# contraction mappings

## Top PDF contraction mappings: ### Fixed point theorems of contraction mappings on zero at infinity varieties

In this paper, we have tried to show that the set of all zero at infinity varieties is a complete metric space and some results and examples about fixed points of contraction mappings on[r] ### Common fixed point results for generalized contraction mappings in b - metric space

Fixed point theory is rapidly moving into the mainstream of Mathematics mainly because of its applications in diverse fields which include numerical methods like Newton-Raphson method, establishing Picard’s existence theorem, existence of solution of integral equations and a system of linear equations. In 1922, S. Banach , The first important and significant result was proved a fixed point theorem for contraction mappings in complete metric space and also called it Banach fixed point theorem / Banach contraction principle which is considered as the mile stone in fixed point theory. This theorem states that, A mapping : → where( , ) is a metric space, is said to be a contraction if there exists ∈ [0,1) such that ### Fixed point of multivalued integral type of contraction mappings

Fixed point theory in the framework of metric spaces is one of the most powerful and useful tools in nonlinear functional analysis. The intrinsic subject of this theory is con- cerned with the conditions for the existence, uniqueness and exact methods of evaluation of ﬁxed point of a mapping. The application of ﬁxed point theorems is remarkable in a wide scale of mathematical, engineering, economic, physical, computer science and other ﬁelds of science. The Banach contraction principle [] is a simplest and limelight result in this direction. In many papers, following the Banach contraction principle, the existence of weaker contractive conditions combined with stronger additional assumptions on the mapping or on the space is investigated. Moreover, since all these results are based on an iteration process, they can be implemented in almost all branches of quantitative sciences. Nadler [] initiated the study of ﬁxed point for multivalued contraction mappings. On the other hand, Branciari [] generalized the Banach contraction principle for a single- valued mapping by using an integral type of contraction. Both of these results were ex- tended and applied by many authors, and we quote some of them [–]. Also, we refer to the paper of Khan et al. [] which improved the metric ﬁxed point theory by introducing a control function called an altering distance function. ### On monotone contraction mappings in modular function spaces

The purpose of this paper is to give an outline of a ﬁxed point theory for monotone- contraction mappings deﬁned on some subsets of modular function spaces which are nat- ural generalizations of both function and sequence variants of many important spaces like the Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and many others []. Recently, the authors in [] presented a series of ﬁxed point results for pointwise contractions and asymptotic pointwise contractions acting in mod- ular functions spaces. The current paper operates within the same framework. ### An optimization problem involving proximal quasi-contraction mappings

Consider a non-self-mapping T : A → B, where (A,B) is a pair of nonempty subsets of a metric space (X, d). In this paper, we study the existence and uniqueness of solutions to the global optimization problem min x∈A d(x,Tx), where T belongs to the class of proximal quasi-contraction mappings. ### On monotone Ćirić quasi-contraction mappings with a graph

As a generalization to the Banach contraction principle, Ćirić [] introduced the concept of quasi-contraction mappings. In this section, we investigate monotone mappings which are quasi-contraction mappings. Throughout this section we assume that (X, d) is a metric space and G is a reﬂexive transitive digraph deﬁned on X. Moreover, we assume that E(G) has property (∗) and G-intervals are closed. Recall that a G-interval is any of the subsets [a, →) = {x ∈ C; (a, x) ∈ E(G)} and (←, b] = {x ∈ C; (x, b) ∈ E(G)} for any a, b ∈ C. ### Best proximity points for Geraghty’s proximal contraction mappings

In this section, we introduce a new class of proximal contractions, the so-called Geraghty’s proximal contraction mappings, and prove best proximity theorems for this class. Deﬁnition . A mapping T : A → B is called Geraghty’s proximal contraction of the ﬁrst kind if, there exists β ∈ S such that ### Fixed points of multivalued contraction mappings in modular metric spaces

As is well known, a ﬁxed point theorem for multivalued contraction mappings was estab- lished by Nadler []. In  Edelstein [] has generalized the Banach contraction principle to mappings satisfying a less restrictive Lipschitz inequality such as local contraction. This result has been generalized to a multivalued version by Nadler []. On the other hand Mi- zoguchi and Takahashi [] have improved Reich’s result [] and proved the existence of ﬁxed points for multivalued maps in the case when values of mappings are closed bounded instead of compact. ### Periodic points for the weak contraction mappings in complete generalized metric spaces

In conclusion, by using the new concepts of (j-φ)-weak contraction mappings and (ψ - φ)-weak contraction mappings, we obtain two theorems (Theorems 1 and 2) which assure the existence of a periodic point for these two types of weak contraction in complete generalized metric spaces. Our results generalize or improve many recent fixed point theorems in the literature. ### Quadruple Fixed Point of Multivalued Nonlinear Contraction Mappings

Abstract. The notion of Quadruple fixed point is introduced by Karapinar E. . Samet and Vetro  established some coupled fixed point theorems for multivalued non linear contraction mapping in partially ordered metric spaces. In this paper, we obtain existence of quadrupled fixed point of multivalued non linear contraction mappings in framework work of partially ordered metric spaces. Also, we give an example. ### Fixed point theorems for cyclic contraction mappings in fuzzy metric spaces

The contraction type mappings in fuzzy metric spaces play a crucial role in ﬁxed point theory. In , Grabiec [] ﬁrst deﬁned the Banach contraction in a fuzzy metric space and extended ﬁxed point theorems of Banach and Edelstein to fuzzy metric spaces. Fol- lowing Grabiec’s approach, Mishra et al. [] obtained some common ﬁxed point theorems for asymptotically commuting mappings in fuzzy metric spaces. In the sequel, Vasuki [] oﬀered a generalization of Grabiec’s fuzzy Banach contraction theorem and proved a com- mon ﬁxed point theorem for a sequence of mappings in a fuzzy metric space. Afterwards, Cho [] presented the concept of compatible mappings of type (α) in fuzzy metric spaces and then studied the ﬁxed point theory. Several years later, Singh and Chauhan [] intro- duced the concept of compatible mapping and proved two common ﬁxed point theorems in the fuzzy metric space with the strongest triangular norm. In , Sharma [] fur- ther extended some known results of ﬁxed point theory for compatible mappings in fuzzy metric spaces. In the same year, Gregori and Sapena [] introduced the notion of fuzzy contractive mapping and presented some ﬁxed point theorems for complete fuzzy metric spaces in the sense of George and Veeramani, and also for Kramosil and Michalek’s fuzzy metric spaces which are complete in Grabiec’s sense. Soon after, Mihet [] proposed a fuzzy Banach theorem for (weak) B-contraction in M-complete fuzzy metric spaces. Later, Mihet [, ] further studied the ﬁxed point theory for the diﬀerent contraction mappings in fuzzy metric spaces, and introduced some new contraction mappings, such as Edelstein fuzzy contractive mappings, fuzzy ψ-contraction of (ε, λ) type, etc. Based on the deﬁni- tion of fuzzy contractive mapping introduced by Mihet [], Abbas et al. [] proposed the notion of ϕ-weak contraction and obtained several results of ﬁxed point in a G-complete metric space. Recently, Shen et al. [] constructed a novel class of ϕ-contractions and proved a ﬁxed point theorem for this kind of mappings in an M-complete fuzzy metric space. ### On best proximity points for multivalued cyclic $F$-contraction mappings

for all x, y ∈ X, then T has at least one fixed point, that is, there exists z ∈ X such that z ∈ T z. In 2003, Kirk, Srinavasan and Veeramani  introduced a concept of cyclic contraction which generalized Banach’s contraction. They also proved fixed point theorems in complete metric spaces, as follows: ### Fixed point theorems for contraction mappings in modular metric spaces

for all x, y Î X, where 0 ≤ k <1. The Banach Contraction Mapping Principle appeared in explicit form in Banach ’ s thesis in 1922 . Since its simplicity and useful- ness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions, see [2-10]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano  and was intensively developed by Koshi, Shimogaki, Yamamuro [11-13] and others. Further and the most complete develop- ment of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [14-18] and their collaborators. A lot of mathematicians are interested fixed points of Modular spaces, for example [4,19-26]. ### Endpoints of multi-valued cyclic contraction mappings

A point x is called a fixed point of T if x ∈ T x. Denote F ix(T ) = {x ∈ X : x ∈ T x}. An element x ∈ X is said to be an endpoint of multi–valued mapping T , if T x = {x}. The set of all endpoints of T denotes by End(T ). Obviously, End(T ) ⊆ F ix(T ). In recent years many authors studied the existence and uniqueness of endpoints for a multi–valued mappings in metric spaces, see for example [1, 5, 6, 11, 12, 14, 15, 19] and references therein. ### Suzuki-type fixed point theorem for fuzzy mappings in ordered metric spaces

Zadeh [] introduced the concept of a fuzzy set. Heilpern [] introduced the concept of fuzzy mappings in a metric space and proved a ﬁxed point theorem for fuzzy contraction mappings as a generalization of the ﬁxed point theorem for multivalued mappings given by Nadler []. Estruch and Vidal [] proved a ﬁxed point theorem for fuzzy contraction mappings in complete metric spaces which in turn generalizes the Heilpern ﬁxed point theorem. Further generalizations of the result given in [] were proved in [, ]. Recently, Suzuki [] generalized the Banach contraction principle and characterized the metric completeness property of an underlying space. Among many generalizations (see [–]) of the results given in [], Dorić and Lazović [] obtained Suzuki-type ﬁxed point results for a generalized multivalued contraction in complete metric spaces. ### The $$(\alpha, \beta)$$-generalized convex contractive condition with approximate fixed point results and some consequence

tence results of approximate ﬁxed points for these mappings on a complete metric space by using the idea of cyclic (α, β)-admissible mappings due to Alizadeh et al. []. We fur- nish an illustrative example to demonstrate the validity of the hypotheses and the degree of utility of our results. Our result extends, uniﬁes, and generalizes various well-known ﬁxed point and approximate ﬁxed point results such as the Banach contraction mapping prin- ciple [], Kannan’s ﬁxed point results [], ﬁxed point and approximate ﬁxed point results for convex contraction mappings due to Istratescu [], and many results in the literature. As a consequence of the presented results, the approximate ﬁxed point results for cyclic mappings are also given in order to illustrate the eﬀectiveness of the obtained results. ### Some Results on Fuzzy Mappings for Rational Expressions

Bose and Sahini extends Heilpern , s result for a pair of generalized fuzzy contraction mappings .Lee and Cho described a fixed point theorem for contractive type fuzzy mappings which is generalization of Heilpern , s  result. Lee, Cho, Lee and Kim  obtained a common fixed point theorem for a sequence of fuzzy mappings satisfying certain conditions, which is generalization of the second theorem of Bose and Sahini . ### Best proximity points and extension of Mizoguchi-Takahashi’s fixed point theorems

extending the Mizoguchi and Takahashi’s contraction for non-self mappings. We also establish a best proximity point for such type contraction mappings in the context of metric spaces. Later, we characterize this result to investigate the existence of best proximity point theorems in uniformly convex Banach spaces. We state some illustrative examples to support our main theorems. Our results extend, improve and enrich some celebrated results in the literature, such as Nadler’s ﬁxed point theorem, Mizoguchi and Takahashi’s ﬁxed point theorem. ### Fixed Points of Different Contractive Type Mappings on Tensor Product Spaces

We have discussed different fixed point theorems with different contractive type mappings on tensor product spaces. Moreover, using a given contraction mapping (with fixed point) on the tensor product space , we have constructed some contraction mappings with fixed points for the individual spaces and . However, many other open problems can be raised regarding different types of contractive mappings on tensor product spaces. In , Alber and Guerre-Delabriere defined weakly contractive maps. In , Rhoades extended some results on weakly contractive maps to arbitrary Banach spaces.For a Banachspace , and a closed convex subset of , a self-map T of is called weakly contractive if for each , 