Operator Equations of the Second Kind: Theorems on the Existence and Uniqueness of the Solution and on the Preservation of Solvability

This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator acting on an arbitrary Banach space , new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used: , where is, generally speaking, another Banach space semi-ordered by the cone . These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov ( , ). The role of an analog of the Lipschitz constant on a given bounded set is played by a bounded linear operator , depending on this set, with spectral radius . In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation , , where is the control parameter from, generally speaking, an arbitrary set . The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation.