Necessary and Sufficient Conditions for the Existence and $\varepsilon$-Uniqueness of Bounded Solutions of the Equation $x'=f(x)-h(t)$

We introduce the notion of $\varepsilon$-unique bounded solution to the nonlinear differential equation $x'=f(x)-h(t)$, where $f\colon\mathbb R\to\mathbb R$ is a continuous function and $h(t)$ is an arbitrary continuous function bounded on $\mathbb R$. We derive necessary and sufficient conditions for the existence and $\varepsilon$-uniqueness of bounded solutions to this equation.