Global behavior of two third order rational difference equations with quadratic terms

In this paper, we determine the forbidden sets, introduce an explicit formula for the solutions and discuss the global behaviors of solutions of the difference equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <m:mstyle> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>a</m:mi> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mi>b</m:mi> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>c</m:mi> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mo>,</m:mo> <m:mspace/> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array xₙ₊₁=axₙxₙ₋₁bxₙ₋₁+ cxₙ₋₂, n=0,1,… array where a , b , c are positive real numbers and the initial conditions x −2 , x −1 , x 0 are real numbers.