On the approximation of fixed points for locally pseudo-contractive mappings

Abstract:

Let X and its dual ${X^ \ast }$ be uniformly convex Banach spaces, D an open and bounded subset of X, T a continuous and pseudo-contractive mapping defined on ${\text {cl}}(D)$ and taking values in X. If T satisfies the following condition: there exists $z \in D$ such that $\left \| {z - Tz} \right \| < \left \| {x - Tx} \right \|$ for all x on the boundary of D, then the trajectory $t \to {z_t} \in D,t \in [0,1)$, defined by ${z_t} = tT({z_t}) + (1 - t)z$ is continuous and converges strongly to a fixed point of T as $t \to {1^ - }$.