Estimates for the Cauchy function of linear singular differential equations and some applications
For higher-order linear singular equations, we find sharp estimates for the Cauchy function and its partial derivatives. We use these estimates to study the properties of solutions of singular differential inequalities and obtain optimal sufficient conditions for the unique solvability of the linear singular Cauchy problem.
In this note we show that a subgradient multifunction of a locally compactly Lipschitzian mapping satisfies a closure condition used extensively in optimisation theory. In addition we derive a chain rule applicable in either separable or reflexive Banach spaces for the class of locally compactly Lipschitzian mappings using a recently derived generalised Jacobian. We apply these results to the derivation of Karush-Kuhn-Tucker and Fritz John optimality conditions for general abstract cone-constrained programming problems. A discussion of constraint qualifications is undertaken in this setting