Abstract
In this paper, some cases on the periodicity of the rational difference equation S_{n+1}=S_{n-p}(((aS_{n-q}+bS_{n-r}+cS_{n-s})/(dS_{n-q}+eS_{ n-r}+fS_{n-s}))), are investigated, where a, b, c, d, e, f ∈(0,∞). The initial conditions S_{-p}, S_{-p+1},...,S_{-q}, S_{-q+1},...,S_{-r}, S_{-r+1},...,S_{-s},...,S_{-s+1},...,S₋₁ and S₀ are arbitrary positive real numbers such that p>q>r>s≥0. Some numerical examples are provided to illustrate the theoretical discussion.
Highlights
Di¤erence equations appear naturally as discrete analogues, numerical solutions of di¤erential and delay di¤erential equations as they are applied in biology, ecology, economy, physics, and so on
Immense e¤ort has been enacted in studying the qualitative analysis of rational di¤erence equations
The main aim of this study is to exhibit some cases on the periodic character of the positive solutions of the rational di¤erence equation
Summary
Di¤erence equations appear naturally as discrete analogues, numerical solutions of di¤erential and delay di¤erential equations as they are applied in biology, ecology, economy, physics, and so on. Di¤erence equations appear simpler in its form, the behaviors of their solutions are hard to be comprehended thoroughly. Immense e¤ort has been enacted in studying the qualitative analysis of rational di¤erence equations. Many researchers have investigated periodic solutions of di¤erence equations, and they have proposed various methods for the existence and qualitative properties of the solutions [7-10]. The main aim of this study is to exhibit some cases on the periodic character of the positive solutions of the rational di¤erence equation. The initial conditions S p, S p+1,...,S q, S q+1,...,S r, S r+1,...,S s,...,S s+1,...,S 1 and S0 are arbitrary positive real numbers such that p > q > r > s 0. Submitted via 2nd International Conference of Mathematical Sciences (ICMS 2018).
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