A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions
Abstract
We study solutions (xn)n ∈ N of nonhomogeneous nonlinear second order difference equations of the type n = xn ( σn,1 xn+1 + σn,0 xn + σn,-1 xn-1 ) + n xn, with given initial data x0 ∈ R, x1 ∈ R+ where (n)n∈N ∈ R+, (σn,0)n∈N ∈ R+ and (n)n∈N ∈ R and the left and right σ-coefficients satisfy either (σn,1)n∈N ∈ R+ and (σn,-1)n∈ N ∈ R+ or (σn,1)n∈N ∈ R+0 and (σn,-1)n∈N ∈ R+0. Depending on one's standpoint, such equations originate either from orthogonal polynomials associated with certain Shohat-Freud-type exponential weight functions or from Painlev\'e's discrete equation \#1.
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