Best simultaneous approximations, asymptotically nonexpansive mappings and variational inequalities in Banach spaces (Q1018303)

Let \(M\) be a subset of a normed space \(X\) and \( f,g,T: M\to M\) be mappings. \(T\) is called a \((f,g)\)-nonexpansive mapping (resp., \((f,g)\)-asymptotically nonexpansive mapping) if \(\| Tx-Ty\| \leq \| x-y\|\) for all \(x,y\in M\) (resp., if there exist a sequence \(\{ k_n \} \) of real numbers with \( k_n \geq 1\) and \( k_n\to 1\) such that \(\| T^n x-T^n y\| \leq \| x-y\|\) for all \(x,y\in M\)). The authors prove, under suitable conditions, the existence of a common fixed point of the mappings \(f,g,T\) for \((f,g)\)-nonexpansive mappings and \((f,g)\)-asymptotically nonexpansive mappings. As applications, some results about best simultaneous approximation are proved and the existence of variational inequalities is obtained