Nonlinear inclusions of elliptic type and methods of lines for evolution problems in Banach spaces

The first part of this work concerns itself with the study of inclusions $$Tx + Cx\ni p,\eqno(*)$$ where $T: X \supset D(T)\to 2\sp{X}$ is usually a (possibly nonlinear) m-accretive operator and $C: \overline{D(T)}\to X$ is usually a (possibly nonlinear) compact operator in a Banach space X. Several new results are given for (*) which involve applications of the Leray-Schauder Degree Theory. In particular, considerable improvements have been made possible of recent results of Zhu for (*) and Yang for a triplet of operators. The second part of this work involves the construction and the proof of the convergence of a method of lines for the quasi-nonlinear problem: $$\eqalign{&x\sp\prime + A(t,x\sb{t})x\ni G(t,x\sb{t}, L\sb{t}x), t\in\lbrack 0, T\rbrack,\cr&x\sb0 = \phi,}$$ where the operators $A(t,\psi)$: $X\supset D(A(t,\psi))\to 2\sp{X}$ are at least m-accretive and the operators $G(t,\psi\sb1,\psi\sb2)$ are Lipschitzian. This method is more general and considerably more direct than the method developed by Ha, Shin and Jin.