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Lakshmi Kanta Dey

Publications and source records attributed to Lakshmi Kanta Dey.

At least 19 recordsLinked to original sources
arXiv2026

Study of Existence and Stability of Fixed Points of Hypotenuse Contracting Mappings with Applications to Parabolic PDE

In this article, we introduce a novel class of mappings identified by their property of contracting the hypotenuse of a right-angled triangle in the setting of metric spaces. We establish sufficient conditions ensuring both the existence and uniqueness of fixed points. A geometric analysis, complemented by illustrative diagrams, is provided to differentiate these mappings from other familiar contraction types, namely perimeter and area contractions, supported by examples. Furthermore, we investigate the Ulam-Hyers stability of the associated fixed point equation, thereby strengthening the robustness of the theoretical framework. Finally, the derived results are applied to demonstrate the existence of solutions for a nonhomogeneous linear parabolic partial differential equation (PDE).

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arXiv2026

Unlocking novel topological structures via rough families

Very recently, the notion of rough family has been introduced in [Leonetti, P., J. Convex Anal. 32(4):1083-1090, 2025] to explore rough ideal convergence in topological spaces where the limit of a sequence may not be unique. This raises the question of whether $T_2$ topological spaces can be characterized using rough families. In this article, we prove that a topological space is $T_2$ if and only if it can never be a rough topological space. In this context, we first introduce the notions of rough interior and rough closure of a set from the perspective of a rough family, which leads to the definition of rough open sets (rough closed sets). As a consequence, we generate a new topology, termed rough topology, as well as rough homeomorphism. Our main contribution presents the novelty of this new class; in particular, we explicitly construct several examples which ensure that two non-homeomorphic spaces can be roughly homeomorphic under certain roughness. Additionally, we extend the concepts of compactness as well as connectedness, where our findings diverge from existing literature in these areas, in a nutshell, providing new insights and perspectives.

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arXiv2026

On the geometry of $G$-norm

Let $X$ and $Y$ be Banach spaces and let $G \in L(X,Y)$ with $\|G\|=1$. We study the geometry of $G$-(semi-)norm on $L(X,Y)$, defined by \[ \|T\|_G := \inf_{δ>0}\sup\{\|Tx\|: \|x\|=1, \|Gx\|>1-δ\}, \] considering it as a norm ($G$-norm), and further explore the associated numerical indices. In particular, we characterize relative spear operators, that is, operators for which the numerical radius with respect to $G$ coincides with the $G$-norm. Relations among the numerical indices and their invariance under isometric isomorphisms are established. We further obtain a description of the dual unit ball of $(L(X,Y),\|\cdot\|_G)$ and characterize smooth points of its unit ball. In finite-dimensional Hilbert spaces, we prove that relative spear operators are partial isometries. Finally, we establish some equivalent criteria for which the $G$-norm is achieved by the norm attainment set of a norm-attaining operator $G$.

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arXiv2026

Best Proximity Point Results for Perimetric Contractions

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.

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arXiv2025

On a novel approach to nonexpansive mappings

This paper seeks to advance the theory of nonexpansive mappings by introducing and exploring a novel class of nonexpansive type mappings, which we aptly designate as perimetric nonexpansive mappings. We establish that the collection of mappings we propose is considerably larger than the existing classes of nonexpansive and quasi-nonexpansive mappings. We also establish fixed point existence findings by examining the connection between periodic points and fixed points in the context of normed linear spaces. Finally, we establish a significant result by proving that every perimetric nonexpansive mapping on a closed bounded convex subset of a Hilbert space necessarily has a fixed point.

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arXiv2024

Perimetric contraction on quadrilaterals and related fixed point results

In this article, we introduce a four-point analogue of Banach-type, Kannan-type, and Chatterjea-type contractions, and examine their properties. We establish sufficient conditions under which these mappings achieve fixed points in a complete metric space. Notably, the classical Banach contraction principle emerges as a special case of our results. To illustrate our theoretical findings, we present several non-trivial examples.

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arXiv2022

On the extended version of Krasnosel'skii's fixed point theorem for Kannan type equicontraction mappings

A sufficient condition is established for the existence of a solution to the equation $\mathcal{T}(u,\mathcal{C}(u))=u$, by considering a class of Kannan type equicontraction mappings $\mathcal{T}:\mathcal{A}\times \overline{\mathcal{C}(\mathcal{A})}\to Ξ$, where $\mathcal{A}$ is a convex, closed and bounded subset of a Banach space $Ξ$ and $\mathcal{C}$ is a compact mapping. To fulfil the desired purpose, we engage the Sadovskii's theorem, involving the measure of noncompactness. The relevance of the acquired results has been illustrated by considering a certain class of initial value problems.

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arXiv2021

Best proximity results for $p$-proximal contractions on topological spaces

In this article, we investigate for some sufficient conditions for the existence and uniqueness of best proximity points for the topological $p$-proximal contractions and $p$-proximal contractive mappings on arbitrary topological spaces. Moreover, our results are authenticated by a few numerical examples and these generalize some of the known results in the literature.

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arXiv2020

On proximal contractions via implicit relations and best proximity points

In this paper, we employ two types of implicit relations to define some new kind of proximal contractions and study about their best proximity points. More precisely, we use two class of functions $\mathcal{A}$ and $\mathcal{A}'$ to explore proximal $\mathcal{A}$, $\mathcal{A}'$-contractions of first and second type and strong proximal $\mathcal{A}$, $\mathcal{A}'$-contractions. We investigate the existence of best proximity point results of the same. It is worth mentioning that the well-known results of Sadiq Basha [J. Approx. Theory, $2011$] on proximal contractions are the special cases of our obtained results. We authenticate our results by suitable examples.

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arXiv2020

Characterization of $M$-compact sets via statistically convergent sequences

In this paper, we study stability of $M$-compactness for $l^p$ sum of Banach spaces for $1\leq p<\infty$. We also obtain a characterization of $M$-compact sets in terms of statistically maximizing sequence, a notion which is weaker than a maximizing sequence. Moreover, we introduce the notion of $\mathcal{I}$-$M$-compactness of a bounded subset $M$ of a normed linear space $X$ with respect to an ideal $\mathcal{I}$ and show that it is equivalent to $M$-compactness for non-trivial admissible ideals.

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arXiv2020

Existence of Solutions to Non-linear Quadratic Integral Equations via Measure of Non-Compactness

The objective of this manuscript is to enquire for the solvability of a specific type of non-linear quadratic integral equations via the interesting notion of measure of non-compactness. Firstly, we inquire into couple of exciting fixed point theorems involving a measure of non-compactness in the setting of a Banach space. Subsequently, bringing into play a suitable measure of non-compactness and the acquired results, we discuss the existence of solution to the aforementioned kind of non-linear quadratic integral equations.

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arXiv2020

Best proximity point results in topological spaces and extension of Banach contraction principle

In this paper, we introduce the notion of topologically Banach contraction mapping defined on an arbitrary topological space X with the help of a continuous function $g:X\times X\rightarrow \mathbb{R}$ and investigate the existence of fixed points of such mapping. Moreover, we introduce two types of mappings defined on a non-empty subset of X and produce sufficient conditions which will ensure the existence of best proximity points for these mappings. Our best proximity point results also extend some existing results from metric spaces or Banach spaces to topological spaces. More precisely, our newly introduced mappings are more general than that of the corresponding notions introduced by Bunlue and Suantai [Arch. Math. (Brno), 54(2018), 165-176]. We present several examples to validate our results and justify its motivation. To study best proximity point results, we introduce the notions of g-closed, g-sequentially compact subsets of X and produce examples to show that there exists a non-empty subset of X which is not closed, sequentially compact under usual topology but is g-closed and g-sequentially compact.

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arXiv2020

On some enriched contractions in Banach spaces

In this paper, we introduce two new types of enriched contractions, viz., enriched $\mathcal{A}$-contraction and enriched $\mathcal{A}'$-contraction. Then we obtain fixed points of mappings satisfying such contractions using the fixed point property of the average operator of the mappings. Further, we study the well-posedness and limit shadowing property of the fixed point problem involving the contractions, and give some examples to validate the results proved. We frame an open question related to the existence of a fixed point of such contractions. We also show that Berinde and Păcurar's recent results on different kinds enriched contractions and some well known classical fixed point results are particular cases of our results.

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arXiv2020

On new existence of a unique common solution to a pair of non-linear matrix equations

The main goal of this article is to study the existence of a unique positive definite common solution to a pair of matrix equations of the form \begin{eqnarray*} X^r=Q_1 + \displaystyle \sum_{i=1}^{m} {A_i}^*F(X)A_i \mbox{ and } X^s=Q_2 + \displaystyle \sum_{i=1}^{m} {A_i}^*G(X)A_i \end{eqnarray*} where $Q_1,Q_2\in P(n)$, $A_i\in M(n)$ and $F,G:P(n)\to P(n)$ are certain functions and $r,s>1$. In order to achieve our target, we take the help of elegant properties of Thompson metric on the set of all $n \times n$ Hermitian positive definite matrices. To proceed this, we first derive a common fixed point result for a pair of mappings utilizing a certain class of control functions in a metric space. Then, we obtain some sufficient conditions to assure a unique positive definite common solution to the said equations. Finally, to validate our results, we provide a couple of numerical examples with diagrammatic representations of the convergence behaviour of iterative sequences.

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arXiv2020

Some remarks on the metrizability of some well known generalized metric-like structures

In \cite[\, An, V.T., Tuyen, Q.L. and Dung, V.N., Stone-type theorem on $b$-metric spaces and applications, Topology Appl. 185-186 (2015), 50-64.]{an}, An et al. had provided a sufficient condition for $b$-metric spaces to be metrizable. However, their proof of metrizability relied on an assumption that the distance function is continuous in one variable. In this short note, we improve upon this result in a more simplified way without considering any assumption on the distance function. Moreover, we provide two shorter proofs of the metrizability of $\mathcal{F}$-metric spaces recently introduced by Jleli and Samet in \cite[\, Jleli, M. and Samet, B., On a new generalization of metric spaces, J. Fixed Point Theory Appl. (2018) 20:128]{JS1}. Lastly, in this short note, we give an alternative proof of the metrizability of $θ$-metric spaces introduced by Khojasteh et al. in \cite[\, Khojasteh, F., Karapinar, E. and Radenovic, S., $θ$-metric space: A Generalization, Math. Probl. Eng. Volume 2013, Article 504609, 7 pages]{ks}.

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arXiv2020

Farthest Point Problem and Partial Statistical Continuity in Normed Linear Spaces

In this paper, we prove that if $E$ is a uniquely remotal subset of a real normed linear space $X$ such that $E$ has a Chebyshev center $c \in X$ and the farthest point map $F:X\rightarrow E$ restricted to $[c,F(c)]$ is partially statistically continuous at $c$, then $E$ is a singleton. We obtain a necessary condition on uniquely remotal subsets of uniformly rotund Banach spaces to be a singleton. Moreover, we show that there exists a remotal set $M$ having a Chebyshev center $c$ such that the farthest point map $F:\mathbb{R}\rightarrow M$ is not continuous at $c$ but is partially statistically continuous there in the multivalued sense.

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arXiv2020

On contractive mappings in $b_v(s)$-metric spaces

The major motives of this paper are to study different types of contractive mappings and also to answer an open question of Garai et al. [The contractive principle for mappings in $b_v(s)$-metric spaces, arXiv:1802.03136]. We first set up some fixed point results associated with two types of contractive mappings in $b_v(s)$-metric spaces and then we give an answer, in positive, to the open question. Most importantly, we characterize the completeness of a $b_v(s)$-metric space via fixed point property of a certain type of contractive mappings. Our results extend and generalized several important results in the literature.

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