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. ℓn=xn(σn,1xn+1+σn,0xn+σn,-1xn-1)+κnxn,n∈N,withgiveninitialdata {x0∈R&x1∈R+} where (ℓn)n∈N∈R+&(σn,0)n∈N∈R+&(κn)n∈N∈R, and the left and right σ-coefficients satisfy either (σn,1)n∈N∈R+&(σn,-1)n∈N∈R+ or (σn,1)n∈N∈R0+&(σn,-1)n∈N∈R0+. Depending on one's standpoint, such equations originate either from orthogonal polynomials associated with certain Shohat-Freud-type exponential weight functions or from Painlevé's discrete equation #1, that is,