New equations of convolution type obtained by replacing the integral by its maximum

Abstract

We study the nonlinear equation max g(γ)|cos(γ - α)| = f(α), γεℝ where f(α) is a given function and g(γ) is the unknown function, to be found in the class of nonnegative continuous π-periodic functions. This equation arose in the context of an applied problem dealing with the construction of a hydrofoil from given pressure envelopes. Necessary and sufficient conditions for the solvability of the equation, an explicit description of the solution set, and a comparison theorem under changes of the right-hand sides are obtained. Some possible ways of generalization are indicated.