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!
Best Answer
For a simple abstract Cauchy problem, for example $u'(t)=f(t)$, $u(t_0)=x$ (1), the classical solution is a continuously differentiable function satisfying the equation (1). Now, if we integrate the equation we get $$u(t)=x+ \int_{t_0}^t f(s)ds.$$ This formula is well defined even if $u$ is not differentiable, which motivate to generalize the classical definition by a "weak" type of solution which only needed to be continuous. This is the same motivation to the solution in distributional sense. In many cases of linear abstract Cauchy problems, the notion of mild solution and the weak solution (in the dual sense, distributions sense) are equivalent.