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In this paper, we introduce a kind of quasipolarity notion for rings, namely, an element a of a ring R is called central quasipolar if there exists p² = p ∈ R such that a+p is central in R, and the ring R is called central quasipolar if every element of R is central quasipolar. We give many characterizations and investigate general properties of central quasipolar rings. We determine the conditions that some subrings of upper triangular matrix rings are central quasipolar. A diagonal matrix over a local ring is characterized in terms of being central quasipolar. We prove that the class of central quasipolar rings lies between the classes of commutative rings and Dedekind finite rings, and a ring R is central quasipolar if and only if it is central clean. Further we show that several results of quasipolar rings can be extended to central quasipolar rings in this general setting. Resumen En este trabajo, se presenta una noción de un tipo de casi-polaridad en anillos, esto es, un elemento a de un anillo R se dice casi-polar central si existe p² = p ∈ R tal que a + p es central en R, y el anillo R es llamado casi-polar central si todo elemento de R es casi-polar central. Se dan algunas caracterizaciones y se investigan propiedades generales de los anillos centrales casi-polares. Se determinan las condiciones bajo las cuales algunos subanillos de anillos de matrices triangulares superiores son casi-polares centrales. Una matriz diagonal sobre un anillo local se caracteriza en términos de ser casi-polar central. Se demuestra que la clase de anillos casi-polares centrales se encuentra dentro de la clase de los anillos conmutativos y los anillos finitos de Dedekind, y un anillo R es casi-polar central si es limpio central. Además se muestra que varios resultados de anillos casi-polares se pueden extender a anillos casi-polares centrales en un contexto general.

We prove a fixed point theorem for cyclic orbital generalized contractions on complete metric spaces from which we deduce, among other results, generalized cyclic versions of the celebrated Boyd and Wong fixed point theorem, and Matkowski fixed point theorem. This is done by adapting to the cyclic framework a condition of Meir–Keeler type discussed in [Jachymski J., Equivalent conditions and the Meir–Keeler type theorems, J. Math. Anal. Appl., 1995, 194(1), 293–303]. Our results generalize some theorems of Kirk, Srinavasan and Veeramani, and of Karpagam and Agrawal, The authors are very grateful to the referee since his/her corrections and suggestions have fairly improved the first version of the paper. Salvador Romaguera also acknowledges the support of the Spanish Ministry of Science and Technology, grant MTM2009-12872-C02-01.