Solution Operators for Caputo-Type Fractional Evolution Equations with Damping

Abstract

This paper investigates an abstract fractional Cauchy problem with damping formulated in the sense of the Caputo derivative, where the derivative orders satisfy $0<\delta <\gamma \le 1$. By introducing the concept of a Caputo fractional $(\gamma ,\delta ,k)$ resolvent and systematically analyzing its fundamental properties, together with key features of the generalized Mittag–Leffler (ML) function, we establish the uniqueness and existence of strong solutions for this class of damped fractional-order evolution equations. Under more restrictive assumptions on the underlying operators, the solution admits an explicit representation in terms of ML-type functions associated with fractional exponents. Furthermore, we demonstrate that the proposed abstract framework can be effectively applied to concrete models, including fractional diffusion equations with damping. These results highlight the relevance and necessity of fractional damping models in accurately describing complex dynamical phenomena, such as vibration processes and anomalous diffusion.

1. Introduction

In both physical and mathematical models, dissipative mechanisms are commonly present that reduce the amplitude of oscillations or motion within a system, ultimately driving it toward a stable state. This phenomenon is known as damping. In the context of dynamical systems and differential equations, damping typically appears as a term involving velocity or the rate of change of the state variable. Such terms account for energy dissipation within the system, thereby attenuating oscillatory behavior and slowing the system’s response.

Damping plays a crucial role in both physical and mathematical formulations of fractional-order systems. Unlike traditional integer-order damping $c{u}^{\prime }\left(t\right)$, fractional damping terms $D_t^{\delta }u\left(t\right)$ introduce memory-dependent dissipation effects through fractional derivatives, allowing for a more accurate description of complex systems with inherent memory characteristics. Such models naturally arise in diverse applications, including multi-damping systems with multiple relaxation mechanisms [1,2], nonlinear vibration and material models [3,4], non-uniform or time-varying dissipation processes [5], fractional-order vibration and wave propagation phenomena with damping [6], and systems in which damping is coupled with external forcing terms [7,8].

It is well established that for the classical abstract Cauchy issue, such as the first-order evolution equation $u^{\prime }\left(t\right)=Au\left(t\right)$, solutions are typically represented by the action of an operator semigroup $U\left(t\right)$ on the initial value x. In contrast, for fractional differential equations, the inherent nonlocality and memory effects associated with fractional derivatives prevent the direct application of semigroup theory.

In the 1990s, research on fractional abstract Cauchy problems began to gain attention, with primary emphasis placed on the existence and representation of solution operators. In ref. [9], a systematic investigation of fractional evolution equations was carried out, introducing the notion of a solution operator, which may be regarded as a precursor to the modern concept of a resolvent. Subsequently, in ref. [10], the concept of a fractional resolvent operator function was explicitly formulated, and its relationship with fractional Cauchy problems was rigorously established. Independently, in ref. [11], the authors introduced the $\gamma$-order fractional resolvent to address Riemann–Liouville (RL) fractional abstract Cauchy problems. This framework was later extended in [12] for incorporating damping effects through the damped RL fractional $(\gamma ,\delta ,k)$ resolvent. In the current work, we study the Caputo fractional $(\gamma ,\delta ,k)$ resolvent, which corresponds to the case $0<\gamma <1$ for the RL derivative modified in the sense of Jumarie [13].

Despite the widespread recognition of fractional models in describing anomalous phenomena, most existing studies remain at the level of mathematical generalization [14,15]. From a physical perspective, the transition from integer-order to fractional-order models is not merely a mathematical extension; it fundamentally alters the memory structure of the system. In classical vibration theory, integer-order derivatives imply that the damping force is proportional to the instantaneous velocity, which corresponds to a Markovian process where the system has no memory of past states. However, real materials such as viscoelastic polymers or biological tissues exhibit power-law memory kernels, meaning that their current state depends strongly on the entire deformation history [16]. Direct comparison with classical integer-order models reveals that fractional derivatives introduce an additional degree of freedom, the fractional order, which allows for a more accurate fitting of experimental attenuation data over broad frequency bands. Therefore, providing a rigorous operator-theoretic framework for such fractionally damped systems is not only mathematically novel but also physically indispensable for accurately characterizing the slow, non-exponential relaxation processes observed in complex materials [17].

In this paper, the core problem studied is the following damped abstract fractional Cauchy issue of the form:

$${(\mathrm{D}–\mathrm{CAFCP})}_x\left\{\begin{array}{c}{}^{C}D_t^{\gamma }u\left(t\right)+k{}^{C}D_t^{\delta }u\left(t\right)=Au\left(t\right),t>0,\hfill \\ u\left(0\right)=x.\hfill \end{array}\right.$$

This equation is suitable for modeling a class of physical processes with well-defined initial states, such as vibrations in viscoelastic materials and wave propagation in heterogeneous media, in which memory-dependent damping effects are present. In this equation, the parameters satisfy $0<\delta <\gamma \le 1$. The state variable $u:[0,\infty )\to X$ is an abstract function taking values in a Banach space (BS) $(X,\parallel ·\parallel )$, $u\in A{C}_{\mathrm{loc}}([0,\infty );D\left(A\right))$, and $A:D\left(A\right)\subset X\to X$ is a densely defined, closed linear operator on ${\parallel x\parallel }_{D\left(A\right)}:=\parallel x\parallel +\parallel Ax\parallel$ whose domain $D\left(A\right)$ is equipped with the graph norm. The operators ${}^{C}D_t^{\gamma }$ denote the Caputo fractional derivatives of orders $\gamma$ and $\delta$, which introduce nonlocal memory effects into the system. The constant $k\ge 0$ represents the damping coefficient, and the term $k{}^{C}D_t^{\delta }u\left(t\right)$ characterizes the history-dependent fractional-order damping mechanism. To systematically analyze the properties of solutions to this problem, this paper develops and investigates the associated Caputo fractional resolvent. By establishing a rigorous resolvent framework, we prove the existence and uniqueness of solutions to the corresponding fractional Cauchy problem, thereby providing a solid mathematical foundation for subsequent numerical approximations and applied analyses.

2. Preliminaries

This section presents vital concepts of fractional integrals and fractional derivatives that will be employed throughout this paper.

X is a complex BS equipped with the norm $\parallel ·\parallel$. Consider a linear operator $A:D\left(A\right)\subset X\to X$. The resolvent set of $\rho \left(A\right)$ is denoted by $R(\lambda ,A):={(\lambda I-A)}^{-1}$ for $\lambda \in \rho \left(A\right)$.

For two functions $f_1,f_2:[0,\infty )\to X$ that are integrable on every compact interval, their convolution is defined by

$$\left(f_1\ast f_2\right)\left(\iota \right):={\int }_0^{\iota }{f}_1\left(\varsigma \right)f_2(\iota -\varsigma )d\varsigma ,\iota \ge 0.$$

Given a locally integrable function $f\in L_{\mathrm{loc}}^1({\mathbb{R}}_{+};X)$, its Laplace transform $\left(LT\right)$ is given by

$$\mathcal{L}\left\{f\right\}\left(\mu \right)\equiv \widehat{f}\left(\mu \right):={\int }_0^{\infty }{e}^{-\mu t}f\left(t\right)dt.$$

The integral converges for the given complex parameter $\mu$. Hereafter, $\mathcal{L}$ denotes the $LT$ with respect to the variable t.

Definition 1.

Let $\nu >0$. The RL fractional integral of order ν is

$$\left(J_t^{\nu }f\right)\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma \left(\nu \right)}}{\int }_0^t{(t-\theta )}^{\nu -1}f\left(\theta \right)d\theta ,\hfill & \nu \in (0,1],\hfill \\ f\left(t\right),\hfill & \nu =0.\hfill \end{array}\right.$$

Definition 2.

Let X be a BS and $u\in A{C}_{\mathrm{loc}}([0,\infty );X)$, so that $u^{\prime }\in L_{\mathrm{loc}}^1({\mathbb{R}}_{+};X)$. For $0<\gamma \le 1$ and $t>0$, the Caputo derivative is defined by

$$\begin{array}{cc}\hfill {}^{C}D_t^{\gamma }u\left(t\right)& =J_t^{\mu }{u}^{\prime }\left(t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\int }_0^t{(t-\tau )}^{-\gamma }{u}^{\prime }\left(s\right)ds.\hfill \end{array}$$

Definition 3.

For a function $u\left(t\right)$, the RL derivative is defined by:

$$\begin{array}{cc}\hfill {}^{\mathit{RL}}D_t^{\gamma }u\left(t\right)& ={\displaystyle \frac{d}{dt}}{J}_t^{1-\gamma }u\left(t\right)\hfill \\ & ={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\displaystyle \frac{d}{dt}}{\int }_0^t{(t-s)}^{-\gamma }u\left(s\right)ds,\hfill \end{array}$$

where $\Gamma (·)$ denotes the Gamma function, and $J_t^{1-\gamma }$ represents the Riemann–Liouville fractional integral operator of order $1-\gamma$.

Lemma 1

([18]). For comparison with the RL derivative, assume $0<\gamma <1$. We recall the Caputo–RL relation:

$${}^{C}D_t^{\gamma }u\left(t\right)={}^{\mathit{RL}}D_t^{\gamma }u\left(t\right)-u\left(0\right).$$

(1)

Lemma 2

([19]). When the transforms exist, the Laplace identities read

$$\mathcal{L}\left\{J_t^{\mu }f\right\}\left(\lambda \right)={\lambda }^{-\mu }\widehat{f}\left(\lambda \right),\mathcal{L}\left\{{}^{C}D_t^{\nu }u\right\}\left(\lambda \right)={\lambda }^{\nu }\widehat{u}\left(\lambda \right)-{\lambda }^{\nu -1}u\left(0^{+}\right).$$

Definition 4

([20]). Let $z,\rho ,\zeta \in \mathbb{C}$ with $Re\left(\rho \right)>0$. The general Mittag–Leffler(ML) function is

$$E_{\rho ,\sigma }^{\zeta }\left(z\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(\zeta \right)}_m}{\Gamma (\rho m+\sigma )}}{\displaystyle \frac{z^m}{m!}},$$

where

$${\left(\zeta \right)}_m={\displaystyle \frac{\Gamma (\zeta +m)}{\Gamma \left(\zeta \right)}}.$$

Remark 1.

By specializing the parameter to $\zeta =1$, the generalized ML function decreases to the classical two-parameter ML function, that is, $E_{\rho ,\sigma }^1\left(z\right)=E_{\rho ,\sigma }\left(z\right)$ for all $z\in \mathbb{C}$. Furthermore, in the special case $\sigma =1$, this simplifies to the one-parameter ML function $E_{\rho }$, namely, $E_{\rho ,1}^1\left(z\right)=E_{\rho }\left(z\right)$.

Definition 5.

Let $\gamma ,\delta ,\zeta \in \mathbb{C}$ with $Re\left(\gamma \right)>0$ and $Re\left(\delta \right)>0$, and let $a\in \mathbb{C}$. The Prabhakar integral kernel $e_{\gamma ,\delta ,a}^{\zeta }:\mathbb{R}\to \mathbb{R}$ is defined as

$$e_{\gamma ,\delta ,a}^{\zeta }\left(t\right)=t^{\delta -1}{E}_{\gamma ,\delta }^{\zeta }\left(a{t}^{\gamma }\right)\mathit{for}t>0,$$

and $e_{\gamma ,\delta ,a}^{\zeta }\left(0\right)=0$. For the special case $\zeta =1$, the kernel simplifies to $e_{\gamma ,\delta ,a}:\mathbb{R}\to \mathbb{R}$, given by $e_{\gamma ,\delta ,a}\left(t\right)=t^{\delta -1}{E}_{\gamma ,\delta }\left(a{t}^{\gamma }\right)\mathit{for}t>0,$ and $e_{\gamma ,\delta ,a}\left(0\right)=0$.

Definition 6

([12]). Let $\gamma ,\delta \in \mathbb{C}$ with $Re\left(\gamma \right)>0$, and $Re\left(\delta \right)>0$, and let $a\in \mathbb{C}$. The ML integral operator ${\mathbb{I}}_t^{\gamma ,\delta ,a}$ acts on locally integrable functions $f\in L_{\mathit{loc}}^1\left({\mathbb{R}}^{+}\right)$ as follows:

$${\mathbb{I}}_t^{\gamma ,\delta ,a}f\left(t\right)=\left(e_{\gamma ,\delta ,a}\ast f\right)\left(t\right)\mathrm{for}t>0,$$

and ${\mathbb{I}}_t^{\gamma ,\delta ,a}f\left(0\right)=0$, where $e_{\gamma ,\delta ,a}$ is the kernel defined in Definition 5.

We repeatedly utilize the LT formula for the generalized ML kernel $e_{\gamma ,\delta ,a}^{\zeta }\left(t\right)$ [21,22]:

$$\mathcal{L}\left\{e_{\gamma ,\delta ,a}^{\zeta }\right\}\left(\lambda \right)={\displaystyle \frac{{\lambda }^{\zeta \gamma -\delta }}{{\left({\lambda }^{\gamma }-a\right)}^{\zeta }}},Re\lambda >{\left|a\right|}^{1/\gamma }.$$

(2)

3. Caputo Fractional $(\mathit{\gamma },\mathit{\delta },\mathit{k})$ Resolvent

This section focuses on the concept of the Caputo fractional $(\gamma ,\delta ,k)$ resolvent and the derivation of its fundamental properties, which form the basis for the subsequent analysis. Throughout this section, we assume that the parameters satisfy $0<\delta <\gamma \le 1$.

Definition 7.

A family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ of bounded linear operators on a BS X is called a Caputo fractional $(\gamma ,\delta ,k)$ resolvent if the following conditions are satisfied:

 (a) 

For each $x\in X$, $U\left(t\right)x\in C\left(\right[0,\infty ),X)$, and

$$U\left(0\right)=I.$$

(3)

 (b) 

The operators commute for all nonnegative arguments, i.e.,

$$U\left(v\right)U\left(\tau \right)=U\left(\tau \right)U\left(v\right),\forall \tau ,vs.\ge 0.$$

 (c) 

For all $\tau ,v>0$, the identity holds:

$$U\left(v\right){\mathbb{I}}_{\tau }^{\gamma -\delta ,\gamma ,-k}U\left(\tau \right)-{\mathbb{I}}_v^{\gamma -\delta ,\gamma ,-k}U\left(v\right)U\left(\tau \right)={\mathbb{I}}_{\tau }^{\gamma -\delta ,\gamma ,-k}U\left(\tau \right)-{\mathbb{I}}_v^{\gamma -\delta ,\gamma ,-k}U\left(v\right).$$

(4)

Remark 2.

By the strong continuity of the operator family, ${sup}_{0\le t\le b}\parallel U\left(t\right)\parallel <\infty$ for every $b>0$, the subsequent integrals and convolution terms, such as those involving ${\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)$ and ${\int }_0^{t}U(t-\sigma )f\left(\sigma \right)\mathrm{d}\sigma$ with $f\in L_{\mathrm{loc}}^1([0,\infty );X)$, are well defined and are used throughout this paper.

Definition 8.

The generator A of a Caputo fractional $(\gamma ,\delta ,k)$ resolvent family ${\left\{U\left(r\right)\right\}}_{r\ge 0}$ is defined through its domain

$$D\left(A\right)=\left\{x\in X:\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)x-x}{r^{\gamma }}}\mathit{exists}\right\},$$

and for each $x\in D\left(A\right)$,

$$Ax={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)x-x}{r^{\gamma }}}.$$

Here, $D\left(A\right)$ denotes the domain of the operator A.

Proposition 1.

Let ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ be a Caputo fractional $(\gamma ,\delta ,k)$ resolvent on a BS X with generator A. Then the following property features hold:

 (1a) 

For all $s\ge 0$, if $x\in D\left(A\right)$, then $U\left(s\right)x$ remains in $D\left(A\right)$ and

$$AU\left(s\right)x=U\left(s\right)Ax.$$

 (1b) 

For any $x\in X$, the action of the operator $U\left(t\right)$ on x satisfies the following relation:

$$U\left(t\right)x=x+A{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)x,t\ge 0.$$

(5)

 (1c) 

If $x\in D\left(A\right)$ and $t>0$, then

$$U\left(t\right)x=x+{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)Ax.$$

(6)

 (1d) 

The operator A is closed and densely defined in X.

Proof. 

(1a) Take any $x\in D\left(A\right)$. From Definition 7 (1b), we know that $U\left(s\right)$ and $U\left(t\right)$ commute. Moreover, since $U\left(t\right)$ is bounded for all $t\ge 0$, it follows that

$$\begin{array}{cc}\hfill AU\left(s\right)x& ={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)U\left(s\right)x-U\left(s\right)x}{r^{\gamma }}}\hfill \\ & =U\left(s\right){\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)x-x}{r^{\gamma }}}\hfill \\ & =U\left(s\right)Ax,s\ge 0.\hfill \end{array}$$

Thus, $U\left(s\right)x\in D\left(A\right)$ and the equality $AU\left(s\right)x=U\left(s\right)Ax$ holds for all $s>0$.

(1b) Observing that $r\mapsto E_{\gamma -\delta ,\gamma }(-r^{\gamma })$ is uniformly continuous on $[0,1]$,

$$E_{\gamma -\delta ,\gamma }\left(0\right)={\displaystyle \frac{1}{\Gamma \left(\gamma \right)}},$$

(7)

and

$$\underset{r\in (0,1]}{sup}{r}^{1-\gamma }=1.$$

(8)

For any given $x\in X$, substituting the definition of the operator ${\mathbb{I}}_s^{\gamma -\delta ,\gamma ,-k}$ and applying the identity ${\displaystyle \frac{\Gamma \left(2\gamma \right)}{{[\Gamma \left(\gamma \right)]}^2}}{\int }_0^1{(1-\tau )}^{\gamma -1}{\tau }^{\gamma -1}d\tau =1$, we obtain:

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}{\displaystyle \frac{{\mathbb{I}}_r^{\gamma -\delta ,\gamma ,-k}U\left(r\right)x}{r^{\gamma }}}-x∥\hfill \\ \hfill & =∥{\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}{\int }_0^r{(r-\tau )}^{\gamma -1}{r}^{-\gamma }{E}_{\gamma -\delta ,\gamma }\left(-k{(r-\tau )}^{\gamma -\delta }\right)U\left(\tau \right)xd\tau -x∥\hfill \\ \hfill & =∥{\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}{\int }_0^1{(r-r\tau )}^{\gamma -1}{r}^{-\gamma }{E}_{\gamma -\delta ,\gamma }\left(-k{(r-r\tau )}^{\gamma -\delta }\right)U\left(r\tau \right)xrd\tau -x∥\hfill \\ \hfill & =∥{\displaystyle \frac{\Gamma \left(2\gamma \right)}{{[\Gamma \left(\gamma \right)]}^2}}{\int }_0^1{(1-\tau )}^{\gamma -1}\Gamma \left(\gamma \right)E_{\gamma -\delta ,\gamma }\left(-k{(r-r\tau )}^{\gamma -\delta }\right)U\left(r\tau \right)xd\tau -x∥\hfill \\ \hfill & =∥{\displaystyle \frac{\Gamma \left(2\gamma \right)}{{[\Gamma \left(\gamma \right)]}^2}}{\int }_0^1{(1-\tau )}^{\gamma -1}{\tau }^{\gamma -1}·{\tau }^{1-\gamma }\Gamma \left(\gamma \right)E_{\gamma -\delta ,\gamma }\left(-k{(r-r\tau )}^{\gamma -\delta }\right)U\left(r\tau \right)xd\tau \hfill \\ \hfill & -{\displaystyle \frac{\Gamma \left(2\gamma \right)}{{[\Gamma \left(\gamma \right)]}^2}}{\int }_0^1{(1-\tau )}^{\gamma -1}{\tau }^{\gamma -1}xd\tau ∥\hfill \\ \hfill & \le {\displaystyle \frac{\Gamma \left(2\gamma \right)}{{[\Gamma \left(\gamma \right)]}^2}}{\int }_0^1{(1-\tau )}^{\gamma -1}{\tau }^{\gamma -1}d\tau \hfill \\ \hfill & ·\underset{\tau \in (0,1]}{sup}∥{\tau }^{1-\gamma }\Gamma \left(\gamma \right)E_{\gamma -\delta ,\gamma }\left(-k{(r-r\tau )}^{\gamma -\delta }\right)U\left(r\tau \right)x-x∥\hfill \\ \hfill & =\underset{\tau \in (0,1]}{sup}∥{\tau }^{1-\gamma }\Gamma \left(\gamma \right)E_{\gamma -\delta ,\gamma }\left(-k{(r-r\tau )}^{\gamma -\delta }\right)U\left(r\tau \right)x-x∥.\hfill \end{array}$$

(9)

Combining Definition 7(a) with Equations (2) and (7)–(9), together with the uniform continuity of $E_{\gamma -\delta ,\gamma }(-r^{\gamma })$, the supremum above converges to zero as $r\to 0^{+}$. Consequently, the following limit relation

$$\underset{r\to 0^{+}}{lim}{\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}{\displaystyle \frac{{\mathbb{I}}_r^{\gamma -\delta ,\gamma ,-k}U\left(r\right)x}{r^{\gamma }}}=x$$

(10)

is verified. Further, by Definitions 7 and 10, we obtain

$$\begin{array}{cc}\hfill A{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)x& ={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right){\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)x-{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)x}{r^{\gamma }}}\hfill \\ & ={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(t\right){\mathbb{I}}_r^{\gamma -\delta ,\gamma ,-k}U\left(r\right)x-{\mathbb{I}}_r^{\gamma -\delta ,\gamma ,-k}U\left(r\right)x}{r^{\gamma }}}\hfill \\ & ={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{[U\left(t\right)-I]{\mathbb{I}}_r^{\gamma -\delta ,\gamma ,-k}U\left(r\right)x}{r^{\gamma }}}\hfill \\ & =U\left(t\right)x-x,\hfill \end{array}$$

which completes the proof of part (1b).

(1c) For $x\in D\left(A\right)$, the existence of the limit in Definition 8 implies that

$$\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)x-x}{r^{\gamma }}}$$

is bounded for small $r>0$. Fix $t\ge 0$. By virtue of the limit definition of the generator A in Definition 8, the operator commutativity in Proposition 1 (1a), the decomposition formula in Proposition 1 (1b), the existence of the Mittag–Leffler integral, and the dominated convergence theorem, we obtain:

$$\begin{array}{cc}\hfill U\left(t\right)x-x& =A{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)x\hfill \\ & ={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)-I}{r^{\gamma }}}·{\int }_0^t{(t-\tau )}^{\gamma -1}{E}_{\gamma -\delta ,\gamma }\left(-k{(t-\tau )}^{\gamma -\delta }\right)U\left(\tau \right)xd\tau \hfill \\ & ={\int }_0^t{(t-\tau )}^{\gamma -1}{E}_{\gamma -\delta ,\gamma }\left(-k{(t-\tau )}^{\gamma -\delta }\right)·U\left(\tau \right){\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)-I}{r^{\gamma }}}xd\tau \hfill \\ & ={\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)Ax.\hfill \end{array}$$

Hence, the identity (6) follows directly.

(1d) Consider a sequence $\left\{x_n\right\}$ in the domain $D\left(A\right)$ such that $x_n\to x$ and $A{x}_n\to y$ in X. Fix an arbitrary $t>0$. From the representation formula (6) in part (1c), using the boundedness of the involved operators, we obtain:

$$\begin{array}{cc}\hfill U\left(t\right)x-x& =\underset{n\to \infty }{lim}\left(U\left(t\right)x_n-x_n\right)\hfill \\ \hfill & =\underset{n\to \infty }{lim}{\int }_0^t{(t-\tau )}^{\gamma -1}{E}_{\gamma -\delta ,\gamma }\left(-k{(t-\tau )}^{\gamma -\delta }\right)U\left(\tau \right)A{x}_{n}d\tau \hfill \\ \hfill & ={\int }_0^t{(t-\tau )}^{\gamma -1}{E}_{\gamma -\delta ,\gamma }\left(-k{(t-\tau )}^{\gamma -\delta }\right)U\left(\tau \right)yd\tau \hfill \\ \hfill & ={\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)y.\hfill \end{array}$$

Now, employing the limit result (10), we compute

$$\begin{array}{cc}\hfill Ax& ={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{U\left(r\right)x-x}{r^{\gamma }}}\hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}\underset{r\to 0^{+}}{lim}{\displaystyle \frac{{\mathbb{I}}_r^{\gamma -\delta ,\gamma ,-k}U\left(r\right)y}{r^{\gamma }}}\hfill \\ \hfill & =y.\hfill \end{array}$$

(11)

Hence, A is a closed operator.

For an arbitrary element $x\in X$ and for any $t>0$, set $x_t={\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}U\left(t\right)x$. According to part (1b), $x_t$ this element belongs to $D\left(A\right)$. Furthermore, the limit relation (10) can be rewritten as ${\displaystyle \frac{\Gamma \left(2\gamma \right)}{\Gamma \left(\gamma \right)}}{t}^{-\gamma }{x}_t\to x$ as $t\to 0^{+}$. This demonstrates that x can be approximated by elements of $D\left(A\right)$, thereby confirming that $\overline{D\left(A\right)}=X$.   □

Remark 3.

Proposition 1 establishes the fundamental properties of the generator A. In order to confirm the uniqueness and existence of solutions to the associated abstract Cauchy problem, a crucial question is whether this generator uniquely determines the resolvent family. The following theorem answers this question affirmatively, thereby guaranteeing the uniqueness of the solution representation derived from the generator.

Theorem 1.

Let ${\left\{U\left(\tau \right)\right\}}_{\tau \ge 0}$ and ${\left\{W\left(\tau \right)\right\}}_{\tau \ge 0}$ be Caputo fractional $(\gamma ,\delta ,k)$ resolvent families on a BS X, with generators A and C, respectively. If the generators coincide, i.e., $A=C$, then the two resolvent families are identical: $U\left(\tau \right)=W\left(\tau \right)$ for all $\tau \ge 0$.

Proof. 

It follows from (1a) and (1c) of Proposition 1 that

$$\begin{array}{cc}\hfill x\ast W\left(\tau \right)x& =\left(U\left(\tau \right)x-e_{\gamma -\delta ,\gamma ,-k}\left(\tau \right)\ast AU\left(\tau \right)x\right)\ast W\left(\tau \right)x\hfill \\ \hfill & =U\left(\tau \right)x\ast W\left(\tau \right)x-e_{\gamma -\delta ,\gamma ,-k}\left(\tau \right)\ast AU\left(\tau \right)x\ast W\left(\tau \right)x\hfill \\ \hfill & =U\left(\tau \right)x\ast W\left(\tau \right)x-e_{\gamma -\delta ,\gamma ,-k}\left(\tau \right)\ast U\left(\tau \right)x\ast AW\left(\tau \right)x\hfill \\ \hfill & =U\left(\tau \right)x\ast \left(W\left(\tau \right)x-e_{\gamma -\delta ,\gamma ,-k}\left(\tau \right)\ast AW\left(\tau \right)x\right)\hfill \\ \hfill & =U\left(\tau \right)x\ast x,x\in D\left(A\right).\hfill \end{array}$$

For $\tau \ge 0$ and $x\in D\left(A\right)$, applying Titchmarsh’s convolution theorem to this equality, we deduce

$$U\left(\tau \right)x=W\left(\tau \right)x.$$

By continuity, together with the density of $D\left(A\right)$ in X, the equality extends to the entire space: $U\left(\tau \right)=W\left(\tau \right)$ for all $\tau \ge 0$.   □

To establish an equivalent characterization for closed linear operators that generate exponentially bounded Caputo fractional $(\gamma ,\delta ,k)$ resolvent families, the following theorem offers an essential and substantial condition that the operator family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ must satisfy. This condition also highlights the intrinsic connection between the LT of the resolvent family and the operator’s resolvent set.

Theorem 2.

For a closed operator A, the following statements are equivalent:

 (i) 

A generates an exponentially bounded Caputo fractional $(\gamma ,\delta ,k)$ resolvent family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ with $\parallel U\left(t\right)\parallel \le M{e}^{\psi t}$.

 (ii) 

The spectral interval $({\psi }^{\gamma }+k{\psi }^{\delta },\infty )\subset \rho \left(A\right)$, and there exists a bounded operator family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ such that:

 1. 

The map $t\mapsto U\left(t\right)x$ is strongly continuous and $U\left(0\right)=I$;

 2. 

$U\left(v\right)U\left(\tau \right)=U\left(\tau \right)U\left(v\right)$ for all $\tau ,v\ge 0$;

 3. 

$\parallel U\left(t\right)\parallel \le M{e}^{\psi t}$ for $t>0$;

 4. 

Its LT satisfies:

$$\mathcal{L}\left\{U(·)x\right\}\left(\lambda \right)=R\left({\lambda }^{\gamma }+k{\lambda }^{\delta },A\right)\left({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}\right)x,\lambda >\psi ,x\in X.$$

(12)

Proof of (i) ⇒ (ii).

We begin by adopting assumption (i). Under this assumption, the family $U(·)$ admits a LT. Define:

$$L\left(\lambda \right)=\mathcal{L}\left\{U\left(t\right)\right\}\left(\lambda \right),\lambda >\psi >0.$$

For every $x\in D\left(A\right)$, statement (1c) of Proposition 1 yields the representation

$$U\left(t\right)x=x+e_{\gamma -\delta ,\gamma ,-k}\left(t\right)\ast U\left(t\right)Ax.$$

(13)

Applying the LT to both sides of the above equality (13) yields:

$$\begin{array}{cc}\hfill L\left(\lambda \right)x& ={\displaystyle \frac{1}{\lambda }}x+{\displaystyle \frac{{\lambda }^{-\delta }}{{\lambda }^{\gamma -\delta }+k}}L\left(\lambda \right)Ax\hfill \\ \hfill & ={\displaystyle \frac{1}{\lambda }}x+{\displaystyle \frac{1}{{\lambda }^{\gamma }+k{\lambda }^{\delta }}}AL\left(\lambda \right)x,\forall x\in D\left(A\right).\hfill \end{array}$$

Here, we use the commutativity between A and $U(·)$. Since, by part (1d), $D\left(A\right)$ is dense in X, the same relation extends to the entire space:

$$L\left(\lambda \right)x={\displaystyle \frac{1}{\lambda }}I+{\displaystyle \frac{1}{{\lambda }^{\gamma }+k{\lambda }^{\delta }}}AL\left(\lambda \right)onX.$$

Hence,

$$\left({\lambda }^{\gamma }+k{\lambda }^{\delta }-A\right)L\left(\lambda \right)x=\left({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}\right)xonX.$$

Therefore,

$$\left({\lambda }^{\gamma }+k{\lambda }^{\delta }-A\right)L\left(\lambda \right)=\left({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}\right).$$

(14)

Then, we prove that the relation holds for all $\lambda >\psi$, $\mu ={\lambda }^{\gamma }+k{\lambda }^{\delta }\in \rho \left(A\right)$. First, we establish injectivity. Assume there exists $y\in D\left(A\right)$ such that $(\mu I-A)y=0$, which implies $Ay=\mu y$.

Employing part (1a) of Proposition 1, we obtain the following:

$$AU\left(t\right)y=U\left(t\right)Ay=U\left(t\right)\mu y=\mu U\left(t\right)y.$$

(15)

Substituting (15) into Proposition 1(1c), we obtain

$$U\left(t\right)y=y+{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}\left({\lambda }^{\gamma }+k{\lambda }^{\delta }\right)U\left(t\right)y.$$

(16)

Taking the LT of (16), we derive that

$$L\left(\lambda \right)y={\displaystyle \frac{1}{\lambda }}y+{\displaystyle \frac{1}{{\lambda }^{\gamma }+k{\lambda }^{\delta }}}·({\lambda }^{\gamma }+k{\lambda }^{\delta })L\left(\lambda \right)y.$$

Thus, $y=0$ since $\lambda >0$, and therefore, the operator $\mu I-A$ is injective. Next, we verify surjectivity. Specifically, we need to show that for any $z\in X$, there exists $y\in D\left(A\right)$ such that $(\mu I-A)y=z$. We construct the candidate solution y as follows:

$$y=L\left(\lambda \right)·{\displaystyle \frac{1}{{\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}}}z.$$

(17)

By the properties of Caputo-type resolvents, $U\left(t\right)x\in D\left(A\right)$ for all $t\ge 0$ and $x\in X$. Since $L\left(\lambda \right)x={\int }_0^{\infty }{e}^{-\lambda t}U\left(t\right)xdt$ is a strongly convergent integral of elements in $D\left(A\right)$ and $D\left(A\right)$ is closed under strong limits, we have $L\left(\lambda \right)x\in D\left(A\right)$. Thus, $y\in D\left(A\right)$. Substituting (17) into $\mu I-A$ and using Equation (14) gives:

$$\begin{array}{cc}\hfill (\mu I-A)y& =(\mu I-A)\left(L\left(\lambda \right)·{\displaystyle \frac{1}{{\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}}}z\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}}}·(\mu I-A)L\left(\lambda \right)z\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}}}·({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1})z\hfill \\ \hfill & =z.\hfill \end{array}$$

Therefore, the operator $\mu I-A$ is surjective.

We have confirmed the bijectivity of $\mu I-A=({\lambda }^{\gamma }+k{\lambda }^{\delta })I-A$. Applying the Closed Graph Theorem, the inverse operator ${(\mu I-A)}^{-1}$ corresponding to this operator is bounded. Accordingly, for all $\lambda >\psi$, $\mu ={\lambda }^{\gamma }+k{\lambda }^{\delta }\in \rho \left(A\right)$, this implies that $({\psi }^{\gamma }+k{\psi }^{\delta },\infty )\subseteq \rho \left(A\right)$. Combining this with 3, it follows that (12) holds.   □

Proof of (ii) ⇒ (i).

By condition (12), we have

$$R\left({\lambda }^{\gamma }+k{\lambda }^{\delta },A\right)\left({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}\right)x=L\left(\lambda \right)x.$$

(18)

The resolvent identity for A gives:

$$R({\lambda }^{\gamma }+k{\lambda }^{\delta },A)-R({\mu }^{\gamma }+k{\mu }^{\delta },A)=\left({\mu }^{\gamma }+k{\mu }^{\delta }-{\lambda }^{\gamma }-k{\lambda }^{\delta }\right)R({\lambda }^{\gamma }+k{\lambda }^{\delta },A)R({\mu }^{\gamma }+k{\mu }^{\delta },A).$$

(19)

Substituting (18) into (19), we obtain:

$${\displaystyle \frac{L\left(\lambda \right)}{{\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}}}-{\displaystyle \frac{L\left(\mu \right)}{{\mu }^{\gamma -1}+k{\mu }^{\delta -1}}}=\left({\mu }^{\gamma }+k{\mu }^{\delta }-{\lambda }^{\gamma }-k{\lambda }^{\delta }\right){\displaystyle \frac{L\left(\lambda \right)L\left(\mu \right)}{\left({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}\right)\left({\mu }^{\gamma -1}+k{\mu }^{\delta -1}\right)}}.$$

Thus,

$${\displaystyle \frac{1}{\mu }}{\displaystyle \frac{L\left(\lambda \right)}{{\lambda }^{\gamma }+k{\lambda }^{\delta }}}-{\displaystyle \frac{1}{\lambda }}{\displaystyle \frac{L\left(\mu \right)}{{\mu }^{\gamma }+k{\mu }^{\delta }}}={\displaystyle \frac{L\left(\lambda \right)L\left(\mu \right)}{\left({\lambda }^{\gamma }+k{\lambda }^{\delta }\right)}}-{\displaystyle \frac{L\left(\lambda \right)L\left(\mu \right)}{\left({\mu }^{\gamma }+k{\mu }^{\delta }\right)}}.$$

Using properties of the LT, we derive:

$$U\left(v\right){\mathbb{I}}_{\tau }^{\gamma -\delta ,\gamma ,-k}U\left(\tau \right)-{\mathbb{I}}_v^{\gamma -\delta ,\gamma ,-k}U\left(v\right)U\left(\tau \right)={\mathbb{I}}_{\tau }^{\gamma -\delta ,\gamma ,-k}U\left(\tau \right)-{\mathbb{I}}_v^{\gamma -\delta ,\gamma ,-k}U\left(v\right)$$

for all $\tau ,vs.>0$.   □

4. Existence and Uniqueness of Strong and Mild Solutions

In this section, we introduce the notions of strong and mild solutions for the damped Caputo abstract fractional Cauchy problem ${(D-CAFCP)}_x$. We first establish the existence and uniqueness of strong solutions for $0<\delta <\gamma \le 1$. The mild solution is then defined and, as a consequence of the properties of the resolvent family, its existence and uniqueness are guaranteed.

Definition 9.

A continuous function is called a strong solution of the Caputo abstract fractional Cauchy problem with damping ${(D-CAFCP)}_x$ if it satisfies the following four conditions:

 (a) 

$u\in C((0,\infty ),D\left(A\right))\cap A{C}_{loc}((0,\infty ),X)$;

 (b) 

The functions defined by

$$I_{\gamma }\left[u\right]\left(t\right)={\int }_0^t{\displaystyle \frac{u\left(s\right)}{{(t-s)}^{\gamma }}}ds\mathit{and}{I}_{\delta }\left[u\right]\left(t\right)={\int }_0^t{\displaystyle \frac{u\left(s\right)}{{(t-s)}^{\delta }}}ds$$

are continuously differentiable on the interval $(0,\infty )$;

 (c) 

There holds ${}^{C}D_t^{\gamma }u\left(t\right)+k{}^{C}D_t^{\delta }u\left(t\right)=Au\left(t\right),t>0$;

 (d) 

u is continuously extendable at $t=0$, and $u\left(0\right)=x\in X$.

To attain the key results of this research, we first present the subsequent lemma.

Lemma 3

([10]). For every positive real number t, the following identity involving ML functions holds:

$$E_{\gamma -\delta }\left(-k{t}^{\gamma -\delta }\right)+k{e}_{\gamma -\delta ,1+\gamma -\delta ,-k}\left(t\right)=1,t>0.$$

(20)

Proof. 

It suffices to verify that the left-hand side of (20) equals 1 for every $t>0$. To do this, we use the series expansions:

$$E_{\gamma -\delta }(-k{t}^{\gamma -\delta })=1+\sum _{k=1}^{\infty }{\displaystyle \frac{{(-k{t}^{\gamma -\delta })}^k}{\Gamma \left((\gamma -\delta )k+1\right)}},$$

$$\begin{array}{cc}\hfill k{e}_{\gamma -\delta ,1+\gamma -\delta ,-k}\left(t\right)& =k{t}^{\gamma -\delta }{E}_{\gamma -\delta ,1+\gamma -\delta }(-k{t}^{\gamma -\delta })=\sum _{k=1}^{\infty }{\displaystyle \frac{-{(-k{t}^{\gamma -\delta })}^k}{\Gamma \left((\gamma -\delta )k+1\right)}}.\hfill \end{array}$$

Adding these two expressions, the infinite sums cancel exactly, leaving only the constant term 1. Hence, (20) holds identically for all $t>0$.   □

Lemma 4

([22]). Let $\gamma ,\delta ,\xi ,\vartheta >0$ and $\delta >\nu >0$. Then:

$$\begin{array}{c}{\displaystyle \frac{t^{\vartheta -1}}{\Gamma \left(\vartheta \right)}}\ast e_{\gamma ,\delta ,a}^{\vartheta }\left(t\right)=e_{\gamma ,\delta +\xi ,a}^{\vartheta }\left(t\right),\end{array}$$

(21)

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{t^{-\nu }}{\Gamma (1-\nu )}}\ast e_{\gamma ,\delta ,a}^{\vartheta }\left(t\right)\right]=e_{\gamma ,\delta -\nu ,a}^{\vartheta }\left(t\right).\end{array}$$

(22)

Remark 4.

Specifically for Equation (21), when $\vartheta =1$,

$${\int }_0^t{e}_{\gamma ,\delta ,a}^{\vartheta }\left(s\right)ds=e_{\gamma ,\delta +1,a}^{\vartheta }\left(t\right).$$

Having introduced the definition of a strong solution, we now present, utilizing Lemmas 3 and 4, a detailed proof of the uniqueness and existence of such a solution for the ${(D-CAFCP)}_x$ problem.

Theorem 3.

Let operator A generate a Caputo fractional $(\gamma ,\delta ,k)$ resolvent family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ in a BS X. Then $U(·)x$ is a strong solution for any $x\in D\left(A\right)$ and $t\ge 0$.

Proof. 

Let x be an arbitrary element of $D\left(A\right)$. Convolving both sides of equality (20) in Lemma 3 with resolvent family $U(·)x$, we obtain

$$\left(E_{\gamma -\delta }(-k{t}^{\gamma -\delta })+k{e}_{\gamma -\delta ,1+\gamma -\delta ,-k}\left(t\right)\right)\ast U\left(t\right)x={\int }_0^{t}U\left(\tau \right)xd\tau .$$

(23)

Substituting the resolvent representation from Proposition 1(1c) into the preceding equation and applying Lemma 4, we obtain:

$$\begin{array}{cc}\hfill {\int }_0^{t}U\left(\varsigma \right)xd\varsigma & ={\int }_0^t\left[E_{\gamma -\delta }(-k{\varsigma }^{\gamma -\delta })x+k{e}_{\gamma -\delta ,1+\gamma -\delta ,-k}\left(\varsigma \right)x+e_{\gamma -\delta ,\gamma ,-k}\left(\varsigma \right)\ast U\left(\varsigma \right)Ax\right]d\varsigma \hfill \\ \hfill & ={\int }_0^t{\displaystyle \frac{d}{d\varsigma }}\left[e_{\gamma -\delta ,\gamma ,-k}\left(\varsigma \right)\ast {\displaystyle \frac{{\varsigma }^{1-\gamma }}{\Gamma (2-\gamma )}}\right]d\varsigma +{\int }_0^t{\displaystyle \frac{d}{d\varsigma }}\left[e_{\gamma -\delta ,\gamma ,-k}\left(\varsigma \right)\ast k{\displaystyle \frac{{\varsigma }^{1-\delta }}{\Gamma (2-\delta )}}\right]d\varsigma \hfill \\ \hfill & +e_{\gamma -\delta ,\gamma +1,-k}\left(t\right)\ast U\left(t\right)Ax,t>0.\hfill \end{array}$$

(24)

Utilizing Equations (21), (23) and (24) collectively, we arrive at the following equation:

$$\begin{array}{cc}& {\int }_0^t{e}_{\gamma -\delta ,\gamma ,-k}\left(t-\varsigma \right)\left[{\int }_0^{\varsigma }{\displaystyle \frac{{(\varsigma -\tau )}^{-\gamma }}{\Gamma (1-\gamma )}}U\left(\tau \right)xd\tau +k{\int }_0^{\varsigma }{\displaystyle \frac{{(\varsigma -\tau )}^{-\delta }}{\Gamma (1-\delta )}}U\left(\tau \right)xd\tau \right]d\varsigma \hfill \\ & ={\int }_0^t{e}_{\gamma -\delta ,\gamma ,-k}\left(t-\varsigma \right)\left[{\displaystyle \frac{{\varsigma }^{1-\gamma }}{\Gamma (2-\gamma )}}+k{\displaystyle \frac{{\varsigma }^{1-\delta }}{\Gamma (2-\delta )}}+{\int }_0^{\varsigma }AU\left(\tau \right)xd\tau \right]d\varsigma ,t>0.\hfill \end{array}$$

By virtue of Titchmarsh’s convolution theorem, the equality of the two LTs implies the pointwise identity for all $t>0$:

$${\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\gamma }}{\Gamma (1-\gamma )}}U\left(\tau \right)xd\tau +k{\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\delta }}{\Gamma (1-\delta )}}U\left(\tau \right)xd\tau ={\displaystyle \frac{t^{1-\gamma }}{\Gamma (2-\gamma )}}+k{\displaystyle \frac{t^{1-\delta }}{\Gamma (2-\delta )}}+{\int }_0^{t}AU\left(\tau \right)xd\tau .$$

Using the expression from Proposition 1 (1c) gives

$$\begin{array}{cc}\hfill {\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\gamma }}{\Gamma (1-\gamma )}}U\left(\tau \right)xd\tau & ={\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\gamma }}{\Gamma (1-\gamma )}}\left[x+\left(e_{\gamma -\delta ,\gamma ,-k}\ast U(·)Ax\right)\left(\tau \right)\right]d\tau \hfill \\ & ={\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\gamma }}{\Gamma (1-\gamma )}}xd\tau +\left(e_{\gamma -\delta ,1+\gamma -\delta ,-k}\ast U(·)Ax\right)\left(t\right).\hfill \end{array}$$

Consequently, $I_{\gamma }\left[u\right]\left(t\right)$ is continuously differentiable over the interval $(0,\infty )$. It is evident that the function ${\displaystyle \frac{t^{1-\gamma }}{\Gamma (2-\gamma )}}+k{\displaystyle \frac{t^{1-\delta }}{\Gamma (2-\delta )}}+{\int }_0^{t}AU\left(\tau \right)xd\tau$ is also continuously differentiable. Therefore, $I_{\gamma }\left[u\right]\left(t\right)$ remains continuously differentiable over the interval $(0,\infty )$. For any $t>0$, we have

$${\displaystyle \frac{d}{dt}}{\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\gamma }}{\Gamma (1-\gamma )}}U\left(\tau \right)xd\tau +k{\displaystyle \frac{d}{dt}}{\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\delta }}{\Gamma (1-\delta )}}U\left(\tau \right)xd\tau ={\displaystyle \frac{d}{dt}}{\int }_0^{t}AU\left(\tau \right)xd\tau +{\displaystyle \frac{t^{-\gamma }}{\Gamma (1-\gamma )}}+k{\displaystyle \frac{t^{-\delta }}{\Gamma (1-\delta )}}.$$

This implies that $t\mapsto D_t^{\gamma }U\left(t\right)x$, $t\mapsto D_t^{\delta }U\left(t\right)x$ are continuous on $(0,\infty )$. Using the semigroup property of fractional integrals, it follows that $U\left(t\right)x\in A{C}_{loc}(0,\infty )$. By (1), we then obtain

$${}^{C}D_t^{\gamma }u\left(t\right)={\displaystyle \frac{d}{dt}}{\int }_0^t{\displaystyle \frac{{(t-\tau )}^{-\gamma }}{\Gamma (1-\gamma )}}u\left(\tau \right)d\tau +{\displaystyle \frac{t^{-\gamma }}{\Gamma (1-\gamma )}}xd\tau .$$

Thus, we get

$${}^{C}D_t^{\gamma }U\left(t\right)x+k{}^{C}D_t^{\delta }U\left(t\right)x=AU\left(t\right)x.$$

Hence, $u\left(t\right)=U\left(t\right)x$ is a strong solution of ${\left(\mathrm{D}-\mathrm{CAFCP}\right)}_x$.   □

Theorem 4.

Assume that A generates a Caputo fractional $(\gamma ,\delta ,k)$ resolvent family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ on BS X. If u is a strong solution of ${\left(D-\mathit{CAFCP}\right)}_x$, then $u\left(t\right)=U\left(t\right)x$.

Proof. 

Suppose u is a strong solution of (D-CAFCP)_x. By the definition of a strong solution, the initial condition and the governing equation are given by

$$u\left(0\right)=x,$$

and

$${}^{C}D_t^{\gamma }u\left(t\right)+k{}^{C}D_t^{\delta }u\left(t\right)=Au\left(t\right).$$

They imply that

$$(g_{1-\gamma }\ast u)\left(0\right)=(g_{1-\delta }\ast u)\left(0\right)=0$$

(25)

and

$${\displaystyle \frac{d}{dt}}{\int }_0^t{\displaystyle \frac{{(t-\sigma )}^{-\gamma }}{\Gamma (1-\gamma )}}u\left(\sigma \right)d\sigma -{\displaystyle \frac{t^{-\gamma }}{\Gamma (1-\gamma )}}+k{\displaystyle \frac{d}{dt}}{\int }_0^t{\displaystyle \frac{{(t-\sigma )}^{-\delta }}{\Gamma (1-\delta )}}u\left(\sigma \right)d\sigma -k{\displaystyle \frac{t^{-\delta }}{\Gamma (1-\delta )}}=Au\left(t\right).$$

(26)

Combining (25) and (26), we obtain:

$${\int }_0^t{\displaystyle \frac{{(t-\sigma )}^{-\gamma }}{\Gamma (1-\gamma )}}u\left(\sigma \right)d\sigma +k{\int }_0^t{\displaystyle \frac{{(t-\sigma )}^{-\delta }}{\Gamma (1-\delta )}}u\left(\sigma \right)d\sigma ={\int }_0^{t}Au\left(\sigma \right)d\sigma +{\displaystyle \frac{t^{1-\gamma }}{\Gamma (2-\gamma )}}x+k{\displaystyle \frac{t^{1-\delta }}{\Gamma (2-\delta )}}x.$$

(27)

Taking the convolution of both sides of (27) with the kernel $e_{\gamma -\delta ,\gamma ,-k}\left(t\right)$ yields:

$$\begin{array}{cc}\hfill & {\int }_0^t{e}_{\gamma -\delta ,\gamma ,-k}(t-r)\left[{\int }_0^r{\displaystyle \frac{{(r-\sigma )}^{-\gamma }}{\Gamma (1-\gamma )}}u\left(\sigma \right)d\sigma +k{\int }_0^r{\displaystyle \frac{{(r-\sigma )}^{-\delta }}{\Gamma (1-\delta )}}u\left(\sigma \right)d\sigma \right]dr\hfill \\ \hfill & ={\int }_0^t{e}_{\gamma -\delta ,\gamma ,-k}(t-r)\left[{\displaystyle \frac{r^{1-\gamma }}{\Gamma (2-\gamma )}}x+k{\displaystyle \frac{r^{1-\delta }}{\Gamma (2-\delta )}}x+{\int }_0^{r}Au\left(\sigma \right)d\sigma \right]dr.\hfill \end{array}$$

Using (2) and (21), the above equation can be rewritten as:

$$\left(E_{\gamma -\delta }(-k{t}^{\gamma -\delta })+k{t}^{\gamma -\delta }{E}_{\gamma -\delta ,1+\gamma -\delta }(-k{t}^{\gamma -\delta })\right)\ast u\left(t\right)={\int }_0^t\left(x+{\mathbb{I}}_s^{\gamma -\delta ,\gamma ,-k}Au\left(s\right)\right)ds.$$

(28)

Direct combination of the identity (20) from Lemma 3 with the above Equation (28) yields:

$${\int }_0^{t}u\left(r\right)dr={\int }_0^t\left(x+{\mathbb{I}}_r^{\gamma -\delta ,\gamma ,-k}Au\left(r\right)\right)dr.$$

Clearly, we have

$$u\left(t\right)=x+{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}Au\left(t\right).$$

(29)

We now use the property established in Proposition 1. For any $t\ge 0$, we have

$$\begin{array}{cc}\hfill x\ast u\left(t\right)& =\left(U\left(t\right)x-e_{\gamma -\delta ,\gamma ,-k}\left(t\right)\ast AU\left(t\right)x\right)\ast u\left(t\right).\hfill \end{array}$$

(30)

Based on Equation (29), we can observe that

$$\begin{array}{cc}\hfill x\ast U\left(t\right)x& =u\left(t\right)-{\mathbb{I}}_t^{\gamma -\delta ,\gamma ,-k}Au\left(t\right)\hfill \\ \hfill & =\left(u\left(t\right)-e_{\gamma -\delta ,\gamma ,-k}\left(t\right)\ast Au\left(t\right)\right)\ast U\left(t\right)x\hfill \\ \hfill & =u\left(t\right)\ast U\left(t\right)x-e_{\gamma -\delta ,\gamma ,-k}\left(t\right)\ast AU\left(t\right)x\ast u\left(t\right).\hfill \end{array}$$

(31)

For $t\ge 0$, from (30) and (31), it can be deduced that

$$x\ast u\left(t\right)=x\ast U\left(t\right)x.$$

Applying Titchmarsh’s theorem to the derived LHS equality, we can conclude that for all $t\ge 0$, we have

$$u\left(t\right)=U\left(t\right)x.$$

Thus, the theorem is proved.   □

Remark 5.

The integration of Theorems 3 and 4 implies that for any $x\in D\left(A\right)$, the strong solution of ${\left(D-\mathit{CAFCP}\right)}_x$ is unique.

Remark 6.

The concluding part of the proof of Theorem 4 shows that for any $x\in D\left(A\right)$, the function defined by $u\left(t\right)=U\left(t\right)x$ for $t\ge 0$ constitutes the unique strong solution to the corresponding Volterra equation

$$u\left(t\right)=x+e_{\gamma -\delta ,\gamma ,-k}\left(t\right)\ast Au\left(t\right).$$

(32)

Here, a function $u\in A{C}_{\mathit{loc}}([0,\infty );D\left(A\right))$ is termed the Volterra equation (32), a strong solution, if $u\in A{C}_{\mathit{loc}}([0,\infty );D\left(A\right))$ and the equality (32) holds.

Motivated by the equivalence of problems ${\left(\mathrm{D}-\mathrm{CAFCP}\right)}_x$ and (32) for strong solutions, we now define the corresponding mild solution as follows.

Definition 10.

Assume that A produces a Caputo fractional $(\gamma ,\delta ,k)$ resolvent ${\left\{U\left(t\right)\right\}}_{t\ge 0}$. For any $x\in X$, we call $u\left(t\right)=U\left(t\right)x$ ($t\ge 0$) the mild solution of ${\left(D-\mathit{CAFCP}\right)}_x$.

It therefore follows from Remark 6 that the mild solution to problem ${\left(\mathrm{D}-\mathrm{CAFCP}\right)}_x$ is guaranteed to exist and is unique.

Example 1.

As a final application, we consider a concrete example of a fractionally damped beam equation. This model employs the Caputo fractional derivative in time to capture memory and damping effects, while the fourth-order spatial derivative describes the beam’s bending. The governing equation, together with homogeneous boundary conditions and a prescribed initial displacement, forms the following initial-boundary value problem

$$\left\{\begin{array}{cc}{}^{C}D_t^{\gamma }u(t,x)+k{}^{C}D_t^{\delta }u(t,x)=a^4{\displaystyle \frac{{\partial }^4}{\partial x^4}}u(t,x),\hfill & x\in (0,1),t>0,\hfill \\ u(t,0)=u(t,1)=0,u_x(t,0)=u_x(t,1)=0,\hfill & t>0,\hfill \\ u(0,x)=f\left(x\right),\hfill & x\in (0,1).\hfill \end{array}\right.$$

(33)

Let the state space be $X=L^2(0,1)$. Define the operator A on X by $A=a^4{\displaystyle \frac{{\partial }^4}{\partial x^4}}$ with domain $D\left(A\right)=\left\{h\in W^{4,2}(0,1):h\left(0\right)=h\left(1\right)=0,h^{\prime }\left(0\right)=h^{\prime }\left(1\right)=0\right\}$. The eigenvalues of A are ${\lambda }_n=a^4{n}^4{\pi }^4$ for $n\in \mathbb{N}$, with corresponding eigenfunctions ${\{sin\left(n\pi x\right)\}}_{n\in \mathbb{N}}$. The resolvent set of A is $\rho \left(A\right)=\mathbb{C}\setminus \left\{a^4{n}^4{\pi }^4\mid n\in \mathbb{N}\right\}$.

We define the operator family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ by:

$$\left(U\left(t\right)h\right)\left(x\right)=\sum _{n=1}^{\infty }\left(\sum _{p=0}^{\infty }{a}^{4p}{n}^{4p}{\pi }^{4p}{e}_{\gamma -\delta ,\gamma p+1,-k}^p\left(t\right)\right)h_{n}sin\left(n\pi x\right),$$

where $h\left(x\right)={\sum }_{n=1}^{\infty }{h}_{n}sin\left(n\pi x\right)$.

We now verify that ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ is a Caputo fractional $(\gamma ,\delta ,k)$ resolvent:

(a)

Obviously, $U(·)h\in C\left(\right[0,\infty ),X)$, and when $t=0$, we have $U\left(0\right)h=h$.

(b)

$U\left(r\right)U\left(v\right)=U\left(v\right)U\left(r\right)$ for all $r,v>0$, which follows directly from the definition of ${\left\{U\left(t\right)\right\}}_{t\ge 0}$.

(c)

By (2), the LT of ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ is obtained by

$$\begin{array}{cc}\hfill \mathcal{L}\left\{U\left(t\right)h\right\}\left(\lambda \right)& =\sum _{n=1}^{\infty }\left(\sum _{p=0}^{\infty }{a}^{4p}{n}^{4p}{\pi }^{4p}{\displaystyle \frac{{\lambda }^{p(\gamma -\delta )-(\gamma p+1)}}{{({\lambda }^{\gamma -\delta }+k)}^p}}\right)h_{n}sin(n\pi ·)\hfill \\ & =\sum _{n=1}^{\infty }{\displaystyle \frac{1}{\lambda }}\left(\sum _{p=0}^{\infty }{a}^{4p}{n}^{4p}{\pi }^{4p}{\displaystyle \frac{1}{{({\lambda }^{\gamma }+k{\lambda }^{\delta })}^p}}\right)h_{n}sin(n\pi ·)\hfill \\ & =\sum _{n=1}^{\infty }{\displaystyle \frac{1}{\lambda }}·{\displaystyle \frac{1}{1-\frac{a^4{n}^4{\pi }^4}{{\lambda }^{\gamma }+k{\lambda }^{\delta }}}}{h}_{n}sin(n\pi ·)\hfill \\ & =\sum _{n=1}^{\infty }{\displaystyle \frac{{\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}}{{\lambda }^{\gamma }+k{\lambda }^{\delta }-a^4{n}^4{\pi }^4}}{h}_{n}sin(n\pi ·).\hfill \end{array}$$

The action of operator A on the eigenfunction basis is given by:

$$Asin\left(n\pi x\right)={\lambda }_{n}sin\left(n\pi x\right).$$

Therefore, the action of the operator ${({\lambda }^{\gamma }+k{\lambda }^{\delta }-A)}^{-1}$ on the function $h(·)$ yields:

$${({\lambda }^{\gamma }+k{\lambda }^{\delta }-A)}^{-1}h=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{{\lambda }^{\gamma }+k{\lambda }^{\delta }-{\lambda }_n}}{h}_{n}sin\left(n\pi x\right).$$

Since $({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1})$ is a scalar multiplier, we have:

$$({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1})h=\sum _{n=1}^{\infty }({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1})h_{n}sin\left(n\pi x\right).$$

Applying the operator ${({\lambda }^{\gamma }+k{\lambda }^{\delta }-A)}^{-1}$, we get:

$${({\lambda }^{\gamma }+k{\lambda }^{\delta }-A)}^{-1}({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1})h=\sum _{n=1}^{\infty }{\displaystyle \frac{{\lambda }^{\gamma -1}+k{\lambda }^{\delta -1}}{{\lambda }^{\gamma }+k{\lambda }^{\delta }-{\lambda }_n}}{h}_{n}sin\left(n\pi x\right).$$

Therefore, we verify that:

$$R({\lambda }^{\gamma }+k{\lambda }^{\delta },A)({\lambda }^{\gamma -1}+k{\lambda }^{\delta -1})h=\mathcal{L}\left\{U\left(t\right)h\right\}\left(\lambda \right).$$

(34)

Based on the foregoing derivation, the integration of the denseness of the domain $D\left(A\right)$, equality (34), and Theorem 2 enables us to conclude that the family ${\left\{U\left(t\right)\right\}}_{t\ge 0}$ constitutes a Caputo fractional $(\gamma ,\delta ,k)$ resolvent generated by A. Consequently, by applying the results of Theorems 3 and 4, we establish that for any initial function $f\in D\left(A\right)$, the fractional differential Equation (33) admits a unique strong solution.

The above model characterizes the fractional damping relaxation behavior of a viscoelastic beam under quasi-static conditions. The fourth-order spatial derivative, scaled by the material parameter $a^4$, captures the bending restoring force, where a is related to the beam’s flexural rigidity. The Caputo derivatives ${}^{C}D_t^{\gamma }$ and ${}^{C}D_t^{\delta }$ jointly describe the material’s slow, memory-dependent relaxation: the primary order $\gamma$ governs the overall relaxation rate, and the damping order $\delta$ reflects the memory intensity of energy dissipation. This model is particularly suitable for describing creep and relaxation experiments in viscoelastic materials such as polymers and biological tissues within the small-deformation regime. It accurately captures power-law decay behavior over wide time scales using only a few parameters. Moreover, the use of the Caputo derivative ensures that the initial deflection $u(0,x)=f\left(x\right)$ possesses a clear physical interpretation, facilitating direct comparison with experimental measurements.

5. Conclusions

This paper establishes an operator-theoretic framework for the following Caputo-type abstract fractional Cauchy problem with damping:

$${}^{C}D_t^{\gamma }u\left(t\right)+k{}^{C}D_t^{\delta }u\left(t\right)=Au\left(t\right),t>0,u\left(0\right)=x.$$

The central contribution is the construction and analysis of an appropriate fractional $(\gamma ,\delta ,k)$ resolvent ${\left\{U\left(t\right)\right\}}_{t\ge 0}$. It is shown that the solution can be explicitly represented as $u\left(t\right)=U\left(t\right)x$. This representation yields the unique strong solution for $x\in D\left(A\right)$ and defines the unique mild solution for all $x\in X$.

Key achievements include the rigorous characterization of the solution operator family $U\left(t\right)$ and its generator A, the establishment of equivalence between this abstract equation and a Volterra integral equation, and the proof of well-posedness via Laplace transform techniques. This framework naturally accommodates the initial condition of the Caputo derivative and can be directly applied to concrete problems such as beam vibration models with fractional damping.

Future work may extend this framework to include nonlinear damping, inhomogeneous terms, and stochastic perturbations. The results presented here provide a solid functional-analytic foundation for analyzing and simulating a broad class of dynamical systems that combine memory effects with energy dissipation.

In the direction of the concrete use of our results, it should be mentioned that our theoretical results and in particular the study of the regularity of the solution can be the starting point for designing accurate and fast numerical methods for more involved time and space-time fractional equations arising in applications, from inverse problems to epidemiological models; see [23,24,25] and references therein.