Modular Spaces Research Articles

Design and Preliminary Ground Experiment for Deployable Sunshade Structures of a Modular Space Telescope.

On-orbit assembling space telescope (OAST) is one of the most feasible methods to implement a large-scale space telescope. Unlike a monolithic space telescope (such as Hubble Space Telescope, HST) or a deployable space telescope (such as James Webb Space Telescope, JWST), OAST can be assembled in the spatial environment. To ensure proper telescope performance, OAST must be equipped with a large deployable sunshade. In order to verify the technology of the OAST, the authors propose a modular space telescope on the China Space Station (CSS) and design a deployable sunshade. The deployable mechanism of the sunshade is made up of a radial deployable mechanism and an axial deployable mechanism. The paper describes the overall design approach, the key component technologies, and the design and preliminary testing of a part of the deployable sunshade assembly.

Modular version of Goebel-Kirk theorem

In this paper we prove a fixed point theorem for asymptotically nonexpansive mappings acting in modular spaces. This result generalises the 1972 fixed point theorem by K. Goebel and W.A. Kirk. In the process, we extend several other results (including the Milman-Pettis theorem) from the class of Banach spaces to the larger class of regular modular spaces.

Fixed point theorem and iterated function system in φ-metric modular space

We introduce and study the concept of φ-metric modular space and, then define φ-α-Meir-Keeler contraction on it and explore its fixed point. Further, we define the Hausdorff distance between two non-empty compact subsets of the considered space. Some topological properties of φ-metric modular space are also explored. Additionally, we prove the existence of the attractor (fractal) of the IFS consisting of φ-α-Meir-Keeler contractions.

Fixed points of $G$-monotone mappings in metric and modular spaces

Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.

Fixed Point Results Under Hausdorff Distance in the Fractal Spaces

We establish a novel notion of Hausdorff distance and explore some of its topological characteristics by using an extended modular metric. We prove a fixed point theorem on generalized modular fractal space from the concept of iterated function system (IFS) and contraction.

Some convergence results in modular spaces with application to a system of integral equations

<p>The paper aimed to achieve three primary objectives. First, it introduced significant common fixed point results in the context of newly proposed partial modular $ b- $metric spaces, thus contributing to the advancement of this field. Second, it presented unique results using a direct approach that did not depend on the strong continuity of the mapping, thereby offering a valuable perspective. Finally, it applied previously established convergence techniques to determine a common solution for a system of Fredholm integral equations, demonstrating the practical implications of the theoretical findings.</p>

Fuzzy stability of mixed type functional equations in Modular spaces

In this article, our aim is to find some stability results for mixed type cubic and quartic functional equations in fuzzy modular spaces using the fundamental results of fixed-point theory. The fixed point method provides one of the effective techniques that can be used to investigate the fuzzy stability of a mixed type cubic and quartic functional equations.

Best proximity point of $ \alpha $-$ \eta $-type generalized $ F $-proximal contractions in modular metric spaces

<abstract><p>The purpose of this paper is to present a study of $ \alpha $-$ \eta $-type generalized $ F $-proximal contraction mappings in the framework of modular metric spaces and to prove some best proximity point theorems for these types of mappings. Some examples are given to show the validity of our results. We also apply our results to establish the existence of solutions for a certain type of non-linear integral equation.</p></abstract>

Modular metric spaces : Some fixed-point theorems and application of secure dynamic routing for WSN

Data exchange within a Wireless Sensor Network (WSN) needs to adhere to high security protocols. The shortest possible route for the transfer of data between two interconnected packets is selected by dynamic routing algorithms, which increases privacy while requesting additional data to be transmitted. Several applications involving remote sensing of environmental requirements, such as WSN, are becoming more prevalent. Highly dense WSNs are evolving into a vital sensor framework enabling the Internet of Things (IoT). WSNs have the potential to synthesize a great deal of actual data, and in order to guarantee that data communication is being performed successfully, they need network architectures that have successful links between Sensor Nodes (SN). The collection of data is a term WSN utilises to define a data fusion/compression technique to minimize the volume of data sent throughout the network’s bandwidth. The process of iteration of Fixed Points (FP) of contractions will be examined in the present paper, which is scheduled to take part under a Modular Metric Space (M-MS) context. The main objective of this research is to arrive at an algorithm for secure dynamic data transmission routing in WSN by integrating the demonstrated results of the Functional Equation (FE) with randomization. An implementation is a tool for evaluating and modelling a network’s architecture in which sensors can be aggregated with a high level of security and least risk in a more secure area (Sensor Field), thereby improving the accuracy of the findings collected by the entire network.

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Triangular ideal relative convergence on modular spaces and Korovkin theorems

"In this paper, we introduce the concept of triangular ideal relative convergence for double sequences of functions defined on a modular space. Based upon this new convergence method, we prove Korovkin theorems. Then, we con- struct an example such that our new approximation results work. Finally, we discuss the reduced results which are obtained by special choices. Keywords: Positive linear operators, the double sequences, triangular ideal relative modular convergence, Korovkin theorem."

Stability results of mixed type functional equations in modular spaces and 2-Banach spaces ∗

In this paper, we investigate the generalized Hyers-Ulam-Stability of additive and quartic functional equations in modular spaces with and without the △2-condition using the direct method and also in 2-Banach Spaces.

Some common fixed point theorems in generalized modular metric spaces with applications

In this paper, we extend the notion of weakly commuting mappings results in modular metric spaces to setting of modular ωG-metric spaces and prove the existence of unique common fixed point of three pairs of weakly commuting self-maps in modular ωG-metric spaces. Moreover, we apply our results in solving certain nonlinear three dimensional integral equations. The results proved in this paper extend, improve and generalize several existing results.

Design and Dynamic Simulation Verification of an On-Orbit Operation-Based Modular Space Robot

Space robots have been playing an important role in space on-orbit operation missions. However, the traditional configuration of space robots only has a single function and cannot meet the requirements of different space missions, and the launch cost of space robots is very high. Thus, the reconfigurable modular space robot system that can carry multiple loads and own mission adaptability is of great significance. Based on the analysis of a robot space mission, combined with the existing reconfigurable robots, this paper develops a configuration design scheme for a modular reconfigurable space robot, and carries out the prototype design. According to the configuration characteristics of the module, the dynamic modeling of the space robot is based on the graph theory analysis and principle of virtual work. Related application scenarios are set up. Function and feasibility of the dynamic modeling methods are verified through assembly experimentation and dynamic simulation.

Examining Nonlinear Fredholm Equations in Lebesgue Spaces with Variable Exponents

We investigate the existence of solutions for the Fredholm integral equation Φ(ϑ)=G(ϑ,Φ(ϑ))+∫01F(ϑ,ζ,Φ(ζ))dζ, for ϑ∈[0,1], in the setting of the modular function spaces Lρ. We also derive an application of this research within the framework of variable exponent Lebesgue spaces Lp(·) subject to specific conditions imposed on the exponent function p(·) and the functions F and G.

Approximation Results by Statistical Convergence Based on a Power Series in Modular Spaces

Approximation Results by Statistical Convergence Based on a Power Series in Modular Spaces

Best Approximation of Fixed-Point Results for Branciari Contraction of Integral Type on Generalized Modular Metric Space

In the realm of generalized modular metric spaces, we substantiate the validity of fixed-point theorems with Branciari contractions. This paper expands and broadens the original theorems in this context. Subsequently, by building upon this foundation, we explore various integral contractions to identify and characterize fixed points within this context. To highlight the practical implications of our work, we introduce the concept of the best proximity pair, thereby culminating in the best approximation theorem. We apply this theoretical construct to a specific example—one that is guided by the best approximation method described in prior research.

Generalized Hyers-Ulam Stability of Additive Functional Inequality in Modular Spaces and β-Homogeneous Banach Spaces

In this work, we investigate the generalised Hyers-Ulam stability of additive functional inequality in modular spaces with ∆2-conditions and in β-homogeneous Banach spaces.

Some Common Fixed Point Results in Modular Ultrametric Space Using Various Contractions and Their Application to Well-Posedness

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a consequence, the examples of fixed point and common fixed point problems are correctly formulated. As an application, the well-posedness of a common fixed point problem is proved. This study expands on prior research in modular ultrametric space to provide a more comprehensive understanding of such spaces using generalized contraction.

Convergence of Perturbed Sampling Kantorovich Operators in Modular Spaces

In the present paper we study the perturbed sampling Kantorovich operators in the general context of the modular spaces. After proving a convergence result for continuous functions with compact support, by using both a modular inequality and a density approach, we establish the main result of modular convergence for these operators. Further, we show several instances of modular spaces in which these results can be applied. In particular, we show some applications in Musielak–Orlicz spaces and in Orlicz spaces and we also consider the case of a modular functional that does not have an integral representation generating a space, which can not be reduced to previous mentioned ones.