On Two Systems of Difference Equations

We give very short and elegant proofs of the main results in the work of Yalcinkaya et al. (2008)

On a solvable class of nonlinear difference equations of fourth order

We consider a class of nonlinear difference equations of the fourth order, which extends some equations in the literature. It is shown that the class of equations is solvable in closed form explaining theoretically, among other things, solvability of some previously considered very special cases. We also present some applications of the main theorem through two examples, which show that some results in the literature are not correct

On a family of nonlinear difference equations of the fifth order solvable in closed form

We present some closed-form formulas for the general solution to the family of difference equations $x_{n+1} = \Phi^{-1}\left(\Phi(x_{n-1})\frac{{\alpha} \Phi(x_{n-2})+{\beta} \Phi(x_{n-4})}{{\gamma} \Phi(x_{n-2})+{\delta} \Phi(x_{n-4})}\right),$ for $n\in{\mathbb N}_0$ where the initial values $x_{-j}$, $j = \overline{0, 4}$ and the parameters ${\alpha}, {\beta}, {\gamma}$ and ${\delta}$ are real numbers satisfying the conditions ${\alpha}^2+{\beta}^2\ne 0,$ ${\gamma}^2+{\delta}^2\ne 0$ and $\Phi$ is a function which is a homeomorphism of the real line such that $\Phi(0) = 0,$ generalizing in a natural way some closed-form formulas to the general solutions to some very special cases of the family of difference equations which have been presented recently in the literature. Besides this, we consider in detail some of the recently formulated statements in the literature on the local and global stability of the equilibria as well as on the boundedness character of positive solutions to the special cases of the difference equation and give some comments and results related to the statements.

On Some Symmetric Systems of Difference Equations

Here we show that the main results in the papers by Yalcinkaya (2008), Yalcinkaya and Cinar (2010), and Yalcinkaya, Cinar, and Simsek (2008), as well as a conjecture from the last mentioned paper, follow from a slight modification of a result by G. Papaschinopoulos and C. J. Schinas. We also give some generalizations of these results

Note on theoretical and practical solvability of a class of discrete equations generalizing the hyperbolic-cotangent class

There has been some recent interest in investigating the hyperbolic-cotangent types of difference equations and systems of difference equations. Among other things their solvability has been studied. We show that there is a class of theoretically solvable difference equations generalizing the hyperbolic-cotangent one. Our analysis shows a bit unexpected fact, namely that the solvability of the class is based on some algebraic relations, not closely related to some trigonometric ones, which enable us to solve them in an elegant way. Some examples of the difference equations belonging to the class which are practically solvable are presented, as well as some interesting comments on connections of the equations with some iteration processes

Representation of solutions of a solvable nonlinear difference equation of second order

We present a representation of well-defined solutions to the following nonlinear second-order difference equation xn+1 = a + b xn c xnxn−1 , n ∈ N0, where parameters a, b, c, and initial values x−1 and x0 are complex numbers such that c 6= 0, in terms of the parameters, initial values, and a special solution to a thirdorder homogeneous linear difference equation with constant coefficients associated to the nonlinear difference equation, generalizing a recent result in the literature, completing the proof therein by using an essentially constructive method, and giving some theoretical explanations related to the method for solving the difference equation. We also give a more concrete representation of the solutions to the nonlinear difference equation by calculating the special solution to the third-order homogeneous linear difference equation in terms of the zeros of the characteristic polynomial associated to the linear difference equation