Faintly compatible mappings and common fixed points

Abstract

In this paper we introduce a generalization of the concept of compatible mappings, and using that condition, we obtain some new fixed point theorems under both contractive and noncontractive conditions, which may allow the existence of a common fixed point or the existence of multiple fixed or coincidence points. We also manifest that the new concept is a necessary condition for the existence of a common fixed point. MSC: 47H10; 54H25

Common fixed points for faintly compatible mappings

“…Now faint compatibility of B and T will imply that there exists a sequence {w n } in X satisfying lim n→∞ Bw n = lim n→∞ T w n = u (say) such that (3) lim n→∞ d(BT w n , T Bw n ) = 0. Again, B is continuous so lim n→∞ BBw n = Bu and lim n→∞ BT w n = Bu .…”

“…Complementing the work of Bisht et al [3], we give following examples for the comparative discussions on the above concepts.…”

“…The pair (f, g) is said to be conditionally compatible iff whenever the set of sequences {x n } satisfying lim n→∞ f x n = lim n→∞ gx n is nonempty, there exists a sequence {y n } such that lim n→∞ f y n = lim n→∞ gy n = t and lim n→∞ d(f gy n , gf y n ) = 0. Over the last two decades, there are a number of common fixed/coincidence point theorems for the pair of mappings under different contractive conditions with compatibility and its weaker versions imposed on the mappings (for more details, see [4,5,6,7,1,3] and refenences therein).…”

“…In a recent work, Bisht and Shahzad [3] gave a new notion of conditionally compatible mappings in a slighty different settings and named it as faintly compatible mappings.…”

“…Bisht et al [3] proved some interesting common fixed point theorems using the concept of faintly compatible mappings on non-complete metric spaces under defferent contractive conditions. Complementing the work of Bisht et al [3], we give following examples for the comparative discussions on the above concepts.…”