Sequential $\Diamond$ Henstock Integral for Locally Convex Space Valued Function on Time Scale

Iluebe, Victor Odalochi; Afariogun, David Adebisi; Iluno, Christiana; Ajilore, Joshua Olugbenga

doi:10.22080/cjms.2025.26691.1680

Sequential $\Diamond$ Henstock Integral for Locally Convex Space Valued Function on Time Scale

Document Type : Review Article

Authors

¹ Department of Mathematics and Computing, Maranatha University, Lagos, Nigeria.

² Mathematics Department, Faculty of Science, Ajayi Crowther University, Oyo, Nigeria

³ Mathematics Department, School of Pure and Applied Sciences, Lagos State University of Science and Technology, lkorodu, Nigeria

10.22080/cjms.2025.26691.1680

Abstract

Let $X$ be a Hausdorff locally convex topological vector space with $\Omega $ and $X^{*}$ as its topology and Topological dual respectively. Suppose $f:[0,1]\rightarrow X$ is a function defined on $X$ and $\rho(X)$, a family of $\rho$-continuous seminorms on $X$ such that the topology is generated by $\rho(X)$. Is f Sequential Mcshane(SMcS) and Sequential Henstock(SH) integrable with respect to the semi-norm on time scale? Do these integrals coincide and relate to other integrals such as Pettis and Bochner for which the Sequential Henstock lemma holds for the characterization of locally Convex space on time scale? It is the purpose of this paper to give affirmative answers to these questions.

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