Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings
C.E. Chidume · Proceedings of the American Mathematical Society · 1987
Suppose $X = {L_p}({\text {or}}\;{l_p}),p \geq 2$, and $K$ is a nonempty closed convex bounded subset of $X$. Suppose $T:K \to K$ is a Lipschitzian strictly pseudo-contractive mapping of $K$ into itself. Let $\{ {C_n}\} _{n = 0}^\infty$ be a real sequence satisfying: (i) $0 < {C_n} < 1$ for all $n \geq 1$, (ii) $\sum olimits _{n = 1}^\infty {{C_n} = \infty }$, and (iii) $\sum olimits _{n = 1}^\infty {C_n^2 < \infty }$. Then the iteration process, ${x_0} \in K$, \[ {x_{n + 1}} = (1 - {C_n}){x_n} + {C_n}T{x_n}\] for $n \geq 1$, converges strongly to a fixed point of $T$ in $K$.