Hemanta Kalita

VIT Bhopal University · Mathematics

In this paper we extend the Theory of Henstock Orlicz space with respect to vector measure.

The School of Advanced Sciences and Languages (SASL) in VIT Bhopal is principally concerned with blending the best knowledge of science and humanities with Engineering to align the expertise of our students with the ever-changing technological requirements. The school offers numerous core and elective courses, developed in collaboration with intern...

In this article, Sobolev spaces on canonical Banach spaces has been discussed. The Hilbert structure of the Sobolev spaces are discussed in this settings. Finally, in application, we discuss the Fourier transform and its relevance for Sobolev spaces on canonical Banach spaces.

In the structure of parametric metric space accompanied by directed graph, we introduce the idea of fuzzy multival-ued F-contractive mappings. Results related to the existence of common fuzzy xed points are introduced. The proven results are supported by example. Our results bring together, sum up and supplement dierent familiar related results in...

This article discusses McShane and strong McShane integration theory in the class of vector-valued functions endowed with the Alexiewicz norm, their relations, and properties as the fundamental theorem of calculus. We extend this theory to some Stieltjes-type integrals via a bounded bilinear operator, such as McShane-Stieltjes, strong McShane-Stiel...

In this article, an application of $\phi$-metric is given via a geometrical example to show how it can help to measure distance for non-planar surfaces where the classical metric becomes incapable. Also, we introduce the concept of best proximity point and proximal contraction for a class of mappings in a $\phi$-metric space and prove a best proxim...

We study several properties of the Banach lattice RL 1 (m, X) of Riemann-Lebesgue integrable function space associated to vector measure m. We also introduce weakly RL-integrable function spaces endowed with vector measure. A representation of the weakly Riemann-Lebesgue integral in terms of unconditionally convergent series is shown. Finally, we d...

In this article, we extend Kluvánek–Lewis–Henstock thegral theory in associate with X-valued measure of [4]. We investigate some properties and convergence theorem of Kluvánek–Lewis–Henstock \(\mu \)-integral for \(\mu \)-measurable functions in associated with X-valued vector measure m. We provide an m-a.e. convergence version of Dominated (resp....

In addition to being enticing in and of itself, the study of mathematics analysis sets the stage for various branches of both pure and applied mathematics. Researchers from all over the world have been actively involved in analysing and developing various mathematical theories that can be used to solve practical issues. The most recent developments...

In this article, we introduce a McShane type integral on a complete metric space, endowed with a Radon measure µ with a family of cells that satisfies the Vitali covering theorem with respect to µ. The Saks-Henstock type lemma in terms of additive functions, some of the fundamental properties of such integrals are investigated. Finally, a relations...

In this article, we introduce Pettis integrability type property for Henstock-Kurzweil-Pettis-integrals. We discuss several necessary conditions that X has Henstock-Kurzweil-Pettis-integrability property for weak Baire measure. Necessary and sufficient conditions of the indefinite integral of any Henstock-Kurzweil-Pettis (respectively, Denjoy-Petti...

In this article we discuss Sobolev spaces on canonical Banach spaces. The completeness of the Sobolev spaces is discussed in these settings. The Hilbert structure of the Sobolev spaces is also discussed. Finally, in application, we discuss the Fourier transform and its relevance for Sobolev spaces on canonical Banach spaces.

In this paper, we study the Riemann-Dunford and Riemann-Pettis integration on time scales. We first provide the definition of the scalarly Riemann ∆-(resp. ∇-) integral, followed by the definition of the Riemann-Dunford ∆-integral and the Riemann-Pettis ∆-integral (respectively the Riemann-Dunford ∇-integral and the Riemann-Pettis ∇-integral). We d...

In this paper we introduce ap-McShane integral of vector valued functions which is a generalization of McShane integral of vector valued functions, and investigate some of its properties, also we characterize ap-McShane integral of vector valued functions by the notion of equi-integrability. Finally, we find the equivalence between the ap-McShane i...

We prove that a closed bounded convex set is uniformly integrable if and only if it has weak drop property. We extract the weakly compact subsets of the Henstock integrable functions on the H-Orlicz spaces with the weak drop property via de la Vallée Poussin Theorem.

Dear Professor Greetings!! We are glad to inform you that an edited book entitled " Fixed point, Summability theory and non absolute integrals" is under process for publication with Springer Nature. This edited book will be indexed in Scopus. Thus we are inviting you to contribute your original research articles to this edited book. Best regards...

We establish the equivalence of the Riemann-Stieltjes ∆-integral as defined in [13, 14] in terms of the Darboux sum definition and the Riemann sum definition, and provide the definition of the Riemann-Stieltjes ∇-integral in terms of the Riemann sum definition and prove its equivalence with the Riemann-Stieltjes ∇-integral as defined in [13] in ter...

The AP-Henstock-Kurzweil-type integral is defined on X, where X is a complete measure metric space. We present some properties of the integral, continuing the study's use of a Radon measure µ. Finally, using locally finite measures, we extend the AP-Henstock-Kurzweil integral theory to second countable Hausdorff spaces that are locally compact. A S...

Kuelbs-Steadman spaces are introduced in this article on a separable metric space having finite diameter together with a finite Borel measure. Kuelbs-Steadman spaces of the Lipschitz type are also discussed. Various inclusion properties are also discussed. In the sequel, we introduce HK-Sobolev spaces on metric mesure space which coincides with HK-...

In this article we develop the theory of H-Orlicz space generated by generalised Young function. Modular convergence of H-Orlicz space for the case of vector-valued functions and norm convergence in H θ (X , μ) where X is any Banach space are discussed. Relationships of modular convergence and norm convergence of H-Orlicz spaces are discussed.

In this article, we discuss few properties of L r-Henstock-Kurzweil (in short L r-HK) integrable functions, introduced by Paul Musial in [8]. We redefined L r-bounded variations. We demonstrated that L r-Henstock-Kurzweil integrable functions are Denjoy integrable.

We study integration of Banach space valued functions with respect to Banach space valued measures. Our main result states that Birkhoff-Dunford integrability implies McShane- Dunford integrability. We also show that a function is measurable and McShane-Dunford inte- grable if and only if Dobrakov-Dunford integrable.

The aim of this chapter is to introduce Zachary space over \(\mathbb {R}^\infty \) and find that this is a Banach space of functions of bounded mean oscillation with order \(p, 1\le p \le \infty \) containing the function of bounded mean oscillation \(BMO[\mathbb {R}_\textrm{I}^\infty ]\) as a dense continuous embedding. As an application of \(\mat...

The functions with bounded mean oscillation (BMO) have been shown to be immense interest in several areas of analysis and probability. We introduce BMO-type space BMO HK (R n) for non-absolute integrable functions. Various properties and completion of BMO HK (R n) are included. Relations between the classical BMO space and BMO HK (R n) are investig...

Kuelbs-Steadman spaces are studied within the framework of Henstock-Kurzweil integrable function spaces with bounded variable exponent. We describe a relationship between the Lebesgue spaces with bounded variable exponents and variable Kuelbs-Steadman spaces. The geometrical properties of the spaces are studied. Finally, we discuss the boundedness...

Our goal in this article is to construct Sobolev spaces over R ∞ I. Completeness of the Sobolev space over R ∞ I are discussed. In application we have constructed the Sobolev spaces on a separable Banach space B.

All the submitted book chapters will be reviewed by the experts of related areas. Moreover, the review process will maintain the strictness in case of originality of the papers along with their qualities and importance in current trends. Publication Submission of abstract Abstract of maximum 200 words (along with maximum five keywords and 2020 AMS...

In this paper we extend the theory of Henstock-Orlicz spaces with respect to vector measure. We study the integral representation of operators. Lastly we study Uniformly convexity, reflexivity and the Radon-Nikodym property of the Henstock-Orlicz spaces

In this paper we extend the theory of Henstock-Orlicz spaces with respect to vector measure. We study the integral representation of operators. Lastly we study Uniformly convexity, reflexivity and the Radon-Nikodym property of the Henstock-Orlicz spaces H θ (µ ∞).

In this survey note we discuss about non absolute integrable functions and we put our view about the question: Why we need Non absolute integral in place of Lebesgue integral? Various areas are discussed, where we can find Henstock-Kurzweil integral in place of Lebesgue integral.

In this paper we discuss the structure of Henstock-Orlicz space with locally Henstock integrable functions. The weak Henstock-Orlicz spaces on R n and some basic properties of the weak Henstock-Orlicz spaces are studied. We obtain some necessary and sufficient conditions for the inclusion properties of these spaces.

We introduce L r-Henstock-Kurzweil integral for finite dimensional Banach spaces. We discuss its properties. In this study we discuss L r-Henstock-Kurzweil integral generalized Henstock-Kurzweil integral for finite dimensional Banach spaces.

The objective of this paper is to construct an extension of the class of Jones distribution Banach spaces SD p [R n ], 1 ≤ p ≤ ∞, which appeared in the book by Gill and Zachary [3] to SD p [R ∞ ] for 1 ≤ p ≤ ∞. These spaces are separable Banach spaces, which contain the Schwartz distributions as continuous dense embedding. These spaces provide a Ba...

The objective of this paper is to construct canonical Orlicz class and study their funda- mental properties. Also, we prove that this space contains Henstock–Kurzweil integrable functions.

In this paper we discuss about the ap?Henstock-Kurzweil integrable functions on a topological vector spaces. Basic results of ap?Henstock-Kurzweil integrable functions are discussed here. We discuss the equivalence of the ap?Henstock-Kurzweil integral on a topological vector spaces and the vector valued ap?Henstock-Kurzweil integral. Finally, sever...

Orlicz spaces contain the basic structural analysis of functions spaces with general Young functions on arbitrary measure spaces. Limitations of the Orlicz spaces is that the spaces C 0 ∞ is not generally dense in Orlicz spaces (see [82]). This limitation encourages us to introduce Orlicz type spaces in short (H-Orlicz spaces) containing Henstock-K...

Suppose \({\mathcal {X}}\) is a real Banach space and \((\varOmega , \varSigma , \mu )\) is a probability space. We characterize the countable additivity of Henstock–Dunford integrable functions taking values in \({\mathcal {X}}\) as those weakly measurable function \( g: \varOmega \rightarrow {\mathcal {X}}, \) for which \(\{y^*g: y^* \in B_{\math...

We investigate some properties and convergence theorem of Kluv\'{a}nek-Lewis-Henstock $\m-$integrability for $\m-$measurable functions that we introduced in \cite{ABH}. We give a $\m-$a.e. convergence version of Dominated (resp. Bounded) Convergence Theorem for $\m.$ We introduce Kluv\'{a}nek-Lewis-Henstock integrable of scalar-valued functions wit...

The objective of this paper is to construct separable Banach spaces S Dp[ℝIn] for 1 ≤ p ≤ ∞, each of which contains all of the standard Lp[ℝIn] spaces, as compact dense embedding.

In T. L. Gill and W. W. Zachary, Functional Analysis and the Feynman Operator Calculus (Springer, New York, 2016), the topology of [Formula: see text] was replaced with a new topology and denoted by [Formula: see text]. This space was then used to construct Lebesgue measure on [Formula: see text] in a manner that is no more difficult than the same...

We construct Zachary space in $\R^\infty$ and find that this is a Banach space of functions of bounded mean oscillation with order $p, 1\leq p \leq \infty$ containing the function of bounded mean oscillation $BMO[\R_I^\infty]$ as a dense continuos embedding. As an application of $\R_I^\infty$ we construction $\mcB,$ where $\mcB $ is separable Banac...

The objective of this paper is the construction of the Banach spaces SD p [R nI ] for 1 ≤ p ≤ ∞, each of which contains all of the standard L p [R nI ] spaces, as well as finitely additive measures, as compact dense embedding.

The motivation of this paper is to introduce Henstock–Orlicz space with non-absolute integrable functions. We prove that [Formula: see text] is dense in the Henstock–Orlicz space, which is not dense in the classical Orlicz space.

The purpose of this paper is to construct a new class of separable Banach spaces K p [B], 1 ≤ p ≤ ∞. Each of these spaces contain the L p [B] spaces, as well as the space M[R ∞ ], of finitely additive measures as dense continuous compact embed-dings. These spaces are of interest because they also contain the Henstock-Kurzweil integrable functions o...

We introduce Kuelbs-Steadman-type spaces for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate their main properties and embeddings in $L^p$-type spaces, considering both the norm associated to norm convergence of the involved integrals and that related to weak convergence of the inte...

We introduce Kuelbs-Steadman-type spaces (KS p spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of L q-type spaces into KS p spaces, considering both the norm associated with the norm convergence of the involved integrals and that relate...

We construct Sobolev-Kuelbs-Steadman spaces on $\mathbb{R}^\infty$ which contains Sobolev spaces as dense embedding. We have seen sequence of weak solution of Sobolev spaces are convergence strongly in Sobolev-Kuelbs-Steadman space. Finally we have shown the Sobolev space through Bessel Potential is densely contained in Sobolev-Kuelbs-Steadman spac...

The purpose of this paper is to construct a new class of separable Banach spaces $KS^p[\mathbb{R}^{\infty}], \; 1\leq p \leq \infty$. Each of these spaces contain the $ L^p[\mathbb{R}^\infty] $ spaces, as well as the space $\mfM[\mathbb{R}^\iy]$, of finitely additive measures as dense continuous compact embeddings. These spaces are of interest beca...

The objective of this paper is to construct separable Banach spaces $S{D^p}[\mathbb{R}^\infty]$ for $1\leq p \leq \infty$, each of which contains the $L^p[\mathbb{R}^\infty] $ spaces, as well as finitely additive measures, as compact dense embedding. Also these spaces contains Henstock-Kurzweil integrable functions.

The motivation of the article is to introduce Henstock-Orlicz space with non absolute integrable functions. We prove $ C_{0}^{\infty} $ is dense in the Henstock-Orlicz space, which is not dense in the classical Orlicz space.

Given a real Banach space $\mathcal{X}$ and probability space $(\Omega, \Sigma, \mu)$ we characterize the countable additivity of Henstock-Dunford integral for Henstock integrable function taking values in $X$ as those weakly measurable function $ g: \Omega \to \mathcal{X} $ for which $\{y^*g~: y^* \in B_\mathcal{X}^* \} $ is relatively weakly comp...