Delay equations in Banach spaces

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 $f\in L^2([-r,0],X)$.

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