On the projection methods for fixed point problems (Q2718027)

Let \(G\) be a subset of a Banach space \(B\). A mapping \(A\) is called weakly contractive if there exists a continuous, nondecreasing function \(\psi(t)\), defined on \(\mathbb{R}^{+}\) such that \(\psi\) is positive on \(\mathbb{R}^{+} \setminus \{0\}, \lim_{t \to \infty} \psi (t) = + \infty\) and, for each \(x, y, \in G, (1) \|Ax - Ay\|\leq \|x - y\|- \psi(\|x - y\|)\). With \(V\) the Lyapunov functional, the authors define a map \(A\) to be strongly suppressive if there exists \(0 < q < 1\) such that, for each \(x, y \in G, V(Ax, Ay) \leq q V(x, y)\). They call a map weakly suppressive if, in (1), one replaces the norm with \(V\); i.e., \(V(Ax, Ay) \leq V(x, y) - \psi(V(x, y))\). The map \(A\) is called nonextensive if \(V(Ax, Ay) \leq V(x, y)\). If \(B\) is a Hilbert space, then strongly contractive and nonextensive operators become strongly contractive and nonexpansive, respectively. The operator \(\prod_G: B \to G \subseteqq B\) is called the generalized projection operator if it associates with an arbitrary point \(x \in B\) the minimum point of the functional \(V(x, \xi)\). Using the iterative-projection method of successive approximations defined by \(x_{n + 1} = \prod_GAx_n\), the authors show that, if \(N(A) \neq \varnothing\) and \(B\) is uniformly convex and uniformly smooth, then \(\{x_n\}\) converges strongly to a fixed point of \(A\) if \(A\) is either strongly or weakly suppressive. They also obtain rate of convergence estimates. Adding conditional conditions they obtain weak convergence for \(A\) nonextensive. Then, with other additional assumptions, they establish several theorems for the norm convergence of the iteration-projection method in Hilbert space, along with convergence rates. Theorem 6.1 of the paper obtains the strong convergence, in a Banach space, of the Mann iteration of a weakly contractive selfmap of \(G\), along with a rate of convergence estimate. Theorem 6.1 is a special case of results obtained independently by the reviewer. [Theorems 3 and 12 of Nonlinear Anal. Theory Methods Appl. 47, 2683-2693 (2001)].