Mappings In Complete Metric Spaces Research Articles

A Comprehensive Review of Fixed Point Theory: Foundations, Applications, and Emerging Trends in Mathematical Spaces

Fixed point theory is a crucial branch of mathematical analysis that investigates the conditions under which a function returns a point to itself, symbolizing stability and equilibrium. This paper delves into various fixed point theorems within the contexts of metric spaces, Banach spaces, and Hilbert spaces, emphasizing their foundational importance and wide-ranging applications. The Banach Fixed Point Theorem guarantees the existence and uniqueness of fixed points for contraction mappings in complete metric spaces, while Brouwer's Fixed Point Theorem states that any continuous function mapping a compact convex set to it will have at least one fixed point. Recent developments have expanded these classical results to encompass new types of contraction mappings and generalized distance functions, enhancing their relevance in dynamic systems, control theory, and optimization challenges. Additionally, the paper discusses the Schauder Fixed Point Theorem in Banach spaces, highlighting its significance in analysing nonlinear operators. In Hilbert spaces, fixed point results are examined in relation to nonlinear integral equations and optimization methods, showcasing their practical implications in engineering and variation techniques. Emerging trends include the study of fixed point results in fuzzy and probabilistic environments, as well as the integration of computational approaches with traditional fixed point methods. This paper illustrates the continuous evolution of fixed point theory, connecting abstract mathematical principles with practical problem-solving across various fields. Finally, it proposes future research directions to further explore the potential of fixed point theory in modern mathematics.

COMMON FIXED POINT THEOREMS FOR WEAKLY COMPATIBLE MAPPINGS IN COMPLETE METRIC SPACES

IJungck and Rhoades [13] introduced the notion of coincidentally commuting (weakly compatible) mappings, which is weaker than compatibility. Many interesting, fixed point theorems for weakly compatible maps satisfying contractive type conditions have been obtained by various authors. Goyal [5,6] prove some common fixed point theorems for six mappings involving rational contractive conditions by using notions of compatibility, weak compatibility and commutativity, in complete metric spaces. In this paper, we prove a common fixed point theorem for three pairs of weakly compatible mappings in complete metric spaces satisfying a rational inequality without any continuity requirement which generalize several previously known results due to Imdad and Ali [7], Goyal [5], Imdad-Khan [8], JeongRhoades [9] and others.

Fixed point theorems for metric tower mappings in complete metric spaces

Fixed point theorems for metric tower mappings in complete metric spaces

Common fixed point theorem for pair of quasi triangular α-orbital admissible mappings in complete metric space with application

In this paper, we establish a common fixed point theorem for pair of quasi triangular α-orbital admissible with an interpolative (φ, ψ)- Banach-Kannan-Chatterjea type Z-contraction mappings with respect to simulation function in complete metric space. An illustrative example is furnished to validate our main result. Our result extends the result of M. S. Khan et al. [15]. As an application, we provide the existence of a solution for a nonlinear Fredholm integral equations.

A Study on Fixed-Point Techniques under the α-ϝ-Convex Contraction with an Application

In this paper, we consider several classes of mappings related to the class of α-ϝ-contraction mappings by introducing a convexity condition and establish some fixed-point theorems for such mappings in complete metric spaces. The present result extends and generalizes the well-known results of α-admissible and convex contraction mapping and many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results. Finally, we derive the existence and uniqueness of a solution to an integral equation to support our main result and give a numerical example to validate the application of our obtained results.

Convergence Theorem for Multivalued Almost Type Contractions via Generalized Simulation Functions

The purpose of this work is to introduce the concepts of generalized multivalued almost type Zcontraction along with C-class functions and generalized Suzuki multivalued almost type Z-contraction along with C-class functions for a pair of mappings, as well as to show that common fixed point theorems for such mappings in complete metric spaces. The results of this study generalize and expand on some established fixed point findings in the literature. We derive several corollaries from our core results and offer examples to support our results.

Fixed Points and Stability for Integral-Type Multivalued Contractive Mappings

The existence and iterative approximations of fixed points concerning two classes of integral-type multivalued contractive mappings in complete metric spaces are proved, and the stability of fixed point sets relative to these multivalued contractive mappings is established. The results obtained in this article generalize and improve some known results in the literature. An illustrative example is given.

A fixed point theorem for (\phi, \shi)-convex contraction in metric spaces

In the present paper, we introduce the notion of (\phi, \shi)-convex contraction mapping of order m and establish a fixed point theorem for such mappings in complete metric spaces. The present result extends and generalizes the well known result of Dutta and Choudhary (Fixed Point Theory Appl. 2008 (2008), Art. ID 406368), Rhoades (Nonlinear Anal., 47(2001), 2683-2693), Istratescu (Ann. Mat. Pura Appl., 130(1982), 89-104) and besides many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

On W-contractions of Jungck-Ćirić-Wardowski-type in metric spaces

In this article, we establish some new results on Jungck-Ciric-Wardowski type mappings in complete metric spaces. Using our new approach, the conditions (W2) and (W3) of Wardowski do not apply for ...

Some common best proximity points theorems for generalised rational α-φ-geraghaty proximal contractions

In this paper, we establish some common best proximity point theorems for generalised rational α-φ-Geraghaty proximal contraction mappings in complete metric spaces. Examples are also given to illustrate our results.

On coincidence and common fixed point theorems of eight self-maps satisfying an FM-contraction condition

In this paper, a new type of contraction for several self-mappings of a metric space, called FM-contraction, is introduced. This extends the one presented for a single map by Wardowski [Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012:94, 2012]. Coincidence and common fixed point of eight self mappings satisfying FM-contraction conditions are established via common limit range property without exploiting the completeness of the space or the continuity of the involved maps. Coincidence and common fixed point of eight self-maps satisfying FM-contraction conditions via the common property (E.A.) are also studied. Our results generalize, extend and improve the analogous recent results in the literature, and some examples are presented to justify the validity of our main results.

Generalized multivalued Khan-type (psi ,phi )-contractions in complete metric spaces

In this article, we study a new generalized multivalued Khan-type (psi ,phi )-contraction. We obtain some fixed point theorems related to the introduced contraction for multivalued mappings in complete metric spaces. Our theorems extend and improve some previous known results with less conditions. Also, we give some illustrative examples.

Common Fixed Point Theorems of Generalized Multivalued (ψ,ϕ)-Contractions in Complete Metric Spaces with Application

The purpose of this paper is to introduce the notion of generalized multivalued ψ , ϕ -type contractions and generalized multivalued ψ , ϕ -type Suzuki contractions and establish some new common fixed point theorems for such multivalued mappings in complete metric spaces. Our results are extension and improvement of the Suzuki and Nadler contraction theorems, Jleli and Samet, Piri and Kumam, Mizoguchi and Takahashi, and Liu et al. fixed point theorems. We provide an example for supporting our new results. Moreover, an application of our main result to the existence of solution of system of functional equations is also presented.

Hardy-Rogers Type Mappings for Fuzzy Metric Space

The evolution of fuzzy mathematics commenced with the introduction of the notion of fuzzy set by Zadeh, where the concept of uncertainty has been introduced in the theory of sets in a non probabilistic manner. The several researchers were conducting the generalization of the concept of fuzzy sets. The present research paper focuses on the existence of fixed points in fuzzy metric space. Hardy-Rogers is to establish a fixed point theorem for three maps of a complete metric space. The contractive condition is generalized and the commuting condition of Jungck is replaced by the concept of weakly commuting. The three Hardy-Rogers type mappings are extended in fuzzy metric space and also extend to generalize non-expansive mapping define over a compact fuzzy metric space. The contractive condition is generalization of Hardy-Rogers and the commuting condition of Jungck is replace by the concept of weakly commuting. Our results deals with mappings satisfying a condition weaker than commutativity in complete fuzzy metric space and is the generalization in complete fuzzy metric space of Hardy-Rogers type mappings in complete metric space. We also provide some illustrative example to support our result. We apply also our main results to derive unique and common fixed point for contractive mappings.

Some fixed point theorems for convex contractive mappings in complete metric spaces with applications

AbstractIn this research work, convexity condition is introduced to some classes of contraction mappings such as Chatterjea and Hardy and Rogers contractive mappings. The fixed points of these maps...

A Unique Common Fixed Point for an Infinity of Set-Valued Maps

Abstract The main purpose of this paper is to establish some common fixed point theorems for single and set-valued maps in complete metric spaces, under contractive conditions by using minimal type commutativity and without continuity. These theorems generalize, extend and improve the result due to Elamrani and Mehdaoui ([2]) and others. Also, common fixed point theorems in metric spaces under strict contractive conditions are given.

Fixed Point Theorems for Classes of Nonlinear Mappings of Contractive Type

For certain classes of contractive mappings in complete metric spaces, we establish the existence of a fixed point which attracts all inexact orbits.

F-Convex Contraction via Admissible Mapping and Related Fixed Point Theorems with an Application

In this paper, we introduce F-convex contraction via admissible mapping in the sense of Wardowski [Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 94 (2012), 6 pages] which extends convex contraction mapping of type-2 of Istrǎţescu [Some fixed point theorems for convex contraction mappings and convex non-expansive mappings (I), Libertas Mathematica, 1(1981), 151–163] and establish a fixed point theorem in the setting of metric space. Our result extends and generalizes some other similar results in the literature. As an application of our main result, we establish an existence theorem for the non-linear Fredholm integral equation and give a numerical example to validate the application of our obtained result.

A Related Fixed Point Theorem for F-Contractions on Two Metric Spaces

Recently, Wardowski in [Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012] introduced the concept of $F$-contraction on complete metric space which is a proper generalization of Banach contraction principle. In the present paper, we proved a related fixed point theorem with $F$-contraction mappings on two complete metric spaces.