On some results of periodic point (Q1092447)

The author gives a simple counterexample to a fixed point theorem of \textit{H. Chatterji} [ibid. 10, 449-450 (1979; Zbl 0403.54046)] but observes that the theorem and proof are valid if the inequality imposed on the distance between images is strict. Using similar techniques and the simpler inequality \[ (I)\quad d(x,y)\geq a[d(x,T^ mx)+d(y,T^ my)]+bd(T^ mx,T^ my) \] for an integer \(m\) and self-mapping \(T\) of a metric space \((X,d)\), he proves the following: each orbit \(0=\{T^{mn}x_ 0: n=0,1,2...\}\) of \(T^ m\) converges to a fixed point of \(T^ m\) (i.e., a point of period \(m\) of \(T\)) if (i) \({\bar 0}\) is complete and (ii) for some \(a,b\) with \(a\leq 1<2a+b\), all distinct \(x, y\) in \({\bar 0}\) satisfy (I). The author's condition \(0<a+b\) is redundant. If \(X\) is only Hausdorff and \(d: X\times X\to [0,\infty)\) a continuous, symmetric mapping where \(d(x,y)=0\) if \(x=y\), he shows that every cluster point \(z\) of 0 is a period \(m\) point of \(T\) if (i') \(T^ m\), \(T^{2m}\) are continuous at \(z\) and (ii') all distinct \(x, y\) in \({\bar 0}\) satisfy (I) strictly where \(a, b\) satisfy \(2a+b=1\) and \(0<a+b\).