Asymptotically almost periodic solutions of nonlinear Volterra difference equations with unbounded delay

The existence of almost periodic solutions of nonlinear Volterra difference equations with unbounded delay is obtained by using uniform stability properties of a bounded solution. An example is also given to illustrate obtained results.     

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.     

Onl p solutions of discrete Volterra equations

Summary In the paper a discrete analog to the Volterra nonlinear integral equation is discussed. Weighted norms are used to find sufficient conditions that all solutions of such equations are elements of anl ^p space.Some generalizations ofl ^p spaces are also considered and the corresponding sufficient conditions are established.     

On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems

In this paper, we obtain a new sufficient condition on the existence of homoclinic solutions of a class of discrete nonlinear periodic systems by using critical point theory in combination with periodic approximations. We prove that it is also necessary in some special cases.     

Multiplicity of periodic solutions for second-order discrete Hamiltonian systems with a small forcing term

Some sufficient conditions for the existence of three periodic solutions are obtained for the second-order discrete Hamiltonian systems with a small forcing term by using a three-critical-point theorem.     

Periodic solutions for a class of higher-dimension functional differential equations with impulses

By employing a fixed-point theorem in cones, we establish some criteria for existence of positive periodic solutions of a class of n$n$-dimension periodic functional differential equations with impulses. We also give some applications to several biomathematical models and new results are obtained.     

Pseudo almost periodic solutions to some non-autonomous neutral functional differential equations with unbounded delay

In this paper we give some sufficient conditions, which ensure the existence of pseudo almost periodic solutions to some classes of non-autonomous first-order partial neutral functional-differential with unbounded delay.     

Applying carvalho's method to find periodic solutions of difference equations

In this paper we apply the method initially developed in [1] for differential-difference equations, to the case of difference equations, in order to find 2 and 3-periodic solutions of some equations that often appear in the literatures as are for instance the case of Applications 2,5 which are examples of population growth models, and Application 4, which is a standard example of nonlinear higher order scalar difference equation depending on two parameters (see, Kocik and Ladas [3]).     

Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical Systems

The local existence and local asymptotic stability of nontrivial p-periodic solutions of p-periodically forced discrete systems are proven using Liapunov-Schmidt methods. The periodic solutions bifurcate transcritically from the trivial solution at the critical value n=n^cr of the bifurcation parameter with a typical exchange of stability. If the trivial solution loses (gains) stability as n is increased through n^cr , then the periodic solutions on the nontrivial bifurcating branch are locally asymptotically stable if and only if they correspond to n>n^cr (n n^cr ).     

Almost periodic solutions of discrete Volterra equations

For discrete Volterra equations with or without delay, we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions. We also investigate the relations among solutions of equations discussed and give an example to illustrate our results.     

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.     

Periodic solutions for dynamic equations on time scales

Massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales. Some interesting properties of the exponential function on time scales are presented. Furthermore, a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented.     

Periodic solutions of nonlinear delay equations

Existence and uniqueness of 2π-periodic solutions of $\text{d}{}^{\mathrm{j}}\text{x(t)}\text{dt}{}^{\mathrm{j}}\text{+}\text{grad}\text{G(x(t ? τ)) = e(t, x(t), x(t ? τ)) (j = 1, 2)}$, where x(t) is in $\text{R}$ⁿ and e(t, u, v) is a given vector function, 2π-periodic in t, are shown under conditions on the spectrum of the Hessian of G. The equation is studied using a fixed point theorem in the space L²(0, 2π). One feature of this approach is that no relationship between the delay and the period is necessary.     

On almost periodic solutions of logistic delay differential equations with almost periodic time dependence

In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458].     

具无穷时滞的中立型Volterra积分微分方程的概周期解

研究了一类具无穷时滞的中立型Volterra积分微分方程的概周期解问题．利用线性系统指数型二分性理论和泛函分析方法,得到了一些关于该方程的概周期解的存在性、唯一性与稳定性的新结果,推广了相关文献的主要结果．     

Periodic solutions for general nonlinear state-dependent delay logistic equations

In this paper, by using the continuation theorem of coincidence degree theory, we investigate the existence of periodic solutions for more general state-dependent delay logistic equations. Several sufficient conditions are given, and the obtained conditions possess important significance in both theories and applications.