Halpern type proximal point algorithm of accretive operators (Q412632)

The main objectives of this paper are to employ a new proof technique to prove the strong convergence of \((x_n)\) and \((y_n)\), defined respectively by NEWLINE\[NEWLINE\begin{aligned} x_{n+1}&=\alpha_nu+\beta_nx_n+(1-\alpha_n-\beta_n)J_{r_n}^A x_n,\\ NEWLINEy_{n+1}&=\beta_ny_n+(1-\beta_n)J_{r_n}^A(\alpha_nu+(1-\alpha_n)y_n),\end{aligned}NEWLINE\]NEWLINE to some zero of an accretive operator \(A\) in a uniformly convex Banach space \(E\) with a uniformly Gâteaux differentiable norm (or with a weakly continuous duality mapping \(J_{\varphi}\)) whenever \((\alpha_n)\) and \((\beta_n)\) are sequences in \((0,1)\) and \((r_n)\subset (0,+\infty)\) satisfying the conditions NEWLINE\[NEWLINE\lim_{n\rightarrow \infty} \alpha_n=0,\quad \sum_{n=1}^{+\infty} \alpha_n=+\infty,\quad \limsup_{n\rightarrow +\infty} \beta_n<1,\quad \liminf_{n\rightarrow +\infty} r_n>0.NEWLINE\]NEWLINENEWLINENEWLINEEditorial remark: See also [\textit{Y.-L. Yu}, ibid., No.~13, 5027--5031 (2012; Zbl 1262.47094)].