Search results for " Integra"
showing 10 items of 2527 documents
Denjoy and P-path integrals on compact groups in an inversion formula for multiplicative transforms
2009
Abstract Denjoy and P-path Kurzweil-Henstock type integrals are defined on compact subsets of some locally compact zero-dimensional abelian groups. Those integrals are applied to obtain an inversion formula for the multiplicative integral transform.
Dual of the Class of HKr Integrable Functions
2019
We define for 1 <= r < infinity a norm for the class of functions which are Henstock-Kurzweil integrable in the L-r sense. We then establish that the dual in this norm is isometrically isomorphic to L-r' and is therefore a Banach space, and in the case r = 2, a Hilbert space. Finally, we give results pertaining to convergence and weak convergence in this space.
A variational henstock integral characterization of the radon-nikodým property
2009
A characterization of Banach spaces possessing the Radon-Nikodym property is given in terms of finitely additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an X-valued function. Due to that characterization several X-valued set functions that are only finitely additive can be represented as integrals.
Approximation of Banach space valued Riemann type integrable functions by step functions
2008
In this talk we consider the possibility to approximate (with respect to some topology induced by the Alexiewicz norm) non absolutely integrable functions defined on the unit interval by step functions. In particular we show that any Henstock (respectively Henstock-Kurzweil-Pettis, Denjoy-Khintchine-Pettis) integrable functions can be scalarly approximate in the Alexiewicz norm by a sequence of step functions. Moreover the approximation may be done in the Alexiewicz norm if and only if the range of the integral is relatively norm compact (property which is automatically satisfied by the Henstock integrable functions). We also provide an example to show that, unlike the Pettis case, Henstock…
MR3093276 Reviewed Naralenkov, K. M. On continuity and compactness of some vector-valued integrals. Rocky Mountain J. Math. 43 (2013), no. 3, 1015–10…
2014
ZBL MS 63/6 Satco, Bianca-Renata; Turcu, Corneliu-Octavian Henstock-Kurzweil-Pettis integral and weak topologies in nonlinear integral equations on t…
2013
The authors prove an existence result for a nonlinear integral equation on time scales under weak topology assumption in the target Banach space. In the setting of vector valued functions on time scales they consider the Henstock-Kurzweil-Pettis $\Delta$-integral which is a kind of Henstock integral recently introduced by Cichon, M. [Commun. Math. Anal. 11 (2011), no. 1, 94�110]. In this framework they show the existence of weakly continuous solutions for an integral equation x(t)= f(t, x(t))+ (HKP)\int_0^t g(t,s,x(s)) \Delta s governed by the sum of two operators: a continuous operator and an integral one. The main tool to get the solutions is a generalization of Krasnosel'skii fixed point…
THE HKr-INTEGRAL IS NOT CONTAINED IN THE Pr-INTEGRAL
2022
We compare a Perron-type integral with a Henstock-Kurzweiltype integral, both having been introduced to recover functions from their generalized derivatives defined in the metric Lr. We give an example of an HKr-integrable function which is not Pr-integrable, thereby showing that the first integral is strictly wider than the second one.
MR2657294 (2011h:28021) Bensimhoun, Michael Change of variable theorems for the KH integral. Real Anal. Exchange 35 (2010), no. 1, 167–194. (Reviewer…
2010
From Reviews: 0 MR2657294 (2011h:28021) Bensimhoun, Michael(IL-HEBR) Change of variable theorems for the KH integral. (English summary) Real Anal. Exchange 35 (2010), no. 1, 167–194. 28B05 (26A42 46G10) PDF Clipboard Journal Article Make Link Let $({\scr E},{\scr F}, {\scr G})$(E,F,G) be a Banach triple and let $f\colon [a,b] \subset \overline{\Bbb R} \rightarrow {\scr E}$f:[a,b]⊂R−−→E, $\varphi \colon [a,b] \rightarrow {\scr F}$φ:[a,b]→F and $\psi\colon [c,d] \subset \overline{\Bbb R} \rightarrow [a,b]$ψ:[c,d]⊂R−−→[a,b] be given. The problem of change of variables in an integral consists in finding the best conditions under which the equality $$ \int_c^d f \circ \psi \cdot d(\varphi \circ …
MR3191427 Naralenkov, Kirill M., A Lusin type measurability property for vector- valued functions. J. Math. Anal. Appl. 417 (2014), no. 1, 293307. 28…
2014
In the paper under review the author introduces the notion of Riemann measurability for vector-valued functions, generalizing the classical Lusin condition, which is equivalent to the Lebesgue measurability for real valued functions. Let X be a Banach space, let f : [a; b] ! X and let E be a measurable subset of [a; b]. The function f is said to be Riemann measurable on E if for each " > 0 there exist a closed set F E with (E n F) < 0 (where is the Lebesgue measure) and a positive number such that k XK k=1 ff(tk) ?? f(t0 k)g (Ik)k < " whenever fIkgKk =1 is a nite collection of pairwise non-overlapping intervals with max1 k K (Ik) < and tk; t0 k 2 Ik T F. The Riemann measurabilit…