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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 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 conditions of functional-differential problem are local or nonlocal?

-In which situation we impose local or nonlocal conditions?

Thank you!

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  • $\begingroup$ It appears that in Byszewski's paper you mention the adjective nonlocal refers to an initial value problem, not to a boundary value problem. $\endgroup$ – user539887 Jan 11 at 9:28
  • $\begingroup$ I quote Byszewski: The functional-differential nonlocal problem, studied in this paper is of the form: ... $\endgroup$ – Motaka Jan 11 at 9:42
  • $\begingroup$ In your question you ask about boundary value problems, whereas in the quoted paper by Byszewski the word boundary never appears. $\endgroup$ – user539887 Jan 11 at 10:10
  • $\begingroup$ I edited my post, I thought that the term local, nonlocal is related to the conditions at the boundary! $\endgroup$ – Motaka Jan 11 at 11:05
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This is not an answer but rather an extended comment.

The derivative in its ordinary sense is local: for any neighborhood of $x$, if $f \equiv g$ on that neighborhood and $f^{(n)}(x)$ exists then $g^{(n)}(x)$ exists and $f^{(n)}(x) = g^{(n)}(x)$. So, equations of the form $$ x^{(n)}(t) = F(t, x(t), x'(t), \dots, x^{(n-1)}(t)) $$ or $$ \frac{\partial u}{\partial t}(t, x) = \frac{\partial^2 u}{\partial x^2}(t,x) + F(t, x, u(t, x)) $$ are referred to as local.

Generally, nonlocal denotes that in an equation in question there is something that does not belong to the above category. For instance, we can replace in an ODE the derivative by a derivative of fractional order: the latter does not have the locality property as described in the first paragraph.

Equations of the form $$ x'(t) = F(t, x(t), x(a(t))), $$ where $a$ is a given function, are, to the best of my knowledge, (almost) never called nonlocal: the standard name appears to be (functional) (ordinary) differential equations with deviating argument (retarded or delayed if $a(t) < t$, and advanced if $a(t) > t$).

The boundary conditions considered in Byszewski's paper look like "usual" multipoint boundary conditions.

It seems that the author just chose to call problems considered by him functional-differential nonlocal problems. I think it would be proper to ask him directly: ludwik.byszewski@pk.edu.pl.

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