arXiv subjects
Publications and source records attributed to Bipan Hazarika.
In this paper, we introduce the RLC-fractal derivative along with the fractal Laplace and fractal Sumudu transforms of it. Then we apply these transformations to solve economic models involving the RLC-fractal derivative.
In this paper, we present a new fractal derivative with a nonsingular kernel and analyze its fundamental properties. The effectiveness of the proposed operator is illustrated through the study of economic models using both the Caputo fractal derivative and the new fractal derivative.
This paper intends to develop a $q$-difference operator $\nabla^{(γ)}_q$ of fractional order $γ$, and give several intriguing properties of this new difference operator. Our main focus remains on the construction of sequence spaces $\ell_p(\nabla^{(γ)})$ and $\ell_\infty (\nabla^{(γ)})$, at the same time comparing these spaces with those already exist in the literature. Apart from obtaining Schauder basis, we determine $α$-, $β$-, and $γ$-duals of the newly defined spaces. A section is also devoted for characterizing matrix classes $(\ell_p(\nabla^{(γ)}),\mathfrak X),$ where $\mathfrak X$ is any of the spaces $\ell_\infty,$ $c,$ $c_0$ and $\ell_1$.
This paper presents the Hausdorff measure of noncompactness (MNC) within the framework of the generalized Hahn sequence space. By applying the MNC, we explore the existence of solutions for nonlinear Caputo fractional differential equations subject to three-point integral boundary conditions in the generalized Hahn sequence space. We apply certain sufficient conditions to ensure the uniqueness of solutions to the aforementioned problem, utilizing the Banach fixed-point theorem, and discuss the Hyers-Ulam stability of the problem. In conclusion, the analytical framework is complemented by illustrative examples that demonstrate the validity and applicability of the main results.
In this paper, we investigate the existence of the asymptotically almost automorphic solution of the following type of abstract nonlinear integro-dynamic equation \begin{eqnarray*} y^Δ(s) &=&Ay(s)+\mathcal{F}\left(s,y(s),\int\limits_{t_0}^{s}{\mathcal{H}(s,τ,y(τ))}Δτ\right),~ s\in\mathbb{T}^k, y(0)&=&y_0 \end{eqnarray*} in the Banach space of continuous function on a time scale $\mathbb{T}$. We apply the Krasnoselskii fixed point theorem to show the existence of an almost automorphic solution of the above dynamic equation.
For a locally compact group $H$ with a left Haar measure, we study variable Lebesgue algebra $\mathcal{L}^{p(\cdot)}(H)$ with respect to a convolution. We show that if $\mathcal{L}^{p(\cdot)}(H)$ has bounded exponent, then it contains a left approximate identity. We also prove a necessary and sufficient condition for $\mathcal{L}^{p(\cdot)}(H)$ to have an identity. We observe that a closed linear subspace of $\mathcal{L}^{p(\cdot)}(H)$ is a left ideal if and only if it is left translation invariant.
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 manuscript we investigate the existence and uniqueness of an impulsive fractional dynamic equation on time scales involving non-local initial condition with help of Caputo nabla derivative. The existency is based on the Scheafer's fixed point theorem along with the Arzela-Ascoli theorem and Banach contraction theorem. The comparison of the Caputo nabla derivative and Riemann-Liouvile nabla derivative of fractional order are also discussed in the context of time scale.
In this {\color{red}{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, several convergence theorems are studied.
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 continuous embedding. As an application of $\R_I^\infty$ we construction $\mcB,$ where $\mcB $ is separable Banach space and finally we construct $\mcZ^p[\mcB]$.
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 $\mcH^θ(X, \barμ)$ where $X$ is any Banach space are discussed. Relationships of modular convergence and norm convergence of $H$-Orlicz spaces are discussed.
In this paper we discuss the structure of Henstock-Orlicz space with locally Henstock integrable functions. The weak Henstock-Orlicz spaces on $\mathbb{R}^n$ 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.
In this paper, we introduce the notion of fuzzy soft numbers. Here defined fuzzy soft number and four arithmetic operations $ \tilde{+}, \tilde{-}, \tilde{\times}, \tilde÷ $ and related properties. Also introduce Hausdorff distance, Fuzzy soft metric space, convergence sequence, Cauchy sequence, Continuity, and uniform continuity of fuzzy soft numbers. At starting of this paper, we study convex and concave fuzzy soft sets and some of their properties.
In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann-Liouville sense. We also introduce the nabla fractional derivative in Grünwald-Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed.
In this paper, we introduce some new double sequence spaces $\mathcal{M}_u(Δ)$ and $\mathcal{C}_{\vartheta}(Δ)$, where $\vartheta\in\{bp,bp0,r,r0\}$ as the domains of the four-dimensional forward difference matrix in the double sequence spaces $\mathcal{M}_u$ and $\mathcal{C}_{\vartheta}$, respectively. Then we investigate some topological and algebraic properties. Moreover, we determine the $α-$, $β(\vartheta)-$, and $γ-$duals of the new spaces $\mathcal{M}_u(Δ)$ and $\mathcal{C}_{\vartheta}(Δ)$. Finally, we characterize four-dimensional matrix classes $(λ(Δ),μ)$ and $(μ,λ(Δ))$, where $λ=\{\mathcal{M}_u,\mathcal{C}_{\vartheta}\}$ and $μ=\{\mathcal{M}_u,\mathcal{C}_{\vartheta}\}$.
Our goal in this article is to construct HK-Sobolev spaces on $\R^\infty$ which contains Sobolev spaces as dense embedding. We discuss that the sequence of weak solution of Sobolev spaces are convergence strongly in HK-Sobolev space. Also, we obtain that the Sobolev space through Bessel Potential is densely contained in HK-Sobolev spaces. Finally we find sufficient condition for the solvability of the divergence equation $\nabla.F= f,$ for $f$ is an element of the subspace $K{S^p}[\R_I^n]$ and $n \in \N$, in the SoboHK-Sobolev space $WS^{k,p}[\R_I^n] $ with the help of Fourier transformation.
We investigate some properties and convergence theorem of Kluvánek-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 Kluvánek-Lewis-Henstock integrable of scalar-valued functions with respect to a set valued measure in a Banach space. Finally we introduce $(KL)-$type Dominated Convergence Theorem for the set-valued Kluvánek-Lewis-Henstock integral.
The purpose of this paper is to construct a new class of separable Banach spaces $\K^p[\mathbb{B}], \; 1\leq p \leq \infty$. Each of these spaces contain the $ \mcL^p[\mathbb{B}] $ spaces, as well as the space $\mfM[\R^\iy]$, of finitely additive measures as dense continuous compact embeddings. These spaces are of interest because they also contain the Henstock-Kurzweil integrable functions on $\mathbb{B}$. Finally, we offer a interesting approach to the Fourier transform on $\K^p[\mathbb{B}].$