Existence of the Solution of an Implicit Integro-Differential Equation of Order One

Abstract

In this study, we establish the existence of the solution of a class of first order implicit integro-differential equation of type F(t, (Kx)(t), x(t), x′(t)) = 0, where F : J × R³ → R, K : C(J) → C(J) defined as (Kx)(t) = ∫₀^tk(t, s)x(s)ds, ∀x ∈ C(J), using the concept of viscosity solution.