Minimum-Norm Fixed Point of Pseudocontractive Mappings

Abstract

Let K be a closed convex subset of a real Hilbert space H and let $T:K\to K$ be a continuous pseudocontractive mapping. Then for $\beta \in (0,1)$ and each $t\in (0,1)$, there exists a sequence $\{y_t\}\subset K$ satisfying $y_t=\beta P_K[(1-t)y_t]+(1-\beta )T(y_t)$ which converges strongly, as $t\to 0^{+}$, to the minimum-norm fixed point of T. Moreover, we provide an explicit iteration process which converges strongly to a minimum-norm fixed point of T provided that T is Lipschitz. Applications are also included. Our theorems improve several results in this direction.