A Common Fixed Point Theorem for a Family of Mappings in Non-Archimedean Menger PM-Spaces

Abstract

In the present paper we prove a unique common fixed point theorem for a family of weakly compatible self maps in non-Archimedean Menger PM-spaces without using the notion of continuity. Our result generalizes and extends some well known previous results.

Common Fixed Point Theorems for Compatible Mappings of Type (A) in Non-Archimedean Menger Pm-Spaces

“…Definition 2.16 [16] Let , ∶ → be mappings. and are said to be compatible if lim n→∞ g(F ASx n SAx n (t)) = 0 For all > 0, whenever { } is a sequence in such that lim n→∞ Ax n = lim n→∞ Sx n = for some ∈ .…”

Fixed Point Theorems via Family of Maps in Weak Non-Archimedean Menger Pm-Spaces

“…In 2009, Khan and Sumitra [13] introduced the concept of R-weakly commuting maps in non-Archimedean Menger PM-spaces and proved a common fixed point theorem for three pointwise R-weakly commuting mappings in complete non-Archimedean Menger PM-spaces. Several authors have already studied fixed point theorems in Non-Archimedean Menger PMspaces for more details, we refer the reader to Singh et al [1], Khan and Sumitra [14], [15], Rashwan and Moustafa [16] and Singh et al [19] In the present paper we prove the fixed point theorem using the pair of semicompatible and reciprocally continuous maps for one pair and R−weakly commuting maps for another pair in Menger PM-spaces. Further we obtain a common fixed point theorems for six maps and one corollary for four maps via rational inequality.…”

Some Fixed Point Theorems via Common Limit Range Property in Non-Archimedean Menger Probabilistic Metric Spaces

“…In 1987, Singh and Pant [34] introduced the notion of weakly commuting mappings on N. A. Menger PM-spaces and proved some common fixed point theorems. Dimri and Pant [11] studied the application of N. A. Menger PMspaces to product spaces.…”