Some fixed point theory results for convex contraction mapping of order 2. (English) Zbl 1491.54041
Summary: Let \(X\) be a nonempty set and \(d\) be a metric. Recall from V. I. Istratescu [Libertas Math. 1, 151–163 (1981; Zbl 0477.54032)] that a map \(T:X\mapsto X\) is called a convex contraction mapping of order 2, if for all \(x,y\in X\), \(a,b\geq 0\), \(a+b<1\), it holds that \(d(T^2x,T^2y)\leq ad(x,y)+bd(Tx,Ty)\). Alternatively, one could say \(T:X\mapsto X\) is a convex type contraction mapping of order 2, if for all \(x,y\in X\) and \(0\leq k<\frac{1}{4}\), it holds that
\[
d(T^2x,T^2y)\leq k[d(x,y)+d(Tx,Ty)].
\]
In this paper, we investigate fixed point results for this type of mapping in various settings.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
Citations:
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