We initially consider systems governed by a partial differential equation of first order. This type of equations describe interesting phenomena such as transport models [46] or population models with age distribution [47]. Here we consider a simplified system with delay described by the equation

$$\begin{aligned}& \frac{\partial w(t, \xi)}{\partial t} + \frac{\partial w(t, \xi )}{\partial\xi} + \alpha w(t, \xi) + \int _{-\infty}^{\infty} g(\xi, \eta) w(t - r, \eta)\,d\eta= \tilde{f}(t, \xi), \end{aligned}$$

(4.1)

$$\begin{aligned}& w(\theta , \xi) = \varphi (\theta , \xi), \end{aligned}$$

(4.2)

for \(\xi\in \mathbb {R}\), \(t \geq0\), and \(-r \leq \theta \leq0\), where \(\alpha , r > 0\) and *g*, \(\tilde{f}\), *φ* are functions that satisfy appropriate conditions that will be specified later. It is well known [48, Example 4.6.1] that the initial value problem

$$\begin{aligned}& \frac{\partial w(t, \xi)}{\partial t} + \frac{\partial w(t, \xi )}{\partial\xi} + \alpha w(t, \xi) = 0,\quad \xi\in \mathbb {R}, t\geq0, \end{aligned}$$

(4.3)

$$\begin{aligned}& w(0,\xi) = h(\xi),\quad \xi\in \mathbb {R}, \end{aligned}$$

(4.4)

can be modeled as an abstract Cauchy problem in the space \(X = L^{2}(\mathbb {R})\). For this reason, in what follows we will assume that \(h \in X\), \(f : \mathbb {R}\to X\) given by \(f(t) = \tilde{f}(t, \cdot)\) is a bounded continuous function, and that \(\varphi \in C([-r, 0], X)\), where as usual we have identified \(\varphi (\theta )(\xi) = \varphi (\theta ,\xi)\). Let *A* be the operator

$$A z(\xi) = - \frac{d z(\xi)}{d \xi} - \alpha z(\xi) $$

on the domain \(D(A) = H^{1}(\mathbb {R})\). The operator *A* is the infinitesimal generator of a strongly continuous group \(T(t)\) on *X* given by

$$T(t) z (\xi) = e^{-\alpha t} z (\xi-t),\quad t, \xi\in \mathbb {R}. $$

Consequently, using the notation \(x(t) = w(t, \cdot)\), the problem (4.3)-(4.4) is reduced to the abstract Cauchy problem

$$\begin{aligned}& x^{\prime}(t) = A x(t),\\& x(0) = h. \end{aligned}$$

Furthermore, the semigroup \(T(t)\), \(t \geq0\), is uniformly stable, and the operator \(T(t)\) is not compact because \(T(t)\) has bounded inverse \(T(-t)\).

We will assume further that \(g : \mathbb {R}^{2} \to \mathbb {R}\) is continuous and

$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \bigl|g(\xi, \eta )\bigr|^{2}\,d\eta\,d \xi< \infty. $$

### Lemma 4.1

*Under the above conditions*, *the linear operator*
\(N : X \to X \)
*given by*

$$N z (\xi) = \int_{-\infty}^{\infty} g(\xi, \eta) z(\eta)\,d\eta $$

*is compact*.

### Proof

For each \(n \in \mathbb {N}\), we define \(N_{n} : X \to X \) by

$$N_{n} z (\xi) = \int_{- n}^{n} g(\xi, \eta) z(\eta)\,d\eta,\quad -n \leq\xi\leq n, $$

and \(N_{n} z (\xi) = 0\) for \(|\xi| > n\). It follows from [49, Proposition 9.5.2] that \(N_{n}\) is a compact operator. Next we prove that \(N_{n} \to N\) as \(n \to\infty\) for the norm of operators. In fact, for \(z \in X\) we have

$$\begin{aligned} \|N z - N_{n} z\|^{2} =& \int_{-\infty}^{\infty} \bigl|N z(\xi) - N_{n} z(\xi)\bigr|^{2}\,d\xi\\ =& \int_{-\infty}^{-n} \biggl| \int_{-\infty}^{\infty} g(\xi, \eta) z(\eta)\,d\eta\biggr|^{2}\,d\xi+ \int_{-n}^{n} \biggl|\int_{-\infty}^{-n} g(\xi, \eta) z(\eta)\,d\eta \\ &{} + \int_{n}^{\infty} g(\xi, \eta) z(\eta)\,d\eta \biggr|^{2}\,d\xi+ \int_{n}^{\infty} \biggl| \int _{-\infty}^{\infty} g(\xi, \eta) z(\eta)\,d\eta \biggr|^{2}\,d\xi \\ \leq& \int_{-\infty}^{-n} \int_{-\infty}^{\infty} \bigl|g(\xi, \eta )\bigr|^{2}\,d\eta\,d \xi\|z\|^{2} + 2 \int _{-n}^{n} \int_{-\infty}^{-n} \bigl|g(\xi, \eta)\bigr|^{2}\,d\eta\,d \xi\|z\|^{2}\\ &{} + 2 \int_{-n}^{n} \int_{n}^{\infty} \bigl|g(\xi, \eta)\bigr|^{2}\,d\eta\,d \xi\|z\|^{2} + \int _{n}^{\infty} \int_{-\infty}^{\infty} \bigl|g(\xi, \eta)\bigr|^{2}\,d\eta\,d\xi\|z\|^{2}. \end{aligned}$$

Let \(D_{n} = \{(\xi, \eta) : \xi, \eta\in[-n, n] \}\). The above estimate shows that

$$\|N - N_{n} \|^{2} \leq2 \int\int_{\mathbb {R}^{2} \setminus D_{n}} \bigl|g(\xi, \eta)\bigr|^{2}\,d\eta\,d \xi\to0,\quad n \to\infty, $$

by Lebesgue’s dominated convergence theorem. This implies the assertion. □

This result also holds for some functions *g* discontinuous (the interested reader can consult [50, Proposition V.4.1]).

We define \(L : C([-r, 0],X) \to X\) by \(L(\psi) = - N \psi(-r)\). By Lemma 4.1 we see that *L* is a compact linear map. With this construction, the original system (4.1)-(4.2) is represented by the abstract system

$$\begin{aligned}& x^{\prime}(t) = A x(t) + L(x_{t}) + f(t),\\& x_{0} = \varphi , \end{aligned}$$

and as a consequence of Theorem 3.1 we get the following property.

### Corollary 4.1

*Under the preceding conditions*, *if*
*f*
*is almost periodic and* (4.1)-(4.2) *has a bounded mild solution on*
\(\mathbb {R}^{+}\), *then it has an almost periodic solution*.

As a second application, we apply our results to study the existence of almost periodic solutions of the wave equation with delay. To establish a general result, we consider an abstract version of the wave equation.

Let *H* be a real Hilbert space. Following [44, Example 2.16] we consider the abstract wave equation

$$\begin{aligned}& x^{\prime\prime}(t) + \beta x^{\prime}(t) + A x(t) = 0, \end{aligned}$$

(4.5)

$$\begin{aligned}& x(0) = x^{0},\qquad x^{\prime}(0) = x^{1}, \end{aligned}$$

(4.6)

where \(x(t) \in H\), \(\beta> 0\) and *A* is a positive self adjoint operator with domain \(D(A)\) such that

$$\langle A x, x \rangle\geq k \|x\|^{2},\quad \forall x \in D(A), $$

for some constant \(k > 0\).

Introducing the Hilbert space \(\mathcal {H}= D(A^{1/2}) \times H\) with inner product

$$\left\langle \begin{bmatrix} x^{1} \\ y^{1} \end{bmatrix}, \begin{bmatrix} x^{2} \\ y^{2} \end{bmatrix} \right\rangle= \bigl\langle A^{1/2} x^{1}, A^{1/2} x^{2} \bigr\rangle + \bigl\langle y^{1}, y^{2} \bigr\rangle , $$

we can write (4.5) as the first order system

$$w^{\prime}(t) = \mathcal {A}w(t), $$

where \(w(t) = \bigl[ {\scriptsize\begin{matrix} x(t) \cr x^{\prime}(t) \end{matrix}}\bigr] \in \mathcal {H}\) and \(\mathcal {A}= \bigl[ {\scriptsize\begin{matrix}0 & I \cr - A & - \beta \end{matrix}} \bigr]\) is defined on \(D(\mathcal {A}) = D(A) \times D(A^{1/2})\). Then \(\mathcal {A}\) generates a strongly continuous group \(G(t)\) on ℋ. Consequently, \(G(t)\) is not compact.

We assume also that \(\mu\in\rho(- A)\) and \(\|R(\mu, -A)\| \leq \frac{C}{|\mu|} \) for \(\operatorname{Re}(\mu) > 0\). Hence, for every \(\lambda\in \mathbb {C}\) with \(\operatorname{Re}(\lambda ) > 0\) we have \(\lambda\in\rho(\mathcal {A})\),

$$(\lambda I - \mathcal {A})^{-1} = \begin{bmatrix} (\lambda+ \beta) R(\lambda(\lambda+ \beta), -A) & R(\lambda(\lambda+ \beta), -A) \\ -A R(\lambda(\lambda+ \beta), -A) & \lambda R(\lambda(\lambda+ \beta), -A) \end{bmatrix}, $$

and \(\|(\lambda I - \mathcal {A})^{-1} \| \leq C\), where \(C > 0\) is a generic constant. Thus, under the above conditions, it follows from [31, Theorem V.1.11] that \(G(t)\) is uniformly stable.

We consider now the inhomogeneous wave equation

$$ x^{\prime\prime}(t) + \beta x^{\prime}(t) + A x(t) = f(t). $$

(4.7)

Using the previous transformation, we can reduce (4.7) to the first order equation

$$w^{\prime}(t) = \mathcal {A}w(t) + \widetilde{f}(t), $$

where \(\widetilde{f}(t) = \bigl[{\scriptsize\begin{matrix} 0 \cr f(t)\end{matrix}} \bigr]\).

Finally we consider the wave equation with delay

$$ x^{\prime\prime}(t) + \beta x^{\prime}(t) + A x(t) = f(t) + L_{1}(x_{t}), $$

(4.8)

where \(L_{1} : C([-r, 0], H) \to H\) is a bounded linear operator. Using the previous transformation, we can reduce (4.8) to the first order equation with delay

$$ w^{\prime}(t) = \mathcal {A}w(t) + L(w_{t}) + \widetilde{f}(t), $$

(4.9)

where \(L: C([-r, 0], \mathcal {H}) \to \mathcal {H}\) is given by

$$L \left ( \begin{bmatrix} \varphi \\ \psi \end{bmatrix} \right ) = \begin{bmatrix} 0 \\ L_{1}(\varphi ) \end{bmatrix}. $$

The following property is a direct consequence of Theorem 3.1.

### Corollary 4.2

*Under the above conditions*, *let*
\(f : \mathbb {R}\to H\)
*be an almost periodic function*. *Assume that*
\(L_{1}\)
*is a compact operator*, *and that* (4.8) *has a bounded mild solution on*
\(\mathbb {R}^{+}\). *Then* (4.8) *has an almost periodic mild solution*.

### Proof

It is clear that *L* is a compact operator so that \(G(t) L\) is also compact for all \(t > 0\). □

To complete this application, next we will present a pair of concrete examples of compact linear operators \(L_{1} : C([-r, 0], H) \to H\).

(i) Let \(K : H \to H\) be a compact linear operator. We fix \(\theta _{0} \in[-r, 0]\) and define \(L_{1} (\varphi ) = K \varphi (\theta _{0})\) for \(\varphi \in C([-r, 0], H)\). It is immediate that \(L_{1}\) is a compact linear operator.

(ii) Let *H* be a separable Hilbert space with orthonormal basis \(\{ z_{n} : n \in \mathbb {N}\}\). Let \(\eta_{n} : [-r, 0] \to \mathbb {C}\) for \(n \in \mathbb {N}\) be a function with bounded variation \(V[\eta_{n}]\). We assume that \(\sum_{n = 1}^{\infty} V[\eta_{n}]^{2} < \infty\). Let \(L_{1}\) given by

$$L_{1}(\varphi ) = \sum_{n = 1}^{\infty} \int_{-r}^{0}\,d_{\theta } \eta _{n}( \theta ) \bigl\langle \varphi (\theta ), z_{n} \bigr\rangle z_{n}, \quad \varphi \in C\bigl([-r, 0], H\bigr). $$

It is not difficult to verify that \(L_{1}\) is a compact linear operator.

Finally, we present an application in which the original semigroup is not uniformly stable. A large number of concrete systems can be formulated in the following abstract form. Let *H* be a separable Hilbert space with orthonormal basis \(\{ z_{n} : n \in \mathbb {N}\}\). Let \((\lambda_{n})_{n}\) be a sequence of complex numbers such that \(\sup_{n \in \mathbb {N}} \operatorname{Re}(\lambda_{n}) < \infty\). We consider the operator \(A : D(A) \subseteq H \to H\) given by

$$A z = \sum_{n = 1}^{\infty} \lambda_{n} \langle z, z_{n} \rangle z_{n},\quad z \in D(A), $$

where \(D(A) = \{z \in H : \sum_{n = 1}^{\infty} |\lambda_{n}|^{2} |\langle z, z_{n} \rangle|^{2} < \infty\}\). It is well known that *A* generates a strongly continuous semigroup \(T(t)\) given by

$$T(t) z = \sum_{n = 1}^{\infty} e^{\lambda_{n} t} \langle z, z_{n} \rangle z_{n},\quad z \in H, $$

which generally is not compact and is not uniformly stable. Let \(\beta > 0\). We decompose \(\mathbb {N}= I \cup J\), where \(I = \{n \in \mathbb {N}: \operatorname{Re}(\lambda _{n}) > - \beta\}\) and \(J = \mathbb {N}\setminus I\). We assume that \(\lambda_{n} \to0\) as \(n \to\infty\), \(n \in I\). We define the operator \(K : H \to H\) by \(K z = \sum_{n \in I} \lambda_{n} \langle z, z_{n} \rangle z_{n}\). It is easy to verify that *K* is a compact linear operator. Moreover, the semigroup \(S(t)\) generated by \(A -K\) is given by

$$S(t) z = \sum_{n \in J} e^{\lambda_{n} t} \langle z, z_{n} \rangle z_{n},\quad z \in H, $$

and \(\|S(t) z\| \leq e^{- \beta t} \|z\|\) for \(t \geq0\), which shows that \(S(t)\) is uniformly stable. Hence, we can apply Corollary 3.1 to conclude that under the preceding conditions if the operator \(T(t) L\) is compact for \(t > 0\), \(f : \mathbb {R}\to H\) is an almost periodic function, and (1.1) has a bounded mild solution on \(\mathbb {R}^{+}\), then it has an almost periodic mild solution.