14 research outputs found
Boundedness character of a max-type system of difference equations of second order
The boundedness character of positive solutions of the next max-type system of difference equations
with , is characterized
Note on the binomial partial difference equation
Some formulas for the "general solution" to the binomial partial difference equation
are known in the literature. However, it seems that there is no such a formula on the most natural domain connected to the equation, that is, on the set By using a connection with the scalar linear first order difference equation we show that the equation on the domain , can be solved in closed form by presenting a formula for the solution in terms of the "side" values , ,
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
for where the initial values , and the parameters and are real numbers satisfying the conditions and is a function which is a homeomorphism of the real line such that 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.
Product-type system of difference equations of second-order solvable in closed form
This paper presents solutions of the following second-order system of difference equations
where , and , in closed form
Solvability of some classes of nonlinear first-order difference equations by invariants and generalized invariants
[[abstract]]We introduce notion of a generalized invariant for difference equations, which naturally generalizes notion of an invariant for the equations. Some motivations, basic examples and methods for application of invariants in the theory of solvability of difference equations are given. By using an invariant, as well as, a generalized invariant it is shown solvability of two classes of nonlinear first-order difference equations of interest, for nonnegative initial values and parameters appearing therein, considerably extending and explaining some problems in the literature. It is also explained how these classes of difference equations can be naturally obtained from some linear second-order difference equations with constant coefficients. [ABSTRACT FROM AUTHOR]
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On a two-dimensional solvable system of difference equations
Here we solve the following system of difference equations xn+1 = ynynâ2 bxnâ1 + aynâ2 , yn+1 = xnxnâ2 dynâ1 + cxnâ2 , n â N0, where parameters a, b, c, d and initial values xâj , yâj , j = 0, 2, are complex numbers, and give a representation of its general solution in terms of two specially chosen solutions to two homogeneous linear difference equations with constant coefficients associated to the system. As some applications of the representation formula for the general solution we obtain solutions to four very special cases of the system recently presented in the literature and proved by induction, without any theoretical explanation how they can be obtained in a constructive way. Our procedure presented here gives some theoretical explanations not only how the general solutions to the special cases are obtained, but how is obtained general solution to the general system
On a two-dimensional solvable system of difference equations
Here we solve the following system of difference equations
where parameters and initial values , are complex numbers, and give a representation of its general solution in terms of two specially chosen solutions to two homogeneous linear difference equations with constant coefficients associated to the system. As some applications of the representation formula for the general solution we obtain solutions to four very special cases of the system recently presented in the literature and proved by induction, without any theoretical explanation how they can be obtained in a constructive way. Our procedure presented here gives some theoretical explanations not only how the general solutions to the special cases are obtained, but how is obtained general solution to the general system
On some classes of solvable difference equations related to iteration processes
We present several classes of nonlinear difference equations solvable in closed form, which can be obtained from some known iteration processes, and for some of them we give some generalizations by presenting methods for constructing them. We also conduct several analyses and give many comments related to the difference equations and 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 xnxnâ1 , n â N0, where parameters a, b, c, and initial values xâ1 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