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Discrete & Continuous Dynamical Systems - B

2018 , Volume 23 , Issue 1

Special issue on dynamical systems and applications

Select all articles


2018, 23(1): i-ii doi: 10.3934/dcdsb.201801i +[Abstract](72) +[HTML](3) +[PDF](76.6KB)
Models of the population playing the rock-paper-scissors game
Włodzimierz Bąk , Tadeusz Nadzieja and  Mateusz Wróbel
2018, 23(1): 1-11 doi: 10.3934/dcdsb.2018001 +[Abstract](185) +[HTML](30) +[PDF](449.2KB)

We consider discrete dynamical systems coming from the models of evolution of populations playing rock-paper-scissors game. Asymptotic behaviour of trajectories of these systems is described, occurrence of the Neimark-Sacker bifurcation and nonexistence of time averages are proved.

Self-similar solutions of fragmentation equations revisited
Weronika Biedrzycka and  Marta Tyran-Kamińska
2018, 23(1): 13-27 doi: 10.3934/dcdsb.2018002 +[Abstract](247) +[HTML](28) +[PDF](406.5KB)

We study the large time behaviour of the mass (size) of particles described by the fragmentation equation with homogeneous breakup kernel. We give necessary and sufficient conditions for the convergence of solutions to the unique self-similar solution.

Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data
Dorota Bors
2018, 23(1): 29-43 doi: 10.3934/dcdsb.2018003 +[Abstract](154) +[HTML](7) +[PDF](425.2KB)

In the paper we consider a boundary value problem involving a differential equation with the fractional Laplacian \begin{document}$ (-Δ)^{α/2}$\end{document} for \begin{document}$ \mathit{\alpha }\in (1,2\rm{)}$\end{document} and some superlinear and subcritical nonlinearity \begin{document}$ G_{z}$\end{document} provided with a nonhomogeneous Dirichlet exterior boundary condition. Some sufficient conditions under which the set of weak solutions to the boundary value problem is nonempty and depends continuously in the Painlevé-Kuratowski sense on distributed parameters and exterior boundary data are stated. The proofs of the existence results rely on the Mountain Pass Theorem.

Dynamical system modeling fermionic limit
Dorota Bors and  Robert Stańczy
2018, 23(1): 45-55 doi: 10.3934/dcdsb.2018004 +[Abstract](68) +[HTML](2) +[PDF](475.2KB)

The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate value of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.

NLS-like equations in bounded domains: Parabolic approximation procedure
Alexandre N. Carvalho and  Jan W. Cholewa
2018, 23(1): 57-77 doi: 10.3934/dcdsb.2018005 +[Abstract](62) +[HTML](6) +[PDF](532.7KB)

The article is devoted to semilinear Schrödinger equations in bounded domains. A unified semigroup approach is applied following a concept of Trotter-Kato approximations.Critical exponents are exhibited and global solutions are constructed for nonlinearities satisfying even a certain critical growth condition in \begin{document}$ H^1_0(Ω)$\end{document}.

Optimal control of normalized SIMR models with vaccination and treatment
Maria do Rosário de Pinho , Helmut Maurer and  Hasnaa Zidani
2018, 23(1): 79-99 doi: 10.3934/dcdsb.2018006 +[Abstract](115) +[HTML](6) +[PDF](1384.1KB)

We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment, \begin{document}$M$\end{document}, to the SIR model. We work with the normalized version of the proposed model. For such model we consider the problem of steering the system to a specified target. We consider both a fixed time optimal control problem with \begin{document}$L^1$\end{document} cost and the minimum time problem to drive the system to the target. In contrast to the literature, we apply different techniques of optimal control to our problems of interest.Using the direct method, we first solve the fixed time problem and then proceed to validate the computed solutions using both necessary conditions and second order sufficient conditions. Noteworthy, we perform a sensitivity analysis of the solutions with respect to some parameters in the model. We also use the Hamiltonian Jacobi approach to study how the minimum time function varies with respect to perturbations of the initial conditions. Additionally, we consider a multi-objective approach to study the trade off between the minimum time and the social costs of the control of diseases. Finally, we propose the application of Model Predictive Control to deal with uncertainties of the model.

Exact asymptotics of positive solutions to Dickman equation
Josef DiblÍk and  Rigoberto Medina
2018, 23(1): 101-121 doi: 10.3934/dcdsb.2018007 +[Abstract](62) +[HTML](2) +[PDF](484.7KB)

The paper considers the Dickman equation

for \begin{document} $t \to \infty $ \end{document}. The number theory uses what is called a Dickman (or Dickman -de Bruijn) function, which is the solution to this equation defined by an initial function \begin{document} $x(t)=1$ \end{document} if \begin{document} $0≤ t≤ 1$ \end{document}. The Dickman equation has two classes of asymptotically different positive solutions. The paper investigates their asymptotic behaviors in detail. A structure formula describing the asymptotic behavior of all solutions to the Dickman equation is given, an improvement of the well-known asymptotic behavior of the Dickman function, important in number theory, is derived and the problem of whether a given initial function defines dominant or subdominant solution is dealt with.

Asymptotic properties of delayed matrix exponential functions via Lambert function
Josef Diblík and  Zdeněk Svoboda
2018, 23(1): 123-144 doi: 10.3934/dcdsb.2018008 +[Abstract](51) +[HTML](6) +[PDF](551.3KB)

In the case of first-order linear systems with single constant delay and with constant matrix, the application of the well-known "step by step" method (when ordinary differential equations with delay are solved) has recently been formalized using a special type matrix, called delayed matrix exponential. This matrix function is defined on the intervals \begin{document} $(k-1)τ≤q t<kτ$ \end{document}, \begin{document} $k=0,1,\dots$ \end{document} (where \begin{document} $τ>0$ \end{document} is a delay) as different matrix polynomials, and is continuous at nodes \begin{document} $t=kτ$ \end{document}. In the paper, the asymptotic properties of delayed matrix exponential are studied for \begin{document} $k\to∞$ \end{document} and it is, e.g., proved that the sequence of values of a delayed matrix exponential at nodes is approximately represented by a geometric progression. A constant matrix has been found such that its matrix exponential is the "quotient" factor that depends on the principal branch of the Lambert function. Applications of the results obtained are given as well.

Sensitivity of combined chemo-and antiangiogenic therapy results in different models describing cancer growth
Marzena Dolbniak , Malgorzata Kardynska and  Jaroslaw Smieja
2018, 23(1): 145-160 doi: 10.3934/dcdsb.2018009 +[Abstract](406) +[HTML](10) +[PDF](7208.9KB)

This paper is concerned with analysis of two anticancer therapy models focused on sensitivity of therapy outcome with respect to model structure and parameters. Realistic periodic therapies are considered, combining cytotoxic and antiangiogenic agents, defined on a fixed time horizon. Tumor size at the end of therapy and average tumor size calculated over therapy horizon are chosen to represent therapy outcome. Sensitivity analysis has been performed numerically, concentrating on model parameters and structure at one hand, and on treatment protocol parameters, on the other. The results show that sensitivity of the therapy outcome highly depends on the model structure, helping to discern a good model. Moreover, it is possible to use this analysis to find a good protocol in case of heterogeneous tumors.

Detecting features of epileptogenesis in EEG after TBI using unsupervised diffusion component analysis
Dominique Duncan , Paul Vespa and  Arthur W. Toga
2018, 23(1): 161-172 doi: 10.3934/dcdsb.2018010 +[Abstract](103) +[HTML](10) +[PDF](541.0KB)

Epilepsy is among the most common serious disabling disorders of the brain, and the global burden of epilepsy exerts a tremendous cost to society. Most people with epilepsy have acquired forms, and the development of antiepileptogenic interventions could potentially prevent or cure these epilepsies [3,13]. The discovery of potential antiepileptogenic treatments is currently a high research priority. Clinical validation would require a means to identify populations of patients at particular high risk for epilepsy after a potential epileptogenic insult to know when to treat and to document prevention or cure. We investigate the development of post-traumatic epilepsy (PTE) following traumatic brain injury (TBI), because this condition offers the best opportunity to know the time of onset of epileptogenesis in patients. Epileptogenesis is common after TBI, and because much is known about the physical history of PTE, it represents a near-ideal human model in which to study the process of developing seizures.

Using scalp and depth EEG recordings for six patients, the goal of our analysis is to find a way to quantitatively detect features in the EEG that could potentially help predict seizure onset post trauma. Unsupervised Diffusion Component Analysis [5], a novel approach based on the diffusion mapping framework [4], reduces data dimensionality and provides pattern recognition that can be used to distinguish different states of the patient, such as seizures and non-seizure spikes in the EEG. This method is also adapted to the data to enable the extraction of the underlying brain activity. Previous work has shown that such techniques can be useful for seizure prediction [6].

Some new results that demonstrate how this algorithm is used to detect spikes in the EEG data as well as other changes over time are shown. This nonlinear and local network approach has been used to determine if the early occurrences of specific electrical features of epileptogenesis, such as interictal epileptiform activity and morphologic changes in spikes and seizures, during the initial week after TBI predicts the development of PTE.

A global inversion theorem for functions with singular points
Piotr Fijałkowski
2018, 23(1): 173-180 doi: 10.3934/dcdsb.2018011 +[Abstract](50) +[HTML](1) +[PDF](321.2KB)

In this paper, we consider a certain theorem on the global invertibility of a \begin{document} $C^{2n+1}$ \end{document}-mapping between Banach spaces with a singular point in which the derivatives of order up to \begin{document} $2n$ \end{document} vanish. The theorem is illustrated by several applications.

Some remarks on the Gottman-Murray model of marital dissolution and time delays
Urszula Foryś
2018, 23(1): 181-191 doi: 10.3934/dcdsb.2018012 +[Abstract](56) +[HTML](1) +[PDF](153.1KB)

In the paper we consider mathematical model proposed by Gottman, Murray and collaborators to describe marital dissolution. This model is described in the framework of discrete dynamical system reflecting emotional states of wife and husband during consecutive rounds of talks between spouses. The model is, however, non-symmetric. To make it symmetric, one need to assume that the husband reacts with delay. Following this idea we consider the influence of time delays in the reaction terms of wife or/and husband. The delay means that one or both of spouses split their attention between present and previous rounds of talks. We study possibility of the change of stability with increasing delay. Surprisingly, it occurs that the delay has no impact on the stability, that is the condition of stability proposed by Murray remains unchanged.

Free boundary problems arising in biology
Avner Friedman
2018, 23(1): 193-202 doi: 10.3934/dcdsb.2018013 +[Abstract](100) +[HTML](1) +[PDF](355.7KB)

The present paper describes the general structure of free boundary problems for systems of PDEs modeling biological processes. It then proceeds to review two recent examples of the evolution of a plaque in the artery, and of a granuloma in the lung. Simplified versions of these models are formulated, and rigorous mathematical results and open questions are stated.

Multiplicity results for discrete anisotropic equations
Marek Galewski , Shapour Heidarkhani and  Amjad Salari
2018, 23(1): 203-218 doi: 10.3934/dcdsb.2018014 +[Abstract](72) +[HTML](1) +[PDF](450.2KB)

In this article we continue the study of discrete anisotropic equations and we will provide a new multiplicity results of the solutions for a discrete anisotropic equation. The procedure viewed here is according to variational methods and critical point theory. In fact, using a consequence of the local minimum theorem due Bonanno and mountain pass theorem we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by mingling two algebraic conditions on the nonlinear term employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

Periodic solutions for nonlinear nonmonotone evolution inclusions
Leszek Gasiński and  Nikolaos S. Papageorgiou
2018, 23(1): 219-238 doi: 10.3934/dcdsb.2018015 +[Abstract](65) +[HTML](1) +[PDF](485.6KB)

We study periodic problems for nonlinear evolution inclusions defined in the framework of an evolution triple \begin{document} $(X,H,X^*)$ \end{document} of spaces. The operator \begin{document} $A(t,x)$ \end{document} representing the spatial differential operator is not in general monotone. The reaction (source) term \begin{document} $F(t,x)$ \end{document} is defined on \begin{document} $[0,b]× X$ \end{document} with values in \begin{document} $2^{X^*}\setminus\{\emptyset\}$ \end{document}. Using elliptic regularization, we approximate the problem, solve the approximation problem and pass to the limit. We also present some applications to periodic parabolic inclusions.

Qualitative properties of solutions of higher order difference equations with deviating arguments
Alina Gleska and  Małgorzata Migda
2018, 23(1): 239-252 doi: 10.3934/dcdsb.2018016 +[Abstract](95) +[HTML](2) +[PDF](403.0KB)

In the paper the general higher order difference equation

with several deviating arguments is considered. According to the kind of the deviations \begin{document} $σ_i$ \end{document} sufficient conditions for the equation to have property A and B are established.

Monotonic solutions of a higher-order neutral difference system
Robert Jankowski , Barbara Łupińska , Magdalena Nockowska-Rosiak and  Ewa Schmeidel
2018, 23(1): 253-261 doi: 10.3934/dcdsb.2018017 +[Abstract](83) +[HTML](1) +[PDF](320.5KB)

A class of a higher-order nonlinear difference system with delayed arguments where the first equation of the system is of a neutral type is considered. A classification of non-oscillatory solutions is given and results on their boundedness or unboundedness are derived. The obtained results are illustrated by examples.

Optimal control problem for a viscoelastic beam and its galerkin approximation
Andrzej Just and  Zdzislaw Stempień
2018, 23(1): 263-274 doi: 10.3934/dcdsb.2018018 +[Abstract](60) +[HTML](3) +[PDF](366.0KB)

This paper is concerned with the optimal control problem of the vibrations of a viscoelastic beam, which is governed by a nonlinear partial differential equation. We discuss the initial-boundary problem for the cases when the ends of the beam are clamped or hinged. We define the weak solution of this initial-boundary problem. Our control problem is formulated by minimization of a functional where the state of a system is the solution of viscoelastic beam equation. We use the Galerkin method to approximate the solution of our control problem with respect to a spatial variable. Based on the finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation points of the optimal solutions of the discrete family control problems exist and each of these points is the solution of the original optimal control problem.

Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals
Igor Kossowski and  Katarzyna Szymańska-Dębowska
2018, 23(1): 275-281 doi: 10.3934/dcdsb.2018019 +[Abstract](139) +[HTML](1) +[PDF](308.2KB)

We study the nonlinear boundary value problem consisting of a system of second order differential equations and boundary conditions involving a Riemann-Stieltjes integrals. Our proofs are based on the generalized Miranda Theorem.

Arzelà-Ascoli's theorem in uniform spaces
Mateusz Krukowski
2018, 23(1): 283-294 doi: 10.3934/dcdsb.2018020 +[Abstract](74) +[HTML](2) +[PDF](386.0KB)

In the paper, we generalize the Arzelà-Ascoli's theorem in the setting of uniform spaces. At first, we recall the Arzelà-Ascoli theorem for functions with locally compact domains and images in uniform spaces, coming from monographs of Kelley and Willard. The main part of the paper introduces the notion of the extension property which, similarly as equicontinuity, equates different topologies on \begin{document}$C(X,Y)$\end{document}. This property enables us to prove the Arzelà-Ascoli's theorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki's theorem, we establish the relative compactness of subfamily of \begin{document}$C({\mathbb{R}},{\mathcal{D}}'({\mathbb{R}}^n))$\end{document}.

The Krasnosel'skii formula for parabolic differential inclusions with state constraints
Wojciech Kryszewski , Dorota Gabor and  Jakub Siemianowski
2018, 23(1): 295-329 doi: 10.3934/dcdsb.2018021 +[Abstract](80) +[HTML](1) +[PDF](671.3KB)

We consider a constrained semilinear evolution inclusion of parabolic type involving an \begin{document}$m$\end{document}-dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the \begin{document}$R_δ$\end{document}-description of the set of solutions surviving in the constraining area and show a relation between the fixed point index of the Krasnosel'skii-Poincaré operator of translation along trajectories associated with the problem and the appropriately defined constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied differential problems.

Optimal control applied to a generalized Michaelis-Menten model of CML therapy
Urszula Ledzewicz and  Helen Moore
2018, 23(1): 331-346 doi: 10.3934/dcdsb.2018022 +[Abstract](75) +[HTML](2) +[PDF](451.6KB)

We generalize a previously-studied model for chronic myeloid leu-kemia (CML) [13,10] by incorporating a differential equation which has a Michaelis-Menten model as the steady-state solution to the dynamics. We use this more general non-steady-state formulation to represent the effects of various therapies on patients with CML and apply optimal control to compute regimens with the best outcomes. The advantage of using this more general differential equation formulation is to reduce nonlinearities in the model, which enables an analysis of the optimal control problem using Lie-algebraic computations. We show both the theoretical analysis for the problem and give graphs that represent numerically-computed optimal combination regimens for treating the disease.

Optimal control of the discrete-time fractional-order Cucker-Smale model
Agnieszka B. Malinowska and  Tatiana Odzijewicz
2018, 23(1): 347-357 doi: 10.3934/dcdsb.2018023 +[Abstract](67) +[HTML](2) +[PDF](379.4KB)

We obtain necessary optimality conditions for the discrete-time fractional-order Cucker-Smale optimal control problem. By using fractional order differences on the left side of nonlinear system we introduce memory effects to the considered problem.

Periodic solutions of a $2$-dimensional system of neutral difference equations
Małgorzata Migda , Ewa Schmeidel and  Małgorzata Zdanowicz
2018, 23(1): 359-367 doi: 10.3934/dcdsb.2018024 +[Abstract](72) +[HTML](1) +[PDF](352.9KB)

The 2-dimensional system of neutral type nonlinear difference equations with delays in the following form

is considered. In this paper we use Schauder's fixed point theorem to study the existence of periodic solutions of the above system.

Existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with $p$-Laplacian
Magdalena Nockowska-Rosiak , Piotr Hachuła and  Ewa Schmeidel
2018, 23(1): 369-375 doi: 10.3934/dcdsb.2018025 +[Abstract](65) +[HTML](1) +[PDF](314.1KB)

This work is devoted to the study of the existence of uncountably many asymptotically constant solutions to discrete nonlinear three-dimensional system with p-Laplacian.

Stability of stochastic semigroups and applications to Stein's neuronal model
Katarzyna PichÓr and  Ryszard Rudnicki
2018, 23(1): 377-385 doi: 10.3934/dcdsb.2018026 +[Abstract](66) +[HTML](19) +[PDF](346.0KB)

A new theorem on asymptotic stability of stochastic semigroups is given. This theorem is applied to a stochastic semigroup corresponding to Stein's neuronal model. Asymptotic properties of models with and without the refractory period are compared.

Numerical treatment of contact problems with thermal effect
Anna Ochal and  Michal Jureczka
2018, 23(1): 387-400 doi: 10.3934/dcdsb.2018027 +[Abstract](51) +[HTML](1) +[PDF](536.4KB)

The paper deals with the formulation and the finite element approximation of a quasi-static thermoviscoelastic problem which describes frictional contact between a deformable body and a rigid foundation. The contact is modeled by normal damped response condition whereas the friction is described by the Coulomb law of dry friction. The weak formulation of the model consists of a coupled system of the variational inequality for the displacement and the parabolic equation for the temperature. The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model.

The role of optimism and pessimism in the dynamics of emotional states
Monika Joanna Piotrowska , Joanna Górecka and  Urszula Foryś
2018, 23(1): 401-423 doi: 10.3934/dcdsb.2018028 +[Abstract](55) +[HTML](1) +[PDF](871.0KB)

In this paper we make an attempt to study the influence of optimism and pessimism into our social life. We base on the model considered earlier by Rinaldi and Gragnani (1998) and Rinaldi et al. (2010) in the context of romantic relationships. Liebovitch et al. (2008) used the same model to describe competition between communicating people or groups of people. Considered system of non-linear differential equations assumes that the emotional state of an actor at any time is affected by the state of each actor alone, rate of return to that state, second actor's emotional state and mutual sympathy. Using this model we describe the change of emotions of both actors as a result of a single meeting. We try to explain who wants to meet whom and why. Interpreting the results, we focus on the analysis of the impact of a person's attitude to life (optimism or pessimism) on establishing emotional relations. It occurs that our conclusions are not always obvious from the psychological point of view. Moreover, using this model, we are able to explain such strange behavior as so-called Stockholm syndrom.

Treatment of glioma with virotherapy and TNF-α inhibitors: Analysis as a dynamical system
Elzbieta Ratajczyk , Urszula Ledzewicz , Maciej Leszczyński and  Heinz Schättler
2018, 23(1): 425-441 doi: 10.3934/dcdsb.2018029 +[Abstract](59) +[HTML](7) +[PDF](1034.4KB)

Oncolytic viruses are genetically altered replication-competent vi-ruses which upon death of a cancer cell produce many new viruses that then infect neighboring tumor cells. A mathematical model for virotherapy of glioma is analyzed as a dynamical system for the case of constant viral infusions and TNF-α inhibitors. Aside from a tumor free equilibrium point, the system also has positive equilibrium point solutions. We investigate the number of equilibrium point solutions depending on the burst number, i.e., depending on the number of new viruses that are released from a dead cancer cell and then infect neighboring tumor cells. After a transcritical bifurcation with a positive equilibrium point solution, the tumor free equilibrium point becomes asymptotically stable and if the average viral load in the system lies above a threshold value related to the transcritical bifurcation parameter, the tumor size shrinks to zero exponentially. Other bifurcation events such as saddle-node and Hopf bifurcations are explored numerically.

Optimal control of a delayed HIV model
Filipe Rodrigues , Cristiana J. Silva , Delfim F. M. Torres and  Helmut Maurer
2018, 23(1): 443-458 doi: 10.3934/dcdsb.2018030 +[Abstract](65) +[HTML](2) +[PDF](620.5KB)

We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function representing the efficiency of reverse transcriptase inhibitors and consider the pharmacological delay associated to the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.

Does assortative mating lead to a polymorphic population? A toy model justification
Ryszard Rudnicki and  Radoslaw Wieczorek
2018, 23(1): 459-472 doi: 10.3934/dcdsb.2018031 +[Abstract](99) +[HTML](46) +[PDF](1511.3KB)

We consider a model of phenotypic evolution in populations with assortative mating of individuals. The model is given by a nonlinear operator acting on the space of probability measures and describes the relation between parental and offspring trait distributions. We study long-time behavior of trait distribution and show that it converges to a combination of Dirac measures. This result means that assortative mating can lead to a polymorphic population and sympatric speciation.

Solution to a stochastic pursuit model using moment equations
Miroslava Růžičková , Irada Dzhalladova , Jitka Laitochová and  Josef Diblík
2018, 23(1): 473-485 doi: 10.3934/dcdsb.2018032 +[Abstract](77) +[HTML](5) +[PDF](1334.4KB)

The paper investigates the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters. The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. Simulations are presented to test effectiveness of the approach.

On homoclinic solutions for a second order difference equation with p-Laplacian
Robert Stegliński
2018, 23(1): 487-492 doi: 10.3934/dcdsb.2018033 +[Abstract](63) +[HTML](3) +[PDF](323.2KB)

In this paper, we obtain conditions under which the difference equation

has infinitely many homoclinic solutions. A variant of the fountain theorem is utilized in the proof of our theorem. Some known results in the literature are extended and complemented.

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