Constancy of Functions via a Complement to Ekeland Variational Principle

This paper establishes new criteria for the constancy of real-valued functions defined on general Banach spaces and on exterior domains in Rn. The main analytical tool is a complement to Ekeland’s variational principle, while several auxiliary lemmas based on convex analysis play a crucial role in extending the argument to the non-convex framework of exterior domains. The obtained results establish constancy criteria under suitable growth assumptions at infinity, both in general Banach spaces and in the Euclidean setting. A key aspect of the analysis is the distinction between the whole-space and exterior-domain frameworks, showing that stronger asymptotic assumptions are required in the latter case. To illustrate the applicability of the general framework, we present an application to differentiable functions satisfying suitable symmetry-type assumptions on their derivatives.