# The difference between the nonlocal and local conditions problems

In some of Boundary value 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 $$I:=[t_0,t_0+T], t_00$$ 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 local conditions: $$u(0)= 0$$ and $$u'(1) = 0.$$

Or

With nonlocal conditions: $$u(0)= 0$$ and $$u(1) = u(\eta)\;\:\eta\in(0,1)$$

My question is:

-When we say that the boundary conditions are local or nonlocal?

-In which situation we impose local or nonlocal conditions?

Thank you!