Pseudo-contractive mappings and the Leray-Schauder boundary condition

Let X be a Banach space and D a bounded closed subset of X with 0 6 intCD). A mapping T:D—> X such that (&-1) ilx-y IU |i(A,I-T)(x)-(AJ-T)(y)ll for all xfy eDf r>0 f is called pseudo-contractive, while T is said to be nonexpansive if l|T(x)-T(y) I) * il x-y )) f xfy €D. It is well-known that if T is nonexpansive, then the Leray-Schauder condition: T(x)^#x for x e 3D, X > lf is sufficient to guarantee that inf 4iix-T(x)li :x€D} » 0. This result is extended here te the wider class of continuous pseudo-contractive mappings under the weaker Leray-Schauder condition: T(x) s ^ x for x e 3 D, %> 1 ==» T(y) a <u,y for some y 6 D and /cte Clf X ). Several related results are also obtained.