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Abstract: Let X be a Banach space whose dual space is uniformly convex. We demonstrate that, for any demicontinuous, weakly Nagumo, k-pseudo-contractive mapping with closed domain, weakly generates a semigroup on . In this paper, we project the consequences of this result on fixed point theory. In particular, we show that if (id est, if T is strongly pseudo-contractive), then T has a unique fixed point. This implies that, if T is pseudo-contractive () and is closed, bounded, and convex, then T has at least one fixed point. Consequently, any demicontinuous pseudo-contractive mapping (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 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 . [Copyright &y& Elsevier]

Abstract: In this paper, we study the local existence and uniqueness of classical solutions to a wide class of systems of chemotaxis equations. These systems are essentially quasi-linear strongly coupled partial differential equations. We also study the maximal interval of existence in time of solutions. The results are illustrated in application to a number of partial differential equation models arising in biology. [Copyright &y& Elsevier]