Controllability of Fractional Integro-Differential Equations with Delays and Singular Kernels in Fréchet Spaces

Abstract

This paper is devoted to the investigation of existence and approximate controllability results for a class of fractional integro-differential equations formulated in Fréchet spaces. The analysis is carried out using a generalized version of Darbo’s fixed point theorem adapted to Fréchet spaces, combined with the concept of the measure of noncompactness. To demonstrate the validity and applicability of the theoretical findings, an illustrative example is presented to demonstrate the applicability and validity of the theoretical findings.

1. Introduction

Fractional differential equations have garnered increasing attention in recent decades due to their capacity to model systems with memory and hereditary properties [1]. When introducing randomness and delays, these models become more realistic for various observed phenomena in physics, biology, and finance. Evolution equations and fractional integro-differential equations with delays, in particular, have emerged as powerful frameworks for analyzing such systems in infinite-dimensional settings. Dai, Xiao, and Bu [2] investigated the well-posedness and numerical approximations of stochastic fractional integro-differential equations featuring weakly singular kernels. Similarly, Asadzade and Mahmudov [3] examined a class of stochastic neutral fractional integro-differential equations with weakly singular kernels using the Euler–Maruyama method. More recent developments include stochastic extensions aimed at enhancing the modeling of random effects; see, for instance, the foundational monograph by Da Prato and Zabczyk [4]. For more details on evolution equations, refer to [5,6].

In particular, the field of semilinear evolution equations has witnessed substantial progress in recent times [7,8,9]. Noteworthy achievements include the establishment of global existence results for evolution equations and inclusions on Fréchet and Banach spaces, employing various approaches [10,11]. It is worth noting that earlier studies often imposed certain assumptions, such as Lipschitz conditions and the compactness of the semigroup. Controllability also holds great significance in control theory and engineering as it relates to pole assignment, quadratic optimal control, observer design, and structural decomposition [12,13,14,15,16].

In [17], the authors discussed the existence of mild solutions for the evolution equation:

$$\left\{\begin{array}{c}{u}^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f(t,u\left(t\right));\mathrm{if}t\in {\mathbb{R}}_{+}:=[0,\infty ),\hfill \\ u\left(0\right)=u_0\in E,\hfill \end{array}\right.$$

where $f:{\mathbb{R}}_{+}\times E\to E$ is a given function, $(E,\parallel ·\parallel )$ is a (real or complex) Banach space, and ${\left\{A\left(t\right)\right\}}_{t>0}$ is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of bounded linear operators ${\left\{U(t,s)\right\}}_{(t,s)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}}$ for $(t,s)\in \Lambda :=\left\{(t,s)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}:0\le s\le t<+\infty \right\}$.

In [5], we examined the possibility that the following nonlinear time fractional non-autonomous evolution equations in Fréchet space have mild solutions to their initial value problem:

$$\left\{\begin{array}{c}{}^{c}D_{0,\delta }^{\varsigma }\chi \left(\delta \right)+\mathcal{Z}\left(\delta \right)\chi \left(\delta \right)=\aleph (\delta ,\chi \left(\delta \right)),\mathrm{a}.\mathrm{e}.\delta \in {\mathbb{R}}_{+},\delta \ne {\delta }_{𝚥},𝚥=1,2,\dots .\hfill \\ \chi \left(0\right)={\chi }_0,\hfill \\ \mho \chi \left({\delta }_{𝚥}\right)=I_{𝚥}\left({\int }_{{\delta }_{𝚥}-{\varkappa }_{𝚥}}^{{\delta }_{𝚥}-{\vartheta }_{𝚥}}\widehat{\aleph }(\epsilon ,\chi \left(\epsilon \right))d\epsilon \right),𝚥=1,2,\dots ,\hfill \end{array}\right.$$

where $(0<\varsigma \le 1),{}^{c}D_{0,\delta }^{\varsigma }$ is the $\varsigma$-order Caputo derivative operator, $(\Im ,\parallel ·\parallel )$ is a Banach space, and ${\left\{\mathcal{Z}\left(\delta \right)\right\}}_{\delta >0}$ is a family of linear closed (not necessarily bounded) operators defined on a dense domain $\mathfrak{H}\left(\mathcal{Z}\right)$ in ℑ into ℑ such that $\mathfrak{H}\left(\mathcal{Z}\right)$ is independent of $\delta$, $\aleph :{\mathbb{R}}_{+}\times \Im \to \Im$ is a given function which will be specified later, ${\chi }_0\in \Im$ and $\widehat{\aleph }:{\mathbb{R}}_{+}\times \Im \to \Im$ is a given function; ${\displaystyle 0<{\delta }_0<{\delta }_1<\dots <{\delta }_{ı}<{\delta }_{ı+1}<\dots <\underset{ı\to \infty }{lim}{\delta }_{ı}=\infty ,I_{𝚥}\in \mathcal{C}(\Im ,\Im )}$ are bounded functions, $0\le {\vartheta }_{𝚥}\le {\varkappa }_{𝚥}\le {\delta }_{𝚥}-{\delta }_{𝚥-1}$ for $𝚥\in \mathrm{I}\mathrm{N}$, $\mho \chi \left({\delta }_{𝚥}\right)=\chi \left({\delta }_{𝚥}^{+}\right)-\chi \left({\delta }_{𝚥}^{-}\right)$ and ${\displaystyle \chi \left({\delta }_{𝚥}^{+}\right)=\underset{\varrho \to 0^{+}}{lim}\chi ({\delta }_{𝚥}+\varrho ),\chi \left({\delta }_{𝚥}^{-}\right)=\underset{\varrho \to 0^{-}}{lim}\chi ({\delta }_{𝚥}-\varrho )}$.

Building upon the aforementioned studies and with the aim of advancing the ongoing development of the field, this article focuses on the approximate controllability of a class of fractional integro-differential equations featuring nonlinear delay and a weakly singular kernel, formulated within a Fréchet space framework. Specifically, we consider the following fractional system:

$${}^{c}D_0^{\alpha }u\left(t\right)+Au\left(t\right)={\int }_0^t{\displaystyle \frac{h\left(u\right(s\left)\right)}{{(t-s)}^{\beta }}}f(s,u\left(s\right),u_s)ds+Bv\left(t\right),t\in {\mathbb{R}}^{+},$$

(1)

with initial condition:

$$u\left(t\right)=\varphi \left(t\right),t\in H:=[-\tau ,0].$$

(2)

where $(F,\parallel ·\parallel )$ is a separable Banach space, ${}^{c}D_0^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha \in \left(0,1\right)$ and $A:D\left(A\right)\subset F\to F$ is a densely defined closed linear operator on F. The control function $v(·)$ belongs to $L^2({\mathbb{R}}^{+},U)$, where U is a Banach space and $B:U\to F$ is a bounded linear operator encoding the control action. The nonlinear function $f:{\mathbb{R}}^{+}\times F\times \mathcal{D}\to F$ incorporates the state and delayed history. The history function is defined by $u_t\left(\theta \right):=u(t+\theta )$ for $\theta \in H$, so that $u_t\in \mathcal{D}:=\mathcal{C}(H,F)$ represents the history segment of the state, and $h:F\to \mathbb{R}$ is a nonlinear continuous function, and the singularity ${(t-s)}^{-\beta }$ is of Riemann-Liouville type with $\beta \in (\alpha ,1)$, whose precise assumptions are detailed in the next section.

The key novelties presented in this study are as follows:

–

The problem (1)–(2) can be viewed as a generalization of several existing works in the literature. In particular, it represents a natural extension of the problem investigated in [17], by replacing the classical derivative with the Caputo fractional derivative, introducing a functional delay, and formulating the problem within the framework of Fréchet space instead of a Banach space. This approach broadens and unifies the frameworks explored in [5,17].

–

We establish both necessary and sufficient conditions ensuring not only the existence of mild solutions to problem (1)–(2) but also its approximate controllability. To the best of our knowledge, these results have not been previously reported in the literature.

–

We also present an illustrative example that demonstrate and validate the theoretical results obtained.

2. Preliminaries

In this section, we present some notations, definitions, and auxiliary results that will be used throughout the paper.

For any interval $[0,b]$, we define:

$$\mathcal{C}(I,F):=\{u:[0,b]\to F\mid u\mathrm{is}\mathrm{continuous}\}$$

equipped with the supremum norm:

$${\parallel u\parallel }_{\infty }:=\underset{t\in [0,b]}{sup}\parallel u\left(t\right)\parallel .$$

We denote by $B\left(F\right)$ the Banach space of all bounded linear operators from F into itself, endowed with the operator norm

$${\parallel N\parallel }_{B\left(F\right)}:=\underset{\parallel y\parallel =1}{sup}\parallel N\left(y\right)\parallel .$$

As usual, $L^1([0,b],F)$ denotes the Banach space of Bochner integrable functions $y:[0,b]\to F$, equipped with the norm

$${\parallel y\parallel }_{L^1}:={\int }_0^b\parallel y\left(t\right)\parallel dt.$$

Let $L^{\infty }([0,b],\mathbb{R})$ be the Banach space of measurable functions $v:[0,b]\to \mathbb{R}$ that are essentially bounded and equipped with the norm

$${\parallel v\parallel }_{L^{\infty }}=inf\{c>0:\parallel v\left(t\right)\parallel \le c,a.e.t\in [0,b]\}.$$

We define the Fréchet space

$$\tilde{F}=\{u\in \mathcal{C}([-\tau ,+\infty ),F)\mid u{\mid }_{[-\tau ,0]}=\varphi \},$$

equipped with the family of seminorms:

$${\parallel u\parallel }_n=\underset{t\in [0,n]}{sup}\parallel u\left(t\right)\parallel \mathrm{for}n\in \mathbb{N}.$$

Definition 1 

(Caputo Derivative [18]). For $\alpha \in (0,1)$, and for any absolutely continuous function f, the Caputo fractional derivative of order α is

$${(}^c{D}^{\alpha }f)\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{\displaystyle \frac{f^{\prime }\left(s\right)}{{(t-s)}^{\alpha }}}ds$$

Lemma 1 

([19]). If Y is a bounded subset of Banach space $X,$ then for each $ϵ>0,$ there is a sequence ${\left\{u_k\right\}}_{k=1}^{\infty }\subset Y$ such that

$${\mu }_n\left(Y\right)\le 2{\mu }_n\left({\left\{u_k\right\}}_{k=1}^{\infty }\right)+ϵ;\mathrm{for}n\in \mathbb{N}.$$

Lemma 2 

(Mönch-Type Estimate [20]). Let ${\left\{u_k\right\}}_{k=1}^{\infty }\subset L^1\left([0,b]\right)$ be uniformly integrable. Then:

1. 

The map $t\mapsto {\mu }_n\left({\left\{u_k\left(t\right)\right\}}_{k=1}^{\infty }\right)$ is measurable.

2. 

For all $t\in [0,b]$,

$$\mu \left({\left\{{\int }_0^t{u}_k\left(s\right)ds\right\}}_{k=1}^{\infty }\right)\le 2{\int }_0^t\mu \left({\left\{u_k\left(s\right)\right\}}_{k=1}^{\infty }\right)ds.$$

Definition 2. 

A multivalued map $f:{\mathbb{R}}^{+}\times F\times \mathcal{D}\to F$ is said to be $L^1$-Carathéodory if

(i) 

$t⟼f(t,y,v)$ is measurable for each $(y,v)\in F\times \mathcal{D};$

(ii) 

$y⟼f(t,y,v)$ and $v⟼f(t,y,v)$ are upper semicontinuous for almost all $t\in {\mathbb{R}}^{+};$

(iii) 

for each $\ell >0$, there exists $p_{\ell }\in L^1({\mathbb{R}}^{+},{\mathbb{R}}_{+})$ such that

$$\begin{array}{c}\hfill \parallel f(t,y,v)\parallel \le p_{\ell }\left(t\right)\mathrm{for}\mathrm{all}\parallel y\parallel \le {\ell ,\parallel v\parallel }_{\mathcal{D}}\le \ell and fora.e.t\in {\mathbb{R}}^{+}.\end{array}$$

The function f is said to be Carathéodory if $\left(i\right)$ and $\left(ii\right)$ hold.

Theorem 1 

(Generalized Darbo’s Theorem [21,22]). Let X be a Fréchet space equipped with the family of seminorms ${\{\parallel ·\parallel }_n{\}}_{n\in \mathbb{N}}$, and let ${\left\{{\mu }_n\right\}}_{n\in \mathbb{N}}$ be a family of measures of noncompactness on X. We consider $\Omega \subset X$ a nonempty, closed, convex subset and $\mathcal{N}:\Omega \to \Omega$ a continuous operator. There exists $k_n\in [0,1)$ such that for any bounded $D\subset \Omega$:

$${\mu }_n\left(\mathcal{N}\left(D\right)\right)\le k_n{\mu }_n\left(D\right)\forall n\in \mathbb{N}$$

(3)

Then $\mathcal{N}$ has at least one fixed point in Ω.

3. Existence and Controllability Results

Definition 3. 

The problem (1)–(2) is said to be infinitely controllable on the interval $[-\tau ;\infty )$ if for every initial function $\varphi \in \mathcal{D}$ and every $u_1\in F$; there is an $n\in \mathbb{N}$, and a control $v\in L^2([0,n];U)$; such that the mild solution u of (1)–(2) satisfies $u\left(n\right)=u_1$.

Definition 4 

(Mild solution). We say that a continuous function $u(·):[-\tau ,\infty )\to F$ is mild solution of the problem (1)–(2), if $u\left(t\right)=\varphi \left(t\right),t\in [-\tau ,0]$ and for each $t\in {\mathbb{R}}^{+}$ satisfies the following integral equation

$$\begin{array}{cc}\hfill u\left(t\right)& =T_{\alpha }\left(t\right)\varphi \left(0\right)+{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\mathcal{F}\left(u\right)\left(s\right)ds\hfill \\ \hfill & +{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)Bv\left(s\right)ds,\hfill \end{array}$$

where the nonlinear term is:

$$\mathcal{F}\left(u\right)\left(s\right)={\int }_0^s{\displaystyle \frac{h\left(u\right(\xi \left)\right)}{{(s-\xi )}^{\beta }}}f(\xi ,u\left(\xi \right),u_{\tau }\left(\xi \right))d\xi ,$$

(4)

and $T_{\alpha }\left(t\right)$ and $S_{\alpha }\left(t\right)$ are defined via the contour integral:

$$\begin{array}{cc}\hfill T_{\alpha }\left(t\right)& ={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{\lambda t}R({\lambda }^{\alpha },A)d\lambda ,\hfill \\ \hfill S_{\alpha }\left(t\right)& ={\displaystyle \frac{1}{2\pi i}}{\int }_{\Gamma }{e}^{\lambda t}{\lambda }^{\alpha -1}R({\lambda }^{\alpha },A)d\lambda ,\hfill \end{array}$$

where Γ is a suitable path in the complex plane around the spectrum of A.

The constants $M_{\alpha }$ and $L_{\alpha }$ are defined by:

$$M_{\alpha }=\underset{t\in [0,n],n\in \mathbb{N}}{sup}\parallel T_{\alpha }\left(t\right)\parallel ,L_{\alpha }=\underset{t\in [0,n],n\in \mathbb{N}}{sup}\parallel S_{\alpha }\left(t\right)\parallel .$$

The hypotheses: To tackle the problem, we introduce the following assumptions, which are needed for establishing our main results:

$\left(H_1\right)$

The operator A generates an equicontinuous $C_0$-semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on F.

$\left(H_2\right)$

$f:{\mathbb{R}}^{+}\times F\times \mathcal{D}\to F$ is $L^1$-Carathéodory.

$\left(H_3\right)$

There exists $L>0$ such that f is globally Lipschitz:

$$\parallel f(t,p_1,q_1)-f(t,p_2,q_2)\parallel \le L(\parallel p_1-p_2\parallel +\parallel q_1-q_2{\parallel }_{\mathcal{D}}),$$

for $t\in {\mathbb{R}}^{+},p_1,p_2\in F$ and $q_1,q_2\in \mathcal{D}$.

$\left(H_4\right)$

The function $h:{\mathbb{R}}^{+}\to F$ satisfies

(i)

For every $R>0$ there exists $M_R$ such that: ${\displaystyle \underset{\parallel p\parallel \le R}{sup}\parallel h\left(p\right)\parallel \le M_R}$.

(ii)

There exists $M>0$ such that

$$\mathrm{for}\mathrm{each}{p}_1,p_2\in F,|h\left(p_1\right)-h\left(p_2\right)|\le M\parallel p_1-p_2\parallel .$$

$\left(H_5\right)$

There exists a function ${\rho }_f\in L^{\infty }({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ and for all $(p,q)\in F\times \mathcal{D}$, we have:

$$\parallel f(t,p,q)\parallel \le {\rho }_f\left(t\right)\left(\parallel p\parallel +{\parallel q\parallel }_{\mathcal{D}}\right).$$

$\left(H_6\right)$

For each bounded and measurable set $Z\subset F$ and $Z_1\subset \mathcal{D}$ and for each $t\in {\mathbb{R}}_{+}$, we have

$$\mu \left(f(t,Z,Z_1)\right)\le {\rho }_f\left(t\right)\left[\mu \left(Z\right)+\underset{-\tau \le \varrho \le 0}{sup}\mu \left(Z_1\left(\varrho \right)\right)\right],$$

where $Z_1\left(\varrho \right)=\{u\left(\varrho \right):u\in Z_1\}$ and $\mu$ is a measure of noncompactness on the Banach space F.

$\left(H_7\right)$

For each $n\in \mathbb{N}$, the linear operator $W:L^2([0,n],F)\to F$ is defined by

$${\displaystyle W_{n}u={\int }_0^n{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)Bv\left(s\right)ds,}$$

has a bounded pseudo inverse operator $W_n^{-1}$ which takes values in $L^2([0,n],F)/ker{W}_n$ and there exist positive constant (uniformly in n) K such that:

$$\parallel B{W}_n^{-1}\parallel \le K.$$

Remark 1. 

For the construction of W, see [23].

Definition 5 

(Control Function). To ensure controllability, we define the control function $v\in L^2([0,n],U)$ by

$$v\left(t\right)=W_n^{-1}\left(u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)-{\int }_0^n{(n-s)}^{\alpha -1}{S}_{\alpha }(n-s)\mathcal{F}\left(u\right)\left(s\right)ds\right)\left(t\right),t\in [0,n].$$

This expression ensures that the mild solution $u\left(t\right)$ satisfies the final condition $u\left(n\right)=u_1$.

For $n\in \mathbb{N}$, let

$${\rho }_n^{\ast }=\underset{t\in [0,n]}{ess\; sup}{\rho }_f\left(t\right)$$

and we define the family of measure of noncompactness ${\left\{{\mu }_n\right\}}_{n\in \mathbb{N}}$ by

$${\mu }_n\left(D\right)=\underset{t\in [0,n]}{sup}\chi \left(D\left(t\right)\right)+{\omega }_0^n\left(D\right),\mathrm{for}D\subset \tilde{F}:=\mathcal{C}\left({\mathbb{R}}^{+}\right)$$

where $\chi$ is the Kuratowski measure and ${\omega }_0^n\left(D\right)$ is the modulus of continuity, and

$$D\left(t\right)=\left\{v\right(t)\in F:v\in D\};t\in [0,n].$$

Let

$$B_{R_n}=\{w\in \mathcal{C}({\mathbb{R}}^{+},F)\mid {\parallel w\parallel }_n\le R_n\}.$$

Theorem 2 

(Controllability Result). If $\left(H_1\right)$–$\left(H_7\right)$ are satisfied and

$$J_n:L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{\displaystyle \frac{B(\alpha ,2-\beta )}{1-\beta }}\left(1+{\displaystyle \frac{L_{\alpha }K}{\alpha }}\right)=<1,$$

then the system (1) and (2) is controllable.

Proof. 

First, we define an operator $\mathcal{N}$ using the solution u of our system (1) and (2). Using Definition 5, we have

$$\begin{array}{cc}\hfill u\left(t\right)=& T_{\alpha }\left(t\right)\varphi \left(0\right)+{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\mathcal{F}\left(u\right)\left(s\right)ds\hfill \\ & +{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)B{W}^{-1}(u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)\hfill \\ & -{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u\right)\left(\sigma \right)d\sigma )\left(s\right)ds.\hfill \end{array}$$

We shall now show that when using this control, the operator $\mathcal{N}:\tilde{F}\to \tilde{F}$ defined by

$$\mathcal{N}\left(u\right)\left(t\right)=\left\{\begin{array}{c}\varphi \left(t\right),\mathrm{if}t\in [-r,0],\hfill \\ \\ {\displaystyle T_{\alpha }\left(t\right)\varphi \left(0\right)+{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\mathcal{F}\left(u\right)\left(s\right)ds}\hfill \\ {\displaystyle +{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)B{W}^{-1}(u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)}\hfill \\ {\displaystyle -{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u\right)\left(\sigma \right)d\sigma )\left(s\right)ds,\mathrm{if}t\ge 0,}\hfill \end{array}\right.$$

has fixed points which are mild solutions of the system (1) and (2).

For any $n\in \mathbb{N},$ let $R_n$ be a positive real number with

$$R_n\ge {\displaystyle \frac{A_n}{1-J_n}},$$

where

$$\begin{array}{cc}\hfill A_n& =M_{\alpha }\parallel \varphi \left(0\right)\parallel +{\displaystyle \frac{L_{\alpha }K{n}^{\alpha }}{\alpha }}\left(\parallel u_1\parallel +M_{\alpha }\parallel \varphi \left(0\right)\parallel \right)\hfill \\ \hfill & +L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{\displaystyle \frac{B(\alpha ,2-\beta )}{1-\beta }}\left(1+{\displaystyle \frac{L_{\alpha }K}{\alpha }}\right){\parallel \varphi \parallel }_{\mathcal{D}}.\hfill \end{array}$$

Consider the ball

$$B_{R_n}:=B(0,R_n)=\{w\in \tilde{F}{:\parallel w\parallel }_n\le R_n\}.$$

The proof will be given in several steps.

Step 1. We will prove that $\mathcal{N}$ transforms the ball $B_{R_n}$ into itself.

The operator $\mathcal{N}$ decomposes as:

$$\begin{array}{cc}\hfill \mathcal{N}\left(u\right)\left(t\right)& =I_1+I_2+I_3,\mathrm{where}\hfill \\ \hfill I_1& =T_{\alpha }\left(t\right)\varphi \left(0\right),\hfill \\ \hfill I_2& ={\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\mathcal{F}\left(u\right)\left(s\right)ds,\hfill \\ \hfill I_3& {\displaystyle ={\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)B{W}^{-1}(u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)}\hfill \\ \hfill & {\displaystyle -{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u\right)\left(\sigma \right)d\sigma )\left(s\right)ds.}\hfill \end{array}$$

For any $n\in \mathbb{N},$ and each $u\in B_{R_n}$ and $t\in [0,n],n\in \mathbb{N}$ we have

$$\parallel I_1{\parallel }_n\le M_{\alpha }\parallel \varphi \left(0\right)\parallel .$$

Using the hypotheses ($H_4$)–($H_5$), we have

$$\begin{array}{cc}\hfill \parallel \mathcal{F}\left(u\right)\left(s\right)\parallel & \le {\int }_0^s{\displaystyle \frac{\left|h\right(u\left(\xi \right)\left)\right|}{{(s-\xi )}^{\beta }}}\parallel f(\xi ,u\left(\xi \right),u_{\tau }\left(\xi \right))\parallel d\xi \hfill \\ \hfill & \le M_{R_n}{\int }_0^s{\displaystyle \frac{{\rho }_f\left(\xi \right)}{{(s-\xi )}^{\beta }}}\left(\parallel u\left(\xi \right)\parallel +\parallel u_{\tau }{\left(\xi \right)\parallel }_{\mathcal{D}}\right)d\xi \hfill \\ \hfill & \le M_{R_n}\left({\parallel u\parallel }_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\int }_0^s{\displaystyle \frac{{\rho }_f\left(\xi \right)}{{(s-\xi )}^{\beta }}}d\xi \hfill \\ \hfill & \le M_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\parallel {\rho }_f\parallel }_{L^{\infty }\left([0,s]\right)}{\displaystyle \frac{s^{1-\beta }}{1-\beta }}\hfill \\ \hfill & \le M_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\rho }_n^{\ast }{\displaystyle \frac{s^{1-\beta }}{1-\beta }}.\hfill \end{array}$$

Using our estimate of $\mathcal{F}\left(u\right)$ we have:

$$\begin{array}{cc}\hfill \parallel I_2\parallel & =∥{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\mathcal{F}\left(u\right)\left(s\right)ds∥\hfill \\ \hfill & \le L_{\alpha }{\int }_0^t{(t-s)}^{\alpha -1}\parallel \mathcal{F}\left(u\right)\left(s\right)\parallel ds\hfill \\ \hfill & \le L_{\alpha }{\int }_0^t{(t-s)}^{\alpha -1}{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\parallel {\rho }_f\parallel }_{L^{\infty }\left([0,s]\right)}{\displaystyle \frac{s^{1-\beta }}{1-\beta }}ds\hfill \\ \hfill & \le L_{\alpha }{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right)\parallel {\rho }_n^{\ast }{\displaystyle \frac{1}{1-\beta }}{\int }_0^t{(t-s)}^{\alpha -1}{s}^{1-\beta }ds\hfill \\ \hfill & \le L_{\alpha }{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right)\parallel {\rho }_n^{\ast }{\displaystyle \frac{1}{1-\beta }}\mathrm{B}(\alpha ,2-\beta ).\hfill \end{array}$$

Then,

$$\parallel I_2{\parallel }_n\le L_{\alpha }{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\rho }_n^{\ast }{\displaystyle \frac{\mathrm{B}(\alpha ,2-\beta )}{1-\beta }}.$$

Also,

$$\begin{array}{cc}\hfill \parallel I_3\parallel & =∥{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)Bv\left(s\right)ds∥\hfill \\ \hfill & \le L_{\alpha }{\int }_0^t{(t-s)}^{\alpha -1}\parallel B\parallel \hfill \\ \hfill & \times ∥W_n^{-1}\left(u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)-{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u\right)\left(\sigma \right)d\sigma \right)\left(s\right)∥ds\hfill \\ \hfill & \le L_{\alpha }K{\int }_0^t{(t-s)}^{\alpha -1}\hfill \\ \hfill & \times ∥\left(u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)-{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u\right)\left(\sigma \right)d\sigma \right)∥ds\hfill \\ \hfill & \le L_{\alpha }K{\int }_0^t{(t-s)}^{\alpha -1}\hfill \\ \hfill & \times \left(\parallel u_1\parallel +\parallel T_{\alpha }\left(n\right)\parallel \parallel \varphi \left(0\right)\parallel +\parallel {\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u\right)\left(\sigma \right)d\sigma \parallel \right)ds\hfill \\ \hfill & \le L_{\alpha }K{\int }_0^t{(t-s)}^{\alpha -1}\left(\parallel u_1\parallel +M_{\alpha }\parallel \varphi \left(0\right)\parallel +L_{\alpha }{\int }_0^n{(n-\sigma )}^{\alpha -1}\parallel \mathcal{F}\left(u\right)\left(\sigma \right)\parallel d\sigma \right)ds\hfill \\ \hfill & \le L_{\alpha }K\left[{\int }_0^t\left({(t-s)}^{\alpha -1}\parallel u_1\parallel +M_{\alpha }\parallel \varphi \left(0\right)\parallel \right)ds\right.\hfill \\ \hfill & +\left.{\int }_0^n{(t-s)}^{\alpha -1}{\int }_0^n{(n-\sigma )}^{\alpha -1}\parallel \mathcal{F}\left(u\right)\left(\sigma \right)\parallel d\sigma ds\right]\hfill \\ \hfill & \le L_{\alpha }K{\displaystyle \frac{n^{\alpha }}{\alpha }}\left(\parallel u_1\parallel +M_{\alpha }\parallel \varphi \left(0\right)\parallel +L_{\alpha }{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\rho }_n^{\ast }{\displaystyle \frac{\mathrm{B}(\alpha ,2-\beta )}{1-\beta }}\right).\hfill \end{array}$$

Then,

$$\parallel I_3{\parallel }_n\le L_{\alpha }K{\displaystyle \frac{n^{\alpha }}{\alpha }}\left(\parallel u_1\parallel +M_{\alpha }\parallel \varphi \left(0\right)\parallel +L_{\alpha }{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\rho }_n^{\ast }{\displaystyle \frac{\mathrm{B}(\alpha ,2-\beta )}{1-\beta }}\right).$$

Combining the individual estimates for $I_1$, $I_2$, and $I_3$ yields the following bound on the operator:

$$\begin{array}{cc}\hfill {\parallel \mathcal{N}\left(u\right)\parallel }_n& \le \parallel I_1{\parallel }_n+\parallel I_2{\parallel }_n+{\parallel I_3\parallel }_n\hfill \\ \hfill & \le M_{\alpha }\parallel \varphi \left(0\right)\parallel +L_{\alpha }{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\rho }_n^{\ast }{\displaystyle \frac{\mathrm{B}(\alpha ,2-\beta )}{1-\beta }}\hfill \\ \hfill & +L_{\alpha }K{\displaystyle \frac{n^{\alpha }}{\alpha }}\left(\parallel u_1\parallel +M_{\alpha }\parallel \varphi \left(0\right)\parallel +L_{\alpha }{M}_{R_n}\left(R_n+{\parallel \varphi \parallel }_{\mathcal{D}}\right){\rho }_n^{\ast }{\displaystyle \frac{\mathrm{B}(\alpha ,2-\beta )}{1-\beta }}\right)\hfill \\ \hfill & \le R_n.\hfill \end{array}$$

This proves that $\mathcal{N}$ transforms the ball $B_{R_n}$ into itself. We shall show that the operator $\mathcal{N}:B_{R_n}\to B_{R_n}$ satisfies all the assumptions of Theorem 1.

Step 2. $\mathcal{N}$ is continuous.

For all $s\in [0,n]$, we have

$$\begin{array}{cc}\hfill & \parallel \mathcal{F}\left(u_k\right)\left(s\right)-\mathcal{F}\left(u\right)\left(s\right)\parallel \hfill \\ \hfill & =∥{\int }_0^s{\displaystyle \frac{1}{{(s-\xi )}^{\beta }}}[h\left(u_k\left(\xi \right)\right)f(\xi ,u_k\left(\xi \right),{\left(u_k\right)}_{\tau }\left(\xi \right))\hfill \\ \hfill & -h\left(u\left(\xi \right)\right)f(\xi ,u\left(\xi \right),u_{\tau }\left(\xi \right))]d\xi ∥\hfill \\ \hfill & \le {\int }_0^s{\displaystyle \frac{1}{{(s-\xi )}^{\beta }}}∥h\left(u_k\left(\xi \right)\right)f(\xi ,u_k\left(\xi \right),{\left(u_k\right)}_{\tau }\left(\xi \right))-h\left(u\left(\xi \right)\right)f(\xi ,u\left(\xi \right),u_{\tau }\left(\xi \right))∥d\xi \hfill \\ \hfill & ={\int }_0^s{\displaystyle \frac{1}{{(s-\xi )}^{\beta }}}(∥\left(h\left(u_k\left(\xi \right)\right)-h\left(u\left(\xi \right)\right)\right)f(\xi ,u_k\left(\xi \right),{\left(u_k\right)}_{\tau }\left(\xi \right))∥\hfill \\ \hfill & +\parallel h\left(u\left(\xi \right)\right)\left(f(\xi ,u_k\left(\xi \right),{\left(u_k\right)}_{\tau }\left(\xi \right))-f(\xi ,u\left(\xi \right),u_{\tau }\left(\xi \right))\right)\parallel )d\xi \hfill \\ \hfill & \le {\int }_0^s{\displaystyle \frac{1}{{(s-\xi )}^{\beta }}}(M\parallel u_k\left(\xi \right)-u\left(\xi \right)\parallel ·{\rho }_f\left(\xi \right)(\parallel u_k\left(\xi \right)\parallel +\parallel {\left(u_k\right)}_{\tau }\left(\xi \right){\parallel }_{\mathcal{D}})\hfill \\ \hfill & +M_{R_n}·L\left(\parallel u_k\left(\xi \right)-u\left(\xi \right)\parallel +\parallel {\left(u_k\right)}_{\tau }\left(\xi \right)-u_{\tau }\left(\xi \right){\parallel }_{\mathcal{D}}\right))d\xi \hfill \\ \hfill & \le \left[M{\rho }_n^{\ast }(R_n+{\parallel \varphi \parallel }_{\mathcal{D}})+M_{R_n}{L(1+\parallel \varphi \parallel }_{\mathcal{D}})\right]{\parallel u_k-u\parallel }_n{\int }_0^s{\displaystyle \frac{d\xi }{{(s-\xi )}^{\beta }}}\hfill \\ \hfill & =\left[M{\rho }_n^{\ast }(R_n+{\parallel \varphi \parallel }_{\mathcal{D}})+M_{R_n}{L(1+\parallel \varphi \parallel }_{\mathcal{D}})\right]{\displaystyle \frac{s^{1-\beta }}{1-\beta }}{\parallel u_k-u\parallel }_n.\hfill \end{array}$$

Since $\parallel {\left(u_k\right)}_{\tau }-u_{\tau }{\parallel }_{\mathcal{D}}\le {\parallel u_k-u\parallel }_n$, then

$$\parallel \mathcal{F}\left(u_k\right)-\mathcal{F}\left(u\right){\parallel }_{L^1[0,n]}\to 0\mathrm{as}k\to \infty .$$

From $\left(H_7\right)$, the control difference satisfies

$$\begin{array}{cc}\hfill \parallel v_k{-v\parallel }_{L^2[0,n]}& ={∥W_n^{-1}\left({\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\left[\mathcal{F}\left(u_k\right)\left(\sigma \right)-\mathcal{F}\left(u\right)\left(\sigma \right)\right]d\sigma \right)∥}_{L^2[0,n]}\hfill \\ \hfill & \le \parallel W_n^{-1}\parallel ∥{\int }_0^n{(n-\sigma )}^{\alpha -1}\parallel S_{\alpha }(n-\sigma )\parallel ·\parallel \mathcal{F}\left(u_k\right)\left(\sigma \right)-\mathcal{F}\left(u\right)\left(\sigma \right)\parallel d\sigma ∥\hfill \\ \hfill & \le \parallel W_n^{-1}\parallel L_{\alpha }{\int }_0^n{(n-\sigma )}^{\alpha -1}\parallel \mathcal{F}\left(u_k\right)\left(\sigma \right)-\mathcal{F}\left(u\right)\left(\sigma \right)\parallel d\sigma \hfill \\ \hfill & \le \parallel W_n^{-1}\parallel L_{\alpha }\left[M{\rho }_n^{\ast }(R_n+{\parallel \varphi \parallel }_{\mathcal{D}})+M_{R_n}{L(1+\parallel \varphi \parallel }_{\mathcal{D}})\right]\hfill \\ \hfill & \times {\displaystyle \frac{1}{1-\beta }}{\parallel u_k-u\parallel }_n{\int }_0^n{(n-\sigma )}^{\alpha -1}{\sigma }^{1-\beta }d\sigma \hfill \\ \hfill & =\parallel W_n^{-1}\parallel L_{\alpha }\left[M{\rho }_n^{\ast }(R_n+{\parallel \varphi \parallel }_{\mathcal{D}})+M_{R_n}{L(1+\parallel \varphi \parallel }_{\mathcal{D}})\right]\hfill \\ \hfill & \times {\displaystyle \frac{1}{1-\beta }}{\parallel u_k-u\parallel }_n·n^{\alpha +1-\beta }\mathrm{B}(\alpha ,2-\beta )\hfill \\ \hfill & \to 0\mathrm{as}k\to \infty .\hfill \end{array}$$

Since $u_k\to u$ as $k\to \infty$, the Lebesgue Dominated Convergence Theorem and the previous estimates imply that:

$$\begin{array}{cc}\hfill & \parallel \mathcal{N}\left(u_k\right)-\mathcal{N}\left(u\right){\parallel }_n\hfill \\ \hfill & \le \parallel I_1\left(u_k\right)-I_1\left(u\right){\parallel }_n+\parallel I_2\left(u_k\right)-I_2\left(u\right){\parallel }_n+{\parallel I_3\left(u_k\right)-I_3\left(u\right)\parallel }_n\hfill \\ \hfill & \le L_{\alpha }{\int }_0^n{(n-s)}^{\alpha -1}\parallel \mathcal{F}\left(u_k\right)\left(s\right)-\mathcal{F}\left(u\right)\left(s\right)\parallel ds\hfill \\ \hfill & +L_{\alpha }K{\int }_0^n{(n-s)}^{\alpha -1}∥{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )(\mathcal{F}\left(u_k\right)\left(\sigma \right)-\mathcal{F}\left(u\right)\left(\sigma \right))d\sigma ∥ds\hfill \\ \hfill & \le L_{\alpha }{\int }_0^n{(n-s)}^{\alpha -1}\parallel \mathcal{F}\left(u_k\right)\left(s\right)-\mathcal{F}\left(u\right)\left(s\right)\parallel ds\hfill \\ \hfill & +L_{\alpha }^{2}K{\int }_0^n{(n-s)}^{\alpha -1}{\int }_0^n{(n-\sigma )}^{\alpha -1}∥\mathcal{F}\left(u_k\right)\left(\sigma \right)-\mathcal{F}\left(u\right)\left(\sigma \right)∥d\sigma ds\hfill \\ \hfill & \to 0\mathrm{as}k\to \infty .\hfill \end{array}$$

Step 3. $\mathcal{N}\left(B_{R_n}\right)$ is bounded.

Since $\mathcal{N}\left(B_{R_n}\right)\subset B_{R_n}$ and $B_{R_n}$ is bounded, then $\mathcal{N}\left(B_{R_n}\right)$ is bounded.

Step 4. For each equicontinuous subset D of $B_{R_n},{\mu }_n\left(\mathcal{N}\left(D\right)\right)\le {\ell }_n{\mu }_n\left(D\right).$

From Lemmas 1 and 2, for any $D\subset B_{R_n}$ and any $ϵ>0,$ there exists a sequence ${\left\{u_k\right\}}_{k=0}^{\infty }\subset D$ such that for all $t\in [0,n]$, under hypotheses $\left(H_4\right)$–$\left(H_7\right)$ we have

$$\begin{array}{cc}\hfill \chi \left(\mathcal{F}\right(D\left)\right(s\left)\right)& \le 2{\int }_0^s\chi \left({\left\{{\displaystyle \frac{h\left(u_k\left(\xi \right)\right)}{{(s-\xi )}^{\beta }}}f(\xi ,u_k\left(\xi \right),{\left(u_{\tau }\right)}_k\left(\xi \right))\right\}}_{k=0}^{\infty }\right)d\xi \hfill \\ \hfill & \le 2{\int }_0^s{\displaystyle \frac{\chi \left({\left\{h\left(u_k\left(\xi \right)\right)f(\xi ,u_k\left(\xi \right),{\left(u_{\tau }\right)}_k\left(\xi \right))\right\}}_{k=0}^{\infty }\right)}{{(s-\xi )}^{\beta }}}d\xi \hfill \\ \hfill & \le 2{\int }_0^s{\displaystyle \frac{{\parallel h\parallel }_{\infty }\chi \left({\left\{f(\xi ,u_k\left(\xi \right),{\left(u_{\tau }\left(\xi \right)\right)}_k)\right\}}_{k=0}^{\infty }\right)}{{(s-\xi )}^{\beta }}}d\xi \hfill \\ \hfill & \le 2{\int }_0^s{\displaystyle \frac{M_{R_n}}{{(s-\xi )}^{\beta }}}{\rho }_f\left(\xi \right)\left[\chi ({\left\{u_k\left(\xi \right)\right\}}_{k=0}^{\infty }+\underset{\omega \in [\xi -\tau ,\xi ]}{sup}\chi \left({\left\{u_k\left(\omega \right)\right\}}_{k=0}^{\infty }\right)\right]d\xi \hfill \\ \hfill & \le 2M_{R_n}{\rho }_n^{\ast }{\int }_0^s{\displaystyle \frac{1}{{(s-\xi )}^{\beta }}}{\rho }_f\left(\xi \right)\left[\chi ({\left\{u_k\left(\xi \right)\right\}}_{k=0}^{\infty }+\underset{\omega \in [\xi -\tau ,\xi ]}{sup}\chi \left({\left\{u_k\left(\omega \right)\right\}}_{k=0}^{\infty }\right)\right]d\xi .\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill & \mu \left(\right(\mathcal{N}\left(D\right)\left)\right(t\left)\right)\hfill \\ \hfill & \le \mu \left(I_1\right)+\mu \left(I_2\right)+\mu \left(I3\right)\hfill \\ \hfill & \le 2\chi \left({\left\{{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\left({\int }_0^s{\displaystyle \frac{h\left(u\right(\xi \left)\right)}{{(s-\xi )}^{\beta }}}f(\xi ,u\left(\xi \right),u_{\tau }\left(\xi \right))d\xi \right)ds\right\}}_{k=1}^{\infty }\right)\hfill \\ \hfill & +2\chi \left(\left\{{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)B{W}_n^{-1}[u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)\right.\right.\hfill \\ \hfill & \left.{\left.-{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\left({\int }_0^{\sigma }{\displaystyle \frac{h\left(u\right(\zeta \left)\right)}{{(\sigma -\zeta )}^{\beta }}}f(\zeta ,u\left(\zeta \right),u_{\tau }\left(\zeta \right))d\zeta \right)d\sigma ]\left(s\right)ds\right\}}_{k=1}^{\infty }\right)+ϵ\hfill \end{array}$$

(5)

$$\begin{array}{ccccccccccccccc}:=2(I_2^{\prime }+I_3^{\prime })+ϵ,\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$$

(6)

where the terms $I_2^{\prime }$ and $I_3^{\prime }$ are estimated as

$$\begin{array}{cc}\hfill I_2^{\prime }& =\chi \left({\left\{{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\mathcal{F}\left(u_k\right)\left(s\right)ds\right\}}_{k=1}^{\infty }\right)\hfill \\ \hfill & \le 2{\int }_0^t{(t-s)}^{\alpha -1}\parallel S_{\alpha }(t-s)\parallel \chi \left({\left\{\mathcal{F}\left(u_k\right)\left(s\right)\right\}}_{k=1}^{\infty }\right)ds\hfill \\ \hfill & =2L_{\alpha }{\int }_0^t{(t-s)}^{\alpha -1}\chi \left({\left\{{\int }_0^s{\displaystyle \frac{h\left(u_k\left(\xi \right)\right)}{{(s-\xi )}^{\beta }}}f(\xi ,u_k\left(\xi \right),{\left(u_{\tau }\right)}_k\left(\xi \right))d\xi \right\}}_{k=1}^{\infty }\right)ds\hfill \\ \hfill & \le 4L_{\alpha }{\int }_0^t{(t-s)}^{\alpha -1}{\int }_0^s{\displaystyle \frac{M_{R_n}}{{(s-\xi )}^{\beta }}}{\rho }_f\left(\xi \right)\left[\chi \left(\left\{u_k\left(\xi \right)\right\}\right)+\underset{\omega \in [\xi -\tau ,\xi ]}{sup}\chi \left(\left\{u_k\left(\omega \right)\right\}\right)\right]d\xi ds\hfill \\ \hfill & \le 8L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{\mu }_n\left(D\right){\int }_0^t{(t-s)}^{\alpha -1}{\int }_0^s{\displaystyle \frac{1}{{(s-\xi )}^{\beta }}}d\xi ds\hfill \\ \hfill & ={\displaystyle \frac{8L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{n}^{1-\beta }}{1-\beta }}{\mu }_n\left(D\right){\int }_0^t{(t-s)}^{\alpha -1}ds\hfill \\ \hfill & ={\displaystyle \frac{8L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{n}^{1-\beta }}{\alpha (1-\beta )}}{t}^{\alpha }{\mu }_n\left(D\right)\hfill \\ \hfill & \le {\displaystyle \frac{8L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{n}^{\alpha +1-\beta }}{\alpha (1-\beta )}}{\mu }_n\left(D\right).\hfill \end{array}$$

Also,

$$\begin{array}{cc}\hfill I_3^{\prime }& =\chi \left(\left\{{\int }_0^t{(t-s)}^{\alpha -1}{S}_{\alpha }(t-s)\right.\right.\hfill \\ \hfill & \left.{\left.\times B{W}_n^{-1}\left[u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)-{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u_k\right)\left(\sigma \right)d\sigma \right]ds\right\}}_{k=1}^{\infty }\right)\hfill \\ \hfill & \le 2{\int }_0^t{(t-s)}^{\alpha -1}\parallel S_{\alpha }(t-s)B{W}_n^{-1}\parallel \hfill \\ \hfill & \times \chi \left({\left\{u_1-T_{\alpha }\left(n\right)\varphi \left(0\right)-{\int }_0^n{(n-\sigma )}^{\alpha -1}{S}_{\alpha }(n-\sigma )\mathcal{F}\left(u_k\right)\left(\sigma \right)d\sigma \right\}}_{k=1}^{\infty }\right)ds\hfill \\ \hfill & \le 2L_{\alpha }K{\int }_0^t{(t-s)}^{\alpha -1}\left(\chi \left(\left\{u_1\right\}\right)+\chi \left(\left\{T_{\alpha }\left(n\right)\varphi \left(0\right)\right\}\right)\right.\hfill \\ \hfill & \left.+2{\int }_0^n{(n-\sigma )}^{\alpha -1}\parallel S_{\alpha }(n-\sigma )\parallel \chi \left(\left\{\mathcal{F}\left(u_k\right)\left(\sigma \right)\right\}\right)d\sigma \right)ds\hfill \\ \hfill & =4L_{\alpha }K{\int }_0^t{(t-s)}^{\alpha -1}{\int }_0^n{(n-\sigma )}^{\alpha -1}{L}_{\alpha }\chi \left(\left\{\mathcal{F}\left(u_k\right)\left(\sigma \right)\right\}\right)d\sigma ds\hfill \\ \hfill & \le 4L_{\alpha }^{2}K{\int }_0^t{(t-s)}^{\alpha -1}{\int }_0^n{(n-\sigma )}^{\alpha -1}\left(2·{\displaystyle \frac{2M_{R_n}{\rho }_n^{\ast }{\sigma }^{1-\beta }}{1-\beta }}{\mu }_n\left(D\right)\right)d\sigma ds\hfill \\ \hfill & ={\displaystyle \frac{16L_{\alpha }^{2}K{M}_{R_n}{\rho }_n^{\ast }}{1-\beta }}{\mu }_n\left(D\right){\int }_0^t{(t-s)}^{\alpha -1}{\int }_0^n{(n-\sigma )}^{\alpha -1}{\sigma }^{1-\beta }d\sigma ds\hfill \\ \hfill & ={\displaystyle \frac{16L_{\alpha }^{2}K{M}_{R_n}{\rho }_n^{\ast }{n}^{2\alpha +1-\beta }}{(1-\beta )\alpha }}B(\alpha ,2-\beta ){\mu }_n\left(D\right).\hfill \end{array}$$

Thus we obtain the final contraction estimate of (5):

$$\begin{array}{cc}\hfill & \mu \left(\right(\mathcal{N}\left(D\right)\left)\right(t\left)\right)\hfill \\ \hfill & \le 2\left({\displaystyle \frac{8L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{n}^{\alpha +1-\beta }}{\alpha (1-\beta )}}{\mu }_n\left(D\right)+{\displaystyle \frac{16L_{\alpha }^{2}K{M}_{R_n}{\rho }_n^{\ast }{n}^{2\alpha +1-\beta }}{(1-\beta )\alpha }}B(\alpha ,2-\beta ){\mu }_n\left(D\right)\right)+ϵ\hfill \\ \hfill & =\left({\displaystyle \frac{16L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{n}^{\alpha +1-\beta }}{\alpha (1-\beta )}}+{\displaystyle \frac{32L_{\alpha }^{2}K{M}_{R_n}{\rho }_n^{\ast }{n}^{2\alpha +1-\beta }}{(1-\beta )\alpha }}B(\alpha ,2-\beta )\right){\mu }_n\left(D\right)+ϵ\hfill \\ \hfill & \le \left[{\displaystyle \frac{16L_{\alpha }{M}_{R_n}{\rho }_n^{\ast }{n}^{\alpha +1-\beta }}{\alpha (1-\beta )}}\left(1+2L_{\alpha }K{n}^{\alpha }B(\alpha ,2-\beta )\right)\right]{\mu }_n\left(D\right)+ϵ.\hfill \end{array}$$

Since $ϵ>0$ is arbitrary, then

$$\mu \left(\left(\mathcal{N}\left(D\right)\right)\left(t\right)\right)\le {\ell }_n\mu \left(D\right).$$

Thus,

$$\mu \left(\mathcal{N}\left(D\right)\right)\le {\ell }_n\mu \left(D\right).$$

As a consequence of the generalized Darbo’s Theorem 1, we conclude that that $\mathcal{N}$ has at least one fixed point in $\tilde{F}$ which is a mild solution of problem (1)–(2). □

4. An Example

We consider the following problem on a bounded domain $\Omega \subset {\mathbb{R}}^d$ with sufficiently smooth boundary:

$$\left\{\begin{array}{c}{\displaystyle {}^{c}D_0^{\alpha }u(t,x)=\Delta u(t,x)+{\int }_0^t\frac{e^{-\lambda u(s,x)}}{{(t-s)}^{\beta }}·\frac{u(s-\tau ,x)}{1+\left|u\right(s-\tau ,x\left)\right|}ds+b\left(x\right)v(t,x),}\hfill \\ (t,x)\in {\mathbb{R}}^{+}\times \Omega ,\hfill \\ u(t,x)=\varphi (t,x),(t,x)\in [-\tau ,0]\times \Omega \hfill \\ u(t,x)=0,(t,x)\in {\mathbb{R}}^{+}\times \partial \Omega ,\hfill \end{array}\right.$$

(7)

where ${}^{c}D_0^{\alpha }$ is the Caputo fractional partial derivative of order $0<\alpha <1$ and $A=\Delta$ (The Laplacian operator) with domain $D\left(A\right)=H^2(\Omega )\cap H_0^1(\Omega )$ generates an analytic semigroup on $F=L^2(\Omega )$.

For $u\in \mathcal{C}({\mathbb{R}}^{+},L^2(\Omega ))$, we have

$$f(t,u,u_{\tau })={\displaystyle \frac{u_{\tau }}{1+|u_{\tau }|}},$$

$$h\left(u\right)=e^{-\lambda u},(\lambda >0).$$

Let the admissible control $v\in L^2([0,n],L^2(\Omega ))$ defined by

$$v(t,x)=\sum _{k=1}^m{\gamma }_k\left(t\right){\psi }_e\left(x\right),$$

and $B=I$ (I is the identity) where ${\left\{{\psi }_k\right\}}_k$ is the orthonormal basis of $L^2(\Omega ).$

• Verification of the assumptions (H₁)–(H₇).

$\left(H_1\right)$

The Laplacian $A=\Delta$ with Dirichlet boundary conditions generates an analytic (hence equicontinuous) $C_0$-semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on $L^2(\Omega )$.

$\left(H_2\right)$

The map ${\displaystyle f(t,p,q)=\frac{q(-\tau )}{1+\left|q\right(-\tau \left)\right|}}$ depends only on the value of the history function q at $\theta =-\tau$. Since the function ${\displaystyle z\mapsto \frac{z}{1+\left|z\right|}}$ is continuous and bounded on $\mathbb{R}$, the mapping $(t,p,q)\mapsto f(t,p,q)$ is continuous in $(p,q)$ and independent of t, hence measurable in t. Moreover, for any bounded sets in $F\times \mathcal{D}$, f is bounded by 1.

Thus, f is $L^1$-Carathéodory.

$\left(H_3\right)$

For any $p_1,p_2\in F$ and $q_1,q_2\in \mathcal{D}$, we have

$${∥f(t,p_1,q_1)-f(t,p_2,q_2)∥}_{L^2}={∥{\displaystyle \frac{q_1(-\tau )}{1+|q_1(-\tau )|}}-{\displaystyle \frac{q_2(-\tau )}{1+|q_2(-\tau )|}}∥}_{L^2}.$$

The scalar function $\xi \mapsto {\displaystyle \frac{\xi }{1+\left|\xi \right|}}$ is globally Lipschitz with constant 1. Hence,

$$\parallel f(t,p_1,q_1)-f(t,p_2,q_2)\parallel \le \parallel q_1(-\tau )-q_2(-\tau )\parallel \le \parallel q_1-q_2{\parallel }_{\mathcal{D}},$$

with Lipschitz constant $L=1$.

$\left(H_4\right)$

The function $h\left(p\right)=e^{-\lambda p}$ satisfies:

(i)

For any $R>0$, if ${\parallel p\parallel }_{L^2}\le R$, then $\left|h\left(p\left(x\right)\right)\right|=e^{-\lambda p\left(x\right)}\le e^{\lambda \left|p\right(x\left)\right|}\le e^{\lambda R}$ a.e., so ${\parallel h\left(p\right)\parallel }_{L^{\infty }}\le e^{\lambda R}$. Hence, ${\parallel h\left(p\right)\parallel }_{L^2}\le {|\Omega |}^{1/2}{e}^{\lambda R}$, so $M_R={|\Omega |}^{1/2}{e}^{\lambda R}$ works.

(ii)

The function h is Lipschitz on bounded subsets of $\mathbb{R}$: for any $a,b\in \mathbb{R}$,

$$|e^{-\lambda a}-e^{-\lambda b}|\le \lambda e^{\lambda max\left(\right|a|,|b\left|\right)}|a-b|.$$

On any ball $\parallel p\parallel \le R$, we have $\left|p\right(x\left)\right|\le R$, so

$$|h\left(p_1\left(x\right)\right)-h\left(p_2\left(x\right)\right)|\le \lambda e^{\lambda R}|p_1\left(x\right)-p_2\left(x\right)|,$$

which implies $\parallel h\left(p_1\right)-h\left(p_2\right){\parallel }_{L^2}\le \lambda e^{\lambda R}{\parallel p_1-p_2\parallel }_{L^2}$. Thus, $\left(H_4\right)$-(ii) holds with $M=\lambda e^{\lambda R}$ locally Lipschitz. However, note that in the proof of Theorem 2, only a global bound $M_R$ and local Lipschitz continuity are used in estimates; this suffices for the fixed point argument on bounded balls $B_{R_n}$.

$\left(H_5\right)$

Since $∥{\displaystyle \frac{u(t-\tau )}{1+\left|u\right(t-\tau \left)\right|}}∥\le \parallel u(t-\tau )\parallel$, we have

$$\parallel f(t,p,q)\parallel \le \parallel q(-\tau )\parallel \le {\parallel q\parallel }_{\mathcal{D}}\le \parallel p\parallel +{\parallel q\parallel }_{\mathcal{D}}.$$

Thus, ${\rho }_f\left(t\right)\equiv 1\in L^{\infty }\left({\mathbb{R}}^{+}\right)$, so ${\rho }_n^{\ast }=1$ for all n.

$\left(H_6\right)$

The function f depends only on the evaluation map $q\mapsto q(-\tau )$, which is linear and continuous from $\mathcal{D}$ to F. The scalar function ${\displaystyle \xi \mapsto \frac{\xi }{1+\left|\xi \right|}}$ is Lipschitz, so it preserves measure of noncompactness up to a constant factor (equal to its Lipschitz constant, which is 1). Therefore, for any bounded sets $Z\subset F$, $Z_1\subset \mathcal{D}$,

$$\mu \left(f(t,Z,Z_1)\right)\le \mu \left(Z_1(-\tau )\right)\le \underset{\vartheta \in [-\tau ,0]}{sup}\mu \left(Z_1\left(\vartheta \right)\right).$$

Since ${\rho }_f\left(t\right)=1$.

$\left(H_7\right)$

We take $B=I$ and assume that the control acts on a finite-dimensional subspace of $L^2(\Omega )$ (e.g., via a finite number of eigenfunctions of $-\Delta$). In this case, the controllability operator $W_n$ has closed range and admits a bounded right inverse $W_n^{-1}$ on its image, with $\parallel W_n^{-1}\parallel \le K$ for some constant $K>0$ independent of n.

Numerical verification. We verify that $J_1<1$ with the following choice of parameters:

$$\alpha =0.6,\beta =0.7,L_{\alpha }=0.18,K=0.5,{\rho }_1^{\ast }=1.$$

Taking a mild solution bounded by $R_1=0.1$ on $[0,1]$, and recalling that $h\left(u\right)=e^{-\lambda u}$ with $\lambda =0.1$, we have

$$M_{R_1}=\underset{\parallel u\parallel \le R_1}{sup}\parallel h\left(u\right)\parallel \le e^{\lambda R_1}=e^{0.01}\approx 1.01.$$

Using $B(0.6,1.3)\approx 1.35$ and $1-\beta =0.3$, we compute

$$J_1=L_{\alpha }{M}_{R_1}{\rho }_1^{\ast }{\displaystyle \frac{B(\alpha ,2-\beta )}{1-\beta }}\left(1+{\displaystyle \frac{L_{\alpha }K}{\alpha }}\right)\approx 0.18·1.01·{\displaystyle \frac{1.35}{0.3}}\left(1+{\displaystyle \frac{0.18·0.5}{0.6}}\right).$$

Since ${\displaystyle \frac{1.35}{0.3}=4.5}$ and ${\displaystyle 1+\frac{0.09}{0.6}=1+0.15=1.15}$, we obtain

$$J_1\approx 0.18·1.01·4.5·1.15\approx 0.94<1.$$

Hence, the contraction condition of Theorem 2 is satisfied and the problem (7) is controllable.

5. Conclusions

In this paper, we studied the existence and controllability of solutions for a class of fractional integro-differential equations defined in Fréchet spaces. The analysis relied on a generalized form of Darbo’s fixed point theorem adapted to Fréchet spaces, combined with the concept of the measure of noncompactness. This approach enabled us to establish the existence of mild solutions under suitable conditions and to examine their controllability features. To illustrate the applicability of the theoretical results, an example was provided confirming that all the required hypotheses and assumptions are satisfied. Future research will focus on extending these results to broader classes of fractional systems, including those with nonlocal and various boundary conditions, fractional stochastic systems, neutral equations, as well as impulsive systems with both instantaneous and non-instantaneous impulses. Moreover, alternative analytical methods will be explored to further advance the study of fractional equations in infinite-dimensional settings.