Abstract metric spaces and Sehgal Guseman-type theorems

Recently, Raja and Vaezpour [P. Raja and S.M. Vaezpour, Some extensions of Banachʹs contraction principle in complete cone metric spaces, Fixed Point Theory Appl. 2008, 11 pages, doi:10.1155/2008/768294. Article ID 768294] proved some results for Sehgal Guseman-type theorems in the framework of abstract (cone) metric spaces over a normal solid cone. The purpose of this paper is to present this in the framework of symmetric spaces which are associated with abstract (cone) metric spaces introduced by Radenovi¢ and Kadelburg [S. Radenovi¢, Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, ISI J. BJMA (electronic) (in press)]. Our results extend and generalize Sehgal Guseman-type theorems from metric and abstract metric spaces to some symmetric spaces. Examples are given to illustrate the results.