Semigroups generated by pseudo-contractive mappings under the Nagumo condition

Let X be a Banach space whose dual space X∗ is uniformly convex. We demonstrate that, for any demicontinuous, weakly Nagumo, k-pseudo-contractive mapping T:D(T)⊆X→X with closed domain, A=T−I weakly generates a semigroup on D(T). In this paper, we project the consequences of this result on fixed point theory. In particular, we show that if k<1 (id est, if T is strongly pseudo-contractive), then T has a unique fixed point. This implies that, if T is pseudo-contractive (k=1) and D(T) is closed, bounded, and convex, then T has at least one fixed point. Consequently, any demicontinuous pseudo-contractive mapping T:C→C (for an appropriate C) has a fixed point, which has been an important open question in fixed point theory for quite some time. In a subsequent paper, we explore the consequences of the semigroup result on the existence of solutions to certain partial differential equations. The semigroup result directly implies the existence of unique global solutions to time evolution equations of the form u′=Au where A is a combination of derivatives. The fixed point results from this paper imply the existence of solutions to partial differential equations of the form Lu=f.