Abstract:

A class of 5-dimensional functions Φ was introduced, and a convergent sequence determined by a family of self-maps {T_i, j }_{i∈N∪{0}, ~j∈N},
 which satisfy some Φ_j-contractive condition in a 2-metric space, was constructed, and then that the limit of the sequence is the unique common fixed point of the maps {T_{i, j ^}i∈N∪{0}, j∈N} was proved when X is complete and the following condition T_α, μ ? T_{β, ν}=T_β, ν? T_α, μ , ∨α, β ∈N∪{0}, μ, ν ∈N, μ≠ν, is satisfied.  Our main theorem improves and generalizes many known unique common fixed point theorems in 2-metric spaces.

Key words: 2-metric spaces, Cauchy sequence, Class of 5-dimensional functions Φ,  Φ_j-quasi contractive condition, Common fixed point