On monotone Ćirić quasi-contraction mappings (Q2790791)

Let \((X,d)\) be a metric space and suppose that a partial order ``\(\leq\)'' is defined on \(X\). Let \(C\) be a nonempty subset of \(X\). A mapping \(T: C \rightarrow C\) is said to be \textit{monotone} if for any \(x,y \in C\) with \(x \leq y\), one has \(T(x) \leq T(y)\). \(T\) is called a \textit{quasi-contraction} if there is a nonnegative fraction \(k\) such that for any \(x,y \in C\) with \(x \leq y\), one has \(d(T(x),T(y)) \leq k \alpha_{x,y}\), where \(\alpha_{x,y}:=\max \{d(x,y), d(x,T(x)), d(y,T(y)), d(x,T(y)), d(y,T(x))\}\). For \(x \in C\), define \(\delta(x):=\sup\{d(T^n(x),T^m(x)): m,n \in \mathbb{N}\}\). Finally, \(u,v \in X\) are said to be \textit{comparable} if either \(u \leq v\) or \(v \leq u\).NEWLINENEWLINEThe main result of the authors is the following result: Let \((X,d)\) be a complete metric space with a partial order ``\(\leq\)'' defined on it. Let \(C\) be a closed nonempty subset of \(X\) and \(T\), a mapping on \(C\) is a monotone quasi-contraction. If \(x \in C\) is such that \(x\) and \(T(x)\) are comparable and \(\delta(x) < \infty\), then the sequence \(\{T^n(x)\}\) converges to a fixed point of \(T\) and is comparable to \(x\). The vector \(x\) is ``unique'' in the following sense: if \(z \in C\) is a fixed point of \(T\) such that \(z\) and \(x\) are comparable, then \(z=x\). An analogue of this result is also proved for a modular metric space.