We consider the Cauchy problem(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{u\sp\prime(t)&= \int\sbsp{0}{t}\ K(t - s)Au(s)ds,\quad t\geq 0\cr u(0)&= f,\cr}\leqno(P)$$(TABLE/EQUATION ENDS)and we are interested in continuous dependence of solutions $u(t) = U(t)f$ on A and K, where $\{U(t)\}\sb{t\geq 0}$ is the resolvent family for (P) Given a family of operators $\{A\sb{n}\}$ and a family of scalar kernels $\{ K\sb{n}\}$, we study the family of Cauchy problems(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{u\sbsp{n}{\prime}(t)&= \int\sbsp{0}{t}\ K\sb{n}(t - s)A\sb{n}u\sb{n}(s)ds,\quad t\geq 0\cr u\sb{n}(0)&= f\sb{n}.\cr}\leqno(Pn)$$(TABLE/EQUATION ENDS)We show that under certain stability conditions for $\{ A\sb{n}\}$ and $\{ K\sb{n}\},$ if $A\sb{n} \to A\sb{o}$ and if $K\sb{n} \to K\sb{o},$ in a certain sense, then $u\sb{n}(t) \to u\sb{o}(t).$ Our result is a partial extension of the Trotter-Neveu-Kato theorem to integro-differential equations / acase@tulane.edu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_24653 |
| Date | January 1991 |
| Contributors | Segalla-Pickett, Gabriella (Author), Goldstein, J. A (Thesis advisor) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Rights | Access requires a license to the Dissertations and Theses (ProQuest) database., Copyright is in accordance with U.S. Copyright law |
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