Controllability of Hammerstein impulsive integro-differential equation driven by Atangana Baleanu Caputo fractional derivative

Abstract

We investigate the existence, uniqueness, and controllability of a class of fractional differential system, specifically system containing semi-linear neutral integro-delay equations with impulsive effects, governed by the Atangana-Baleanu Caputo fractional derivatives. Using the Banach fixed-point theorem, we establish sufficient conditions for the well-posedness of the system. Furthermore, by combining semi-group theory with Darbo’s fixed-point theorem, the controllability results are obtained. The chosen fractional derivative effectively captures memory and hereditary properties, providing a refined representation of the system’s complex dynamics. To address analytical challenges arising from impulsive effects and time delays, we employ the Kuratowski measure of non-compactness. A numerical example, given at the end, illustrates the practical applicability of the theoretical findings. Overall, this work offers a robust and comprehensive analytical framework for modeling and controlling intricate fractional-order dynamics, by supplying valuable tools for addressing problems in engineering, physics, and related applied sciences.

Keywords

resolvent family measure of non-compactness controllability Darbo’s fixed-point theorem impulse Atangana-Baleanu Caputo fractional derivative delay system 34A08 34K45 93B05 47H09

1. Introduction

The notion of fractional calculus dates back to 1695, when Newton and Leibniz introduced the concept, and L’Hopital posed the question regarding the meaning of a derivative of order 1/2. In recent past, fractional derivatives (FDs) have gained significant attention due to their effectiveness in modeling different physical phenomena. Traditional mathematical approaches based on integer-order differential equations often fail to correctly describe systems depending on memory or hereditary effects. Thus, fractional calculus has become an essential tool in a wide range of disciplines such as image analysis, mechanical systems, population modeling, viscoelastic materials, electrochemical processes, fluid dynamics, medical sciences, and various branches of engineering (Kilbas et al., 2006; Miller and Ross, 1993; Syam and Al-Refai, 2019). Although many forms of FD have been introduced, the Riemann-Liouville (RL) and Caputo definitions remain the most widely used in practical applications. These two derivatives are strongly connected, but the Caputo form is often preferred in time-fractional models because it allows the use of standard initial conditions, whereas the RL derivative requires initial values involving fractional integrals. Another useful feature is that the Caputo derivative (CD) of a constant is zero, which aligns with the behavior of the classical Fréchet derivative on $R$ . In view of these observations, the Caputo FD naturally incorporates the idea of a memory dependent rate of change, meaning that the present behavior of the system is influenced by its past states. For this reason, CD is often considered more suitable than other definitions for modeling different physical processes, that exhibit memory effects.

The benefits of RL and Caputo FD have been extensively examined in Agarwal et al. (2017) and Balasubramaniam (2021). However, both of these derivatives, as well as the Grünwald-Letnikov formulation, involve singular kernels, as noted in Kilbas et al. (2006) and Podlubny (1999). Despite this limitation, a substantial body of work has emerged that establishes existence, controllability, and stability results for dynamical systems modeled with the RL and Caputo FDs formulated within Banach spaces (BS) of both finite and infinite dimensions; detailed discussions can be found in the related literature. To address some of these issues, Hilfer (Hernández and O’Regan, 2013) introduced a generalized form of the RL derivative, known as the Hilfer FD and is denoted as $D_{0^{+}}^{η, ν}, η \in (n - 1, n), ν \in (0,1)$ . This operator unifies the RL derivative for ν = 0 and the CD for ν = 1, but also retains a singular kernel. In contemporary studies, extensive qualitative analysis of dynamical systems involving Hilfer FD have been developed in both $R^{n}$ and abstract BS (Bedi et al., 2020; Saravanakumar and Balasubramaniam, 2021). As all the RL, Caputo and Hilfer FDs involve kernels that are singular at the origin, as their solution formulas typically include exponential functions or Mittag-Leffler (ML) functions. This singular behavior reduces their practical applicability (Balasubramaniam, 2021). In addition, employing CD requires the computation of the classical derivative of the underlying function, which may not always be convenient or feasible. This requires smoothness in differentiation. Memory can be represented as a function through the kernel of an integro-differential operator. In essence, memory refers to the fact that the current output of a system is determined by all previous values of the input over a certain time interval, which may be finite or infinite. To establish the physical foundation of fractional operators with various memory effects, Caputo and Fabrizio in 2015 proposed a definition of FD of order ∝ on a BS, which does not involve a singular kernel (Caputo and Fabrizio, 2015).

{}^{C F}D_{0^{+}}^{\propto} χ (ϖ) = \frac{B (\propto)}{1 - \propto} \int_{0}^{ϖ} χ^{'} (s) e x p (- G (ϖ - s)) d s,

where

\propto \in (0,1), χ \in H^{1} (0, q)

is a Sobolev space defined by

{χ \in L^{2} (0, q) : D^{Z} χ \in L^{2} (0, q) \forall | Z | \leq 1}, q > 0, G = \propto / 1 - \propto

and the normalization function

B (\propto)

with

B (0) = 1 = B (1)

. Shortly thereafter, numerous researchers extended its applications, demonstrating its usefulness in thermodynamics, engineering, and groundwater studies. For further details, see Bas and Ozarslan (2018) and Bastos (2018).

Atangana and Baleanu extended the earlier established mathematical models by proposing a new fractional derivative whose kernel is based on the ML function (Atangana and Baleanu, 2016). This approach naturally blends features of both exponential behavior and power-law decay, focusing on a more flexible representation of memory effects. The kernel (memory) function plays an important role in the analysis of different systems, containing economic dynamics, the continuous-time Harrod-Domar growth model, and phenomena involving nonlocal effects in electromagnetic and thermodynamics (Pfitzenreiter, 2004; Wang and Li, 2011). Recently, considerable attention has been paid to the qualitative behavior of AB FDEs, mainly due to their extensive applicability in different phenomena (Atangana and Baleanu, 2016; Atangana and Koca, 2016; Jarad et al., 2018; Ravichandran et al., 2019). As a result, the development of fractional derivative definitions incorporating non-singular kernels such as the Caputo-Fabrizio and Atangana-Baleanu (AB) FDs has become particularly important. The motivation for introducing non-singular kernels also arises from phenomena involving fractional operators characterized by dissipativity (Hristov, 2019), for more recent applications of FDEs, we recommend Khan et al. (2025) and Shah et al. (2025).

Hammerstein-type integro-differential equations represent an important class of nonlinear models that combine Volterra or Fredholm integral operators with nonlinear composition terms. The mentioned equations arise naturally in different applications, such as heat transfer, population dynamics, and control theory. The investigation of these equations in fractional form has attracted considerable research interest in recent years. Dadsetadi et al. analyzed the existence and uniqueness (EU) of solutions for a class of nonlinear fractional integro-differential equations of Hammerstein type in BS (Dadsetadi et al., 2022). They utilized Schauder’s fixed point theorem for the existence and the general form of Banach fixed point theorem for uniqueness, with the Caputo fractional derivative. In a related work (Dadsetadi and Nouri, 2020), they developed analytical and numerical methods for fractional Hammerstein equations, transforming the fractional equations into Volterra integral equations using differential transformation methods and obtaining approximate solutions through modified block pulse functions. Shah and Zada examined the existence, uniqueness, and various stability properties, specifically Ulam-Hyers (UH) and Ulam-Hyers-Rassias (UHR) stability, for nonlinear impulsive Hammerstein integro-differential dynamic systems on time scales (Shah and Zada, 2019). More recently, the authors in Shah et al. (2024) analyzed the stability and controllability of nonlinear Volterra-Fredholm Hammerstein impulsive integro-dynamic systems with delay on time scales, employing fixed point techniques and control functions to establish their results. In Tunç et al. (2023), the authors studied the EU of solutions for nonlinear Hammerstein-type functional integral equations using Burton’s method of progressive contractions in BS with the Chebyshev norm. In Ahmed et al. (2025), the authors developed numerical methods based on quadratic B-spline functions for solving nonlinear systems of Volterra integro-FDEs of Hammerstein type in the Caputo sense, transforming the system into nonlinear algebraic equations solvable by Newton’s method.

The framework of impulsive differential equations naturally emerges in models that involve abrupt changes in the system’s state at specific moments (Shu et al., 2011; Wang et al., 2015). Depending on the duration of the impulsive effect, impulses can be classified into two categories. The first type involves instantaneous impulses, where the impact occurs in an extremely short time. The second type corresponds to Non-instantaneous (NI) impulses, where the effects persist over a finite time interval. The authors in Hernández and O’Regan (2013) introduced NI impulsive effects that frequently appear in the modeling of evolution systems. The qualitative behavior of both instantaneous and NI fractional differential systems has been extensively studied in the literature (Agarwal et al., 2017; Bas and Ozarslan, 2018; Saravanakumar and Balasubramaniam, 2021), often using fixed point theorems. Significant contributions in this area include the work (Jolić and Konjik, 2023) on time-varying systems, as well as the studies by Dineshkumar et al. (2022) and Johnson et al. (2023) on stochastic differential models involving AB CDs. Furthermore, in Hussain et al. (2023), the authors investigated the fractional evolution inclusions. Additional developments include the analysis by Algolam et al. (2025) of stochastic impulsive systems driven by Rosenblatt noise, along with studies by Raghavendran et al. (2024) on neural network models and Jothilakshmi and Vadivoo (2025) on systems that incorporate infinite delays.

The present work significantly extends the controllability results established in Aimene et al. (2019) for neutral systems subject to NI impulses. Moreover, the existence results reported in Hernández and O’Regan (2013); Ravichandran et al. (2019); and Shu et al. (2011) are further extended to encompass neutral systems. The obtained results are derived under relaxed Lipschitz and linear growth conditions imposed on the nonlinear neutral terms and the NI impulsive components. Additionally, as far as the author is aware, currently no work has addressed the controllability of neutral Non-instantaneous impulsive functional differential equations (NIFDEs) involving the AB Caputo FD. Motivated by the preceding considerations, we formulate the required conditions to ensure the EU and controllability of the below model using the measure of non-compactness.

\{\begin{cases} \begin{aligned} {}^{A B C}D^{\propto} & (℘ (ϖ) - δ (ϖ, ℘_{ϖ}) = A ℘ (ϖ) + B u (ϖ) + ϝ (ϖ, ℘ (ϖ), ℘ (p (ϖ))) \\ \times \int_{0}^{ϖ} G (ϖ, γ) K (γ, ℘ (γ), ℘ (p (γ))) d γ, ϖ \in Q = [0, q], ϖ \neq ϖ_{ς}, \end{aligned} \\ ℘ (ϖ_{ς}^{+}) = ℘ (ϖ_{ς}^{-}) + J_{ς} (ϖ, ℘ (ϖ_{ς}^{-})), ϖ \in (ϖ_{ς}, γ_{ς}], ς = 1,2,3, \dots, m, \\ ℘ (ϖ) = φ (ϖ) \in Z, for ϖ \in [- ℏ, 0], \\ ℘ (0) = φ_{0}, for ϖ \in [0, q], \end{cases}

(1)

while ℘ is the variable state, and ${}^{A B C}D^{\propto}$ is the Caputo FD AB with lower limit at “0” the order ∝ ∈ (0, 1). $A : D (A) \subset N \to N$ is the infinitesimal generator of a ∝-resolvent family ${S_{\propto} (ϖ)}_{ϖ \geq 0}$ , and ${T_{\propto} (ϖ)}_{ϖ \geq 0}$ is the solution operator defined on a BS $(N, ‖ \cdot ‖)$ . Assume that the control function $u (\cdot) \in L^{2} (Q, W)$ , where this space consists of square-integrable functions taking values in the BS $W$ . Moreover, let B denote a bounded linear operator acting from $W$ to $N$ .

Let $Z = \{κ : [- ℏ, 0] \to N, “ κ^{''}$ be continuous throughout, except at a finite set of points γ, where κ(γ⁻), κ(γ⁺) exists and $κ (γ^{-}) = κ (γ)\}$ . $δ \in C (Q \times Z, N), ϝ \in C (Q \times Z \times N, N)$ , $φ \in C (Q, Q^{+})$ where $Q^{+} = [- ℏ, q], Q^{'} = Q - {ϖ_{1}, ϖ_{2}, \dots, ϖ_{m}}, ϖ - ℏ < φ (ϖ) < ϖ, ϖ > 0, ϖ \in Q$ and $ϖ_{k} < φ (ϖ) < ϖ, ϖ \in (ϖ_{ς}, ϖ_{ς + 1}], J_{ς} \in C (N, N), Δ ℘ (ϖ_{ς}) |_{ϖ = ϖ_{ς}} = ℘ (ϖ_{ς}^{+}) - ℘ (ϖ_{ς}^{-})$ , 0 = ϖ₀ = γ₀ < ϖ₁ < γ₁ < ϖ₂… < ϖ_m < γ_m < ϖ_m+1 = q for ς = 1, 2, 3, …, m, respectively, and $℘ (ϖ_{ς}^{-})$ , $℘ (ϖ_{ς}^{+})$ denote left and right-hand limits of the function ℘(ϖ) at $ϖ = ϖ_{ς}, ℘_{ϖ} \in Z$ that satisfies ℘_ϖ(a) = ℘(ϖ + a), a ∈ [−ℏ, 0], ℘_ϖ(.) is the state history up to ϖ, from ϖ − a. Moreover, $G : Q \times Q \to N$ , $K : Q \times Z \times N \to N$ and $p : Q \to Q \cup [ϖ - Q, ϖ]$ with p(ϖ) ≤ ϖ, is the right-dense continuous delay function. Section 2 establishes the foundational concepts by presenting the necessary definitions. In Section 3, the EU results for equation (1) are derived and proved under the given assumptions. Section 4 presents the controllability results for (1). The theoretical developments are illustrated in Section 5 with a detailed example. Section 6, offers the concluding remarks, and Section 7 provides a glossary of abbreviations.

Very recent developments have expanded the applications of fractional operators and computational techniques in novel directions. Ghafoor et al. developed a numerical method for solving one- and two-dimensional Burgers’ equations involving the time-fractional AB CD with a non-singular kernel (Ghafoor et al., 2024). Their approach combines a quadrature rule for the fractional derivative with Haar wavelet approximations, reducing the fractional problems to systems of linear equations through a collocation procedure. The approach presents second-order convergence and stability with the help of Lax-Richtmyer method, by numerical examples appreciating its effectiveness. In different applications area, Sher et al. employed deep neural networks for the computational analysis of coupled systems of fractional integro-differential equations, demonstrating the potential of machine learning techniques in dealing fractional-order complex systems (Sher et al., 2025). They also developed a neural network and fractals-based, analysis for the fractional cases to predict infections in eye diseases caused by the conjunctivitis viruses, illustrating the need of combining fractional calculus with new computational intelligence for diverse applications. Also, in Abdulwasaa et al. (2024), the authors conducted the statistical and computational analysis for a corruption and poverty model using Caputo-type FDEs, emphasizing how fractional operators can effectively capture the complex dynamics of socioeconomic phenomena.

2. Basics

Before delving into the main discussion, we first introduce some fundamental definitions, theorems, and lemmas that will serve as the basis for this article.

By $C (Q, N)$ we denote a BS consisting of all continuous functions from $Q$ to $N$ . Define the norm κ on $Z$ as $‖ κ ‖_{Z} = \sup {κ (γ) : γ \in [- ℏ, 0]}$ , $X = P C (Q^{+}, N) = \{℘ : Q^{+} \to N, ℘ / Q_{ς} \in C (Q_{ς}, N)$ and $\exists ℘ (ϖ_{ς}^{+}), ℘ (ϖ_{ς}^{-})$ such that $℘ (ϖ_{ς}^{-}) = ℘ (ϖ_{ς}), ℘_{0} = φ (ϖ) \in Z, ς = 1,2,3, \dots m\}$ , where $℘ / Q_{ς}$ is the restriction of $℘ \to Q_{ς} = (ϖ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m$ . Also, $X$ with $‖ ℘ ‖_{X} = \sup {‖ ℘ (ϖ) ‖, ϖ \in Q^{+}}$ , is a BS for each $℘ \in X, ℘_{ϖ} \in Z$ for $ϖ \in Q$ .

Definition 2.1

(Banaś, 1980) Let $ϝ \subset N$ be a bounded set. The Kuratowski measure of noncompactness of ϝ, denoted by $H (ϝ)$ , is defined as: $m a t h b b H (ϝ) = \inf \{ζ > 0 : ℵ can be covered by a finite number of sets with diameter \leq ζ\}$ , where the diameter of $Z \subset N$ is $| Z | = \sup \{‖ ℘ - ς ‖ : ℘, ς \in Z\}$ .

Lemma 2.2

( Banaś, 1980 ) Let $N$ be a real BS, and let $U$ and $P$ be bounded subsets of $N$ . Then

• If $U \subseteq P$ then $H (U) \subseteq H (P)$ .

• $H (U) = H (\bar{U})$ while $\bar{U}$ represents the closure of $U$ .

• $H (U) = 0 ⟺ U$ is relatively compact in $N$ .

• $H (ϑ U) = | ϑ | H (U), ϑ \in R$ .

• $H (U + P) \leq H (U) + H (P)$ ,where $U + P = {℘ = q : ℘ \in U, q \in P}$ .

• $H (U \cup P) = \max {H (U), H (P)}$ .

• $H (U + \hat{c}) = H (\hat{c})$ for any $\hat{c} \in N$ .

Lemma 2.3

(Aimene et al., 2019) Let $P_{0} = {F_{ℏ}}_{ℏ = 1}^{\infty} \subset L^{1} (Q, N)$ is uniformly integrable and there exists $\overset{ˇ}{z} \in L^{1} (Q, N)$ satisfying $‖ F_{ℏ} ‖ \leq m (ϖ)$ , for almost everywhere $ϖ \in Q$ then $H ({F_{ℏ = 0}^{\infty}})$ is Lebesgue integrable on $Q$ and $H ({\{\int_{Q} F_{ℏ} (ϖ) d ϖ\}}_{ℏ = 1}^{\infty}) \leq 2 \int_{Q} H ({F_{ℏ} (ϖ)}_{ℏ = 1}^{\infty}) d ϖ$ .

Lemma 2.4

(Aimene et al., 2019) For BS $N$ , the boundedness of $P \subset N$ implies that ∃ a countable subset $P_{0} \subset P$ where $H (P) \leq 2 H (P_{0})$ .

Lemma 2.5

(Aimene et al., 2019) Assume $N$ be a BS and $P$ be equi-continuous and bounded in $C (Q, N)$ , then $H (P) = \sup_{ϖ \in Q} H (P (ϖ))$ , where $P (ϖ) = {℘ (ϖ) : ℘ \in P} \subset N$ .

Definition 2.6

(Atangana and Baleanu, 2016) The AB Caputo FD is defined by

{}^{A B C}D_{γ^{+}}^{ρ^{*}} G (ϖ) = \frac{Q (ρ^{*})}{1 - ρ^{*}} \int_{γ}^{ϖ} G^{'} (\hat{y}) E_{ρ^{*}} (\frac{- ρ^{*}}{1 - ρ^{*}} {(ϖ - \hat{y})}^{ρ^{*}}) d \hat{y},

for ρ* ∈ (0, 1), ϖ ∈ [γ, q], and

G

differentiable on [γ, q],

E_{ρ^{*}}

denote the ML function. The normalization function Q (ρ*) = 1 − ρ* + ρ*/Γ(ρ*) satisfies Q (0) = Q (1) = 1.

Definition 2.7

(Atangana and Baleanu, 2016) The fractional integral associated with the AB FD is

{}^{A B}I_{a}^{ρ^{*}} G (ϖ) = \frac{1 - ρ^{*}}{Q (ρ^{*})} G (ϖ) + \frac{ρ^{*}}{Q (ρ^{*}) Γ (ρ^{*})} \int_{γ}^{ϖ} G (\hat{y}) {(ϖ - \hat{y})}^{ρ^{*} - 1} d \hat{y} .

The AB Caputo FD is formulated using both the one- and two-parameter ML function (Kilbas et al., 2006):

E_{ρ^{*}, β} (F) = \sum_{v = 0}^{\infty} \frac{F^{v}}{Γ (v ρ^{*} + β)},

where F is a bounded linear operator.

If β = 1, then

E_{ρ^{*}} (F) = E_{ρ^{*}, 1} (F) = \sum_{v = 0}^{\infty} \frac{F^{v}}{Γ (v ρ^{*} + 1)} .

We now outline several of properties Laplace transform associated with the AB CD and the generalized ML function (Atangana and Baleanu, 2016; Bahaa and Hamiaz, 2018):

• F{ϑ(ϖ)∗p(ϖ)}(ϱ) = F{ϑ(ϖ)}(ϱ)∗F{p(ϖ)}(ϱ),

• F{ϖ^∝}(ϱ) = $\frac{Γ (\propto + 1)}{ϱ^{\propto + 1}}$ ,

• F{ϖ^∝−1E_∝,β(−dϖ^∝)}(ϱ) = $\frac{ϱ^{\propto - β}}{ϱ^{\propto + d}}$ ,

• $F {{}^{A B C}D_{β^{+}}^{\propto} ϑ (ϖ)} (ϱ) = \frac{Q (\propto)}{1 - \propto} F {E_{\propto} (- d ϖ^{\propto})} (ϱ) * (ϱ F {ϑ (ϖ)} (ϱ) - ϑ (0))$ .

Definition 2.8

(Pazy, 2012) Let $X$ be a linear operator on a BS $N$ . The resolvent set of $X$ , denoted by $ρ^{*} (X)$ , is the set of all complex numbers $ℏ \in C$ such that:

• $ℏ I - X$ is bijective,

• its inverse ${(ℏ I - X)}^{- 1}$ is bounded.

Formally,

ρ^{*} (X) = {ℏ \in C ∣ {(ℏ I - X)}^{- 1} exists and is bounded} .

The complement of $ρ^{*} (X)$ in $C$ is called the spectrum of $X$ , denoted by $σ (X)$ .

Definition 2.9

(Pazy, 2012) A closed linear operator A is called sectorial if $\exists ı \in [π / 2, π], \tilde{E} > 0$ and $ȷ \in R$ , such that:

• $\sum_{ı, ȷ} = {ℏ \in C; ℏ \neq ȷ, \arg (ℏ - ȷ) < ı} \subset ρ^{*} (A)$ ,

• $‖ I (ℏ, X) ‖ \leq \tilde{E} / | ℏ - ȷ |, ℏ \in \sum_{ı, ȷ}$ .

Remark 2.10

To obtain the mild solution of (1), it is essential to establish the mild solution of its associated Cauchy problem.

\{\begin{cases} {}^{A B C}D^{\propto} ℘ (ϖ) = A ℘ (ϖ) + G (ϖ), ϖ \in Q = [0, q], \propto \in (0,1), \\ ℘ (0) = ℘_{0} \in N . \end{cases}

Using the established framework, the mild solution can be written based on (Bahaa and Hamiaz, 2018; Shu et al., 2011).

℘ (ϖ) = K T_{\propto} (ϖ) ℘_{0} + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} G (γ) d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{ϖ} S_{\propto} (ϖ - γ) G (γ) d γ,

where the linear operators K = n/nI − A, G = −dA/nI − A and n = Q (∝)(1−∝)⁻¹,

\begin{aligned} T_{\propto} (ϖ) = E_{\propto} (- G ϖ^{\propto}) = \frac{1}{2 π ι} \int_{U} e^{γ ϖ} γ^{\propto - 1} {(γ^{\propto} I - G)}^{- 1} d γ, \\ S_{\propto} (ϖ) = ϖ^{\propto - 1} E_{\propto, \propto} (- G ϖ^{\propto}) = \frac{1}{2 π ι} \int_{U} e^{γ ϖ} {(γ^{\propto} I - G)}^{- 1} d γ, \end{aligned}

U

is specific on ∑_ı,ȷ and

G \in C (Q, N)

In view of the above, we have.

℘ (ϖ) = \{\begin{cases} φ (ϖ), ϖ \in [- ℏ, 0], \\ K T_{\propto} (ϖ) [φ_{0} - δ (0, φ_{0})] + δ (ϖ, ℘_{ϖ}) + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} [B u_{℘} (γ) \\ + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{ϖ} S_{\propto} (ϖ - γ) \\ \times [B u_{℘} (γ) + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ, ϖ \in [0, ϖ_{1}], \\ ⋮ \\ K T_{\propto} (ϖ - ϖ_{ς}) [J_{ς} (ϖ, ℘ (ϖ_{ς}^{-})) - δ (γ_{ς}, ℘_{γ_{ς}})] + δ (ϖ, ℘_{ϖ}) + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \\ \times [B u_{℘} (γ) + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{γ_{ς}}^{ϖ} S_{\propto} (ϖ - γ) \\ \times [B u_{℘} (γ) + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(2)

Definition 2.11

A function $℘ (.) \in X$ is said to be a mild solution of (1) if ℘(ϖ) = φ(ϖ) on $ϖ \in [- ℏ, 0], ℘ (ϖ) \in N, ϖ \in Q$ satisfies the impulsive condition $℘ (ϖ) = J_{ς} (℘ (ϖ_{ς})), ϖ \in (ϖ_{ς}, γ_{ς}), ς = 1,2,3, \dots, m$ and then,

3. Results

We now investigate the EU of mild solutions corresponding to (1). Assuming A ∈ A^∝(ı₀, ȷ₀), the inequalities $‖ T_{\propto} (ϖ) ‖ \leq W e^{v ϖ}$ and $‖ S_{\propto} (ϖ) ‖ \leq C e^{v ϖ} (1 + e^{\propto - 1})$ are valid ∀ ϖ > 0 and v > v₀. So, $\sup_{ϖ \geq 0} ‖ T_{\propto} (ϖ) ‖ = R, \sup_{ϖ \geq 0} C e^{v ϖ} (1 + e^{\propto - 1}) = R_{1}$ . Hence, we obtain $‖ T_{\propto} (ϖ) ‖ \leq R; ‖ S_{\propto} (ϖ) ‖ \leq ϖ^{\propto - 1} R_{1}$ ; for more details, see Shu et al. (2011). From this point onward, we will use the following notation.

(1 + L_{2} L_{B} L_{M}) [L_{K} R (L_{J} + 2 L_{δ} + ϱ) + L_{δ} (β^{*} + 1)] + L_{2} L_{B} L_{M} (L_{1} + ‖ ℘_{ϖ} ‖) + \frac{L_{2} L_{f} L_{g}}{1 + \propto} \leq β^{*}

(3)

\begin{matrix} L_{1} = \frac{L_{K} L_{f} L_{g} L_{k}}{Q (\propto) (1 + \propto)} b^{\propto + 1} (\frac{L_{G} (1 - \propto)}{Γ (1 + \propto)} + L_{K} R_{1}) \\ L_{2} = \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (1 + \propto)} + L_{K} R_{1}) \\ ϱ = \max \sup_{ϖ \in Q} ||J_{ς} (ϖ, 0)||, ς = 1, 2, 3, \dots, m . \end{matrix}

Next, we give some assumptions which are required to establish the results of this paper as follows:

(S1) The function $ϝ : Q \times Z \times N \to N$ which satisfies Caratheodory conditions and $\exists L_{f_{1}} > 0, L_{f_{2}} > 0$ such that $‖ ϝ (ϖ, {\hat{n}}^{'}, {\hat{r}}^{'}) - ϝ (ϖ, \hat{n}, \hat{r}) ‖ \leq L_{f_{1}} ‖ {\hat{n}}^{'} - \hat{n} ‖ + L_{f_{2}} ‖ {\hat{r}}^{'} - \hat{r} ‖$ . Also ϝ satisfies $‖ ϝ (ϖ, \hat{n}, \hat{r}) ‖ \leq L_{f}, L_{f} > 0$ for $ϖ \in Q, \hat{n} \in Z, \hat{r} \in N$ and $L_{f^{*}} = \max {L_{f_{1}}, L_{f_{2}}, L_{f}}$ .

(S2) $‖ K (ϖ, {\hat{x}}_{1}, {\hat{y}}_{1}) - K (ϖ, {\hat{x}}_{2}, {\hat{y}}_{2}) ‖ \leq L_{k_{1}} ‖ {\hat{x}}_{1} - {\hat{x}}_{2} ‖ + L_{k_{2}} ‖ {\hat{y}}_{1} - {\hat{y}}_{2} ‖, L_{k_{1}}, L_{k_{2}} \geq 0, \forall ϖ \in Q$ , ${\hat{x}}_{i} \in Z$ and ${\hat{y}}_{i} \in N, i = 1,2$ . Also ∃ L_k > 0 such that $‖ K (ϖ, \hat{x}, \hat{y}) ‖ \leq L_{k}$ and $L_{k^{*}} = \max {L_{k_{1}}, L_{k_{2}}, L_{k}}$ .

(S3) The function $δ : Q \times Z \to N$ is continuous and for $L_{δ_{1}} > 0$ such that $‖ δ (ϖ, {\hat{x}}_{1} (ϖ)) - δ (ϖ, {\hat{x}}_{2} (ϖ)) ‖ \leq L_{δ_{1}} ‖ {\hat{x}}_{1} (ϖ) - {\hat{x}}_{2} (ϖ) ‖$ and $‖ δ (ϖ, {\hat{x}}_{ϖ}) ‖ \leq L_{δ} (‖ \hat{x} ‖ + 1), L_{δ} > 0$

(S4) The continuous functions $J_{ς} : (ϖ_{ς}, γ_{ς}] \times Z \to N$ satisfy $‖ J_{ς} (ϖ, \hat{x}) - J_{ς} (ϖ, \hat{y}) ‖ \leq L_{J_{1}} ‖ \hat{x} - \hat{y} ‖, \forall \hat{x}, \hat{y} \in N, ϖ \in (ϖ_{ς}, γ_{ς}]$ and L_J > 0. Also $‖ J_{ς} (ϖ, \hat{x} (ϖ_{ς})) ‖ \leq L_{J}, L_{J} > 0$ .

(S5) $G, G$ and K are bounded $‖ G ‖ \leq L_{G}, ‖ G (ϖ, γ) ‖ \leq L_{G}$ and ‖K‖ ≤ L_K, where $L_{G} > 0, L_{G} > 0 & L_{K} > 0$ .

(S6) $£ : L^{2} (Q, W) \to N$ , the linear operator spacified as $£_{u} = G K (1 - \propto) / Q (\propto) Γ (\propto) \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} B u (γ) d γ + \propto K^{2} / Q (\propto) \int_{γ_{ς}}^{ϖ} S_{\propto} (ϖ - γ) B u (γ) d γ, ς = 1,2,3, \dots, m$ has an inverse bounded operator $£^{- 1} : N \to L^{2} (Q, W) / k e r (£)$ , so ∃ L_B > 0 & L_£ > 0 such that ‖B‖ ≤ L_B and ‖£⁻¹‖ ≤ L_£.

(S6) For every bounded set $V \subseteq Z$ and $V^{'} \subseteq N, \exists L_{1}, L_{2}, L_{δ}, L_{J} > 0$ such that $E (ϝ (ϖ, V, V^{'})) \leq L_{1} E (V) + L_{2} E (V^{'})$ , $E (δ (ϖ, V)) \leq L_{δ} E (V), \max E (J_{ς} (ϖ, V)) \leq L_{J} E (V), ς = 1,2,3, \dots, m$ .

L e t m = L_{K} R (L_{J} + L_{δ}) + 2 (L_{B} N_{p} + L_{f} L_{g} L_{k}) L_{1},

where

N_{p} = L_{£} L_{K} R (L_{J} + L_{δ}) + L_{δ} + 2 (L_{£} L_{1} L_{f} L_{g} L_{k}) .

Theorem 3.1

Let the hypotheses (S₁)–(S₆) hold and m≤1.

Then (1) has a solution in $Q$ .

Proof. Assume $F : X \to X$ is an operator given as

F (℘) (ϖ) = \{\begin{cases} φ (ϖ), ϖ \in [- ℏ, 0], \\ K T_{\propto} (ϖ) [φ_{0} - δ (0, φ_{0})] + δ (ϖ, ℘_{ϖ}) + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} [B u_{℘} (γ) \\ + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{ϖ} S_{\propto} (ϖ - γ) [B u_{℘} (γ) \\ + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ, ϖ \in [0, ϖ_{1}], \\ ⋮ \\ K T_{\propto} (ϖ - ϖ_{ς}) [J_{ς} (ϖ, ℘ (ϖ_{ς}^{-})) - δ (γ_{ς}, ℘_{γ_{ς}})] + δ (ϖ, ℘_{ϖ}) + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} [B u_{℘} (γ) \\ + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{γ_{ς}}^{ϖ} S_{\propto} (ϖ - γ) [B u_{℘} (γ) \\ + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(4)

Employing (S₆), we introduce the control function below for $℘ (.) \in X$ by

u_{℘} (ϖ) = £^{- 1} \{\begin{cases} ℘_{ϖ} - K T_{\propto} (b) [φ_{0} - δ (0, φ_{0})] - δ (b, ℘_{b}) - \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{b} {(b - γ)}^{\propto - 1} [ϝ (γ, ℘_{γ}, ℘ (p (γ))) \\ \times \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ - \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{b} S_{\propto} (b - γ) [ϝ (γ, ℘_{γ}, ℘ (p (γ))) \\ \times \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ, ϖ \in [0, ϖ_{1}], \\ ⋮ \\ ℘_{ϖ} - K T_{\propto} (b - γ_{ς}) [J_{ς} (γ_{ς}, ℘_{γ_{ς}}) δ (γ_{ς}, ℘_{γ_{ς}})] - δ (b, ℘_{b}) - \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} \\ \times [ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ - \frac{\propto K^{2}}{Q (\propto)} \int_{γ_{ς}}^{b} S_{\propto} (b - γ) \\ \times [ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ] d γ, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(5)

To proceed, we decompose the proof into four parts.

Part:1 We begin by showing that $F$ is continuous. Let {℘_n} be a sequence in $X_{σ}$ with ℘_n → ℘. Then, for every $ϖ \in Q$ , using (4), we obtain

\begin{aligned} ‖F (℘_{n}) (ϖ) & - F (℘) (ϖ)‖ \\ \leq & ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{‖ G ‖ ‖ K ‖ (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} \\ \times (‖ B ‖ ‖ u_{n ℘} (γ) - u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \\ \times \int_{0}^{γ} ‖ G (γ, ζ) ‖ ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ + \frac{\propto ‖ K ‖^{2}}{Q (\propto)} \int_{0}^{ϖ} ‖S_{\propto} (ϖ - γ)‖ (‖ B ‖ ‖ u_{n ℘} (γ) - u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} ‖ G (γ, ζ) ‖ ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ \leq & ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{0}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖ K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d b + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{0}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d b) \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖ K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{0}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d b + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{0}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d b) \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ \\ \to 0 as ℘_{n} \to ℘, ϖ \in (0, ϖ_{1}] . \end{aligned}

Furthermore, for every ϖ ∈ (γ_ς, ϖ_ς+1], ς = 1, 2, 3, …, m, we have

\begin{aligned} ‖F (℘_{n}) (ϖ) & - F (℘) (ϖ)‖ \\ \leq & L_{K} R [‖J_{ς} (γ_{ς}, ℘_{n} (γ_{ς}^{-})) - δ (γ_{ς}, ℘_{n γ_{ς}}) - J_{ς} (γ_{ς}, ℘ (γ_{ς}^{-})) + δ (γ_{ς}, ℘_{γ_{ς}})‖] \\ + ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{‖ G ‖ ‖ K ‖ (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \\ \times (‖ B ‖ ‖ u_{n ℘} (γ) - u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \\ \times \int_{γ_{ς}}^{γ} ‖ G (γ, ζ) ‖ ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ + \frac{\propto ‖ K ‖^{2}}{Q (\propto)} \int_{γ_{ς}}^{ϖ} ‖S_{\propto} (ϖ - γ)‖ (‖ B ‖ ‖ u_{n ℘} (γ) - u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} ‖ G (γ, ζ) ‖ ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ \leq & L_{K} R [‖J_{ς} (γ_{ς}, ℘_{n} (γ_{ς}^{-})) - J_{ς} (γ_{ς}, ℘ (γ_{ς}^{-}))‖ + ‖δ (γ_{ς}, ℘_{n γ_{ς}}) - δ (γ_{ς}, ℘_{γ_{ς}})‖] \\ + ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d b + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d b) \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘ (ζ), ℘ (p (ζ)))‖ d ζ d b + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d b) \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ, \end{aligned}

\lim_{n \to \infty} F (℘_{n}) (ϖ) = F (℘) (ϖ)

X_{σ}

, since the functions

ϝ, G, δ, K

and

J_{ς}, ς = 1,2,3, \dots, m

are continuous. Thus,

F

is continuous on

X_{σ}

Part:2 $F$ is bounded in $P C ([- ℏ, q])$ .

It is enough to show that for any β* > 0, then $℘ \in X_{β^{*}} = {℘ \in X : ‖ ℘ ‖ \leq β^{*}}$ , where β* satisfies (3). Using (3), we have

‖ u_{℘} (ϖ) ‖ \leq L_{£} \{\begin{cases} ‖℘_{ϖ}‖ + L_{K} R (‖ φ_{0} ‖ + L_{δ} (‖ φ_{0} ‖ + 1)) + L_{δ} (‖ ℘_{b} ‖ + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} R_{1} L_{f} L_{g} L_{k}}{Q (\propto) (\propto + 1)} b^{\propto + 1}, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ ‖℘_{ϖ}‖ + L_{K} R ((L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}) + L_{δ} (β^{*} + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} L_{f} L_{g} L_{k} R_{1}}{Q (\propto) (\propto + 1)} b^{\propto + 1}, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

Let $W_{0} = L_{K} R (‖ φ_{0} ‖ + L_{δ} (‖ φ_{0} ‖ + 1))$ , then $\exists Z_{1} > 0$ and $Z_{2} > 0$ be such that

‖ u_{℘} (ϖ) ‖ \leq L_{£} \{\begin{cases} ‖℘_{ϖ}‖ + W_{0} + L_{δ} (‖ ℘_{b} ‖ + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} R_{1} L_{f} L_{g} L_{k}}{Q (\propto) (\propto + 1)} b^{\propto + 1} = Z_{1}, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ ‖℘_{ϖ}‖ + L_{K} R [(L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}] + L_{δ} (β^{*} + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} L_{f} L_{g} L_{k} R_{1}}{Q (\propto) (\propto + 1)} b^{\propto + 1} = Z_{2}, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(6)

Then, for each $℘ \in X_{σ}$ and for (3.1), we have

‖ F (℘) (ϖ) ‖ \leq \{\begin{cases} W_{0} + L_{δ} (β^{*} + 1) + \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) \\ \times (L_{B} Z_{1} + \frac{L_{f} L_{g}}{\propto + 1} b) \leq β^{*}, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ L_{K} R ((L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}) + L_{δ} (β^{*} + 1) \\ + \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) \\ \times (L_{B} Z_{2} + \frac{L_{f} L_{g}}{\propto + 1} b) \leq β^{*}, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(7)

∴ F

is bounded in

P C ([- ℏ, q])

Part:3 $F$ is equi-continuous $X$ . Let l₁, l₂ ∈ (0, ϖ₁] with l₁ < l₂, and we have

\begin{aligned} ‖(F ℘) (l_{2}) - (F ℘) (l_{1})‖ \leq & L_{k} ‖T_{\propto} (l_{2}) - T_{\propto} (l_{1})‖ [‖ φ_{0} - δ (0, φ_{0}) ‖] + ‖δ (l_{2}, ℘_{l_{2}}) - δ (l_{1}, ℘_{l_{1}})‖ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \int_{0}^{l_{1}} {(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1} d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{k} \int_{0}^{l_{1}} γ {(l_{2} - γ)}^{\propto - 1} - γ {(l_{1} - γ)}^{\propto - 1} d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \int_{0}^{l_{1}} {(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1} d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} L_{f} L_{g} L_{k} R_{1} \int_{0}^{l_{1}} γ {(l_{2} - γ)}^{\propto - 1} - γ {(l_{1} - γ)}^{\propto - 1} d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{k} \int_{l_{1}}^{l_{2}} γ {(l_{2} - γ)}^{\propto - 1} d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ \\ + L_{f} L_{g} L_{k} R_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ . \end{aligned}

Further, for l₁, l₂ ∈ (γ_ς, ϖ_ς+1] with l₁ < l₂, ς = 1, 2, 3, …, m, we have

\begin{aligned} ‖(F ℘) (l_{2}) - (F ℘) (l_{1})‖ \leq & L_{K} ‖T_{\propto} (l_{2} - γ_{ς}) - T_{\propto} (l_{1} - γ_{ς})‖ ‖J_{ς} (γ_{ς}, ℘_{γ_{ς}}) - δ (γ_{ς}, ℘_{γ_{ς}})‖ \\ + ‖δ (l_{2}, ℘_{l_{2}}) - δ (l_{1}, ℘_{l_{1}})‖ + \frac{L_{G} L_{ς} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \\ \times \int_{0}^{l_{1}} [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ + \frac{L_{G} L_{ς} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{ς} \\ \times \int_{0}^{l_{1}} γ [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \\ \times \int_{0}^{l_{1}} [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} L_{f} L_{g} L_{k} R_{1} \\ \times \int_{0}^{l_{1}} γ [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \\ \times \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{k} \\ \times \int_{l_{1}}^{l_{2}} γ {(l_{2} - γ)}^{\propto - 1} d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \end{aligned}

\begin{aligned} \times \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} + L_{f} L_{g} L_{k} R_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ \\ \to 0 as l_{2} \to l_{1}, \end{aligned}

we proved

F

is equi-continuous. As a consequence of part

(1) - (3), F

is completely continuous.

Part:4 A priori bounds.

The final step is to establish that the set $Δ = \{ϖ \in X : ℘ = λ^{*} F (℘)$ , for some $λ^{*} \in (0,1)\}$ is bounded. Let ℘ ∈ Δ, for some λ* ∈ (0, 1), for each ϖ ∈ (0, q], we have $℘ (ϖ) = λ^{*} F ℘ (ϖ)$ ,

\begin{aligned} ‖℘ (ϖ)‖ & = ‖ λ^{*} F ℘ (ϖ) ‖ \\ \leq & λ^{*} ‖ K ‖ ‖ T_{\propto} (ϖ - ϖ_{ς}) ‖ ‖ J_{ς} (ϖ, ℘ (ϖ_{ς}^{-})) - δ (γ_{ς}, ℘_{γ_{ς}}) ‖ + ‖ δ (ϖ, ℘_{ϖ}) ‖ \\ + \frac{‖ G ‖ ‖ K ‖ (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} [‖ B ‖ ‖ u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \\ \times \int_{γ_{ς}}^{γ} ‖ G (γ, ζ) ‖ ‖ K (ζ, ℘_{ζ}, ℘ (p (ζ))) ‖ d ζ] d γ \\ + \frac{\propto ‖ K^{2} ‖}{Q (\propto)} \int_{γ_{k}}^{ϖ} ‖ S_{\propto} (ϖ - γ) ‖ [‖ B ‖ ‖ u_{℘} (γ) ‖ \\ + ‖ ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{k}}^{γ} ‖ G (γ, ζ) ‖ ‖ K (ζ, ℘_{ζ}, ℘ (p (ζ))) ‖ d ζ] d γ \\ \leq & λ^{*} (L_{K} R [(L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}] + L_{δ} (β^{*} + 1) + \frac{L_{G} L_{K} L_{B} Z_{1} (1 - \propto)}{Q (\propto) Γ (\propto + 1)} q^{\propto} \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} q^{\propto + 1} + \frac{L_{K}^{2} R_{1} L_{B} Z_{1}}{Q (\propto)} q^{\propto} + \frac{L_{K}^{2} R_{1} L_{B} L_{f} L_{g} L_{k}}{Q (\propto) (1 + \propto)} q^{\propto + 1}) . \end{aligned}

Thus, by using (3), for every ϖ ∈ (0, q] and for λ* < 1, we have

\begin{aligned} \leq λ^{*} (L_{K} R [(L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}] + L_{δ} (β^{*} + 1) \\ + \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) (L_{B} Z_{2} + \frac{L_{f} L_{g}}{\propto + 1} b)), \\ \leq β^{*} . \end{aligned}

As it is true for every ℘ ∈ Δ, Δ is bounded. By Schaefer’s fixed point theorem, we claim that $F$ has a fixed point which is a solution of (1).

Theorem 3.2

If assumptions (S₁)–(S₆) hold, then (1) admits a unique solution, provided that

\begin{aligned} L_{K} R (m L_{J_{1}} + L_{δ_{1}}) + L_{δ_{1}} + \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) \\ \times (\frac{L_{B} L_{K} L_{f} L_{g} L_{k}}{Q (\propto) (\propto + 1)} + \frac{L_{f} L_{g} Γ (\propto + 1)}{L_{G} (1 {- \propto}^{2}) + L_{K} R_{1} Γ (\propto + 2)}) \leq 1, \end{aligned}

(8)

where

\sum_{ς = 1}^{m} ‖J_{ς} (ϖ, ℘_{1} (ϖ)) - J_{ς} (ϖ, ℘_{2} (ϖ))‖ = m L_{J_{1}} ‖℘_{1} - ℘_{2}‖

Proof. To demonstrate uniqueness, we proceed by contradiction, let ℘₁ and ℘₂ be two solutions of (1).

Case 1: For $ϖ \in [- ℏ, 0], ‖ F ℘_{1} - F ℘_{2} ‖ \to 0$ as ℘₁ → ℘₂.

Case 2: For ϖ ∈ (γ_ς, ϖ_ς+1],

\begin{aligned} ‖F ℘_{1} - F ℘_{2}‖ \leq & L_{K} R [‖J_{ς} (γ_{ς}, ℘_{1} (γ_{ς}^{-})) - J_{ς} (γ_{ς}, ℘_{2} (γ_{ς}^{-}))‖ + ‖δ (γ_{ς}, ℘_{1 γ_{ς}}) - δ (γ_{ς}, ℘_{2 γ_{ς}})‖] \\ + ‖δ (ϖ, ℘_{1 t}) - δ (ϖ, ℘_{2 ϖ})‖ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} (‖ ϝ (γ, ℘_{1 γ}, ℘_{1} (p (γ))) - ϝ (γ, ℘_{2 γ}, ℘_{2} (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{1 ζ}, ℘_{1} (p (ζ))) \\ - K (ζ, ℘_{2 ζ}, ℘_{2} (p (ζ)))‖ d ζ) d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} (‖ ϝ (γ, ℘_{1 γ}, ℘_{1} (p (γ))) \\ - ϝ (γ, ℘_{2}, ℘_{2} (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{1 ζ}, ℘_{1} (p (ζ))) - K (ζ, ℘_{2 ζ}, ℘_{2} (p (ζ)))‖ d ζ) d γ \\ + ‖ ϝ (γ, ℘_{1 γ}, ℘_{1} (p (γ))) - ϝ (γ, ℘_{2 γ}, ℘_{2} (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{1 ζ}, ℘_{1} (p (ζ))) \\ - K (ζ, ℘_{2 ζ}, ℘_{2} (p (ζ)))‖ d ζ\} d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{1 γ}, ℘_{1} (p (γ))) - ϝ (γ, ℘_{2 γ}, ℘_{2} (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{1 ζ}, ℘_{1} (p (ζ))) \\ - K (ζ, ℘_{1} (ζ), ℘_{2} (p (ζ)))‖ d ζ d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} (‖ ϝ (γ, ℘_{1 γ}, ℘_{1} (p (γ))) \\ - ϝ (γ, ℘_{2 γ}, ℘_{2} (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{1 ζ}, ℘_{1} (p (ζ))) - K (ζ, ℘_{2 ζ}, ℘_{2} (p (ζ)))‖ d ζ) d γ \\ + ‖ ϝ (γ, ℘_{1 γ}, ℘_{1} (p (γ))) - ϝ (γ, ℘_{2 γ}, ℘_{2} (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{1 ζ}, ℘_{1} (p (ζ))) \\ - K (ζ, ℘_{2 ζ}, ℘_{2} (p (ζ)))‖ d ζ\} d γ, \end{aligned}

\begin{aligned} \leq & L_{K} R (m L_{J_{1}} + L_{δ_{1}}) ‖℘_{1} - ℘_{2}‖ + L_{δ_{1}} ‖℘_{1} - ℘_{2}‖ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} (L_{f^{*}} L_{g} L_{k^{*}} ‖ ℘_{1} - ℘_{2} ‖ \int_{γ_{ς}}^{γ} ‖℘_{1} - ℘_{2}‖ d ζ) d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} (L_{f^{*}} L_{g} L_{k^{*}} ‖℘_{1} - ℘_{2}‖ \int_{0}^{γ} ‖℘_{1} - ℘_{2}‖ d ζ) d γ \\ + L_{f^{*}} L_{g} L_{k^{*}} ‖ ℘_{1} - ℘_{2} ‖ \int_{γ_{ς}}^{γ} ‖℘_{1} - ℘_{2}‖ d ζ\} d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} L_{f^{*}} L_{g} L_{k^{*}} ‖ ℘_{1} - ℘_{2} ‖ \int_{γ_{ς}}^{γ} ‖℘_{1} - ℘_{2}‖ d ζ d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} (L_{f^{*}} L_{g} L_{k^{*}} ‖℘_{1} - ℘_{2}‖ \int_{γ_{ς}}^{γ} ‖℘_{1} - ℘_{2}‖ d ζ) d γ \\ + L_{f^{*}} L_{g} L_{k^{*}} ‖℘_{1} - ℘_{2}‖ \int_{γ_{ς}}^{γ} L_{g} ‖℘_{1} - ℘_{2}‖ d ζ\} d γ, \\ \leq & L_{K} R (m L_{J_{1}} + L_{δ_{1}}) ‖℘_{1} - ℘_{2}‖ + L_{δ_{1}} ‖℘_{1} - ℘_{2}‖ + \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) \\ \times (\frac{L_{B} L_{K} L_{f} L_{g} L_{k}}{Q (\propto) (\propto + 1)} + \frac{L_{f} L_{g} Γ (\propto + 1)}{L_{G} (1 {- \propto}^{2}) + L_{K} R_{1} Γ (\propto + 2)}) ‖℘_{1} - ℘_{2}‖ . \end{aligned}

By assumptions (S₁)–(S₆) and (8), $F$ is a contraction. Then, by Banach fixed-point theorem, (1) has a unique solution.

4. Controllability results

This section is devoted to the controllability of (1).

Definition 4.1

(Aimene et al., 2019) The (1) is said to be controllable if for every $φ \in Z$ and $℘_{1} \in N$ , $\exists u \in L^{2} (Q, W)$ such that ℘(.) of (1) satisfies ℘ (q) = ℘₁ in $Q$ .

Theorem 4.2

( Banaś, 1980 ) Suppose $S \neq \emptyset$ subset of a BS $N$ that is convex, closed, and bounded. The mapping $F : S \to S$ is said to be a $D$ -set contraction if

H (F (X)) \leq D H (X),

(9)

where

X

is a closed subset of

S

and

D \in [0,1]

. The

F

has a fixed point in

S

Theorem 4.3

Let assumptions (S₁)–(S₇) hold and

L_{K} R (L_{J} + L_{δ}) + 2 (L_{B} N_{S} + L_{f} L_{g} L_{k}) L_{1} \leq 1

(10)

where

N_{S} = L_{£} L_{K} R (L_{J} + L_{δ}) + L_{δ} + 2 (L_{£} L_{1} L_{f} L_{g} L_{k}) .

Then (1) is controllable on $Q$ .

Proof. Assume $F : X \to X$ is an operator given as

Employing S₆, we introduce the control function given below for $℘ (.) \in X$ by

F (℘) (ϖ) = \{\begin{cases} φ (ϖ), ϖ \in [- ℏ, 0] \\ K T_{\propto} (ϖ) (φ_{0} - δ (0, φ_{0})) + δ (ϖ, ℘_{ϖ}) + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} (B u_{℘} (γ) \\ + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ) d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{ϖ} S_{\propto} (ϖ - γ) \\ \times (B u_{℘} (γ) + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ) d γ, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ K T_{\propto} (ϖ - ϖ_{ς}) (J_{ς} (ϖ, ℘ (ϖ_{ς}^{-})) - δ (γ_{ς}, ℘_{γ_{ς}})) + δ (ϖ, ℘_{ϖ}) + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \\ \times (B u_{℘} (γ) + ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ) d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{γ_{ς}}^{ϖ} S_{\propto} (ϖ - γ) \\ \times (B u_{℘} (γ) + ϝ (γ, ℘_{γ}, ℘ (p (γ)))] \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ) d γ, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(11)

u_{℘} (ϖ) = £^{- 1} \{\begin{cases} ℘_{ϖ} - K T_{\propto} (b) [φ_{0} - δ (0, φ_{0})] - δ (b, ℘_{b}) - \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{b} {(b - γ)}^{\propto - 1} \\ \times ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ d γ - \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{b} S_{\propto} (b - γ) \\ \times ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ d γ, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ ℘_{ϖ} - K T_{\propto} (b - γ_{ς}) [J_{ς} (γ_{ς}, ℘_{γ_{ς}}) - δ (γ_{ς}, ℘_{γ_{ς}})] - δ (b, ℘_{b}) - \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} \\ \times ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ d γ - \frac{\propto K^{2}}{Q (\propto)} \int_{γ_{ς}}^{b} S_{\propto} (b - γ) \\ \times ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ d γ, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(12)

Let $X_{σ} = {℘ \in X : ‖ ℘ ‖ \leq β^{*}}$ , where β* satisfies (3).

It is clear that $X_{σ}$ forms a convex, closed, and bounded subset of $X$ . We now proceed to establish the controllability of (1) that satisfies $F (℘) (b) = ℘_{ϖ}$ .

To proceed, we decompose the proof into the following four parts.

Part:1 We begin by showing that $F$ is continuous. Let {℘_n} be a sequence in $X_{σ}$ with ℘_n → ℘. Then, for every $ϖ \in Q$ , using (11), we obtain

\begin{aligned} ‖F (℘_{n}) (ϖ) & - F (℘) (ϖ)‖ \\ \leq & ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{‖ G ‖ ‖ K ‖ (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} (‖ B ‖ ‖ u_{n ℘} (γ) \\ - u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} ‖ G (γ, ζ) ‖ \\ \times ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ + \frac{\propto ‖ K ‖^{2}}{Q (\propto)} \\ \times \int_{0}^{ϖ} ‖S_{\propto} (ϖ - γ)‖ (‖ B ‖ ‖ u_{n ℘} (γ) - u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} ‖ G (γ, ζ) ‖ ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ \leq & ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{0}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{0}^{b} {(b - γ)}^{\propto - 1} (‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d γ) \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{0}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘ (ζ), ℘ (p (ζ)))‖ d ζ d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{0}^{b} {(b - γ)}^{\propto - 1} (‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ, \\ \to 0 as ℘_{n} \to ℘ ϖ \in (0, ϖ_{1}] . \end{aligned}

Furthermore, for every ϖ ∈ (γ_&, ϖ_&+1], & = 1, 2, 3, …, m, we have

\begin{aligned} ‖F (℘_{n}) (ϖ) & - F (℘) (ϖ)‖ \\ \leq & L_{K} R [‖J_{ς} (γ_{ς}, ℘_{n} (γ_{ς}^{-})) - δ (γ_{ς}, ℘_{n γ_{ς}}) - J_{ς} (γ_{ς}, ℘ (γ_{ς}^{-})) + δ (γ_{ς}, ℘_{γ_{ς}})‖] \\ + ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{‖ G ‖ ‖ K ‖ (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \\ \times (‖ B ‖ ‖ u_{n ℘} (γ) - u_{℘} (γ) ‖ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \\ \times \int_{γ_{ς}}^{γ} ‖ G (γ, ζ) ‖ ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ + \frac{\propto ‖ K ‖^{2}}{Q (\propto)} \int_{γ_{ς}}^{ϖ} ‖S_{\propto} (ϖ - γ)‖ (‖ B ‖ ‖ u_{n ℘} (γ) - u_{℘} (γ) ‖ \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} ‖ G (γ, ζ) ‖ \\ \times ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ) d γ \\ \leq & L_{K} R [‖J_{ς} (γ_{ς}, ℘_{n} (γ_{ς}^{-})) - J_{ς} (γ_{ς}, ℘ (γ_{ς}^{-}))‖ + ‖δ (γ_{ς}, ℘_{n γ_{ς}}) - δ (γ_{ς}, ℘_{γ_{ς}})‖] \\ + ‖δ (ϖ, ℘_{n ϖ}) - δ (ϖ, ℘_{ϖ})‖ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{0}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ d γ) \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} \{L_{B} (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \\ \times \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘ (ζ), ℘ (p (ζ)))‖ d ζ d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} R_{1} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) - K (ζ, x_{ζ}, x (p (ζ)))‖ d ζ d γ) \\ + ‖ ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) - ϝ (γ, ℘_{γ}, ℘ (p (γ))) ‖ \int_{γ_{ς}}^{γ} L_{g} ‖K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) \\ - K (ζ, ℘_{ζ}, ℘ (p (ζ)))‖ d ζ\} d γ . \end{aligned}

We obtain that $\lim_{n \to \infty} F (℘_{n}) (ϖ) = F (℘) (ϖ)$ in $X_{σ}$ , since the functions $ϝ, G, δ, K$ and $J_{ς}, ς = 1,2,3, \dots, m$ are continuous. Thus $F$ is continuous on $X_{σ}$ .

Part:2 $F$ is bounded in $P C ([- ℏ, q])$ .

It is enough to show that $F (X_{σ}) \subset X_{σ}$ . Assume that $℘ \in X_{σ}$ , we estimate ‖u_℘(ϖ)‖ from (12) and by hypotheses, as

‖ u_{℘} (ϖ) ‖ \leq L_{£} \{\begin{cases} ‖℘_{ϖ}‖ + L_{K} R (‖ φ_{0} ‖ + L_{δ} (‖ φ_{0} ‖ + 1)) + L_{δ} (‖ ℘_{b} ‖ + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} R_{1} L_{f} L_{g} L_{k}}{Q (\propto) (\propto + 1)} b^{\propto + 1}, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ ‖℘_{ϖ}‖ + L_{K} R ((L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}) + L_{δ} (β^{*} + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} L_{f} L_{g} L_{k} R_{1}}{Q (\propto) (\propto + 1)} b^{\propto + 1}, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

Let $W_{0} = L_{K} R (‖ φ_{0} ‖ + L_{δ} (‖ φ_{0} ‖ + 1))$ , then $\exists Z_{1} > 0$ and $Z_{2} > 0$ such that

‖ u_{℘} (ϖ) ‖ \leq L_{£} \{\begin{cases} ‖℘_{ϖ}‖ + L_{0} + L_{δ} (‖ ℘_{b} ‖ + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} R_{1} L_{f} L_{g} L_{k}}{Q (\propto) (\propto + 1)} b^{\propto + 1} = Z_{1}, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ ‖℘_{ϖ}‖ + L_{K} R [(L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}] + L_{δ} (β^{*} + 1) \\ + \frac{L_{G} L_{K} L_{f} L_{g} L_{k} (1 - \propto)}{Q (\propto) Γ (\propto + 2)} b^{\propto + 1} + \frac{L_{K}^{2} L_{f} L_{g} L_{k} R_{1}}{Q (\propto) (\propto + 1)} b^{\propto + 1} = Z_{2}, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(13)

Then, for each $℘ \in X_{σ}$ and from (11), we have

‖ F (℘) (ϖ) ‖ \leq \{\begin{cases} W_{0} + L_{δ} (β^{*} + 1) + \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) \\ \times (L_{B} Z_{1} + \frac{L_{f} L_{g}}{\propto + 1} b) \leq β^{*}, ϖ \in [0, ϖ_{1}] \\ ⋮ \\ L_{K} R ((L_{J} + L_{δ}) β^{*} + ϱ + L_{δ}) + L_{δ} (β^{*} + 1) \\ + \frac{L_{K}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) \\ \times (L_{B} Z_{2} + \frac{L_{f} L_{g}}{\propto + 1} b) \leq β^{*}, ϖ \in [γ_{ς}, ϖ_{ς + 1}], ς = 1,2, \dots, m . \end{cases}

(14)

Consequently, $‖ F (℘) (ϖ) ‖ \leq β^{*}$ ; hence, $F$ is bounded in $P C ([- ℏ, q])$ .

Part:3 The set ${F (℘) (ϖ) : ℘ \in X_{σ}}$ is equi-continuous. Let $℘ \in X_{σ}$ and l₁, l₂ ∈ (0, ϖ₁] with l₁ < l₂, we have

\begin{aligned} ‖(F ℘) (l_{2}) - (F ℘) (l_{1})‖ \leq & L_{k} ‖T_{\propto} (l_{2}) - T_{\propto} (l_{1})‖ [‖φ_{0} - δ (0, φ_{0})‖] + ‖δ (i_{2}, ℘_{l_{2}}) - δ (l_{1}, ℘_{l_{1}})‖ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \int_{0}^{l_{1}} [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{k} \int_{0}^{l_{1}} γ [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \int_{0}^{l_{1}} [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} L_{f} L_{g} L_{k} R_{1} \int_{0}^{l_{1}} γ [{(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1}] d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{k} \int_{l_{1}}^{l_{2}} γ {(l_{2} - γ)}^{\propto - 1} d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ \\ + L_{f} L_{g} L_{k} R_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ . \end{aligned}

Further for l₁, l₂ ∈ (γ_&, ϖ_&+1] with l₁ < l₂, & = 1, 2, 3, …, m, we have

\begin{aligned} ‖(F ℘) (l_{2}) - (F ℘) (l_{1})‖ \leq & L_{K} ‖T_{\propto} (l_{2} - γ_{ς}) - T_{\propto} (l_{1} - γ_{ς})‖ ‖J_{ς} (γ_{ς}, ℘_{γ_{ς}}) - δ (γ_{ς}, ℘_{γ_{ς}})‖ \\ + ‖δ (l_{2}, ℘_{l_{2}}) - δ (l_{1}, ℘_{l_{1}})‖ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \int_{0}^{l_{1}} {(l_{2} - γ)}^{\propto - 1} \\ - {(l_{1} - γ)}^{\propto - 1} d γ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{k} \int_{0}^{l_{1}} γ {(l_{2} - γ)}^{\propto - 1} \\ - γ {(l_{1} - γ)}^{\propto - 1} d γ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \int_{0}^{l_{1}} {(l_{2} - γ)}^{\propto - 1} - {(l_{1} - γ)}^{\propto - 1} d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} L_{f} L_{g} L_{k} R_{1} \int_{0}^{l_{1}} γ {(l_{2} - γ)}^{\propto - 1} - γ {(l_{1} - γ)}^{\propto - 1} d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{B} Z_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ \\ + \frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} L_{f} L_{g} L_{k}) \int_{l_{1}}^{l_{2}} γ {(l_{2} - γ)}^{\propto - 1} d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} Z_{1} L_{B} R_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} \\ + L_{f} L_{g} L_{k} R_{1} \int_{l_{1}}^{l_{2}} {(l_{2} - γ)}^{\propto - 1} d γ \end{aligned}

\to 0 as l_{2} \to l_{1} .

This means that $F (X_{σ})$ is equi-continuous.

Part:4 $F : X_{σ} \to X_{σ}$ is a strict set contraction.

By 2.4, ${℘^{n}}_{n = 1}^{\infty} \subset P$ there exists $\forall P \subseteq X_{σ}$ , also $H (F (P) (ϖ)) \leq 2 H ({℘^{n}}_{n = 1}^{\infty})$ by the equi-continuous on $X_{σ}$ . If $ϖ \in [- ℏ, 0], H (F (P) (ϖ)) = H ({F (℘^{n}) (ϖ), ℘^{n} \in P}_{n = 1}^{\infty}) = H ({φ (ϖ), ℘^{n} \in P}_{n = 1}^{\infty}) = 0 \forall ϖ \in [0, ϖ_{1}]$ , by S₇, we get

H ({δ (ϖ, ℘_{ϖ}^{n})}_{n = 1}^{\infty}) \leq L_{δ} H (P)

(15)

\begin{aligned} H (\{\frac{K G (1 - \propto)}{Q (\propto) Γ (\propto)} & {\int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \int_{0}^{γ} G (γ, z) K (z, ℘_{n z}, ℘_{n} (p (z))) d z d γ\}}_{n = 1}^{\infty}) \\ \leq & 2 \frac{L_{K} L_{G} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} [(L_{f_{1}} \sup_{- ℏ \leq ı \leq 0} H ({℘_{n} (ı + ϖ)}_{n = 1}^{\infty}) \\ + L_{f_{2}} \sup_{0 \leq γ \leq ϖ} H ({℘_{n} (p (γ))}_{n = 1}^{\infty})) \int_{0}^{γ} L_{g} (L_{k_{1}} \sup_{- ℏ \leq ȷ \leq 0} H ({℘_{n} (ȷ + z)}_{n = 1}^{\infty}) \\ + L_{k_{2}} \sup_{0 \leq z \leq ϖ} H ({℘_{n} (p (z))}_{n = 1}^{\infty})) d z] d γ \\ \leq & 2 \frac{L_{K} L_{G} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} [(L_{f_{1}} \sup_{0 \leq τ \leq γ} H ({℘_{n} (τ)}_{n = 1}^{\infty}) \\ + L_{f_{2}} \sup_{0 \leq γ \leq ϖ} H ({℘_{n} (p (γ))}_{n = 1}^{\infty})) \int_{0}^{γ} L_{g} (L_{k_{1}} \sup_{0 \leq ζ \leq z} H ({℘_{n} (ζ)}_{n = 1}^{\infty}) \\ + L_{k_{2}} \sup_{0 \leq z \leq ϖ} H ({℘_{n} (p (z))}_{n = 1}^{\infty})) d z] d γ \\ \leq & 2 \frac{L_{K} L_{G} (1 - \propto)}{Q (\propto) Γ (\propto)} q^{\propto + 1} L_{g} (L_{f_{1}} + L_{f_{2}}) (L_{k_{1}} + L_{k_{2}}) H (P) . \end{aligned}

(16)

Similarly

\begin{aligned} H (\{\frac{\propto K^{2}}{Q (\propto)} & {\int_{0}^{ϖ} S_{\propto} (ϖ - γ) ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \int_{0}^{γ} G (γ, z) K (z, ℘_{n z}, ℘_{n} (p (z))) d z d γ\}}_{n = 1}^{\infty}) \\ \leq 2 \frac{\propto L_{K}^{2}}{Q (\propto)} q^{\propto + 1} L_{g} R_{1} (L_{f_{1}} + L_{f_{2}}) (L_{k_{1}} + L_{k_{2}}) H (P) . \end{aligned}

(17)

Now, by applying (16) and (17), from (12), we have

\begin{aligned} H ({u_{℘^{n}} (ϖ)}_{n = 1}^{\infty}) = & H (\{£^{- 1} [℘_{ϖ} - K T_{\propto} (b) (φ_{0} - δ (0, φ_{0})) - δ (b, ℘_{n b}) - \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{b} {(b - γ)}^{\propto - 1} \\ \times ϝ (γ, ℘_{γ}, ℘ (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ d γ - \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{b} S_{\propto} (b - γ) \\ \times {ϝ (γ, ℘_{γ}, ℘ (p (γ)))] \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{ζ}, ℘ (p (ζ))) d ζ d γ]\}}_{n = 1}^{\infty}) \\ \leq & [L_{δ} + \frac{2 L_{£} L_{g} L_{K}}{(\propto + 1) Q (\propto)} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1})] \\ \times (L_{f_{1}} + L_{f_{2}}) (L_{k_{1}} + L_{k_{2}}) H (P) \\ \leq & γ^{*} H (P), where γ^{*} = L_{δ} + 2 L_{£} L_{1} . \end{aligned}

(18)

Hence, by (16), (17), and (18), we have

\begin{aligned} H (F (P) (ϖ)) = & H ({F (℘_{n}) (ϖ)}_{n = 1}^{\infty}) \\ = & H (\{K T_{\propto} (ϖ) [φ_{0} - δ (0, φ_{0})] + δ (ϖ, ℘_{n ϖ}) + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} (B u_{℘_{n}} (γ) \\ + ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) d ζ) d γ + \frac{\propto K^{2}}{Q (\propto)} \int_{0}^{ϖ} S_{\propto} (ϖ - γ) \\ {\times (B u_{n ℘} (γ) + ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \int_{0}^{γ} G (γ, ζ) K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) d ζ) d γ\}}_{n = 1}^{\infty}) \\ \leq & L_{δ} H ({℘_{n ϖ}}_{n = 1}^{\infty}) + 2 (\frac{L_{G} L_{K} (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{0}^{ϖ} {(ϖ - γ)}^{\propto - 1} (L_{B} H ({u_{℘_{n}} (γ)}_{n = 1}^{\infty}) \\ + H ({ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ)))}_{n = 1}^{\infty}) \int_{0}^{γ} L_{g} H {(\{K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))))\}}_{n = 1}^{\infty} d ζ) d γ \\ + \frac{\propto L_{K}^{2}}{Q (\propto)} \int_{0}^{ϖ} S_{\propto} (ϖ - γ) (L_{B} H ({u_{℘_{n}} (γ)}_{n = 1}^{\infty}) + H ({ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ)))}_{n = 1}^{\infty}) \\ \times \int_{0}^{γ} L_{g} H {(\{K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))))\}}_{n = 1}^{\infty} d ζ) d γ) \\ \leq & [L_{δ} + 2 \{\frac{L_{K} L_{B}}{Q (\propto)} b^{\propto} (\frac{L_{G} (1 - \propto)}{Γ (\propto + 1)} + L_{K} R_{1}) + \frac{L_{K} L_{g}}{Q (\propto) (1 + \propto)} b^{\propto + 1} (\frac{L_{G} (1 - \propto)}{Γ (1 + \propto)} \\ + L_{k} R_{1}) (L_{f_{1}} + L_{f_{2}}) (L_{k_{1}} + L_{k_{2}})\}] H (P) \\ \leq & [L_{δ} + 2 L_{B} (L_{1} + L_{2})] H (P) . \end{aligned}

(19)

For ϖ ∈ (γ_ς, ϖ_ς], ς = 1, 2, 3, …, m, we know that $H (J_{ς} (γ_{ς}, ℘_{γ_{ς}})) \leq L_{J} H (P)$ . By using the above calculation for ϖ ∈ (0, ϖ₁], we can extend the above for ϖ ∈ (γ_ς, ϖ_ς+1], ς = 1, 2, 3, …, m by using (S₇), (16)–(18), and from (12)

\begin{aligned} H ({u_{℘^{n}} (t)}_{n = 1}^{\infty}) = & H (\{£^{- 1} [℘_{ϖ} - K T_{\propto} (b - γ_{ς}) (J_{ς} (γ_{ς}, ℘_{n γ_{ς}}) - δ (γ_{ς}, ℘_{n γ_{ς}})) - δ (b, ℘_{n b}) \\ - \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{b} {(b - γ)}^{\propto - 1} ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ \times \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) d ζ d γ - \frac{\propto K^{2}}{Q (\propto)} \int_{γ_{ς}}^{b} S_{\propto} (b - γ) \\ \times {ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) d ζ d γ\}}_{n = 1}^{\infty}) \\ \leq & 2 (L_{£} L_{K} R (L_{J} + L_{δ}) + L_{δ} + L_{£} L_{1}) H (P) \\ \leq & X H (P), \end{aligned}

(20)

where

X = 2 (L_{£} L_{K} R (L_{J} + L_{δ}) + L_{δ} + L_{£} L_{1})

Hence, by (S₇), (16), (17), and (20), we get

\begin{aligned} H (F (P) (ϖ)) = & H ({F (℘_{n}) (ϖ)}_{n = 1}^{\infty}) \\ = & H (\{K T_{\propto} (ϖ) (b - γ_{ς}) (J_{ς} (γ_{ς}, ℘_{n γ_{ς}}) - δ (γ_{ς}, ℘_{n γ_{ς}})) + δ (ϖ, ℘_{n ϖ}) \\ + \frac{G K (1 - \propto)}{Q (\propto) Γ (\propto)} \int_{γ_{ς}}^{ϖ} {(ϖ - γ)}^{\propto - 1} (B u_{℘_{n}} (γ) + ϝ (γ, ℘_{n γ}, ℘_{n} (p (γ))) \\ \times \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) d ζ) d γ \\ + \frac{\propto K^{2}}{Q (\propto)} \int_{γ_{ς}}^{ϖ} S_{\propto} (ϖ - γ) (B u_{n ℘} (γ) + ϝ (γ, ℘_{n γ}, w p_{n} (p (γ))) \\ {\times \int_{γ_{ς}}^{γ} G (γ, ζ) K (ζ, ℘_{n ζ}, ℘_{n} (p (ζ))) d ζ) d γ\}}_{n = 1}^{\infty}) \\ \leq & (L_{K} R (L_{J} + L_{δ}) + L_{δ} + 2 L_{1} L_{B} X) H (P) . \end{aligned}

(21)

Thus, the set contraction $F$ fulfills all the conditions of 4.2 and (4.3). Therefore, (1) is controllable on $Q$ because $F ℘ (q) = ℘_{1}$ .

5. Example

Consider the following semi-linear impulsive HIDDE for ζ ∈ [0, π],

\begin{aligned} {}^{A B C}D^{\propto} [Q (t, ζ), - \frac{1}{9} Q (t - τ, ζ)] = & \frac{\partial^{2}}{\partial ζ^{2}} [Q (t, ζ) - \frac{1}{9} e^{- t} Q (t - τ, ζ)] + S (t, ζ) \\ + \frac{1}{10} (\frac{e^{- t}}{1 + e^{t}}) (\frac{| Q (t - τ, ζ) |}{1 + | Q (t - τ, ζ) |} i n t_{0}^{t} e^{- 12} (ε + \sin Q (ε)) d ε, \\ t \in (0,1 / 3] \cup (2 / 3, 1] \\ Q (t, 0) & = Q (t, π) = 0, t \in [0,1] \end{aligned}

(22)

Q (t, ζ) = φ (t, ζ), t \in [- ℏ, 0], Q (t, ζ) = \sum_{i = 1}^{2} \frac{1}{4^{i}} Q (t - \frac{1}{3^{i}}, ζ), t \in (1 / 3,2 / 3)

where ∝ ∈ (0, 1), τ > 0 and

φ \in Z = \{φ : [- ℏ, 0] \times [0, π] \to R, φ

is discontinuous on

{}^{'}s^{'}

where

φ (s^{-}) = φ (s)\}

A : H (A) \subset L^{2} [0, π]

defined by

A p = p^{″}, p \in H (A)

with

H (A) = \{p \in N, p, p^{″}

are absolutely continuous,

p^{″} \in N, p (0) = 0 = p (1)\}

. Then

A p = \sum_{ν = 0}^{\infty} ν^{2} (p, p_{ν}) p_{ν}, p \in H (A)

. Here,

p_{ν} (ζ) = \sqrt{2 / π \sin (ν ζ)}, ν \in N

is an orthogonal set of the eigenvector

A

A

is a generator of an analytic semi-group

{T (t)}_{t} \geq 0

N

given by

T (t) p = \sum_{ν = 1}^{\infty} e^{- ν^{2} t} (p, p_{ν}), p \in N, t > 0

R (r, A) = {(r I - A)}^{- 1}

an operator

\forall r \in e (A), {T (t)}_{t} \geq 0

is a uniformly bounded compact semi-group such that

A \in K^{\propto} (ζ_{0}, p_{0})

. Also, the subordination principle of solution operator

{T (t)}_{t} \geq 0

such that

‖ T (t) ‖ \leq R

for t ∈ [0, 1].

Thus, for (t, ζ) ∈ [0, 1] × [0, π], σ ∈ [−ℏ, 0], one can define the following $℘ (t) (ζ) = Q (t, ζ), δ (t, φ) = 1 / 9 e^{- t} φ (σ, ζ), G (γ, ζ) = e^{- 12}, K (t, ℘ (t), ℘ (p (t))) = ε + \sin Q (ε), ϝ (s, ℘ (s), ℘ (p (s))) = 1 / 10 e^{- t} / 1 + e^{t} (ϕ (σ, ζ) / 1 + | ϕ (σ, ζ) |)$ , $J (Q (t, ζ)) = \sum_{ν = 1}^{2} 1 / 4^{ν} Q (t - 1 / 3 ν, ζ), B u (t) (ζ) = S (t, ζ)$ , and $φ (t) (ζ) = Q (t, ζ), t \in [- ℏ, 0]$ .

Based on the above framework, it can be directly checked that the hypotheses S₁–S₇ together with (8) are fulfilled when estimates $L_{f} = 1 / 10, L_{δ} = 1 / 9, L_{J} = 1 / 2$ and L_k = 1 = Lg are used. Consequently, the abstract form of (22) coincides with (1), and therefore (22) is controllable on $Q$ . The visual analysis of the example is shown in Figures 1 and 2.

Figure 1.

The 3D plot illustrates the evolution of u (t, ζ) for (22) governed by impulsive fractional-order PDE. This visualizes the distribution of time-spatial to (22) controllability. In this setup, t ∈ [0, 1] is the time variable, while ζ ∈ [0, π] corresponds to the spatial domain.

Figure 2.

The figure displays the behavior of $Q (t, ζ)$ at different selected moments t = 0, t = 0.17, t = 0.33, t = 0.50, t = 0.67, t = 0.84 and t = 1. These plots describe how the system evolves under the influence of the AB FD, the nonlinear components, the delay term, and the imposed impulsive condition. A noticeable jump appears at t = 0.5, which reflects the action of the impulse defined by $Δ Q = | Q^{-} | / 25 + Q^{-}$ , causing an instantaneous change in the trajectory of the solution.

Given the chosen operators, nonlinear, terms and parameter values, all assumptions (S₁)–(S₇) together with inequality (10) are satisfied. Consequently, (4.3) guaranties that (1) is controllable on $Q$ . Figures 3 and 4 provide a graphical representation of the visual analysis corresponding to the example. The numerical simulation results of the proposed example are visually demonstrated in Figures 5–7.

Figure 3.

The 3D plot of $Q (t, s)$ on [0, 1] × [0, π] show how the $Q (t, s)$ develops in both space and time when governed by ∝ = 0.7, δ = 0.15, and nonlinear interaction terms. The color shading on the surface indicates the amplitude of $Q$ , revealing the intricate, wave-like patterns that naturally arise in systems driven by fractional diffusion effects.

Figure 4.

The spatial distribution of $Q (1, s)$ at the final time $T_{final} = 1$ is plotted over s ∈ [0, π]. This one-dimensional slice highlights the stabilized profile reached by (22), illustrating the long-term behavior shaped by interplay of the fractional time operator, the incorporated delay effects, and the imposed impulsive conditions.

Figure 5.

State trajectory of $S (t, ζ)$ for ∝ = 0.7, illustrating the system’s evolution and the distinct changes produced by the applied impulse actions.

Figure 6.

Breakdown of the system’s behavior into its fundamental components, showing the individual contributions from the linear dynamics, control input, delay effects, and nonlinear impulsive terms.

Figure 7.

Controllability investigation of (22) with impulses, featuring the state trajectory, the corresponding control input, and the associated phase portrait.

6. Conclusion

This paper established results concerning the existence, uniqueness, and controllability of a class of semi-linear impulsive neutral HIDDEs, incorporating non-instantaneous impulses and the AB Caputo FD. To address the complexities introduced by impulsive effects and time delays, the analysis employed conditions formulated using measures of non-compactness. By combining Darbo’s fixed point theorem, the Kuratowski measure of non-compactness, and semi-group theory, controllability criteria were derived. A numerical example was provided to support the theoretical findings and to demonstrate the practical relevance of the results. Future research may extend this framework to stochastic systems influenced by random noise, apply the developed controllability conditions to optimal control problems, adapt the methodology to fractional derivatives of variable order or to problems with general boundary conditions, and construct numerical schemes for efficient simulations. Such extensions would enhance the applicability of the theoretical framework across engineering and the applied sciences.

Footnotes

ORCID iD

Akbar Zada

Author contributions

Akbar Zada: Conceptualization, Methodology, Formal analysis, Investigation, Writing—original draft, and Supervision. Sohail Khan: Conceptualization, Methodology, Formal analysis, Investigation, and Writing–original draft. Sultan Hussain: Conceptualization, Methodology, Formal analysis, and Investigation.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2603).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

No new data were generated in this study. All analysis are based on previously published data, which are cited in the references.*

Appendix

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