One-sided derivative at the boundary of ODE

Question

Suppose we are solving the Sturm-Liouville problem $$y'' + \lambda y = 0$$ on some interval $x \in [a,b]$. Now, since $a$ and $b$ are the boundaries of the interval, we cannot define the usual derivatives $y'$ and $y''$ there, only the one-sided derivatives. Therefore, I believe that the ODE should be restricted to the open interval $(a,b)$ and the solution should be extended to the boundary points via continuity. However, I have never seen this explicitly stated. Is my reasoning correct?

Answer 1

It is common with equations of this type to study the equation in $L^2(0,1)$. For example, consider $Lf=-f''$ on the domain $\mathcal{D}(L)$ consisting of all twice absolutely continuous functions $f : (0,1)\rightarrow\mathbb{C}$ such that $f,Lf$ are square integrable on $(0,1)$. You have to be a bit careful in talking about the functions as having pointwise limits, but it makes sense because there can be at most one continuous function in the equivalence class for $f$, $f'$. Because $f'' \in L^2$, then the following has limits as $a\downarrow 0$ and $b\uparrow 1$: $$ \int_{a}^{b}f''(t)dt $$ So there is a natural and unique way to define $f'(a),f'(b)$ as limits of $f'$ as you near the endpoints. Likewise, there is a unique way to define $f(a)$, $f(b)$ as limits of $f$ as you near the endpoints.

The existence of endpoint values for $f,f'$ are deduced from knowing properties of $f\in\mathcal{D}(L)$ in $(a,b)$. So your suspicions about how the property should be formulated coincide with the conventions adopted to deal with ODEs in the setting of $L^2$. The endpoint values are defined through the set of linear functionals $\Phi$ that are continuous on the graph of $L : \mathcal{D}(L)\subset L^2\rightarrow L^2$, such that $\Phi$ vanishes on the compactly supported $C^{\infty}$ functions on $(a,b)$. In this case, the boundary functionals form a $4$ dimensional subspace spanned by the limits of $f,f'$ at $a,b$, and these limits can be shown to exist knowing only a formulation in $(a,b)$.

Answer 2

If $f:[a,b] \to \mathbb R$ is a function, then $f$ is called differentiable in $a$ if

$ \lim_{x \to a+0} \frac{f(x)-f(a)}{x-a}$ exists

and $f$ is called differentiable in $b$ if

$ \lim_{x \to b-0} \frac{f(x)-f(b)}{x-b}$ exists.