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The following difference equationxn=xn−lxn−k−1, n∈ℕ0,where k,l∈ℕ, k<l, gcd(k,l)=1, and the initial values x-l,…,x-2,x-1 arereal numbers, has been investigated so far only for some particular values of kand l. To get any general result on the equation is turned out as a not so easyproblem. In this paper, we give the first result on the behaviour of solutions of thedifference equation of general character, by describing the long-term behaviorof the solutions of the equation for all values of parameters k and l, wherethe initial values satisfy the following condition min{x-l,…,x-2,x-1}

abstract and applied analysis

Unbounded Solutions of the Difference Equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>

Unbounded Solutions of the Difference Equationxn=xn−lxn−k−1