the quantity of bacteria in rivers. In this paper, we investigate properties of the mathematical model estimating the number of bacteria in a river presented by Lawrence, Mummert and Somerville. Global analysis of equilibria is presented, using a Lyapunov function. Moreover, the existence of positive periodic solutions is proven. Copyright © 2013 John Wiley & Sons, Ltd.

We obtain conditions for eradication and permanence of infection for a nonautonomous SIQR model with time-dependent parameters that are not assumed to be periodic. The incidence is given by functions of all compartments, and the threshold conditions are given by some numbers that play the role of the basic reproduction number. We obtain simple threshold conditions in the autonomous, asymptotically autonomous and periodic settings and show that our thresholds coincide with the ones already established. Additionally, we obtain threshold conditions for the general nonautonomous model with mass-action, standard and quarantine-adjusted incidence. Copyright © 2013 John Wiley & Sons, Ltd.

]]>We give a sharp upper bound on the vanishing order of solutions to the Schrödinger equation with electric and magnetic potentials on a compact smooth manifold. Our main result is that the vanishing order of nontrivial solutions to Δ*u* + *V* · ∇ *u* + *Wu* = 0 is everywhere less than . Our method is based on quantitative Carleman type inequalities, and it allows us to show the following uniform doubling inequality

which implies the desired result. Copyright © 2013 John Wiley & Sons, Ltd.

]]>Sobolev type nonlinear equations with time fractional derivatives are considered. Using the test function method, limiting exponents for nonexistence of solutions are found. Copyright © 2013 John Wiley & Sons, Ltd.

It is well known that the least-squares QR-factorization (LSQR) algorithm is a powerful method for solving linear systems *Ax* = *b* and unconstrained least-squares problem min_{x} | | *Ax* − *b* | | . In the paper, the LSQR approach is developed to obtain iterative algorithms for solving the generalized Sylvester-transpose matrix equation

the minimum Frobenius norm residual problem

and the periodic Sylvester matrix equation

Numerical results are given to illustrate the effect of the proposed algorithms. Copyright © 2013 John Wiley & Sons, Ltd.

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