Generalized hybrid mappings in Hilbert spaces and Banach spaces (Q763723)

Let \(C\) be a closed convex subset of a real Banach space \(X\) and \(\alpha,\beta\) be two real numbers. A mapping \(T:C\to C\) is called an \((\alpha,\beta)\)-generalized hybrid mapping if \(\alpha\|Tx-Ty\|^2+(1-\alpha)\|x-Ty\|^2\leq\beta\|Tx-y\|^2+(1-\beta)\|x-y\|^2\) for all \(x,y\in C\). Obviously, \((1,0)\)-, \((2,1)\)- and \((\frac32,\frac12)\)-generalized hybrid mappings are nonexpansive, nonspreading, and hybrid; and fixed point theorems for such mappings have been studied recently. In this paper, the authors prove fixed point theorems for \((\alpha,\beta)\)-generalized hybrid mappings and duality theorems for some nonlinear mappings.