Implicit and explicit algorithms for minimum-norm fixed points of pseudocontractions in Hilbert spaces
Let \(C\) be a nonempty convex subset of a real Hilbert space \(H\) with inner product \(\left\langle\cdot,\cdot\right\rangle\) and norm \(\left\| \cdot \right\|\). A mapping \(T:C \rightarrow C\) is said to be pseudocontractive (or a pseudocontraction) if \[ \left\langle Tx-Ty,x-y\right\rangle \leq \left\| x-y \right\|^2 \text{ for all }x,y \in C. \] In order to approximate fixed points of pseudocontractions, there exist various iterative methods, e.g., Mann iteration, Ishikawa iteration, etc. In the paper under review, the authors consider two new iterative methods (one implicit and another one explicit) for approximating the fixed points of Lipschitzian pseudocontractions \(T\). The main results of the paper show that these algorithms converge strongly to the minimum-norm fixed point of \(T\). Examples are also given to illustrate the theoretical results and to show that the new results are effective improvements of those existing in the literature.
scientific article; zbMATH DE number 6137979