The authors study (generalized) Hyers-Ulam and (generalized) Hyers-Ulam-Rassias stability of nonlinear Volterra delay integro-differential equations of the form: \[ x'(t) = f\left(t, x(t), x(g(t)), \int_0^t h(t, s, x(s), x(g(s)))\,ds\right), \] with \(t\in I=[0, b]\), \(b>0\), and also by considering the initial condition \(x(t)=\phi(t)\), \(t\in [-r, 0]\), where, for \(0 < r <\infty\), \(\phi\in C([-r, 0], {\mathbb R})\), \(f\in C(I\times {\mathbb R}^3,{\mathbb R})\), \(h\in C(I\times I\times {\mathbb R}^2,{\mathbb R})\) and \(g\in C(I, [-r, b])\) satisfies \(g(t)\leq t\). Sufficient conditions for (generalized) Hyers-Ulam and (generalized) Hyers-Ulam-Rassias stability of that class of equations are obtained based on the use of Pachpatte’s inequality and Picard operator theory.
Concrete examples to illustrate the obtained results are included. For the case of the infinite interval \(I=[0, \infty)\), an example is also included to illustrate that the obtained sufficient conditions (for the finite interval case) do not ensure the Hyers-Ulam stability of the corresponding Volterra delay integro-differential equations (on infinite intervals).