In the article, we introduce a new concept of contraction and prove a fixed point theorem which generalizes Banach contraction principle in a different way than in the known results from the literature. The article includes an example which shows the validity of our results, additionally there is delivered numerical data which illustrates the provided example. MSC: 47H10; 54E50

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Fixed points of a new type of contractive mappings in complete metric spaces

We present a new generalization of the Banach contraction principle in the setting of Branciari metric spaces.

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A new generalization of the Banach contraction principle

We prove some common fixed point results for four mappings satisfying generalized weak contractive condition in partially ordered complete b-metric spaces. Our results extend and improve several comparable results in the existing literature

https://www.degruyter.com/downloadpdf/j/ms.2014.64.issue-4/s12175-014-0250-6/s12175-014-0250-6.xml

Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces

Using an old M. Krein’s result and a result concerning symmetric spaces from [S. Radenovic, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J. Math. Anal. 5 (1) (2011), 38–50], we show in a very short way that all fixed point results in cone metric spaces obtained recently, in which the assumption that the underlying cone is normal and solid is present, can be reduced to the corresponding results in metric spaces. On the other hand, when we deal with non-normal solid cones, this is not possible. In the recent paper [M.A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl. 2010, 7 pages, Article ID 315398, doi:10.1115/2010/315398 ] the author claims that most of the cone fixed point results are merely copies of the classical ones and that any extension of known fixed point results to cone metric spaces is redundant; also that underlying Banach space and the associated cone subset are not necessary. In fact, Khamsi’s approach includes a small class of results and is very limited since it requires only normal cones, so that all results with non-normal cones (which are proper extensions of the corresponding results for metric spaces) cannot be dealt with by his approach.

On cone metric spaces: A survey

Common coupled fixed point theorems in cone metric spaces for w-compatible mappings

Several fixed point and common fixed point theorems are obtained in the setting of metric-type spaces introduced by M. A. Khamsi in 2010.

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Common Fixed Point Results in Metric-Type Spaces

Suzuki’s fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) and (Suzuki, Nonlinear Anal. 71:5313-5317, 2009) are extended to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones. MSC:47H10, 54H25.

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Suzuki-type fixed point results in metric type spaces

A note on the equivalence of some metric and cone metric fixed point results

Recent results of Doric [D. Doric, Common fixed point for generalized (@j,@f)-weak contractions, Appl. Math. Lett. 22 (2009) 1896-1900] on generalized weakly contractive mappings are extended to the setting of partially ordered metric spaces. Thus, generalization of fixed point results of Harjani and Sadarangani [J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (2009) 3403-3410] is obtained. Some applications are presented. Examples are given to show that our results are proper generalizations of the existing ones.

Zoran Kadelburg

Papers

On cone metric spaces: A survey

Common Fixed Point Results in Metric-Type Spaces

Suzuki-type fixed point results in metric type spaces

A note on the equivalence of some metric and cone metric fixed point results

Generalized weak contractions in partially ordered metric spaces