The generalized Banach contraction conjecture in \(b\)-metric spaces
The generalized Banach contraction conjecture on \(b\)-metric spaces states as follows:\ Let \((X,D,K)\) be a complete \(b\)-metric space, \(T: X\to X\) a map, and \(J\) a positive integer. If \(\min\,\{ D(T^kx,T^ky): k=1,\dots,J\} \leq M D(x,y)\) for all \(x,y\in X\) and for some \(M\in (0,1)\), then \(T\) has a fixed point. When \(M\in (0,1/K)\), partial answers to the above conjecture are given, by using combinatorial arguments. First, the conjecture is true in the special case where \(X\) is a complete metric-type space. Next, it is true for the case \(J=2\) and, moreover, also for the case \(J=3\) if \(T\) is assumed to be continuous.
scientific article; zbMATH DE number 6858693