The delay equations are an important class of differential equations. In fact, these equations are studied by both mathematicians and engineers because most systems are affected by a delay.
Imagine a basketball game transmission from Italy to the United States. Sometimes a lag between the voice and the television images occurs. Indeed, the voice arrives before the image, so there is a delay for the image to coincide with the voice.
In the following, we will see that the translation operator plays a key role in the delay equation existence of solutions.
Functions of bounded variations
Let $X$ be a Banach space, and denote by $\mathcal{L}(X)$ the Banach algebra of all linear bounded operators from $X$ to $X,$ endowed with uniform topology.
A function $\mu:[-r,0]\to \mathcal{L}(X)$ is called a bounded variation functions if \begin{align*}|\mu|([-r,0])=\sup\left\{\sum_{i=1}^N |\mu(\tau_i)-\mu(\tau_{i-1})|, 0=\tau_0>\tau_1\cdots>\tau_N=-r\right\}\end{align*} is finite. We not $|\mu|$ is a Borel measure on $[-r,0]$.
For a continuous function $f:[-r,0]\to X,$ we define the following Riemann-Stieltjes integral \begin{align*}L f=\int^0_{-r}d\mu(\theta)f(\theta).\end{align*} We note that \begin{align*}\|Lf\|&\le \int^0_{-r}|f(\theta)|d|\mu|( \theta )\cr & \le \gamma \|f\|_\infty, \end{align*} where $\gamma:= |\mu|([-r,0]) $ and $\|f\|_\infty:=\sup_{\theta\in [-r,0]}|f(\theta)|$.
The heat delay equations
Here we consider a classical example of delay equations. We select $X=L^2([0,L])$ and define the operator \begin{align*} Au=u”,\quad D(A)=\{u\in W^{2,2}([0,L]):u(0)=u(L)=0\}.\end{align*} We recall from semigroup theory, Lumer-Phillips theorem, that $(A,D(A))$ generates a strongly continuous semigroup $(T(t))_{t\ge 0}$ on $X$. That is \begin{align*} & T(t)\in\mathcal{L}(X),\quad T(0)=Id,\cr & T(t+s)=T(t)T(s),\qquad \forall t,s\ge 0,\cr & \lim_{t\to 0}\|T(t)f-f\|=0,\qquad \forall f\in X.\end{align*} Moreover, $f\in D(A)$ if and only if the following limit \begin{align*}Af=\lim_{t\to 0} \frac{T(t)f-f}{t}\end{align*}exists.
Consider the delayed heat equation \begin{align*}\tag{Eq} \begin{cases}\dot{u}(t)=Au(t)+\displaystyle\int^0_{-r}d\mu(\theta)u(t+\theta),& t\ge 0,\cr u(0)=h,\cr u(t)=f(t),& -r\le t\le 0.\end{cases}\end{align*} Here $h\in X$ and $f\in L^2([-r,0],X)$.
