In this paper, we study a class of fractional nonlinear impulsive switched coupled evolution equations. Existence and uniqueness of solutions as well as Hyers-Ulam stability results are presented. An example is provided for the verification of our results.

]]>In this paper, we are mainly concerned with 2 types of constrained matrix equation problems of the form *A**X**B*=*C*, the least squares problem and the optimal approximation problem, and we consider several constraint matrices, such as general Toeplitz matrices, upper triangular Toeplitz matrices, lower triangular Toeplitz matrices, symmetric Toeplitz matrices, and Hankel matrices. In the first problem, owing to the special structure of the constraint matrix
, we construct special algorithms; necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions. In the second problem, we use von Neumann alternating projection algorithm to obtain the solutions of problem. Then we give 2 numerical examples to demonstrate the effectiveness of the algorithms.

In this paper, we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual-phase-lag and 3-phase-lag theories, reflecting Saint-Venant's principle. In a recent paper, 2 families of cases for high-order partial differential equations were studied. Here, we investigate a third family of cases, which corresponds to the fact that a certain condition on the time derivative must be satisfied. We also study the spatial behavior of a thermoelastic problem. We obtain a Phragmén-Lindelöf alternative for the solutions in both cases. The main tool to handle these problems is the use of an exponentially weighted Poincaré inequality.

]]>This paper investigates the asymptotic behavior of a delayed Nicholson-type system involving patch structure and nonlinear density-dependent mortality terms. Based on the fluctuation lemma and some differential inequality techniques, delay-dependent criteria are deduced for the global attractivity of the addressed system, which indicate that small delays have no effect on the attractivity of the system. Meanwhile, an illustrative numerical example is given to illustrate the feasibility of the theoretical results.

]]>The aim of this paper is to construct generating functions for *m*-dimensional unification of the Bernstein basis functions. We give some properties of these functions. We also give derivative formulas and a recurrence relation of the *m*-dimensional unification of the Bernstein basis functions with help of their generating functions. By combining the *m*-dimensional unification of the Bernstein basis functions with *m* variable functions on simplex and cube, we give *m*-dimensional unification of the Bernstein operator. Furthermore, by applying integrals method including the Riemann integral, the *q*-integral, and the *p*-adic integral to some identities for the (*q*-) Bernstein basis functions, we derive some combinatorial sums including the Bernoulli numbers and Euler numbers and also the Stirling numbers and the Cauchy numbers (the Bernoulli numbers of the second kind).

The detection of image edges is of great importance in image processing. One of the efficient implementations for this image recovery problem is based on the identification of sharp jumps of the gray function of the image. Mathematically, this problem can be modeled by the numerical differentiation of the gray function with 2 variables. For this ill-posed problem with nonsmooth solution, we investigate the regularization schemes with total variation and *L*^{1} penalty term, respectively. We prove that the regularizing parameter under the Tikhonov regularization framework can be uniquely chosen in terms of the Morozov's discrepancy principle and then establish the convergence rate of the regularizing solutions in terms of the Bregman distance. The discrete schemes are performed by the lagged diffusivity fixed point iteration, with numerical implementations showing the validity of the proposed scheme.

In this paper, we propose a free final time optimal control approach applied to 4 ordinary differential equations which describe the tumor-immune interactions after the injection of the bacillus Calmette-Guérin (BCG) in the bladder of a hypothetical patient. The main goal of this optimal control strategy is to find the optimal dosage amount needed in each instillation of BCG for stimulating the immune-system and then killing superficial bladder tumors and to determine the optimal duration of treatment, adequate for stopping the intravesical therapy with lesser side-effects. For this, we introduce into the model of interest, a control function which represents the dose of BCG immunotherapy procedure and we formulate a minimization problem where the final time is considered free (nonfixed). The characterization of the sought optimal control noted *u*^{∗} is derived based on Pontryagin's maximum principle, while the formulation of the sought optimal final time noted
is based on formulae of sensitivity which are obtained conditions from the derivative of the objective function with respect to
. We investigate the resolution of the free final time optimal control problem in 3 possible cases: (a) when
is quadratic in the final gain function, (b) when the final gain function does not depend on
, and (c) when
is linear in the final gain function. Finally, we obtain the sought optimal dose of BCG, and we conclude that in case (a), we obtain an optimal duration which is more beneficial regarding the activation of immune cells while cases (b) and (c) both provide an optimal duration which is more adequate for the minimization of the tumoral population.

In this note, we study a type of nonlinear Riemann-Liouville fractional *q*–difference equations. By using the Krasnoselskii fixed-point theorem, we investigate the existence of solutions of the addressed equation. Prior to the main results and unlike previous theorems, we provide the variation of parameters formula in case the solution starts at some *a*>0, which belongs to the quantum time scale
where the solution is only defined. The case when the solution starts at *a*=0 is also considered. To apply our result, we provide an existence theorem for the Riemann-Liouville fractional *q*–logistic type model.

This paper is devoted to the reconstruction of the conductivity coefficient for a nonautonomous hyperbolic operator an infinite cylindrical domain. Applying a local Carleman estimate, we prove the uniqueness and a Hölder stability in the determination of the conductivity using a single measurement data on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in 3 dimensions.

]]>This paper introduces and studies a new class of functions between topological spaces, namely, Γ-continuous functions, which properly contains the class of s-continuous functions. Finally, we study some of its properties.

]]>The saddle point equilibrium problem is of great importance in differential game theory. In this paper, an optimistic value model for the saddle point equilibrium problem under uncertain environment is investigated. The equilibrium equation for the proposed model is presented. Then a linear quadratic model is discussed. Finally, a counter terror problem is analyzed by the results obtained in the paper.

]]>This paper is concerned with the stability of essential spectra of singular Sturm-Liouville differential operators with complex-valued coefficients. It is proved that the essential spectrum of the corresponding minimal operator is preserved by perturbations small at infinity with respect to the unperturbed operator. Based on it, 1-dimensional Schrödinger operators under local dilative perturbations are studied.

]]>In this paper, we discuss the qualitative behavior of a discrete host-parasitoid model with the host subject to refuge and strong Allee effects. More precisely, we study the local and global asymptotic stability, stable manifolds and unstable manifolds of boundary equilibrium points, existence and unique positive equilibrium point, local and global behavior of the positive equilibrium point, and the uniform persistence for the model with the host subject to the refuge or both refuge and strong Allee effects. It is also proved that the model undergoes a transcritical bifurcation in a small neighborhood of the boundary equilibrium point. Some numerical simulations are given to support our theoretical results. We can obtain that the addition of the refuge may make the parasitoids go extinct while the hosts survive or may stabilize the host-parasitoid interaction; the addition of both refuge and strong Allee effects has either a negative or positive impact on the coexistence of both populations.

]]>A generalized Fourier series method is constructed to approximate the solution of the Neumann problem in a finite domain for the system of equations governing the bending of elastic plates with transverse shear deformation. The method is illustrated by an example with computation performed by 3 different techniques that are contrasted and compared for efficiency, accuracy, and stability.

]]>We consider a quantum particle in a potential *V*(*x*) subject to a time-dependent (and spatially homogeneous) electric field *E*(*t*) (the control). Boscain, Caponigro, Chambrion, and Sigalotti proved that, under generic assumptions on *V*, this system is approximately controllable on the
unit sphere, in sufficiently large time *T*. In the present article, we show that, for a large class of initial states (dense in
unit sphere), approximate controllability does not hold in arbitrarily small time. This generalizes our previous result for Gaussian initial conditions. Furthermore, we prove that the minimal time can in fact be arbitrarily large.

We find multiplicity results for forced oscillations of a periodically perturbed autonomous second-order equation, the perturbing term possibly depending on the whole history of the system. The techniques that we use are topological in nature, but the technical details are hidden in the proofs and completely transparent to the reader only interested in the results.

]]>A triangular splitting implementation of Runge–Kutta–Nyström–type Fourier collocation methods is presented and analyzed in this paper. The proposed implementation relies on a reformulation of the method and on the Crout factorization of a corresponding matrix associated with the method. The excellent behavior of the splitting implementation is confirmed by its performance on a few numerical tests.

]]>In this article, we present several results on global exponential stability of a fractional-order cellular neural network with impulses and with time-varying and distributed delay. By using the Lyapunov-like function methods in conjunction with the Razumikhin techniques, we derive sufficient condition for the exponential stability with an exponential convergence rate. The obtained outcomes of our present investigation significantly extend and generalize the corresponding results existing in the current literature. Finally, we give 2 illustrative examples to demonstrate the theoretical findings.

]]>In this paper, a generalized variable-coefficient Gardner equation which involves a linear damping term and a dissipative term is studied. This equation broadens out a large number of equations previously considered. The equivalence group is used to simplify the analysis of the equation. The gauging of arbitrary elements of the family allows us to perform an exhaustive study of the Lie symmetries admitted by the family. Finally, by means of the multiplier method, we classify all low-order conservation laws of the equations that have been obtained by applying the gauging.

In this paper, we propose a host-vector model for malaria transmission by incorporating infection age in the infected host population and nonlinear incidence for transmission from infectious vectors to susceptible hosts. One novelty of the model is that the recovered hosts only have temporary immunity and another is that successfully recovered infected hosts may become susceptible immediately. Firstly, the existence and local stability of equilibria is studied. Secondly, rigorous mathematical analyses on technical materials and necessary arguments, including asymptotic smoothness and uniform persistence of the system, are given. Thirdly, by applying the fluctuation lemma and the approach of Lyapunov functionals, the threshold dynamics of the model for a special case were established. Roughly speaking, the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than one and otherwise the endemic equilibrium is globally asymptotically stable when no reinfection occurs. It is shown that the infection age and nonlinear incidence not only impact on the basic reproduction number but also could affect the values of the endemic steady state. Numerical simulations were performed to support the theoretical results.

]]>In this paper, we study the stability of a 1-dimensional Bresse system with infinite memory-type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. When the thermal effect vanishes, the system becomes elastic with memory term acting on one equation. We consider the interesting case of fully Dirichlet boundary conditions. Indeed, under equal speed of propagation condition, we establish the exponential stability of the system. However, in the natural physical case when the speeds of propagation are different, using a spectrum method, we show that the Bresse system is not uniformly exponentially stable. In this case, we establish a polynomial energy decay rate. Our study is valid for all other mixed boundary conditions.

]]>In recent years, many authors have introduced some fixed point theorems for multivalued quasi-contraction whose contractivity constants *α* belonged to the interval
. However, as Amini-Harandi pointed out, it is not clear whether such results also hold when
. Up to now, efforts in this direction either have failed or have been lessened to a lighter version. The main result of the current manuscript gives a partial positive answer to the above-mentioned problem by adding a necessary and sufficient condition to guarantee existence of strict fixed points of quasi-contractive multivalued mappings. This problem has remained open for many years. In addition to this, 2 examples are shown to support given results.

The time fractional Fokker-Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine pseudospectral method based on Gegenbauer polynomials and Chebyshev spectral differentiation matrix to solve numerically a class of initial-boundary value problems of the time fractional Fokker-Planck equation on a finite domain. The presented method reduces the main problem to a generalized Sylvester matrix equation, which can be solved by the global generalized minimal residual method. Some numerical experiments are considered to demonstrate the accuracy and the efficiency of the proposed computational procedure.

]]>We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two-by-two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric-to-magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.

]]>The final goal of control policies in neglected vector-borne diseases in developing countries is to protect humans. These vector-borne diseases include leishmaniasis, dengue, chagas, and malaria. The traditional control measures for vector-borne diseases, as with any other illnesses, suggest to reduce the basic reproduction number below the value 1. This strategy is not necessarily sufficient when a backward bifurcation occurs. Because of its worldwide relevance, we are interested in modeling cutaneous leishmaniasis with Peru as a specific example. We use a vector-host model with an extrinsic incubation period, which gives evidence that a backward bifurcation can occur under certain conditions. We estimate some parameters for the cutaneous leishmaniasis model in Peru. The uncertainty of the parameters suggests that we cannot guarantee the avoidance of a backward bifurcation range. It is important to be attentive to the appearance of phenomena that could make eradication more difficult. Local and global sensitivity analyses agree that is most sensitive to the number of bites by a female sandfly and its natural mortality rate. The former dependency suggests very practical control policies.

]]>In this paper, we study global existence and asymptotic stability of solutions for the initial value problem of the three-dimensional (3-D) generalized incompressible micropolar system in Fourier-Besov spaces. Besides, we also establish some regularizing rate estimates of the higher-order spatial derivatives of solutions, which particularly imply the spatial analyticity and the temporal decay of global solutions.

]]>The notion of *L*_{p}(*h*,*k*) solutions of linear impulsive differential equations in Banach spaces is introduced. Sufficient conditions for existence of such solutions are derived. Possible applications to linear control systems with impulses are considered. An illustrative example is given.

Inverse problems to recover a space-dependent factor of a source term and an initial condition in a perturbed time fractional diffusion equation containing an additional convolution term from final data are considered. Existence, uniqueness, and stability of solutions to these problems are proved.

]]>The immersed boundary (IB) method is an elegant way to fully couple the motion of a fluid and deformations of an immersed elastic structure. In that vein, the *IB2d* software allows for expedited explorations of fluid-structure interaction for beginners and veterans to the field of computational fluid dynamics. While most open-source computational fluid dynamics codes are written in low-level programming environments, *IB2d* was specifically written in high-level programming environments to make its accessibility extend beyond scientists with vast programming experience. Although previously introduced by Battista et al. 2006 many improvements and additions have been made to the software to allow for even more robust models of material properties for the elastic structures, including a data analysis package for both the fluid and immersed structure data, an improved time-stepping scheme for higher accuracy solutions, and functionality for modeling slight fluid density variations as given by the Boussinesq approximation.

This paper is concerned with a delayed Nicholson's blowflies model with discontinuous harvesting, which is described by an almost periodic nonsmooth dynamical system. Under some reasonable assumptions on the discontinuous harvesting function, by using the Filippov regulation techniques and the theory of dichotomy, together with the Halanay inequality, we establish some new criteria on the existence of positive almost periodic solution and its convergence. An example with numerical simulation is also presented to support the theoretical results.

]]>In the paper, we provide sufficient conditions assuring existence of viable solutions of differential inclusions with fractional derivative without singular kernel, namely, Caputo-Fabrizio derivative of order *α*∈(0,1). A construction of an approximate solution is presented. A modified condition of tangency, according to specifity of the system with this new fractional derivative, is given.

We present a systematic mathematical analysis of the qualitative steady-state response to rate perturbations in large classes of reaction networks. This includes multimolecular reactions and allows for catalysis, enzymatic reactions, multiple reaction products, nonmonotone rate functions, and non-closed autonomous systems. Our *structural sensitivity analysis* is based on the stoichiometry of the reaction network, only. It does not require numerical data on reaction rates. Instead, we impose mild and generic nondegeneracy conditions of algebraic type. From the structural data, only, we derive which steady-state concentrations are sensitive to, and hence influenced by, changes of any particular reaction rate—and which are not. We also establish transitivity properties for influences involving rate perturbations. This allows us to derive an *influence graph* which globally summarizes the influence pattern of any given network. The influence graph allows the computational, but meaningful, automatic identification of functional subunits in general networks, which hierarchically influence each other. We illustrate our results for several variants of the glycolytic citric acid cycle. Biological applications include enzyme knockout experiments and metabolic control.

We propose a new mathematical model for the spread of Zika virus. Special attention is paid to the transmission of microcephaly. Numerical simulations show the accuracy of the model with respect to the Zika outbreak occurred in Brazil.

]]>In this study, we propose a modified Laguerre collocation method based on operational matrix technique to solve 1-dimensional parabolic convection-diffusion problems arising in applied sciences. The method transforms the equation and mixed conditions of problem into a matrix equation with unknown Laguerre coefficients by means of collocation points and operational matrices. The solution of this matrix equation yields the Laguerre coefficients of the solution function. Thereby, the approximate solution is obtained in the truncated Laguerre series form. Also, to illustrate the usefulness and applicability of the method, we apply it to a test problem together with residual error estimation and compare the results with existing ones. Besides, the algorithm of the present method is given to represent the calculation of approximate solution.

]]>We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by

with the initial condition *u*|_{(t=0)}=*u*_{0}, *u*_{t}|_{(t=0)}=*u*_{1}, and *θ*|_{(t=0)}=*θ*_{0} in Ω and the boundary condition *u*=*∂*_{ν}*u*=*θ*=0 on Γ, where *u*=*u*(*x*,*t*) denotes a vertical displacement at time *t* at the point *x*=(*x*_{1},⋯,*x*_{n})∈Ω, while *θ*=*θ*(*x*,*t*) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half-space case. We prove the existence of
-bounded solution operators of the corresponding resolvent problem. Then, the generation of *C*_{0} analytic semigroup and the maximal *L*_{p}-*L*_{q}-regularity of time-dependent problem are derived.

In this paper, we extend proper efficiency concepts for a vector optimization problem in complex space. The relationships among the concepts due to Benson, Borwein, Kuhn-Tucker, and Geoffrion have been discussed under some convexity and constraints qualification conditions. The necessary and sufficient conditions for a feasible point to be a properly efficient solution are established. The results are generalizations of the concepts of proper efficiency and their related theorems from real to complex space and complete the existing ones.

]]>An efficient algorithm for the computation of a *C*^{2} interpolating clothoid spline is herein presented. The spline is obtained following an optimisation process, subject to continuity constraints. Among the 9 various targets/problems considered, there are boundary conditions, minimum length path, minimum jerk, and minimum curvature (energy). Some of these problems are solved with just a couple of Newton iterations, whereas the more complex minimisations are solved with few iterations of a nonlinear solver. The solvers are warmly started with a suitable initial guess, which is extensively discussed, making the algorithm fast. Applications of the algorithm are shown relating to fonts, path planning for human walkers, and as a tool for the time-optimal lap on a Formula 1 circuit track.

In this paper we solve an initial-boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first-order pde; the stochastic version yields a second-order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first-order case, the analysis does not readily extend to the second-order case. We develop a method for solving the second-order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.

]]>In this paper, a new generalized 5×5 matrix spectral problem of Ablowitz-Kaup-Newell-Segur type associated with the enlarged matrix Lie superalgebra is proposed, and its corresponding super soliton hierarchy is established. The super variational identities are used to furnish super Hamiltonian structures for the resulting super soliton hierarchy.

]]>We consider the integration of the special second-order initial value problem of the form . A recently introduced family of 7 stages, eighth-order methods, sharing constant coefficients, is used as base. This family is properly modified to derive phase fitted and zero dissipative methods (ie, trigonometric fitted) that are best suited for integrating oscillatory problems. Numerical tests over a set of problems shows enhanced performance when the purely linear part of the problems is rather large in comparison with the rest of nonlinear parts. An appendix implementing a MATLAB listing with the coefficients of the new method is also given.

]]>The aim of this paper is to establish a global existence result for a nonlinear reaction diffusion system with fractional Laplacians of different orders and a balance law. Our method of proof is based on a duality argument and a recent maximal regularity result due to Zhang.

]]>In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit-point case at *a*(*b*) and limit-circle case at *b*(*a*)) acting in the Hilbert space
In terms of boundary conditions at *a* and *b*, all maximal dissipative, accumulative, and self-adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at *a*” and “dissipative at *b*.” For 2 cases, a self-adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl-Titchmarsh function of the corresponding self-adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.

This paper studies the effects of the time change on the frequencies of specific terms connected to the document field in a given period. These specific terms are the field association (FA) terms. The paper also suggests a new method for automatic evaluation of the stabilization classes of FA terms to improve the precision of decision tree. The stabilization classes point out the popularity of list of FA terms depending on time change. Moreover, the suggested method manipulates the problem of the scattering of data numbers among classes to improve the performance of decision tree precision. The presented method is evaluated through conducting experiments by simulating the result of 1245 files, which are equivalent to 4.15 MB. The F-measure for increment, fairly steady, and decrement classes achieves %90.4, %99.3, and %38.6, sequentially.

]]>This paper is devoted to discuss a multidimensional backward heat conduction problem for time-fractional diffusion equation with inhomogeneous source. This problem is ill-posed. We use quasi-reversibility regularization method to solve this inverse problem. Moreover, the convergence estimates between regularization solution and the exact solution are obtained under the a priori and the a posteriori choice rules. Finally, the numerical examples for one-dimensional and two-dimensional cases are presented to show that our method is feasible and effective.

]]>In this paper, the problem of a nonlinear beam equation involving an integral term of the deformation energy, which is unknown before the solution, under different boundary conditions with simply supported, 2-end fixed, and cantilevered is investigated. We transform the governing equation into an integral equation and then solve it by using the sinusoidal functions, which are chosen both as the test functions and the bases of numerical solution. Because of the orthogonality of the sinusoidal functions, we can find the expansion coefficients of the numerical solution that are given in closed form by using the Drazin inversion formula. Furthermore, we introduce the concept of fourth-order and fifth-order boundary functions in the solution bases, which can greatly raise the accuracy over 4 orders than that using the partial boundary functions. The iterative algorithms converge very fast to find the highly accurate numerical solutions of the nonlinear beam equation, which are confirmed by 6 numerical examples.

]]>In this paper, we give a simple proof for the convergence of the deterministic particle swarm optimization algorithm under the weak chaotic assumption and remark that the weak chaotic assumption does not relax the stagnation assumption in essence. Under the spectral radius assumption, we propose a convergence criterion for the deterministic particle swarm optimization algorithm in terms of the personal best and neighborhood best position of the particle that incorporates the stagnation assumption or the weak chaotic assumption as a special case.

]]>In this paper, we study the zero-flux chemotaxis-system

where Ω is a bounded and smooth domain of
, *n*≥1, and where
, *k*,*μ*>0 and *α*≤1. For any *v*≥0, the chemotactic sensitivity function is assumed to behave as the prototype *χ*(*v*)=*χ*_{0}/(1+*a**v*)^{2}, with *a*≥0 and *χ*_{0}>0. We prove that for any nonnegative and sufficiently regular initial data *u*(*x*,0), the corresponding initial-boundary value problem admits a unique global bounded classical solution if *α*<1; indeed, for *α*=1, the same conclusion is obtained provided *μ* is large enough. Finally, we illustrate the range of dynamics present within the chemotaxis system in 1, 2, and 3 dimensions by means of numerical simulations.

A partial inverse problem for an integro-differential Sturm-Liouville operator on a star-shaped graph is studied. We suppose that the convolution kernels are known on all the edges of the graph except one and recover the kernel on the remaining edge from a part of the spectrum. We prove the uniqueness theorem for this problem and develop a constructive algorithm for its solution, based on the reduction of the inverse problem on the graph to the inverse problem on the interval by using the Riesz basis property of the special system of functions.

]]>This paper deals with the behavior of positive solutions to a nonautonomous reaction-diffusion system with homogeneous Neumann boundary conditions, which describes a two-species predator-prey system in which there is an infectious disease in prey. The sufficient condition on the permanence of the prey and the predator is established by combining the comparison principle with the results related to the corresponding ODE system. Some sufficient conditions for the spreading and vanishing of the disease are obtained. The global attractivity is also discussed by constructing a Lyapunov functional. Our results show that the disease is spreading if the transmission rate is suitably large, while if the transmission rate is small, the disease must be vanishing.

]]>In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher-order moduli of smoothness and of best approximation quantity are obtained.

]]>We are concerned with a family of dissipative active scalar equation on
. By using similar methods from the previous paper of Y. Giga et al. (see Introduction below), we construct a unique real, spatially almost periodic mild solution *θ* of satisfying . In this paper, we consider some countable sum-closed frequency sets (see Remark ). We show that the property of the solution is rather different from Chae et al and obtain that
with some initial data *θ*_{0} for all *t*≥0,
and 0≤*α*≤*ω*, where *ω* is a fixed constant. Furthermore, arranging the elements of a countable sum-closed frequency set *F*_{δ} as in Remark , we have for any 0≤*α*≤*ω* that
belongs to
, where *F*_{δ} is defined in or .

In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô-Volterra integral equations driven by fractional Brownian motion with Hurst parameter . The proposed method is based on the operational matrices of modification of hat functions (MHFs) and the collocation method. In this approach, by approximating functions that appear in the integral equation by MHFs and using Newton's-Cotes points, nonlinear integral equation is transformed to nonlinear system of algebraic equations. This nonlinear system is solved by using Newton's numerical method, and the approximate solution of integral equation is achieved. Some theorems related to error estimate and convergence analysis of the suggested scheme are also established. Finally, 2 illustrative examples are included to confirm applicability, efficiency, and accuracy of the proposed method. It should be noted that this scheme can be used to solve other appropriate problems, but some modifications are required.

]]>The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro-differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro-differential equation and examines their convergence, stability, and accuracy.

]]>A Hamilton-Poisson system is an approach for the motion of a spacecraft around an asteroid or for the motion of an underwater vehicle. We construct a coordinate chart on the symplectic leaf which contains a specific generic equilibrium point and we establish stability conditions for this equilibrium point.

]]>In this paper, the Cauchy problem for the 3D diffusion approximation model in radiation hydrodynamics is considered. By using the embedding theorem and interpolation technique, we establish the global well-posedness of strong solutions in *H*^{2}.

Foreground segmentation is a critical early step in most human-computer interaction applications notably in action and gesture recognition domain. In this paper, an approach to model background which based on luminance-invariant color with an adaptive Gaussian mixture is proposed to discriminate foreground object from their background in complex scene. Firstly, the background model is learned based on the spectral properties of shadows and scene activity. Secondly, the shadow with the hypotheses on color invariance is adaptively set up and updated. Finally, the log-likelihood measurement is to conduct the adaptation. Our experiments are performed on a wide range of practical applications of gesture and action recognition videos. Additionally, the proposed approach is efficient and more robust than premature state-of-the-art with no sacrificing real-time performance.

]]>This paper studies the output tracking problem of Boolean control networks (BCNs) with impulsive effects via the algebraic state-space representation approach. The dynamics of BCNs with impulsive effects is converted to an algebraic form. Based on the algebraic form, some necessary and sufficient conditions are presented for the feedback output tracking control of BCNs with impulsive effects. These conditions contain constant reference signal case and time-varying reference signal case. The study of an illustrative example shows that the obtained new results are effective.

]]>In this article, we are interested by a system of heat equations with initial condition and zero Dirichlet boundary conditions. We prove a finite-time blow-up result for a large class of solutions with positive initial energy.

]]>The inverse scattering transform for the derivative nonlinear Schrödinger-type equation is studied via the Riemann-Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann-Hilbert problem is established for the derivative nonlinear Schrödinger-type equation. In the inverse scattering process, *N*-soliton solutions of the derivative nonlinear Schrödinger-type equation are obtained by solving Riemann-Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.

We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green's function methods, energy methods, Fourier analysis, and frequency decomposition, we obtain the global existence and the optimal time-decay estimate of solutions.

]]>No abstract is available for this article.

]]>In this paper, the authors established a unified framework for deriving and analyzing a posteriori error estimators for finite volume methods for the Stokes equations. The a posteriori error estimators are residual based and are applicable to various finite volume methods for the Stokes equations. In particular, the unified theoretical analysis works well for finite volume schemes arising from using trial functions of conforming, nonconforming, and discontinuous finite element functions, yielding new results that are not seen in the existing literature. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, the Schrödinger operator

where the potential *q*(*x*) is single-well with transition point
is considered. Suppose {*λ*_{n}}_{n ≥ 1} is the set of eigenvalues for the previous Schrödinger operator and the potential *q*(*x*) possesses an additional condition, we show that the first two eigenvalues *λ*_{2} and *λ*_{1} satisfy

Equality holds if and only if *q*(*x*) is a constant. Copyright © 2014 John Wiley & Sons, Ltd.

This paper studies a new preconditioning technique for sparse systems arising from discretized partial differential equations in computational fluid dynamics problems. This preconditioning technique exploits the multilevel sequentially semiseparable (MSSS) structure of the system matrix. MSSS matrix computations give a data-sparse way to approximate the *LU* factorization of a sparse matrix from discretized partial differential equations in linear computational complexity with respect to the problem size. In contrast to the standard block diagonal and block upper-triangular preconditioners, we exploit the global MSSS structure of the 2×2 block system from the discretized Stokes equation and linearized Navier-Stokes equation. This avoids approximating the Schur complement explicitly, which is a big advantage over standard block preconditioners. Through numerical experiments on standard computational fluid dynamics benchmark problems in Incompressible Flow and Iterative Solver Software, we show the performance of the MSSS preconditioners. They indicate that the global MSSS preconditioner not only yields mesh size independent convergence but also gives viscosity parameter and Reynolds number independent convergence. Compared with the algebraic multigrid (AMG) method and the geometric multigrid (GMG) method for block preconditioners, the MSSS preconditioning technique is more robust than both the AMG method and GMG method, and considerably faster than the AMG method. Copyright © 2015 John Wiley & Sons, Ltd.

In this study, the nonlinear fractional partial differential equations have been defined by the modified Riemann–Liouville fractional derivative. By using this fractional derivative and traveling wave transformation, the nonlinear fractional partial differential equations have been converted into nonlinear ordinary differential equations. The modified trial equation method is implemented to obtain exact solutions of the nonlinear fractional Klein–Gordon equation and fractional clannish random walker's parabolic equation. As a result, some exact solutions including single kink solution and periodic and rational function solutions of these equations have been successfully obtained. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In the present paper, we consider the Bézier variant of an operator involving Laguerre polynomials of degree k, with *α*≥1, for bounded functions *f* defined on the interval [0,1]. In particular, by using the Chanturia modulus of variation, we estimate the rate of pointwise convergence of (*P*_{n,α}*f*)(*x*,*t*) at those points *x*∈(0,1) at which the one-sided limits *f*(*x*+),*f*(*x*−) exist. To prove our main result, we have used some methods and techniques of probability theory. Our result extends and generalizes the very recent results of very recent results of the authors to more general classes of functions. Copyright © 2016 John Wiley & Sons, Ltd.

Based on the fourth-order compact finite difference scheme, new extrapolation cascadic multigrid methods for two-dimensional Poisson problem are presented. In these new methods, a new extrapolation operator and a spline interpolation operator are used to provide a better initial value on refined grid. Numerical experiments show the new methods have higher accuracy and better efficiency. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, an infinite integral concerning numerical computation in crystallography is investigated, which was studied in two recent articles, and integration by parts is employed for calculating this typical integral. A variable transformation and a single integration by parts lead to a new formula for this integral, and at this time, it becomes a completely definite integral. Using integration by parts iteratively, the singularity at the points near three points *a* = 0,1,2 can be eliminated in terms containing obtained integrals, and the factors of amplifying round-off error are released into two simple fractions independent of the integral. Series expansions for this integral are obtained, and estimations of its remainders are given, which show that accuracy 2^{−n} is achieved in about 2*n* operations for every value in a given domain. Finally, numerical results are given to verify error analysis, which coincide well with the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

The initial-boundary value problems encountered in the regime of the dynamic theory of gradient elasticity are characterized by several crucial and intrinsic issues. One of the most important characteristics is that the involving differential equation is of fourth order and mainly that the second order time derivative – appearing in it – is not given explicitly but in the contrary is incorporated implicitly in mixed-type differential terms (via spatial-time derivatives). The aim of the present work is the investigation of the emerged initial-boundary value problems of gradient elasticity in one spatial dimension and the establishment of the suitable functional setting assuring existence and uniqueness of weak (or strong) solutions. Furthermore, the spectral analysis of the gradient elastic operator is investigated and compared with the well-known results of classical elasticity. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The Melan beam equation modeling suspension bridges is considered. A slightly modified equation is derived by applying variational principles and by minimizing the total energy of the bridge. The equation is nonlinear and nonlocal, while the beam is hinged at the endpoints. We show that the problem always admits at least one solution whereas the uniqueness remains open although some numerical results suggest that it should hold. We also emphasize the qualitative difference with some simplified models. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Wave propagation in lossy nonlinear metamaterials is analytically investigated by means of perturbation methods. In the left-handed band of the nonlinear metamaterial, a higher-order nonlinear Schrödinger equation is obtained, while in the frequency band gap, a dissipative short-pulse equation is derived. In both cases, dissipation is described by linear terms, which lead to an exponential decay of the solutions. The decay rates, that is, the inverses of the linear loss coefficients in these two models, are found in terms of the dielectric and magnetic properties of the metamaterial. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper presents a cylindrical multipole expansion for periodic sources with applications for three-phase power cables. It is the aim of the contribution to provide some analytical solutions and techniques that can be useful in the calculation of cable losses. Explicit analytical results are given for the fields generated by a three-phase helical current distribution and which can be computed efficiently as an input to other numerical methods such as, for example, the Method of Moments. It is shown that the field computations are numerically stable at low frequencies (such as 50 Hz) as well as in the quasi-magnetostatic limit provided that sources are divergence-free. The cylindrical multipole expansion is furthermore used to derive an efficient analytical model of a measurement coil to measure and estimate the complex valued permeability of magnetic steel armour in the presence of a strong skin-effect. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We prove optimal lower bounds for the growth of the potential energy over balls of minimizers to the vectorial Allen–Cahn energy in two spatial dimensions, as the radius tends to infinity. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The well-known low-frequency expansion of the total acoustic field in the exterior of a penetrable spherical scatterer is revisited in view of the Atkinson Wilcox theorem. The corresponding low-frequency approximations of any order are calculated following a purely algebraic algorithmic procedure, based on the spectral decomposition of the problem's far field pattern. As an indication of its accuracy and effectiveness, the proposed algebraic procedure is shown to recover already known low-frequency coefficients and also to deduce higher order of approximations. The proposed track of calculations leads to a closed-form expression of any such coefficient and has been recently applied on impenetrable spherical scatterers as well, offering equally accurate results. The effectiveness of the proposed procedure indicates an underlying general efficient method applicable to a wider class of starshaped scatterers. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the triharmonic operator subject to homogeneous boundary conditions of intermediate type on a bounded domain of the N-dimensional Euclidean space. We study its spectral behaviour when the boundary of the domain undergoes a perturbation of oscillatory type. We identify the appropriate limit problems that depend on whether the strength of the oscillation is above or below a critical threshold. We analyse in detail the critical case that provides a typical homogenization problem leading to a strange boundary term in the limit problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate the behavior of the solution of a mixed problem for the Poisson equation in a domain with two moderately close holes. If *ϱ*_{1} and *ϱ*_{2} are two positive parameters, we define a perforated domain Ω(*ϱ*_{1},*ϱ*_{2}) by making two small perforations in an open set: the size of the perforations is *ϱ*_{1}*ϱ*_{2}, while the distance of the cavities is proportional to *ϱ*_{1}. Then, if *r*_{∗} is small enough, we analyze the behavior of the solution for (*ϱ*_{1},*ϱ*_{2}) close to the degenerate pair (0,*r*_{∗}). Copyright © 2016 John Wiley & Sons, Ltd.

Recent developments in mathematical modeling provide today a new theoretical framework and state-of-the-art tools for supporting both the corresponding academic research and important socio-economical applications/activities. Within this framework, the present work focuses on advances in differential, and in particular information, geometry towards the development of non-conventional models able to support applications in environmental simulation and forecasting. The latter are the scientific areas where a great number of institutes and operational centers worldwide are focusing, providing solutions and new ideas for applications related with meteorology, natural hazards, renewable energy, ship and aviation safety, and a variety of other corresponding activities. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In the present work, the scattering problem of an elastic wave by a penetrable thermoelastic body in an isotropic and homogeneous elastic medium is considered. The corresponding scattering problem is formulated in a suitable compact form, and taking into account the physical parameters of the thermoelastic body and integral representations for the total exterior elastic and the total interior thermoelastic field are presented. Using asymptotic analysis of the fundamental solution of the Navier equation, expressions of the far-field patterns are obtained, and reciprocity theorems for plane and spherical wave incidence are established. Finally, a general scattering theorem for plane wave incidence is also presented. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio *σ*. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to *σ*∈[0,1[and that all the Neumann eigenvalues tend to zero as *σ*1^{−}. Moreover, we show that the Neumann problem defined by setting *σ* = 1 admits a sequence of positive eigenvalues of finite multiplicity that are not limiting points for the Neumann eigenvalues with *σ*∈[0,1[as *σ*1^{−} and that coincide with the Dirichlet eigenvalues of the biharmonic operator. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we perform a numerical study on the interesting phenomenon of soliton reflection of solid walls. We consider the 2D cubic nonlinear Schrödinger equation as the underlying mathematical model, and we use an implicit–explicit type Crank–Nicolson finite element scheme for its numerical solution. After verifying the perfect reflection of the solitons on a vertical wall, we present the imperfect reflection of a dark soliton on a diagonal wall. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the problem
among curves connecting two given wells of *W*≥0, and we reduce it, following a standard method, to a geodesic problem of the form
with
. We then prove existence of curves minimizing this new action just by proving that the distance induced by *K* is proper (i.e., its closed balls are compact). The assumptions on *W* are minimal, and the method seems robust enough to be applied in the future to some PDE problems. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we revisit the recently proposed results for a general class of linear stochastic degenerate Sobolev systems with additive noise by using a different approach keeping, however, the main assumptions unchanged for the purpose of comparison. In particular, the mild and strong well-posedness for the initial and final value problems are presented and studied by applying a suitable transformation that formulates the degenerate stochastic system as a pseudoparabolic one. Based on the derived results for the forward and backward cases, under this new framework, the conditions for the exact controllability are revisited for a particular class of degenerate Sobolev systems. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The problem of model aggregation from various information sources of unknown validity is addressed in terms of a variational problem in the space of probability measures. A weight allocation scheme to the various sources is proposed, which is designed to lead to the best aggregate model compatible with the available data and the set of prior measures provided by the information sources. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A layered sphere is excited by an acoustic point source arbitrarily located inside or outside the sphere. First, the direct scattering problem is solved analytically by developing a transitions matrix methodology. Then, approximations of the acoustic far-field in the low-frequency regime are derived, by using asymptotic techniques, and are subsequently utilized in inverse scattering algorithms concerning either the determination of the sphere's characteristics or the parameters of an internal point source. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate sufficient conditions for existence of multiple solutions to a coupled system of fractional-order differential equations with three-point boundary conditions. By coupling the method of upper and lower solutions together with the method of monotone iterative technique, we develop conditions for iterative solutions. Based on these conditions, we study maximal and minimal solutions to the problem under consideration. We also study error estimates and provide an example. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the model equations for the Timoshenko beam as a first-order system in the framework of evolutionary equations. The focus is on boundary damping, which is implemented as a dynamic boundary condition. A change of material laws allows the inclusion of a large class of cases of boundary damping. By choosing a particular material law, it is shown that the first-order approach to Sturm–Liouville problems with boundary damping is also covered. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Completing previous work, a new class of interior solutions for compact static fluid spheres exhibiting pressure anisotropy, admitting conformal motion, and having 7, 8, 9, and 10 spacetime dimensions, respectively, is presented. Einstein's field equations without cosmological constant are solved for a particular energy density distribution function, assuming non-commutative geometry of spacetime. The behavior of the physical quantities obtained does not exclude the possible existence of ultra-compact, though rather exotic, stars in higher spacetime dimensions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Tsunamis are rare events compared with other extreme natural hazards, but the growth of population along coastlines has increased their potential impact. Tsunamis are most often generated by earthquake-induced dislocations of the seafloor, which displace large water masses. They can be simulated effectively as long waves whose propagation is modeled by the nonlinear shallow water equations. In this note, we present a brief assessment of earthquake-generated tsunami hazards for the city of Heraklion, Crete. We employ numerical hydrodynamic simulations, including inundation computations with the model MOST, and use high-resolution bathymetry and topography data for the area of interest. MOST implements a splitting method in space to reduce the system of shallow water equations in two successive systems, one for each spatial variable, and it uses a dispersive, Godunov-type finite difference method to solve the equations in characteristic form. We perform probabilistic analysis to assess the effects of the earthquake epicenter location on the tsunami, for time windows of 100, 500, and 1000years. The tsunami hazard is assessed through computed values of the maximum inundation range and maximum flow depth. Finally, we present a brief vulnerability analysis for the city of Heraklion, Crete. The data needed to identify tsunami-vulnerable areas are obtained by combining remote sensing techniques and geographic information system technology with surveyed observations and estimates of population data. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In the current paper, we consider a stochastic parabolic equation that actually serves as a mathematical model describing the operation of an electrostatic actuated micro-electro-mechanical system. We first present the derivation of the mathematical model. Then after establishing the local well posedeness of the problem, we investigate under which circumstances a *finite-time quenching* for this stochastic partial differential equation, corresponding to the mechanical phenomenon of *touching down*, occurs. For that purpose, the Kaplan's eigenfunction method adapted in the context of stochastic partial differential equations is employed. Copyright © 2016 John Wiley & Sons, Ltd.

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

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.

]]>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.

]]>We study nonlinear Volterra-type evolution integral equations of the form:
in a *C*^{∗}-algebra
or in a Hilbert algebra of Dixmier-Segal type, acting on a Hilbert space tensor product
, where
denotes a Hilbert space and
is the Boson-Einstein (Fermion-Dirac) Fock space, over a complex Hilbert space
. Under suitable Carathéodory-type conditions on the corresponding Nemytskii operator Φ of *f* and assuming that *k* is a quantum dynamical-type semigroup, we obtain exactly one classical global solution in the space
of bounded continuous (operator-valued) quantum stochastic processes. Moreover, we prove the existence of exactly one positive (respectively completely positive) classical global solution in
(respectively in
, applying a positivity (respectively completely positivity preserving) quantum stochastic integration process and assuming that *k* is a quantum dynamical semigroup acting on
, where Φ defines a positive (respectively completely positive) quantum stochastic process.

In the current study, we consider the approximate solutions of fractional-order PDEs with initial-boundary conditions based on the shifted Chebyshev polynomials. The proposed method is combined with the operational matrix of fractional-order differentiation described in the Caputo's sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations with constant coefficients by dispersing unknown variables. The validity and effectiveness of the approach are demonstrated via some numerical examples. Lastly, the error analysis of the proposed method has been investigated. 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.

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

where *a*>0, *b*>0, *λ*>0, and *p*≥6. Under some suitable conditions, we prove that the equation has a nontrivial solution for small *λ*>0 by variational method. Moreover, we regard *b* as a parameter and obtain a convergence property of the nontrivial 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 paper, we consider the following nonlinear Choquard equation driven by fractional Laplacian

where *V*(*x*) is a nonnegative continuous potential function, 0<*s*<1, *N*≥2, (*N*−4*s*)^{+}<*α*<*N*, and *λ* is a positive parameter. By variational methods, we prove the existence of least energy solution which localizes near the bottom of potential well *i**n**t*(*V*^{−1}(0)) as *λ* large enough.

This paper is concerned with laminated beams modeled from the well-established Timoshenko system with time delays and boundary feedbacks. By using semigroup method, we prove the global well-posedness of solutions. Assuming the weights of the delay are small, we establish the exponential decay of energy to the system by using an appropriate Lyapunov functional.

]]>The qualifier “ephemeral” was proposed for continuous models of bodies, such as gases, for which the generally tacit axiom of permanence of material elements fails to apply. Consequently, to their scrutiny, a Eulerian (local) approach is mandatory, such as one adopted, eg, in molecular dynamics. Within the scheme of ephemeral continua, we discuss here 3 essential subclasses of bodies: (1) those undergoing energy-preserving processes (in this sense hyperelastic), (2) hypoelastic bodies inspired by a type proposed by Truesdell, and (3) a number of minor ones. We re-examine the essential issues of the general format focusing on the proposal of appropriate concepts of strains and strainings.

]]>Lie group classification for 2 Burgers-type systems is obtained. Systems contain 2 arbitrary elements that depend on the 2 dependent variables. Equivalence transformations for the systems are derived. Examples of nonclassical reductions are given. A Hopf-Cole–type mapping that linearizes a nonlinear system is presented.

]]>This paper is devoted to the study of the differential systems in arbitrary Banach spaces that are obtained by mixing nonlinear evolutionary equations and generalized quasi-hemivariational inequalities (EEQHVI). We start by showing that the solution set of the quasi-hemivariational inequality associated to problem EEQHVI is nonempty, closed, and convex. Furthermore, we establish upper semicontinuity and measurability properties for this solution set. Then, based on them, we prove the existence of solutions for problem EEQHVI and the compactness of the set of corresponding trajectories of EEQHVI. These statements extend previous results in several directions, for instance, by dropping the boundedness requirement for the set of constraints and substantially relaxing monotonicity hypotheses.

]]>Bidirectional associative memory models are 2-layer heteroassociative networks. In this work, we prove the existence and the global exponential stability of the unique weighted pseudo–almost periodic solution of bidirectional associative memory neural networks with mixed time-varying delays and leakage time-varying delays on time-space scales. Some sufficient conditions are given for the existence, the convergence, and the global exponential stability of the weighted pseudo–almost periodic solution by using fixed-point theorem and differential inequality techniques. The results of this paper complement the previous outcomes. An example is given to show the effectiveness of the derived results via computer simulations.

]]>This work addresses the study of the *L*^{p}-boundedness and compactness of abstract linear and nonlinear fractional integro-differential equations. The analysis is performed for the whole range of values of *p*, ie,
. In addition, theoretical results are complemented with illustrating particular cases of systems modeled by fractional evolution equations as heat conduction problems and problems arising in the theory of viscoelastic materials.

In this paper, we investigate the well-posedness and stability of mild solutions for a class of neutral impulsive stochastic integro-differential equations in a real separable Hilbert space. By the inequality technique combined with theory of resolvent operator, some sufficient conditions are established for the concerned problems. The obtained conclusions are completely new, which generalize and improve some existing results. An example is given to illustrate the effectiveness of our results.

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