Abstract
Asymptotic properties of solutions of a difference equation of the form are studied. We present sufficient conditions under which, for any polynomial of degree at most and for any real , there exists a solution x of the above equation such that . We give also sufficient conditions under which, for given real , all solutions x of the equation satisfy the condition for some polynomial of degree at most . MSC:39A10.
Highlights
Let N, Z, R denote the set of positive integers, the set of all integers and the set of real numbers, respectively
The purpose of this paper is to study the asymptotic behavior of solutions of equation (E)
In [ ], Popenda and Drozdowicz presented necessary and sufficient conditions under which the equation mxn = anf has a convergent solution
Summary
Let N, Z, R denote the set of positive integers, the set of all integers and the set of real numbers, respectively. In [ ], Popenda and Drozdowicz presented necessary and sufficient conditions under which the equation mxn = anf (xn) has a convergent solution (i.e., a solution that is asymptotically polynomial of degree zero). In [ ] sufficient conditions under which, for any φ ∈ Pol(m – ), there exists a solution x of the equation mxn = anf (xn) + bn such that xn = φ(n) + o( ) are presented. In [ ] sufficient conditions under which every solution x of the equation mxn = anF(n, xg(n)) + bn has the property xn = φ(n) + o( ) for some φ ∈ Pol(m – ) are presented. Sufficient conditions under which, for every φ ∈ Pol(m – ) there exists a solution x of this equation such that xn = φ(n) + o( ), are presented.
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