On some degenerate differential operators on weighted function spaces

[摘要] We deal with the degenerate differential operator Au(x) := alpha(x)u''(x) (x greater than or equal to 0) defined for every u is an element of W-2(0) boolean AND C-2(]0, + infinity) satisfying lim(x --> 0)+ alpha (x)u''(x) = lim(x --> + infinity)(alpha(x)/(1 + x(2)))u''(x) = 0. Here W-2(0) denotes the Banach space of all continuous functions f:[0, + infinity[ --> R such that f(x)/(1 + x(2)) vanishes at infinity, endowed with the weighted norm parallel to f parallel to(2) (sic) sup(x greater than or equal to 0) (\f(x)\/(1 + x(2))) (f is an element of W-2(0)). Moreover, we assume that the function alpha is continuous and positive on [0, +infinity[, it is differentiable at 0, and satisfies the inequalities 0 < alpha(0) less than or equal to alpha(x)/x less than or equal to alpha(1) (x > 0) for suitable constants alpha(0) and alpha(1). We show that the operator A generates a C-0-semigroup (T(t))(t greater than or equal to 0) of positive operators on W-2(0). Moreover, we prove that every T(t) can be represented as a limit of powers of suitable discrete-type positive linear operators that are constructed by means of the coefficient alpha. (C) 1997 Academic Press.

[发布日期] 1997-09-01