Approximation of Fixed Points of Strongly Pseudocontractive Mappings

Abstract

Let be a real Banach space with a uniformly convex dual, and let be a nonempty closed convex and bounded subset of . Let be a continuous strongly pseudocontractive mapping of into itself. Let <!-- MATH $\{ {c_n}\} _{n = 1}^\infty$ --> be a real sequence satisfying: (i) <!-- MATH $0 < {c_n} < 1$ --> <img width="98" height="37" align="MIDDLE" border="0" src="images/img7.gif" alt="$ 0 < {c_n} < 1$"> for all <!-- MATH $n \geqslant 1$ --> ; (ii) <!-- MATH $\sum\nolimits_{n = 1}^\infty {{c_n} = \infty }$ --> ; and (iii) <!-- MATH $\sum\nolimits_{n = 1}^\infty {{c_n}b({c_n}) < \infty }$ --> <img width="171" height="41" align="MIDDLE" border="0" src="images/img10.gif" alt="$ \sum\nolimits_{n = 1}^\infty {{c_n}b({c_n}) < \infty } $">, where <!-- MATH $b:[0,\infty ) \to [0,\infty )$ --> is some continuous nondecreasing function satisfying <!-- MATH $b(0) = 0,\,b(ct) \leqslant cb(t)$ --> for all <!-- MATH $c \geqslant 1$ --> . Then the sequence <!-- MATH $\{ {x_n}\} _{n = 1}^\infty$ --> generated by <!-- MATH ${x_1} \in K$ --> , <!-- MATH \begin{displaymath} {x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1, \end{displaymath} -->