DefinePK

Abstract

Consider the following general three dimensional system of difference equations<br>\nwhere⁢ 𝑛∈ℕ0,𝑝,𝑞,𝑟∈ℕ,𝑓,𝑔,ℎ :D→ℝ⁢ are continuous one-to-one functions on D⊆ℝ, the coefficients ⁡(𝑎𝑛)⁢𝑛∈ℕ0,(𝑏𝑛)⁢𝑛∈ℕ0⁡(𝑐𝑛)⁢𝑛∈ℕ0,(𝑑𝑛)⁢𝑛∈ℕ0,(𝑠𝑛)⁢𝑛∈ℕ0,(𝑡𝑛)⁢𝑛∈ℕ0⁢ are non-zero real numbers and the initial values 𝑥−𝑖,𝑦−𝑖,𝑧−𝑖,𝑖=0,1,2, are real numbers. We will give explicit formulas for well-defined solutions of the aforementioned system in both variable and constant cases of the coefficients. As an application, we will deduce the formulas of the solutions of the particular system obtained from the general one by taking 𝑓⁡(𝑥)=𝑔⁡(𝑥)=ℎ⁡(𝑥)=𝑥.