t-Osculating Operators in a Space of Continuous Functions and Applications Original Research Article

We introduce a linearization property for parameter dependent operators from a space of continuous functions into itself. This notion leads to a new implicit function theorem. As an application, we study the stability of the solutions of the integro-differential equation x′(t) = f(λ, t, x(t)) + ∫t0 g(t, s, x(s)) ds, x(0) = 0, with respect to the parameter λ.