an:02229132
Zbl 1084.41023
O'Regan, Donal; Shahzad, Naseer
Invariant approximations for generalized \(I\)-contractions
EN
Numer. Funct. Anal. Optimization 26, No. 4-5, 565-575 (2005).
0163-0563 1532-2467
2005
j
41A50 47H10 54H25
best approximation; common fixed point; normed space; \(R\)-subweakly commuting map
Let \(X\) be a normed vector space. Let \(S\) be a subset of \(X\) and let \({T, I}\) be self-mappings of \(X\). Then \(T\) is called \(I\)-contraction on \(S\) if there exists \({k\in [0,1)}\) such that \({\| Tx-Ty\| \leq k \| Ix-Iy\| }\) for all \({x,y\in S}\). By \({[a,b]}\) we denote a linear segment in \(X\) joining points \({a,b\in X}\). Suppose \(p\) is a point in \(S\) and let \(S\) be \(p\)-star-shaped. The mappings \(T\) and \(I\) are called \(R\)-subweakly commuting on \(S\) if there exists \({R\in (0,\infty)}\) such that \({\| TIx-ITx\| \leq R\cdot dist(Ix,[Tx,p])}\) for all \({x\in S}\). The purpose of the paper under review is to obtain some results on common fixed points for generalized \(I\)-contractions and \(R\)-subweakly commuting maps. As applications, various invariant approximation results are derived.
Victor Milman (Minsk)