Setting Of Cone Metric Spaces Research Articles

Some multi-valued contraction theorems on \(\mathcal{H}\)-cone metric

In this paper we define some new type of multi-valued contraction maps on \(\mathcal{H}\)-cone metric and proved some fixed point and common fixed point theorems in the setting of cone metric spaces.

Sequences of contractions on cone metric spaces over Banach algebras and applications to nonlinear systems of equations and systems of differential equations

It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems of differential equations and coupled functional equations, respectively, that are linked to the concept of sequences of contractions.

Common fixed point theorems in cone metric spaces under general contractive conditions

In this article, some common fixed point theorems for two pairs of compatible mappings together with sub-sequential continuity (alternately sub-compatible mappings together with reciprocal continuity) are proved in the setting of cone metric spaces. Our results are new in this setting especially in view of the note given in Imdad et al. [15]. On the other hand some fixed point results for faintly compatible mappings are also established. Illustrative examples along with their pictorial representation are furnished to highlight the validity of the hypothesis of our results.

Relation-theoretic contraction principle in cone metric spaces with Banach algebra

In this work, we introduce the notion of relation-theoretic contractions in cone metric spaces with Banach algebra and prove some fixed point results for such contractions. Our results generalize and unify several known results in the setting of cone metric spaces with Banach algebra. An example is provided which illustrate the results proved herein and shows that how the new results are different from existing ones.

Fixed point and common fixed point results in cone metric space and application to invariant approximation

In this work, the concept of almost contraction for multi-valued mappings in the setting of cone metric spaces is defined and then we establish some fixed point and common fixed point results in the set-up of cone metric spaces. As an application, some invariant approximation results are obtained. The results of this paper extend and improve the corresponding results of multi-valued mapping from metric space theory to cone metric spaces. Further our results improve the recent result of Arshad and Ahmad (Sci. World J. 2013:481601, 2013).

Common fixed point theorems for generalized multivalued contractions on cone metric spaces over a non-normal solid cone

Abstract In this paper we define a new class of multivalued generalized contractions on cone metric spaces. Then, by using a necessary new technique, we prove two common fixed point theorems for a pair of those mappings on complete cone metric spaces over solid, not necessarily normal cone. Our main theorems are generalizations of the theorem of Wardowski (Appl. Math. Lett. 24:275-278, 2011) and many existing theorems in the literature. By using our main theorems, we can obtained some important corollaries which are generalizations of the well-known metric fixed point theorems to setting of cone metric spaces over a solid non-normal cone. MSC:47H10, 54H25.

Common fixed point theorem for two pairs of non-self-mappings satisfying generalized Ćirić type contraction condition in cone metric spaces

AbstractIn this paper, we prove a common fixed point theorem for two pairs of non-self-mappings satisfying the generalized contraction condition of Ćirić type in cone metric spaces. Our result generalizes and extends some recent results related to non-self-mappings in the setting of cone metric spaces.MSC:47H10, 54H25.

Iterative approximation of fixed points of quasi-contraction mappings in cone metric spaces

In this paper, we establish a full statement of Ćirić’s fixed point theorem in the setting of cone metric spaces. More exactly, we obtain a priori and a posteriori error estimates for approximating fixed points of quasi-contractions in a cone metric space. Our result complements recent results of Zhang (Comput. Math. Appl. 62:1627-1633, 2011), Ding et al. (J. Comput. Anal. Appl. 15:463-470, 2013) and others.MSC: Primary 54H25; secondary 47H10, 46A19.

Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings

In this paper we introduce the concept of cone metric spaces with Banach algebras, replacing Banach spaces by Banach algebras as the underlying spaces of cone metric spaces. With this modification, we shall prove some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants. An example shows that our main results concerning the fixed point theorems in the setting of cone metric spaces with Banach algebras are more useful than the standard results in cone metric spaces presented in the literature.

Common fixed point theorems for generalized contractive mappings on cone metric spaces

In this paper, we present the generalization of contractive type mappings in the setting of cone metric spaces.

On Multivalued Contractions in Cone Metric Spaces without Normality

Wardowski (2011) in this paper for a normal cone metric space (X, d) and for the family 𝒜 of subsets of X established a new cone metric H : 𝒜 × 𝒜 → E and obtained fixed point of set-valued contraction of Nadler type. Further, it is noticed in the work of Janković et al., 2011 that the fixed-point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is nonnormal. In the present paper we improve Wardowski's result by proving the same without the assumption of normality on cones.

Some fixed and coincidence point theorems for expansive maps in cone metric spaces

In this article, we establish some common fixed and common coincidence point theorems for expansive type mappings in the setting of cone metric spaces. Our results extend some known results in metric spaces to cone metric spaces. Also, we introduce some examples the support the validity of our results.Mathematics Subject Classification: 54H25; 47H10; 54E50.

On Approximate Coincidence Point Properties and Their Applications to Fixed Point Theory

We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde‐Berinde′s fixed point theorem, Mizoguchi‐Takahashi′s fixed point theorem, and some well‐known results in the literature.

The Existence of Cone Critical Point and Common Fixed Point with Applications

We first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš‐Hegedüs‐Medvegyev′s principle and the existence theorem related with Ekeland′s variational principle, Caristi′s common fixed point theorem for multivalued maps, Takahashi′s nonconvex minimization theorem, and common fuzzy fixed point theorem. We also obtain some fixed point theorems for weakly contractive maps in the setting of cone metric spaces and focus our research on the equivalence between scalar versions and vectorial versions of some results of fixed point and others.

Critical Point Theorems and Ekeland Type Variational Principle with Applications

AbstractWe introduce the notion of "Equation missing"-spaces which is much weaker than cone metric spaces defined by Huang and X. Zhang (2007). We establish some critical point theorems in the setting of "Equation missing"-spaces and, in particular, in the setting of complete cone metric spaces. Our results generalize the critical point theorem proposed by Dancs et al. (1983) and the results given by Khanh and Quy (2010) to "Equation missing"-spaces and cone metric spaces. As applications of our results, we characterize the completeness of "Equation missing"-space (cone metric spaces and quasimetric spaces are special cases of "Equation missing"-space) and studying the Ekeland type variational principle for single variable vector-valued functions as well as for multivalued bifunctions in the setting of cone metric spaces.

Compatible and weakly compatible mappings in cone metric spaces

In this paper we extend and generalize common fixed point theorems for six self-maps of Singh and Jain [B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439–448] from Menger and metric spaces to cone metric spaces. We also extend the notions of compatible and weakly compatible mappings from the setting of Menger and metric spaces to the setting of cone metric spaces. We do not impose the normality property on the cone, but suppose only that the cone P , in the ordered Banach space E , has a nonempty interior. Examples are given to illustrate the results.

Common fixed points in cone metric spaces

In this paper we consider a notion of g-weak contractive mappings in the setting of cone metric spaces and we give results of common fixed points. This results generalize some common fixed points results in metric spaces and some of the results of Huang and Zhang in cone metric spaces.