Fixed point theorems for (φ,k,i,j)(φ,k,i,j)-mappings

We introduce a new notion of “gauge function” and establish a general fixed point theorem for the new class of not necessary contractive, the (φ,k,i,j)(φ,k,i,j)-mappings.The convergence speed of the iterative sequence (f(n)x)(f(n)x) to the fixed point in this theorem is at least exponential, but (f(n)x)(f(n)x) does not satisfy, in general, any inequality of the form d(f(n)x,f(n+1)x)⩽λ d(f(n−1)x,f(n)x),withλ<1. Moreover, we can have d(f(n)x,f(n+1)x)> d(f(n−1)x,f(n)x)d(f(n)x,f(n+1)x)> d(f(n−1)x,f(n)x) for infinitely many integers nn.The theorem is given in the more general setting of quasi-metric spaces, special cases of which are the metric spaces. Some special theorems are also given.