Representation and solution of two fifth-order nonlinear difference equations

Difference equations appear as an approximation to differential equations (in numerical analysis). These equations appear in nature in modeling many situations in biology and ecology as well as in economics and engineering.Solvable difference equations and systems of difference equations occur in many areas of science and mathematics. In this paper, we represent and study the well-defined solutions of the difference equationω_(n+1)=(ω_(n-3) ω_(n-4))/(ω_n+ω_(n-4) ),n∈N_0,where the initial values ω_0,ω_(-1),ω_(-2),ω_(-3) and ω_(-4) are real numbers. We give a representation to the above-mentioned equation using a sequence {σ_n }_(n=0)^∞ that satisfies the linear second-order difference equationσ_(n+2)-σ_(n+1) -σ_n=0,n∈N_0,with σ_0=0 ,σ=1, and give the solution of the difference equationω_(n+1)=(ω_(n-3) ω_(n-4))/(〖-ω〗_n+ω_(n-4) ),n∈N_0,where the initial values ω_0,ω_(-1),ω_(-2),ω_(-3) and ω_(-4) are real numbers.The important result in this paper is that every well-defined solution 〖{ω_n}〗_(n=-4)^∞ to the first above-mentioned equations converges to zero and every well-defined solution 〖{ω_n}〗_(n=-4)^∞ to the second above-mentioned equation is periodic.A very important tool in studying difference equations is the forbidden set. We introduce here the forbidden set for the above-mentioned equations. We give some examples to show the theoretical results.