Existence, uniqueness and controllability results of nonlinear neutral implicit $\mathcal{ABC}$ fractional integro-differential equations with delay and impulses

1.   Introduction

Fractional calculus presents numerous applications in modeling complex problems, and analyses using fixed-point techniques are highly effective in fractional integro-differential equations (FIDEs). Models using FIDEs are better for real-world problems compared to those using local derivatives ^[2,3,4]. The results obtained from these FIDEs, with various definitions of non-local (fractional) derivatives, demonstrate their unique applications in scientific and non-scientific fields. Among these derivatives, Atangana and Baleanu introduced a non-local derivative with a non-singular kernel based on the Mittag-Leffler function in the sense of Caputo ^[6,7,8]. This definition highlights the importance of the Mittag-Leffler function, and together, they present numerous applications in different areas. The $\mathcal{ABC}$ derivative involves the Mittag-Leffler kernel; as such, it is not affected by the singularities of FIDEs compared to other fractional derivatives. Also, it effectively captures the memory effect of the system, performing better than classical derivatives. The ultimate merit of the $\mathcal{ABC}$ derivative is to maintain physical phenomena while evaluating the existence and uniqueness of mild solutions for fractional differential equations.

Numerous problems in the biomedical field involve sudden state changes. Impulsive differential equations of non-integer orderprovide a clear frameworks for addressing such problems in future investigation. Many research works have been conducted in this area ^[9,10], and remarkable results have been obtained. In particular, researchers have analyzed impulsive fractional integro-differential equations (IFIDEs) ^[9,10,11] using semigroup theory and fixed-point techniques (FPTs). The study of impulsive problems involving non-integer order derivatives is particularly noteworthy due to their distinctiveness.

Several models for physical phenomena completely rely on historical data. In such cases, delay differential equations (DDEs) ^[7,12,13] are used to model scenarios in fields such as control systems, oceanography, and geography. Researchers have investigated the approximate mild solution of the multi-pantograph DDE of second order with singularity ^[14,15]. Models involving DDEs account only for past states but not past rates ^{[12,13,14,15]}.

Many researchers are working on coupled delayed fractional systems ^[16]. For the first time, the fractional adaptive sliding mode control method is being used to study the projective synchronization of uncertain fractional-order reaction-diffusion systems. Adaptive sliding mode control laws are derived by creating a fractional-order integral-type switching function, which makes the fractional-order sliding mode surface reachable in a finite amount of time. In ^[17], the Lyapunov functional approach and Fillipov's theory were used to derive a novel algebraic necessary condition for the global ML synchronization of fractional-order memristor neural networks (FOMNNs) with leakage delay via a hybrid adaptive controller. In ^[18], researchers investigated global Mittag-Leffler synchronization by designing a new fractional integral sliding mode surface and its associated control law. In ^[19,20], authors examined the well-posedness of systems of incommensurate delay fractional differential equations (DFDEs) of retarded type with non-vanishing constant delay in the space of continuous functions. The behavior of dynamical systems can occasionally vary due to impulses and abrupt process changes. These changes can be modeled ^[21] using short-memory fractional differential equations

The existence of mild solutions for the given problem and their stability was discussed by Reunsumrit et al. ^[11]:

where $^{\mathcal{ABC}}_{0}{D}^{\alpha}_{t}$- is the $\mathcal{ABC}$ fractional derivative of order $\alpha$ and $\mathfrak{U, S}: \mathfrak {J}^{'} \times \mathscr{R} \rightarrow \mathscr{R}$ and $\mathfrak{V}, g: \mathfrak {J}^{'} \times \mathscr{R}^{2} \rightarrow \mathscr{R}$ is a continuous function.

Here, $\mathfrak{L} {s}(t) = \int_{0}^t g(t, \tau, \phi(\tau)) d\tau$, and $I_{i} : \mathscr{R}\rightarrow \mathscr{R}, \ i = 1, 2, ... q$, $0 = t_{0} < t_{1} < t_{2} < ... < t_{q} = \top$, ${\Delta} {s}|_{t = t_{i}} = {s} (t^{+}_{i})- {s}(t^{-}_{i})$, and ${s}(t^{+}_{i}) = \lim\limits_{\texttt{h} \rightarrow 0^{+}} {s}(t_{i}+\texttt{h})$ and ${s}(t^{-}_{i}) = \lim\limits_{\texttt{h} \rightarrow 0^{-}} {s}(t_{i}-\texttt{h})$ represents the limits from the left and right sides of ${s}(t)$ at $t = t_{i}$.

Benchohra et al. ^[12] inspected the existence and stability of the mild solution for the below implicit fractional differential equations (FDEs) involving neutrality and impulses

where $^c {D}^{\alpha}_{t_{q}}$ represents the fractional derivative in Caputo sense, $\mathcal{I}:[0, \top]\times PC([-\mathcal{r}, 0], \Re) \times \Re \rightarrow \Re$, $\mathcal{N}:[0, \mathcal{\top}]\times PC([-\mathcal{r}, 0], \Re) \rightarrow \Re \ $ are the given functions with $\mathcal{I}(0, \varphi) = 0$, $I_{q}:PC([-\mathcal{r}, 0], \Re) \rightarrow \Re, \ \varphi \in PC([-\mathcal{r}, 0], \Re)$, $0 = t_0 < t_1 < \cdots < t_n < t_{n+1} = \mathcal{\top}$. $\Delta {s}\vert_{{t} = {t_{q}}} = {s}(t_{q}^{+}) - {s}(t_{q}^{-})$, where ${s}(t_{q}^{+}) = \lim\limits_{h\rightarrow 0^{+}} {s}(t_{q}+h)$ and ${s}({t}_{q}^{-}) = \lim\limits_{h\rightarrow 0^{-}} {s}(t_{q}-h)$ denote the limit values approaching from the right and left side of ${s}$ at $t = t_{q}$, respectively. Here, ${s}_{t}(\theta) = {s}(t+\theta)$.

Gul et al. ^[1] researched the existence of the mild solution for BVPs, using the $\mathcal{ABC}$ non-integer order derivative

Here, $^{\mathcal{ABC}}_{0}{D}^{\alpha}_{t}$- is the $\mathcal{ABC}$ derivative of non-integer order $\alpha$ and $\mathcal{N, U, I}: \mathcal {J}^{'} \times \Re \rightarrow \Re$.

Karthikeyan et al. ^[14] studied the existence of the mild solution for implicit FIDEs using $\mathcal{ABC}$ derivatives as mentioned below:

where $^{\mathcal{ABC}}_{0}{D}^{\alpha}_{t}$- is the $\mathcal{ABC}$ fractional derivative of order $\alpha$, $\mathfrak{P, E}: \mathcal{J} \times \Re \rightarrow \Re$ and $\mathcal{I}: \mathcal{J} \times \Re^{2} \rightarrow \Re$ are continuous functions. Where $I_{i} : \Re\rightarrow \Re, \ \mathfrak{ z} = 1, 2, ...\ell, 0 = t_{0} < t_{1} < {t}_{2} < ... < {t_{n}} = \top, \; {\Delta} {s}|_{t} = {t}_{i} = {s}({t}^{+}_{i})- {s}({t}^{-}_{i})$, ${s}({t}^{-}_{i}) = \lim\limits_{{\mathfrak{r}} \rightarrow 0^{-}} {s}({t}_{i})-\mathfrak{r})$ and ${s}({t}^{+}_{i}) = \lim\limits_{{\mathfrak{r}} \rightarrow 0^{+}} {s}({t}_{i}+{\mathfrak{r}})$ represent the limit of ${s}({t})$ from right and left respectively, at ${t} = {t}_{i}$. For any $t\in \mathcal{J}$, we represent ${s}_{t}$ by ${s}_{t}(\theta) = {s}(t+\theta) \ \text{and} \ -\mathfrak{r} \leq s \leq 0$.

Nowadays, many researchers are analyzing the exact and approximate controllability of the above mentioned systems involving different non-local derivatives. Aimene et al. ^[7] verified the controllability for semi-linear FDEs involving the $\mathcal{ABC}$ derivative and delay.

Here, $^{\mathcal{ABC}}_{0}{D}^{\alpha}_{t}$- denotes the $\mathcal{ABC}$ non-integer derivative with order $\alpha$ and $A:D(A)\subset \Omega \rightarrow \Omega$ is an infinitesimal generator of $\alpha-$ resolvent family $T_{\alpha}\; \& \; \; \mathcal{S}_{\alpha}$ for $t \geq 0$ forming the solution operator on the Banach space $(\Omega, \Vert.\Vert).$ Let $c \in \mathcal{L}^{2}([0, \top]; \complement)$, here $\complement$ is also a Banach space and $B$ is a bounded linear operator such that $\mathcal{B} = \{g:[-\mathfrak{r}, 0] \mapsto \Omega\}$, g is continuous everywhere except for a finite number of points r at which $g(r^{-}), \; \; g(r^{+})$ exists and $g(r^{-}) = g(r), g\in\complement(\mathcal{J}\; x\; \mathcal{B}\; x\; \Omega, \; \; \Omega), \; \; \chi \in \complement(\mathcal{J}, \; \; \mathcal{J}^{+})$ where $\mathcal{J} = \{t_{1}, ...t_{\ell}\}, \; \; \mathcal{J}^{+} = [-\mathfrak{r}, \; \top]$ and $t-\mathfrak{r} < \chi(t) < t, \; \mathfrak{r} > 0.\; \; t \in \mathcal{J}\; \; and\; \; t_{i} < \chi(t) < t, \; \; t \in (t_{i}, t_{i+1}], \; \; I \in \complement(\Omega, \; \Omega), \; \; \Delta({s})\; \Bigr|_{\substack{t} = t_{i}} = {s}(t_{i}^{+})- {s}(t_{i}^{-}), \; \; 0 = t_{0} < t_{1} < {t}_{2} < ... < {t_{n}} = \top, \; for\; n = 1, 2, ..., \ell,$ with respect to the right and left side approach, respectively, at $t = t_{i}$ is ${s}({t}_{i}^{+}) \; \; and\; \; {s}({t}_{i}^{-}). \; {s}_{t}\in \mathcal{B}$ satisfies ${s}_{t}(\theta) = {s}(t+\theta), \theta \in [-\mathfrak{r}, 0]. \; \; {s}_{t}(.)$ is the history of the state from $t-\mathfrak{r}$ to $t$.

The controllability of the above defined system has been derived by researchers with the help of k-set contraction mapping.

Inspired by the above mentioned research articles, we aim to investigate nonlinear implicit fractional systems involving impulses and delay in terms of $\mathcal{ABC}$, the non-local derivative of the form,

where $^{\mathcal{ABC}}_{0}{D}^{\alpha}_{t}$- denotes the $\mathcal{ABC}$ non-integer derivative with order $\alpha$ and $A:D(A)\subset \Omega \rightarrow \Omega$ is an infinitesimal generator of $\alpha-$ resolvent family $T_{\alpha}\; \& \; \; \mathcal{S}_{\alpha}$ for $t \geq 0$ forming the solution operator on the Banach space $(\Omega, \Vert.\Vert).$ Let $c \in \mathcal{L}^{2}([0, \top]; \complement)$, here $\complement$ is also a Banach space and $B$ is a bounded linear operator such that $B:\complement \mapsto \Omega$. The functions $\mathfrak{N, G}: \mathcal{J} \times \Re \rightarrow \Re$ and $\mathcal{I}: \mathcal{J} \times \Re^{3} \rightarrow \Re$ are continuous. Also, $^{\mathcal{ABC}}_{0}{D}^{\alpha}_{t}$- denotes the $\mathcal{ABC}$ non-integer derivative with order $\alpha$, and $\mathfrak{N, G}: \mathcal{J} \times \Re \rightarrow \Re$ are continuous functions. Also, $I_{i} : \Re \rightarrow \Re, \ {i} = 1, 2, ...n, 0 = t_{0} < t_{1} < {t}_{2} < ... < {t_{n}} = \top, \; {\Delta} {s}|_{t} = {t}_{i} = {s}({t}^{+}_{i})- {s}({t}^{-}_{i})$, ${s}({t}^{-}_{i}) = \lim\limits_{{\mathfrak{r}} \rightarrow 0^{-}} {s}({t}_{i})-\mathfrak{r})$ and ${s}({t}^{+}_{i}) = \lim\limits_{{\mathfrak{r}} \rightarrow 0^{+}} {s}({t}_{i}+{\mathfrak{r}})$ denotes the limit of ${s}({t})$ with respect to the right and left side approach, respectively, at ${t} = {t}_{i}$. For any $t\in \mathcal{J}$, we represent ${s}_{t}$ by ${s}_{t}(h) = {s}(t+h) \ \text{and} \ -\mathfrak{r} \leq s \leq 0$.

The remaining of this paper is organized as follows: rudimentary concepts, like definitions and lemmas, are given in Section 2. The existence of the mild solution for nonlinear neutral implicit impulsive FIDEs involving $\mathcal{ABC}$ fractional derivative with delay is verified in Section 3. The controllability of the nonlinear neutral implicit impulsive FIDEs involving $\mathcal{ABC}$ with delay is examined in Section 4. An example is provided in Section 5 to demonstrate the applicability of the proposed problem.

2.   Preliminaries

Let us define $\grave{\mathcal{PC}}([-\mathfrak{r}, 0], \Re) = \{ {s}:[-\mathfrak{r}, 0] \rightarrow \Re: {s} \in \grave{\mathcal{C}}((t_{i}, t_{i+1}], \Re), \ i = 0, 1, ...\ell,$ and $\exists \ {s}(t_{i}^{-})$ and ${s}(t_{i}^{+}), \ i = 1....\ell,$ with ${s}(t_{i}^{-}) = {s}(t_{i}^{+})\}.$

$\grave{\mathcal{PC}}([-\mathfrak{r}, 0], \Re)$ denotes the Banach space, having norm $\| {s}\|_{\grave{\mathcal{PC}}} = \sup\limits_{t\in [-\mathfrak{r}, 0]} \Vert {s}(t) \Vert.$

$\grave{\mathcal{PC}_{1}}([0, \top], \Re) = \{ {s}:[0, \top] \rightarrow \Re: {s} \in \grave{\mathcal{C}}((t_{i}, t_{i+1}], \Re), \ i = 0, 1, ...\ell,$ and $\exists \ {s}(t_{i}^{-})$ and ${s}(t_{i}^{+}), \ i = 1, ..., \ell, \ \text{with} \ {s}(t_{i}^{-}) = {s}(t_{i}^{+})\}. \; \grave{\mathcal{PC}_{1}}([0, \top], \Re)$ represents the Banach space, having norm $\| {s}\|_{\grave{\mathcal{PC}}_{1}} = \sup\limits_{t \in [0, \top]} \Vert {s}(t) \Vert,$

$\Omega$ holds the properties of Banach space with norm $\| {s}\|_{\Omega} = \sup\limits_{t\in [-\mathfrak{r}, \top]} \Vert {s}(t)\Vert.$

Remark 1 ([^[7,8,10,12]]). $\grave{\mathcal{PC}}([0, \top], \Re)$ is the Banach space, which is a complete normed vector space ($\grave{\mathcal{PC}}, \Vert. \Vert$) with the following properties:

(1) $\Vert f \Vert \geq 0$ and $\Vert f \Vert = 0$ if and only if $f = 0, \; \; \; \forall \; f\; \in \grave{\mathcal{PC}}([0, \top], \Re)$.

(2) $\Vert \beta f \Vert = \vert \beta \vert\; \; \; \Vert f \Vert, \; \; where\; \; \beta$ is a scalar, $\forall f \in \grave{\mathcal{PC}}([0, \top], \Re) \; \; and\; \; \beta \in \Re.$

(3) $\Vert f+g \Vert \leq \Vert f \Vert + \Vert g \Vert, \forall \; \; f\; \; and\; \; g\in \grave{\mathcal{PC}}([0, \top], \Re).$

Definition 1 ([^[7,8,10,12]]). The non-integer order $\mathcal{ABC}$ derivative of $f(t)$ is

where, $\alpha \in (0, 1]$ and $\alpha \in \mathbb{E}^1(0, \top). \; \mathbb{N}(\alpha)$ is the normalization function satisfying $\mathbb{N}(0) = \mathbb{N}(1) = 1$ and $\mathbb{E}_{\alpha} = {\sum\limits^{\infty}_{i = 0}} \frac{t^{i \alpha}}{(\alpha i+1)}$ is a special function, introduced by Mittag-Leffler.

Definition 2 ([^[7,8,10,12]]). The non-integer order $\mathcal{ABC}$ integral of $f$ is

where ${I}^{\alpha}$ represents the R-L fractional integral.

Remark 2 ([^[7,8,10,12]]). Some important properties of $\mathcal{ABC}$ derivative and the generalized Mittag-Leffler function during the implementation of Laplace transform are as follows:

$(1) \; \mathcal{L}[^{\mathcal{ABC}}{D}^{\alpha}_{a^{+}} f(t)]({s}) = \frac{\mathbb{N}\alpha}{1-\alpha}\mathcal{L}[\mathbb{E}_{\alpha}(-\lambda t^{\alpha})({s})[ {s}\mathcal{L}(f(t))({s})-f(0)]].$

$(2) \; \mathcal{L}[t^(\alpha-1)\mathbb{E}_{\alpha, \alpha}(-\lambda t^{\alpha})({s})] = \frac{ {s}^{\alpha}-\alpha}{ {s}^{\alpha}+\lambda}.$

$(3) \; \mathcal{L}[t^(\alpha)]({s}) = \frac{\Gamma(\alpha+1)}{ {s}^{\alpha+1}}.$

$(4) \; \mathcal{L}[f(t)*\Psi(t)]({s}) = \mathcal{L}[f(t)](s)\mathcal{L}[\Psi(t)](s).$

Definition 3 ([^[7,8,10,12]]). The Kurtawoski measure of non-compactness $\Upsilon$ on a bounded set $\mathcal{B}\subset \mathbb{Y}$ is considered as follows:

with the following properties:

(1) $\grave{\mathbb{L}}_{1} \subset \grave{\mathbb{L}}_{2}$ gives $\Upsilon(\grave{\mathbb{L}}_{1})\leq \Upsilon(\grave{\mathbb{L}}_{2})$ where $\grave{\mathbb{L}}_{1}, \grave{\mathbb{L}}_{2}$ are bounded subsets of $\mathbb{Y}$.

(2) $\Upsilon(\grave{\mathbb{L}}) = 0$ iff $\grave{\mathbb{L}}$ is relatively compact in $\mathbb{Y}$.

(3) $\Upsilon(\{z\}\bigcup \grave{\mathbb{L}}) = \Upsilon(\grave{\mathbb{L}})$ for all $z \in \mathbb{Y}, \grave{\mathbb{L}}\subseteq \mathbb{Y}$.

(4) $\Upsilon(\grave{\mathbb{L}}_{1}\bigcup \grave{\mathbb{L}}_{2}) \leq \text{max}\{ \Upsilon(\grave{\mathbb{L}}_{1}), \Upsilon(\grave{\mathbb{L}}_{2})\}$.

(5) $\Upsilon(\grave{\mathbb{L}}_{1}+\grave{\mathbb{L}}_{2}) \leq \Upsilon(\grave{\mathbb{L}}_{1})+\Upsilon(\grave{\mathbb{L}}_{2})$.

(6) $\Upsilon((\mathfrak{z})\grave{\mathbb{L}}) \leq |(\mathfrak{z})| \Upsilon(\grave{\mathbb{L}})$ for $(\mathfrak{z}) \in R$.

Let $\grave{\mathcal{M}} \subset C(I, \mathbb{Y})$ and $\grave{\mathcal{M}}((\mathfrak{z})) = \Big\{ \upsilon(r) \in \mathbb{Y} | \upsilon \in \grave{\mathcal{M}} \}$. We define

Proposition 1 (^[7,8,12]). If $\grave{\mathbb{M}}\subset C(\grave{\mathbb{J}}, \mathbb{Y})$ is equi-continuous and bounded, then $t\rightarrow \Upsilon(\grave{\mathbb{M}}(t))$ is continuous on $I$, also

Proposition 2 (^[7,8,12]). Let the functions $\{\upsilon_{n}:\grave{\mathbb{J}}\rightarrow \mathbb{Y}, n \in {N}\}$ be Bochner integrable. For $n\in N, \|\upsilon_{n}\|\leq m(t)$ a.e $m \in \mathcal{L}^{1}(\mathcal{I}, \mathcal{R}^{+})$ and $\xi(t) = \Upsilon(\{\upsilon_{n}(t)\}_{n = 1}^{\infty}) \in \mathcal{L}^{1}(\mathcal{I}, \mathcal{R}^{+})$, then it satisfies

Proposition 3 (^[7,8,12]). Let $\grave{\mathcal{M}}$ be a bounded set. Then, for each $\zeta > 0$, there exists a sequence $\{\upsilon_{n}\}_{n = 1}^{\infty}\subset \grave{\mathcal{M}}$, such that

Definition 4 ([^[7,8,12]]). Let $0 < \mu < \pi$ and $-1 < \beta < 0$. We define $S_{\mu}^{0} = \{\grave{\upsilon}\in \mathcal{C}\setminus \{0\}\; \text{ that is}\; |arg \grave{\upsilon}| < \mu\}$ and the closure of the form $S_{\mu}$, that is

Definition 5 (^[7,8,12]). For $-1 < \beta < 0, 0 < \omega < \frac{\pi}{2}$, we define $\{ {\odot}_{\omega}^{\beta}\}$ as a family of all closed linear operators $A:D(A)\subset \Omega \rightarrow \Omega$; this implies

(1) $\sigma({A}) \in S_{\omega}$, where $\sigma({A})$ is the spectrum, which is a complement of the resolvent set.

(2) For all $\mu \in (\omega, \pi), \exists M_{\mu}$ implies $\|\mathscr{R}(z, A)\|_{L(X)}\leq M_{\mu}|z|^{\beta}$, where $\mathscr{R}(z, A) = (zI-A)^{-1}$ is the resolvent operator and $A \in {\odot}_{\omega}^{\beta}$ is said to be an Almost Sectorial operator on $\Omega$.

Proposition 4 (^[7,8,12]). Let $A \in {\odot}_{\omega}^{\beta}$ for $-1 < \beta < 0$ and $0 < \omega < \frac{\pi}{2}$ and we define $\{ {\odot}_{\omega}^{\beta}\}$ as a family of all closed linear operators $A:D(A)\subset \Omega \rightarrow \Omega$. Then, the following properties are fulfilled:

(1) $\grave{\mho}(t)$ is analytic and $\frac{d^{n}}{d{t}^{n}}\grave{\mho}(t) = (-A^{n}\grave{\mho}(t)(t\in S_{\frac{\pi}{2}}^{0})$;

(2) $\grave{\mho}(t+s) = \grave{\mho}(t)\grave{\mho}(s)\; \; \; \forall\; \; t, s \in S_{\frac{\pi}{2}}^{0}$;

(3) $\|\grave{\mho}(t)\|_{L(\Omega)}\leq C_{0}t^{-\beta-1}(t > 0)$; where $C_{0} = C_{0}(\beta) > 0$ is a constant;

(4) Let $\mathcal{\sum}_{\grave{\mho}} = \{x \in \Omega \; \; :lim_{t\rightarrow 0_{+}} \grave{\mho}(t)x = x \}$. Then $\mathsf{D}(A^{\Upsilon})\subset \mathcal{\sum}_{\grave{\mho}}$ if $\Upsilon > 1+\beta$;

(5) $\mathscr{R}(z, -A) = \int_{0}^{\infty}e^{-zs}\grave{\mho}(s)ds$, $z\in \mathbb{C}$ with $Re(z) > 0$;

(6) The range $\mathcal {R}(\grave{\mho}(t))$ of $\grave{\mho}(t)$, $t \in S_{\frac{\pi}{2}-{\omega}}^{0}$ is contained in $\mathsf {D(A)}^{\infty}$. Particularly, $\mathcal {R}(\grave{\mho}(t)$ is contained in $\mathsf {D(A)}^{\beta}$ for all ${\beta}\in C$ with $Re{\beta} > 0$,

for all $x\in X$, and hence there exists a constant $C' = C'(\varphi, \beta) > 0$, such that

for all $t > 0$.

Remark 3. $\grave{\mho}(t)$ is a $C_{0}$ semi-group operator of an infinitesimal generator $A.$

Definition 6 ([^[7,8,12]]). Observe the system represented by the problem given below:

The mild solution of the given problem is of the form,

Proof. From Definition 2, we obtain

□

Theorem 1 (^[7,8,12]). Let $(\Omega, d)$ be a complete metric space. Then, a function $\Psi : \Omega \rightarrow \Omega$ is said to be a contraction mapping if there is a constant $\alpha$ with $0\leq\alpha < 1$ such that for all $x, y \in \Omega$, $d(\Psi(x), \Psi(y))\leq \alpha d(x, y).$

Theorem 2 (Banach contraction principle). Let $\Psi : \Omega \rightarrow \Omega$ represent the completely continuous operator on the Banach space $\Omega$. Consider that the set $\Finv = \{ {s} \in \Omega : {s} = \lambda \Xi {s}, \mathit{\text{for some}} \ \lambda \in (0, 1)\}$ is bounded, then $\Psi$ has fixed points.

Theorem 3 (Arzel$\grave{a}$–Ascoli theorem). Let $\Omega, d$ be a compact space. A subset $\Omega_{\mu_{b}}$ of $\complement(\Omega)$ is relatively compact if and only if $\Omega_{\mu_{b}}$is uniformly bounded and equi-continuous.

Theorem 4 (Schauder's fixed-point theorem). Let $\Omega, d$ be a complete metric space. Let $\Omega_{\mu_{b}}$ be a closed convex subset of $\Omega$ and let $\Psi : \Omega_{\mu_{b}} \rightarrow \Omega_{\mu_{b}}$ be a mapping such that the set $\{\Psi {s}: {s} \in \Omega_{\mu_{b}}\}$ is relatively compact in $\Omega$, then $\Psi$ has at least one fixed point.

Lemma 1 (^[12]). Let the real function $\nu(.):[0, \top] \mapsto (0, \infty)$ and $\rho(t)$ be a non-negative, locally integrable on $[0, \top]$, and assume the constants $c_1 > 0$ & $0 < c_2 \leq 1$ such that

which implies a constant $\complement = \complement(c_2)$ such that

Lemma 2 (^[7,8,12]). Let the BVP with nonlinear integral boundary conditions, if $f \in L(\mathcal{J})$,

then, the mild solution ${s} \in \mathcal{AC}(\mathcal{J})$ is,

Here, $P$ and $Q$ represents the linear operators, $P = \kappa(\kappa I-\mathcal{A})^{-1}$ and $Q = -\eta A(\kappa I-A)^{-1}$, where $\kappa = \frac{\mathbb{N}(\alpha)}{1-\alpha}$,

Proof. We easily prove result (2.1) through Lemma 2 directly by substituting ${s}_0$ as the boundary condition. □

Definition 7 ([^[7,8,12]]). Let the BVP with nonlinear integral boundary conditions, if $f \in L(\mathcal{J})$,

then, the mild solution ${s} \in \grave{\mathcal{PC}}(\mathcal{J})$ is,

Here, $P$ and $\mathbb{Q}$ represents the linear operators. $P = \kappa(\kappa I-\mathcal{A})^{-1}$ and $\mathbb{Q} = -\eta A(\kappa I-A)^{-1}$ where $\kappa = \frac{\mathbb{N}(\alpha)}{1-\alpha}$ and

Definition 8 ([^[7,8,12]]). The equivalent fractional solution integral for the prescribed system (1.1) is

Definition 9 ([^[7,8,12]]). Let $\phi \in \Omega$ be an initial function and ${s}_{a} \in \grave{\mathcal{PC}}([0, \top], \Re) \subset \Omega$, then there exists a control $c\in L^{2}(\mathcal{J}, \complement)$, corresponding to the mild solution ${s}(t)$ of (1.1), that fulfills ${s}(\top) = {s}_{a}$; then the system is controllable on $[0, \top].$

Remark 4 ([^[7,8,10,12]]). The readers may verify the mild solution and the solution operator in ^[7,8,12].

Remark 5 ([^[7,8,10,12]]). If $A \in A^{\alpha}(\alpha_{0}, \beta_{0})$, then $\Vert T_{\alpha} (t) \Vert \leq R e^{\beta t}$ and $\Vert \mathcal{S}_{\alpha} (t) \leq Q e^{\beta t}(1+t^{\alpha-1})\Vert$ for all $t > 0, \; \; \beta > \beta_{0}$. Therefore, we get $\hat{R} = \sup_{t \geq 0} \Vert T_{\alpha} (t) \Vert, \hat{R}_{1} = \sup_{t \geq 0} Q e^{\beta t}(1+t^{\alpha-1})$ and so $\Vert T_{\alpha} (t) \Vert \leq \hat{R}; \Vert \mathcal{S}_{\alpha} (t)\Vert \leq t^{\alpha-1}\hat{R}_{1}.$

3.   Existence and uniqueness results

We examine the existence and uniqueness of the mild solutions of the proposed system by assuming that

(P1) For $\grave{\mathcal{K}}_{u} > 0$ and for any ${s}, {\rho}\in \Omega$

and

(P2) For $\grave{\mathcal K}_{v}, \grave{\mathcal{L}}_{v} \& \grave{\mathcal{M}}_{v}$ and for any ${s}_{1}, {s}_{2}, {s}_{3}, \rho_{1}, \rho_{2}, \rho_{3}(t)\in \Omega$

and

(P3) For $\grave{\mathcal {K}}_{t} > 0$ and for any ${s}, \rho \in \Omega$

and

(P4) For $\grave{\mathcal{K}}_{s} > 0$ and for any ${s}, \rho \in \Omega$

and

(P5) There exists $c_1, c_2, c_3, c_4 \in \complement (\mathcal{J}, \Re_{+})$ with $c_3^{*} = \sup_{t\in \mathcal{J}} c_3(t) < 1$ and $c_4^{*} = \sup_{t\in \mathcal{J}} c_4(t) < 1$ such that

for $t \in \mathcal{J}, {s} \in \grave{\mathcal{PC}}([-\mathfrak{r}, 0], \Re)$ and $\rho, \mu \in \Re$.

(P6) There are constants $\complement_1^{*}, \complement_2^{*} > 0$ such that

for each ${s} \in \grave{\mathcal{PC}}([-\mathfrak{r}, 0], \Re)$, $i = 1, ..., \ell.$

(P7) Let the completely continuous function be $\mathfrak{N}$, and for each bounded set $B_{\top^{*}}$ in $\Omega$, the set $t \rightarrow \mathfrak{N}(t, {s}_{t}): {s}\in B_{\top^{*}}$ is equi-continuous in $\grave{\mathcal{PC}}(\mathcal{J}, \Re)$ and hence, there are constants $p_{1} > 0$, $p_{2} > 0$ with $\ell\complement_1^{*}+p_{1} < 1$ such that

(P8) The control operator is a bounded linear operator, and for each bounded set $B_{\top^{*}}$ in $\Omega$,

(P9) The linear operator $\mathcal{W}:\grave{\mathcal{PC}}(\mathcal{J}) \mapsto \Re$ is

Here, we get an invertible operator $\mathcal{W}^{-1} : \mathcal{B} \mapsto L^{2}((0, \top], C)/ker(\mathcal{W}), \mathcal{W}^{-1}$ is also bounded and hence we have $\Vert B \Vert \leq \bar{R}\; \; \text{and}\; \; \Vert \mathcal{W}^{-1} \Vert \leq \bar{R}_{1}$.

(P10) $P \; \& \; \; Q$ are linear operators that are bounded on $\mathcal{B}$ and hence $\Vert P \Vert \leq \zeta_{1} \; \& \; \; \Vert Q \Vert \leq \zeta_{2}$.

Theorem 5. Let us consider that hypotheses (P1)–(P8) hold, then the proposed problem (1.1) has at least one mild solution.

Proof. Consider the set,

$\Omega$ holds the properties of Banach space with norm

We define the operator $\Psi_{1}: \Omega \rightarrow \Omega$ defined by

The operator $\Psi_{1}$ represented in (3.1) can be formed as $\Psi = \mathfrak{N}(t, {s}(t)) + \Psi_{1},$ for all $t \in \mathcal{J}.$

With the help of Schauder's FPT, we derive the existence of a fixed point of $\Psi$. First, we show that $\Psi$ is completely continuous. Due to the postulate (P7) of $\mathfrak{N}$, it is enough to show that $\Psi_{1}$ is completely continuous. □

Step 1: $\Psi_{1}$ is continuous. Consider the sequence $\{ {s}_{\ell}\}$ such that ${s}_{\ell} \rightarrow {s}$ in $\Omega$. If $t \in [-\mathfrak{r}, 0]$, then

For $t \in \mathcal{J}$, we have

here, $\mathfrak{q^{*}}, \mathfrak{{q}^{*}_{\ell}} \in \complement(\mathcal{J}, \Re)$ such that

and

By $(P2)$, we have

Due to the result ${s}_{\ell} \rightarrow {s}$, it gives $\mathfrak{q^{*}}_{\ell}(t) \rightarrow \mathfrak{q^{*}}(t)$ as $\ell \rightarrow \infty$ for all $t \in \mathcal{J}$.

Now, consider $\nu > 0$, for each $t \in \mathcal{J}$, we write $\vert \mathfrak{q^{*}}_{\ell} (t) \vert \leq \nu$ and $\vert \mathfrak{q^{*}}(t)\vert\leq \nu$.

Hence, we get

and

For all $t \in \mathcal{J}$, the maps $\mathfrak{z} \rightarrow 2 \nu (t-\mathfrak{z})^{\beta-1}$ and $\mathfrak{z} \rightarrow 2 \nu (t_{k}-\mathfrak{z})^{\beta-1}$ are integrable on $[0, t]$; hence, by applying the Lebesgue dominated convergence theorem and (3.2), we get

which results in the continuity of $\Psi_{1}$.

Step 2: The bounded sets of $\Omega$ will be mapped in to bounded sets of $\Omega$ by the function $\Psi_{1}$. To show this, it is sufficient to prove that for any $\top^{*} > 0, \exists \wp$ such that for each ${s} \in B_{\top^{*}} = \{ {s} \in \Omega: \Vert {s}\Vert_{\Omega} \leq \top^{*}\}$, we have $\Vert \Psi_{1}({s}) \vert_{\Omega} \leq \wp$.

For each $t \in \mathcal{J}$, we get

here, $\mathfrak{q^{*}} \in \complement(\mathcal{J}, \Re)$ such that $\mathfrak{q^{*}}(t) = \mathcal{I}(t, {s}_{t}, \mathfrak{q^{*}}(t), c(t)).$ From $(P5)$, for each $t \in \mathcal{J}$, we can write

where $c_{1}^{*} = \sup_{t\in \mathcal{J}} c_{1}(t)$, and $c_{2}^{*} = \sup_{t\in \mathcal{J}} c_{2}(t).$ Then

Thus from (3.3),

For $t \in [-\mathfrak{r}, 0]$, then

Hence,

Step 3: The function $\Psi_{1}$ maps the bounded sets of $\Omega$ into equi-continuous sets of $\Omega$.

Let $t_{i-1}, t_{i} \in (0, \top], \ t_{i-1} < t_{i}, \ B_{\top^{*}}$ be a bounded set of $\Omega$ as in Step 2, and let ${s} \in B_{\top^{*}}$. Then

As $t_{\ell} \rightarrow t_{\ell-1}$, the RHS of the above inequality converges to 0. Hence, $\Psi_{1}$ is completely continuous.

Step 4: A priori estimates. We show that

is bounded. Consider ${s} \in \Finv$, then ${s} = \kappa \Psi_{1} ({s})$ for some $\kappa \in (0, 1)$. Now let, for each $t \in \mathcal{J}$,

Hence, for each $t \in \mathcal{J}$ and from $(P5)$, we get,

For each $t \in \mathcal{J}$ and by (3.4), (P6), and (P7), we have

Define $\nu$ by

Then there exists $t^{*} \in [-\mathfrak{r}, \top]$ such that $\nu(t) = \vert {s}(t^{*})\vert$. If $t \in [0, \top]$, then by the previous inequality, we have for $t \in \mathcal{J}$

Applying Lemma 1, we get

where $\complement(c_{2})$ is a constant. If $t^{*}\in [-\mathfrak{r}, 0],$ then $\nu(t) = \Vert \phi \Vert_{\grave{\mathcal{PC}}}$, thus for any $t\in \mathcal{J}$, $\Vert {s} \Vert_{\Omega} \leq \nu(t)$, we get

Hence the set $\Finv$ is bounded. By Theorems 3 and 4, $\Psi$ has at least one fixed point in $\Omega$ which is a mild solution of the problem (1.1).

Theorem 6. Under hypotheses (P1)–(P8), the considered problem (1.1) has a unique mild solution if

Proof. Define a set,

$\Omega$ holds the properties of Banach space with the norm

Consider the operator $\Psi_{1}: \Omega \rightarrow \Omega$ by

where $\mathfrak{q^{*}}(t) \in \mathfrak{C}(\mathcal{J}, \Re)$ and

If ${s}, \rho \in \Omega$, for $t\in [-\mathfrak{r}, 0]$, which implies

For $t \in \mathcal{J}$ and from (3.5), we have

here, $\mathfrak{q^{*}}, \mathfrak{\bar{q}^{*}} \in \mathfrak{C}(\mathcal{J}, \Re)$ is

and

By $(P2)$, we prove

Hence, we obtain

Therefore, $\Psi$ is a contraction and (1.1) has a unique mild solution by Theorem 2 in $\Omega$. □

4.   Controllability

Theorem 7. Let us consider that hypotheses (P1)–(P10) hold, then the proposed problem (1.1) is controllable if

Proof. Let us define the set

where

$\Omega_{\mu_{b}}\subset \Omega$ is closed, bounded, and convex. We observe that the fixed points of the operator $\Psi_{1}$ are the mild solutions of the formulated problem (1.1) with $\Psi_{1}({s})(\top) = {s}_{1}$. This implies that the system is controllable. Now, we derive the postulates of Theorem 2.

We define the operator $\Psi_{1}: \Omega \rightarrow \Omega$ defined by,

By $(P2)$, we define the control, $c_{s}(t)$

Step 1: $\Psi_{1}$ is continuous.

For, $t \in (t_{j-1}, t_{j}]$, we get

We easily observe that $\Psi_{1}({s}_{r}) \mapsto \Psi_{1}({s})$ in $\Omega_{\mu_{b}}$ due to the continuity of the functions $\mathfrak{q^{*}} \; \; and\; \; I$. This implies the proof of continuity of $\Psi_{1}$.

Step 2: $\Psi_{1}$ maps the bounded sets into bounded sets.

By the postulates (P1)–(P10),

Let the two constants be

for all $t \in (0, t_{1}]$, where

where

for all $t \in (t_{j}, t_{j-1}]$, where $\eta = max(\top-t_{j})^{\alpha}.$

And therefore, for $t \in (0, t_{1}]$

for $t \in (t_{j}, t_{j-1}]$,

Hence we have,

This implies that, $\Vert \Psi_{1}({s})\Vert\leq \mu_{b}.$ So, $\Psi_{1}(\Omega_{\mu_{b}}) \subset \Omega_{\mu_{b}}.$

Step 3: Verify the equi-continuity of $\Psi_{1}$. Consider ${s} \in \Omega_{\mu_{b}}$ and $\rho_{1}, \rho_{2} \in (t_{j-1}, t_{j}]$. Here, $\zeta_{1} < \zeta_{2}, j = 1, 2, ...\ell$,

Now,

This result converges to 0 during $\rho_{1}\; \; \text{tending to}\; \; \rho_{2}$. By the compactness and the strong continuity of the operators $T_{\alpha}(t)$ and $\mathcal{S}_{\alpha}(t)$, we easily get that $\Psi_{1}$ is continuous in uniform operator topology. Hence, $\Psi(\Omega_{\mu_{b}})$ satisfies the condition of equi-continuous.

Step 4: $\Psi_{1}$ is a contraction on $\Omega_{\mu_{b}}$. For $t \in (0, t_{1}]$,

For $t \in (t_{j-1}, t_{j}]$,

This implies that $\Psi_{1}$ is a contraction on $\Omega_{\mu_{b}}$ for

Hence, $\Psi_{1}$ possesses a fixed point and so it is a mild solution of the proposed system (1.1) based on the defined control function $c_{s}(t)$ given in (4.3) by Theorem 2. By the definition of controllability (Definition 9), the proposed problem is controllable. □

5.   Implementation

The following application is provided for evidencing the theoretical results:

Here, $A:D(A) \subset \Omega \mapsto \Omega$ is an infinitesimal generator $A\chi = \chi^{\ell}$ where, $\Omega = \mathcal{L}^{2}[0, \pi]$ and the domain is defined by $D(A) = \{ \chi \in \Omega:\chi \text{ and }\; \chi'$ are absolutely continuous, $\chi'' \in \Omega, \chi(0) = 1 = \chi(1) \},$

Whence the eigenvectors that are orthogonal are

Hence, the corresponding analytic semi-group $\mathcal{S}(t)$ related to $A$ in $\Omega$ is $\mathcal{S}(t)\chi = \sum_{\ell = 1}^{\infty}{e^-{\ell^{2}}} < \chi, \chi \in \Omega \text{ and }\; \Vert (\mathcal{S}(t) \Vert \leq 1.$ The resolvent operator $Q(\grave{\mu}, A) = (\grave{\mu}\mathcal{I}-A)^{-1}$ where $\grave{\mu} \in \rho(A).$ So the proposed system (5.1) will take the form of (1.1) by replacing

where

We can easily verify that (5.1) fulfills the postulates (P1)–(P10) and so the proposed system is controllable by (4.1) on $[0, \pi]$.

6.   Discussion

This research article gathers the results of existence, uniqueness, and controllability. In previous studies, authors either developed the results of existence and uniqueness or controllability. However, this article verifies the controllability results, being sufficient to verify the existence of a mild solution for the proposed system. Additionally, we have shown the uniqueness results using the Banach contraction principle to some extent. Due to this uniqueness, a single trajectory can be obtained for a unique control input. Also, researchers can design control strategies according to the system due to the uniqueness of the mild solution of the problem. We can ensure the well-defined controls, making the study of controllability results more straightforward. A new researcher can improve the system or include some delays in state space or control, obtaining new results. Highlighting the stability results of the problem is a key focus in current research scenarios. The comparative analysis of numerical solutions and theoretical results are gaining significant attention among researchers.

7.   Conclusions

This work has successfully investigated the existence results for the nonlinear neutral fractional implicit impulsive differential equation with impulses, delay, and integro initial conditions by means of semi-group theory and fixed-point techniques. These types of problems have numerous applications, namely to the mathematical modeling of human diseases and complex problems. Based on Arzel$\grave{a}$ Ascoli theorem and Schauder's fixed-point theorem, we established the adequate results for at least one mild solution. Banach contraction principle helped to derive the uniqueness and controllability results of the defined system. The derived results were justified by providing a suitable illustration. Researchers can establish the stability results of the given problem as a future work. Also, changing the initial condition and including state delay, control delay, or both will obtain innovative results. Future work may be extended to non-instantaneous impulses and comparative analysis with numerical techniques.

Author contributions

Sivaranjani Ramasamy: Writing original draft, Conceptualization and Methodology, and Validation; Thangavelu Senthilprabu: Writing original draft, Validation and Resources; Kulandhaivel Kathikeyan: Investigation and Validation; Palanisamy Geetha: Investigation and Validation; Saowaluck Chasreechai: Writing original draft, Conceptualization and Methodology, and Investigation; Thanin Sitthiwirattham: Investigation and Validation. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was funded by National Science, Research and Innovation Fund (NSRF), and King Mongkut's University of Technology North Bangkok with Contract No. KMUTNB-FF-68-B-25. Moreover, we would like to thank for our management to provide KPR Pride Fellowship for conducting the research.

Conflict of interest

The authors declare that there is no conflict of interest.