## LSU Historical Dissertations and Theses

1997

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Jimmie D. Lawson

#### Abstract

Let M be a manifold modeled on a Banach space B, and let U be an open subset of M containing the domain of some chart $\varphi$. The aim of this work is to set mathematical foundations (topological, algebraic geometrical) for the theory of pseudosemigroups of local transformations S on M and their infinitesimal generators $L\sp{r}(S,\ U).$ In the first part of this dissertation we define the topology of local uniform convergence, the most suitable in this case, similar to the compact open topology in the finite dimensional case, and show what relationship it has with different topologies. In the second par we show that $L\sp{r}(S,\ U)$ form a cone in the tangent space TM, and derive an explicit form of the exponential mapping of a Lipschitz infinitesimal generator, which turn out to be a duality form. In the last section we generalize the Trotter Product Formula to this case: $(\exp({t\over n}X) {\rm o}\exp({t\over n}Y))\sp{n} \to \exp t(X + Y)$ locally uniformly on U and for all $t\in\lbrack 0, J\rbrack$, for some $J\in \IR,$ where X and Y are Lipschitz vector fields. This useful theorem lead us to the solution of a class of Lipschitz regulated partial differential equation.

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