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This paper is concerned with interval general bidirectional associative memory (BAM) neural networks with proportional delays. Using appropriate nonlinear variable transformations, the interval general BAM neural networks with proportional delays can be equivalently transformed into the interval general BAM neural networks with constant delays. The sufficient condition for the existence and uniqueness of equilibrium point of the model is established by applying Brouwer's fixed point theorem. By constructing suitable delay differential inequalities, some sufficient conditions for the global exponential stability of the model are obtained. Two examples are given to illustrate the effectiveness of the obtained results. This paper ends with a brief conclusion. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper we prove the existence and uniqueness of a strong solution to a 3-D model of phase separation in elastic solids. The model has the form of an initial-boundary-value problem for a nonlinear coupled system of hyperbolic–parabolic type. The key idea of the proof is based on the analysis of the system once- and twice differentiated with respect to time variable. The paper develops results of the previous work (Top. Meth. Nonlin. Anal., in press). Copyright © 2007 John Wiley & Sons, Ltd.

In this paper we consider a mathematical model describing the shock of a rigid body and a linear viscoelastic bar, resting on a rigid base, in the frames of the Mikusinski operator calculus. We determine and analyse the character of the exact solution of the corresponding problem in the field of Mikusinski operators. We construct an approximate solution of this problem, and estimate the error of approximation. The mentioned problem is taken from the papers (J. Mech. NCSR Vietnam Tom 1991; 13(2):1–7; Nonlinear Anal. Theory Meth. Appl. 2001; 43:547–561). Copyright © 2007 John Wiley & Sons, Ltd.

This paper is concerned with the unique global solvability of a three-dimensional (3-D) non-linear thermoelasticity system arising from the study of shape memory materials. The system consists of the coupled evolutionary problems of viscoelasticity with non-convex elastic energy and non-linear heat conduction with mechanical dissipation. The present paper extends the previous 2-D existence result of the authors Reference [1] to 3-D case. This goal is achieved by means of the Leray–Schauder fixed point theorem using technique based on energy arguments and DeGiorgi method. Copyright © 2004 John Wiley & Sons, Ltd.

In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided

In this paper, a generalized fractional $(q,h)$-Gronwall inequality is investigated. Based on this inequality, we derive the uniqueness theorem and the finite-time stability criterion of nonlinear fractional delay $(q,h)$-difference systems. Several examples are given to illustrate our theoretical result.

This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel. Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem. Then, by means of the Dhage fixed-point principle, we discuss the existence of mild solutions. Finally, we study the Ulam–Hyers stability of the introduced fractional hybrid problem.

We study the global stability of a multistrain SIS model with superinfection and patch structure. We establish an iterative procedure to obtain a sequence of threshold parameters. By a repeated application of a result by Takeuchi et al. [Nonlinear Anal Real World Appl. 2006 7:235-247], we show that these parameters completely determine the global dynamics of the system: for any number of patches and strains with different infectivities, any subset of the strains can stably coexist depending on the particular choice of the parameters. Finally, we return to the special case of one patch examined in [Math Biosci Eng. 2017 14:421-35] and give a correction to the proof of Theorem 2.2 of that paper.

This paper deals with multiplicity results of solutions for a class of impulsive fractional differential systems. The approach is based on variational methods and critical point theory. Indeed, we establish existence results for our system under some algebraic conditions on the nonlinear part with the classical Ambrosetti–Rabinowitz (AR) condition on the nonlinear and the impulsive terms. Moreover, by combining two algebraic conditions on the nonlinear term which guarantee the existence of two weak solutions, applying the mountain pass theorem we establish the existence of third weak solution for our system.

In this paper we provide a detailed convergence analysis for fully discrete second order (in both time and space) numerical schemes for nonlocal Allen-Cahn (nAC) and nonlocal Cahn-Hilliard (nCH) equations. The unconditional unique solvability and energy stability ensures $\ell^4$ stability. The convergence analysis for the nAC equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nCH equation, due to the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken and an $H^{-1}$ inner product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a-priori $W^{1,\infty}$ bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (e.g., 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an $O( s^3 + h^4)$ convergence in $\ell^\infty (0, T; \ell^2)$ norm, which leads to the necessary bound under a standard constraint $s \le C h$. Here, we also prove convergence of the scheme in the maximum norm under the same constraint.

This paper investigates the properties of the p-mean Stepanov-like doubly weighted pseudo almost automorphic (SpDWPAA) processes and its application to Sobolev-type stochastic differential equations driven by G-Brownian motion. We firstly prove the equivalent relation between the SpDWPAA and Stepanov-like asymptotically almost automorphic stochastic processes based on ergodic zero set. We further establish the completeness of the space and the composition theorem for SpDWPAA processes. These results obtained improve and extend previous related conclusions. As an application, we show the existence and uniqueness of the Sp DWPAA solution for a class of nonlinear Sobolev-type stochastic differential equations driven by G-Brownian motion and present a decomposition of this unique solution. Moreover, an example is given to illustrate the effectiveness of our results.

In this paper, we investigate a Vector-Borne disease model with nonlinear incidence rate and 2 delays: One is the incubation period in the vectors and the other is the incubation period in the host. Under the biologically motivated assumptions, we show that the global dynamics are completely determined by the basic reproduction number R0. The disease-free equilibrium is globally asymptotically stable if R0≤1; when R0>1, the system is uniformly persistent, and there exists a unique endemic equilibrium that is globally asymptotically. Numerical simulations are conducted to illustrate the theoretical results.

In this paper, the approximate analytical solutions of Lotka–Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka–Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd.

This paper concerns new continuum phenomenological model for epitaxial thin-film growth with three different forms of the Ehrlich–Schwoebel current. Two of these forms were first proposed by Politi and Villain 1996 and then studied by Evans, Thiel, and Bartelt 2006. The other one is completely new. Energy structure and properties of the new model are studied. Following the techniques used in Li and Liu 2003, we present rigorous analysis of the well-posedness, regularity, and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. By using a convex–concave time-splitting scheme, one can naturally build unconditionally stable semi-implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published before which indicates that the new model correctly captures the essential morphological states in the thin-film growth process. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider a generalized Kirchhoff equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem, we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools. Copyright © 2016 John Wiley & Sons, Ltd.

Ductal carcinoma in situ – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with the existence of positive solutions of the discrete counterpart of nonlinear elliptic problems. We apply our methods to the study of positive solutions under different hypotheses about the nonlinearities. For example, we consider the cases that are superlinear and sublinear at infinity. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a nonperiodic discrete nonlinear equation with Jacobi operators is considered. By using the critical point theory, we establish some new sufficient conditions on the existence and multiplicity of homoclinic solutions. Recent results are generalized and significantly improved. Furthermore, our results greatly improve some existing ones even for some special cases. Copyright (c) 2016 John Wiley & Sons, Ltd.

This paper is concerned with the relative approximate controllability of functional systems with infinite delay and delayed control in Hilbert spaces. In particular, we begin with studying some criteria of the controllability of linear systems. Based on those results, sufficient conditions are derived for the relative approximate controllability of nonlinear functional systems. Finally, an example is included to illustrate the effectiveness of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. The approach we use to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems is new in this context. Moreover, a new method based on a generalization of the Mittag–Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate the effectiveness of our main result. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we model the growths of populations by means of local fractional calculus. We formulate the local fractional rate equation and the local fractional logistic equation. The exact solutions of local fractional rate equation and local fractional logistic equation with the Mittag-Leffler function defined on Cantor sets are presented. The obtained results illustrate the accuracy and efficiency for modeling the complexity of linear and nonlinear population dynamics (PD). Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources. After establishing the necessary local existence theorems of strong solutions, we investigate the blow-up and global existence profile. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a numerical method for solving Lane-Emden type equations, which are nonlinear ordinary differential equations on the semi-infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane-Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is devoted to global asymptotic stability of cellular neural networks with impulses and multi-proportional delays. First, by means of the transformation vi(t) = ui(et), the impulsive cellular neural networks with proportional delays are transformed into impulsive cellular neural networks with the variable coefficients and constant delays. Second, we prove the global exponential stability of the latter by nonlinear measure, and that the exponential stability of the latter implies the asymptotic stability of the former. We furthermore provide a sufficient condition to the existence, uniqueness, and the global asymptotic stability of the equilibrium point of the former. Our results are generalizations of some existing ones. Finally, an example and its simulation are presented to illustrate effectiveness of our method. Copyright © 2015 John Wiley & Sons, Ltd.

In a fairly recent paper (2008 American Control Conference, June 11-13, 1035-1039), the problem of dealing with trading in optimal pairs was treated from the viewpoint of stochastic control. The analysis of the subsequent nonlinear evolution partial differential equation was based upon a succession of Ansätze, which can lead to a solution of the terminal-value problem. Through an application of the Lie Theory of Continuous Groups to this equation, we show that the Ansätze are based upon the underlying symmetries of the equation (their (14)). We solve the problem in a more general context by allowing the parameters to be explicitly time dependent. The extension means thatmore realistic problems are amenable to the samemode of solution. Copyright © 2014 John Wiley & Sons, Ltd. 38 17 4448 4460

In this paper, we investigate the existence of solutions to nonlinear fractional order differential coupled systemswith the classical nonlocal initial conditions.We introduce a useful vector norm, named β·B-vector norm,which is not only a novelty but also provides another way to deal with a large number of problems not limit to integer and noninteger differential systems and singular integral systems. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study a class of nonlinear obstacle problems with nonstandard growth. We obtain the L∞ estimate on the solutions and prove the existence and uniqueness of solutions to such problems. Our results are generalizations of the corresponding results in the constant exponent case. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, a novel 3D nonlinear system with chaotic behavior is considered. Our proposed system is a generalized Lorenz-like system, with no reflections about coordinates axes. Although the considered system depends on seven real parameters, its stability is completely analyzed. Also, the presence of Hopf bifurcation is pointed out. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study the multiplicity results of positive solutions for a semi-linear elliptic system involving critical growth terms. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient h(x) of the critical nonlinearity affects the number of positive solutions of that problem and get a relationship between the number of positive solutions and the topology of the global maximum set of h. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the homogenization of a diffusion-reaction process in a domain undergoing an evolution of the microstructure. The main novelty is the consideration of a chemical process on a hypersurface, where the surface itself is evolving. After transformation to a description in a fixed domain, we employ methods of periodic homogenization to arrive at a limit problem in the bulk coupled with problems in a reference cell. Here, the structural evolution takes place only in the reference structure. Additional problems come from nonlinear reaction rates, which require the Kolmogoroff compactness criterion for the derivation of suitable a priori estimates. The results can be used as a modeling tool and for the investigation of complex biological and chemical systems in porous media, for example, carbon sequestration or biological enhanced oil recovery. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with a class of fourth-order nonlinear difference equations. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for Dirichlet boundary value problems and give some new results. Our results successfully complement the existing ones. Copyright (c) 2014 John Wiley & Sons, Ltd.

In this paper, we consider a nonlinear sublinear Schrodinger equation at resonance in . By using bounded domain approximation technique, we prove that the problem has infinitely many solutions. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we propose a nonlinear fractional order model in order to explain and understand the outbreaks of influenza A(H1N1). In the fractional model, the next state depends not only upon its current state but also upon all of its historical states. Thus, the fractional model is more general than the classical epidemic models. In order to deal with the fractional derivatives of the model, we rely on the Caputo operator and on the Grunwald–Letnikov method to numerically approximate the fractional derivatives. We conclude that the nonlinear fractional order epidemic model is well suited to provide numerical results that agree very well with real data of influenza A(H1N1) at the level population. In addition, the proposed model can provide useful information for the understanding, prediction, and control of the transmission of different epidemics worldwide. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we establish necessary and sufficient conditions for the solutions of a second-order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright (c) 2013 John Wiley & Sons, Ltd.

In this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p-Laplacian boundary value problem where φp(s) = | s | p − 2s, p > 1, is continuous, T is an integer with T ≥ 5 and . By virtue of Jensen's discrete inequalities, we use fixed point index theory to establish our main results. Copyright © 2013 John Wiley & Sons, Ltd.

The piecewise continuous function space is an important state space in various nonlinear problems. In this paper, we establish some new fixed point theorems for the perturbed contraction operators in piecewise continuous function spaces. Our results can be applied to various integral operators. Some previous results are improved in this literature. As applications, the existence and uniqueness of solutions of impulsive anti-periodic boundary value problems and the Lasota–Wazewska models with delays are exhibited at last two sections. Copyright © 2013 John Wiley & Sons, Ltd.

This paper aims to present a general framework of the quadratic spline functions to develop a numerical method for solving the nonlinear space fractional Fisher's equation. Using Von Neumann method, the proposed method is shown to be conditionally stable. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm. The results reveal that the proposed approach is very effective, convenient, and quite accurate to such considered problems. Copyright © 2013 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 H3 initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper we study the existence, uniqueness, multiplicity and stability of positive solution of a non-linear elliptic problem that combines local and non-local terms taking the form of an integral in space. The proofs are mainly based on fixed point theorems, bifurcation techniques, sub-supersolutions and continuation arguments. Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brasil) Fondo Europeo de Desarrollo Regional Ministerio de Ciencia e Innovación

This paper studies the Cauchy problem of the 3D Navier–Stokes equations with nonlinear damping term | u | β−1u (β ≥ 1). For β ≥ 3, we derive a decay rate of the L2-norm of the solutions. Then, the large time behavior is given by comparing the equation with the classic 3D Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we use homotopy analysis method (HAM) to solve two-point nonlinear boundary value problems that have at least one solution. The new approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. The scheme shows importance of choice of convergence-control parameter ℏ to guarantee the convergence of the solutions of nonlinear differential equations. This scheme is tested on three nonlinear exactly solvable differential equations. Two of the examples are practical in science and engineering. The results demonstrate reliability, simplicity and efficiency of the algorithm developed. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we study the multiple solutions for the semilinear elliptic equation where , 1 1 for N = 2. We will prove that the problem possesses infinitely many solutions under some assumptions on Q(x). Copyright © 2011 John Wiley & Sons, Ltd.

In this paper we consider a doubly nonlinear Volterra equation related to the p-Laplacian with a nonsmooth kernel. By exploiting a suitable implicit time-discretization technique we obtain the existence of global strong solution. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, a new technique of homotopy analysis method (HAM) is proposed for solving high-order nonlinear initial value problems. This method improves the convergence of the series solution, eliminates the unneeded terms and reduces time consuming in the standard homotopy analysis method (HAM) by transform the nth-order nonlinear differential equation to a system of n first-order equations. Second- and third- order problems are solved as illustration examples of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve the Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations., Comment: 18 pages, 5 figures; Published online in the journal of "Mathematical Methods in the Applied Sciences"

Finite volume method and characteristics finite element method are two important methods for solving the partial differential equations. These two methods are combined in this paper to establish a fully discrete characteristics finite volume method for fully nonlinear convection-dominated diffusion problems. Through detailed theoretical analysis, optimal order H1 norm error estimates are obtained for this fully discrete scheme. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper a reaction–diffusion model describing two interacting pioneer and climax species is considered. The role of diffusivity and forcing (stocking or harvesting of the species) on the nonlinear stability of a coexistence equilibrium is analysed. The study is performed in the context of a new approach to nonlinear L2-stability based on the analysis of stability of the zero solution of a suitable linear system of ordinary differential equations. Theorems concerning the effect of forcing and diffusivity on the dynamics are established and stability–instability thresholds for the system are obtained. An example to illustrate the practical use of the results is also provided. Copyright © 2008 John Wiley & Sons, Ltd.

This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a $p$-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local H\"older regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller-Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model.

The subject of the present paper consists in proving the convergence of a phase-field model, based on the entropy equation with memory, to phase relaxation. The well-posedness and the long-time behaviour of solutions for the non-linear and singular phase-field system have been recently shown by Bonetti et al. (Preprint IMATI-CNR, 2005; Discrete Contin. Dyn. Syst. Ser. B, in press). Here, we study the asymptotic behaviour of such solutions as the interfacial energy coefficient tends to zero. The limit problem is a phase relaxation problem with memory, which is new. We prove well-posedness results through convergence under rather general assumptions. However, the case of a quadratic non-linearity for the latent heat is excluded. Such a situation is dealt for the problem without memory in a generalized setting by introducing an ad hoc logarithm. Copyright © 2006 John Wiley & Sons, Ltd.