(F, G, C)-Resolvent Operator Families and Applications

Abstract

In this paper, we introduce and investigate several new classes of $(F,G,C)$-regularized resolvent operator families subgenerated by multivalued linear operators in locally convex spaces. The known classes of $(a,k)$-regularized C-resolvent operator-type families are special cases of the classes introduced in this paper. We provide certain applications of $(F,G,C)$-regularized resolvent operator families and $(a,k)$-regularized C-resolvent families to abstract fractional differential–difference inclusions and abstract Volterra integro-difference inclusions.

1. Introduction and Preliminaries

The theory of abstract Volterra integro-differential equations is still a very active field of research of many authors (cf. the monographs [1,2,3] and references cited therein for more details on the subject). In this paper, we continue our recent investigations of abstract Volterra integro-differential inclusions by examining several new classes of abstract fractional differential–difference inclusions and abstract Volterra integro-difference inclusions.

The use of the vector-valued Laplace transform and $(a,k)$-regularized C-resolvent families is almost inevitable in any important research of abstract non-scalar Volterra integro-differential equations. The main purpose of this research article is to extend the notion of an exponentially equicontinuous $(a,k)$-regularized C-resolvent family (mild $(a,k)$-regularized $C_1$-existence family, mild $(a,k)$-regularized $C_2$-uniqueness family) by introducing the concept of an $(F,G,C)$-regularized resolvent family (mild $(F,G,C_1)$-regularized existence family, mild $(F,G,C_2)$-regularized uniqueness family); cf. Definition 1 and Proposition 1 for more details in this direction. The introduced solution operator families are subgenerated by multivalued linear operators (briefly, MLOs) in locally convex spaces.

The organization and main ideas of this research study can be briefly summarized as follows. After explaining the main notation and terminology used, we introduce and analyze various classes of $(F,G,C)$-regularized resolvent operator-type families in Section 2. Many structural characterizations of $(a,k)$-regularized C-resolvent operator-type families continue to hold for $(F,G,C)$-regularized resolvent operator-type families; for the sake of brevity, we will only quote the corresponding results and explain the basic differences in our new framework (cf. [2] (Chapter 3) and [4] for more details on the subject; before proceeding further, we would like to say that it is far from being clear how we can introduce our new concepts in the local framework). We introduce the notion of the integral generator of a mild $(F,G,C_2)$-regularized uniqueness family ($(F,G,C)$-regularized resolvent family) and clarify the main structural features of subgenerators of $(F,G,C)$-regularized resolvent operator-type families. In Theorems 1 and 2, we prove the existence of a very specific example of an $(F,G)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$, which cannot be an $(a,k)$-regularized resolvent family for any choice of a Laplace transformable function $a\in L_{\mathrm{loc}}^{\infty }\left([0,\infty )\right)$ ($a\in L_{\mathrm{loc}}^1\left([0,\infty )\right)$) and a continuous Laplace transformable function $k(·)$. The notion of an (exponentially equicontinuous) analytic $(F,G,C)$-regularized resolvent family of angle $\alpha$ is introduced in Definition 2. After that, we examine the subordination principles, the generation results for (exponentially equicontinuous, analytic) $(F,G,C)$-regularized resolvent families and the smoothing properties of $(F,G,C)$-regularized resolvent families.

We continue by recalling that S. Bochner [5] has analyzed the linear difference–differential operators

$$L_h=\sum _{i=0}^p\sum _{j=0}^q{a}_{ij}\frac{d^i}{d{x}^i}h\left(·+h_j\right),$$

where $a_{ij}$ are complex numbers ($0\le i\le p$, $0\le j\le q$) and $h={\left(h_j\right)}_{0\le j\le q}\subseteq {\mathbb{R}}^q$; some results about the existence and uniqueness of generalized almost periodic solutions of the above equation can be found in [6]. Concerning the applications of $(F,G,C)$-regularized resolvent operator families to the abstract fractional differential–difference inclusions (cf. Section 3), we would like to emphasize first that C. Lizama and his coauthors have examined, in a series of important research papers, the well-posedness of the following abstract fractional difference equation:

$$\begin{array}{c}\hfill {\Delta }^{\alpha }u\left(k\right)=Au(k+1)+f\left(k\right),k\in \mathbb{Z},\end{array}$$

where A is a closed linear operator on Banach space X, $0<\alpha <1$ and ${\Delta }^{\alpha }u\left(k\right)$ denotes the Caputo fractional difference operator of order $\alpha$; cf. [7,8,9] and references cited therein for more details about this problem. The results established in the aforementioned research papers have strongly influenced us to examine what happens with the well-posedness of the following abstract degenerate fractional differential–difference inclusions:

$${\left(\mathrm{DFP}\right)}_{R,a,b}:\left\{\begin{array}{c}{\mathbf{D}}_t^{\alpha }Bu(t+a)\in \mathcal{A}u(t+b)+f\left(t\right),t\ge 0,\hfill \\ \left(Bu\right)\left(t\right)=B{u}_0\left(t\right),0\le t\le c\hfill \end{array}\right.$$

and

$${\left(\mathrm{DFP}\right)}_{L,a,b}:\left\{\begin{array}{c}\mathcal{B}{\mathbf{D}}_t^{\alpha }u(t+a)\subseteq \mathcal{A}u(t+b)+f\left(t\right),t\ge 0,\hfill \\ u\left(t\right)=u_0\left(t\right),0\le t\le c,\hfill \end{array}\right.$$

where $0\le a,b<\infty$, $c=max(a,b)$, $f:[0,\infty )\to X$ is Lebesgue measurable, B is a closed linear operator, $\mathcal{A}$ and $\mathcal{B}$ are closed MLOs. In Example 3, which opens Section 3, we perceive that the solutions of problems ${\left(\mathrm{DFP}\right)}_{R,a,b}$ and ${\left(\mathrm{DFP}\right)}_{L,a,b}$ can be simply calculated step by step in many concrete situations as well as that it is almost impossible to apply the vector-valued Laplace transform in the deeper analysis of the abstract fractional Cauchy inclusions ${\left(\mathrm{DFP}\right)}_{R,a,b}$ and ${\left(\mathrm{DFP}\right)}_{L,a,b}$, provided that $0\le a<b$; on the other hand, if $0\le b<a$, then $(a,k)$-regularized C-resolvent operator families can be employed to give the simple form of solutions of these problems (it would be very difficult to summarize here the basic results concerning the Volterra integro-functional equations and their applications, even in the scalar-valued setting; for further information in this direction, we refer the reader to the lectures of H. Brunner [10], the research article [11] by C. Corduneanu, the research article [12] by J.R. Haddock, M.N. Nkashama, J.H. Wu, the important research monographs [13] by L. Erbe, Q. Kong, B.G. Zhang, [14] by J.K. Hale and the references cited therein).

In the continuation of Section 3, we analyze the well-posedness of the following abstract Volterra initial value problem

$$\begin{array}{c}\hfill u\left(t\right)\in f\left(t\right)+\sum _{i=1}^m\left(a_i\ast \mathcal{A}u\left(·+t_i\right)\right)\left(t\right),t\ge 0;u\left(t\right)=u_0\left(t\right),0\le t\le \tau ,\end{array}$$

where $m\in \mathbb{N}$, $0\le t_1,t_2,\dots ,t_m<+\infty$, $\tau :=max(t_1,t_2,\dots ,t_m)$, $f:[0,\infty )\to X$ is Lebesgue measurable, $\mathcal{A}$ is a closed MLO on X and $a_1\left(t\right),\dots ,a_m\left(t\right)$ are locally integrable scalar-valued functions defined for $t\ge 0$; in contrast to many other recent research papers concerning the abstract delay integro-differential equations, the terms $u(·+t_i)$ considered here are under the action of the multivalued linear operator $\mathcal{A}$ (the problem seems to be new even for the single-valued linear operators).

The solution of the above problem can be very difficultly calculated step by step, with the exception of some extremely peculiar situations (for example, this can be done provided that $a_1\left(t\right)=a_2\left(t\right)=\dots =a_m\left(t\right)=g_{\alpha }\left(t\right)$ for some $\alpha >0$ and $0\in \rho \left(\mathcal{A}\right)$; details can be left to interested readers). As a result of that, we will follow here the abstract theoretical approach based on the use of vector-valued Laplace transforms and $(F,G,C)$-regularized resolvent families; in many concrete situations, the theory of $(a,k)$-regularized C-resolvent families can be used, as well. We also consider the following abstract Volterra initial value problem

$$u\left(t\right)\in f\left(t\right)+\sum _{i=1}^m\left(a_i\ast \mathcal{A}u\left(·+t_i\right)\right)\left(t\right),t\in \mathbb{R};u\left(t\right)=u_0\left(t\right),t\in \left[{\tau }_1,{\tau }_2\right],$$

where $m\in \mathbb{N}$, $t_1,t_2,\dots ,t_m\in \mathbb{R}$ and there exists an index $i\in {\mathbb{N}}_m\equiv \{1,\dots ,m\}$ with $t_i<0$, $f:[0,\infty )\to X$ is Lebesgue measurable, $\mathcal{A}$ is a closed MLO on X and $a_1\left(t\right),\dots ,a_m\left(t\right)$ are locally integrable scalar-valued functions defined for $t\in \mathbb{R}$, ${\tau }_1:=min(t_1,t_2,\dots ,t_m)$, ${\tau }_2:=max(t_1,t_2,\dots ,t_m)$ if there exists an index $j\in {\mathbb{N}}_m$ with $t_j>0$ and ${\tau }_2:=0$, otherwise.

Before proceeding any further, we would like to emphasize that there exist some unpleasant situations in which the unique solution of the above-mentioned Cauchy inclusions exists only if the multivalued linear operator $\mathcal{A}$ satisfies certain very exceptional spectral conditions; for example, the C-resolvent set of $\mathcal{A}$ must exist outside a compact set $K\subseteq \mathbb{C}$ sometimes. Concerning this issue, we would like to recall that if $\mathcal{A}=A$ is single valued with a polynomially bounded resolvent existing outside a compact set $K\subseteq \mathbb{C}$ and X is a Banach space, then A must be bounded (see, e.g., the proof of [15], Theorem 6.5, pp. 135–138). On the other hand, there exists a non-densely defined operator A on a Banach space X with ultra-polynomially bounded resolvent existing on $\mathbb{C}$, which will be extremely important for some applications (for example, we can employ this operator in the analysis of problem (4), since the term $\parallel {\lambda }^{\alpha -1}{e}^{-\lambda }{({\lambda }^{\alpha }{e}^{-\lambda }-A)}^{-1}\parallel$ is ultra-polynomially bounded for $\Re \lambda \ge 0$; this cannot be done in the analysis of problem (5) below).

Example 1 

(See [1], Example 2.6.10, for more details). Let $s>1$,

$$E:=\left\{f\in C^{\infty }[0,1];\parallel f\parallel :=\underset{p\ge 0}{sup}\frac{\parallel f^{\left(p\right)}{\parallel }_{\infty }}{p{!}^s}<\infty \right\}$$

and

$$A:=-d/ds,D\left(A\right):=\left\{f\in E;f^{\prime }\in E,f\left(0\right)=0\right\}.$$

Let $P\left(z\right)={\sum }_{j=0}^N{a}_j{z}^j$, $z\in \mathbb{C}$, $a_N\ne 0$ be a complex non-zero polynomial, and let $N=dg\left(P\right)\ge 2$. We define the operator $P\left(A\right)$ in the usual way; then we know that $P\left(A\right)$ is non-densely defined as well as that there exist real numbers $M>0$, $b>0$ and $c>0$ such that

$$\rho \left(P\left(A\right)\right)=\mathbb{C} and \parallel R(\lambda :P\left(A\right))\parallel \le M{e}^{{b\left|\lambda \right|}^{1/Ns}+c{\left|\lambda \right|}^{1/N}},\lambda \in \mathbb{C}.$$

Notation and Terminology

By X we denote a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short. If Y is also an SCLCS over the same field of scalars as X, then the shorthand $L(X,Y)$ denotes the vector space consisting of all continuous linear mappings from X into Y; $L\left(X\right)\equiv L(X,X)$. By ⊛ (${⊛}_Y$) we denote the fundamental system of seminorms which defines the topology of X (Y). The symbol $\mathrm{I}$ denotes the identity operator on X. Let $0<\tau \le \infty$, then a strongly continuous operator family ${\left(W\left(t\right)\right)}_{t\in [0,\tau )}\subseteq L(X,Y)$ is said to be locally equicontinuous if and only if, for every $T\in (0,\tau )$ and $p\in {⊛}_Y$, there exist $q_p\in ⊛$ and $c_p>0$ such that $p\left(W\left(t\right)x\right)\le c_p{q}_p\left(x\right)$, $x\in X$, $t\in [0,T]$; the notions of equicontinuity of ${\left(W\left(t\right)\right)}_{t\in [0,\tau )}$ and the (exponential) equicontinuity of ${\left(W\left(t\right)\right)}_{t\ge 0}$ are defined similarly.

By $\mathcal{B}$ we denote the family consisting of all bounded subsets of X. Define $p_{\mathbb{B}}\left(T\right):={sup}_{x\in \mathbb{B}}p\left(Tx\right)$, $p\in {⊛}_Y$, $\mathbb{B}\in \mathcal{B}$, $T\in L(X,Y)$. Then, $p_{\mathbb{B}}(·)$ is a seminorm on $L(X,Y)$ and the system ${\left(p_{\mathbb{B}}\right)}_{(p,\mathbb{B})\in {⊛}_Y\times \mathcal{B}}$ induces the Hausdorff locally convex topology on $L(X,Y)$. Suppose that A is a closed linear operator acting on X. Then, we denote the domain, kernel space and range of A by $D\left(A\right)$, $N\left(A\right)$ and $R\left(A\right)$, respectively. In general, we set $\check{f}(·):=f(-·)$; by ${\chi }_D(·)$ we denote the characteristic function of a set $D\subseteq \mathbb{C}$. We define $g_{\alpha }\left(t\right):=t^{\alpha -1}/\Gamma \left(\alpha \right)$, $t>0$ ($\alpha >0$), where $\Gamma (·)$ denotes the Euler Gamma function.

Let us consider now the existence of the Laplace integral

$$\left(\mathcal{L}f\right)\left(\lambda \right):=\tilde{f}\left(\lambda \right):={\int }_0^{\infty }{e}^{-\lambda t}f\left(t\right)dt:=\underset{\tau \to \infty }{lim}{\int }_0^{\tau }{e}^{-\lambda t}f\left(t\right)dt,$$

for $\lambda \in \mathbb{C}$. If $\tilde{f}\left({\lambda }_0\right)$ exists for some ${\lambda }_0\in \mathbb{C}$, then we define the abscissa of convergence of $\tilde{f}(·)$ by ${\mathrm{abs}}_X\left(f\right):=inf\left\{\Re \lambda :\tilde{f}\left(\lambda \right)\mathrm{exists}\right\};$ otherwise, we set ${\mathrm{abs}}_X\left(f\right):=+\infty$. We say that $f(·)$ is Laplace transformable, or equivalently, that $f(·)$ belongs to the class (P1)-X, if and only if ${\mathrm{abs}}_X\left(f\right)<\infty$. Furthermore, we abbreviate ${\mathrm{abs}}_X\left(f\right)$ to $\mathrm{abs}\left(f\right)$, so there is no risk for confusion. The vector-valued Laplace transform has many interesting features; in the next section, we will particularly use the following ones (see, e.g., [2], Theorem 1.4.2).

Lemma 1. 

Let $f\in$(P1)-X, $s<0$ and $f_0\in L^1[s,0]$.

(i) 

Put $f_s\left(t\right):=f(t-s)$, $t\ge 0$, $h_s\left(t\right):=f(t+s)$, $t\ge -s$ and $h_s\left(t\right):=f_0(t+s)$, $t\in [0,-s]$. Then, $\mathrm{abs}\left(f_s\right)=\mathrm{abs}\left(h_s\right)=\mathrm{abs}\left(f\right)$, $\widetilde{f_s}\left(\lambda \right)=e^{-\lambda s}[\tilde{f}\left(\lambda \right)-{\int }_0^{-s}{e}^{-\lambda t}f\left(t\right)dt]$ and $\widetilde{h_s}\left(\lambda \right)=e^{\lambda s}[\tilde{f}\left(\lambda \right)-{\int }_0^s{e}^{-\lambda t}{f}_0\left(t\right)dt]$ ($\lambda \in \mathbb{C}$, $\Re \lambda >\mathrm{abs}\left(f\right))$.

(ii) 

Let $f\in$(P1)-X, $h\in L_{\mathrm{loc}}^1\left([0,\infty )\right)$ and abs $\left(\right|h\left|\right)<\infty$. Suppose, in addition, that $f\in C\left(\right[0,\infty ):X)$. Put

$$(h\ast f)\left(t\right):={\int }_0^{t}h(t-s)f\left(s\right)ds,t\ge 0.$$

Then, the mapping $t\mapsto (h\ast f)\left(t\right)$, $t\ge 0$ is continuous, $h\ast f\in$(P1)-X, and

$$\widetilde{h\ast f}\left(\lambda \right)=\tilde{h}\left(\lambda \right)\tilde{f}\left(\lambda \right),\lambda \in \mathbb{C},\Re \lambda >max\left(\mathrm{abs}\left(\right|h\left|\right),\mathrm{abs}\left(f\right)\right).$$

For more details about multivalued linear operators in locally convex spaces, $(a,k)$-regularized C-resolvent solution operator families subgenerated by multivalued linear operators and the Laplace transform of functions with values in SCLCSs, we refer the reader to [2] as well as to [15,16,17,18]; we will use the same notion and notation as in the easily accessible monograph [2]. Concerning fractional calculus and fractional differential equations, we will only recommend here the research monographs [19,20,21] and the doctoral dissertation [22]. If $\zeta >0$, then we define the Caputo fractional derivative ${\mathbf{D}}_t^{\zeta }u$ through Equation ([22], (1.18)). A basic source of information about abstract Volterra degenerate integro-differential equations can be found by consulting monographs [2,18,23] and the references cited therein.

2. $(\mathit{F},\mathit{G},\mathit{C})$-Resolvent Operator Families: Definitions and Main Results

We will always assume henceforth that $\mathcal{A}:X\to P\left(X\right)$ is an MLO, $C_1\in L(Y,X)$, $C_2\in L\left(X\right)$ is injective, $C\in L\left(X\right)$ is injective and $C\mathcal{A}\subseteq \mathcal{A}C$.

The following notion will be essentially important in our further work.

Definition 1. 

Suppose that $\omega \in \mathbb{R}$, $F:\{\lambda \in \mathbb{C}|\Re \lambda >\omega \}\to \mathbb{C}$ and $G:\{\lambda \in \mathbb{C}|\Re \lambda >\omega \}\to \mathbb{C}$.

(i) 

A strongly continuous operator family ${\left(R_1\left(t\right)\right)}_{t\ge 0}\subseteq L(Y,X)$ is said to be a mild $(F,G,C_1)$-regularized existence family with a subgenerator $\mathcal{A}$ if and only if for each $y\in Y$ the mapping $t\mapsto R_1\left(t\right)y$, $t\ge 0$ is Laplace transformable, $\mathrm{abs}\left(R_1(·)y\right)\le \omega$ and

$$\begin{array}{c}\hfill F\left(\lambda \right)C_{1}y\in \left(G\left(\lambda \right)-\mathcal{A}\right){\int }_0^{\infty }{e}^{-\lambda t}{R}_1\left(t\right)ydt,y\in Y,\Re \lambda >\omega .\end{array}$$

(ii) 

A strongly continuous operator family ${\left(R_2\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ is said to be a mild $(F,G,C_2)$-regularized uniqueness family with a subgenerator $\mathcal{A}$ if and only if for each $x\in D\left(\mathcal{A}\right)\cup R\left(\mathcal{A}\right)$ the mapping $t\mapsto R_2\left(t\right)x$, $t\ge 0$ is Laplace transformable, $\mathrm{abs}\left(R_2(·)x\right)\le \omega$ and

$$\begin{array}{c}\hfill F\left(\lambda \right)C_{2}y=G\left(\lambda \right){\int }_0^{\infty }{e}^{-\lambda t}{R}_2\left(t\right)xdt-{\int }_0^{\infty }{e}^{-\lambda t}{R}_2\left(t\right)ydt,\end{array}$$

(1)

whenever $(x,y)\in \mathcal{A}$ and $\Re \lambda >\omega$.

(iii) 

A strongly continuous operator family ${\left(R\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ is said to be an $(F,G,C)$-regularized resolvent family with a subgenerator $\mathcal{A}$ if and only if for each $x\in X$ the mapping $t\mapsto R\left(t\right)x$, $t\ge 0$ is Laplace transformable, $\mathrm{abs}\left(R\right(·\left)x\right)\le \omega$, $G\left(\lambda \right)\in {\rho }_C\left(\mathcal{A}\right)$ for $\Re \lambda >\omega$ and

$$\begin{array}{c}\hfill F\left(\lambda \right){\left(G\left(\lambda \right)-\mathcal{A}\right)}^{-1}Cx={\int }_0^{\infty }{e}^{-\lambda t}R\left(t\right)xdt,x\in X,\Re \lambda >\omega .\end{array}$$

If $C=\mathrm{I}$, then we omit the term “C” from the notation, with the meaning clear.

Keeping in mind the introduced notion, we can almost immediately clarify the following result.

Proposition 1. 

Suppose $\mathcal{A}$ is a closed MLO in X, $C_1\in L(Y,X),C_2\in L\left(X\right)$, $C_2$ is injective, ${\omega }_0\ge 0$, $\omega \ge max({\omega }_0,\mathrm{abs}\left(\right|a\left|\right),\mathrm{abs}\left(k\right))$ and $\tilde{a}\left(\lambda \right)\ne 0$ for $\Re \lambda >\omega$. Then, the following holds:

(i) 

If ${\left(R_1\left(t\right)\right)}_{t\ge 0}\subseteq L(Y,X)$ is a mild $(a,k)$-regularized $C_1$-existence family with a subgenerator $\mathcal{A}$ and the family $\{e^{-\omega t}{R}_1\left(t\right):t\ge 0\}\subseteq L(Y,X)$ is equicontinuous, then ${\left(R_1\left(t\right)\right)}_{t\ge 0}$ is a mild $(F,G,C_1)$-regularized existence family with a subgenerator $\mathcal{A}$, where

$$\begin{array}{c}\hfill F\left(\lambda \right):=\tilde{k}\left(\lambda \right)/\tilde{a}\left(\lambda \right),\Re \lambda >\omega andG\left(\lambda \right):=1/\tilde{a}\left(\lambda \right),\Re \lambda >\omega .\end{array}$$

(2)

(ii) 

If ${\left(R_2\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ is a mild $(a,k)$-regularized $C_2$-uniqueness family with a subgenerator $\mathcal{A}$ and the family $\{e^{-\omega t}{R}_2\left(t\right):t\ge 0\}\subseteq L\left(X\right)$ is equicontinuous, then ${\left(R_2\left(t\right)\right)}_{t\ge 0}$ is a mild $(F,G,C_2)$-regularized uniqueness family with a subgenerator $\mathcal{A}$, where $F(·)$ and $G(·)$ are given through (2).

(iii) 

If ${\left(R\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ is an $(a,k)$-regularized C-resolvent family with a subgenerator $\mathcal{A}$ and the family $\{e^{-\omega t}R\left(t\right):t\ge 0\}\subseteq L\left(X\right)$ is equicontinuous, then ${\left(R\left(t\right)\right)}_{t\ge 0}$ is an $(F,G,C)$-regularized resolvent family with a subgenerator $\mathcal{A}$, where $F(·)$ and $G(·)$ are given through (2).

The case in which $G\left(\lambda \right)=\tilde{a}\left(\lambda \right)$ for some Laplace transformable function $a(·)$ (see also (2)) is not important for us since, in this case, the corresponding abstract Volterra problem is equivalent to the problem ([2], (262)) with $\mathcal{B}=A$ and $A=\mathrm{I}$ therein.

If ${\left(R_1\left(t\right)\right)}_{t\ge 0}\subseteq L(Y,X)$ is a mild $(F,G,C_1)$-regularized existence family with a subgenerator $\mathcal{A}$, then it is clear that any extension of $\mathcal{A}$ is also a subgenerator of ${\left(R_1\left(t\right)\right)}_{t\ge 0}$. The integral generator ${\mathcal{A}}_{\mathrm{int}}$ of a mild $(F,G,C_2)$-regularized uniqueness family ${\left(R_2\left(t\right)\right)}_{t\ge 0}$ is defined by

$${\mathcal{A}}_{\mathrm{int}}:=\left\{(x,y)\in X\times X:\left(1\right) \mathrm{holds} \mathrm{for} \Re \lambda >\omega \right\}.$$

It is clear that the integral generator ${\mathcal{A}}_{\mathrm{int}}$ of ${\left(R_2\left(t\right)\right)}_{t\ge 0}$ is an extension of any subgenerator of ${\left(R_2\left(t\right)\right)}_{t\ge 0}$ as well that ${\mathcal{A}}_{\mathrm{int}}$ is likewise a subgenerator of ${\left(R_2\left(t\right)\right)}_{t\ge 0}$. After dividing (1) by $\lambda$, it is not difficult to prove that ${\mathcal{A}}_{\mathrm{int}}$ must be closed. Furthermore, if ${\left(R\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ is an $(F,G,C)$-regularized resolvent family with a closed subgenerator $\mathcal{A}$, then we can prove that $R\left(t\right)\mathcal{A}\subseteq \mathcal{A}R\left(t\right)$ for all $t\ge 0$, so that ${\left(R\left(t\right)\right)}_{t\ge 0}\subseteq L\left(X\right)$ is a mild $(F,G,C)$-regularized uniqueness family with subgenerator $\mathcal{A}$. In this case, we define the integral generator ${\mathcal{A}}_{\mathrm{int}}$ as above. The interested reader may try to clarify the main properties of subgenerators of $(F,G,C)$-regularized resolvent families.

The following simple example shows that it is not necessarily true that $F\equiv F_1$ and $G\equiv G_1$ if ${\left(R\left(t\right)\right)}_{t\ge 0}$ is both an $(F,G,C)$-regularized resolvent family with the integral generator $\mathcal{A}$ and an $(F_1,G_1,C)$-regularized resolvent family with the integral generator $\mathcal{A}$.

Example 2. 

Suppose that $\mathcal{A}=c\mathrm{I}$, where $c\ne 1$, $C=\mathrm{I}$ and $R\left(t\right)x:=(1/(1-c))e^{c/(1-c)t}$, $t\ge 0$, $x\in X$. Then, ${\left(R\left(t\right)\right)}_{t\ge 0}$ is both a $(1/(\lambda +1),\lambda /(\lambda +1),C)$-resolvent family with the integral generator $\mathcal{A}$ and a $\left(\right(2\lambda -c)/((1-c)\lambda -c),2\lambda ,C)$-resolvent family with the integral generator $\mathcal{A}$, as easily proven.

We continue by stating the following result:

Theorem 1. 

There exists a real number $\omega \ge 0$, an analytic function $F:\{\lambda \in \mathbb{C}|\Re \lambda >\omega \}\to \mathbb{C}$, a function $G:\{\lambda \in \mathbb{C}|\Re \lambda >\omega \}\to \mathbb{C}$, an infinite-dimensional Banach space X and an exponentially bounded $(F,G)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$ with the integral generator $\mathcal{A}=A\in L\left(X\right)$, having the empty point spectrum, such that there does not exist a Laplace transformable function $a\in L_{\mathrm{loc}}^{\infty }\left([0,\infty )\right)$ and a continuous Laplace transformable function $k(·)$ such that ${\left(R\left(t\right)\right)}_{t\ge 0}$ is an exponentially bounded $(a,k)$-regularized resolvent family generated by A.

Proof. 

It is well known that there exists an infinite-dimensional Banach space X and an operator $\mathcal{A}=A\in L\left(X\right)$ having the empty point spectrum. Then, it is very simple to show the following:

(i1)

If $Ø\ne \Omega \subseteq \mathbb{C}$ and $c_i:\Omega \to \mathbb{C}$ are given functions ($1\le i\le 4$) such that $c_1\left(\lambda \right)\mathrm{I}-c_2\left(\lambda \right)A=c_3\left(\lambda \right)\mathrm{I}-c_4\left(\lambda \right)A$ for all $\lambda \in \Omega$, then $c_1\left(\lambda \right)=c_3\left(\lambda \right)$ and $c_2\left(\lambda \right)=c_4\left(\lambda \right)$ for all $\lambda \in \Omega$.

(i2)

If $Ø\ne \Omega \subseteq \rho \left(A\right)$ and $c_i:\Omega \to \mathbb{C}$ are given functions ($1\le i\le 4$), then the assumption $c_1\left(\lambda \right){(c_2\left(\lambda \right)-A)}^{-1}=c_3\left(\lambda \right){(c_4\left(\lambda \right)-A)}^{-1}$ for all $\lambda \in \Omega$ implies $c_1\left(\lambda \right)=c_3\left(\lambda \right)$ and $c_2\left(\lambda \right)=c_4\left(\lambda \right)$ for all $\lambda \in \Omega$.

Furthermore, due to the famous counterexample by W. Desch and J. Prüss [24] (Proposition 4.1), we know that there exists an analytic function $H:\{\lambda \in \mathbb{C}:\Re \lambda >0\}\to \mathbb{C}$ and a finite real constant $M>0$ such that $\left|H\right(\lambda \left)\right|\le M/(1+|\lambda \left|\right)$, $\Re \lambda >0$ and there does not exist a Laplace transformable function $a\in L_{\mathrm{loc}}^{\infty }\left([0,\infty )\right)$ such that $\tilde{a}\left(\lambda \right)=H\left(\lambda \right)$ for $\Re \lambda >0$. Put $F\left(\lambda \right):={\lambda }^{-1}H\left(\lambda \right)$, $\Re \lambda >0$, $G\left(\lambda \right):=1/H\left(\lambda \right)$ if $\Re \lambda >0$ and $H\left(\lambda \right)\ne 0$, and $G\left(\lambda \right):={\lambda }_0\in \rho \left(A\right)$, if $\Re \lambda >0$ and $H\left(\lambda \right)=0$. Suppose that there exist two functions $a(·)$ and $k(·)$ with the prescribed assumptions such that A generates the exponentially bounded $(a,k)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$. Then, we have two possibilities:

(1)

$H\left(\lambda \right)\ne 0$ for all $\Re \lambda >0$. Then, a very simple analysis involving the inequality ${\parallel (\lambda -A)}^{-1}\parallel \le 1/\left(\right|\lambda |-\parallel A\parallel )$ for $\left|\lambda \right|>\parallel A\parallel$ implies that the function $\lambda \mapsto F\left(\lambda \right){(G\left(\lambda \right)-A)}^{-1}$, $\Re \lambda >0$ is analytic and bounded by $M^{\prime }/{\left|\lambda \right|}^2$ on the right half-plane, where $M^{\prime }>0$ is finite. Using the complex characterization theorem for the Laplace transform, we prove the existence of an exponentially bounded $(F,G)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$ with the integral generator A. Due to (i1)–(i2), we obtain that $H\left(\lambda \right)=\tilde{a}\left(\lambda \right)$, $\Re \lambda >0$, which is a contradiction.

(2)

There exists a discrete sequence ${\left({\lambda }_k\right)}_{k\in \mathbb{N}}$ in the right half-plane such that $H\left({\lambda }_k\right)=0$ for all $k\in \mathbb{N}$. Then, the mapping $\lambda \mapsto F\left(\lambda \right){(G\left(\lambda \right)-A)}^{-1}$, $\Re \lambda >0$, $\lambda \ne {\lambda }_k$ for all $k\in \mathbb{N}$; $\lambda \mapsto 0$, $\lambda ={\lambda }_k$ for some $k\in \mathbb{N}$, is analytic and bounded by $M^{″}/{\left|\lambda \right|}^2$ on some right half-plane, where $M^{″}>0$ is finite, as easily proven. Using the complex characterization theorem for the Laplace transform, we prove the existence of an exponentially bounded $(F,G)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$ with the integral generator A. Then, the final conclusion follows similar to in part (i), with the help of the uniqueness theorem for the analytic functions and the issues (i1)–(i2).

□

The proof of [24] (Proposition 4.1) is not given in a constructive way and it is not clear whether the function $H(·)$ has infinitely many zeros in the right half-plane. Furthermore, if the Laplace transform of a function $b\in L_{\mathrm{loc}}^1\left([0,\infty )\right)$ is equal to $H\left(\lambda \right)$, then the complex characterization theorem for the Laplace transform implies that the function $t\mapsto (g_{\alpha }\ast b)\left(t\right)$, $t\ge 0$ must be exponentially bounded for each $\alpha >0$; however, this does not imply the local essential boundedness of $b(·)$ as a class of very simple counterexamples shows (consider, for example, the function $b\left(t\right)={|t-1|}^{-1/2}$, $t\ne 1$). In the following extension of Theorem 1, we will use the well-known identities from the theory of Laplace transforms of distributions:

$$\begin{array}{c}\hfill \left[\mathcal{L}\left(\delta (·-a)\right)\right]=e^{-a\lambda }\mathrm{and}\left[\mathcal{L}\left({\delta }^{\prime }(·-a)\right)\right]=\lambda e^{-a\lambda },\end{array}$$

(3)

where $\delta (·-a)$ denotes the Dirac delta distribution centered at the point $t=a>0$:

Theorem 2. 

There exists a real number $\omega \ge 0$, an analytic function $F:\{\lambda \in \mathbb{C}|\Re \lambda >\omega \}\to \mathbb{C}$, a function $G:\{\lambda \in \mathbb{C}|\Re \lambda >\omega \}\to \mathbb{C}$, an infinite-dimensional Banach space X and an exponentially bounded $(F,G)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$ with the integral generator $\mathcal{A}=A\in L\left(X\right)$, having the empty point spectrum, such that the conclusions of Theorem 1 hold with a Laplace transformable function $a\in L_{\mathrm{loc}}^1\left([0,\infty )\right)$ and a continuous Laplace transformable function $k(·)$.

Proof. 

Let $\omega =0$ and let $X:=c_0$, the Banach space of all numerical sequences vanishing at infinity, equipped with the sup-norm. Define

$$A⟨x_k⟩:=⟨0,e^{-e^1}{x}_1,e^{-e^2}{x}_2,\dots ,e^{-e^k}{x}_k,\dots ⟩,⟨x_k⟩\in c_0.$$

Then, $A\in L\left(X\right)$, the spectrum of A is equal to $\left\{0\right\}$ and the point spectrum of A is empty, which can be simply shown. Let $\sigma <-1$, $F\left(\lambda \right):={\lambda }^{1+\sigma }{e}^{-\lambda }$ and $G\left(\lambda \right):=\lambda e^{-\lambda }$ ($\Re \lambda >0$). Since there exists a finite real constant $M>0$ such that $\parallel {(\lambda e^{-\lambda }-A)}^{-1}\parallel \le M$ if $|\lambda e^{-\lambda }|\le 2\parallel A\parallel$ and

$$∥{(\lambda e^{-\lambda }-A)}^{-1}∥\le \frac{|\lambda e^{-\lambda }|}{|\lambda e^{-\lambda }|-\parallel A\parallel }\le M$$

if $|\lambda e^{-\lambda }|>2\parallel A\parallel$, the complex characterization theorem for the Laplace transform implies that A is the integral generator of an exponentially bounded $(F,G)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$. Keeping in mind (3) and the uniqueness theorem for the Laplace transform of distributions (see, e.g., [25,26]), the argument contained in the proof of Theorem 1 shows that there does not exist a Laplace transformable function $a\in L_{\mathrm{loc}}^1\left([0,\infty )\right)$ and a continuous Laplace transformable function $k(·)$ such that ${\left(R\left(t\right)\right)}_{t\ge 0}$ is an exponentially bounded $(a,k)$-regularized resolvent family generated by A. □

We continue by introducing the following notion.

Definition 2. 

(i) 

Suppose that $\mathcal{A}$ is an MLO in X. Let $\alpha \in (0,\pi ]$, and let ${\left(R\left(t\right)\right)}_{t\ge 0}$ be an $(F,G,C)$-regularized resolvent family with subgenerator $\mathcal{A}$. Then, it is said that ${\left(R\left(t\right)\right)}_{t\ge 0}$ is an analytic $(F,G,C)$-regularized resolvent family of angle α if and only if there exists a function $\mathbf{R}:{\Sigma }_{\alpha }\to L\left(X\right)$ which satisfies that, for every $x\in X$, the mapping $z\mapsto \mathbf{R}\left(z\right)x$, $z\in {\Sigma }_{\alpha }$ is analytic, as well as that:

(a) 

$\mathbf{R}\left(t\right)=R\left(t\right),t>0$;

(b) 

${lim}_{z\to 0,z\in {\Sigma }_{\gamma }}\mathbf{R}\left(z\right)x=k\left(0\right)Cx$ for all $\gamma \in (0,\alpha )$ and $x\in X$.

(ii) 

Let ${\left(R\left(t\right)\right)}_{t\ge 0}$ be an analytic $(F,G,C)$-regularized resolvent family of angle $\alpha \in (0,\pi ]$. Then, it is said that ${\left(R\left(t\right)\right)}_{t\ge 0}$ is an exponentially equicontinuous, analytic $(F,G,C)$-regularized resolvent family of angle α (equicontinuous analytic $(F,G,C)$-regularized resolvent family of angle α), if and only if for every $\gamma \in (0,\alpha )$, there exists ${\omega }_{\gamma }\ge 0$ (${\omega }_{\gamma }=0$), such that the family $\{e^{-{\omega }_{\gamma }\Re z}\mathbf{R}\left(z\right):z\in {\Sigma }_{\gamma }\}\subseteq L\left(X\right)$ is equicontinuous. To avoid risk of confusion, we will identify in the following $R(·)$ and $\mathbf{R}(·)$.

The statements of [2] (Theorems 3.2.19, 3.2.25 and 3.2.26) can be straightforwardly formulated in our new framework. Furthermore, the subordination principle for abstract time-fractional inclusions clarified in [2] (Theorem 3.1.8) can be extended to $(F,G,C)$-regularized families, for example, if $\mathcal{A}$ is a closed subgenerator of an $(F,G,C)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$ satisfying certain extra properties, then for each number $\gamma \in (0,1)$, we can define the operator family ${\left(R_{\gamma }\left(t\right)\right)}_{t\ge 0}$ by $R_{\gamma }\left(0\right):=R\left(0\right)$ and

$$R_{\gamma }\left(t\right):={\int }_0^{\infty }{t}^{-\gamma }{\Phi }_{\gamma }\left(s{t}^{-\gamma }\right)R\left(s\right)xds,x\in X,t>0,$$

where ${\Phi }_{\gamma }(·)$ denotes the Wright function. Then, we have $\left[\widetilde{R_{\gamma }\left(t\right)}\right]\left(\lambda \right)={\lambda }^{\gamma -1}\left[\widetilde{R\left(t\right)}\right]\left({\lambda }^{\gamma }\right)$ for sufficiently large $\Re \lambda >\omega$, so that ${\left(R_{\gamma }\left(t\right)\right)}_{t\ge 0}$ is a $({·}^{\gamma -1}F\left({·}^{\gamma }\right),G\left({·}^{\gamma }\right),C)$-regularized resolvent family with subgenerator $\mathcal{A}$. Moreover, it can be proven that ${\left(R_{\gamma }\left(t\right)\right)}_{t\ge 0}$ is an exponentially equicontinuous analytic $({·}^{\gamma -1}F\left({·}^{\gamma }\right),G\left({·}^{\gamma }\right),C)$-regularized resolvent family of certain angles (for more details, see the issues (i)–(iii) in the formulation of the last mentioned result).

In [2] (Proposition 3.1.8 (i)), we have proven that the functional equality $C_2{R}_1\left(t\right)=R_2\left(t\right)C_1$, $t\in [0,\tau )$ holds for the local $(a,k)$-regularized $(C_1,C_2)$-existence and uniqueness families; in our new framework, we must assume certain extra conditions ensuring the validity of this equality. An attempt should be made to extend the real characterization theorem ([2], Theorem 3.2.12) for $(F,G,C)$-regularized resolvent families as well. On the other hand, the statement of [2] (Proposition 3.2.3) and the complex characterization theorem ([2], Theorem 3.2.10) can be straightforwardly extended to $(F,G,C)$-regularized resolvent families.

3. Some Applications

In this section, we provide some applications of $(F,G,C)$-regularized resolvent families and $(a,k)$-regularized C-resolvent families to the abstract Volterra integro-difference inclusions. We mainly use the vector-valued Laplace transform here.

First of all, we will provide the following illustrative example with $C=\mathrm{I}$.

Example 3. 

(i) 

Suppose that $0<\alpha <1$ and $\mathcal{A}$ is a closed MLO. Consider the following abstract fractional Cauchy inclusion

$$\begin{array}{c}\hfill {\mathbf{D}}_t^{\alpha }u\left(t\right)\in \mathcal{A}u(t+1)+f\left(t\right),t\ge 0;u\left(t\right)=u_0\left(t\right),0\le t<1.\end{array}$$

(4)

In some very simple situations (for example, if $\mathcal{A}=c\mathrm{I}$ for some $c\in \mathbb{C}\backslash \left\{0\right\}$), the unique solution of (4) can be directly calculated step by step; on the other hand, the theory of vector-valued Laplace transforms is completely inapplicable here. To explain this in more detail, let us try to apply the Laplace transform identity ([22], (1.23)) and Lemma 1 (i). Then, we get

$$\begin{array}{c}\hfill {\lambda }^{\alpha }\tilde{u}\left(\lambda \right)-{\lambda }^{\alpha -1}u\left(0\right)\in \mathcal{A}\left[e^{\lambda }\left\{\tilde{u}\left(\lambda \right)-{\int }_0^1{e}^{-\lambda t}{u}_0\left(t\right)dt\right\}\right]+\tilde{f}\left(\lambda \right),\Re \lambda >0sufficientlylarge,\end{array}$$

provided that all terms are well defined and exponentially bounded, with the meaning clear. This simply implies

$$\begin{array}{c}\hfill u\left(t\right)={\mathcal{L}}^{-1}\left[{\left({\lambda }^{\alpha }-e^{\lambda }\mathcal{A}\right)}^{-1}\left\{{\lambda }^{\alpha -1}u\left(0\right)+\tilde{f}\left(\lambda \right)-e^{\lambda }\mathcal{A}{\int }_0^1{e}^{-\lambda t}{u}_0\left(t\right)dt\right\}\right]\left(t\right),t\ge 0,\end{array}$$

provided that all terms are well defined. Consider now the case $u_0\left(t\right)\equiv 0$, with $\mathcal{A}=c\mathrm{I}$ for some $c\in \mathbb{C}\backslash \left\{0\right\}$, then we obtain

$$\begin{array}{c}\hfill u\left(t\right)=\left({\mathcal{L}}^{-1}\left[{\lambda }^{\alpha -1}{\left({\lambda }^{\alpha }-c{e}^{\lambda }\right)}^{-1}\right]\ast f\right)\left(t\right),t\ge 0.\end{array}$$

However, there does not exist an exponentially bounded $(F,G)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$ with the integral generator $c\mathrm{I}$ such that

$${\lambda }^{\alpha -1}{\left({\lambda }^{\alpha }-c{e}^{\lambda }\right)}^{-1}x={\int }_0^{\infty }{e}^{-\lambda t}R\left(t\right)xdt,\Re \lambda >0suff.large,x\in X.$$

Speaking matter-of-factly, the mapping $\lambda \mapsto {\lambda }^{\alpha }-c{e}^{\lambda }$ has infinitely many zeros in any right half-plane and the function $\lambda \mapsto {\lambda }^{\alpha -1}{({\lambda }^{\alpha }-c{e}^{\lambda })}^{-1}$ cannot be analytic there (the situation in which there exists a complex number $c\in \mathbb{C}\backslash \left\{0\right\}$ such that $c\notin \rho \left(\mathcal{A}\right)$ cannot be considered as well; on the other hand, the situation in which $\sigma \left(A\right)=\left\{0\right\}$ can be simply considered following the lines of the proof of Theorem 2).

As mentioned in the introductory part, the above example indicates that it is very difficult to apply the vector-valued Laplace transform in the analysis of the abstract fractional Cauchy inclusions ${\left(DFP\right)}_{R,a,b}$ and ${\left(DFP\right)}_{L,a,b}$, where $0\le a<b$, $f:[0,\infty )\to X$ is exponentially bounded, B is a closed linear operator and $\mathcal{A}$ and $\mathcal{B}$ are closed MLOs (we can use the substitution $t\mapsto t+a$ and a similar calculation).

(ii) 

Suppose now that $0<\alpha <1$ and consider the following abstract fractional Cauchy inclusion

$$\begin{array}{c}\hfill {\mathbf{D}}_t^{\alpha }u(t+1)\in \mathcal{A}u\left(t\right)+f\left(t\right),t\ge 0;u\left(0\right)=x,{\mathbf{D}}_t^{\alpha }u\left(t\right)=u_0\left(t\right),0\le t<1.\end{array}$$

(5)

The unique solution of (5) can be directly calculated step by step in some situations, but much better results can be established by applying the vector-valued Laplace transform and the theory of $(a,k)$-regularized C-resolvent families. In fact, the unique solution $u\left(t\right)$ of (5) is given by:

$$\begin{array}{c}\hfill u\left(t\right)={\mathcal{L}}^{-1}\left[{\left({\lambda }^{\alpha }{e}^{\lambda }-\mathcal{A}\right)}^{-1}\left\{\tilde{f}\left(\lambda \right)+e^{\lambda }{\int }_0^1{e}^{-\lambda t}{u}_0\left(t\right)dt+{\lambda }^{\alpha -1}{e}^{\lambda }u\left(0\right)\right\}\right]\left(t\right),t\ge 0.\end{array}$$

If $u_0\left(t\right)\equiv 0$ and $x=0$, then it can be simply verified that we have $u\left(t\right)=(R\ast f)\left(t\right)$, $t\ge 0$, where ${\left(R\left(t\right)\right)}_{t\ge 0}$ is an exponentially equicontinuous $(a,a)$-regularized resolvent family generated by $\mathcal{A}$ with $a\left(t\right)$ being equal to zero for $0\le t\le 1$ and $g_{2-\alpha }(t-1)$ for $t>1$. However, some serious problems occur again since the complement of the set $\{{\lambda }^{\alpha }{e}^{\lambda }:\lambda \in \mathbb{C},\Re \lambda >\omega \}$ is compact for any $\omega \ge 0$, which implies that the resolvent set of $\mathcal{A}$ must contain the set $\mathbb{C}\backslash K$ for some compact set $K\subseteq \mathbb{C}$. If $\mathcal{A}=A$ is single-valued with polynomially bounded resolvent and X is a Banach space, then the above implies that A must be bounded, when some applications can be given.

Usually, in order to study abstract (multi-term) fractional differential inclusions, we convert them into the corresponding abstract (multi-term) Volterra integro-differential inclusions. Abstract multi-term Volterra integro-functional inclusions will be our main subject in the remainder of this section, which will be broken down into two individual parts:

1. In the first part, we consider the following abstract Volterra initial value problem

$$\begin{array}{c}\hfill u\left(t\right)\in f\left(t\right)+\sum _{i=1}^m\left(a_i\ast \mathcal{A}u\left(·+t_i\right)\right)\left(t\right),t\ge 0;u\left(t\right)=u_0\left(t\right),0\le t\le \tau ,\end{array}$$

(6)

where $m\in \mathbb{N}$, $0\le t_1,t_2,\dots ,t_m<+\infty$, $\tau :=max(t_1,t_2,\dots ,t_m)$, $f:[0,\infty )\to X$ is Lebesgue measurable, $u_0\in L^1([0,\tau ]:X)$, $\mathcal{A}$ is a closed MLO on X and for each $i\in {\mathbb{N}}_m$ we have that $a_i(·)$ is the sum of a locally integrable scalar-valued function defined for $t\ge 0$ and a scalar-valued distribution $a_i(·)=c_i\delta (·-b_i)$, where $c_i\in \mathbb{C}$ and $0\le b_i<+\infty$ (if $a_i(·)=c_i\delta (·-b_i)$, then according to Lemma 1 and (3), we define $(a_i\ast u)\left(t\right):=c_{i}u(t-b_i)$, $t>b_i$ and $(a_i\ast u)\left(t\right):=0$, $0\le t\le b_i$).

We will use the following concepts of solutions.

Definition 3. 

(i) 

By a mild solution of (6), we mean any continuous function $u:[0,\infty )\to X$ such that $(a_i\ast u(·+t_i))\left(t\right)\in D\left(\mathcal{A}\right)$ for all $t\ge 0$, $i\in {\mathbb{N}}_m$ and

$$\begin{array}{c}\hfill u\left(t\right)\in f\left(t\right)+\sum _{i=1}^m\mathcal{A}\left(a_i\ast u\left(·+t_i\right)\right)\left(t\right),t\ge 0;u\left(t\right)=u_0\left(t\right),0\le t\le \tau .\end{array}$$

(7)

(ii) 

By an LT-mild solution of (6), we mean any Laplace transformable mild solution of (6).

(iii) 

By a strong solution of (6), we mean any continuous function $u:[0,\infty )\to X$ of (6) which satisfies that, for every $i\in {\mathbb{N}}_m$, there exists a continuous function $u_i:[0,\infty )\to X$ such that $u_i\left(t\right)\in \mathcal{A}\left(u(t+t_i)\right)$ for all $t\ge 0$ and

$$\begin{array}{c}\hfill u\left(t\right)=f\left(t\right)+\sum _{i=1}^m\left(a_i\ast u_i(·)\right)\left(t\right),t\ge 0;u\left(t\right)=u_0\left(t\right),0\le t\le \tau .\end{array}$$

(8)

(iv) 

By a strong LT-solution of (6), we mean any continuous Laplace transformable function $u:[0,\infty )\to X$ of (6) which satisfies that, for every $i\in {\mathbb{N}}_m$, there exists a continuous Laplace transformable function $u_i:[0,\infty )\to X$ such that $u_i\left(t\right)\in \mathcal{A}\left(u(t+t_i)\right)$ for all $t\ge 0$ and (8) holds.

Since $\mathcal{A}$ is closed, any strong (LT-)solution of (6) is also a mild (LT-)solution of (6); on the other hand, it is clear that any LT-mild solution of (6) is a mild solution of (6) and any strong LT-solution of (6) is a strong solution of (6). Unfortunately, it is very difficult to say anything relevant about the existence and uniqueness of mild (strong) solutions of problem (6).

In the following, we will always assume that the following condition holds.

(Q):

There exists $\omega \in \mathbb{R}$ such that $\omega \ge max\left(\mathrm{abs}\right(|a_1\left|\right),\dots ,\mathrm{abs}\left(\right|a_m\left|\right))$ and

$$\begin{array}{c}\hfill \widetilde{a_1}\left(\lambda \right)e^{\lambda t_1}+\dots +\widetilde{a_m}\left(\lambda \right)e^{\lambda t_m}\ne 0,\Re \lambda >\omega ,\end{array}$$

where we put $\mathrm{abs}\left(\right|a_i\left|\right):=-\infty$ if $a_i(·)=c_i\delta (·-b_i)$ with some $c_i\in \mathbb{C}$ and $0\le b_i<+\infty$ as well as $\mathrm{abs}\left(\right|a_i\left|\right):=\mathrm{abs}\left(\right|b_i\left|\right)$ if $a_i(·)=b_i(·)+c_i\delta (·-b_i)$ with some Laplace transformable function $b_i(·)$, $c_i\in \mathbb{C}$ and $0\le b_i<+\infty$ ($1\le i\le m$).

Define

$$\begin{array}{c}\hfill G\left(\lambda \right):=\frac{1}{\widetilde{a_1}\left(\lambda \right)e^{\lambda t_1}+\dots +\widetilde{a_m}\left(\lambda \right)e^{\lambda t_m}},\Re \lambda >\omega .\end{array}$$

The main result concerning the well-posedness of problem (6) reads as follows.

Theorem 3. 

Suppose that $\mathcal{A}$ is a closed subgenerator of an exponentially equicontinuous $(\tilde{k}(·)G,G,C)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$, where $k(·)$ is Laplace transformable or $k(·)=\delta (·)$, the Dirac delta distribution.

(i) 

If $f(·)$ is Laplace transformable and $k\ne 0$, then the solutions of (6) are unique.

(ii) 

If there exists a function $v_0\in L^1([0,\tau ]:X)$ such that $v_0\left(t\right)\in \mathcal{A}{u}_0\left(t\right)$ for almost all $t\in [0,\tau ]$ and a Laplace transformable function $u_1:[0,\infty )\to X$ such that

$$\begin{array}{c}\hfill {\int }_0^{\infty }{e}^{-\lambda t}{u}_1\left(t\right)dt=\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}{v}_0\left(t\right)dt,\Re \lambda >0 suff.large,\end{array}$$

then $u(·)$ is a strong LT-solution of (6) if and only if

$$\begin{array}{c}\hfill (k\ast Cu)\left(t\right)=\left(R\ast \left[f-u_1\right]\right)\left(t\right),t\ge 0.\end{array}$$

(9)

(iii) 

Suppose that there exists a Laplace transformable function $u_2:[0,\infty )\to X$ such that

$$\begin{array}{c}\hfill {\int }_0^{\infty }{e}^{-\lambda t}{u}_2\left(t\right)dt=G\left(\lambda \right)\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}{u}_0\left(t\right)dt,\Re \lambda >0 suff.large.\end{array}$$

Then, $u(·)$ is a strong LT-solution of (6) if and only if

$$\begin{array}{c}\hfill (k\ast Cu)\left(t\right)=\left(R\ast \left[f-u_2\right]\right)\left(t\right)+\left(kC\ast u_2\right)\left(t\right),t\ge 0.\end{array}$$

Proof. 

We have

$$\tilde{k}\left(\lambda \right){\left[I-\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}\mathcal{A}\right]}^{-1}Cx={\int }_0^{\infty }{e}^{-\lambda t}R\left(t\right)xdt,x\in X,\Re \lambda >{\omega }_0>\omega \mathrm{suff}.\mathrm{large},$$

cf. also (3). Suppose first that $u(·)$ is a strong LT-solution of (6). Then, $\widetilde{u_i}\left(\lambda \right)\in e^{\lambda t_i}\mathcal{A}[\tilde{u}\left(\lambda \right)-{\int }_0^{t_i}{e}^{-\lambda t}{u}_0\left(t\right)dt]$, $\Re \lambda >{\omega }_0$ ($1\le i\le m$). Applying the Laplace transform to (8), we get

$$\tilde{u}\left(\lambda \right)\in \tilde{f}\left(\lambda \right)+\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}\mathcal{A}\left[\tilde{u}\left(\lambda \right)-{\int }_0^{t_i}{e}^{-\lambda t}{u}_0\left(t\right)dt\right],\Re \lambda >{\omega }_0.$$

The prescribed assumptions imply

$$\begin{array}{c}\hfill \left[I-\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}\mathcal{A}\right]C\tilde{u}\left(\lambda \right)\in C\tilde{f}\left(\lambda \right)-C\mathcal{A}\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}{u}_0\left(t\right)dt,\Re \lambda >{\omega }_0\end{array}$$

(10)

and

$$\begin{array}{cc}\hfill C\tilde{u}\left(\lambda \right)& \in {\left[I-\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}\mathcal{A}\right]}^{-1}C\tilde{f}\left(\lambda \right)\hfill \\ & -{\left[I-\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}\mathcal{A}\right]}^{-1}C\mathcal{A}\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}{u}_0\left(t\right)dt,\Re \lambda >{\omega }_0.\hfill \end{array}$$

Since the operator ${[I-{\sum }_{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}\mathcal{A}]}^{-1}C\mathcal{A}$ is single-valued on account of [2] (Theorem 1.2.4 (i)), we get that

$$\begin{array}{c}\tilde{k}\left(\lambda \right)C\tilde{u}\left(\lambda \right)=\tilde{k}\left(\lambda \right)G\left(\lambda \right){\left(G\left(\lambda \right)-\mathcal{A}\right)}^{-1}C\tilde{f}\left(\lambda \right)\hfill \\ -\tilde{k}\left(\lambda \right)G\left(\lambda \right)\left[G\left(\lambda \right){\left(G\left(\lambda \right)-\mathcal{A}\right)}^{-1}C-C\right]\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}{u}_0\left(t\right)dt,\Re \lambda >{\omega }_0.\hfill \end{array}$$

(11)

If $u_0\equiv 0$, then the Titchmarsh convolution theorem, the injectivity of C and the above arguments together imply (9) with $u_1\equiv 0$, finishing the proof of (i). On the other hand, a simple argument involving [2] (Theorem 1.2.3) and (10) shows that

$$\begin{array}{cc}\hfill \widetilde{(k\ast Cu)(·)}\left(\lambda \right)& =\tilde{k}\left(\lambda \right){\left[I-\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}\mathcal{A}\right]}^{-1}C\tilde{f}\left(\lambda \right)\hfill \\ & -\tilde{k}\left(\lambda \right){\left[I-\sum _{i=1}^m{e}^{\lambda t_i}\mathcal{A}\right]}^{-1}C\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}{v}_0\left(t\right)dt,\Re \lambda >{\omega }_0.\hfill \end{array}$$

Performing the inverse Laplace transform, we get (9). The proof of sufficiency in part (ii) can be deduced by following the inverse procedure. The proof of (iii) follows from a similar argument involving Equation (11) and therefore can be omitted. □

We continue by providing some illustrative examples.

Example 4. 

(i) 

Suppose that $m=1$, $t_1=1$ and $a_1\left(t\right)\equiv 1$. If $\mathcal{A}=A$ is the operator from the formulation of Theorem 2, then the theory of $(a,k)$-regularized C-resolvent families cannot be applied in the analysis of the well-posedness of problem (6). On the other hand, we can apply Theorem 3, with $F\left(\lambda \right)={\lambda }^{1+\sigma }{e}^{-\lambda }$ for arbitrary $\sigma <-1$ and $G\left(\lambda \right)=\lambda e^{-\lambda }$, in the analysis of the well-posedness of problem (6).

(ii) 

Suppose that $a(·)$ and $k(·)$ are Laplace transformable, $k(·)$ is continuous, $\tilde{a}(·)$ does not vanish on some right half-plane and $\mathcal{A}$ is a subgenerator of an exponentially equicontinuous $(a,k)$-regularized C-resolvent operator family ${\left(R\left(t\right)\right)}_{t\ge 0}$. If $m=1$ and $t_1>0$, then we define $a_1\left(t\right):=a(t-t_1)$, $t\ge t_1$ and $a_1\left(t\right):=0$, $0\le t<t_1$. Then, $\widetilde{a_1}\left(\lambda \right)=e^{-\lambda t_1}\tilde{a}\left(\lambda \right)$ for sufficiently large $\Re \lambda >0$ and Theorem 3 can be applied with $G\left(\lambda \right)={\left[\tilde{a}\left(\lambda \right)\right]}^{-1}$, $u_1(·)=(a\ast v_0{\chi }_{[0,t_1]})(·)$ and $u_2(·)=\left(u_0{\chi }_{[0,t_1]}\right)(·)$, which is the most symptomatic and easiest way to use this result. If this is the case and $u(·)$ is a strong LT-solution of (6), then we have

$$\begin{array}{cc}\hfill {\int }_0^t& a_1\left(s\right)u_1\left(t+t_1-s\right)ds={\int }_{t_1}^t{a}_1\left(s\right)u_1\left(t+t_1-s\right)ds\hfill \\ & ={\int }_{t_1}^{t}a\left(s-t_1\right)u_1\left(t-s+t_1\right)ds={\chi }_{[t_1,+\infty )}\left(t\right){\int }_0^{t-t_1}a\left(s\right)u_1(t-s)ds,t\ge 0,\hfill \end{array}$$

so that $u_0\left(t\right)=f\left(t\right)$, $0\le t\le t_1$ and $u(·)$ solves, in a certain sense, the abstract Cauchy inclusion

$$\begin{array}{c}\hfill v\left(t\right)\in f\left(t\right)+\mathcal{A}{\int }_0^{t-t_1}a\left(s\right)v(t-s)ds,t>t_1;v\left(t\right)=f\left(t\right),0\le t\le t_1.\end{array}$$

(12)

Moreover, if the requirement from Theorem 3(i) holds and $u_{\mathrm{cl}}(·)$ is a strong solution of the abstract Cauchy inclusion

$$\begin{array}{c}\hfill u\left(t\right)\in f\left(t\right)+\mathcal{A}{\int }_0^{t}a(t-s)u\left(s\right)ds,t\ge 0,\end{array}$$

then the use of (9) and [2] (Proposition 2.3.8 (ii)) implies

$$\begin{array}{c}\hfill \left(k\ast C\left[u_{\mathrm{cl}}-u\right]\right)\left(t\right)=\left(R\ast a\ast v_0{\chi }_{[0,t_1]}\right)\left(t\right),t\ge 0.\end{array}$$

(13)

As an application, we can consider the fractional-functional Poisson heat equations of the form (12), for example, cf. [2,18] for more details about the subject.

(iii) 

The situation in which there exist positive real numbers ${\alpha }_i>0$ such that $\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}={\lambda }^{-{\alpha }_i}$ for $1\le i\le m$ is also important. In this situation, $\mathcal{A}$ needs to be a subgenerator of an exponentially equicontinuous $(a,k)$-regularized C-resolvent operator family ${\left(R\left(t\right)\right)}_{t\ge 0}$, where $a\left(t\right)=g_{{\alpha }_1}\left(t\right)+\dots +g_{{\alpha }_m}\left(t\right)$; additionally, in place of kernels $g_{{\alpha }_1}\left(t\right)$, …, $g_{{\alpha }_m}\left(t\right)$, we can consider the general Laplace transformable kernels $a_1\left(t\right)$, …, $a_m\left(t\right)$.

In many concrete situations, which are not similar to the situations explored in Example 4(ii)–(iii), the complement of the set $\left\{G\right(\lambda ):\Re \lambda >\omega \}$ is compact in $\mathbb{C}$; without going into further details concerning this question and many other important questions concerning the range and zeros of complex exponential polynomials (see, e.g., [27,28] and the references cited therein for more details about these issues), it seems very plausible that the above happens if

$$G\left(\lambda \right)=\frac{1}{{\lambda }^{-{\alpha }_1}{e}^{\lambda t_1^{\prime }}+\dots +{\lambda }^{-{\alpha }_m}{e}^{\lambda t_m^{\prime }}},\Re \lambda >\omega ,$$

for some positive real numbers ${\alpha }_i>0$ and non-zero real numbers $t_i^{\prime }\ne 0$ ($1\le i\le m$). Therefore, the operators from Example 1 are sometimes essentially important in the analysis of the existence and uniqueness of convoluted solutions of the abstract Cauchy problem (6).

Example 5. 

Suppose that A is the operator from Example 1, ${\omega }_0\ge 0$, $a_1(·)=\dots =a_m(·)=a(·)$ satisfy the general assumptions of Theorem 3, $0<t_1<t_2<\dots <t_m$, the numbers $t_2,\dots ,t_m$ are integer multiples of $t_1$ and there exists a complex polynomial $S(·)$ such that $S\left(z\right)\ne 0$ for $\Re z\ge {\omega }_0$ and

$$\begin{array}{c}\hfill e^{\lambda t_1}+\dots +e^{\lambda t_m}=S\left(e^{\lambda t_1}\right),\Re \lambda >{\omega }_0.\end{array}$$

Then, there exists $\omega \ge {\omega }_0$ such that condition (Q) holds. Suppose, further, that there exist ${\omega }_1\ge {\omega }_0$, $M>0$ and $\beta \in (0,N)$ such that $|\tilde{a}{\left(\lambda \right)|}^{-\sigma }\le M{\left|\lambda \right|}^{\beta }$ for $\Re \lambda \ge {\omega }_1$. Since

$$\left|e^{\lambda t_1}+\dots +e^{\lambda t_m}\right|\ge e^{\Re \left(\lambda \right)t_m}-e^{\Re \left(\lambda \right)t_{m-1}}-\dots -e^{\Re \left(\lambda \right)t_1},\lambda \in \mathbb{C},$$

there exists a sufficiently large real number $\omega \ge {\omega }_1$ such that the assumptions of Theorem 3 are satisfied with a kernel $k(·)$ whose Laplace transform decays ultra-polynomially on some right half-space. For example, Theorem 3(ii) can be applied provided that $a(·)=c\delta (·-t_{m+1})$, where $c\in \mathbb{C}$ and $+\infty >t_{m+1}>t_m$, while Theorem 3(iii) can be applied, with $u_2(·)\equiv u_0(·)$, provided that $u_0\left(t\right)=0$, $t>t_1$. Concerning the corresponding abstract Cauchy problems, the main applications can be given in the case that the function $(f-u_1)(·)$ [$(f-u_2)(·)$] belongs to the range of the convolution transform $k\ast ·$, which can really come off.

Without going into full details, we will only note that Example 1 can be reconsidered in the degenerate setting as well as that certain applications can be given to the fractional-functional analogues of the linearized Benney–Luke type equation

$${\left(\lambda u-\Delta u\right)}_t=\alpha \Delta u-\beta {\Delta }^{2}u(\alpha >0,\beta >0;\lambda \in \mathbb{R});$$

see [18] (Theorem 1.14, p. 28) and [2] (Example 2.2.18) for further information in this direction.

2. In the second part, we consider the following abstract Volterra initial value problem

$$\begin{array}{c}\hfill u\left(t\right)\in f\left(t\right)+\sum _{i=1}^m\left(a_i\ast \mathcal{A}u\left(·+t_i\right)\right)\left(t\right),t\in \mathbb{R};u\left(t\right)=u_0\left(t\right),t\in \left[{\tau }_1,{\tau }_2\right],\end{array}$$

(14)

where $m\in \mathbb{N}$, $t_1,t_2,\dots ,t_m\in \mathbb{R}$ and there exists an index $i\in {\mathbb{N}}_m$ where $t_i<0$, $f:[0,\infty )\to X$ is Lebesgue measurable, $\mathcal{A}$ is a closed MLO on X and $a_1(·),\dots ,a_m(·)$ are locally integrable scalar-valued functions defined for $t\in \mathbb{R}$ such that the functions ${a_i}_{\left|\right[0,\infty )}(·)$ and ${\check{a_i}}_{\left|\right[0,\infty )}(·)$ are Laplace transformable for $1\le i\le m$ (for the sake of simplicity, we will not consider the case in which some $a_i(·)$ contain a non-vanishing distribution part), ${\tau }_1:=min(t_1,t_2,\dots ,t_m)$, ${\tau }_2:=max(t_1,t_2,\dots ,t_m)$ if there exists an index $j\in {\mathbb{N}}_m$ with $t_j>0$ and ${\tau }_2:=0$, otherwise.

We define a mild (strong) solution of (14) in the same way as in Definition 3. By a mild LT-solution of (14) we mean any mild solution of (14) such that the functions $u_{[0,\infty )}(·)$ and $\check{u_{[0,\infty )}}(·)$ are Laplace transformable; by a strong LT-solution of (14) we mean any continuous function $u:\mathbb{R}\to X$ such that the functions $u_{[0,\infty )}(·)$ and $\check{u_{[0,\infty )}}(·)$ are Laplace transformable and which satisfies that, for every $i\in {\mathbb{N}}_m$, there exists a continuous function $u_i:\mathbb{R}\to X$ such that the functions $u_{i;[0,\infty )}(·)$ and $\check{u_{i;[0,\infty )}}(·)$ are Laplace transformable, $u_i\left(t\right)\in \mathcal{A}\left(u(t+t_i)\right)$ for all $t\in \mathbb{R}$ and

$$\begin{array}{c}\hfill u\left(t\right)=f\left(t\right)+\sum _{i=1}^m\left(a_i\ast u_i(·)\right)\left(t\right),t\in \mathbb{R};u\left(t\right)=u_0\left(t\right),{\tau }_1\le t\le {\tau }_2.\end{array}$$

Keeping in mind the introduced notion, we have

$$\begin{array}{cc}\hfill & \check{u}\left(t\right)=u(-t)\in f(-t)+\sum _{i=1}^m{\int }_0^{-t}{a}_i\left(s\right)\mathcal{A}u(-t-s+t_i)ds\hfill \\ & \in f(-t)+\sum _{i=1}^m{\int }_0^{-t}{a}_i\left(s\right)\mathcal{A}\check{u}(t+s-t_i)ds\in f(-t)-\sum _{i=1}^m{\int }_0^t{a}_i(-s)\mathcal{A}\check{u}(t-s-t_i)ds\hfill \\ & \in f(-t)+\sum _{i=1}^m{\int }_t^0{a}_i(s-t)\mathcal{A}\check{u}(s-t_i)ds\in f(-t)-\sum _{i=1}^m{\int }_0^t\check{a_i}(t-s)\mathcal{A}\check{u}(s-t_i)ds,t\in \mathbb{R},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \check{u}\left(t\right)=u(-t)=f(-t)+\sum _{i=1}^m{\int }_0^{-t}{a}_i\left(s\right)u_i(-t-s)ds\hfill \\ & =f(-t)+\sum _{i=1}^m{\int }_0^{-t}{a}_i\left(s\right)\check{u_i}(t+s)ds=f(-t)-\sum _{i=1}^m{\int }_0^t{a}_i(-s)\check{u_i}(t-s)ds\hfill \\ & =f(-t)-\sum _{i=1}^m{\int }_0^t\check{a_i}(-s)\check{u_i}(t-s)ds=f(-t)-\sum _{i=1}^m{\int }_0^t\check{a_i}(t-s)\check{u_i}(t-s)ds,t\in \mathbb{R}.\hfill \end{array}$$

Since $\check{u_i}\left(t\right)\in \mathcal{A}\check{u}(t-t_i)$, $t\in \mathbb{R}$, $1\le i\le m$, this implies the following.

Proposition 2. 

Suppose that $u:\mathbb{R}\to X$ is a continuous function. Then, $u(·)$ is a mild (strong) solution of (14) (mild LT- (strong LT-) solution of (14)) if and only if the function $v(·)=\check{u}(·)$ is a mild (strong) solution of (15) (mild LT- (strong LT-) solution of (15)), where

$$\begin{array}{c}\hfill v\left(t\right)\in \check{f}\left(t\right)+\sum _{i=1}^m\left(\check{a_i}\ast (-\mathcal{A})u\left(·-t_i\right)\right)\left(t\right),t\in \mathbb{R};v\left(t\right)=v_0\left(t\right)=\check{u_0}\left(t\right),t\in \left[-{\tau }_2,-{\tau }_1\right].\end{array}$$

(15)

If the functions $a(·)$ and $\check{a}(·)$ are Laplace transformable, then we have

$${\int }_0^{\infty }{e}^{-\lambda t}\check{a}\left(t\right)dt={\int }_{-\infty }^0{e}^{\lambda t}a\left(t\right)dt,\Re \lambda >0 \mathrm{suff}.\mathrm{large}.$$

Keeping this in mind, Lemma 1(i), Proposition 2, Theorem 3 and its proof together imply, after considering the functions $u_{\left|\right[0,\infty )}$ and $u_{\left|\right(-\infty ,0]}$ separately, the following result.

Theorem 4. 

Suppose that $\mathcal{A}$ is a closed subgenerator of an exponentially equicontinuous $(\tilde{k}(·)G,G,C)$-regularized resolvent family ${\left(R\left(t\right)\right)}_{t\ge 0}$, where $k(·)$ is Laplace transformable or $k(·)=\delta (·)$, the Dirac delta distribution. Further suppose that $-A$ is a subgenerator of an exponentially equicontinuous $(\widetilde{k_1}(·)G_1,G_1,C^{\prime })$-regularized resolvent family ${\left(R_1\left(t\right)\right)}_{t\ge 0}$, where $C^{\prime }\in L\left(X\right)$ is injective and commutes with A as well as that $k_1(·)$ is Laplace transformable or $k_1(·)=\delta (·)$, and

$$G_1\left(\lambda \right)=\frac{1}{\left[{\int }_{-\infty }^0{e}^{\lambda t}{a}_1\left(t\right)dt\right]e^{-\lambda t_1}+\dots +\left[{\int }_{-\infty }^0{e}^{\lambda t}{a}_m\left(t\right)dt\right]e^{-\lambda t_m}},$$

where we assume that $G_1\left(\lambda \right)$ is well defined on some right half-plane.

(i) 

If $f(·)$ and $\check{f}(·)$ are Laplace transformable, $k\ne 0$ and $k_1\ne 0$, then the solutions of (14) are unique.

(ii) 

If there exists a function $v_0\in L^1([{\tau }_1,{\tau }_2]:X)$ such that $v_0\left(t\right)\in \mathcal{A}{u}_0\left(t\right)$ for almost all $t\in [{\tau }_1,{\tau }_2]$, there exists a Laplace transformable function $u_1:[0,\infty )\to X$ such that

$${\int }_0^{\infty }{e}^{-\lambda t}{u}_1\left(t\right)dt=\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}{v}_0\left(t\right)dt,\Re \lambda >0 suff.large,$$

and there exists a Laplace transformable function $u_2:[0,\infty )\to X$ such that

$${\int }_0^{\infty }{e}^{-\lambda t}{u}_2\left(t\right)dt=\sum _{i=1}^m\left[{\int }_{-\infty }^0{e}^{\lambda t}{a}_i\left(t\right)dt\right]e^{-\lambda t_i}{\int }_0^{-t_i}{e}^{-\lambda t}{v}_0(-t)dt,\Re \lambda >0 suff.large.$$

Then, $u(·)$ is a strong LT-solution of (14) if and only if

$$\begin{array}{c}\hfill (k\ast Cu)\left(t\right)=\left(R\ast \left[f-u_1\right]\right)\left(t\right),t\ge 0and\left(k_1\ast C^{\prime }\check{u}\right)\left(t\right)=\left(R_1\ast \left[\check{f}-u_2\right]\right)\left(t\right),t\ge 0.\end{array}$$

(iii) 

Suppose that there exists a Laplace transformable function $u_3:[0,\infty )\to X$ such that

$$\begin{array}{c}\hfill {\int }_0^{\infty }{e}^{-\lambda t}{u}_3\left(t\right)dt=G\left(\lambda \right)\sum _{i=1}^m\widetilde{a_i}\left(\lambda \right)e^{\lambda t_i}{\int }_0^{t_i}{e}^{-\lambda t}C{u}_0\left(t\right)dt,\Re \lambda >0 suff.large,\end{array}$$

and there exists a Laplace transformable function $u_4:[0,\infty )\to X$ such that

$$\begin{array}{cc}\hfill {\int }_0^{\infty }& e^{-\lambda t}{u}_4\left(t\right)dt\hfill \\ & =G_1\left(\lambda \right)\sum _{i=1}^m\left[{\int }_{-\infty }^0{e}^{\lambda t}{a}_i\left(t\right)dt\right]e^{-\lambda t_i}{\int }_0^{-t_i}{e}^{-\lambda t}{C}^{\prime }{u}_0\left(t\right)dt,\Re \lambda >0 suff.large.\hfill \end{array}$$

Then, $u(·)$ is a strong LT-solution of (14) if and only if

$$\begin{array}{c}\hfill (k\ast Cu)\left(t\right)=\left(R\ast \left[f-u_3\right]\right)\left(t\right)+\left(kC\ast u_2\right)\left(t\right),t\ge 0\end{array}$$

and

$$\begin{array}{c}\hfill (k_1\ast C_1\check{u})\left(t\right)=\left(R_1\ast \left[\check{f}-u_4\right]\right)\left(t\right)+\left(k_1{C}^{\prime }\ast u_4\right)\left(t\right),t\ge 0.\end{array}$$

The illustrative applications in Examples 4 and 5 can be repeated for the abstract Volterra initial value problem (14), for example, we have the following.

Example 6. 

Suppose that $m=1$, $t_1<0$, $a(·)$ and $k(·)$ are Laplace transformable, $k(·)$ is continuous, $\tilde{a}(·)$ does not vanish on some right half-plane and $\pm \mathcal{A}$ are subgenerators of exponentially equicontinuous $(a,k)$-regularized C-resolvent operator families ${\left(R_{\pm }\left(t\right)\right)}_{t\ge 0}$. Define $a(-t):=a\left(t\right)$, $t\ge 0$. If $\widetilde{a_1}\left(\lambda \right)=e^{-\lambda t_1}\tilde{a}\left(\lambda \right)$ for sufficiently large $\Re \lambda >0$, then Theorem 4 can be applied. Furthermore, an analogue of Formula (13) can be given in our new framework.

4. Conclusions and Final Remarks

In this research paper, we have introduced and briefly analyzed various classes of $(F,G,C)$-regularized resolvent families and their applications in the study of well-posedness for certain classes of the abstract fractional functional inclusions and the abstract Volterra functional inclusions. The introduced classes of regularized resolvent families generalize the already known classes of $(a,k)$-regularized C-resolvent families, whose applications are also analyzed here. Introducing new classes of operator families may increase the complexity of the mathematical framework, making it challenging for readers to grasp and apply the concepts easily (see, e.g., the recent research article [29], where the authors have investigated the numerical solutions for a class of linear fuzzy Volterra integral equations of the second kind with weakly singular kernels).

We close the paper by quoting some topics not considered in our previous work:

1.

In [2] (Section 3.2.2), we have investigated the case in which some of the operators $C_2$ or C are not injective. We will skip all related details concerning the corresponding classes of $(F,G,C_2)$-regularized uniqueness families and $(F,G,C)$-regularized resolvent families.

2.

Let ${\mathcal{A}}_i$ be a closed MLO ($1\le i\le m$). The abstract multi-term Volterra integro-difference inclusion

$$\begin{array}{c}\hfill u\left(t\right)\in f\left(t\right)+\sum _{i=1}^m\left(a_i\ast {\mathcal{A}}_{i}u\left(·+t_i\right)\right)\left(t\right),t\ge 0;u\left(t\right)=u_0\left(t\right),0\le t\le \tau \end{array}$$

and the abstract multi-term Volterra integro-difference inclusion

$$\begin{array}{c}\hfill u\left(t\right)\in f\left(t\right)+\sum _{i=1}^m\left(a_i\ast {\mathcal{A}}_{i}u\left(·+t_i\right)\right)\left(t\right),t\in \mathbb{R};u\left(t\right)=u_0\left(t\right),t\in \left[{\tau }_1,{\tau }_2\right]\end{array}$$

will be considered somewhere else.

3.

The abstract degenerate higher-order Cauchy problems of the form

$$B{u}^{\left(n\right)}\left(t\right)=A_{n-1}{u}^{(n-1)}\left(t+t_{n-1}\right)+\dots +A_{0}u\left(t+t_0\right);u\left(t\right)\equiv u_0\left(t\right),0\le t\le \tau ,$$

deserve special attention and will be considered somewhere else ($0\le t_0,\dots ,t_{n-1}<+\infty$, $\tau \equiv max\left(t_0,\dots ,t_{n-1}\right)$; $B,A_{n-1},\dots ,A_0$ are closed linear operators).