Henstock-Kurzweil nd McShne product integrtion Chrles University, Prgue slvik@krlin.mff.cuni.cz CDDE 2006
Motivtion Consider the differentil eqution y (x) = A(x)y(x) y() = y 0 where x [, b], y : [, b] R n, y 0 R n, A : [, b] R n n. Insted of it we cn solve Y (x) = A(x)Y (x) Y () = I, where x [, b], Y, A : [, b] R n n. (The solution of the originl problem is then y(x) = Y (x)y 0.)
Peno series solution An equivlent formultion of our eqution is x Y (x) = I + A(t)Y (t) dt. Using the method of successive pproximtions we find the Peno series solution Y (x) = I + x A(t 1 ) dt 1 + x t1 A(t 1 )A(t 2 ) dt 2 dt 1 + x t1 t2 + A(t 1 )A(t 2 )A(t 3 ) dt 3 dt 2 dt 1 +
Another pproch (Vito Volterr, 1887) Eqution Y (x) = A(x)Y (x) implies tht for smll x Y (x + x) =. Y (x) + Y (x) x = (I + A(x) x)y (x). Therefore we expect (using the fct Y () = I) tht Y (b) = lim ν(d) 0 i=m 1 (I + A(x i 1 )(x i x i 1 )), where D is prtition of [, b] with division points = x 0 x 1 x m = b.
Product integrl definition For ny function A : [, b] R n n denote P(A, D) = 1 (I + A(ξ i )(x i x i 1 )), i=m where D is tgged prtition of [, b] with division points = x 0 x 1 x m = b nd tgs ξ i [x i 1, x i ]. If the limit lim ν(d) 0 P(A, D) exists, we cll it the product integrl of A over [, b] nd use the nottion b lim P(A, D) = (I + A(x) dx). ν(d) 0
Existence of product integrl If A : [, b] R n n is Riemnn integrble function, then the product integrl of A over [, b] exists nd b (I + A(t) dt) = I + b b t1 A(t 1 ) dt 1 + A(t 1 )A(t 2 ) dt 2 dt 1 + b t1 t2 + A(t 1 )A(t 2 )A(t 3 ) dt 3 dt 2 dt 1 +
Indefinite product integrl If A : [, b] R n n is Riemnn integrble function, then Y (x) = x (I + A(t) dt) exists for x [, b] nd x Y (x) = I + A(t)Y (t) dt. If A is continuous, then Y (x) = A(x)Y (x) for every x [, b].
K-prtitions nd M-prtitions A finite collection of point-intervl pirs D = {(ξ i, [x i 1, x i ])} m i=1 is clled n M-prtition of intervl [, b] if = x 0 x 1 x m = b nd ξ i [, b], i = 1,..., m. A K -prtition is M-prtition which stisfies ξ i [x i 1, x i ], i = 1,..., m. Given function : [, b] (0, ), the prtition D is clled -fine if [x i 1, x i ] (ξ i (ξ i ), ξ i + (ξ i )), i = 1,..., m.
Henstock-Kurzweil nd McShne product integrls (1) For every A : [, b] R n n nd for every M-prtition D = {(ξ i, [x i 1, x i ])} m i=1 of [, b] put P(A, D) = 1 (I + A(ξ i )(x i x i 1 )). i=m The function A is Henstock-Kurzweil product integrble, if there exists mtrix P A R n n such tht for every ε > 0 we cn find : [, b] (0, ) so tht P(A, D) P A < ε for every -fine K-prtition D of [, b].
Henstock-Kurzweil nd McShne product integrls (2) The mtrix P A is clled the Henstock-Kurzweil product integrl of A over [, b]. The definition of McShne product integrl is obtined by replcing K -prtitions by M-prtitions. Nottion: (HK ) b (I + A(t) dt), (M) b (I + A(t) dt)
Indefinite Henstock-Kurzweil product integrl J. Kurzweil, J. Jrník (1987): Consider function A : [, b] R n n such tht the integrl (HK ) b (I + A(t) dt) exists nd is invertible. Then the function Y (x) = (HK ) x (I + A(t) dt) is well defined for every x (, b), it is ACG nd stisfies Y (x) = A(x)Y (x) lmost everywhere on [, b].
Indefinite McShne product integrl Š. Schwbik, A. Slvík (2006): Consider function A : [, b] R n n such tht the integrl (M) b (I + A(t) dt) exists nd is invertible. Then the function Y (x) = (M) x (I + A(t) dt) is well defined for every x (, b), it is bsolutely continuous nd stisfies Y (x) = A(x)Y (x) lmost everywhere on [, b]. Corollry: The McShne product integrl coincides with the Lebesgue/Bochner product integrl.
History of product integrtion 1887 V. Volterr Riemnn-type product integrl 1931 L. Schlesinger Lebesgue product integrl 1938 B. Hostinský product integrtion of opertor-vlued functions 1947 P. R. Msni Riemnn product integrtion in Bnch lgebrs 1987 J. Kurzweil, J. Jrník H.-K. product integrl 1994 Š. Schwbik McShne product integrl In preprtion: Product integrtion, its history nd pplictions.