Fixed point theorems for an elastic nonlinear mapping in Banach spaces (Q1668920)

Summary: Let \(E\) be a smooth Banach space with a norm \(\|\cdot\|\). Let \(V(x,y)=\| x\|^2+\| y\|^2-2\langle x,Jy\rangle\) for any \(x,y\in E\), where \(\langle\cdot,\cdot\rangle\) stands for the duality pair and \(J\) is the normalized duality mapping. We define a \(V\)-strongly nonexpansive mapping by \(V(\cdot,\cdot)\). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a \(V\)-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give an existence theorem.