Fixed point theorems for Meir-Keeler type contractions in metric spaces (Q2823163)

Let \((X,d)\) be a metric space and \(T:X\to X\) be a map. Further, let \(M:X\times X\to \mathbb R_+\) be a function. The following is the main result in this paper.NEWLINENEWLINETheorem. Suppose thatNEWLINENEWLINE(i) \(T\) is an asymptotically regular mapNEWLINENEWLINE(ii) for each \(\varepsilon> 0\), there exist \(\delta> 0\), \(k\in N\), such that \(M(T^kx,T^ky)< \varepsilon+\delta\) implies \(d(T^{k+1}x,T^{k+1}y)\leq \varepsilon\)NEWLINENEWLINE(iii) for each \(x\in X\), \(\limsup_n M(x_{p(n)},x_{q(n)})\leq \limsup_n d(x_{p(n)},x_{q(n)})\), for any two subsequences \((x_{p(n)})\) and \((x_{q(n)})\) of \((x_n:=T^nx)\).NEWLINENEWLINEThen, the following conclusions holdNEWLINENEWLINE(A): for each \(x\in X\), \((T^nx)\) is CauchyNEWLINENEWLINE(B): if \(X\) is complete, \(T\) is continuous and \(d(T^nx,T^ny)\to 0\) for all \(x,y\in X\), then \(z:=\lim_n T^nx\) is a fixed point of \(T\); and is uniquely determined under such a property.NEWLINENEWLINESome particular versions of this result are also given, to verify its unifying character.