On Some Degenerate Differential Operators on Weighted Function Spaces

Abstract

We deal with the degenerate differential operatorAu(x)≔α(x)u″(x) (x≥0) defined for everyu∈W20∩C2(]0,+∞[) satisfying limx→0+α(x)u″(x)=limx→+∞(α(x)/(1+x2))u″(x)=0. HereW20denotes the Banach space of all continuous functionsf:[0,+∞[→R such thatf(x)/(1+x2) vanishes at infinity, endowed with the weighted norm ‖f‖2≔supx≥0(|f(x)|/(1+x2)) (f∈W20). Moreover, we assume that the function α is continuous and positive on [0,+∞[, it is differentiable at 0, and satisfies the inequalities 0<α0≤α(x)/x≤α1(x>0) for suitable constants α0and α1. We show that the operatorAgenerates aC0-semigroup (T(t))t≥0of positive operators onW20. Moreover, we prove that everyT(t) can be represented as a limit of powers of suitable discrete-type positive linear operators that are constructed by means of the coefficient α.