In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem **(1)**:

$(1)\left\{\begin{matrix} u'(t)=f(t,u(t),u(a(t))),\:\:t\in I \\ u(t_0)+\sum_{k=1}^{p}c_ku(t_k)=x_0 \end{matrix}\right.$

With $I:=[t_0,t_0+T], t_0<t_1<...<t_p\leq t_0+T, T>0$ and $f:I\times E^2\rightarrow E \:$ and $\:a:I\rightarrow I \:$are given functions satisfying some assumptions; $E$ is a Banach space with norm $\:\left \| . \right \|; x_0\in E, c_k\neq 0 \:\:(k=1,...,p)\: p \in \mathbb N$.

Now let suppose that $\sum_{k=1}^{p}c_k\neq -1 $ then a function $u \in \mathcal C(I,E)$ , satisfying the integral equation:

$u(t)=\frac {\big(x_0 -\sum_{k=1}^{p}c_k \int_{t_0}^{t_k} f(\tau ,u(\tau ),u(a(\tau )))d\tau\big)}{\big(1+\sum_{k=1}^{p}c_k\big)}+\int_{t_0}^{t} f(\tau ,u(\tau ),u(a(\tau )))d\tau \:\:t\in I\:,$

is said to be a **mild solution** of the nonlocal problem **(1)**.

My question is:

- What does mild solution means, it's interpretation?

PS:But I would like to have a very useful reference about this notion.

Thank you!