Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.
We show that the system of three difference equationsxn+1=an(1)xn-2/(bn(1)ynzn-1xn-2+cn(1)),yn+1=an(2)yn-2/(bn(2)znxn-1yn-2+cn(2)), andzn+1=an(3)zn-2/(bn(3)xnyn-1zn-2+cn(3)),n∈N0, where all elements of the sequencesan(i),bn(i),cn(i),n∈N0,i∈{1,2,3}, and initial valuesx-j,y-j,z-j,j∈{0,1,2}, are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.
A representation formula for the general solution to a higher-order rational difference equation has been given recently in this journal. The formula was proved by the method of induction without giving any theoretical explanation related to it. Here we show how the corresponding representation formula for the general solution to a general higher-order rational difference equation, that is, to the bilinear difference equation with a delay, is obtained in an elegant way. We also give some theoretical explanations related to the representation, as well as some explanations related to such types of difference equations. The corresponding representation for a system of bilinear difference equations with delay is also presented.
KEYWORDS bilinear difference equation, difference equation with interlacing indices, general solutionMath Meth Appl Sci. 2018;41:9349-9360.wileyonlinelibrary.com/journal/mma
A solvable two-dimensional product-type system of difference equations of interest is presented. Closed form formulas for its general solution are given.MSC: Primary 39A10; 39A20
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.