Non-normal Cone Research Articles

Ciric type fixed point theorems under c-distance on non-normal cone metric spaces over Banach algebras

In this paper, we obtain Ciric type fixed point theorems for continuous or non-continuous mappings under c-distance on mapping-orbitally complete cone metric spaces over Banach algebras without normalities.

Topological Properties of E-Metric Spaces with Applications to Fixed Point Theory

The purpose of this paper is to present some topological properties in E-metric spaces such as the properties of e-sequences, the decision conditions of e-Cauchy sequences, the characteristics of non-normal cones, and so on. Moreover, the theorem of nested closed-balls in such spaces is displayed. In addition, some principal applications to fixed point theory are also given.

Best Proximity Point Results for Quasi Contractions of Perov Type in Non-Normal Cone Metric Space

In this paper, we study the notion of Ciric-Perov quasi contraction and Fisher-Perov quasi contraction and prove some best proximity point theorems for such contractions in the frame work of non-normal regular cone metric spaces. We give an example to support our result. Our results extend and generalized many existing results in literature.

Unique common fixed points for two generalized expansive mappings on non-normal cone metric spaces with Banach algebras

By using some elementary results concerning cone metric spaces over Banach algebras and the related ones about c-sequence on cone metric spaces, some new coincidence point and common fixed point theorems for two generalized expansive mappings were discussed and obtained on cone metric spaces over Banach algebras without the assumption of normality and some unique fixed point theorems were given. Also, One of the main results is supported with a relevant example.

Generalizations of Banach-Kannan-Chatterjea type fixed point theorems on non-normal cone metric spaces with Banach algebras

In this paper, we obtain generalizations of Banach contraction principle, Kannan fixed point theorem and Chatterjea fixed point theorem for mappings satisfying Suzuki type contractive conditions on cone metric spaces over Banach algebras without the assumption of normality. The obtained results generalize and improve the corresponding conclusions in the literature.

Unique common fixed points for mixed contractive mappings on non-normal cone metric spaces over Banach algebras

In this paper, we discuss and obtain some new unique common fixed point theorems for two mappings satisfying Kannan type mixed contractive conditions and Chatterjea type mixed contractive conditions respectively on cone metric spaces over Banach algebras without the assumption of normality and give some generalizations of Kannan type and Chatterjea type fixed point theorems.

Some critical remarks on the paper “A note on the metrizability of tvs-cone metric spaces”

This short and concise note provides a detailed exposition of the approach and results established by (Lin et al, 2015, pp.271-279). We show that the obtained results are not particularly surprising and new. Namely, using an old result due to K. Deimling it is indicated that tvs-cone metric spaces over solid cones are actually cone metric spaces over normal solid cones. Hence, there are only cone metric spaces over normal solid cones or over normal non-solid cones. One question still unanswered is whether an ordered topological vector space with a non-normal non-solid cone exists.

Some Fixed Point Theorems for Expansive Mappings in Cone Pentagonal Metric Spaces

In this paper, we prove some xed point theorems for mappings satisfying expansive conditions in non-normal cone pentagonal metric spaces. Our results extend and improve the recent results announced by Patil and Salunke [Fixed point theorems for Expansion mappings in Cone rectangular metric spaces, Gen. Math. Notes, 29(1), (2015), 30-39], Shatanawi and Awawdeh, [Some xed and coincidence point theorems for Expansive maps in Cone metric spaces, Fixed Point Theory and Applications, 1(2012), 1-10], Huang, Zhu and Wen, [Fixed point theorems for Expanding mappings in Cone metric spaces, Math. Reports 14(64), 2(2012), 141-148], Kadelburg, Murthy and Radenovic, [Common xed points for Expansive mappings in Cone metric spaces, Int. J. Math. Anal, 5(27), (2011), 1309-1319], Aage and Salunke, [Some xed point theorems for Expansion onto mappings on Cone metric spaces, Acta Mathematica Sinica, 27(6), (2011), 1101-1106], Kumar and Garg, [Common xed points for Expansion mappings Theorems in metric spaces, Int. J. Contemp. Math. Sciences, 4(36), (2009), 1749-1758], and many others in the literature.

KANNAN--TYPE FIXED POINT THEOREM IN CONE PENTAGONAL METRIC SPACES

In this paper, we prove Kannan - type fixed point theorem for two self mappings in non-normal cone pentagonal metric spaces. Our results extend and improve the recent results announced by many authors.

A Note on Banach Contraction Mapping principle in Cone Hexagonal Metric Space

In this paper, we prove fixed point theorem of a self mapping in non-normal cone hexagonal metric spaces. Our result extend and improve some recent results of Azam et al., [Banach contraction principle on cone rectangular metric spaces, Applicable Analysis and Discrete Mathematics, 3 (2), 236 241, 2009], Rashwan and Saleh [Some Fixed Point Theorems in Cone Rectangular Metric Spaces, Mathematica Aeterna, 2 (6): 573 587, 2012], Garg and Agarwal, [Banach Contraction Principle on Cone Pentagonal Metric Space, J. Adv. Studies Topol., 3 (1), 12 18, 2012], Garg, [Banach Contraction Principle on Cone Hexagonal Metric Space, Ultra Scientist, 26 (1), 97 103, 2014], and others. *Corresponding author: E-mail: abba.auwalu@neu.edu.tr, abbaauwalu@yahoo.com; Auwalu and Hincal; BJMCS, 16(1), 1-12, 2016; Article no.BJMCS.25172

Non-normal cone metric and cone b-metric spaces and fixed point results

We show that most fixed point results obtained so far in cone metric spaces over solid non-normal cones can be easily reduced to the case of solid normal cones and, hence, their proofs can be made much simpler. Also, cone tvs-valued spaces over solid cones are not an essential generalization of cone metric spaces. These results are consequences of the simple fact that each solid cone in a topological vector space is in fact normal under a suitably defined norm. The proof follows by using the technique of Minkowski functional. As an application of these results, we prove an extension of the classical Nemytzki-Edelstein fixed point result to (tvs)-(b)-cone metric spaces over solid cones.

Sub-compatible maps, weakly commuting maps and common fixed points in cone metric spaces

The purpose of this paper is to obtain some common fixed point theorems under weaker conditions such as sub compatible mappings and weakly commuting with respect g in the setting of non-normal cone metric space.

Common fixed point results for mappings of Perov type

Fixed point results for mappings f and g are obtained where f and g are related in the sense of Das and Naik, Ciric, Hardy and Rogers, and all are of Perov type. We consider both normal and non-normal cones. The theory is illustrated with some examples.

Extensions of Perov theorem

[Perov, A. I., On Cauchy problem for a system of ordinary diferential equations, (in Russian), Priblizhen. Metody Reshen. Difer. Uravn., 2 (1964), 115-134] used the concept of vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article we study fixed point results for the new extensions of Banach’s contraction principle to cone metric space, and we give some generalized versions of the fixed point theorem of Perov. As corollaries some results of [Zima, M., A certain fixed point theorem and its applications to integral-functional equations, Bull. Austral. Math. Soc., 46 (1992), 179–186] and [Borkowski, M., Bugajewski, D. and Zima, M., On some fixed-point theorems for generalized contractions and their perturbations, J. Math. Anal. Appl., 367 (2010), 464–475] are generalized for a Banach cone space with a non-normal cone. The theory is illustrated with some examples.

Erratum to: Nonlinear quasi-contractions in non-normal cone metric spaces

In the note we correct some errors that appeared in the article (Jiang and Li in Fixed Point Theory Appl. 2014:165, 2014).

Nonlinear quasi-contractions in non-normal cone metric spaces

In the paper, we prove a new fixed point theorem of nonlinear quasi-contractions in non-normal cone metric spaces, which partially improve the recent results of Arandelović and Kečkić’s and of Li and Jiang since some of the essential conditions therein are removed. A suitable example is presented to show the usability of our theorem. It is worth mentioning that the results in this paper could not be derived from the corresponding results in the setting of metric spaces by using a scalarization function or a Minkowski functional. MSC:06A07, 47H10.

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 Theorems of Contractions in Partial Cone Metric Spaces over Nonnormal Cones

We prove some common fixed point theorems of contractions restricted with variable positive linear bounded mappings inθ-complete partial cone metric spaces over nonnormal cones and present some examples to support the usability of our results.

Common Fixed Point Theorems of Expanding Mappings in Cone Metric Spaces

In this paper,the existence and uniqueness of common fixed points for expanding mappings satisfying certain expansive conditions,without appealing to continuity,in a nonnormal cone metric space are obtained.The results generalize and improve some well-known comparable results.

An extension of the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Huang and Zhang [1] established the results of xed point theorems using contractive method in cone metric space. Later on, Rezapour and Hamlbarani[2] generalize the results [1] with non-normal cones and omitted the assumptionof normality. In the present paper we improve these results.