MINIMUM-NORM FIXED POINT OF FINITE FAMILY OF $\LAMBDA-$STRICTLY PSEUDOCONTRACTIVE MAPPINGS

Abstract

Let $K$ be a nonempty closed and convex subset of a real Hilbert space $H$ and for each $1\leq i\leq N$, let $T_i: K\rightarrow K$ be $\lambda_i$-strictly pseudocontractive mapping. Then for $\beta \in (0,2\lambda]$, where $\lambda:=\min\{ \lambda_i:i=1,2,...,N\}$, and each $t \in (0,1)$, it is proved that, there exists a sequence $ \{y_t\} \subset K$ satisfying$ y_t= P_K\big[(1-t)(\beta Ty_t+(1-\beta) y_t)\big],$where $T:=\theta_1T_1+\theta_2T_2+...+\theta_NT_N$, for $\theta_1+\theta_2+...+\theta_N=1$, which converges strongly, as $t\to 0^+$, to the common minimum-norm fixed point of $\{T_i: i=1,2,...,N\}$.Moreover, we provide an explicit iteration process which converges strongly to a common minimum-norm fixed point of $\{T_i:i=1,2,...,N\}$. Corresponding results, for a common minimum-norm solution of finite family of $\alpha-$inverse strongly monotone mappings are also discussed. Our theorems improve several results in this direction.