Perturbations in a Class of Nonlinear Abstract Equations
John E. Lagnese · SIAM Journal on Mathematical Analysis · 1975
Let V be a real reflexive Banach space and denote the dual of V by $V'$. Let $\{ B_\varepsilon :0 < \varepsilon \leqq \varepsilon _0 \} $ be a family of (possibly) nonlinear operators from V into $V'$ and $\{ A_\varepsilon :0 < \varepsilon \leqq \varepsilon _0 \} \subset \mathcal{L}(V,V')$, and let $\Lambda $ be an unbounded linear operator in $V'$. Consider the equation $\Lambda A_\varepsilon u_\varepsilon + B_\varepsilon u_\varepsilon = f_\varepsilon \in V'$. It is shown that $\{ {u_\varepsilon } \}$ converges in a certain sense to a solution of $\Lambda A + Bu = f$ provided $A_\varepsilon \to 1$, $B_\varepsilon \to B$ and $f_\varepsilon \to f$ in some appropriate sense, and provided certain other conditions on the operators involved are satisfied. This result is shown to apply to certain nonlinear evolution equations. Two examples are discussed the first concerns a nonlinear, pseudoparabolic partial differential equation with a small parameter the second concerns a nonlinear, degenerate parabolic equation with a small parameter.