Theorems of the Alternative for Systems of Convex Inequalities

Systems of convex inequalities in function spaces are considered. Solvability conditions are obtained in the form of a theorem of the alternative. We revisit some results from the literature where such theorems are incorrect. We present counterexamples concerning these results and by introducing some regularity conditions we obtain new theorems which are dedicated to solvability of systems of convex inequalities. © 2017, Springer Science+Business Media Dordrecht.