Abstract Degenerate Non-Scalar Volterra Equations on the Line

Abstract

The main aim of this paper is to investigate the existence and uniqueness of solutions for some classes of abstract degenerate non-scalar Volterra equations on the line. In order to achieve our aims, we essentially apply the vector-valued Fourier transform. We use the class of $(A,k,B)$-regularized C-pseudoresolvent families in our analysis as well, and present several useful remarks and illustrative applications of the established results.

1. Introduction and Preliminaries

In ([1], Section 11, Section 12), J. Prüss investigated the solvability of the following problem on the line:

$$\begin{array}{c}\hfill u\left(t\right)=f\left(t\right)+{\int }_0^{+\infty }A\left(s\right)u(t-s)ds,t\in \mathbb{R},\end{array}$$

(1)

where X and Y are complex Banach spaces such that Y is densely embedded into X, $A\in L_{loc}^1([0,+\infty ):L(Y,X))$ and $f\in L_{loc}^1(\mathbb{R}:X).$ The author introduced the notion of a vector-valued homogeneous space $\mathcal{H}\left(X\right)$ and analyzed the well-posedness of the equation

$$\begin{array}{c}\hfill u^{\prime }\left(t\right)={\int }_0^{+\infty }{A}_0\left(s\right)u^{\prime }(t-s)ds+{\int }_0^{+\infty }d{A}_1\left(s\right)u(t-s)ds+f\left(t\right),t\in \mathbb{R},\end{array}$$

(2)

where $A_0\in L^1([0,+\infty ):L(Y,X))$ and $A_1\in BV([0,+\infty ):L(Y,X)).$ The existence of strong solutions to (2) has been proved in the case that the forcing term $f(·)$ belongs to a subspace ${\mathcal{H}}_{\Lambda }^{\sigma }\left(X\right)$ of $\mathcal{H}\left(X\right).$ In particular, for such inhomogeneities $f(·)$, the Fourier–Carleman spectrum $\sigma \left(f\right)\subseteq \Lambda$ is compact (cf. ([1], Subsection 0.5, Subsection 0.6) for more details on the subject), where the real spectrum of (2) is defined through

$$\Lambda :=\left\{\xi \in \mathbb{R}:i\xi -i\xi \widehat{A_0}\left(i\xi \right)-\widehat{d{A}_1}\left(i\xi \right)\in L(Y,X)is not invertible\right\},$$

and ${\mathcal{H}}_{\Lambda }^{\sigma }\left(X\right)$ is a proper subspace of the space consisting of all functions $f(·)$, which admits an extension to an entire function of exponential growth (the strong solution $u(·)$ of (2) also enjoys this feature; see ([1], Theorem 11.1). For some other references concerning the abstract Volterra integro-differential equations on the line, one may refer, e.g., to the paper [2] by V. E. Fedorov and N. M. Skripka and [3] by R. Ponce. The abstract Volterra integro-differential inclusions on the real line with generalized Weyl fractional derivatives have recently been investigated in [4]; cf. also [5,6] for some other important references worth mentioning.

Here, we briefly analyze the following extension of problem (1):

$$\begin{array}{c}\hfill Bu\left(t\right)=Cf\left(t\right)+{\int }_0^{+\infty }A\left(s\right)u(t-s)ds,t\in \mathbb{R},\end{array}$$

(3)

where X and Y are complex Banach spaces, Y is embedded into $X,$ B is a closed linear operator with a domain and range contained in $X,$ $Y\subseteq D\left(B\right)$, $A\in L_{loc}^1([0,+\infty ):L(Y,X))$, $f\in L_{loc}^1(\mathbb{R}:X)$ and the operator $C\in L\left(X\right)$ is injective. For simplicity, we will not consider the perturbations of term $A(·)$ which are bounded in variation, and we will apply the usually considered Fourier transform in place of the Fourier–Carleman transform considered in [1].

The structure of this paper is as follows: We first explain the basic notation and terminology used throughout the paper. After that, we recall the basic definitions and results on $(A,k,B)$-regularized C-pseudoresolvent families (Section 1.1) and generalized Weyl integro-differential operators (Section 1.2).

In Section 2, we investigate the well-posedness of abstract degenerate non-scalar Volterra integral Equation (3) on the line. The first structural result of Section 2 is Proposition 2, where we consider the situation in which ${\left(S\left(t\right)\right)}_{t\ge 0}\subseteq L(X,\left[D\left(B\right)\right])$ is a global $(A,k,B)$-regularized C-pseudoresolvent family. Here, we prove the existence of solutions to the Cauchy problem

$$\begin{array}{c}\hfill Bu\left(t\right)={\int }_{-\infty }^{t}k(t-s)Cf\left(s\right)ds+{\int }_{-\infty }^{t}A(t-s)u\left(s\right)ds,t\in \mathbb{R}.\end{array}$$

In Proposition 3, we clarify a uniqueness result for the above problem, where we assume the existence of a global $(A,k,B)$-regularized C-uniqueness family ${\left(V\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ for (3); cf. [7] for the notion and more details in this direction. In Theorem 1, we apply the Fourier transform in the space of tempered vector-valued distributions in order to analyze the existence and uniqueness of solutions to Problem (3); cf. also Example 1, where we provide some applications to the kernels

$$A\left(t\right):=a_1\left(t\right)A_1+....+a_m\left(t\right)A_m,t\ge 0,$$

where $m\in \mathbb{N}$, $a_j\in L^1\left([0,+\infty )\right)$, $A_j$ is a closed linear operator on X ($1\le j\le m$) and the vector space $Y:=D\left(B\right)\cap D\left(A_1\right)\cap ...\cap D\left(A_m\right)$ is equipped with the graph norm ${\parallel y\parallel }_Y:=\parallel y\parallel +\parallel By\parallel +\parallel A_{1}y\parallel +...+\parallel A_{m}y\parallel ,$ $y\in Y.$ In Theorem 2, we consider the differential and analytical properties of mapping

$$\xi \mapsto {\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\in L(X,Y),\xi \in \mathbb{R},$$

which is incredibly important in our analysis. After that, we are ready to apply the operator-valued version of Mikhlin’s theorem ([8], Proposition 8.2.3) in the analysis of the existence of a solution $u(·)$ of (3) which satisfies $u\in L^1(\mathbb{R}:Y)$ and $Bu\in L^1(\mathbb{R}:X)$; cf. Theorem 3 for more details. The existence and uniqueness of almost-periodic-type solutions to (3) are briefly considered in Section Almost-Periodic-Type Solutions to (3).

For convenience, we will work in the setting of complex Banach spaces. We will reconsider the results established by J. Prüss in ([1], Section 11, Section 12) for a degenerate abstract Cauchy problem on the line (3) somewhere else. We will not consider here the abstract semilinear Volterra integral equations of non-scalar type on the line as well.

Before explaining the notation and terminology used throughout the paper, we feel it is our duty to emphasize that we have not been able to find any practical applications of the established results in physical or engineering models and also that our theoretical findings are not accompanied here by any numerical illustrations or concrete examples. Although some notations probably could be refined, we have done our best to increase the clarity and readability of this paper as well as the exposition of the material.

• Notation and preliminaries. If $\zeta >0$, then we set $g_{\zeta }\left(t\right):=t^{\zeta -1}/\Gamma \left(\zeta \right),$ $t>0,$ where $\Gamma (·)$ denotes the Euler Gamma function and $g_0\left(t\right):=$ the Dirac delta distribution. If $z\in \mathbb{C}$ and $r>0,$ then we set $L(z,r):=\{w\in \mathbb{C}:|z-w|<r\}.$

Unless specified otherwise, we assume henceforward that $(X,\parallel ·\parallel )$ and $(Y,\parallel ·{\parallel }_Y)$ are complex Banach spaces such that Y is embedded into X as well as that $A\in L_{loc}^1([0,\tau ):L(Y,X)),$ where $L(Y,X)$ denotes the space consisting of all continuous linear mappings from Y into X and $L\left(X\right)\equiv L(X,X).$ We assume that the operator $C\in L\left(X\right)$ is injective and B is a closed linear operator in X such that $Y\subseteq D\left(B\right),$ by $\left[D\right(B\left)\right].$ We denote the Banach space $(D\left(B\right),\parallel ·{\parallel }_B),$ where the graph norm ${\parallel ·\parallel }_B$ is defined by ${\parallel x\parallel }_B:=\parallel x\parallel +\parallel Bx\parallel ,$ $x\in D\left(B\right).$ The symbol $\mathrm{I}$ stands for the identity operator on $X.$ For more details about the integration of functions with values in Banach spaces, we refer the reader to [7,8]. We will use the same notion and notation as in [7]. Finally, let us recall that the Fourier transform on the real line and its inverse transform are defined by

$$\left(\mathcal{F}f\right)\left(\xi \right):={\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt \mathrm{and} \left({\mathcal{F}}^{-1}f\right)\left(t\right):={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{+\infty }{e}^{i\xi t}f\left(\xi \right)d\xi (t,\xi \in \mathbb{R}),$$

respectively; see [8,9,10,11,12] and the references quoted therein.

1.1. $(A,k,B)$-Regularized C-Pseudoresolvent Families

We need the following notion (cf. ([7], Definition 2.9.2)):

Definition 1.

Let $\tau \in (0,\infty ]$, $k\in C\left(\right[0,\tau \left)\right)$, $k\ne 0$ and $A\in L_{loc}^1([0,\tau ):L(Y,X))$. A family ${\left(S\left(t\right)\right)}_{t\in [0,\tau )}$ in $L(X,[D\left(B\right)\left]\right)$ is called an $(A,k,B)$-regularized C-pseudoresolvent family if the following holds:

(S1) 

The mappings $t\mapsto S\left(t\right)x$, $t\in [0,\tau )$ and $t\mapsto BS\left(t\right)x$, $t\in [0,\tau )$ are continuous in X for every fixed $x\in X$, $BS\left(0\right)=k\left(0\right)C$ and $S\left(t\right)C=CS\left(t\right)$, $t\in [0,\tau )$.

(S2) 

Put $U\left(t\right)x:={\int }_0^{t}S\left(s\right)xds$, $x\in X$, $t\in [0,\tau )$. Then (S2) means $U\left(t\right)Y\subseteq Y$, $U{\left(t\right)}_{\mid Y}\in L\left(Y\right)$, $t\in [0,\tau )$ and ${\left(U{\left(t\right)}_{\mid Y}\right)}_{t\in [0,\tau )}$ is locally Lipschitz continuous in $L\left(Y\right)$.

(S3) 

The resolvent equations

$$BS\left(t\right)y=k\left(t\right)Cy+{\int }_0^{t}A(t-s)dU\left(s\right)y,t\in [0,\tau ),y\in Y,$$

(4)

$$BS\left(t\right)y=k\left(t\right)Cy+{\int }_0^{t}S(t-s)A\left(s\right)yds,t\in [0,\tau ),y\in Y,$$

(5)

hold, and (4), resp. (5), is called the first resolvent equation, resp. the second resolvent equation.

Let us recall that (S3) can be equivalently written as

$$\begin{array}{cc}\hfill \left(\mathrm{S}3\right)\prime & BU\left(t\right)y=\Theta \left(t\right)Cy+{\int }_0^{t}A(t-s)U\left(s\right)yds,t\in [0,\tau ),y\in Y,\hfill \\ \hfill & BU\left(t\right)y=\Theta \left(t\right)Cy+{\int }_0^{t}U(t-s)A\left(s\right)yds,t\in [0,\tau ),y\in Y,\hfill \end{array}$$

where $\Theta \left(t\right):={\int }_0^{t}k\left(s\right)ds,$ $t\in [0,\tau );$ cf. also ([1], p. 153).

By continuity, we mean continuity in $X.$ We need to recall the notion from ([7], Definition 2.9.3):

Definition 2.

Let $\tau \in (0,\infty ]$, $k\in C\left(\right[0,\tau \left)\right)$, $k\ne 0$ and $A\in L_{loc}^1([0,\tau ):L(Y,X))$. A strongly continuous operator family ${\left(V\left(t\right)\right)}_{t\in [0,\tau )}\subseteq L\left(X\right)$ is said to be an $(A,k,B)$-regularized C-uniqueness family if

$$V\left(t\right)By=k\left(t\right)Cy+{\int }_0^{t}V(t-s)A\left(s\right)yds,t\in [0,\tau ),y\in Y.$$

For further information, the reader may consult [7] and the references quoted therein.

1.2. Generalized Weyl Fractional Derivatives

For Weyl fractional calculus, we refer to the research monographs [13] by K. S. Miller and B. Ross and [14] by S. G. Samko, A. A. Kilbas and O. I. Marichev; cf. also the research articles quoted in [4]. In ([15], Definition 3), we recently introduced the notion of a generalized Weyl $(\alpha ,a)$-fractional derivative $D_W^{\alpha ,a}u$ of a locally integrable function $u:\mathbb{R}\to X,$ where X is a complex Banach space. A special case of $D_W^{\alpha ,a}u$ is the usually considered Weyl fractional derivative $D_W^{\alpha }u:$

Definition 3.

Suppose that $a\in L_{loc}^1\left([0,\infty )\right)$, $u:\mathbb{R}\to X$ is a locally integrable function, $\alpha >0$ and $m=\lceil \alpha \rceil .$ The generalized Weyl fractional derivative $D_W^{\alpha ,a}u$ of function $u(·)$ is well defined if the mapping $t\mapsto {\int }_{-\infty }^{t}a(t-s)u\left(s\right)ds,$ $t\in \mathbb{R}$ is well defined and m-times continuously differentiable by

$$\left[D_W^{\alpha ,a}u\right]\left(t\right):={\displaystyle \frac{d^m}{d{t}^m}}{\int }_{-\infty }^{t}a(t-s)u\left(s\right)ds,t\in \mathbb{R}.$$

The function $t\mapsto I_{W,a}\left(t\right):={\int }_{-\infty }^{t}a(t-s)u\left(s\right)ds,t\in \mathbb{R}$ is said to be a generalized Weyl a-integral of function $u(·).$ If $a\left(t\right)=g_{\zeta }\left(t\right)$ for some $\zeta \in (0,1),$ then the class of functions for which the above integral absolutely converges was first considered by M. J. Lighthill in [16].

We need the following auxiliary results from [4]:

Lemma 1.

(i) Suppose that $a,b\in L_{loc}^1\left([0,+\infty )\right)$, $u\in L_{loc}^1(\mathbb{R}:X)$ and $t\in \mathbb{R}.$ If

$$\begin{array}{c}\hfill {\int }_{-t}^{+\infty }|\left(a{\ast }_{0}b)(s-t)\right|·\parallel u(-s)\parallel ds<+\infty ,\end{array}$$

(6)

 

then the term $\left(I_{W,a}{I}_{W,b}u\right)\left(t\right)$ is well defined, the term $\left(I_{W,a{\ast }_{0}b}u\right)\left(t\right)$ is well defined and we have $\left(I_{W,a}{I}_{W,b}u\right)\left(t\right)=\left(I_{W,a{\ast }_{0}b}u\right)\left(t\right).$

(ii) 

Suppose that $a\in L_{loc}^1\left([0,+\infty )\right)$, Z is a complex Banach space, the operator family ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(Z,X)$ is strongly continuous, for each $x\in Z$ we have ${\int }_0^1\parallel R\left(t\right)x\parallel dt<+\infty ,$ $u\in L_{loc}^1(\mathbb{R}:Z)$ and $t\in \mathbb{R}$. If

$$\begin{array}{c}\hfill {\int }_{-t}^{+\infty }∥\left(a{\ast }_{0}R\right)(s-t)u(-s)∥ds<+\infty ,\end{array}$$

then the term $\left(I_{W,a}{I}_{W,R}u\right)\left(t\right)$ is well defined, the term $\left(I_{W,a{\ast }_{0}R}u\right)\left(t\right)$ is well defined and we have $\left(I_{W,a}{I}_{W,R}u\right)\left(t\right)=\left(I_{W,a{\ast }_{0}R}u\right)\left(t\right).$

We can similarly prove the next proposition:

Proposition 1.

Suppose that $A\in L_{loc}^1([0,+\infty ):L(Y,X))$, Z is a complex Banach space, the operator family ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(Z,Y)$ is strongly continuous, ${\int }_0^1{\parallel R\left(t\right)\parallel }_{L(Z,Y)}dt<+\infty ,$ $u\in L_{loc}^1(\mathbb{R}:Z)$ and $t\in \mathbb{R}$. If

$$\begin{array}{c}\hfill {\int }_{-t}^{+\infty }∥\left(A{\ast }_{0}R\right)(s-t)u(-s)∥ds<+\infty ,\end{array}$$

then the term $\left(I_{W,A}{I}_{W,R}u\right)\left(t\right)$ is well defined, the term $\left(I_{W,A{\ast }_{0}R}u\right)\left(t\right)$ is well defined and we have $\left(I_{W,A}{I}_{W,R}u\right)\left(t\right)=\left(I_{W,A{\ast }_{0}R}u\right)\left(t\right).$

2. Well-Posedness of Abstract Degenerate Non-Scalar Volterra Integral Equations on the Line

In the following proposition, all considered integrals are taken with respect to the topology of space X (cf. also ([7], Proposition 11.4)):

Proposition 2.

Assume that $k\in C\left(\right[0,\infty \left)\right)$, $k\ne 0$, $A\in L_{loc}^1([0,\infty ):L(Y,X)),$ $f\in L_{loc}^1(\mathbb{R}:Y)$ and ${\left(S\left(t\right)\right)}_{t\ge 0}\subseteq L(X,\left[D\left(B\right)\right])$ is a global $(A,k,B)$-regularized C-pseudoresolvent family. Suppose that the functions

$$\begin{array}{c}\hfill u\left(t\right):={\int }_{-\infty }^{t}S(t-s)f\left(s\right)ds,t\in \mathbb{R}andv\left(t\right):={\int }_{-\infty }^{t}U(t-s)f\left(s\right)ds,t\in \mathbb{R}\end{array}$$

are well defined and Lebesgue measurable on the real line, the integrals ${\int }_{-\infty }^{t}BU(t-s)f\left(s\right)ds$ and ${\int }_{-\infty }^t\Theta (t-s)Cf\left(s\right)ds$ are convergent and

$${\int }_0^{+\infty }∥\left(A{\ast }_{0}U\right)\left(s\right)f(t-s)∥ds<+\infty ,t\in \mathbb{R}.$$

Then, we have

$$\begin{array}{c}\hfill Bv\left(t\right)={\int }_{-\infty }^t\Theta (t-s)Cf\left(s\right)ds+{\int }_{-\infty }^{t}A(t-s)v\left(s\right)ds,t\in \mathbb{R}.\end{array}$$

(7)

Furthermore, if the function $u(·)$ is continuous, the functions ${\int }_{-\infty }^{·}k(·-s)Cf\left(s\right)ds$ and ${\int }_{-\infty }^{·}A(·-s)u\left(s\right)ds$ are well defined and continuous on the real line and the integrals defining the functions $v(·)$ and ${\int }_{-\infty }^{·}\Theta (·-s)Cf\left(s\right)ds$ are absolutely convergent, then the function $Bu(·)$ is also continuous on the real line and we have

$$\begin{array}{c}\hfill Bu\left(t\right)={\int }_{-\infty }^{t}k(t-s)Cf\left(s\right)ds+{\int }_{-\infty }^{t}A(t-s)u\left(s\right)ds,t\in \mathbb{R}.\end{array}$$

(8)

Proof. 

It is clear that $v\left(t\right)={\int }_0^{+\infty }U\left(s\right)f(t-s)ds,$ $t\in \mathbb{R}.$ The prescribed assumptions imply, together with ([7], Theorem 1.2.3), Proposition 1 and (S3)’, that

$$\begin{array}{cc}\hfill Bv\left(t\right)& ={\int }_0^{+\infty }BU\left(s\right)f(t-s)ds={\int }_0^{+\infty }\left[\Theta \left(s\right)C+\left(A{\ast }_{0}U\right)\left(s\right)\right]f(t-s)ds\hfill \\ & ={\int }_{-\infty }^t\Theta (t-s)Cf\left(s\right)ds+{\int }_{-\infty }^{t}A(t-s)v\left(s\right)ds,t\in \mathbb{R},\hfill \end{array}$$

as claimed. Suppose now that the requirements in the second part of the proposition hold. To show that $Bu(·)$ is continuous on the real line and (8) holds, observe first that the integral which defines the function $v(·)$ is absolutely continuous and the function $u(·)$ is continuous. It can be simply proved by means of Lemma 1(ii) that

$$\begin{array}{c}\hfill v\left(t\right)={\int }_{-\infty }^t\left(g_1{\ast }_{0}S\right)(t-s)f\left(s\right)ds={\int }_{-\infty }^t{\int }_{-\infty }^{s}S(s-r)f\left(r\right)drds,t\in \mathbb{R},\end{array}$$

so that $v^{\prime }\left(t\right)=u\left(t\right)$, $t\in \mathbb{R}.$ Since the functions ${\int }_{-\infty }^{·}k(·-s)Cf\left(s\right)ds$ and ${\int }_{-\infty }^{·}A(·-s)u\left(s\right)ds$ are continuous on the real line and the integral which defines the function $v(·)={\int }_{-\infty }^{·}\left(g_1{\ast }_{0}A\right)(·-s)u\left(s\right)ds$ is absolutely convergent, we can similarly prove with the help of Lemma 1(ii) that

$${\displaystyle \frac{d}{dt}}{\int }_{-\infty }^t\Theta (t-s)Cf\left(s\right)ds={\int }_{-\infty }^{t}k(t-s)Cf\left(s\right)ds,t\in \mathbb{R}$$

and

$${\displaystyle \frac{d}{dt}}{\int }_{-\infty }^{t}A(t-s)v\left(s\right)ds={\int }_{-\infty }^{t}A(t-s)u\left(s\right)ds,t\in \mathbb{R}.$$

This simply implies the desired assertion by differentiation of (7). □

In the commonly considered case $k(·)\equiv 1,$ the integral ${\int }_{-\infty }^{t}Cf\left(s\right)ds$ must be defined for all $t\in \mathbb{R}$ so that the boundedness of function $Cf(·)$ is not sufficient enough for applications of Proposition 2.

The uniqueness of solutions to Problem (8) with $f\equiv 0$ is examined in the following result:

Proposition 3.

Assume that $k\in C\left(\right[0,\infty \left)\right),$ $k\ne 0$ and ${\left(V\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ is an $(A,k,B)$-regularized C-uniqueness family for (3). Suppose, further, that $u:\mathbb{R}\to X$ is a continuous function, $u\left(t\right)\in Y$ for a.e. $t\in \mathbb{R}$, there exist $b\in L_{loc}^1\left([0,\infty )\right)$ and $m\in \mathbb{N}$ such that $k{\ast }_{0}b=g_m,$ the integral ${\int }_{-\infty }^t{g}_m(t-s)Cu\left(s\right)ds$ is absolutely convergent in X for each $t\in \mathbb{R},$ the integral ${\int }_{-\infty }^t\left(V{\ast }_{0}A\right)(t-s)Cu\left(s\right)ds$ is absolutely convergent in X for each $t\in \mathbb{R}$ and

$$\begin{array}{c}\hfill Bu\left(t\right)={\int }_{-\infty }^{t}A(t-s)u\left(s\right)ds,t\in \mathbb{R}.\end{array}$$

(9)

Then, $u\left(t\right)=0$ for all $t\in \mathbb{R}.$

Proof. 

Since we have assumed that the integral ${\int }_{-\infty }^t\left(V{\ast }_{0}A\right)(t-s)Cu\left(s\right)ds$ is absolutely convergent in X, Proposition 1 and (9) together imply that the integral ${\int }_{-\infty }^{t}V(t-s)Bu\left(s\right)ds$ is convergent in X for each $t\in \mathbb{R}$ and

$$\begin{array}{cc}\hfill {\int }_{-\infty }^{t}V(t-s)Bu\left(s\right)ds& ={\int }_{-\infty }^{t}V(t-s){\int }_{-\infty }^{s}A(s-r)u\left(r\right)drds\hfill \\ & ={\int }_{-\infty }^t\left(V{\ast }_{0}A\right)(t-s)u\left(s\right)ds,t\in \mathbb{R}.\hfill \end{array}$$

Using the functional equality of ${\left(V\left(t\right)\right)}_{t\ge 0},$ the above yields

$$\begin{array}{c}\hfill {\int }_{-\infty }^{t}V(t-s)Bu\left(s\right)ds={\int }_{-\infty }^t\left[V(t-s)Bu\left(s\right)-k(t-s)Cu\left(s\right)\right]ds,t\in \mathbb{R},\end{array}$$

so that ${\int }_{-\infty }^{t}k(t-s)Cu\left(s\right)ds=0,$ $t\in \mathbb{R}.$ Since $k{\ast }_{0}b=g_m$ and the integral ${\int }_{-\infty }^t{g}_m(t-s)Cu\left(s\right)ds$ is absolutely convergent in X for each $t\in \mathbb{R},$ Lemma 1(i) implies ${\int }_{-\infty }^t{g}_m(t-s)Cu\left(s\right)ds=0,$ $t\in \mathbb{R}.$ Keeping in mind that the function $u(·)$ is continuous, we can differentiate m-times the last equality to show that $Cu\left(t\right)=0$, $t\in \mathbb{R}.$ By the injectiveness of regularizing operator C, we finally obtain $u\left(t\right)=0,$ $t\in \mathbb{R}.$ □

Concerning possible applications of Proposition 2 and Proposition 3 with $C\ne \mathrm{I},$ we would like to say that these results can be successfully applied to the $(A,1)$-regularized C-pseudoresolvent families constructed in ([17], Theorem A.12, Corollary A.13) and $(A,1,B)$-regularized C-pseudoresolvent families constructed in ([7], Theorem 2.9.7). In a degenerate setting, possible applications can be given to the abstract degenerate fractional integro-differential equations considered on p. 221 in [7].

Suppose now that, for every $\xi \in \mathbb{R}$ and $x\in X,$ there is a unique continuous function $u:\mathbb{R}\to X$ such that (3) holds with $f\left(t\right)=e^{i\xi t}Cx,$ $t\in \mathbb{R},$ the function $Bu(·)$ is differentiable and $(d/dt){\int }_0^{+\infty }A\left(s\right)u(t-s)ds={\int }_0^{+\infty }A\left(s\right)u^{\prime }(t-s)ds$, $t\in \mathbb{R}.$ Then, the closedness of B implies $(d/dt)Bu\left(t\right)=B{u}^{\prime }\left(t\right),$ $t\in \mathbb{R}$, and we have

$$B{u}^{\prime }\left(t\right)={\int }_0^{+\infty }A\left(s\right)u^{\prime }(t-s)ds+i\xi e^{i\xi t}Cx,t\in \mathbb{R}.$$

Multiplying (3) with $i\xi$ and using the uniqueness of the solutions, it follows that $u^{\prime }\left(t\right)=i\xi u\left(t\right),$ $t\in \mathbb{R}$ so that there is a unique element $y\in Y$ s.t. $u\left(t\right)=e^{i\xi t}y,$ $t\in \mathbb{R}.$ Coming back to (3), we obtain

$$B{e}^{i\xi t}y={\int }_0^{+\infty }A\left(s\right)e^{i\xi (t-s)}yds+e^{i\xi t}Cx,t\in \mathbb{R}.$$

Therefore, the integral ${\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)yds$ is convergent for every $\xi \in \mathbb{R}$ and $(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)y=Cx.$ Consequently, we have

$$\begin{array}{c}\hfill {\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\in L(X,Y),\xi \in \mathbb{R}.\end{array}$$

(10)

If $h(·)$ is a vector-valued function and the integral

$${\int }_{-\infty }^{+\infty }{e}^{-i\xi t}h\left(t\right)dt:=\underset{T\to +\infty }{lim}{\int }_{-T}^T{e}^{-i\xi t}h\left(t\right)dt$$

is convergent for some $\xi \in \mathbb{R}$, then we tacitly assume henceforth that $h(·)$ is locally integrable. By a Fourier-transformable function, we mean any locally integrable function $h(·)$ such that the integral ${\int }_{-\infty }^{+\infty }{e}^{-i\xi t}h\left(t\right)dt$ is convergent for all $\xi \in \mathbb{R}.$

The condition (10) has an important role in the formulation of the subsequent theorem, which seems to be new even for the equations of scalar type, with $B=\mathrm{I}$:

Theorem 1.

(i) Suppose that the operator $B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds$ is well defined and injective for all $\xi \in \mathbb{R}$. Then, there exists a unique function $u\in L^1(\mathbb{R}:Y)$ such that the integral ${\int }_{-\infty }^{+\infty }{e}^{-i\xi t}Bu\left(t\right)dt$ is convergent in X for every $\xi \in \mathbb{R}$, (3) holds with $f\equiv 0$ and ${\int }_{-\infty }^{+\infty }{\int }_0^{+\infty }\parallel A\left(s\right)u(t-s)\parallel dsdt<+\infty .$

(ii) 

Suppose that (10) holds, $f\in L^1(\mathbb{R}:X)$, the function

$$\begin{array}{c}\hfill \xi \mapsto G\left(\xi \right):={\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt,\xi \in \mathbb{R}\end{array}$$

(11)

is locally integrable in Y and the regular distribution determined by $G(·)$ belongs to the space of tempered Y-valued distributions. If $u:={\mathcal{F}}^{-1}G\in u\in L^1(\mathbb{R}:Y)$, then $u(·)$ is a solution of (3) for a.e. $t\in \mathbb{R},$ provided that $Bu\in L^1(\mathbb{R}:X)$ and ${\int }_{-\infty }^{+\infty }{\int }_0^{+\infty }\parallel A\left(s\right)u(t-s)\parallel dsdt<+\infty .$

Proof. 

Suppose that the function $Bu(·)$ is Fourier transformable, (3) holds with $f\equiv 0$ and ${\int }_{-\infty }^{+\infty }{\int }_0^{+\infty }\parallel A\left(s\right)u(t-s)\parallel dsdt<+\infty .$ Since B is closed, it follows that

$$B{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}u\left(t\right)dt={\int }_{-\infty }^{+\infty }{e}^{-i\xi t}Bu\left(t\right)dt,\xi \in \mathbb{R};$$

Furthermore, the Fubini theorem yields that the function $t\mapsto {\int }_0^{+\infty }A\left(s\right)u(t-s)ds$, $t\in \mathbb{R}$ is Fourier transformable and

$$\begin{array}{cc}\hfill B{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}u\left(t\right)dt& ={\int }_{-\infty }^{+\infty }{e}^{-i\xi t}{\int }_0^{+\infty }A\left(s\right)u(t-s)dsdt\hfill \\ & ={\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right){\int }_{-\infty }^{+\infty }{e}^{-i\xi (t-s)}u(t-s)dtds\hfill \\ & =\left({\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}u\left(t\right)dt,\xi \in \mathbb{R},\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill \left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}u\left(t\right)dt=0,\xi \in \mathbb{R}.\end{array}$$

The injectivity of operator $B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds$ implies ${\int }_{-\infty }^{+\infty }{e}^{-i\xi t}u\left(t\right)dt=0,$ $\xi \in \mathbb{R}.$ Since $u\in L^1(\mathbb{R}:Y)$, the Fourier inversion formula ([8], Theorem 1.8.1 d) yields that $u\left(t\right)=0$ for a.e. $t\in \mathbb{R},$ which completes the proof of (i). The proof of (ii) is quite similar. First of all, it is clear that

$$\left(\mathcal{F}u\right)\left(\xi \right)={\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt,\xi \in \mathbb{R}.$$

Taking into account the prescribed assumptions, the above simply implies by the reverse procedure that

$${\int }_{-\infty }^{+\infty }{e}^{-i\xi t}Bu\left(t\right)dt={\int }_{-\infty }^{+\infty }{e}^{-i\xi t}{\int }_0^{+\infty }A\left(s\right)u(t-s)dsdt+{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}Cf\left(t\right)dt,$$

for any $\xi \in \mathbb{R}.$ Since $Bu(·)\in L^1(\mathbb{R}:X)$, ${\int }_0^{+\infty }A\left(s\right)u(·-s)ds\in L^1(\mathbb{R}:X)$ and $Cf(·)\in L^1(\mathbb{R}:X),$ the Fourier inversion formula ([8], Theorem 1.8.1 d) yields that (3) holds for a.e. $t\in \mathbb{R}$. □

Unfortunately, we cannot expect that there exists a function $H\in L^1(\mathbb{R}:L(X,Y))$ such that

$$\begin{array}{c}\hfill \left(\mathcal{F}H\right)\left(\xi \right)=G\left(\xi \right)={\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\in L(X,Y),\xi \in \mathbb{R},\end{array}$$

(12)

so the solution $u\in L^p(\mathbb{R}:Y)$ of (3), where $1\le p\le \infty ,$ cannot be found in the form $u=H\ast f$ with $H\in L^1(\mathbb{R}:L(X,Y))$ and $f\in L^p(\mathbb{R}:X).$ More precisely, the following holds:

Proposition 4.

Suppose that $A\in L^1([0,\infty ):L(Y,X))$, $H\in L^1(\mathbb{R}:L(X,Y))$ and (12) holds. Then, $X=Y=\left\{0\right\}.$

Proof. 

Suppose the contrary. Take any $x\in X$ such that $x\ne 0.$ Then, $Cx\ne 0$ and

$$\begin{array}{c}\hfill \left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)\left(\mathcal{F}H\right)\left(\xi \right)x=Cx,\xi \in \mathbb{R}.\end{array}$$

(13)

Define $\tilde{A}:\mathbb{R}\to L(Y,X)$ by $\tilde{A}\left(t\right):=A\left(t\right),$ $t\ge 0$ and $\tilde{A}\left(t\right):=0,$ $t<0.$ By ([8], Theorem 1.8.1 a), the equality (13) implies

$$\begin{array}{c}\hfill B\left(\mathcal{F}H\right)\left(\xi \right)x=\left(\mathcal{F}\left(\tilde{A}\ast H\right)\right)\left(\xi \right)x+Cx.\end{array}$$

(14)

It is clear that $\tilde{A}\ast H\in L^1(\mathbb{R}:L\left(X\right))$ so that the Riemann–Lebesgue lemma ([8], Theorem 1.8.1 c) yields that ${lim}_{\left|\xi \right|\to +\infty }\left(\mathcal{F}H\right)\left(\xi \right)x=0$ and ${lim}_{\left|\xi \right|\to +\infty }\left(\mathcal{F}(\tilde{A}\ast H)\right)\left(\xi \right)x=0$. Keeping in mind these facts, the closedness of B and (14) together imply $0=B0=Cx,$ which is a contradiction. □

Now we will illustrate Theorem 1(ii) with the following example:

Example 1.

The most symptomatic case for applications to multi-term problems is the case in which we have

$$A\left(t\right):=a_1\left(t\right)A_1+....+a_m\left(t\right)A_m,t\ge 0,$$

where $m\in \mathbb{N}$, $a_j\in L^1\left([0,+\infty )\right)$ and $A_j$ is a closed linear operator on X ($1\le j\le m$). We endow the vector space $Y:=D\left(B\right)\cap D\left(A_1\right)\cap ...\cap D\left(A_m\right)$ with the graph norm ${\parallel y\parallel }_Y:=\parallel y\parallel +\parallel By\parallel +\parallel A_{1}y\parallel +...+\parallel A_{m}y\parallel ,$ $y\in Y,$ under which Y is complete. In our concrete situation, we have

$$\begin{array}{cc}\hfill & {\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\hfill \\ & ={\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}{a}_1\left(s\right)ds·A_1-...-{\int }_0^{+\infty }{e}^{-i\xi s}{a}_m\left(s\right)ds·A_m\right)}^{-1}C,\xi \in \mathbb{R}.\hfill \end{array}$$

Assume now that there exist closed linear operator A on X and complex polynomials $P_j(·),$ where $0\le j\le m,$ such that $B:=P_0\left(A\right)$ and $A_j:=P_j\left(A\right)$ for $1\le j\le m;$ cf. [7] for the notion and more details. Despite the negative result established in Proposition 4, the requirements necessary for applications of Theorem 1 are satisfied in many concrete situations, especially in those situations where the forcing term $f(·)$ belongs to the Schwartz space of rapidly decreasing functions $\mathcal{S}\left(X\right)$, and the functions $\xi \mapsto {(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L(X,Y)$, $\xi \in \mathbb{R}$ and $\xi \mapsto {\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\in L(Y,X)$, $\xi \in \mathbb{R}$ admit polynomially bounded holomorphic extensions to the strip $\{z\in \mathbb{C}:|\Im z|<ϵ\}$, where a sufficiently small number $ϵ>0.$ Take, for instance, $a_j\left(t\right)=exp(-{ϵ}_{j}t)g_{{\alpha }_j}\left(t\right),$ $t>0,$ where ${ϵ}_j>0$ and ${\alpha }_j>0$ for $1\le j\le m.$ Then, the functions $\xi \mapsto {(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L(X,Y)$, $\xi \in \mathbb{R}$ and $\xi \mapsto B{(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L\left(X\right)$, $\xi \in \mathbb{R}$ are infinitely differentiable and all their derivatives are polynomially bounded on the real line, which follows by means of the product rule, Cauchy integral theorem and equality

$$\begin{array}{cc}\hfill B(B& -{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds{)}^{-1}C\hfill \\ & =C+\left[{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right]·{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C,\xi \in \mathbb{R},\hfill \end{array}$$

so that the solution $u={\mathcal{F}}^{-1}G$ of (3) and the function $Bu(·)$ belong to the space $\mathcal{S}\left(Y\right).$ For some concrete examples, we may refer to ([7], Theorem 1.2.7, Example 2.5.8). We can also provide many illustrative applications of Theorem 1(ii) to the abstract (degenerate) Volterra equations of the scalar type on the line.

Let us note that Theorem 1(ii) can be also applied to the abstract degenerate non-scalar Volterra equations on the line, provided that the forcing term $f(·)$ belongs to the space of rapidly decreasing ultra-differentiable functions of Beurling type ${\mathcal{S}}^{\left(M_p\right)}\left(X\right)$ as well as that the functions $\xi \mapsto {(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L(X,Y)$, $\xi \in \mathbb{R}$ and $\xi \mapsto B{(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L\left(X\right)$, $\xi \in \mathbb{R}$ admit ultra-polynomially bounded holomorphic extensions to the strip $\{z\in \mathbb{C}:|\Im z|<ϵ\};$ see ([18], Proposition 4.1) and ([7], Example 2.2.18) for more details in this direction.

For the sequel, we need the next result:

Theorem 2.

Suppose that the mapping $\xi \mapsto {(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L(X,Y)$, $\xi \in \mathbb{R}$ is continuous and ${\int }_0^{+\infty }s{\parallel A\left(s\right)\parallel }_{L(Y,X)}ds<+\infty .$ Then, the following holds:

(i) 

If the assumption $x\in D\left(B\right)$ implies $Cx\in D\left(B\right)$ and $BCx=CBx$ and the assumptions $y\in Y$ and $t\ge 0$ imply $Cy\in Y$ and $A\left(t\right)Cy=CA\left(t\right)t$ for $t\ge 0,$ then the following holds:

$$\begin{array}{c}{\displaystyle \frac{d}{d\xi }}{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}{C}^2={\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\hfill \\ \times \left(B-{\int }_0^{+\infty }{e}^{-i\xi s}(-is)A\left(s\right)ds\right){\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\in L(X,Y),\xi \in \mathbb{R}.\hfill \end{array}$$

(15)

Furthermore, if for each $\xi \in \mathbb{R}$ there exists $ϵ>0$ such that there exist analytic mappings $F:L(\xi ,ϵ)\to L(X,Y)$ and $G:L(\xi ,ϵ)\to L\left(X\right)$ such that $F\left(t\right)={(B-{\int }_0^{+\infty }{e}^{-its}A\left(s\right)ds)}^{-1}C$, $t\in (\xi -ϵ,\xi +ϵ)$ and

$$\begin{array}{c}\hfill G\left(t\right)=\left(B-{\int }_0^{+\infty }{e}^{-its}(-is)A\left(s\right)ds\right){\left(B-{\int }_0^{+\infty }{e}^{-its}A\left(s\right)ds\right)}^{-1}C,t\in (\xi -ϵ,\xi +ϵ),\end{array}$$

then the mapping $F^{\prime }:L(\xi ,ϵ)\to L(X,Y)$ is analytic, the mapping $\lambda \mapsto C^{-1}F\left(\lambda \right)G\left(\lambda \right)\in L(X,Y),$ $\lambda \in L(\xi ,ϵ)$ is analytic, $F^{\prime }\left(\lambda \right)x=C^{-1}F\left(\lambda \right)G\left(\lambda \right)x$, $\lambda \in L(\xi ,ϵ),$ $x\in X$, ${\displaystyle \frac{d}{d\xi }}{(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L(X,Y)$, $\xi \in \mathbb{R},$

$$\begin{array}{c}{C}^{-1}{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\hfill \\ \times \left(B-{\int }_0^{+\infty }{e}^{-i\xi s}(-is)A\left(s\right)ds\right){\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\in L(X,Y),\xi \in \mathbb{R},\hfill \end{array}$$

(16)

and

$$\begin{array}{c}{\displaystyle \frac{d}{d\xi }}{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C=C^{-1}{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\hfill \\ \times \left(B-{\int }_0^{+\infty }{e}^{-i\xi s}(-is)A\left(s\right)ds\right){\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\in L(X,Y),\xi \in \mathbb{R}.\hfill \end{array}$$

(17)

(ii) 

If ${(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}\in L(X,Y)$, $\xi \in \mathbb{R}$, then the following holds:

$$\begin{array}{c}{\displaystyle \frac{d}{d\xi }}{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C={\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}\hfill \\ \times \left(B-{\int }_0^{+\infty }{e}^{-i\xi s}(-is)A\left(s\right)ds\right){\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\in L(X,Y),\xi \in \mathbb{R}.\hfill \end{array}$$

(18)

Proof. 

Denote $I_x:={(B-{\int }_0^{+\infty }{e}^{-ixs}A\left(s\right)ds)}^{-1}C,$ $x\in \mathbb{R}.$ Then we have

$$\begin{array}{cc}\hfill B\left[I_x-I_y\right]& =\left(B-{\int }_0^{+\infty }{e}^{-ixs}A\left(s\right)ds\right)\left[I_x-I_y\right]\hfill \\ & +\left[{\int }_0^{+\infty }{e}^{-ixs}A\left(s\right)ds-{\int }_0^{+\infty }{e}^{-iys}A\left(s\right)ds\right]·I_y,x,y\in \mathbb{R}.\hfill \end{array}$$

Let us fix now a number $x\in \mathbb{R}.$ Since the operator $B-{\int }_0^{+\infty }{e}^{-ixs}A\left(s\right)ds$ is injective, the above equality implies after a simple computation that

$$\begin{array}{cc}\hfill {\displaystyle \frac{I_x-I_y}{x-y}}& ={\left(B-{\int }_0^{+\infty }{e}^{-ixs}A\left(s\right)ds\right)}^{-1}\hfill \\ & ·\left[{\int }_0^{+\infty }{\displaystyle \frac{e^{-ixs}-e^{-iys}}{x-y}}A\left(s\right)ds\right]·I_y\in L(X,Y),y\in \mathbb{R}\setminus \left\{x\right\}.\hfill \end{array}$$

Since ${\int }_0^{+\infty }s{\parallel A\left(s\right)\parallel }_{L(Y,X)}ds<+\infty$, the dominated convergence theorem implies

$$\underset{y\to x}{lim}{\int }_0^{+\infty }{\displaystyle \frac{e^{-ixs}-e^{-iys}}{x-y}}A\left(s\right)ds={\int }_0^{+\infty }{e}^{-ixs}(-is)A\left(s\right)ds\in L(Y,X).$$

Moreover, the mapping $\xi \mapsto {(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L(X,Y)$, $\xi \in \mathbb{R}$ is continuous so that ${lim}_{y\to x}{I}_y=I_x$ in $L(X,Y).$ Taken together with the above equality, we obtain

$$\begin{array}{c}\hfill \underset{y\to x}{lim}\left[{\int }_0^{+\infty }{\displaystyle \frac{e^{-ixs}-e^{-iys}}{x-y}}A\left(s\right)ds\right]·I_y=\left[{\int }_0^{+\infty }{e}^{-ixs}(-is)A\left(s\right)ds\right]·I_x\in L\left(X\right).\end{array}$$

This simply yields (15) and (18). The second statement in (i) remains to be proved. Towards this end, observe first that the mapping $F^{\prime }:L(\xi ,ϵ)\to L(X,Y)$ is analytic. Let $x\in X$ be fixed. From the first part of (i) and the identity theorem for holomorphic functions ([8], Proposition A.2), it follows that $F^{\prime }\left(\lambda \right)Cx=F\left(\lambda \right)G\left(\lambda \right)x,$ $\lambda \in L(\xi ,ϵ).$ Using the last equality, prescribed commutativity assumptions and ([8], Proposition A.3), we find that the mapping $\lambda \mapsto C^{-1}F\left(\lambda \right)G\left(\lambda \right)\in L(X,Y),$ $\lambda \in L(\xi ,ϵ)$ is analytic and $F^{\prime }\left(\lambda \right)=C^{-1}F\left(\lambda \right)G\left(\lambda \right)\in L(X,Y),$ $\lambda \in L(\xi ,ϵ)$. This simply yields that $(d/d\xi ){(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds)}^{-1}C\in L(X,Y)$, $\xi \in \mathbb{R}$ as well as that (16) and (17) hold true, finishing the proof of the theorem. □

Observe that the statements of Bernstein’s lemma ([8], Lemma 8.2.1) and Mikhlin’s theorem ([8], Proposition 8.2.3) continue to hold for the vector-valued functions. Using this fact, it readily follows that there exists a function $u\in L^1(\mathbb{R}:Y)$ such that $\left(\mathcal{F}u\right)\left(\xi \right)=G\left(\xi \right)$, $\xi \in \mathbb{R},$ where $G(·)$ is given by (11), provided that $G\in C^1(\mathbb{R}:Y)$ and there exists a sufficiently small real number $ϵ>0$ such that

$$\begin{array}{c}\hfill \underset{\xi \in \mathbb{R}}{sup}\left[{\left(1+\left|\xi \right|\right)}^{ϵ}{\parallel G\left(\xi \right)\parallel }_Y+{\left(1+\left|\xi \right|\right)}^{1+ϵ}{∥G^{\prime }\left(\xi \right)∥}_Y\right]<+\infty .\end{array}$$

(19)

Now we would like to state the following result:

Theorem 3.

Suppose that (10) holds, ${\int }_0^{+\infty }{\parallel A\left(s\right)\parallel }_{L(Y,X)}ds<+\infty ,$ $f\in L^1(\mathbb{R}:X)$, $G\in C^1(\mathbb{R}:Y),$ $BG\in C^1(\mathbb{R}:X)$ and there exists $ϵ>0$ such that (19) holds and

$$\begin{array}{c}\hfill \underset{\xi \in \mathbb{R}}{sup}\left[{\left(1+\left|\xi \right|\right)}^{ϵ}\parallel BG\left(\xi \right)\parallel +{\left(1+\left|\xi \right|\right)}^{1+ϵ}∥B{G}^{\prime }\left(\xi \right)∥\right]<+\infty .\end{array}$$

(20)

Then, there exists a function $u\in L^1(\mathbb{R}:Y)$ such that $Bu\in L^1(\mathbb{R}:X)$, $\left(\mathcal{F}u\right)\left(\xi \right)=G\left(\xi \right)$, $\xi \in \mathbb{R}$ and (3) holds for a.e. $t\in \mathbb{R}.$

Proof. 

Clearly, there exists a function $u\in L^1(\mathbb{R}:Y)$ such that $\left(\mathcal{F}u\right)\left(\xi \right)=G\left(\xi \right)$, $\xi \in \mathbb{R}.$ Since $BG\in C^1(\mathbb{R}:X)$ and (20) holds, we similarly obtain $Bu\in L^1(\mathbb{R}:X).$ Furthermore,

$$\begin{array}{cc}\hfill {\int }_{-\infty }^{+\infty }{\int }_0^{+\infty }\parallel A\left(s\right)u(t-s)\parallel dsdt& \le {\int }_0^{+\infty }{\parallel A\left(s\right)\parallel }_{L(Y,X)}{\int }_{-\infty }^{+\infty }{\parallel u(t-s)\parallel }_{Y}dtds\hfill \\ & ={\parallel u\parallel }_{L^1(\mathbb{R}:Y)}·{\int }_0^{+\infty }{\parallel A\left(s\right)\parallel }_{L(Y,X)}ds<+\infty .\hfill \end{array}$$

Now the statement follows Theorem 1(ii). □

Under certain reasonable assumptions, we have the following identity:

$$\begin{array}{c}{\displaystyle \frac{d}{d\xi }}\left[{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt\right]\hfill \\ =\left[{\displaystyle \frac{d}{d\xi }}{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\right]·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt\hfill \\ +\left[{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}C\right]·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}(-i\xi )f\left(t\right)dt,\xi \in \mathbb{R},\hfill \end{array}$$

so that Theorem 2 can be used to state a straightforward corollary of Theorem 3. To illustrate this result, let us assume that $C=\mathrm{I}$ and $A\left(t\right)=a\left(t\right)A,$ $t\ge 0$, where $a\in L^1\left([0,+\infty )\right)$ and A is a closed linear operator on $X.$ Then, we have

$$\begin{array}{c}{\displaystyle \frac{d}{d\xi }}\left[{\left(B-{\int }_0^{+\infty }{e}^{-i\xi s}A\left(s\right)ds\right)}^{-1}·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt\right]\hfill \\ ={\left(B-\left[{\int }_0^{+\infty }{e}^{-i\xi s}a\left(s\right)ds\right]·A\right)}^{-1}·\left(B+\left[{\int }_0^{+\infty }{e}^{-i\xi s}\left(i\xi \right)a\left(s\right)ds\right]·A\right)\hfill \\ ·{\left(B-\left[{\int }_0^{+\infty }{e}^{-i\xi s}a\left(s\right)ds\right]·A\right)}^{-1}·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt\hfill \\ +{\left(B-\left[{\int }_0^{+\infty }{e}^{-i\xi s}a\left(s\right)ds\right]·A\right)}^{-1}·{\int }_{-\infty }^{+\infty }{e}^{-i\xi t}(-i\xi )f\left(t\right)dt,\xi \in \mathbb{R},\hfill \end{array}$$

so that Theorem 2 can be applied in many concrete situations in which the terms ${\int }_{-\infty }^{+\infty }{e}^{-i\xi t}f\left(t\right)dt$ and ${\int }_{-\infty }^{+\infty }{e}^{-i\xi t}(-i\xi )f\left(t\right)dt$ decay sufficiently enough as $\left|\xi \right|\to +\infty .$

Finally, let us note that it would be very tempting to state some notable results on the existence of $L^p$-solutions of (3) in the case of $1<p<\infty .$

Almost-Periodic-Type Solutions to (3)

The notion of almost periodicity was introduced by H. Bohr around 1924–1926 ([19,20,21]); cf. also [22,23] and the references quoted therein. Suppose that $f:\mathbb{R}\to X$ is continuous. If $ϵ>0,$ then a number $\tau \in \mathbb{R}$ is said to be an $ϵ$-period for $f(·)$ if $\parallel f(t+\tau )-f\left(t\right)\parallel \le ϵ,$ $t\in \mathbb{R}.$ By $\vartheta (f,ϵ),$ we denote the set consisting of all $ϵ$-periods for $f(·)$. Let us recall that $f(·)$ is almost periodic if, for each $ϵ>0$, the set $\vartheta (f,ϵ)$ is relatively dense in $\mathbb{R},$ i.e., there exists $l>0$ such that any subinterval of $\mathbb{R}$ of length l meets $\vartheta (f,ϵ)$. Any almost-periodic function $f:\mathbb{R}\to X$ is bounded and uniformly continuous, and the set of all almost-periodic functions $f:\mathbb{R}\to X$, denoted by $AP(\mathbb{R}:X),$ is a vector space with the usual operations. Equipped with the sup-norm, $AP(\mathbb{R}:X)$ is a Banach space.

Let $1\le p<+\infty .$ Then, we say that a function $f\in L_{loc}^p(\mathbb{R}:X)$ is Stepanov p-almost periodic if there is $l>0$ such that any subinterval of $\mathbb{R}$ of length l contains a point $\tau$ such that

$${\int }_t^{t+1}\parallel f(s+\tau )-f\left(s\right)\parallel ds<ϵ,t\in \mathbb{R}.$$

Further on, a function $f\in L_{loc}^p(\mathbb{R}:X)$ is said to be

(i)

equi-Weyl-p-almost periodic, if, for each $ϵ>0$, we can find $l>0$ and $L>0$ such that any interval $I\subseteq \mathbb{R}$ of length L contains a point $\tau \in I$ such that

$$\begin{array}{c}\hfill \underset{x\in \mathbb{R}}{sup}{\left[{\displaystyle \frac{1}{l}}{\int }_x^{x+l}{∥f(t+\tau )-f\left(t\right)∥}^{p}dt\right]}^{1/p}\le ϵ;\end{array}$$

(ii)

Weyl-p-almost periodic, if, for each $ϵ>0$, we can find $L>0$ such that any interval $I\subseteq \mathbb{R}$ of length L contains a point $\tau \in I$ such that

$$\begin{array}{c}\hfill \underset{l\to \infty }{lim}\underset{x\in \mathbb{R}}{sup}{\left[{\displaystyle \frac{1}{l}}{\int }_x^{x+l}{∥f(t+\tau )-f\left(t\right)∥}^{p}dt\right]}^{1/p}\le ϵ.\end{array}$$

Any almost-periodic function $f:\mathbb{R}\to X$ is Stepanov p-almost periodic, any Stepanov-p-almost-periodic function $f:\mathbb{R}\to X$ is equi-Weyl-p-almost periodic and any equi-Weyl-p-almost-periodic function $f:\mathbb{R}\to X$ is Weyl-p-almost periodic. All these inclusions are strict. Let us also note that the set of all Stepanov-p-almost-periodic functions $f:\mathbb{R}\to X$, resp. equi-Weyl-p-almost-periodic functions $f:\mathbb{R}\to X$, is a vector space with the usual operations, which is no longer true for Weyl-p-almost-periodic functions.

We will first observe that the Banach contraction principle can be successfully applied in the analysis of the existence and uniqueness of almost periodic solutions to the Problem (1), provided that $Y=X$, ${\int }_0^{+\infty }{\parallel A\left(s\right)\parallel }_{L\left(X\right)}ds<1$ and $f:\mathbb{R}\to X$ is almost periodic. Towards this end, it suffices to observe that the mapping $\Pi :AP(\mathbb{R}:X)\to AP(\mathbb{R}:X),$ given by

$$(\Pi g)\left(t\right):=f\left(t\right)+{\int }_0^{+\infty }A\left(s\right)g(t-s)ds,t\in \mathbb{R},g\in AP(\mathbb{R}:X),$$

is a well-defined contraction.

Concerning Proposition 2, we will only observe here that the solution $u(·)$ inherits certain Weyl almost-periodic behavior from the forcing term $f(·);$ cf. [22]. It is also worth noting that, for every $\omega \in \mathbb{R},$ the abstract Cauchy problem (3) is equivalent to the problem

$$\begin{array}{c}\hfill Bh\left(t\right)=e^{-\omega t}Cf\left(t\right)+{\int }_0^{+\infty }\left[e^{-\omega s}A\left(s\right)\right]h(t-s)ds,t\in \mathbb{R},\end{array}$$

where $h\left(t\right):=e^{-\omega t}u\left(t\right),$ $t\in \mathbb{R}$, as well as that the problem (8) is equivalent with the problem

$$\begin{array}{c}\hfill Bh\left(t\right)={\int }_{-\infty }^t\left[e^{-\omega (t-s)}k(t-s)\right]\left[e^{-\omega s}Cf\left(s\right)\right]ds+{\int }_{-\infty }^t\left[e^{-\omega (t-s)}A(t-s)\right]h\left(s\right)ds,t\in \mathbb{R}.\end{array}$$

(21)

This observation allows us to consider the situations in which the kernel $A(·)$ is not uniformly integrable in $L(Y,X)$ but only exponentially bounded. Here, it is worth noting that, if the operator family ${\left(S\left(t\right)\right)}_{t\ge 0}$ in $L(X,[D\left(B\right)\left]\right)$ is a global $(A,k,B)$-regularized C-pseudoresolvent family (the operator family ${\left(V\left(t\right)\right)}_{t\ge 0}$ in $L\left(X\right)$ is a global $(A,k,B)$-regularized C-uniqueness family), then ${\left(e^{-\omega t}S\left(t\right)\right)}_{t\ge 0}$ is a global $(e^{-\omega ·}A,e^{-\omega ·}k,B)$-regularized C-pseudoresolvent family (${\left(e^{-\omega t}V\left(t\right)\right)}_{t\ge 0}$ is a global $(e^{-\omega ·}A,e^{-\omega ·}k,B)$-regularized C-uniqueness family). For example, if we assume that the forcing term $e^{-\omega ·}Cf(·)$ is almost periodic and the operator family ${\left(e^{-\omega t}S\left(t\right)\right)}_{t\ge 0}$ is exponentially decaying, then the solution $h(·)$ of (21) will be almost periodic as well ([22]). For many important examples and applications of $(A,1,\mathrm{I})$-regularized $\mathrm{I}$-pseudoresolvent families, we refer the reader to [1].

We close this subsection by stating the next proposition:

Proposition 5.

Let $B\in L\left(X\right).$ Then, the following holds:

(i) 

Suppose that $A\in L^1([0,+\infty ):L(Y,X)).$ If $u:\mathbb{R}\to X$ is almost periodic and (3) holds, then the function $Cf:\mathbb{R}\to X$ is almost periodic.

(ii) 

Suppose that $1\le p<\infty ,$ $1/p+1/q=1$ and ${\sum }_{k=0}^{\infty }{\parallel A(·)\parallel }_{L^q[k,k+1]}<\infty .$ If $u:\mathbb{R}\to X$ is Stepanov p-almost periodic and (3) holds, then the function $Cf:\mathbb{R}\to X$ is Stepanov p-almost periodic.

(iii) 

Let $1/p+1/q=1$ and let

$$\begin{array}{c}\hfill {\parallel A\left(t\right)\parallel }_{L(Y,X)}\le {\displaystyle \frac{M{t}^{\beta -1}}{1+t^{\gamma }}},t>0\mathrm{for} some finite constants \gamma >\beta ,\beta \in (0,1],M>0.\end{array}$$

Suppose that $u:\mathbb{R}\to Y$ is equi-Weyl-p-almost periodic, resp. Weyl-p-almost periodic and Weyl p-bounded, $q(\beta -1)>-1$ provided that $p>1$, resp. $\beta =1,$ provided that $p=1$, and (3) holds. Then, the function $Cf:\mathbb{R}\to X$ is equi-Weyl-p-almost periodic, resp. $Cf:\mathbb{R}\to X$ can be represented as the difference of two Weyl-p-almost-periodic functions.

Proof. 

We will prove only (iii), because the proofs of (i) and (ii) are quite similar to the proof of (iii); cf. also ([22], Proposition 2.6.11). Using the argumentation given in the proof of ([22], Theorem 2.11.4), we can show that the mapping $t\mapsto {\int }_0^{+\infty }A\left(s\right)u(t-s)ds,$ $t\in \mathbb{R}$ is equi-Weyl-p-almost periodic, resp. Weyl-p-almost periodic. Since $B\in L\left(X\right),$ the function $Bu(·)$ is also equi-Weyl-p-almost periodic, resp. Weyl-p-almost periodic, which simply implies the required conclusion. □

3. Conclusions and Final Remarks

In this paper, we have investigated the existence and uniqueness of solutions for some classes of abstract degenerate non-scalar Volterra equations on the line. We have provided several new results and illustrative applications in this direction considering also the existence and uniqueness of almost-periodic-type solutions.

The admissibility of homogeneous spaces of functions defined on the real line and the Euclidean space ${\mathbb{R}}^n$, some connections between the abstract Volterra integro-differential equations on the half-line and the abstract Volterra integro-differential equations on the real line and the abstract multi-term Cauchy problems with generalized Weyl integro-differential operators will be analyzed somewhere else.