Generalized Contraction Mappings Research Articles

Further generalized contraction mapping principle and best proximity theorem in metric spaces

AbstractThe aim of this paper is to prove a more generalized contraction mapping principle. By using this more generalized contraction mapping principle, a further generalized best proximity theorem was established. Some concrete results have been derived by using the above two theorems. The results of this paper improve many important results published recently in the literature.

Generalized contraction mapping principle and generalized best proximity point theorems in probabilistic metric spaces

The purpose of this paper is to introduce some basic definitions about fixed point and best proximity point in two classes of probabilistic metric spaces and to prove contraction mapping principle and relevant best proximity point theorems. The first class is the so-called S-probabilistic metric spaces. In S-probabilistic metric spaces, the generalized contraction mapping principle and generalized best proximity point theorems have been proved by authors. These results improve and extend the recent results of Su and Zhang (Fixed Point Theory Appl. 2014:170, 2014). The second class is the so-called Menger probabilistic metric spaces. In Menger probabilistic metric spaces, the contraction mapping principle and relevant best proximity point theorems have been proved by authors. These results also improve and extend the results of many authors. In order to get the results of this paper, some new methods have been used. Meanwhile some error estimate inequalities have been established.

FIXED POINT THEOREM FOR CONTRACTIVE TYPE MAPPING IN DISLOCATED METRIC SPACE

The objective of this paper is to obtain some fixed point theorem for Dislocated metric space for generalized contraction mappings.

Fixed-point Theorems in G-complete Fuzzy Metric Spaces

In the present paper we introduce generalized contraction mapping in fuzzy metric space and some fixed-point theorems for G-complete fuzzy metric space are proved. Our results generalize and extend many known results in metric spaces to a (non-Archimedean) fuzzy metric space in the in the sense of George and Veeramani [George A, Veeramani P, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 1994;64:395-9].

Existence and data dependence of the fixed points of generalized contraction mappings with applications

The aim of this paper is to introduce a new type of generalized multivalued contraction mappings and to present some results regarding fixed points of new class of multivalued contractions. As applications we obtain some basic results in fixed point theory like characterization of metric completeness, data dependence of fixed points and homotopy result. We prove the existence and uniqueness of bounded solution of functional equation arising in dynamic programming. Our results generalize, extend and unify various comparable results in the existing literature.

Some fixed point results for generalized contraction mappings with cyclic ( α , β ) -admissible mapping in multiplicative metric spaces

The purpose of this work is to introduce new types of contraction mappings in the sense of a multiplicative metric space. Fixed point results for these contraction mappings in multiplicative metric spaces are obtained. Our presented results generalize, extend, and improve results on the topic in the literature. Moreover, our results cannot be directly obtained as a consequence from the corresponding results in metric spaces. We also state some illustrative examples to claim that our results properly generalize some results in the literature. We apply our main results for proving a fixed point theorem involving a cyclic mapping.

Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations

The aim of this paper is to present the definition of a generalized altering distance function and to extend the results of Yan et al. (Fixed Point Theory Appl. 2012:152, 2012) and some others, and to prove a new fixed point theorem of generalized contraction mappings in a complete metric space endowed with a partial order by using generalized altering distance functions. The results of this paper can be used to investigate a large class of nonlinear problems. As an application, we discuss the existence of a solution for a periodic boundary value problem.

Ulam-Hyers Stability and Well-Posedness of Fixed Point Problems forα-λ-Contraction Mapping in Metric Spaces

We study Ulam-Hyers stability and the well-posedness of the fixed point problem for new type of generalized contraction mapping, so calledα-λ-contraction mapping. The results in this paper generalize and unify several results in the literature such as the Banach contraction principle.

Generalized Ulam-Hyers Stability, Well-Posedness, and Limit Shadowing of Fixed Point Problems forα-β-Contraction Mapping in Metric Spaces

We study the generalized Ulam-Hyers stability, the well-posedness, and the limit shadowing of the fixed point problem for new type of generalized contraction mapping, the so-called α-β-contraction mapping. Our results in this paper are generalized and unify several results in the literature as the result of Geraghty (1973) and the Banach contraction principle.

Viscosity approximation methods based on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces

In this paper, we introduce viscosity approximation methods based on generalized contraction mappings for finding a set of common fixed points of a countable family of strict pseudo-contraction mappings, the common element of the set solutions of a general system of variational inequalities with Lipschitzian and relaxed cocoercive mappings and the set of solutions of a generalized mixed equilibrium problem. Furthermore, strong convergence theorems of the purposed iterative process are established in the framework of Banach spaces. The results presented in this paper improve and extend many recent important results.

Fixed point theorems in generalized contraction mappings in M–fuzzy metric spaces

In this paper, we introduce generalized contraction mapping in � - fuzzy metric space. Some fixed point theorems for fuzzy metric space in the sense of George and Veeramani [2]. Mathematics Subject Classification: 47H10, 54A40.

Periodic points of T-Ciric generalized contraction mappings in ordered metric spaces

The aim of this paper is to study the existence of fixed points of T -Ciric type mappings in the setup of partially ordered spaces. Moreover, the T -generalized weakly contractive mappings are introduced and also sufficient conditions for obtaining fixed points of such mappings in ordered spaces are studied. As an application of our results, periodic points of T -Banach operators are obtained. Finally, we provide some examples to illustrate the results presented herein.

Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg-de Vries-Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg-de Vries-Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall-Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.

The Equivalence between the Mann and Ishikawa Iterations for Generalized Contraction Mappings in a Cone

In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping in cone subset of a real Banach space is discussed.

Study of the classical solution to the one-dimensional mixed problem for a class of semilinear long-wave equations

Many problems in mathematical physics are reduced to one- or multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear Sobolev-type equations. In this work, local and global classical solvability is studied for the one-dimensional mixed problem with homogeneous Riquier-type boundary conditions for a class of semilinear long-wave equations $$ U_{tt} \left( {t,x} \right) - U_{xx} \left( {t,x} \right) - \alpha U_{ttxx} \left( {t,x} \right) = F\left( {t,x,U\left( {t,x} \right),U_x \left( {t,x} \right),U_{xx} \left( {t,x} \right),U_t \left( {t,x} \right),U_{tx} \left( {t,x} \right),U_{txx} \left( {t,x} \right)} \right) $$ , where α > 0 is a fixed number, 0 ≤ t ≤ T, 0 ≤ x ≤ π, 0 < T < +∞, F is a given function, and U(t, x) is the sought function. A uniqueness theorem for the mixed problem is proved using the Gronwall-Bellman inequality. A local existence result is proved by applying the generalized contraction mapping principle combined with the Schauder fixed point theorem. The method of a priori estimates is used to prove the global existence of a classical solution to the mixed problem.

Viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping

The purpose of this paper is to introduce and construct the implicit and explicit viscosity iterative processes by a generalized contraction mapping f and a nonexpansive semigroup { T ( t ) : t ⩾ 0 } , and to prove that under suitable conditions these iterative processes converge strongly to a unique common fixed point of { T ( t ) : t ⩾ 0 } in reflexive Banach spaces which admits a weakly sequentially continuous duality mapping.

Viscosity approximations by generalized contractions for resolvents of accretive operators in Banach spaces

In this paper, we prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method with a generalized contraction mapping. The proximal point algorithm in a Banach space is also considered. The results extend some very recent theorems of W. Takahashi.

Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings

Let X be a reflexive and smooth real Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation scheme x n + 1 = λ n + 1 f ( x n ) + ( 1 − λ n + 1 ) T n + 1 x n (where f is a generalized contraction mapping) for infinitely many nonexpansive self-mappings T 1 , T 2 , T 3 , … in X . We establish a strong convergence result which generalizes some results in the literature.

Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations

Using the idea of Atanassov, we define the notion of intuitionistic Menger spaces as a netural generalizations of Menger spaces due to Menger. We also obtain a new generalized contraction mapping and utilize this contraction mapping to prove the existence theorems of solutions to differential equations in intuitionistic Menger spaces.

Fixed point theorems for set-valued [formula omitted]-generalized contractions on gauge spaces

In this paper, we present a common fixed point theorem for a family of set-valued Φ -generalized contraction mappings on gauge spaces, which generalizes the main result from Espínola and Kirk [R. Espínola, W.A. Kirk, Set-valued contractions and fixed points, Nonlinear Anal. 54 (2003) 485–494].