Common fixed points from best approximation (Q2433601)

Let \((E,\tau )\) be a Hausdorff locally convex topological vector space and let \(A^{\ast }(\tau )=\{p_{\alpha }:\alpha \in I\}\) be an augmented family of seminorm on \(E\) associated to \(\tau .\) Let \(M\subset E,\;\)and let \(T,\) \(I\) be selfmaps on \(M.\) Then \((i)\) \(T\) is nonexpansive if for all \(x,y\in M,\) \( p_{\alpha }(Tx-Ty)\leq p_{\alpha }(x-y)\), for each \(p_{\alpha }\in A^{\ast }(\tau );\) \((ii)\) \(T\) is \(I\)-nonexpansive if for all \(x,y\in M,\) \(p_{\alpha }(Tx-Ty)\leq p_{\alpha }(Ix-Iy)\) for each \(p_{\alpha }\in A^{\ast }(\tau )\); \((iii)\) If \(M\) is \(q\)-starshaped where \(q\) is a fixed point of \(I,\) \( T(M)\subset M,\) \(I(M)\subset M,\) then \(I\) and \(T\) are \(R\)-subcommutative iff there exists \(R>0\) such that \(p_{\alpha }(ITx-TIx)\leq (\frac{R}{k} )p_{\alpha }((kTx+(1-k)q)-Ix),\) for all \(x\in M,\) \(p_{\alpha }\in A^{\ast }(\tau )\) and \(k\in (0,1).\) The author gives some results concerning the common fixed points \((x=Tx=Ix,\) \(x\in M)\) and coincidence points \((Tx=Ix,\) \(x\in M)\) of \(I\) and \(T.\) Sample: Let \(M\) be a nonempty \(\tau \)-bounded, \(\tau \)-sequentially complete and \(q\) -starshaped subset of \((E,\tau ).\) Let \(T\) and \(I\) be selfmaps on \(M\) such that \(T\) is \(I\)-nonexpansive, \(I(M)=M,\) \(q\;\)is a fixed point of \(I,\) \(I\) is nonexpansive and affine, and \(T\) and \(I\) are 1-subcommutative on \(M.\) Then \( T \) and \(I\) have a common fixed point provided one of the following conditions holds: (i) \(M\) is \(\tau \)-sequentially compact; (ii) \(T\) is a compact map; (iii) \(M\) is weakly compact, \(I\) is weakly continuous and \(I\)-\( T\) is demiclosed in \(0;\) \((iv)\) \(M\) is weakly compact in an Opial space \( (E,\tau )\) and \(I\) is weakly continuous. Also some results concerning common random fixed points and random coincidence points for random operators are presented.