Fixed Point Theorems for Pseudo-Contractive Mappings in Banach Spaces
Let X be a Banach space, X* its dual and J: X → X* be an duality mapping. Suppose T is any self-mapping on X which satisfies
<(I − T)x − (I − T)y> ≥ 0 for all x, y ∈ X.
Then we say T is pseudo-contractive.
We have established sufficient conditions for the existence of a fixed point for continuous pseudo-contractive mappings defined on a subset D of a Banach space X under the following boundary condition:
There is z ∈ D so that Tx − z ≠ λ(x − z) for x ∈ D, λ > 1.