Vladimir Rakočević

QUASI-CONTRACTION ON A CONE METRIC SPACE

In this work we define and study quasi-contraction on a cone metric space. For such a mapping we prove a fixed point theorem. Among other things, we generalize a recent result of H. L. Guang and Z. Xian, and the main result of Ciric is also recovered.

Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces

We prove several fixed point theorems on cone metric spaces in which the cone does not need to be normal. These theorems generalize the recent results of Huang and Zhang (2007), Abbas and Jungck (2008), and Vetro (2007). Furthermore as corollaries, we obtain recent results of Rezapour and Hamlborani (2008).

Some new extensions of Banach’s contraction principle to partial metric space☆

Abstract In S.G. Matthews [S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, in: Ann. New York Acad. Sci., vol. 728, 1994, pp. 183–197], the author introduced and studied the concept of partial metric space, and obtained a Banach type fixed point theorem on complete partial metric spaces. In this work we study fixed point results for new extensions of Banach’s contraction principle to partial metric space, and we give some generalized versions of the fixed point theorem of Matthews. The theory is illustrated by some examples.

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.

On matrix domains of triangles

We prove some general results for the determination of the β-duals of, and the characterisations of matrix transformations on matrix domains of arbitrary triangles in FK spaces. Our results contain almost all recently published ones as special cases. We also study the measure of noncompactness of several matrix transformations. In particular, we obtain some known results of Sember, give the corrected version of a recent result by Djolovic, and study compact operators on some spaces of Altay, Basar and Mursaleen.

A weighted Drazin inverse and applications

Abstract In this paper the notation of the Cline–Greville W-weighted Drazin inverse of a rectangular matrix is extended to bounded linear operators between Banach spaces. We give new characterizations of the W-weighted Drazin inverse, and we study the perturbations and the the continuity of the W-weighted Drazin inverse.

Extensions of the Zamfirescu theorem to partial metric spaces

Abstract Zamfirescu [T. ZamfirescuFix point theorems in metric spaces Arch. Math. (Basel) 23 (1972) 292-298], obtained a very interesting fixed point theorem on complete metric spaces by combining the results of S. Banach, R. Kannan and S.K. Chatterjea. In [S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, in: Ann. New York Acad. Sci., vol. 728, 1994, pp. 183-197], the author introduced and studied the concept of partial metric spaces, and obtained a Banach type fixed point theorem on complete partial metric spaces. In this paper, we study new extensions of the Zamfirescu theorem to the context of partial metric spaces, and among other things, we give some generalized versions of the fixed point theorem of Matthews. The theory is illustrated by some examples.

Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems

We develop the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of (normed-valued) cone metric spaces. Examples are given to distinguish our results from the known ones.

Invertibility of the difference of idempotents

We study conditions equivalent to the invertibility of f m g when f and g are idempotents in a unital ring, and give applications to bounded linear operators in Banach and Hilbert spaces. In the setting of rings we are able to show that many conditions previously linked to finite dimensionality, rank equalities, norm topology of bounded linear operators or to properties of C *-algebras can be in fact proved by simple algebraic arguments.