Zeros ofm-accretive operators and abstract evolution equations in Banach spaces

In the first part of this dissertation, we approximate the solution of the equation $Tx \ni 0$ via use of differential equations associated with it. The operator $T : D(T) \subset X \to 2\sp{X}$ is at least m-accretive, where $X$ is a real Banach space.

The method of lines is used for the approximants and results of Browder are extended to general Banach spaces.

The second part of the work is devoted to the existence of solutions of perturbed abstract functional differential equation of the form:(UNFORMATTED TABLE OR EQUATION FOLLOWS) $$\left(\eqalign{&x\sp\prime + A(t) \ni G(t, x\sb t),\cr&x(s)=\phi(s),\cr}\qquad\eqalign{&t \in \lbrack 0, T\rbrack,\cr&s\in \lbrack-r,0\rbrack,\cr}\right.$$ (TABLE/EQUATION ENDS)where $A(t) : D(A(t))\subset X\to2\sp{X}$ is m-accretive and $G : \lbrack 0, T\rbrack \times C(\lbrack -r, 0\rbrack, \overline{D(A(t))})\to X$ is continuous. The dual space of $X$ is assumed to be uniformly convex and results are given under compactness or equicontinuity conditions on the evolution operator generated by $A(t)$.

Results of Pavel, Vrabie, Gutman and other authors are extended.