Let E be a Banach space and D a
subset of E. A mapping f : D → E such that ∥u−v∥≦∥(1 + r)(u−v) −r(f(u) −f(v))∥
for all u,v ∈ D, r > 0 is called pseudo-contractive. The basic result is the following:
Let X be a bounded closed subset of E, suppose f : X → E is a continuous
pseudo-contractive mapping such that f[X] is bounded, and suppose there
exists z ∈ X such that ∥z − f(z)∥ < ∥x − f(x)∥ for all x ∈ boundary (X).
Then inf{∥x − f(x)∥ : x ∈ X} = 0. If in addition X has the fixed point
property with respect to nonexpansive self-mappings, then f has a fixed
point in X. It follows from this result that if T : E → E is continuous and
accretive with ∥T(x)∥→∞ as ∥x∥→∞, then T[E] is dense in E, and if in
addition it is assumed that the closed balls in E have the fixed-point property
with respect to nonexpansive self-mappings, then T[E] = E. Also included
are some theorems for continuous pseudo-contractive mappings f which
involve demi-closedness of I − f and consequently require uniform convexity of
E.
