Continuous dependence of nonmonotonic discontinuous differential equations
Daniel C. Biles · Transactions of the American Mathematical Society · 1993
Continuous dependence of solutions for a class of nonmonotonic, discontinuous differential equations is studied. First, a local existence theorem due to Z Z . Wu is extended to a larger class. Then, a result concerning continuous dependence for this larger class is proven. This employs a type of convergence similar to Gihman’s Convergence Criterion, which is defined to be \[ For all a , b and y lim n → ∞ ∫ a b f n ( s , y ) d s = ∫ a b f ∞ ( s , y ) d s . {\text {For all}}\;a,b\;{\text {and}}\;y\quad \lim \limits _{n \to \infty } \int _a^b {{f_n}(s,y)ds = } \int _a^b {{f_\infty }(s,y)\,ds}. \] The significance of Gihman’s Convergence Criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence. Finally, examples are presented to motivate and clarify this continuous dependence result.