Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces with Applications | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 979586 10 pages doi 2011 979586 Research Article Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces with Applications Cristian Chifu and Gabriela Petrusel Faculty of Business Babeậ-Bolyai University Horia Street no. 7 400174 Cluj-Napoca Romania Correspondence should be addressed to Cristian Chifu cochifu@ Received 6 December 2010 Accepted 31 December 2010 Academic Editor Jen Chih Yao Copyright 2011 C. Chifu and G. Petrusel. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. The purpose of this paper is to present some fixed-point results for single-valued -contractions on ordered and complete gauge space. Our theorems generalize and extend some recent results in the literature. As an application existence results for some integral equations on the positive real axis are given. 1. Introduction Throughout this paper E will denote a nonempty set E endowed with a separating gauge structure D da aeA where A is a directed set see 1 for definitions . Let N 0 1 2 . and N N 0 . We also denote by R the set of all real numbers and by R 0 x . A sequence xn of elements in E is said to be Cauchy if for every e 0 and a e A there is an N with da xn xn p e for all n N and p e N . The sequence xn is called convergent if there exists an x0 e X such that for every e 0 and a e A there is an N e N with da xo xn e for all n N. A gauge space E is called complete if any Cauchy sequence is convergent. A subset of X is said to be closed if it contains the limit of any convergent sequence of its elements. See also Dugundji 1 for other definitions and details. If f E E is an operator then x e E is called fixed point for f if and only if x f x . The set Ff x e E x f x denotes the fixed-point set of f. On .
