Fixed points of generalized contractive mappings in ordered metric spaces

A. Amini-Harandi
2014 Filomat  
Existence theorem for fixed point of mappings satisfying a new generalized contractive condition, involving some well-known contractive conditions of rational type, in ordered metric spaces is proved. Some examples are given which illustrate the value of the obtained results in comparison to some of the existing ones in literature. Theorem 1.1. (Jaggi [11]) Let (X, d) be a complete metric space and let T : X → X be a continuous mapping such that there exist α, β ≥ 0 with α + β < 1 satisfying
more » ... β < 1 satisfying d(Tx, Ty) ≤ α d(x, Tx)d(y, Ty) d(x, y) + βd(x, y), (1.1) for all x, y ∈ X, x y. Then T has a unique fixed point. Theorem 1.2. (Dass and Gupta [12]) Let (X, d) be a complete metric space and T : X → X a mapping such that there exist α, β ≥ 0 with α + β < 1 satisfying d(Tx, Ty) ≤ α d(y, Ty)[1 + d(x, Tx)] 1 + d(x, y) + βd(x, y), (1.2) for all x, y ∈ X. Then T has a unique fixed point.
doi:10.2298/fil1406247a fatcat:zypy5y5xyvhwzdu5kpf6fxwhs4