On three principles of solvability of operator equations

Three principles of solvability of operator equations are considered. The first is connected with the existence of solutions of equations in partially ordered sets and generalizes the Birkhoff–Tarski theorem and certain other results on this topic. The second is a result of the development of the Pokhozhaev–Krasnosel'skii–Zabreiko method, as applied to normal cones, connected with a covering of a Banach space with the help of a Gâteaux-differentiable mapping with closed range. The third generalizes ideas of Plastock, Krasnosel'skii, Zabreiko, and Cristea on global solvability of operator equations to the case of mappings of quasisemimetric spaces into normed cones. The results are illustrated by examples from the theory of integro-differential and differential equations.