Fixed Point Theorems in G-Metric Spaces With W Maps

The main aim of this paper is to prove that the existence and uniqueness of solutions for systems of simultaneous Volterra Hammerstein and Urysohn nonlinear integral equations in G-metric spaces and partially ordered G-metric spaces settings by using common fixed point theorems satisfying generalized contractive conditions.

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.

We introduce a new hybrid contraction condition in the setting of G-metric spaces that unifies Banach-, Kannan-, and Chatterjea-type contractions applied to an iterate Tp of a self-map T. Under a natural coefficient constraint, we prove that such a map admits a unique fixed point in a complete G-metric space. An illustrative example is provided to demonstrate the applicability of the result beyond classical contractions.

In the context of G-metric spaces we prove a common fixed point theorem for a sequence of self mappings using a new concept of α-series.

In this article, we prove some coincidence and common fixed point theorems under the relation-theoretic Meir-Keeler contractions in a metric space endowed with a locally finitely T -transitive binary relation. Our newly proved results generalize, extend, and sharpen some existing coincidence point as well as fixed point theorems existing in the literature. Moreover, we give some examples to affirm the efficacy of our results.

In this paper we prove some fixed point theorems for weakly compatible mappings with the notation of A-distance and E-distance in uniform space.

In this present paper, we establish Cantor's intersection like theorem in a locally convex topological vector spaces. Some fixed point and common fixed point theorems are proved for Reich and Caccioppoli type contractive mappings in such a locally convex topological vector space. Also in this setting we prove a fixed point theorem for some mapping which is the uniform limit of a sequence of Reich type contractive mappings therein. <br><br> Встановлена теорема, подібна теоремі Кантора про перетин, у випадку локально опуклих векторних просторів. Для стискуючих відображень типу Райха і Каччіополі відповідних просторів доведені теореми про нерухому точку та спільну нерухому точку. Також у цій постановці доведена теорема про нерухому точку для відображення, яке є рівномірною границею послідовності стискуючих відображень типу Райха.

&lt;p&gt;In this paper, we prove fixed point theorems for weakly compatible self mappings satisfying certain contractive conditions of integral type in 𝐺 - metric spaces.&lt;/p&gt;

We deduce coincidence and fixed point theorems under generalized weakly contractive conditions in G-metric spaces equipped with partial order. We furnish examples to demonstrate the usage of the results and to distinguish them from the known ones.

In this paper, we introduce weakly generalized contraction conditions on G-metric space and prove some common fixed point theorems for the proposed contractions. The results in this paper differ from the recent corresponding results given by some authors in literature.

Fixed point theory in cone metric spaces obtained via the scalarization method

Common fixed points of nonlinear hybrid mappings under strict contractions in semi-metric spaces

Recently Suzuki (2008) and then Kikkawa and Suzuki (2008) gave a new generalization of the Banach contraction principle. Then, Mot and Petrusel (2009), Dhompapangasa and Yingtaweesttikulue (2009), Bose and Roychowd- hury (2011), Singh and Mishra (2010), and Doric and Lazovic (2011) further extended their work. Recently Bose (2012) obtained some Suzuki-type common fixed point theorems for generalized contractive multivalued mappings using a result of Bose and Mukherjee(1977) which extend the previously obtained re- sults. Recently Damjanovic and Doric(2011) obtained a multivalued generaliza- tion of theorems Kikkawa and Suzuki (2008) concerning Kannan mappings. Also Singh and Mishra (2010) considered coincidence and fixed point theorems for a class of hybrid pair of single-valued and multi-valued maps in metric space setting and Singh et al (2012) prsented a common fixed point theorem for a pair of multi-valued maps in a complete metric space extending a recent theorem of Doric and Lazovic. First we generalize the theorem of Singh et al which also extend the theorem of Singh and Mishra(2010). Then we extend the theorem of Damjanovic and Doric to a common fixed point theorem of a pair of mul- tivalued mappings. As an application, we consider the existence of a common solution for a class of functional equations arising in dynamic programming.

In this paper, we prove some common fixed point theorems using (CLR) property in KM and GV-fuzzy metric spaces. Our results extend and unify several fixed point theorems present in the literature.

&lt;abstract&gt;&lt;p&gt;We introduce a Ćirić type contraction principle in a vector-valued $ b $-metric space that generalizes Perov's contraction principle. We investigate the possible conditions on the mappings $ W, E:G\rightarrow G $ ($ G $ is a non-empty set), for which these mappings admit a unique common fixed point in $ G $ subject to a nonlinear operator $ {\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m} $. We illustrate the hypothesis of our findings with examples. We consider an infectious disease model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.&lt;/p&gt;&lt;/abstract&gt;