Zoran Kadelburg

Generalized weak contractions in partially ordered metric spaces

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.

Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces

In the first part of this paper we generalize results on common fixed points in ordered cone metric spaces obtained by I. Altun and G. Durmaz [I. Altun, G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo, 58 (2009) 319-325] and I. Altun, B. Damnjanovic and D. Djoric [I. Altun, B. Damnjanovic, D. Djoric, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett. (2009) doi:10.1016/j.aml.2009.09.016] by weakening the respective contractive condition. Then, the notions of quasicontraction and g-quasicontraction are introduced in the setting of ordered cone metric spaces and respective (common) fixed point theorems are proved. In such a way, known results on quasicontractions and g-quasicontractions in metric spaces and cone metric spaces are extended to the setting of ordered cone metric spaces. Examples show that there are cases when new results can be applied, while old ones cannot.

Suzuki-type 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.

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

Abstract In the present work, using Minkowski functionals in topological vector spaces, we establish the equivalence between some fixed point results in metric and in (topological vector space) cone metric spaces. Thus, a lot of results in the cone metric setting can be directly obtained from their metric counterparts. In particular, a common fixed point theorem for f -quasicontractions is obtained. Our approach is even easier than that of Du [Wei-Shih Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010) 2259–2261] where similar conclusions were obtained using scalarization functions.

Remarks on “Quasi-contraction on a cone metric space”

Abstract Recently, D. Ilic and V. Rakocevic [D. Ilic, V. Rakocevic, Quasi-contraction on a cone metric space, Appl. Math. Lett. (2008) doi:10.1016/j.aml.2008.08.011 ] proved a fixed point theorem for quasi-contractive mappings in cone metric spaces when the underlying cone is normal. The aim of this paper is to prove this and some related results without using the normality condition.

Common fixed point results in metric-type spaces.

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

Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces

Abstract Common fixed point theorems for T -weakly isotone increasing mappings satisfying a generalized contractive type condition under a continuous function φ : [ 0 , + ∞ ) → [ 0 , + ∞ ) with φ ( t ) t for each t > 0 and φ ( 0 ) = 0 in complete ordered partial metric spaces are proved. To illustrate our results and to distinguish them from the existing ones, we equip the paper with examples.

Assad-Kirk-Type Fixed Point Theorems for a Pair of Nonself Mappings on Cone Metric Spaces

New fixed point results for a pair of non-self mappings defined on a closed subset of a metrically convex cone metric space (which is not necessarily normal) are obtained. By adapting Assad-Kirks method the existence of a unique common fixed point for a pair of non-self mappings is proved, using only the assumption that the cone interior is nonempty. Examples show that the obtained results are proper extensions of the existing ones.

Fixed Points of Geraghty-Type Mappings in Various Generalized Metric Spaces

Fixed point theorems for mappings satisfying Geraghty-type contractive conditions are proved in the frame of partial metric spaces, ordered partial metric spaces, and metric-type spaces. Examples are given showing that these results are proper extensions of the existing ones.

Coupled Coincidence Points of Mappings in Ordered Partial Metric Spaces

New coupled coincidence point and coupled fixed point results in ordered partial metric spaces under the contractive conditions of Geraghty, Rakotch, and Branciari types are obtained. Examples show that these results are distinct from the known ones.