Convergence results for the Abstrct Kurzweil-Stieltjes integrl: survey Giselle A. Monteiro Mthemticl Institute, Slovk Acdemy Sciences Seminr on ordinry differentil equtions nd integrtion theory - Specil session in honor of the 90th birthdy of Jroslv Kurzweil - Prgue, My 2016
Some time go, not fr from here...
Some time go, not fr from here... Some time go, right here!
The Abstrct Kurzweil-Stieltjes integrl J. Kurzweil, Generlized ordinry differentil eqution nd continuous dependence on prmeter. Czech. Mth. J. (1957). Let U : [, b] [, b] R. The generlized Perron integrl DU(τ, s) exists if there is I R such tht for every ε > 0 there exists guge δ : [, b ] R + such tht I provided the prtition m [U(τ j, s j ) U(τ j, s j 1 )] < ε j=1 = s 0 < s 1 < < s m = b, τ j [s j 1, s j ] is δ-fine, i.e. [s j 1, s j ] ( τ j δ(τ j ), τ j + δ(τ j ) )
The Abstrct Kurzweil-Stieltjes integrl Š. Schwbik, Abstrct Perron-Stieltjes integrl. Mth. Bohem. (1996). X is Bnch spce, L(X ) is the spce of liner bounded opertors in X Definition Let F : [, b ] L(X ) nd g : [, b ] X. The bstrct Kurzweil-Stieltjes integrl F d[g] exists if there is I X such tht for every ε > 0 there exists guge δ : [, b ] R + such tht I provided the prtition m F (τ j )[g(s j ) g(s j 1 )] < ε X j=1 = s 0 < s 1 < < s m = b, τ j [s j 1, s j ] is δ-fine, i.e. [s j 1, s j ] ( τ j δ(τ j ), τ j + δ(τ j ) )
The Abstrct Kurzweil-Stieltjes integrl Š. Schwbik, Abstrct Perron-Stieltjes integrl. Mth. Bohem. (1996). X is Bnch spce, L(X ) is the spce of liner bounded opertors in X Definition Let F : [, b ] L(X ) nd g : [, b ] X. The bstrct Kurzweil-Stieltjes integrl d[f ]g exists if there is I X such tht for every ε > 0 there exists guge δ : [, b ] R + such tht I provided the prtition m [F (s j ) F (s j 1 )]g(τ j ) < ε X j=1 = s 0 < s 1 < < s m = b, τ j [s j 1, s j ] is δ-fine, i.e. [s j 1, s j ] ( τ j δ(τ j ), τ j + δ(τ j ) )
The Abstrct Kurzweil-Stieltjes integrl Importnt clsses of functions: G([, b ], X ) (regulted), BV ([, b ], X ) (bounded vrition)
The Abstrct Kurzweil-Stieltjes integrl Importnt clsses of functions: G([, b ], X ) (regulted), BV ([, b ], X ) (bounded vrition) Existence of F d[g] F G([, b ], L(X )) nd g BV ([, b ], X ) [Schwbik (1996)]
The Abstrct Kurzweil-Stieltjes integrl Importnt clsses of functions: G([, b ], X ) (regulted), BV ([, b ], X ) (bounded vrition) Existence of F d[g] F G([, b ], L(X )) nd g BV ([, b ], X ) [Schwbik (1996)] Uniform convergence theorem [Schwbik (1996)] Assume F, F k G([, b ], L(X )) for k N, g BV ([, b ], X ), Then: F k F. F k d[g] F d[g] on [, b ].
The Abstrct Kurzweil-Stieltjes integrl Importnt clsses of functions: G([, b ], X ) (regulted), BV ([, b ], X ) (bounded vrition) Existence of F d[g] F G([, b ], L(X )) nd g BV ([, b ], X ) [Schwbik (1996)] F BV ([, b ], L(X )) nd g G([, b ], X ) [Monteiro & Tvrdý (2012)] G. Monteiro nd M. Tvrdý, On the Kurzweil-Stieltjes integrl in Bnch spce. Mth. Bohem. (2012). Uniform convergence theorem [Schwbik (1996)] Assume F, F k G([, b ], L(X )) for k N, g BV ([, b ], X ), Then: F k F. F k d[g] F d[g] on [, b ].
The Abstrct Kurzweil-Stieltjes integrl Existence of F d[g] F G([, b ], L(X )) nd g BV ([, b ], X ) [Schwbik (1996)] F BV ([, b ], L(X )) nd g G([, b ], X ) [Monteiro & Tvrdý (2012)] Uniform convergence theorem [Schwbik (1996)] Assume F, F k G([, b ], L(X )) for k N, g BV ([, b ], X ), F k F. Then: F k d[g] F d[g] on [, b ]. Uniform convergence theorem [Monteiro & Tvrdý (2012)] Assume F BV ([, b ], L(X )), g, g k G([, b ], X ) for k N, Then: g k g. F d[g k ] F d[g] on [, b ].
Other convergence results G. Monteiro nd M. Tvrdý, Generlized liner differentil equtions in Bnch spce: Continuous dependence on prmeter. Discrete Contin. Dyn. Syst. (2013). G. Monteiro nd A. Slvík, Liner mesure functionl differentil equtions with infinite dely, Mth. Nchrichten (2014). G. Monteiro nd M. Tvrdý, Continuous dependence of solutions of bstrct generlized liner differentil equtions with potentil converging uniformly with weight Boundry Vlue Problem (2014). G. Monteiro, U. Hnung nd M. Tvrdý, Bounded convergence theorem for bstrct Kurzweil-Stieltjes integrl. Montsh. Mth. (2015).
Other convergence results G. Monteiro nd M. Tvrdý, Generlized liner differentil equtions in Bnch spce: Continuous dependence on prmeter. Discrete Contin. Dyn. Syst. (2013). [Monteiro & Tvrdý (2013)] Assume F, F k BV ([, b ], L(X )), g, g k G([, b ], X ) for k N, Then: F k F, g k g vr b F k γ for k N. d[f k ] g k d[f ] g on [, b ].
Other convergence results G. Monteiro nd M. Tvrdý, Generlized liner differentil equtions in Bnch spce: Continuous dependence on prmeter. Discrete Contin. Dyn. Syst. (2013). [Monteiro & Tvrdý (2013)] Assume F, F k BV ([, b ], L(X )), g, g k G([, b ], X ) for k N, Then: F k F, g k g vr b F k γ for k N. d[f k ] g k d[f ] g on [, b ]. G. Monteiro nd A. Slvík, Liner mesure functionl differentil equtions with infinite dely, Mth. Nchrichten (2014).
Other convergence results [Monteiro & Tvrdý (2013)] Assume F, F k BV ([, b ], L(X )), g, g k G([, b ], X ) for k N, F k F, g k g vr bf k γ for k N. Then: d[f k ] g k d[f ] g on [, b ]. [Monteiro & Slvík (2014)] Assume F, F k BV ([, b ], L(X )), g, g k G([, b ], X ) for k N, g k g, lim k sup t [,b] [F k (t) F (t)]x = 0 for every x X, Then: there exists γ > 0 such tht vr b F k γ, for k N. d[f k ] g k d[f ] g on [, b ].
Other convergence results G. Monteiro nd M. Tvrdý, Generlized liner differentil equtions in Bnch spce: Continuous dependence on prmeter. Discrete Contin. Dyn. Syst. (2013). G. Monteiro nd M. Tvrdý, Continuous dependence of solutions of bstrct generlized liner differentil equtions with potentil converging uniformly with weight Boundry Vlue Problem (2014). x k (t) = x k + d [A k ] x k + f k (t) f k (), t [, b ] x(t) = x + d [A] x + f (t) f (), t [, b ] A, A k BV ([, b ], L(X )), f k G([, b ], X ), x k X for k N, ssuming: f BV ([, b ], X ) lim A ( ) k A 1 + vr b A k = 0 k lim f ( ) k f 1 + vr b A k = 0 k
Other convergence results G. Monteiro nd M. Tvrdý, Generlized liner differentil equtions in Bnch spce: Continuous dependence on prmeter. Discrete Contin. Dyn. Syst. (2013). G. Monteiro nd M. Tvrdý, Continuous dependence of solutions of bstrct generlized liner differentil equtions with potentil converging uniformly with weight Boundry Vlue Problem (2014). x k (t) = x k + d [A k ] x k + f k (t) f k (), t [, b ] x(t) = x + d [A] x + f (t) f (), t [, b ] A, A k BV ([, b ], L(X )), f k G([, b ], X ), x k X for k N, ssuming: f BV ([, b ], X ) lim A ( ) k A 1 + vr b A k = 0 k lim f ( ) k f 1 + vr b A k = 0 k There exists f G[0, 1], A k BV [0, 1], k N, such tht ( ) lim 1 + vr 1 0 A k Ak A = 0 BUT k 1 0 d[a k ] f 1 0 d[a] f
Other convergence results G. Monteiro, U. Hnung nd M. Tvrdý, Bounded convergence theorem for bstrct Kurzweil-Stieltjes integrl. Montsh. Mth. (2015). Bounded Convergence Theorem Assume F, F k G([, b ], L(X )), k N, g BV ([, b ], X ), Then: F k (t) F (t) for t [, b ], F k γ for k N. F k d[g] F d[g].
Other convergence results Bounded Convergence Theorem [Montsh. Mth. (2015)] Assume F, F k G([, b ], L(X )), k N, g BV ([, b ], X ), F k (t) F (t) for t [, b ], F k γ for k N. Then: F k d[g] F d[g]. [Kurzweil (1980)] Assume g BV [, b] nd f k : [, b] R, k N such tht f k(s) dg(s) exists, (i) f k (t) f (t) for t [, b] (ii) There exists K > 0 such tht for every division = σ 0 < σ 1 < < σ l = b of [, b] nd every finite sequence m 1,..., m l N l σj f mj (s) dg(s) K. j=1 σ j 1 Then: f k d[g] f d[g]. J. Kurzweil, Nichtbsolut konvergent Integrle 1980.
References B. Bongiorno, L. Di Pizz, Convergence theorems for generlized Riemnn Stieltjes integrls, Rel Anl. Exchnge 17 (1991/92) S. S. Co, The Henstock integrl for Bnch-vlued functions, Southest Asin Bull. Mth. 16 (1992) J. Kurzweil, J. Jrnik, Equiintegrbility nd controlled convergence of Perron-type integrble functions, Rel Anl.Exchnge 17 (1991/92) S. A. Tikre, M. S. Chudhry, Henstock-Stieltjes integrl for Bnch spce-vlued functions, Bulletin of Kerl Mthemtics Assocition 6 (2010) J. H. Yoon,B. M. Kim, The Convergence Theorems For the McShne-Stieltjes integrl, J. Kore Soc. Mth. Educ. 7 (2000)
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