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 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.

In this paper, a computational technique based on the pseudo-spectral method is presented for the solution of the optimal control problem constrained with elliptic variational inequality. In fact, our aim in this paper is to present a direct approach for this class of optimal control problems. By using the pseudo-spectral method, the infinite dimensional mathematical programming with equilibrium constraint, which can be an equivalent form of the considered problem, is converted to a finite dimensional mathematical programming with complementarity constraint. Then, the finite dimensional problem can be solved by the well-developed methods. Finally, numerical examples are presented to show the validity and efficiency of the technique. 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.

]]>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.

]]>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.

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.

]]>A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface with the boundary is investigated. The main objective is to trace what happens in Γ-limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ-limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space . 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.

]]>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.

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.

]]>We consider smooth compactly supported solution to the classical Vlasov–Poisson system in three space dimensions in the electrostatic case. An estimate on velocities is derived, showing a growth rate at most like the power 1/8 of the time variable. As a consequence, a better decay estimate is obtained for the electric field in the norm. 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.

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.

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate random forcing. The existence and pathwise uniqueness of strong solutions with *H*^{1}-initial data is established, and then the existence of an invariant measure for the Feller semigroup is shown by Krylov–Bogoliubov theorem. Because of the coupled items in the stochastic Ginzburg–Landau–Newell equations, the higher order momentum estimates can be only obtained in the *L*^{2}-norm. We show the ergodicity of invariant measure for the transition semigroup by asymptotically strong Feller property and the support property. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we study the global well-posedness and scattering theory of the solution to the Cauchy problem of a generalized fourth-order wave equation [[EQUATION1]] where
if *d*⩽4, and
if *d*⩾5. The main strategy we use in this paper is concentration-compactness argument, which was first introduced by Kenig and Merle to handle the scattering problem vector so as to control the momentum. 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.

]]>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 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 prove the existence of weak solutions to a one-dimensional initial-boundary value problem for a model system of partial differential equations, which consists of a sub-system of linear elasticity and a nonlinear non-uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function in that work by . The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. 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.

In this paper, we prove the existence and uniqueness of a solution for a class of backward stochastic differential equations driven by *G*-Brownian motion with subdifferential operator by means of the Moreau–Yosida approximation method. Moreover, we give a probabilistic interpretation for the viscosity solutions of a kind of nonlinear variational inequalities. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the non-autonomous Navier–Stokes equations with discontinuous initial data. We prove the global existence of solutions, the decay rate of density, and the equilibrium state of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider interactions of smooth and discontinuous germs as generalized integrations over non-rectifiable paths with applications in theory of boundary value problems of complex analysis. 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.

]]>We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the nonexistence result for the weighted Lane–Emden equation:

- (0.1)

and the weighted Lane–Emden equation with nonlinear Neumann boundary condition:

- (0.2)

where *f*(|*x*|) and *g*(|*x*|) are the radial and continuously differential functions,
is an upper half space in
, and
. Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems (0.1) and (0.2) under appropriate assumptions on *f*(|*x*|) and *g*(|*x*|). Copyright © 2017 John Wiley & Sons, Ltd.

Synchronization of complex networks with time-varying coupling matrices is studied in this paper. Two kinds of time-varying coupling are taken into account. One is the time-varying inner coupling in the node state space and the other is the time-varying outer coupling in the network topology space. By respectively setting linear controllers and adaptive controllers, time-varying complex networks can be synchronized to a desired state. Meanwhile, different influences of the control parameters of linear controllers and adaptive controllers on the synchronization have also been investigated. Based on the Lyapunov function theory, we construct appropriate positive-definite functions, and several sufficient synchronization criteria are obtained. Numerical simulations further illustrate the effectiveness of conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A discrete multi-group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>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.

]]>The seasonality of conception for populations of the past using no contraception has remained a *terra incognita*. First, the influence of that of marriages on the seasonality of births is highlighted, taking into account the different stages in women's reproductive lives and the presence of successive cohorts of unequal size. Second, the age-dependent and time-dependent monthly distribution of conception is disentangled from monthly marriage and birth time series by means of stochastic optimization under a Leslie recursion with time-varying and age-varying probability of conception. The application to Armenian-Gregorians in the Don Army Territory (South Russia) from 1889 to 1912 reveals strong consistency between reconstructed conception, mean age at marriage, and fertility time series. Copyright © 2017 John Wiley & Sons, Ltd.

We present an analytic approach to solve a degenerate parabolic problem associated with the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second-order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half-plane. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation to the canonical form **y**^{(n)}=0 consists of copies of the same iterative scalar equation. It is also shown that contrary to the scalar case, an iterative vector equation need not be reducible to the canonical form by an invertible point transformation. Other properties of iterative linear systems are also derived, as well as a simple algebraic formula for their general solution. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative:

- (FD)

where *α*∈(1/2,1), *λ*>0,
are the left and right tempered fractional derivatives,
is the fractional Sobolev spaces, and
. Assuming that *f* satisfies the Ambrosetti–Rabinowitz condition and another suitable conditions, by using mountain pass theorem and minimization argument over Nehari manifold, we show that (FD) has a ground state solution. Furthermore, we show that this solution is a radially symmetric solution. Copyright © 2017 John Wiley & Sons, Ltd.

This paper shows the existence and the uniqueness of the nonnegative viscosity solution of the singular boundary value problem
for *t*>0,
, where *f* is a continuous non-decreasing function such that *f*(0)⩾0, and *h* is a nonnegative function satisfying the Keller–Osserman condition. Moreover, when *h*(*u*)=*u*^{p} with *p*>3, we obtain the global estimates for the classic solution *u*(*t*) and the exact blow-up rate of it at *t*=0. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we present a novel strategy to face the problem of dimensionality within datasets involved in conversational and feature selection systems. We base our work on a sound and complete logic along with an efficient attribute closure method to manage implications. All of them together allow us to reduce the overload of information we encounter when dealing with these kind of systems. An experiment carried out over a dataset containing real information comes to expose the benefits of our design. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The group analysis method is applied to the two-dimensional nonlinear Klein–Gordon equation with time-varying delay. Determining equations for equations with a time-varying delay are derived. A complete group classification of the studied equation with respect to the function involved into the equation is obtained. All admitted Lie algebras are classified. By using the classifications, representations of all invariant solutions are found. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, a second-order fast explicit operator splitting method is proposed to solve the mass-conserving Allen–Cahn equation with a space–time-dependent Lagrange multiplier. The space–time-dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the asymptotic stability of a composite wave consisting of two traveling waves to a Keller–Segel chemotaxis model with logarithmic sensitivity and nonzero chemical diffusion. We show that the composite wave is asymptotically stable under general initial perturbation, which only be needed small in *H*^{1}-norm. This improves previous results. Copyright © 2017 John Wiley & Sons, Ltd.

We demonstrate how the problem of finding the effective property of quasiperiodic constitutive relations can be simplified to the periodic homogenization setting by transforming the original quasiperiodic material structure to a periodic heterogeneous material in a higher dimensional space. The characterization of two-scale cut-and-projection convergence limits of partial differential operators is presented. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Green's function technique serves as a powerful tool to find the particle displacements due to SH-wave propagation in layer of a shape different from the space between two parallel planes. Therefore, the present paper undertook to study the propagation of SH-wave in a transversely isotropic piezoelectric layer under the influence of a point source and overlying a heterogeneous substrate using Green's function technique. The coupled electromechanical field equations are solved with the aid of Green's function technique. Expression for displacements in both layer and substrate, scalar potential and finally the dispersion relation is obtained analytically for the case when wave propagates along the direction of layering. Numerical computations are carried out and demonstrated with the aid of graphs for six different piezoelectric materials namely PZT-5H ceramics, Barium titanate (BaTiO_{3}) ceramics, Silicon dioxide (SiO_{2}) glass, Borosilicate glass, Cobalt Iron Oxide (CoFe_{2}*O*_{4}), and Aluminum Nitride (AlN). The effects of heterogeneity, piezoelectric and dielectric constants on the dispersion curve are highlighted. Moreover, comparative study is carried out taking the phase velocity for different piezoelectric materials on one hand and isotropic case on the other. Dispersion relation is reduced to well-known classical Love wave equation with a view to illuminate the authenticity of problem. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the following fractional Schrödinger–Poisson problem:

where *s*,*t*∈(0,1],4*s*+2*t*>3,*V*(*x*),*K*(*x*), and *f*(*x*,*u*) are periodic or asymptotically periodic in *x*. We use the non-Nehari manifold approach to establish the existence of the Nehari-type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions
uniformly in
with
and

with constant *θ*_{0}∈(0,1), instead of
uniformly in
and the usual Nehari-type monotonic condition on *f*(*x*,*τ*)/|*τ*|^{3}. Our results unify both asymptotically cubic or super-cubic nonlinearities, which are new even for *s*=*t*=1. Copyright © 2017 John Wiley & Sons, Ltd.

We consider the long time behavior of solutions for the non-autonomous stochastic *p*-Laplacian equation with additive noise on an unbounded domain. First, we show the existence of a unique
-pullback attractor, where *q* is related to the order of the nonlinearity. The main difficulty existed here is to prove the asymptotic compactness of systems in both spaces, because the Laplacian operator is nonlinear and additive noise is considered. We overcome these obstacles by applying the compactness of solutions inside a ball, a truncation method and some new techniques of estimates involving the Laplacian operator. Next, we establish the upper semi-continuity of attractors at any intensity of noise under the topology of
. Finally, we prove this continuity of attractors from domains in the norm of
, which improves an early result by Bates *et al.*(2001) who studied such continuity when the deterministic lattice equations were approached by finite-dimensional systems, and also complements Li *et al.* (2015) who discussed this approximation when the nonlinearity *f*(·,0) had a compact support. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the shape inverse problem of a body immersed in the incompressible fluid governed by thermodynamic equations. By applying the domain derivative method, we obtain the explicit representation of the derivative of solution with respect to the boundary, which plays an important role in the inverse design framework. Moreover, according to the boundary parametrization technique, we present a regularized Gauss–Newton algorithm for the shape reconstruction problem. Finally, numerical examples indicate the proposed algorithm is feasible and effective for the low Reynolds numbers. Copyright © 2017 John Wiley & Sons, Ltd.

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

in a bounded domain
with smooth boundary under no-flux boundary conditions, where
satisfies
for all
with *l*⩾2 and some nondecreasing function
on [0,*∞*). Here, *f*(*v*)∈*C*^{1}([0,*∞*)) is nonnegative for all *v*⩾0. It is proved that when
, the system possesses at least one global bounded weak solution for any sufficiently smooth nonnegative initial data. This extends a recent result by Wang (Math. Methods Appl. Sci. 2016 **39**: 1159–1175) which shows global existence and boundedness of weak solutions under the condition
. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we deal with discrete-time linear control systems in which the state is constrained to lie in the positive orthant independently of the inputs involved, that is, the inputs can take negative values. Such (positive state) systems appear, for example, in ecology models where the removal of individuals from a population is described. Controllability and reachability are fundamental properties of a system that show its ability to move in space, which are analyzed from an algebraic point of view throughout the text, paying special attention to the single-input case. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we prove that if the initial data is small enough, we obtain an explicit *L*^{∞}(*Q*_{T})-estimate for a two-dimensional mathematical model of cancer invasion, proving an explicit bound with respect to time *T* for the estimate of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

This paper deals with Lasota–Wazewska red blood cell model with perturbation on time scales. By applying the fixed point theorem of decreasing operator, we establish sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by means of Gronwall inequality. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we use the arrowhead matrices as a tool to study graph theory. More precisely, we deal with an interesting class of directed multigraphs, the hub-directed multigraphs. We associate the arrowhead matrices with the adjacency matrices of a class of directed multigraph, and we obtain new properties of the second objects by using properties of the first ones. The hub-directed multigraphs with potential use in applications are also defined. As main result, we show that a hub-directed multigraph *G*(*H*) with adjacency matrix *H*^{∗} is a dominant hub-directed multigraph if and only if *H*^{∗}=*C**E*, where *C* is the adjacency matrix of another directed multigraph and *E* is the adjacency matrix of a particular elementary dominant hub-directed pseudo-graph. Another decomposition of its Gram (arrowhead) matrix
is also given. Copyright © 2017 John Wiley & Sons, Ltd.

Through solving the problem step by step and by applying the method of a *C*_{0} semigroup of operators combined with the Banach contraction theorem, we investigate the existence and uniqueness of a mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces. In addition, an explicit iterative approximation sequence of the mild solution is derived. The assumed conditions in the present theorems are weaker and more general, and the results obtained are the generalizations and improvements of some known results. Examples are also given to illustrate our main results. Copyright © 2017 John Wiley & Sons, Ltd.

Many works on hybrid projective synchronization (or simply ‘HPS’ for short) of nonlinear real dynamic systems have been performed, while the HPS of chaotic complex systems and its application have not been extensively studied. In this paper, the HPS of complex Duffing–Holmes oscillators with known and unknown parameters is separately investigated via nonlinear control. The adaptive control methods and explicit expressions are derived for controllers and parameters estimation law, which are respectively used to achieve HPS. These expressions on controllers are tested numerically, which are in excellent agreement with theory analysis. The proposed synchronization scheme is applied to image encryption with exclusive or (or simply ‘XOR’ for short). The related security analysis shows the high security of the encryption scheme. Concerning the complex Duffing–Holmes oscillator, we also discuss its chaotic properties via the maximum Lyapunov exponent. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion-acoustic waves by including a quasi-potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the following Kirchhoff-type equations

where *a*>0,*b*⩾0,4<*p*<2^{∗}=6, and
. Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, this problem has infinitely many sign-changing solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we are concerned with the general decay result of the quasi-linear wave equation with a time-varying delay in the boundary feedback and acoustic boundary conditions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, a new approach for pseudo almost periodic solution under the measure theory, under Acquistpace-Terreni conditions. We make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of *μ*-pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations. For illustration, we propose some application to a nonautonomous heat equation. Copyright © 2017 John Wiley & Sons, Ltd.

On the basis of the ideas of non-traditional biomanipulation control in fresh water body, a kind of nutrient–algae fish model is presented to investigate the effects of constant releasing fish on the nutrient and the algae. The threshold conditions for the extinction of the algae are obtained by discussing the stability of boundary equilibrium. The conditions for the coexistence of the algae and the fish are obtained by discussing the existence and stability of positive equilibrium. Besides, Hopf bifurcation is also analyzed by considering the parameter about the amount of the released fish. Furthermore, a kind of optimal problem is presented, and the necessary condition for the existence of the optimal solution is given by Pontryagin maximum principle. Finally, the mathematical results are verified by numerical simulations. The mathematical results show that there is a threshold amount of the released fish, above which the activity of releasing fish can control the growth of the algae and further reduce the probability of the algae bloom, but can not decrease the eutrophication level of the water body. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of **R**^{3} bounded with two coaxial cylinders that present the solid thermoinsulated walls. In the thermodynamical sense, the fluid is perfect and polytropic. We assume that the initial density and temperature are bounded from below with a positive constant, and that the initial data are sufficiently smooth cylindrically symmetric functions. The starting problem is transformed into the Lagrangian description on the spatial domain ]0,*L*[. In this work, we prove that our problem has a generalized solution for any time interval [0,*T*], *T*∈**R**^{+}. The proof is based on the local existence theorem and the extension principle. Copyright © 2017 John Wiley & Sons, Ltd.

We consider a non-stationary Stokes system in a thin porous medium of thickness *ε* that is perforated by periodically distributed solid cylinders of size *ε*, and containing a fissure of width *η*_{ε}. Passing to the limit when *ε* goes to zero, we find a critical size
in which the flow is described by a 2D quasi-stationary Darcy law coupled with a 1D quasi-stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we apply the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations on the cone Sobolev spaces. We not only give some results of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions. Finally, we show blow-up in finite time of solutions on a manifold with conical singularities. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we establish exact solutions of the Cauchy problem for the 3D cylindrically symmetric incompressible Navier–Stokes equations and further study the global existence and asymptotic behavior of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary-domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary-domain integral equation system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.

]]>Two-dimensional time-fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time-fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag-Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An infection-age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values *R*_{0}, the basic reproductive number for viral infection, and *R*_{1}, the viral reproductive number at the immune-free infection steady state (*R*_{1}<*R*_{0}). If *R*_{0}<1, the uninfected steady state *E*^{0} is globally asymptotically stable; if *R*_{0}>1 > *R*_{1}, the immune-free infected steady state *E*^{∗} is globally asymptotically stable; while if *R*_{1}>1, the antibody immune infected steady state
is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we establish a blow-up criterion for the three-dimentional viscous, compressible magnetohydrodynamic flows. It is shown that for the Cauchy problem and the initial-boundary-value problem with initial density allowed to vanish, the strong or smooth solution for the three-dimentional magnetohydrodynamic flows exists globally if the density, temperature, and magnetic field is bounded from above. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem

where 0≤*λ* < 1,^{C}D^{α} is the Caputo's differential operator of order *α*, and *f*:[0,1] × [0,*∞*)[0,*∞*) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary *α*, 1≤*n* < *α*≤*n* + 1:

Problem 1:

where
, 0≤*λ* < *k* + 1;

Problem 2:

where 0≤*λ*≤*α* and *D*^{α} is the Riemann–Liouville fractional derivative of order *α*.

For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval

with the integral boundary conditions

By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases and , are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity-loss type, which causes difficulty in high-frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with the following Schrödinger–Poisson systems

where *λ*, *ν* are positive parameters and *V*(*x*) is sign-changing and may vanish at infinity. Under some suitable assumptions, the existence of positive ground state solutions is obtained by using variational methods. Our main results unify and improve the recent ones in the literatures. Copyright © 2016 John Wiley & Sons, Ltd.

An asymptotic expansion of the contrasting structure-like solution of the generalized Kolmogorov–Petrovskii–Piskunov equation is presented. A generalized maximum principle for the pseudoparabolic equations is developed. This, together with the generalized differential inequalities method, allows to prove the consistence and convergence of the asymptotic series method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The Jacobi system on a full-line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is performed by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schrödinger equation on the full line. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we develop an *h**p*-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an *h**p*-version adaptive finite element discretization (based on a robust *a posteriori* residual analysis), thereby leading to a fully *h**p*-adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity-loss type, which causes the difficulty in high-frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for *n*≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for *n*≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for *n* = 1. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with a certain condenser capacity in an anisotropic environment. More precisely, we are going to investigate a free boundary problem for a class of anisotropic equations on a ring domain
*N*≥2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff-shaped ring. The proof makes use of a maximum principle for an appropriate P-function, in the sense of L. E. Payne, a Rellich type identity and some geometric arguments involving the anisotropic mean curvature of the free boundary. Copyright © 2016 John Wiley & Sons, Ltd.

We study the magnetic Bénard problem in two-dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first utilize the vanishing diffusivity method to prove the existence of global quasi-strong solutions and get some higher order estimates, and then prove the global well-posedness of the two-dimensional Boussinesq system with variable viscosity for *H*^{3} initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the elliptic boundary blow-up problem

where Ω is a bounded smooth domain of
are positive continuous functions supported in disjoint subdomains Ω_{+},Ω_{−} of Ω, respectively, *a*_{+} vanishes on the boundary of
satisfies *p*(*x*)≥1 in Ω,*p*(*x*) > 1 on *∂*Ω and
, and *ε* is a parameter. We show that there exists *ε*^{∗}>0 such that no positive solutions exist when *ε* > *ε*^{∗}, while a minimal positive solution *u*_{ε} exists for every *ε*∈(0,*ε*^{∗}). Under the additional hypotheses that
is a smooth *N* − 1-dimensional manifold and that *a*_{+},*a*_{−} have a convenient decay near Γ, we show that a second positive solution *v*_{ε} exists for every *ε*∈(0,*ε*^{∗}) if
, where *N*^{∗}=(*N* + 2)/(*N* − 2) if *N* > 2 and
if *N* = 2. Our results extend that of Jorge Garcá-Melián in 2011, where the case that *p* > 1 is a constant and *a*_{+}>0 on *∂*Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon-type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon-type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far-field operator that is defined on a unit sphere and contains the far-field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic-type gradient estimates for positive solutions of the nonlinear parabolic equation:

on
for *α* ≥ 1 and *α* ≤ 0. As applications, related Liouville-type theorem is exported. Our results are complement of known results. Copyright © 2016 John Wiley & Sons, Ltd.

This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non-interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split-quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we consider the question of controllability of a class of integrodifferential equations on Hilbert space with measures as controls. We assume that the linear part has a resolvent operator in the sense given by R. Grimmer. We generalize the original work of N. Ahmed on vector measures, and we use it to develop necessary and sufficient conditions for weak and the exact controllability of the integrodifferential equation. Using the latter, we prove that exact controllability of the integrodifferential equation implies exact controllability of a perturbed integrodifferential equation. Controllability problem for the perturbed system is formulated fixed point problem in the space of vector measures. Our results cover impulsive controls as well as regular controls. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the following quasilinear chemotaxis–haptotaxis system

- (⋆)

in a bounded smooth domain
under zero-flux boundary conditions, where the nonlinearities *D*,*S*_{1}, and *S*_{2} are supposed to generalize the prototypes

with
, and *f*∈*C*^{1}([0,+*∞*) × [0,+*∞*)) satisfies

with *r* > 0 and *b* > 0. If the nonnegative initial data *u*_{0}(*x*)∈*W*^{1,∞}(Ω),*v*_{0}(*x*)∈*W*^{1,∞}(Ω), and
for some *α*∈(0,1), it is proved that

- For
*n*= 1, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. - For
*n*= 2, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. - For
*n*≥3, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded.

Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main purpose of this article is to investigate the optimal wholesaler's replenishment decisions for deterioration items under two levels of the trade credit policy and two storage facilities in order to reflect the supply chain management situation within the economic order quantity framework. In this study, each of the following assumptions have been made: (1) The own warehouse with limited capacity always is not sufficient to store the order quantity, so that a rented warehouse is needed to store the excess units over the capacity of the own warehouse; (2) The wholesaler always obtains the partial trade credit, which is independent of the order quantity offered by the supplier, but the wholesaler offers the full trade credit to the retailer; (3) The wholesaler must take a loan to pay his or her supplier the partial payment immediately when the order is received and then pay off the loan with the entire revenue. Under these three conditions, the wholesaler can obtain the least costs. Furthermore, this study models the wholesaler's optimal replenishment decisions under the aforementioned conditions in the supply chain management. Two theorems are developed to efficiently determine the optimal replenishment decisions for the wholesaler. Finally, numerical examples are given to illustrate the theorems that are proven in this study, and the sensitivity analysis with respect to the major parameters in this study is performed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the global solvability to the mixed problem involving the wave equation with memory term and acoustic boundary conditions for non-locally reacting boundary is considered. Moreover, the general decay of the energy functionality is established by the techniques of Messaoudi. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a multistage susceptible-infectious-recovered model with distributed delays and nonlinear incidence rate is investigated, which extends the model considered by Guo *et al*. *[H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261–279]*. Under some appropriate and realistic conditions, the global dynamics is completely determined by the basic reproduction number *R*_{0}. If *R*_{0}≤1, then the infection-free equilibrium is globally asymptotically stable and the disease dies out in all stages. If *R*_{0}>1, then a unique endemic equilibrium exists, and it is globally asymptotically stable, and hence the disease persists in all stages. The results are proved by utilizing the theory of non-negative matrices, Lyapunov functionals, and the graph-theoretical approach. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the following generalized quasilinear Schrödinger equation

where *N*≥3,
is a *C*^{1} even function, *g*(0) = 1 and *g*^{′}(*s*) > 0 for all *s* > 0. Under some suitable conditions, we prove that the equation has a ground state solution and infinitely many pairs ±*u* of geometrically distinct solutions. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with a class of neutral-type BAM neural networks with distributed leakage delays. By applying the exponential dichotomy of linear differential equations, Lyapunov functional method and contraction mapping principle, we establish some sufficient conditions which ensure the existence and exponential stability of almost periodic solutions for such BAM neural networks. An example is given to illustrate the effectiveness of the theoretical findings. The results obtained in this article are completely new and complement the previously known studies. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term-by-term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto-dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2*r*th order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady-state solutions bifurcating from the nonzero steady-state solution. Moreover, we illustrate our general results by applications to models with a one-dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the attraction–repulsion chemotaxis system with rotational flux terms

where
is a bounded domain with smooth boundary. Here, *S*_{1} and *S*_{2} are given parameter functions on [0,*∞*)^{2}×Ω with values in
. It is shown that for any choice of suitably regular initial data (*u*_{0},*v*_{0},*w*_{0}) fulfilling a smallness condition on the norm of *v*_{0},*w*_{0} in *L*^{∞}(Ω), the corresponding initial-boundary value problem possesses a global bounded classical solution. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we prove existence and regularity results for weak solutions in the framework of anisotropic Sobolev spaces for a class of nonlinear anisotropic elliptic equations in the whole with variable exponents and locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. The functional setting involves Lebesgue–Sobolev spaces with variable exponents. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we study the two-mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the strongly nonlinear Nernst–Planck equations coupled with the quasi-linear Poisson equation under inhomogeneous, moreover, nonlinear boundary conditions. This system describes joint multi-component electrokinetics in a pore phase. The system is supplemented by the force balance and by the volume and positivity constraints. We establish well-posedness of the problem in the variational setting. Namely, we prove the existence theorem supported by the energy and the entropy a-priori estimates, and we provide the Lyapunov stability of the solution as well as its uniqueness in special cases. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the existence of periodic solutions for the Newtonian equation of motion with *p*-Laplacian operator by asymptotic behavior of potential function, establish some new sufficient criteria of existence of periodic solutions for the differential system under the frame of Fuc̆ik spectrum, generalize and improve some known works, and give an example to illustrate the application of the theorems. Copyright © 2017 John Wiley & Sons, Ltd.