Iterative sequence for asymptotically demicontractive maps in Banach spaces (Q1763818)

Let \(E\) be a real Banach space. A mapping \(T: E\to E\) is called (i) uniformly \(L\)-Lipschitzian if there exists a constant \(L> 0\) such that for all \(x,y\in E\) and \(n\in\mathbb{N}\), \(\| T^nx- T^ny\|\leq L\| x-y\|\), (ii) asymptotically demicontractive if there exists a sequence \(\{a_n\}^\infty_{n=0}\) with \(\lim_{n\to\infty} a_n= 1\) such that \(\| T^nx- x^*\|^2\leq a^2_n\| x- x^*\|^2+ k\| x- T^n x\|^2\) for some \(k\in [0,1)\) and for all \(x\in E\), \(x^*\in F(T)\) and \(n\in\mathbb{N}\), where \(F(T):= \{x\in E: x= Tx\}\) denotes the set of fixed points of \(T\). In this paper, the authors prove necessary and sufficient conditions for the strong convergence of the Mann iteration sequence to a fixed point of an asymptotically demicontractive map in real Banach spaces. The main result goes as follows. Let \(E\) be a real Banach space and \(T: E\to E\) be a uniformly \(L\)-Lipschitzian asymptotically demicontractive map with sequence \(\{a_n\}\) and \(F(T)\neq \varnothing\). Let \(\{c_n\}_{n\geq 0}\subset [0,1]\) be a real sequence such that \(\sum_{n\geq 0} c^2_n<\infty\) and \(\sum_{n\geq 0} c_n(a^2_n- 1)< \infty\). Let \(\{x_n\}_{n\geq 0}\) be the sequence generated from an arbitrary \(x_0\in E\) by \(x_{n+1}= (1- c_n) x_n+ c_n T^n x_n\), \(n\geq 0\). Then \(\{x_n\}_{n\geq 0}\) converges strongly (i) to a fixed point of \(T\) if and only if \(\liminf_{n\to\infty}\, d(x_n, F(T))= 0\), (ii) to \(x^*\in F(T)\) if and only if there exists an infinite subsequence of \(\{x_n\}_{n\geq 0}\) which converges strongly to \(x^*\in F(T)\).