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The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, cross-dispersal is considered in a predator–prey model with a patchy environment. A new predator–prey model with cross-dispersal among patches is constructed. A new cross-dispersal matrix is established by the coupling relationship between vertices. First, an existence theorem of the positive equilibrium for the new model is obtained. Secondly, based on the idea of constructing Lyapunov functions and a graph-theoretical approach for coupled systems, sufficient conditions that the positive equilibrium of the new model is globally asymptotically stable in R + 2 n are derived on a network with strongly connected graphs. Thirdly, based on the theory of asymptotically autonomous systems, Lyapunov functions method and graph theory, a stability theorem for the positive equilibrium of the new model is established on a complex network without strongly connected graphs. Finally, two examples are given to illustrate main results. [ABSTRACT FROM AUTHOR]

In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system's parameters. Some numerical simulations are presented to verify the obtained mathematical results. [ABSTRACT FROM AUTHOR]

The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper. [ABSTRACT FROM AUTHOR]

In this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence theorem is proved under some continuity and convexity assumptions on the bifunction and the nonemptiness of the common solution set of Minty equilibrium problems. Furthermore, we demonstrate the quality of our inertial parallel hybrid algorithm by using image restoration, as well as its superior efficiency when compared with previously considered parallel algorithms. [ABSTRACT FROM AUTHOR]

This paper provides iterative construction of a common solution associated with the classes of equilibrium problems (EP) and split convex feasibility problems. In particular, we are interested in the EP defined with respect to the pseudomonotone bifunction, the fixed point problem (FPP) for a finite family of -demicontractive operators, and the split null point problem. From the numerical standpoint, combining various classical iterative algorithms to study two or more abstract problems is a fascinating field of research. We, therefore, propose an iterative algorithm that combines the parallel hybrid extragradient algorithm with the inertial extrapolation technique. The analysis of the proposed algorithm comprises theoretical results concerning strong convergence under a suitable set of constraints and numerical results. [ABSTRACT FROM AUTHOR]

This paper deals with a split equality equilibrium problem for pseudomonotone bifunctions and a split equality hierarchical fixed point problem for nonexpansive and quasinonexpansive mappings. We suggest and analyze an iterative scheme where the stepsizes do not depend on the operator norms, the so-called simultaneous projected subgradient-proximal iterative scheme for approximating a common solution of the split equality equilibrium problem and the split equality hierarchical fixed point problem. Further, we prove a weak convergence theorem for the sequences generated by this scheme. Furthermore, we discuss some consequences of the weak convergence theorem. We present a numerical example to justify the main result. [ABSTRACT FROM AUTHOR]

A class of inertial neural networks (INNs) with reaction-diffusion terms and distributed delays is studied. The existence and uniqueness of the equilibrium point for the considered system is obtained by topological degree theory, and a sufficient condition is given to guarantee global exponential stability of the equilibrium point. Finally, an example is given to show the effectiveness of the results in this paper. [ABSTRACT FROM AUTHOR]

This paper is motivated by the series of research papers that consider parasitoids' external input upon the host–parasitoid interactions. We explore a class of host–parasitoid models with variable release and constant release of parasitoids. We assume that the host population has a constant rate of increase, but we do not assume any density dependence regulation other than parasitism acting on the host population. We compare the obtained results for constant stocking with the results for proportional stocking. We observe that under a specific condition, the release of a constant number of parasitoids can eventually drive the host population (pests) to extinction. There is always a boundary equilibrium where the host population extinct occurs, and the parasitoid population is stabilized at the constant stocking level. The constant and variable stocking can decrease the host population level in the unique interior equilibrium point; on the other hand, the parasitoid population level stays constant and does not depend on stocking. We prove the existence of Neimark–Sacker bifurcation and compute the approximation of the closed invariant curve. Then we consider a few host–parasitoid models with proportional and constant stocking, where we choose well-known probability functions of parasitism. By using the software package Mathematica we provide numerical simulations to support our study. [ABSTRACT FROM AUTHOR]

We propose a modified version of the classical Cesáro means method endowed with the hybrid shrinking projection method to solve the split equilibrium and fixed point problems (SEFPP) in Hilbert spaces. One of the main reasons to equip the classical Cesáro means method with the shrinking projection method is to establish strong convergence results which are often required in infinite-dimensional functional spaces. As a consequence, the convergence analysis is carried out under mild conditions on the underlying shrinking Cesáro means method. We emphasize that the results accounted in this manuscript can be considered as an improvement and generalization of various existing exciting results in this field of study. [ABSTRACT FROM AUTHOR]

In this paper, we build a mathematical model to study the impact of external removable devices on a network with weakly- and strongly-protected computers. The model describes the dynamics between weak, strong, infected computers and susceptible, infected removable media. Analytical investigations of the model produce two equilibrium points: virus-free and endemic. Moreover, we investigate the local and global stability of both equilibria. The existence and stability conditions of the equilibrium points depend primarily on the basic reproduction number ( R 0 ) of the model. Furthermore, we perform numerical simulations to substantiate the analytical results. Also, a sensitivity analysis is carried out to examine the critical parameters that lead to strategies to control the dissipation of viruses. [ABSTRACT FROM AUTHOR]

A Lotka–Volterra predator–prey system incorporating fear effect of the prey and the predator has other food resource is proposed and studied in this paper. It is shown that the trivial equilibrium and the predator free equilibrium are both unstable, and depending on some inequalities, the system may have a globally asymptotically stable prey free equilibrium or positive equilibrium. Our study shows the fear effect is one of the most important factors that lead to the extinction of the prey species. Such a finding is quite different from the known result. Numeric simulations are carried out to show the feasibility of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey metapopulation model with a ring-structured configuration of an arbitrary and finite number of patches. The prey are assumed to disperse between the connected patches with a constant dispersal delay. We show that the dispersal delay can induce stability switches exhibiting both stabilizing and destabilizing roles in the stability of the symmetric coexistence equilibrium. Numerical simulations are presented to further illustrate the effects of the dispersal delay, the dispersal rate, the fraction of dispersal due to predation avoidance and the network topology on the number of stability switches. [ABSTRACT FROM AUTHOR]

There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes. [ABSTRACT FROM AUTHOR]

We propose and study a discrete competitive system of the following form: We obtain some conditions for the local stability of the equilibria. Using the iterative method and the comparison principle of a difference equation, we also obtain a set of sufficient conditions that ensure the global stability of the interior equilibrium. Numeric simulations show the feasibility of the main results. Our results supplement and complement some known results. [ABSTRACT FROM AUTHOR]

The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model. [ABSTRACT FROM AUTHOR]

In this paper, we propose to study a class of discrete schistosomiasis models with general incidence function. This model is derived from a continuous schistosomiasis model (in Appl. Math. 4:1682-1693, 2003) by using the backward Euler discretization method with step size $h=1$ . We visit some basic properties of this discrete model and we study the stabilities of the equilibria by constructing some appropriate Lyapunov functional for the endemic equilibrium. [ABSTRACT FROM AUTHOR]

In this article we consider the global behavior of the system of first order piecewise linear difference equations: $x_{n+1} = \vert x_{n}\vert - y _{n} +b$ and $y_{n+1} = x_{n} - \vert y_{n}\vert - d$ where the parameters b and d are any positive real numbers. We show that for any initial condition in $R^{2}$ the solution to the system is eventually the equilibrium, $(2b + d, b)$ . Moreover, the solutions of the system will reach the equilibrium within six iterations. [ABSTRACT FROM AUTHOR]

In this work, we study Hopf bifurcations in the extended Lorenz system, $\dot{x}=y$ , $\dot{y}=mx-ny-mxz-px^{3}$ , $\dot{z}=-az+bx^{2}$ , with five parameters $m,n,p,a,b\in \mathbb{R}$ . For some values of the parameters, this system can be transformed to the classical Lorenz system. In this paper, we give conditions for occurrence of Hopf bifurcation at the equilibrium points. We find totally three limit cycles, each of them located around one of the three equilibria of the system. Numerical simulations illustrate the validity of these conditions and the existence of limit cycles. [ABSTRACT FROM AUTHOR]

We consider a discrete competition model of plankton allelopathy with infinite deviating arguments of the form By using an iterative method we investigate the global attractivity of the interior equilibrium point of the system. [ABSTRACT FROM AUTHOR]

This paper concerns the dynamics of a stochastic SIVR epidemic model with imperfect vaccine where, differently from the epidemic model with perfect vaccine, the vaccinated is perturbed by the noise. This difference is the main difficulty to be conquered to give the threshold $R_{0}^{S}$ . Firstly, $R_{0}^{S}>1$ is proved to be sufficient for persistence in mean of the system. Then, three conditions for the disease to die out are given, which improve the previously known results on extinction of the disease. In case that the disease goes extinct, we show that the disease-free equilibrium is almost surely stable by using the nonnegative semimartingale convergence theorem. [ABSTRACT FROM AUTHOR]

We study a discrete competition model of the form where $x_{i}(k)$ ( $i=1,2$ ) are the population density of the ith species at k-generation. $r_{i}$ , $K_{i}$ , $i=1, 2$ , are all positive constants such that $K_{i}>\alpha_{i}$ , $i=1, 2$ . By using the iterative method and the comparison principle of difference equations we obtain sufficient conditions that ensure the global attractivity of the interior equilibrium of the system. Our result supplements and complements the main result of Yang et al. (Abstr. Appl. Anal. 2014:709124, 2014). [ABSTRACT FROM AUTHOR]

A discrete cooperative model incorporating harvesting that takes the form is proposed and studied in this paper. By using the iterative method and the comparison principle of difference equations, a set of sufficient conditions which ensure the global attractivity of the interior equilibrium of the system is obtained. Numeric simulations show the feasibility of the main result. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic disease dynamics of an SEIS epidemic model with latent patients and active patients. The two parameters $R_{0}^{s}$ and $R_{0}^{*}$ are identified as the disease-free and endemic dynamics of the model. More specifically, we give the almost surely exponential stability of the disease-free equilibrium in terms of $R_{0}^{s}$ , and stochastic endemic dynamics in terms of $R_{0}^{*}$ . The theoretical and numerical results may be useful for studying the dynamics of disease spreading in a randomly fluctuating environment. [ABSTRACT FROM AUTHOR]

We discuss the dynamical properties of a SIRS computer virus propagation model with impulsive detoxication and saturation effect. By the Internet, new antivirus software can be released immediately and take effect quickly after it is running. This leads to the circumstance that many infected computers can be cured in a short time. So impulsive detoxication is a vitally important way for prohibiting the spread of network viruses. The theoretical results show that: (a) the virus-free equilibrium is globally stable when the basic reproduction ratio (BRR) is less than unity, (b) the system is uniformly permanent when BRR exceeds unity, and (c) a supercritical bifurcation occurs when BRR equals unity. Several numerical examples also clearly display the results obtained. Finally, some feasible strategies of eradicating electronic viruses are advised. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator-prey-mutualist system with digestion delay. First, we calculate the threshold value of delay and prove that the positive equilibrium is locally asymptotically stable when the delay is less than the threshold value and the system undergoes a Hopf bifurcation at the positive equilibrium when the delay is equal to the threshold value. Second, by applying the normal form method and center manifold theorem, we investigate the properties of Hopf bifurcation, such as the direction and stability. Finally, some numerical simulations are carried out to verify the main theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we extend the SIR model with vaccination into a fractional-order model by using a system of fractional ordinary differential equations in the sense of the Caputo derivative of order $\alpha\in(0,1]$. By applying fractional calculus, we give a detailed analysis of the equilibrium points of the model. In particular, we analytically obtain a certain threshold value of the basic reproduction number $R_{0}$ and describe the existence conditions of multiple equilibrium points. Moreover, it is shown that the stability region of the equilibrium points increases by choosing an appropriate value of the fractional order α. Finally, the analytical results are confirmed by some numerical simulations for real data related to pertussis disease. [ABSTRACT FROM AUTHOR]

We consider a time delay predator-prey model with Holling type-IV functional response and stage-structured for the prey. Our aim is to observe the dynamics of this model under the influence of gestation delay of the predator. We obtain sufficient conditions for the local stability of each of feasible equilibria of the system and the existence of a Hopf bifurcation at the coexistence equilibrium. By using the normal form theory and center manifold theory we also derive some explicit formulae determining the bifurcation direction and the stability of the bifurcated periodic solutions. Finally, numerical simulations are given to explain the theoretical results. [ABSTRACT FROM AUTHOR]

We investigate the global character of the difference equation of the form with several period-two solutions, where f is increasing in all its variables. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. As an application of our results we give the global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types. [ABSTRACT FROM AUTHOR]

In this paper, a May cooperative system with feedback controls is proposed and studied. The dynamic behaviors of the system are discussed by using the Lyapunov function method. If $b_{i}\neq0$, $i=1,2$, we show that feedback control variables have no influence on the global stability of the unique positive equilibrium of the system, which means that feedback control variables only change the position of the positive equilibrium and retain its global stability property. If $b_{i}= 0$, $i=1,2$, we can make the system which has a unique globally stable equilibrium or has unboundedly large solutions become globally stable. Some examples are given to illustrate the feasibility of the main results. [ABSTRACT FROM AUTHOR]

We generalize the deterministic and the stochastic single-group SIRS epidemic models with saturated incidence rate introduced by Lahrouz, Omari, and Kiouach to the multi-group versions. In the deterministic multi-group model, the fact is highlighted that if the threshold $\mathscr{R}_{0}\leq1$, then the infective condition disappears and it means the extinction of the disease. If $\mathscr{R}_{0}>1$, then there exists an endemic equilibrium in a feasible region. Allowing the noise perturbation, for the stochastic version, we utilize stochastic Lyapunov functions to show the stability of the disease-free equilibrium of system. A detailed analysis is performed on almost surely exponential stability and pth moment exponential stability of the disease-free equilibrium. We also go into several numerical simulations to illustrate how exactly the theoretical results are verified. Good agreement was observed between our theoretical results and numerical simulations. A comprehensive conclusion is provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce stochasticity into a model of SIR with density dependent birth rate. We show that the model possesses non-negative solutions as desired in any population dynamics. We also carry out the globally asymptotical stability of the equilibrium through the stochastic Lyapunov functional method if $R_{0}\le1$. Furthermore, when $R_{0}>1$, we give the asymptotic behavior of the stochastic system around the endemic equilibrium of the deterministic model and show that the solution will oscillate around the endemic equilibrium. We consider that the disease will prevail when the white noise is small and the death rate due to disease is limited. [ABSTRACT FROM AUTHOR]

In this paper, we present a time-delayed echinococcosis transmission model to explore effective control and prevention strategies. We first give the basic reproduction number $R_{0}$. It is shown that if $R_{0}<1$, the disease-free equilibrium is globally asymptotically stable, and if $R_{0}>1$, the disease persists. We further show that the endemic equilibrium is globally asymptotically stable for a special case. Numerical simulations are performed to illustrate our analytic results. We give some sensitivity analysis of some parameters and give some useful comments on controlling the transmission of echinococcosis. [ABSTRACT FROM AUTHOR]

We investigate the local and global character of the equilibrium and the local stability of the period-two solution of the difference equation xn+1 = βxnxn-1+γxn-1²+δxn/Bxnxn-1+Cxn-1²+Dxn where the parameters β, γ, δ, B, C, D are nonnegative numbers which satisfy B + C + D > 0 and the initial conditions xn-1 and x0 are arbitrary nonnegative numbers such that Bxnxn-1 + Cxn-1² + Dxn > 0 for all n ≥ 0. [ABSTRACT FROM AUTHOR]

An epidemic model which describes vector-borne plant diseases is proposed with the aim to investigate the effect of insect vectors on the spread of plant diseases. Firstly, the analytical formula for the basic reproduction number R0 is obtained by using the next generation matrix method, and then the existence of disease-free equilibrium and endemic equilibrium is discussed. Secondly, by constructing a suitable Lyapunov function and employing the theory of additive compound matrices, the threshold for the dynamics is obtained. If R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable, which means that the plant disease will disappear eventually; if R0 > 1, then the endemic equilibrium is globally asymptotically stable, which indicates that the plant disease will persist for all time. Finally some numerical investigations are provided to verify our theoretical results, and the biological implications of the main results are briefly discussed in the last section. [ABSTRACT FROM AUTHOR]

The dynamical behaviors of a discrete-time SIS epidemic model are investigated in this paper. The result indicates that the model undergoes a flip bifurcation and a Hopf bifurcation, as found by using the center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors, such as the period-doubling bifurcation in period-2, -4, -8, quasi-periodic orbits and the chaotic sets. Specifically, when the parameters A, d1, d2, r, λ are fixed at some values and the bifurcation parameter h changes with different values, there exist local stability, Hopf bifurcation, 3-periodic orbits, 7-periodic orbits, period-doubling bifurcation and chaotic sets. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models although the discrete epidemic model is simple. Finally, the feedback control method is used to stabilize chaotic orbits at an unstable endemic equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, we study the qualitative behavior of a Nicholson-Bailey host-parasitoid model given by ..., where R, a and initial conditions x0, y0 are positive real numbers. More precisely, we investigate the local asymptotic stability of the unique positive equilibrium point of this system. Some numerical examples are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and potential uniqueness of equilibrium points and some stability, instability and oscillatory properties of a discrete nonlinear epidemic model, which generalises previous Stevic’s model, whose solutions possess memory from a finite chain of preceding samples. An application example is provided where the proposed model is ‘ad hoc’ adapted to a class of SIS models widely used in epidemiology. [ABSTRACT FROM AUTHOR]

In this paper, we consider the oscillations of numerical solutions for the nonlinear delay differential equations in a hematopoiesis model. Using two θ-methods, namely the linear θ-method and the one-leg θ-method, several conditions, under which the numerical solutions oscillate, are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to an equilibrium point of the original system. Some numerical experiments are provided to support the theoretical results. MSC: 65L05, 65L20. [ABSTRACT FROM AUTHOR]

We consider the family of nonlinear difference equations: xn+1 = (Σ³ i=1fi(xn,...,xn-k) + f4(xn,..., xn-k)f5(xn,...,xn-k))/(f1(xn,...,xn-k)f2 (xn,...,xn-k) + Σ5 i=3fi(xn,...,xn-k)), n = 0,1,..., where fi ε C((0,-∞)k+1, (0,+∞)), for i ε {1,2,4,5}, f3 ε C([0,+∞)k+1, (0,+∞)), k ε {1,2,...} and the initial values x-k, x-k+1,...,x0 ε (0,+∞). We give sufficient conditions under which the unique equilibrium x̄ = 1 of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references. [ABSTRACT FROM AUTHOR]

It is supposed that the fractional difference equation xn+1 = (μ + &Sigmaj=0k ajxn-j)/λ + &Sigmaj=0k bjxn-j), n = 0,1,..., has an equilibrium point x̂ and is exposed to additive stochastic perturbations type of σ(xn - x̂)ζn+1 that are directly proportional to the deviation of the system state xn from the equilibrium point x̂. It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted. [ABSTRACT FROM AUTHOR]

We investigate some qualitative behavior of the solutions of the difference equation xn+1 = axn -bxn/(cxn -dxn-1), n = 0, 1, … , where the initial conditions x-1, x0 are arbitrary real numbers and a, b, c, d are positive constants. [ABSTRACT FROM AUTHOR]

For a delay difference equation N(n + 1) - N(n) = N(n)Σ k=1m ak (n)(1 - N(gk(n))/K), ak(n) ≥ 0, gk(n) ≥ n, K > 0, a connection between oscillation properties of this equation and the corresponding linear equations is established. Explicit nonoscillation and oscillation conditions are presented. Positiveness of solutions is discussed. [ABSTRACT FROM AUTHOR]

We study the global behavior of positive solutions of the system of rational difference equations xn+1 = f (yn-q,xn-s), yn+1 = g(xn-t , yn-p), n = 0, 1, 2, …, where p,q, s, t ∈ {0, 1, 2, …} with s ≥ t and p ≥ q, the initial values x-s,x-s+1, … ,x0, y-p, y-p+1, … , y0 ∈ (0,+∞). We give sufficient conditions under which every positive solution of this system converges to the unique positive equilibrium. [ABSTRACT FROM AUTHOR]

We investigate in this paper the global behavior of the following difference equation: xn+1 = (Pk(xn-i0 ,xn-i1, …, xn-i2k) + b)/(Qk(xn-i0 ,xn-i1, …, xn-i2k) + b), n = 0, 1, …, under appropriate assumptions, where b ∈ [0,∞), k ≥ 1, i0, i1, … , i2k ∈ {0, 1, …} with i0 < i1 < … < i2k, the initial conditions xi-2k, xi-2k+1, … ,x0 ∈ (0,∞).We prove that unique equilibrium x̄ = 1 of that equation is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+xn)/(b + yn), yn+1 = (d + yn)/(e +xn), n = 0, 1, …, where the parameters a,b,d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x. [ABSTRACT FROM AUTHOR]

We study the global asymptotic behavior of the positive solutions of the system of rational difference equations xn+1 = f (xn, yn-k), yn+1 = f (yn,xn-k), n = 0, 1, 2, …, under appropriate assumptions, where k ∈ {1, 2, …} and the initial values x-k,x-k+1, … ,x0, y-k, y-k+1, … , y0 ∈ (0,+∞). We give sufficient conditions under which every positive solution of this equation converges to a positive equilibrium. The main theorem in [1] is included in our result. [ABSTRACT FROM AUTHOR]

In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We introduce a new iterative method for finding a common element of the set of fixed points of pseudo-contractive mapping, the set of solutions to a variational inclusion and the set of solutions to a generalized equilibrium problem in a real Hilbert space. We provide some results about strongly and weakly convergent of the iterative scheme sequence to a point p ∈ Ω which is the unique solution of a variational inequality, where Ω is an intersection of set as given by Ω = F (S) ∩ (A + B) − 1 (0) ∩ N − 1 (0) ∩ GEP (F , M) ≠ ∅ . This gives us a common solution. Also, We show that our results extend some published recent results in this field. Finally, we provide an example to illustrate our main result. [ABSTRACT FROM AUTHOR]

Iterative algorithms are widely applied to solve convex optimization problems under a suitable set of constraints. In this paper, we develop an iterative algorithm whose architecture comprises a modified version of the forward-backward splitting algorithm and the hybrid shrinking projection algorithm. We provide theoretical results concerning weak and strong convergence of the proposed algorithm towards a common solution of the monotone inclusion problem and the split mixed equilibrium problem in Hilbert spaces. Moreover, numerical experiments compare favorably the efficiency of the proposed algorithm with the existing algorithms. As a consequence, our results improve various existing results in the current literature. [ABSTRACT FROM AUTHOR]