Sulle equazioni alle differenze con incrementi variabili.

Borelli Forti, Constanza ; Fenyö, István

Stochastica, Tome 4 (1980), p. 93-101 / Harvested from Biblioteca Digital de Matemáticas

Résumé

Let X be an arbitrary Abelian group and E a Banach space. We consider the difference-operators ∆n defined by induction:

(∆f)(x;y) = f(x+y) - f(x), (∆nf)(x;y1,...,yn) = (∆n-1(∆f)(.;y1)) (x;y2,...,yn)

(n = 2,3,4,..., ∆1=∆, x,yi belonging to X, i = 1,2,...,n; f: X --> E).

Considering the difference equation (∆nf)(x;y1,y2,...,yn) = d(x;y1,y2,...,yn) with independent variable increments, the most general solution is given explicitly if d: X x Xn --> E is a given bounded function. Also the most general form of bounded functions in the range of ∆n is determined.

Another type of operator, designed by ∆2 n is defined by

(∆2f)(x;y) = f(x+2y) - 2f(x+y) + f(x),

(∆2 nf)(x;y1,...,yn) = (∆2 n-1(∆2f)(.;y1)) (x;y2,...,yn),

(n = 2,3,4,..., ∆2 1=∆2, x,yi belonging to X, i = 1,2,...,n) and under the same conditions as above the most general solution of the equation ∆2 nf = d is established.

Détail

Publié le : 1980-01-01

MR MR0599135 | Zbl 0456.39002

DMLE-ID : 1610