Common Fixed Point Theorems of Single and Set-Valued Mappings on 2-Metric Spaces

The concept of a 2-metric space is a natural generalization of a metric space. It has been investigated initially by Gähler [4]. Iseki [5] studied the fixed point theorems in 2-metric spaces. Sessa [17] defined weak commutativity and proved common fixed point theorem for weakly commuting maps. In [7] Jungck introduced more generalized commuting mappings, called compatible mappings, which are more general than commuting and weakly commuting mappings. This concept has been useful for obtaining more comprehensive fixed point theorems. In [8,9] Jungck and Rhoades defined the concepts of δ-compatible and weakly compatible mappings, which extend the concept of compatible mappings in the single-valued setting on metric spaces. Several authors used these concepts to prove some common fixed point theorems (See, e.g., [13-16]). In this paper we generalized some definitions on 2metric spaces and studied common fixed point theorems for four mappings on 2metric spaces.