Show me the steps of deremine blue and inf is here i need evey I need all the details step by step and inf is here Theorem 5 Suppose that (5) holds. If x (n) is a solution of (4), then eitherx (n)= 0 eventually orlim sup (x; (n)|)/" = xj.where A1,, Ak are the (not necessarily distinct) roots of the characteristic..equation (7).Firstly, we take the change of the variables for Eq.(2) as follows ynFrom this, we obtain the following difference equationYnYn+1 = 1 +p-2Уп-тВwhere p = . From now on, we handle the difference equation (8). The uniquepositive equilibrium point of Eq.(8) isA² •1+ v1+ 4p2 Motivated by the above studies, we study the dynamics of following higherorder difference equationXn+1 =A + B(2)where A, B are positive real numbers and the initial conditions are positivenumbers. Additionally, we investigate the boundedness, periodicity, oscillationbehaviours, global asymptotically stability and rate of convergence of relatedhigiorder difference equations.Consider the scalar kth-order linear difference equationx (n + k) + P1 (n)x (n + k – 1) + .+ Pk (n)x (n) = 0,(4)where k is a positive integer and p; : Z+ → C for i = 1,,k. Assume thatqi = lim p;(n), i = 1, ... , k,(5)exist in C. Consider the limiting equation of (4):x (n + k) + q1x (n + k – 1) ++ qkx (n) = 0.(6)Theorem 4 (Poincaré's Theorem) Consider (4) subject to condition (5).Let A1,, Ak be the roots of the characteristic equation入*+ 91入+ 9k(7)= 0..of the limiting equation (6) and suppose that |Ai| # |A;| for i # j. If x (n) isa solution of (4), then either x (n) = 0 for all large n or there exists an indexje {1,... , k} such thatx (n + 1)x (n)lim

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Show me the steps of deremine blue and inf is here i need evey I need all the details step by step and inf is here Theorem 5 Suppose that (5) holds. If x (n) is a solution of (4), then eitherx (n)= 0 eventually orlim sup (x; (n)|)/&#34; = xj.where A1,, Ak are the (not necessarily distinct) roots of the characteristic..equation (7).Firstly, we take the change of the variables for Eq.(2) as follows ynFrom this, we obtain the following difference equationYnYn+1 = 1 +p-2Уп-тВwhere p = . From now on, we handle the difference equation (8). The uniquepositive equilibrium point of Eq.(8) isA² •1+ v1+ 4p2 Motivated by the above studies, we study the dynamics of following higherorder difference equationXn+1 =A + B(2)where A, B are positive real numbers and the initial conditions are positivenumbers. Additionally, we investigate the boundedness, periodicity, oscillationbehaviours, global asymptotically stability and rate of convergence of relatedhigiorder difference equations.Consider the scalar kth-order linear difference equationx (n + k) + P1 (n)x (n + k – 1) + .+ Pk (n)x (n) = 0,(4)where k is a positive integer and p; : Z+ → C for i = 1,,k. Assume thatqi = lim p;(n), i = 1, ... , k,(5)exist in C. Consider the limiting equation of (4):x (n + k) + q1x (n + k – 1) ++ qkx (n) = 0.(6)Theorem 4 (Poincaré's Theorem) Consider (4) subject to condition (5).Let A1,, Ak be the roots of the characteristic equation入*+ 91入+ 9k(7)= 0..of the limiting equation (6) and suppose that |Ai| # |A;| for i # j. If x (n) isa solution of (4), then either x (n) = 0 for all large n or there exists an indexje {1,... , k} such thatx (n + 1)x (n)lim

Accepted Answer

Given : difference equation 
                     yn+1=1+pynyn-m2               ----- (8)
where…