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.
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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
where \(K:\mathbb {R}^+ \rightarrow \mathbb {C}^{m\times m}\) is a measurable matrix-valued function satisfying the following standing assumptions
and
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
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
where \(f:\mathbb {R}^+\times X\rightarrow \mathbb {C}^m\) is a continuous function satisfying the global Lipschitz condition
where \(\gamma \ge 0\). Equations (1.1) and (1.2) can be written equivalently as functional equations in X of the form
and
respectively, where \(L:X\rightarrow \mathbb {C}^m\) is a linear functional defined by
As shown in [7, p. 493], L satisfies
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
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
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
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]),
Moreover, if a function \(\zeta :\mathbb {R}^+\rightarrow X\) satisfies the relation
then \(\zeta \) is a continuous on \(\mathbb {R}^+\), the function \(z:\mathbb {R}\rightarrow \mathbb {C}^m\) defined by
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
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])
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])
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
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 \(E^s\),
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
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
Let
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
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
satisfying the initial condition \(y(s)=\alpha (s)\) for \(s\in \mathbb {R}^+\). We have that \(\sigma (A)=\sigma (A^\#)\), and if we define
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])
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
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
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
From this and the duality (2.9), it follows that \(G_c^\#=G_c\). Indeed, from (2.9) and (2.11), we obtain
This, together with (2.10) and (2.11), yields
Moreover, we have that
The spectral projection \(\varPi ^c:X\rightarrow E^c\) can be given explicitly by
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
satisfies the ordinary differential equation
called the center equation, where \(H_c\) is the \(d_c\times m\) matrix given by
(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
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
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
It follows by a similiar argument that the limit
for \(t\ge 0\). This, combined with the continuity of \(T^u(-t)\), implies that the limit
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
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
exists in X. This, combined with the continuity of \(T^u(t)\), implies the existence of the limit
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 \(t\ge 0\),
Hence
From (3.2), we find for \(t\ge 0\),
Substracting the last equation from (3.2), we obtain for \(t\ge 0\),
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
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 \(t\ge 0\),
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
where \(\tilde{K}(s):=K(s)e^{-\lambda s}\) for \(s\in \mathbb {R}^+\). This follows from the relation
which, together with \({\text {Re}}\lambda =0\), implies that
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
This implies that
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
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
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
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
From this and (3.10), we obtain
This implies that the scalar function \(h:[0,\infty )\rightarrow \mathbb {C}\) defined by
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
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
From this, letting \(n\rightarrow \infty \) in (3.12) and using the Lebesgue dominated convergence theorem, we conclude that
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
This, combined with (3.9) and (3.13), yields
the last inequality being a consequence of (3.8). From this, we conclude that
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
so that
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
Define
Clearly, S is a nonempty and closed subset of B. For \(\xi \in S\) and \(t\ge 0\), define
where
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 \(t\ge 0\),
This, combined with the definition of S, implies that
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 \(\xi \in S\),
Hence
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 \(t\ge 0\),
Hence
for \(t\ge 0\) and \(n\in \mathbb {N}\). From this, we find for \(t\ge 0\),
Hence
where
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
with the property \(z_t=\zeta (t)\) in X for all \(t\ge 0\). Hence
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
for \(t\ge 0\). Thus, x is a solution of Eq. (1.2) with the desired property (1.4).
\(\square \)
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Dedicated to the memory of Professor István Győri.
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Davor Dragičević was supported in part by Croatian Science Foundation under the Project IP-2019-04-1239 and by the University of Rijeka under the Projects uniri-prirod-18-9 and uniri-prprirod-19-16. Mihály Pituk was supported in part by the Hungarian National Research, Development and Innovation Office Grant No. K139346.
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Dragičević, D., Pituk, M. Hyers–Ulam stability of integral equations with infinite delay. Aequat. Math. 98, 1265–1280 (2024). https://doi.org/10.1007/s00010-024-01080-2
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DOI: https://doi.org/10.1007/s00010-024-01080-2