Global Behaviour of Solutions of Nonlinear Delay Difference Equations*

In this paper global asymptotic stability and asymptotic behaviour of solutions of nonlinear delay difference equations has been studied and a few sets of sufficient conditions for global asymptotic stability are derived.     

On the Behaviour of Solutions of a Class of Third Order Difference Equations

In this paper, the structure of the solution space of y n +3 + ry n +2 + qy n +1 + py n =0, n S 0, is studied, keeping oscillatory/nonoscillatory behaviour of solutions of the equation in view, where p , q and r are constants. Some of these results are generalized partially to hold for y n +3 + r n y n +2 + q n y n +1 + p n y n =0, n S 0, where { p n }, { q n } and { r n } are sequences of real numbers.     

On Perturbed Discrete Boundary Value Problems

In this paper, we study nonlinear discrete boundary value problems of the form x ( t +1)= A ( t ) x ( t )+ h ( t )+ k f ( t , x ( t ), k ) subject to Bx (0)+ Dx ( J )= u + k g ( x (0), x ( J ), k ) where k is a "small" parameter. Our main concern is the case of resonance, that is, the situation where the associated linear homogeneous boundary value problem x ( t +1)= A ( t ) x ( t ), Bx (0)+ Dx ( J )=0 admits nontrivial solutions. We establish conditions for the solvability of the nonlinear boundary value problem when k is "small". We also establish qualitative properties of these solutions.     

On Detecting Solutions of Polynomial Nonlinear Difference Equations

In this paper a method for discovering solutions of nonlinear polynomial difference equations is presented. It is based on the concepts of i -operator and star-product. These notions create a proper algebraic background by means of which we can find linear equations "included" into the original nonlinear one and to seek for solutions among them. A corresponding algorithm and some examples are also provided.     

Existence and structure results on almost periodic solutions of difference equations

We study the almost periodic solutions of Euler equations and of some more general Difference Equations. We consider two different notions of almost periodic sequences, and we establish some relations between them. We build suitable sequences spaces and we prove some properties of these spaces. We also prove properties of Nemytskii operators on these spaces. We build a variational approach to establish existence of almost periodic solutions as critical points, We obtain existence theorems fornonautonomous linear equations and for an Euler equation with a concave and coercive Lagrangian. We also use a Fixed Point approach to obtain existence results for quasi-linear Difference Equations.     

Types of convergence for sequences of probability densities and attractors

Some necessary and sufficient conditions on densities convergence for the existence of an ergoidc mixing, or exact dynamical system on a probability space given in [8] are extended to a measure space to obtain an ergodic, mixing or exact dynamical system on this measure space. More general sufficient conditions are given for the existence of those kinds of dynamical systems; as a consequence of these conditions it is obtained an ergodic, exact attractor for the orbits of almost every phase state. This orbits’ behaviour recall the thermodynamical evolution of systems from nonequilibrium to equilibrium states     

Statistical Dynamics of Solitons in Optical Fibers

The soliton perturbation theory is used to study and analyze the stochastic perturbation of optical solitons in addition to deterministic perturbations of optical solitons that are governed by the nonlinear Schro¨dinger's equation. The Langevin equations are derived and analyzed. The deterministic perturbations that are considered here are of both Hamiltonian as well as of non-Hamiltonian type.     

Application of variational iteration method for Hamilton–Jacobi–Bellman equations

In this paper, we use the variational iteration method (VIM) for optimal control problems. First, optimal control problems are transferred to Hamilton–Jacobi–Bellman (HJB) equation as a nonlinear first order hyperbolic partial differential equation. Then, the basic VIM is applied to construct a nonlinear optimal feedback control law. By this method, the control and state variables can be approximated as a function of time. Also, the numerical value of the performance index is obtained readily. In view of the convergence of the method, some illustrative examples are presented to show the efficiency and reliability of the presented method.     

Higher-order Nonlinear System of Equations of Changing Type

In practical problems there appears higher-order equations of changing type([1]). But, there are only a few of papers concerning this type of equations ([2]-[6]). In this paper, the global existence of regular solutions to the Initial-boundary value problem for a class of higher-order system of equations of changing type with a strong nonlinear term is studied.