Hyers–Ulam stability of integral equations with infinite delay

Abstract

Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay.

1 Introduction and the main results

Let $\mathbb {R}=(-\infty ,\infty )$, $\mathbb {R}^+=[0,\infty )$ and $\mathbb {R}^-=(-\infty ,0]$. As usual, $\mathbb {C}$ denotes the set of complex numbers. Given a positive integer m, $\mathbb {C}^m$ and $\mathbb {C}^{m\times m}$ denote the m-dimensional space of complex column vectors and the space of $m\times m$ matrices with complex entries, respectively. The $l_2$-norm on $\mathbb {C}^m$ and the associated induced norm on $\mathbb {C}^{m\times m}$ will be denoted by the same symbol $|\cdot |$.

Roughly speaking, a differential equation or difference equation is Hyers–Ulam stable if in a neighborhood of an approximate solution we can always find a true solution. The study of Hyers–Ulam stability has received much attention in the literature (see [4] and the references therein). In this paper, we will study this problem for a general class of integral equations with infinite delay.

Consider the linear homogeneous integral equation with infinite delay

$$\begin{aligned} x(t)=\int _{-\infty }^t K(t-s)x(s)\, ds, \qquad t\ge 0, \end{aligned}$$

(1.1)

where $K:\mathbb {R}^+ \rightarrow \mathbb {C}^{m\times m}$ is a measurable matrix-valued function satisfying the following standing assumptions

$$\begin{aligned} \Vert K\Vert _{1, \rho }:=\int _0^\infty |K(t)|e^{\rho t}\,dt <\infty \end{aligned}$$

and

$$\begin{aligned} \Vert K\Vert _{\infty , \rho }:={\text {ess}} \, {\text {sup}}\{|K(t)|e^{\rho t} : t\ge 0\}<\infty , \end{aligned}$$

where $\rho >0$ is fixed. The phase space for Eq. (1.1) is $X:=L_{1,\rho }(\mathbb {R}^-,\mathbb {C}^m)$, the space of equivalent classes of measurable functions $\phi :\mathbb {R}^-\rightarrow \mathbb {C}^m$ such that

$$\begin{aligned} \Vert \phi \Vert _{X}:=\int _{-\infty }^0|\phi (\theta )|e^{\rho \theta }\, d\theta <\infty . \end{aligned}$$

Clearly, $(X,\Vert \cdot \Vert _X)$ is a Banach space.

For any function $x:\mathbb {R}\rightarrow \mathbb {C}^m$ and $t\in \mathbb {R}$, we define the t-segment $x_t:\mathbb {R}^-\rightarrow \mathbb {C}^m$ by $x_t(\theta )=x(t+\theta )$ for $\theta \in \mathbb {R}^-$. We shall also consider nonlinear perturbations of Eq. (1.1) of the form

$$\begin{aligned} x(t)=\int _{-\infty }^t K(t-s)x(s)\, ds +f(t,x_t), \qquad t\ge 0, \end{aligned}$$

(1.2)

where $f:\mathbb {R}^+\times X\rightarrow \mathbb {C}^m$ is a continuous function satisfying the global Lipschitz condition

$$\begin{aligned} |f(t,\phi )-f(t,\psi )|\le \gamma \Vert \phi -\psi \Vert _X \qquad \text{ for } \text{ all } \ t\ge 0\ \text{ and } \ \phi ,\psi \in X, \end{aligned}$$

(1.3)

where $\gamma \ge 0$. Equations (1.1) and (1.2) can be written equivalently as functional equations in X of the form

$$\begin{aligned} x(t)=L(x_t),\qquad t\ge 0, \end{aligned}$$

and

$$\begin{aligned} x(t)=L(x_t)+f(t,x_t),\qquad t\ge 0, \end{aligned}$$

respectively, where $L:X\rightarrow \mathbb {C}^m$ is a linear functional defined by

$$\begin{aligned} L(\phi )=\int _{-\infty }^0 K(-\theta )\phi (\theta )\, d\theta ,\qquad \phi \in X. \end{aligned}$$

As shown in [7, p. 493], L satisfies

$$\begin{aligned} |L(\phi )|\le \Vert K\Vert _{\infty , \rho }\Vert \phi \Vert _{X},\qquad \phi \in X. \end{aligned}$$

Thus, L is a bounded linear functional on X.

According to [7, Lemma 1 and Proposition 3], under the above hypotheses, for every $\phi \in X$, there exists a unique (measurable) function $x:\mathbb {R}\rightarrow \mathbb {C}^m$ satisfying the following three conditions:

• (i) $x(\theta )=\phi (\theta )$ for all $\theta <0$ so that $x_0=\phi$ in X;

• (ii) x is continuous on $[0,\infty )$;

• (iii) $x(t)=L(x_t)+f(t,x_t)$ for all $t\ge 0$, i.e., x satisfies (1.2).

As usual, the continuity of x on $[0,\infty )$ means that x is continuous at each $t\in (0,\infty )$ and x is continuous from the right at $t=0$. We shall call x the (unique global) solution of Eq. (1.2) with initial value $\phi \in X$ at zero. Throughout the paper, by a solution of Eq. (1.2), we mean a solution with initial value $\phi$ at zero for some $\phi \in X$. It should be noted that the above requirements for the solutions are slightly different from those in [7]. Namely, instead of the condition $x(0)=\phi (0)$ imposed in [7], we require that x is continuous from the right at $t=0$, which is motivated by a later use of the variation of constants formula in the phase space. In view of [7, Lemma 1], this requirement can certainly be guaranteed. We shall also consider pseudosolutions (approximate solutions) of Eq. (1.2) which are defined in a similar manner. More precisely, if $\delta >0$, then by a $\delta$-pseudosolution of Eq. (1.2), we mean a function $y:\mathbb {R}\rightarrow \mathbb {C}^m$ such that

• (i)$^{*}$ $y_0\in X$;

• (ii)$^{*}$ y is continuous on $[0,\infty )$;

• (iii)$^{*}$ $|y(t)-L(y_t)-f(t,y_t)|\le \delta$ for all $t\ge 0$.

Equation (1.2) is called Hyers–Ulam stable if there exists $\kappa >0$ such that if y is a $\delta$-pseudosolution of Eq. (1.2) for some $\delta >0$, then Eq. (1.2) has a solution x satisfying

$$\begin{aligned} \sup _{t\ge 0}\Vert x_t-y_t\Vert _{X} \le \kappa \delta . \end{aligned}$$

(1.4)

The corresponding definitions of a solution, pseudosolution and Hyers–Ulam stability for the unperturbed equation (1.1) are obtained as special cases when f is identically zero.

In this paper, we will establish two types of stability criteria. First, we will show that the Hyers–Ulam stability of the linear homogeneous equation (1.1) can be completely characterized in terms of its characteristic values. Second, we will prove that the Hyers–Ulam stability of the linear equation (1.1) is preserved for the nonlinear equation (1.2) whenever $\gamma$ is sufficiently small. Recall that the characteristic values of Eq. (1.1) are the complex roots of the characteristic equation

$$\begin{aligned} \det \varDelta (z)=0,\qquad \varDelta (z):=I_m-\int _{0}^\infty K(s)e^{-z s}\,ds,\quad {\text {Re}}z>-\rho , \end{aligned}$$

(1.5)

where $I_m$ is the $m\times m$ identity matrix. Our main results are following two theorems.

Theorem 1.1

Equation (1.1) is Hyers–Ulam stable if and only if it has no characteristic value with zero real part.

Theorem 1.2

Suppose that Eq. (1.1) is Hyers–Ulam stable and $f:\mathbb {R}^+\times X\rightarrow \mathbb {C}^m$ is a continuous fuction satisfying condition (1.3). If the Lipschitz constant $\gamma \ge 0$ is sufficiently small, then the perturbed equation (1.2) is also Hyers–Ulam stable.

Remark 1.3

The smallness condition on $\gamma$ in Theorem 1.2 can be given explicitly by condition (3.14) below in terms of the dichotomy constants of the unperturbed equation (1.1) which will be specified in Sec. 2.

Remark 1.4

Theorems 1.1 and 1.2 have their analogues for delay differential equations (see [1, Theorem 2.3] and [2, Theorem 2.3 ($\gamma =0$)]).

Remark 1.5

In contrast with Theorems 1.1 and 1.2, most of the results on Hyers–Ulam stability of integral equations with delay available in the literature are restricted to scalar equations with finite delay (see, e.g., [3, Theorem 2.1], [4, Chap. 5, Sec. 9], [8] and the references therein).

The proofs of Theorems 1.1 and 1.2 will be given in Sec. 3. They will be based on the decomposition theory of linear integral equations with infinite delay, a dynamical system approach, which has been developed recently in a series of papers by Matsunaga, Murakami, Nagabuchi and Van Minh (see [5,6,7]).

2 Preliminaries

In this section, we introduce the notations and recall some known results which will be needed in the proofs of our main theorems. For the proofs and more details, see [5,6,7].

For any $t\ge 0$ and $\phi \in X$, define $T(t)\phi \in X$ by $T(t)\phi =x_t(\phi )$, where $x(\phi )$ is the unique solution of Eq. (1.1) with initial value $\phi$ at zero. It is known (see [7, Sec. 3]) that $\{T(t)\}_{t\ge 0}$ is a strongly continuous semigroup of bounded linear operators on X, called the solution semigroup of Eq. (1.1). We will also consider the nonhomogeneous linear equation

$$\begin{aligned} z(t)=L(z_t)+p(t),\qquad t\ge 0, \end{aligned}$$

(2.1)

where $p:\mathbb {R}^+\rightarrow \mathbb {C}^m$ is continuous. For each $n\in \mathbb {N}$, we choose a nonnegative function $\varGamma ^n:\mathbb {R}^-\rightarrow \mathbb {R}^+$ of compact support with ${\text {supp}}\varGamma ^n\subset [-1/n,0]$ such that $\int _{-\infty }^0\varGamma ^n(\theta )\,d\theta =1$. Clearly, $\varGamma ^n x\in X$ and $\Vert \varGamma ^n x\Vert _X\le |x|$ for $x\in \mathbb {C}^m$. If $p:\mathbb {R}^+\rightarrow \mathbb {C}^m$ is a continuous function, then every solution z of (2.1) satisfies the following representation formula in X, called the variation of constants formula in the phase space (see [7, Theorem 4]),

$$\begin{aligned} z_t=T(t)z_0+\lim _{n\rightarrow \infty }\int _0^t T(t-s)(\varGamma ^n p(s))\, ds\qquad \text{ in } X\ \text{ for } \text{ all } \ t\ge 0. \end{aligned}$$

(2.2)

Moreover, if a function $\zeta :\mathbb {R}^+\rightarrow X$ satisfies the relation

$$\begin{aligned} \zeta (t)=T(t)\zeta (0)+\lim _{n\rightarrow \infty }\int _0^t T(t-s)(\varGamma ^n p(s))\,ds\qquad \text{ for } \text{ all } \ t\ge 0, \end{aligned}$$

(2.3)

then $\zeta$ is a continuous on $\mathbb {R}^+$, the function $z:\mathbb {R}\rightarrow \mathbb {C}^m$ defined by

(2.4)

is a solution of (2.1) with initial value $\zeta (0)$ at zero, and $z_t=\zeta (t)$ in X for all $t\ge 0$. Set

$$\begin{aligned} \tilde{X}:=\bigl \{\, \tilde{\phi }\in X: \tilde{\phi }\in AC_\text {loc}(\mathbb R^-,\mathbb C^m), \bigl (\tilde{\phi }\bigr )' \in X \ { \text{ and } }\ \tilde{\phi }(0)=L(\tilde{\phi })\,\bigr \}, \end{aligned}$$

where the symbol $AC_\textrm{loc}(\mathbb {R}^-,\mathbb {C}^m)$ denotes the set of $\mathbb {C}^m$-valued functions which are locally absolutely continuous on $\mathbb {R}^-$. The infinitesimal generator $(A,\mathcal {D}(A))$ of the solutions semigroup $\{T(t)\}_{t\ge 0}$ is given by (see [7, Proposition 4])

$$\begin{aligned} \mathcal D(A)= & {} {} \{\, \phi \in X: \phi (\theta )=\tilde{\phi }(\theta )\ \text{ for } \text{ a.e. } \ \theta \in \mathbb R^- \ \text{ for } \text{ some } \ \tilde{\phi }\in \tilde{X}\,\},\\ {}{} & {} {} A\phi = \bigl (\tilde{\phi }\bigr )'. \end{aligned}$$

Let $\sigma (A)$ and $P\sigma (A)$ denote the spectrum and the point spectrum of A, respectively, and define $\mathbb {C}_{-\rho }:=\{\,\lambda \in \mathbb {C}\mid {\text {Re}}\lambda >-\rho \,\}$. Between the spectra of A and the characteristic values of Eq. (1.1), we have the following relation (see [7, Proposition 5])

$$\begin{aligned} \sigma (A)\cap \mathbb {C}_{-\rho }=P\sigma (A)\cap \mathbb {C}_{-\rho }=\{\, \lambda \in \mathbb {C}_{-\rho }: \ \det \Delta (\lambda )=0\,\}. \end{aligned}$$

For ${\text {ess}}(A)$, the essential spectrum of A, we have that $\sup _{\lambda \in {\text {ess}}(A)}{\text {Re}}\lambda \le -\rho$ (see [7, Corollary 2]). Set

$$\begin{aligned} \varSigma ^u:&=\{\lambda \in \sigma (A): \ {\text {Re}} \lambda >0\},\\ \varSigma ^c:&=\{ \lambda \in \sigma (A): \ {\text {Re}} \lambda =0\},\\\varSigma ^s:&=\sigma (A)\setminus (\varSigma ^c\cup \varSigma ^u). \end{aligned}$$

Then $(\varSigma ^c\cup \varSigma ^u)\,\cap \,{\text {ess}}(A)=\emptyset$, $\varSigma ^c$ and $\varSigma ^u$ are finite sets, and we have a decomposition of the phase space X into the direct sum of closed subspaces $E^u$, $E^c$ and $E^s$,

$$\begin{aligned} X=E^u\oplus E^c\oplus E^s, \end{aligned}$$

(2.5)

with the following properties (see [6, Theorem 3]):

• (a) $\dim (E^u \oplus E^c)<\infty$;

• (b) $T(t)E^u \subset E^u$, $T(t)E^c \subset E^c$, and $T(t)E^s\subset E^s$ for $t\in \mathbb {R}^+$;

• (c) $\sigma (A|_{E^u})=\varSigma ^u$, $\sigma (A|_{E^c})=\varSigma ^c$ and $\sigma (A|_{E^s \cap \mathcal {D}(A)})=\varSigma ^s$;

• (d) $T^u(t):=T(t)|_{E^u}$ and $T^c(t):=T(t)|_{E^c}$ can be extended to all $t\in \mathbb {R}$ as groups of bounded linear operators on $E^u$ and $E^c$, respectively;

• (e) there exist positive constants a and $\epsilon$, $a>\epsilon$, and a constant $C\ge 1$ such that

  $$\begin{aligned} \Vert T^s(t)\Vert _{\mathcal {L}(E^s)}&\le \quad Ce^{-at}\quad \text{ for } \ t\in \mathbb {R}^+;\\ \Vert T^u(t)\Vert _{\mathcal {L}(E^u)}&\le \quad Ce^{a t}\quad \text{ for } \ t\in \mathbb {R}^-;\\ \Vert T^c(t)\Vert _{\mathcal {L}(E^c)}&\le \quad Ce^{\epsilon |t|}\quad \text{ for } \ t\in \mathbb {R}, \end{aligned}$$

where $\mathcal {L}(X)$ denotes the Banach space of bounded linear operators on X equipped with the operator norm $\Vert \cdot \Vert _{\mathcal {L}(X)}$. We will use the notation $\varPi ^u$ for the projection of X onto $E^u$ along $E^c\oplus E^s$ associated with the decomposition (2.5) and similarly for $\varPi ^c$ and $\varPi ^s$. In view of the invariance property (b), the projections $\varPi ^u$, $\varPi ^c$ and $\varPi ^s$ commute with the solution operator T(t) for $t\ge 0$. We set

$$\begin{aligned} D:= \max \left\{ \Vert \varPi ^u\Vert _{\mathcal {L}(X)}, \Vert \varPi ^c\Vert _{\mathcal {L}(X)}, \Vert \varPi ^s\Vert _{\mathcal {L}(X)}\right\} . \end{aligned}$$

(2.6)

The $E^c$-component of a solution of the nonhomogeneous equation (2.1) satisfies an ordinary differential equation which can be described using the formal adjoint theory developed in [5]. Let $\mathbb {C}^{m*}$ be the m-dimensional space of complex row vectors with the $l_2$-norm, denoted by the same symbol $|\cdot |$ as the $l_2$-norm on $\mathbb {C}^m$. Thus, $|z^*|=|z|$ and $|z^*z|=|z|^2$ for $z\in \mathbb {C}^m$, where $z^*$ denotes the conjugate transpose of $z\in \mathbb {C}^m$. Define $X^{\#}:=L_{1,\rho }(\mathbb {R}^+,\mathbb {C}^{m*})$, the Banach space of equivalent classes of measurable functions $\alpha :\mathbb {R}^+\rightarrow \mathbb {C}^{m*}$ such that

$$\begin{aligned} \Vert \alpha \Vert _{X^\#}:=\int _{0}^\infty |\alpha (s)|e^{-\rho s}\, ds <\infty . \end{aligned}$$

Let

$$\begin{aligned} \tilde{X}^\#:=\bigl \{\,\tilde{\alpha }\in X^\#: \tilde{\alpha }\in AC_\text {loc}(\mathbb R^+,\mathbb C^{m*}), \bigl (\tilde{\alpha }\bigr )' \in X^\# \ {\hbox { and }} \ \tilde{\alpha }(0)=L^T(\tilde{\alpha })\,\bigr \}, \end{aligned}$$

where $AC_\textrm{loc}(\mathbb {R}^+,\mathbb {C}^{m*})$ denotes the set of $\mathbb {C}^{m*}$-valued functions which are locally absolutely continuous on $\mathbb {R}^+$ and $L^T(\tilde{\alpha })=\int _{0}^{\infty }\tilde{\alpha }(s)K(s)\,ds$. As shown in [5, Sec. 3.2], the pair $(A^\#,\mathcal {D}(A^\#))$ given by

$$\begin{aligned} \mathcal D(A^\#)= & {} \{\,\alpha \in X^\#: \,\alpha (s)=\tilde{\alpha }(s)\ \text{ for } \text{ a.e. }\ s\in \mathbb R^+ \ \text{ for } \text{ some } \ \tilde{\alpha }\in \tilde{X}^\#\,\},\\ A^\#\alpha= & {} -\bigl (\tilde{\alpha }\bigr )', \end{aligned}$$

is the infinitesimal generator of the semigroup $\{T^\#(t)\}_{t\ge 0}$ of bounded linear operators on $X^\#$ defined by $[T^\#(t)\alpha ](s):=y(-t+s)$ for $\alpha \in X^\#$ and $s\in \mathbb {R}^+$, where $y=y(\alpha )$ is the solution of the formal adjoint equation

$$\begin{aligned} y(\theta )=\int _{\theta }^\infty y(s)K(-\theta +s)\,ds, \qquad \theta \le 0, \end{aligned}$$

(2.7)

satisfying the initial condition $y(s)=\alpha (s)$ for $s\in \mathbb {R}^+$. We have that $\sigma (A)=\sigma (A^\#)$, and if we define

$$\begin{aligned} \langle \alpha , \phi \rangle :=\int _{-\infty }^0\int _{\theta }^0\alpha (\zeta -\theta )K(-\theta )\phi (\zeta )\, d\zeta \, d\theta ,\qquad (\alpha ,\phi )\in X^\#\times X, \end{aligned}$$

(2.8)

then this pairing is a bounded bilinear form on $X^\#\times X$ (see [5, pp. 817, 818]). Moreover, we have the duality (see [5, Proposition 3.3])

$$\begin{aligned} \langle \alpha , A\phi \rangle = \langle A^{\#}\alpha , \phi \rangle , \qquad (\alpha ,\phi )\in \mathcal {D}(A^\#)\times \mathcal {D}(A). \end{aligned}$$

(2.9)

Suppose that $\varSigma ^c\ne \emptyset$. Then the center subspace $E^c$ from the decomposition (2.5) coincides with the generalized eigenspace of A associated with the finite, nonempty set of eigenvalues $\varSigma ^c$ (see [5, Sec. 2] for definition). If $(E^c)^\#$ is the generalized eigenspace of $A^\#$ associated with $\varSigma ^c$, then

$$\begin{aligned} d_c:=\dim E^c=\dim (E^c)^\#<\infty . \end{aligned}$$

Let $\varPhi _c=(\phi _1,\dots ,\phi _{d_c})$ and $\varPsi _c= {\text {col}}(\psi _1,\dots ,\psi _{d_c})$ be bases for $E^c$ and $(E^c)^\#$, respectively. In view of the characterization of $\mathcal {D}(A)$ and $\mathcal {D}(A^{\#})$, we may (and do) assume that the elements of the bases $\varPhi _c$ and $\varPsi _c$ belong to $\tilde{X}$ and $\tilde{X}^{\#}$, respectively, so that

$$\begin{aligned} A\varPhi _c=(\varPhi _c)' \qquad \text {and}\qquad A^\#\varPsi _c=-(\varPsi _c)'. \end{aligned}$$

(2.10)

It is known that the matrix $\langle \varPsi _c, \varPhi _c\rangle :=\left( \langle \psi _i, \phi _j\rangle \right) _{i,j=1,\dots d_c}$ is nonsingular and hence we can assume that $\langle \varPsi _c, \varPhi _c\rangle =I_{d_c}$. Since $A(E^c)\subset E^c$ and $A^\#((E^c)^{\#})\subset (E^c)^{\#}$, there exist $d_c\times d_c$ constant matrices $G_c$ and $G_c^\#$ with complex entries such that $\sigma (G_c)=\sigma (G_c^\#)=\varSigma ^c$ and

$$\begin{aligned} A\varPhi _c=\varPhi _c G_c,\qquad A^\#\varPsi _c=G_c^\#\varPsi _c. \end{aligned}$$

(2.11)

From this and the duality (2.9), it follows that $G_c^\#=G_c$. Indeed, from  (2.9) and (2.11), we obtain

$$\begin{aligned} G_c^\#&= G_c^\# \langle \varPsi _c, \varPhi _c\rangle =\langle G_c^\# \varPsi _c, \varPhi _c\rangle =\langle A^\# \varPsi _c, \varPhi _c\rangle \\&= \langle \varPsi _c, A\varPhi _c \rangle = \langle \varPsi _c, \varPhi _c G_c\rangle =\langle \varPsi _c, \varPhi _c\rangle G_c=G_c. \end{aligned}$$

This, together with (2.10) and (2.11), yields

$$\begin{aligned} \varPhi _c(\theta )=\varPhi _c(0)e^{G_c\theta }\quad \text{ for } \ \theta \in \mathbb {R}^-, \qquad \varPsi _c(s)=e^{-G_c s}\varPsi _c(0)\quad \text{ for } s\ \in \mathbb {R}^+. \end{aligned}$$

(2.12)

Moreover, we have that

$$\begin{aligned} T(t)\varPhi _c=\varPhi _c e^{G_c t}\quad \text {and}\quad T^\#(t)\varPsi _c=e^{-G_c t}\varPsi _c\quad \text {for }t\in \mathbb {R}^+. \end{aligned}$$

(2.13)

The spectral projection $\varPi ^c:X\rightarrow E^c$ can be given explicitly by

$$\begin{aligned} \varPi ^c\phi =\varPhi _c \langle \varPsi _c, \phi \rangle , \qquad \phi \in X, \end{aligned}$$

where $\langle \Psi _c, \phi \rangle :={\text {col}}( \langle \psi _1, \phi \rangle , \dots , \langle \psi _{d_c}, \phi \rangle )\in \mathbb {C}^{d_c}$. Finally, if z is a solution of the nonhomogeneous equation (2.1), then its $E^c$-component given by

$$\begin{aligned} \varPi ^c z_t=\varPhi _c u(t),\qquad u(t)= \langle \varPsi _c, z_t\rangle ,\qquad t\ge 0, \end{aligned}$$

satisfies the ordinary differential equation

$$\begin{aligned} u'(t)=G_c u(t)+H_c p(t)\qquad \text {for a.e. }t\ge 0, \end{aligned}$$

(2.14)

called the center equation, where $H_c$ is the $d_c\times m$ matrix given by

$$\begin{aligned} H_c x=\lim _{n\rightarrow \infty }\langle \varPsi _c, \varGamma ^n x \rangle , \qquad x\in \mathbb C^m \end{aligned}$$

(see [7, Theorems 7 and 8]).

3 Proof of the main results

As a preparation for the proofs of Theorems 1.1 and 1.2, we remark that conditions (i)$^{*}$ and (ii)$^{*}$ imply in the same manner as in the proof of [7, Lemma 1] that if $\delta >0$, then for every $\delta$-pseudosolution y of Eq.  (1.2), we have that $y_t\in X$ for $t\ge 0$ and $y_t$ depends continuously on $t\in [0,\infty )$.

Now we are in a position to give a proof of Theorem 1.1.

Proof of Theorem 1.1

Throughout the proof, we will use the notations of Sec. 2. In order to prove the “if” part, suppose that the characteristic equation (1.5) has no root with zero real part. In this case, we have that $E^c=\{0\}$ and hence $\varPi ^c=0$. We will show that if y is a $\delta$-pseudosolution of Eq. (1.1) for some $\delta >0$, then Eq. (1.1) has a solution x such that

$$\begin{aligned} \Vert x_t-y_t\Vert _X\le \frac{2CD}{a}\delta ,\qquad t\ge 0, \end{aligned}$$

(3.1)

where C, a and D have the meaning from condition (e) and (2.6), respectively. Evidently, this implies that Eq. (1.1) is Hyers–Ulam stable with $\kappa =\frac{2CD}{a}$. Given $\delta >0$, let y be an arbitrary $\delta$-pseudosolution of Eq. (1.1). Then the function $p:\mathbb {R}^+\rightarrow \mathbb {C}^m$ defined by $p(t):=L(y_t)-y(t)$ for $t\in \mathbb {R}^+$ satisfies $|p(t)|\le \delta$ for all $t\ge 0$. As noted before, $y_t\in X$ depends continuously on $t\in [0,\infty )$. This, together with the continuity of L, implies that $L(y_t)$ is a continuous function of $t\in [0,\infty )$. From this and the continuity of y (see condition (ii)$^{*}$), we conclude that p is continuous on $[0,\infty )$. Define

$$\begin{aligned} \begin{aligned} \zeta (t):&=\lim _{n\rightarrow \infty }\int _0^t T^s(t-s)\varPi ^s(\varGamma ^n p(s))\, ds\\&\qquad -\lim _{n\rightarrow \infty }\int _t^\infty T^u(t-s)\varPi ^u(\varGamma ^n p(s))\, ds \end{aligned} \end{aligned}$$

(3.2)

for $t\in \mathbb {R}^+$. The existence of the first limit in X follows from the variation of constants formula (2.2) and the continuity of the projection $\varPi ^s$ since

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^t T^s(t-s)\varPi ^s(\varGamma ^n p(s))\, ds=\varPi ^s\biggl (\lim _{n\rightarrow \infty }\int _0^t T(t-s)(\varGamma ^n p(s))\, ds\biggr ). \end{aligned}$$

It follows by a similiar argument that the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^t T^u(t-s)\varPi ^u(\varGamma ^n p(s))\, ds \qquad \text {exists in}~X \end{aligned}$$

(3.3)

for $t\ge 0$. This, combined with the continuity of $T^u(-t)$, implies that the limit

$$\begin{aligned} \begin{aligned}&\lim _{n\rightarrow \infty }\int _0^t T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds\\&\qquad = T^u(-t)\biggl (\lim _{n\rightarrow \infty }\int _0^t T^u(t-s)\varPi ^u(\varGamma ^n p(s))\, ds\biggr ) \end{aligned} \end{aligned}$$

(3.4)

also exists for every $t\ge 0$.

If $0\le t_1<t_2$ and $n\in \mathbb {N}$, then, using conditions (e) and (2.6) and the estimates $\Vert \varGamma ^n x\Vert _X\le |x|$ for $x\in \mathbb {C}^m$ and $|p(t)|\le \delta$ for $t\in \mathbb {R}^+$, we obtain

$$\begin{aligned}&\biggl \Vert \int _{0}^{t_1} T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds -\int _{0}^{t_2} T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds \biggr \Vert _X\\ {}&\quad =\biggl \Vert \int _{t_1}^{t_2}T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds\biggr \Vert _X\\ {}&\quad \le \int _{t_1}^{t_2}\Vert T^u(-s)\Vert _{\mathcal {L}(E^u)}\Vert \varPi ^u\Vert _{\mathcal {L}(X)}\Vert \varGamma ^n p(s)\Vert _X\, ds\\ {}&\quad \le \int _{t_1}^\infty Ce^{-as}D|p(s)|\,ds \le \delta \frac{CD}{a}e^{-at_1}. \end{aligned}$$

Since the last expression tends to zero uniformly in $n\in \mathbb {N}$ as $t_1\rightarrow \infty$, we conclude that the limit $\lim _{t\rightarrow \infty }\int _0^t T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds$ exists uniformly in $n\in \mathbb {N}$. From this and (3.4), it follows by a Moore–Osgood type theorem (see, e.g., the Appendix in [2]) that the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^\infty T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds = \lim _{n\rightarrow \infty }\lim _{t\rightarrow \infty }\int _0^t T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds \end{aligned}$$

exists in X. This, combined with the continuity of $T^u(t)$, implies the existence of the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^\infty T^u(t-s)\varPi ^u(\varGamma ^n p(s))\, ds = T^u(t)\biggl (\lim _{n\rightarrow \infty }\int _0^\infty T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds\biggr ) \end{aligned}$$

in X. From this and (3.3), we conclude that the second limit in (3.2) also exists. Thus, $\zeta$ is well-defined. From (3.2), it follows by similar estimates as before that for all $t\ge 0$,

$$\begin{aligned} \Vert \zeta (t)\Vert _X&\le \limsup _{n\rightarrow \infty }\int _0^t \Vert T^s(t-s)\Vert _{\mathcal {L}(E^s)}\Vert \varPi ^s\Vert _{\mathcal {L}(X)}\Vert \varGamma ^n p(s)\Vert _X\, ds\\ {}&\quad +\limsup _{n\rightarrow \infty } \int _t^\infty \Vert T^u(t-s)\Vert _{\mathcal {L}(E^u)}\Vert \varPi ^u\Vert _{\mathcal {L}(X)}\Vert \varGamma ^n p(s)\Vert _X\, ds\\ {}&\le CD\delta \biggl (\int _0^t e^{-a(t-s)}\,ds+\int _t^\infty e^{a(t-s)}\,ds\biggr ). \end{aligned}$$

Hence

$$\begin{aligned} \Vert \zeta (t)\Vert _X\le \frac{2CD}{a}\delta ,\qquad t\ge 0. \end{aligned}$$

(3.5)

From (3.2), we find for $t\ge 0$,

$$\begin{aligned} T^u(t)\zeta (0)&=- T^u(t)\biggl (\lim _{n\rightarrow \infty }\int _0^\infty T^u(-s)\varPi ^u(\varGamma ^n p(s))\, ds\biggr )\\&=-\lim _{n\rightarrow \infty }\int _0^\infty T^u(t-s)\varPi ^u(\varGamma ^n p(s))\, ds. \end{aligned}$$

Substracting the last equation from (3.2), we obtain for $t\ge 0$,

$$\begin{aligned} \zeta (t)-T^u(t)\zeta (0)&=\lim _{n\rightarrow \infty }\int _0^t T^s(t-s)\varPi ^s(\varGamma ^n p(s))\, ds\\&\quad +\lim _{n\rightarrow \infty }\int _0^t T^u(t-s)\varPi ^u(\varGamma ^n p(s))\, ds. \end{aligned}$$

Taking into account that $\varPi ^s+\varPi ^u=I_X$, the identity on X, from this we conclude that $\zeta$ satisfies the limit relation (2.3). As noted in Sec. 2, this implies that the function z defined by (2.4) is a solution of Eq. (2.1) such that $z_t=\zeta (t)$ in X for all $t\ge 0$. From this and (3.5), we conclude that the function $x:=y+z$ satisfies

$$\begin{aligned} \Vert x_t-y_t\Vert _X=\Vert z_t\Vert _X=\Vert \zeta (t)\Vert _X\le \frac{2CD}{a}\delta \end{aligned}$$

for $t\ge 0$. Thus, (3.1) holds. Moreover, from Eq. (2.1), taking into account that $p(t)=L(y_t)-y(t)$ for $t\ge 0$, we obtain for $t\ge 0$,

$$\begin{aligned} z(t)=L(z_t)+p(t)=L(z_t)+L(y_t)-y(t)=L(z_t+y_t)-y(t). \end{aligned}$$

This implies that $x=y+z$ is a solution of Eq. (1.1) with the desired property (3.1). Since $\delta >0$ was arbitrary, this implies that Eq. (1.1) is Hyers–Ulam stable. It remains to prove the “only if” part. Assume that Eq. (1.1) is Hyers–Ulam stable. We need to show that the characteristic equation (1.5) has no root on the imaginary axis. Suppose, for the sake of contradiction, that Eq. (1.5) has a root $\lambda \in \mathbb {C}$ with ${\text {Re}}\lambda =0$. Without loss of generality, we may (and do) assume that $\lambda =0$. Otherwise, letting $\tilde{x}(t):=e^{-\lambda t}x(t)$ for $t\in \mathbb {R}^+$, we transform the solutions ($\delta$-pseudosolutions) of Eq. (1.1) into the solutions ($\delta$-pseudosolutions) of the equation

$$\begin{aligned} \tilde{x}(t)=\int _{-\infty }^t \tilde{K}(t-s)\tilde{x}(s)\, ds, \qquad t\ge 0, \end{aligned}$$

(3.6)

where $\tilde{K}(s):=K(s)e^{-\lambda s}$ for $s\in \mathbb {R}^+$. This follows from the relation

$$\begin{aligned} x(t)-\int _{-\infty }^t K(t-s)x(s)\, ds=e^{\lambda t}\biggl (\tilde{x}(t)-\int _{-\infty }^t\tilde{K}(t-s)\tilde{x}(s)\, ds\biggr ), \end{aligned}$$

which, together with ${\text {Re}}\lambda =0$, implies that

$$\begin{aligned} \biggl |\,x(t)-\int _{-\infty }^t K(t-s)x(s)\, ds\,\biggr |=\biggl |\,\tilde{x}(t)-\int _{-\infty }^t\tilde{K}(t-s)\tilde{x}(s)\, ds\,\biggr | \end{aligned}$$

for $t\ge 0$. Thus, Eq. (1.1) is Hyers–Ulam stable if and only if so is Eq. (3.6). Moreover, it is easy to verify that $\lambda$ is a characteristic value of Eq. (1.1) if and only if 0 is a characteristic value of Eq. (3.6). Therefore, from now on we will assume that 0 is a characteristic value of Eq. (1.1), i.e. $0\in \varSigma ^c$ with $\varSigma ^c$ as in Sec. 2. Since $0\in \varSigma _c=\sigma (G_c)$, there exists a nonzero vector $v\in \mathbb {C}^{d_c*}$ such that $vG_c=0$. From $vG_c=0$, it follows by the definition of the matrix exponential that $ve^{G_c s}=v$ for all $s\in \mathbb {R}$. This, together with the second relation in (2.12), yields

$$\begin{aligned} v\varPsi _c(s)=v\varPsi _c(0),\qquad s\in \mathbb {R}^+. \end{aligned}$$

(3.7)

This implies that

$$\begin{aligned} v\varPsi _c(0)\ne 0. \end{aligned}$$

(3.8)

Indeed, if $v\varPsi _c(0)=0$, then (3.7) implies that $v\varPsi _c$ is identically zero on $\mathbb {R}^+$. On the other hand, since v is a nonzero vector, $v\varPsi _c$ is a nontrivial linear combination of the basis functions $\psi _1,\dots ,\psi _{d_c}$ of $(E^c)^\#$ which cannot be identically zero on $\mathbb {R}^+$. This contradiction proves that  (3.8) hods. Define

$$\begin{aligned} p(t)=w:=(v\varPsi _c(0))^{*},\qquad t\in \mathbb {R}^+. \end{aligned}$$

(3.9)

Let y be the unique solution of the nonhomogeneous equation (2.1) with p as in (3.9) and initial value $y_0=0$. From (3.8) and (3.9), we obtain that $|p(t)|\le \delta$ for $t\in \mathbb {R}^+$, where $\delta :=|(v\varPsi _c(0))^{*}|=|v\varPsi _c(0)|>0$. Thus, y is a $\delta$-pseudosolution of Eq. (1.1). The Hyers–Ulam stability of Eq. (1.1) implies the existence of a solution x of (1.1) satisfying (1.4). Define $z(t):=y(t)-x(t)$ for $t\in \mathbb {R}$. By virtue of (1.4), we have that

$$\begin{aligned} \Vert z_t\Vert _X\le \kappa \delta ,\qquad t\ge 0. \end{aligned}$$

(3.10)

Since x is a solution of (1.1), y and z satisfy the same nonhomogeneous equation (2.1) with p given by (3.9). As noted in Sec. 2, the function u defined by

$$\begin{aligned} u(t)=\langle \varPsi _c,z_t\rangle ,\qquad t\ge 0, \end{aligned}$$

is a solution of the ordinary differential equation (2.14). The boundedness of the bilinear form given by (2.8) implies the existence of a constant $M>0$ such that

$$\begin{aligned} |\langle \varPsi _c,\phi \rangle |\le M\Vert \phi \Vert _X,\qquad \phi \in X. \end{aligned}$$

From this and (3.10), we obtain

$$\begin{aligned} |u(t)|=|\langle \varPsi _c,z_t\rangle |\le M\Vert z_t\Vert _X\le M\kappa \delta ,\qquad t\ge 0. \end{aligned}$$

(3.11)

This implies that the scalar function $h:[0,\infty )\rightarrow \mathbb {C}$ defined by

$$\begin{aligned} h(t)=vu(t),\qquad t\ge 0, \end{aligned}$$

is bounded on $[0,\infty )$. Multiplying Eq. (2.14) by v from left, taking into account that $vG_c=0$ and using (3.7) and (3.9), we have that

$$\begin{aligned} \begin{aligned} h'(t)&=\lim _{n\rightarrow \infty }\langle v\varPsi _c, \varGamma ^n p(t) \rangle = \lim _{n\rightarrow \infty }\langle v\varPsi _c(0), \varGamma ^n w \rangle \\ {}&=v\varPsi _c(0)\lim _{n\rightarrow \infty }\int _{-\infty }^0 K(-\theta )\int _{\theta }^0 \varGamma ^n(\zeta )\, d\zeta \, d\theta \,w \end{aligned} \end{aligned}$$

(3.12)

for a.e. $t\ge 0$. For every $\theta <0$, we have that $\theta <-1/n$ for all large n. Therefore, if $\theta <0$, then, for all large n, we have that ${\text {supp}}\varGamma ^n\subset [-1/n,0]\subset [\theta ,0]$ and hence $0\le \int _{\theta }^0\varGamma ^n(\zeta )\,d\zeta =\int _{-\infty }^0\varGamma ^n(\zeta )\,d\zeta =1$. This implies

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\theta }^0 \varGamma ^n(\zeta )\, d\zeta =1,\qquad \theta <0. \end{aligned}$$

From this, letting $n\rightarrow \infty$ in (3.12) and using the Lebesgue dominated convergence theorem, we conclude that

$$\begin{aligned} h'(t)=v\varPsi _c(0)\int _{-\infty }^0 K(-\theta )\,d\theta \,w =v\varPsi _c(0)\int _0^\infty K(s)\,ds\,w \end{aligned}$$

(3.13)

for a.e. $t\ge 0$. As noted before, $vG_c=0$ implies that $ve^{-G_c t}=v$ for all $t\in \mathbb {R}$. This, together with the second relation in (2.13), implies that $v\varPsi _c(0)$ is a constant solution of the formal adjoint equation (2.7). Hence

$$\begin{aligned} v\varPsi _c(0)=v\varPsi _c(0)\int _0^\infty K(s)\,ds. \end{aligned}$$

This, combined with (3.9) and (3.13), yields

$$\begin{aligned} h'(t)=v\varPsi _c(0)w=(v\varPsi _c(0))(v\Psi _c(0))^{*}=|v\varPsi _c(0)|^2>0\qquad \text{ for } \text{ a.e. } \ t\ge 0, \end{aligned}$$

the last inequality being a consequence of (3.8). From this, we conclude that

$$\begin{aligned} h(t)=h(0)+\int _0^t h'(s)\,ds=h(0)+|v\varPsi _c(0)|^2 t\longrightarrow \infty ,\qquad t\rightarrow \infty , \end{aligned}$$

contradicting the boundedness of h on $[0,\infty )$. This contradiction shows that Eq. (1.1) cannot have a characteristic value with zero real part. $\square$

Now we give a proof of Theorem 1.2.

Proof of Theorem 1.2

According to Theorem 1.1, the Hyers–Ulam stability of Eq. (1.1) implies that the characteristic equation (1.5) has no root with zero real part. Therefore, $E^c=\emptyset$ and $\varPi ^c=0$. Suppose that

$$\begin{aligned} \gamma <\frac{a}{2CD} \end{aligned}$$

(3.14)

so that

$$\begin{aligned} \kappa :=\frac{2CD}{a}\biggl (1-\frac{2CD}{a}\gamma \biggr )^{-1}>0, \end{aligned}$$

(3.15)

where the positive constants a, C and D have the meaning from conditions (e) and (2.6), respectively. We will show that Eq. (1.2) is Hyers–Ulam stable with $\kappa$ as in (3.15). Given $\delta >0$, let y be an arbitrary $\delta$-pseudosolution of Eq. (1.2). Let B denote the Banach space of those functions $\xi :\mathbb {R}^+\rightarrow X$ which are continuous and bounded, equipped with the norm

$$\begin{aligned} \Vert \xi \Vert _B:=\sup _{t\ge 0}\Vert \xi (t)\Vert _X, \qquad \xi \in B. \end{aligned}$$

Define

$$\begin{aligned} S:=\{\,\xi \in B:\Vert \xi \Vert _B\le \kappa \delta \,\}. \end{aligned}$$

Clearly, S is a nonempty and closed subset of B. For $\xi \in S$ and $t\ge 0$, define

$$\begin{aligned} (\mathcal {F}\xi )(t)&= \lim _{n\rightarrow \infty }\int _0^t T^s(t-s)\varPi ^s(\varGamma ^n p_\xi (s))\,ds\\&\quad -\lim _{n\rightarrow \infty }\int _t^\infty T^u(t-s)\varPi ^u(\varGamma ^n p_\xi (s))\,ds, \end{aligned}$$

where

$$\begin{aligned} p_\xi (t)=f(t,y_t+\xi (t))+L(y_t)-y(t) \end{aligned}$$

(3.16)

for $\xi \in S$ and $t\ge 0$. Both y(t) and $y_t$ depend continuously on $t\in [0,\infty )$. This, together with the continuity of f, $\xi$ and L, implies that, for every $\xi \in S$, the function $p_\xi :[0,\infty )\rightarrow \mathbb {C}^m$ is continuous. Moreover, by virtue of (1.3), we have for $\xi \in S$ and $t\ge 0$,

$$\begin{aligned} |p_\xi (t)|&=|f(t,y_t+\xi (t))-f(t,y_t)+f(t,y_t)+L(t)y_t-y(t)|\\&\le |f(t,y_t+\xi (t))-f(t,y_t)|+|f(t,y_t)+L(t)y_t-y(t)| \\&\le \gamma \Vert \xi (t)\Vert _X+|f(t,y_t)+L(t)y_t-y(t)|\le \gamma \Vert \xi \Vert _B+\delta . \end{aligned}$$

This, combined with the definition of S, implies that

$$\begin{aligned} |p_\xi (t)|\le (1+\gamma \kappa )\delta \qquad \text{ for } \ \xi \in S\ \text{ and } \ t\ge 0. \end{aligned}$$

(3.17)

As shown in the first part of the proof of Theorem 1.1, this implies that, for every $\xi \in S$, the function $\zeta =\mathcal {F}\xi$ satisfies the limit relation (2.3) in X with $p=p_\xi$. Consequently, $\mathcal {F}\xi :\mathbb {R}^+\rightarrow X$ is a continuous function for every $\xi \in S$. From (3.17), it follows by a similar argument as in the proof of the inequality (3.5) that for every $\xi \in S$,

$$\begin{aligned} \Vert (\mathcal {F}\xi )(t)\Vert _X\le \frac{2CD}{a}(1+\gamma \kappa )\delta ,\qquad t\ge 0. \end{aligned}$$

Hence

$$\begin{aligned} \Vert \mathcal {F}\xi \Vert _B\le \frac{2CD}{a}(1+\gamma \kappa )\delta =\kappa \delta ,\qquad \xi \in S, \end{aligned}$$

the last equality being a consequence of (3.15). Thus, $\mathcal {F}(S)\subset S$.

Let $\xi _1,\xi _2\in S$. By virtue of (1.3) and (3.16), we have for $t\ge 0$,

$$\begin{aligned} |p_{\xi _1}(t)-p_{\xi _2}(t)|&=|f(t,y_t+\xi _1(t)-f(t,y_t+\xi _2(t)|\\&\le \gamma \Vert \xi _1(t)-\xi _2(t)\Vert _X\le \gamma \Vert \xi _1-\xi _2\Vert _B. \end{aligned}$$

Hence

$$\begin{aligned} \Vert \varGamma ^n(p_{\xi _1}(t)-p_{\xi _2}(t))\Vert _X\le |p_{\xi _1}(t)-p_{\xi _2}(t)| \le \gamma \Vert \xi _1-\xi _2\Vert _B \end{aligned}$$

for $t\ge 0$ and $n\in \mathbb {N}$. From this, we find for $t\ge 0$,

$$\begin{aligned}&\Vert (\mathcal {F}\xi _1)(t)-(\mathcal {F}\xi _2)(t)\Vert _X\\ {}&\quad \le \limsup _{n\rightarrow \infty }\int _0^t \Vert T^s(t-s)\Vert _{\mathcal {L}(E^s)}\Vert \Vert \varPi ^s\Vert _{\mathcal {L}(X)} \Vert \varGamma ^n (p_{\xi _1}(s)-p_{\xi _2}(s))\Vert _X\,ds\\ {}&\qquad + \limsup _{n\rightarrow \infty }\int _t^\infty \Vert T^u(t-s)\Vert _{\mathcal {L}(E^u)}\Vert \Vert \varPi ^u\Vert _{\mathcal {L}(X)} \Vert \varGamma ^n (p_{\xi _1}(s)-p_{\xi _2}(s))\Vert _X\,ds\\ {}&\quad \le \gamma \Vert \xi _1-\xi _2\Vert _B CD \biggl (\int _0^t e^{-a(t-s)}\,ds+\int _t^\infty e^{a(t-s)}\,ds\biggr )\\ {}&\quad \le \gamma \frac{2CD}{a}\Vert \xi _1-\xi _2\Vert _B. \end{aligned}$$

Hence

$$\begin{aligned} \Vert \mathcal {F}\xi _1-\mathcal {F}\xi _2\Vert _B\le q\Vert \xi _1-\xi _2\Vert _B\qquad \text {whenever } \xi _1, \xi _2\in S, \end{aligned}$$

where

$$\begin{aligned} q:=\gamma \frac{2CD}{a}<1 \end{aligned}$$

by (3.14). Thus, $\mathcal {F}:S\rightarrow S$ is a contraction. Let $\zeta$ be the unique fixed point of $\mathcal {F}$ in S. As noted before, $\zeta =\mathcal {F}\zeta$ satisfies the limit relation (2.3) in X with $p=p_\zeta$. This implies that the function $z:\mathbb {R}\rightarrow \mathbb {C}^m$ defined by (2.4) is a solution of the nonhomogeneous equation

$$\begin{aligned} z(t)=L(z_t)+p_\zeta (t)=L(z_t)+f(t,y_t+\zeta (t))+L(y_t)-y(t),\qquad t\ge 0, \end{aligned}$$

with the property $z_t=\zeta (t)$ in X for all $t\ge 0$. Hence

$$\begin{aligned} z(t)=L(z_t)+f(t,y_t+z_t)+L(y_t)-y(t)=L(y_t+z_t)+f(t,y_t+z_t)-y(t) \end{aligned}$$

for $t\ge 0$. From this, we conclude that $x:=y+z$ is a solution of Eq. (1.2). Since $\zeta \in S$, we have that

$$\begin{aligned} \Vert x_t-y_t\Vert _X=\Vert z_t\Vert _X=\Vert \zeta (t)\Vert _X\le \Vert \zeta \Vert _B\le \kappa \delta \end{aligned}$$

for $t\ge 0$. Thus, x is a solution of Eq. (1.2) with the desired property (1.4).

$\square$