In some problems involving ordinary differential equations, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.

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:

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

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 $\:\left \| . \right \|; x_0\in E, c_k\neq 0 \:\:(k=1,...,p)\: p \in \mathbb N$.

And here, in the classical Robin problem: $$u''(t) + f(t,u(t),u'(t)) = 0$$

With

localconditions: $u(0)= 0$ and $u'(1) = 0.$

Or

With

nonlocalconditions: $u(0)= 0$ and $u(1) = u(\eta)\;\:\eta\in(0,1)$

My question is:

-When we say that The conditions of functional-differential problem are **local** or **nonlocal**?

-In which situation we impose **local** or **nonlocal** conditions?

*Thank you!*

nonlocalrefers to aninitialvalue problem, not to a boundary value problem. $\endgroup$ – user539887 Jan 11 at 9:28The functional-differential nonlocal problem, studied in this paper is of the form: ...$\endgroup$ – Motaka Jan 11 at 9:42boundaryvalue problems, whereas in the quoted paper by Byszewski the wordboundarynever appears. $\endgroup$ – user539887 Jan 11 at 10:10