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_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 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!
