G-metric Spaces Research Articles

Fixed Point Theorems in G-Metric Vector Spaces and their applications in Machine Learning and Optimization Algorithms

This research aims to develop a novel mathematical formulation of fixed point theorems in G-metric vector spaces and explore its applications in machine learning and optimization algorithms. The study begins by defining a complete G-metric space and establishing a generalized contraction condition for an operator representing iterative updates in optimization processes. Utilizing gradient descent with regularization as a case study, a numerical example is presented to validate the proposed formulation. The methodology involves iterative calculations demonstrating the convergence of a sequence of parameters toward a fixed point, highlighting the effectiveness of the auxiliary function in promoting sparsity while accommodating the non-Euclidean characteristics of high-dimensional data. The main results reveal that the proposed framework not only enhances the convergence properties of iterative algorithms but also aligns well with contemporary regularization techniques in machine learning. The conclusions drawn emphasize the significance of integrating generalized G-metric spaces and auxiliary functions in fixed point theories, suggesting that this approach provides a robust foundation for developing adaptive optimization algorithms. The findings indicate substantial implications for future research, highlighting the need for broader exploration of G-metric spaces and their applicability in various machine learning paradigms to address complex data challenges.

Common Attractors for Generalized F-Iterated Function Systems in G-Metric Spaces

In this paper, we study the generalized F-iterated function system in G-metric space. Several results of common attractors of generalized iterated function systems obtained by using generalized F-Hutchinson operators are also established. We prove that the triplet of F-Hutchinson operators defined for a finite number of general contractive mappings on a complete G-metric space is itself a generalized F-contraction mapping on a space of compact sets. We also present several examples in 2-D and 3-D for our results.

Approximation of fixed points in convex G-metric spaces

In this paper, we extend some fixed point results for various classes of mappings to approximating fixed points, using Mann iterative process in the context of convex $G$-metric spaces.

Fixed point results for G-F-Contractive mappings of Hardy-Rogers type

In this paper, we present the notation of G-F-Contractive mappings of Hardy-Rogers type and give some fixed point results of Hardy-Rogers type for self-mappings in complete G-metric spaces.

Convergence theorems in G-metric space and its application

Convergence theorems in G-metric space and its application

ON L-FUZZY FIXED POINT RESULTS IN G-METRIC SPACE

Among numerous efforts in progressing fuzzy mathematics, a lot of attention has been paid to studying new L-fuzzy analogues of the conventional fixed point results and their various applications. In this note, novel concepts of L-fuzzy contractions in G-metric space is examined, and sufficient conditions for the existence of L-fuzzy fixed points for such mappings are investigated. Nontrivial examples are provided to support the assumptions of our obtained theorems. It is highlighted that the main ideas studied herein refine and include some known results in the corresponding literature. Some of these special cases of our results are pointed out and analyzed.

On Ideal Convergence in Generalized Metric Spaces

Our main goal in this paper is to introduce the concept of ideal convergence in G-metric spaces. We give definitions of GI-convergence and GI*-convergence in G-metric spaces. We also extend the I-convergence concept's properties to GI-convergence. Then we demonstrate that GI-convergence and GI*-convergence are equivalent by giving the property (AP) definition. Additionally we introduce GI-Cauchy and GI*-Cauchy sequences and adapt the classically stated theorems to G-metric spaces.

Rational type cyclic contraction in G-metric spaces

Rational type cyclic contraction in G-metric spaces

A Novel Approach for Digital Image Compression in Some Fixed Point Results on Complete G-metric Space Using Comparison Function

In the present time world, digital images are crucial for various applications, that includes the medical industry, aircraft and satellite imaging, underwater imaging and so on. For this huge quantities of digital images are produced and used by these applications. For a variety of reasons, these images also need to be transmitted and stored. Therefore, a technique known as compression is applied to resolve this storage issue while transmitting these images. In this article, by extending some unique fixed point theorem results for comparison function on a complete symmetric G-metric space are used and it is a new approach. Moreover, this paper focuses on a compression method using the new structure of extended G-contraction mapping as it assists in compressing the size of the image. Thus, grayscale images are compressed using extended G-contraction mapping. And thus, grayscale images can be represented as matrices in this structure (pixel values). Also, similar images of reduced size can be obtained using an appropriate matrix G-metric and extended G-contraction mapping. The size of the matrix can be substantially reduced without losing any quality by controlling the order of sub matrices. These images are easy to store and transmit, with little variation between the original and contracted image.

Some Common Fuzzy Fixed Point Theorems on G - Metric Space

The motive of this paper is to give a few not unusual place constant factor theorems in G-Metric spaces, with the aid of using the perception of Common restrict with inside the variety belongings and to illustrate appropriate examples. These outcomes make bigger and generalizes numerous widely known outcomes with inside the literature.

Generalized G-Hausdorff space and applications in fractals

In this paper, we introduce the concept of a G-Hausdorff space and show how the results established in the usual metric space can be generalized to the G-metric space. The proven results are used to propose an iterated function system (IFS) called G-IFS. Additionally, the existence of the fractal associated with this construction is demonstrated. This paper shows how non-affine transformations and fractal interpolation functions (FIFs) can be used to approximate fractals by G-IFS. This paper contributes to the understanding of fractal geometry and its applications in mathematics and other fields.

(H, Ωb)-Interpolative Contractions in Ωb-distance Mappings with Application

Interpolative Kannan contractions are a refinement of Kannan contraction, which is considered as one of the significant notions in fixed point theory. Gb-metric spaces is considered as a generalized concept of both concepts b-metric and G-metric spaces therefore, the significant fixed and common fixed point results of the contraction based on this concept is generalized resultsfor both concepts. The purpose of this manuscript, is to take advantage to interpolative Kannan contraction together with the notion of Ωb which equipped with Gb-metric spaces and H simulation functions to formulate two new interpolative contractions namely, (H, Ωb)-interpolative contraction for self mapping f and generalized (H, Ωb)-interpolative contraction for pair of self mappings (f1, f2). We discuss new fixed and common fixed point theorems. Moreover, to demonstrate the applicability and novelty of our theorems, we formulate numerical examples and applications to illustrate the importance of fixed point theory in applied mathematics and other sciences.

Some Observations on g-Metric Spaces in Light of Generalized Statistical Convergence of Double Sequences via Ideals

In this article, we delve into the notions of I2-statistical convergence and I2-lacunary statistical convergence for sequences in general metric spaces, specifically g metric spaces. We thoroughly explore these concepts within the realm of g-metric spaces.

Some applications of fixed point results for monotone multivalued and integral type contractive mappings

The motivation of the present paper is to introduce and establish some new fixed point results for monotone multivalued functions in partially ordered complete D^{*}-metric spaces, where the partial ordered set (X,leq ) is obtained via a pair of functions ( Upsilon,Omega ). Moreover, several existence and uniqueness coupled fixed point theorems of mappings satisfying contractive conditions have been investigated and verified in the setting of partially ordered complete D^{*}-metric spaces by using the concept of integral type contractions with respect to partially ordered D^{*}-metric space. Furthermore, we present appropriate examples as an application for our main results. Our results generalize the work of Ghasab, Majani, and Rad on the study of integral type contraction and coupled fixed point theorems in the ordered G-metric spaces.

A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces

This paper contains the equivalence between tvs-G cone metric and G-metric using a scalarization function $\zeta_p$, defined over a locally convex Hausdorff topological vector space. This function ensures that most studies on the existence and uniqueness of fixed-point theorems on G-metric space and tvs-G cone metric spaces are equivalent. We prove the equivalence between the vector-valued version and scalar-valued version of the fixed-point theorems of those spaces. Moreover, we present that if a real Banach space is considered instead of a locally convex Hausdorff space, then the theorems of this article extend some results of G-cone metric spaces and ensure the correspondence between any G-cone metric space and the G-metric space.

Advancements in Hybrid Fixed Point Results and F-Contractive Operators

The aim of this manuscript is to introduce a novel concept called Jaggi-type hybrid (ϕ -F)-contraction and establish some fixed point results for this class of contractions in the framework of G-metric space. The validity of the main result is shown by a suitable example and the realized improvements with respect to the corresponding literature are highlighted. By using the constructed example, it is observed that the results established herein cannot be deduced from their analogs in previously announced results in the literature. As an application, the existence and uniqueness of solutions to certain nonlinear Volterra integral equations are investigated to illustrate the utility of our obtained results.

Common Fixed Point Theorems for Weakly Compatible Maps Satisfying Integral Type Contraction in G-Metric Spaces

Common Fixed Point Theorems for Weakly Compatible Maps Satisfying Integral Type Contraction in G-Metric Spaces

Fg-Metric Spaces

This paper introduces the concept of Fg-metric Space. We generalize the concept of G-metric space. Some supporting examples are given.

Some New Estimates of Fixed Point Results under Multi-Valued Mappings in G-Metric Spaces with Application

It is well known fact that fixed point results are very useful for finding the solution of different types of differential equations. In this paper, some new results of multi-valued functions involving rational inequality in G−metric spaces have been obtained. Some of the new concepts are also defined over G−metric spaces that are open set, interior of set, and limit point of set. Moreover, we have presented the application of our main results in the field of Homotopy. Some non-trivial examples are also provided to discuss the validation of the main finding.

Best Proximity Point for G-Generalized ζ-β-T Contraction

In this paper, we find the best proximity point in G-metric spaces for G-generalized ζ-β-T contraction mappings and verify the existence and uniqueness of the best proximity point in the complete G metric space using the idea of an approximatively compact set. In addition, an example is provided to illustrate the outcome.