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Existence and Approximate Controllability of Random Impulsive 

Neutral Functional Differential Equation with Finite Delay  

 

Tharmalingam Gunasekar1,2, Srinivasan Madhumitha1, Prakaash A. S3, Sakthi R4, Ganapathy 

G5, M. Suba6 

1Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai - 

600062, Tamil Nadu, India. tguna84@gmail.com1,2, smadhumitha2410@gmail.com1 

2School of Artificial Intelligence and Data Science, Indian Institute of Technology (IIT), Jodhpur 342030, India. 

3Department of Mathematics, Panimalar Engineering College Chennai, Tamil Nadu, India, prakaashphd333@gmail.com  

4Department of Science and Humanities, R.M.K. College of Engineering Technology, Puduvoyal - 601 206, Tamil Nadu, 

India, rsakth@gmail.com 

5Department of Mathematics, R.M.D. Engineering College, Kavaraipettai - 601 206, Tamil Nadu, India. 

barathganagandhi@gmail.com 

6Department of Mathematics S.A. Engineering College (Autonomous) Chennai, Tamil Nadu, India. 

suba.hari87@gmail.com 

 

Article History: 

Received: 06-07-2024 

Revised: 21-08-2024 

Accepted: 03-09-2024 

Abstract:  

This study investigates second-order neutral functional differential equations with delays, 

prevalent in various scientific and engineering fields. These equations, characterized by their 

neutral nature and delays, present unique challenges within Banach spaces. The research 

focuses on the existence and approximate controllability of solutions, using advanced 

mathematical tools like cosine family theory and the Leray-Schauder theorem to establish 

rigorous solution conditions. These theoretical results are empirically validated through 

practical examples, enhancing understanding of real-life behavior and bridging theory with 

practice. The study’s findings advance the understanding of delayed feedback systems, 

facilitating effective control strategies and practical engineering solutions, thereby 

contributing significantly to dynamical systems and control theory. 

Keywords: Differential equation; Lerray-Schauder fixed point; mild solution; finite delay; 

semigroup theory; approximate controllability. 

MSC 2010: : 45J05; 34K30; 47G20; 34K20, 93B05. 

 

1  Introduction 

In the area of mathematical analysis and its interdisciplinary applications, a diverse array of theories, 

techniques, and models has emerged to address complex phenomena across various scientific 

domains. This introduction highlights a selection of seminal works and recent research contributions 

that delve into the intricate landscapes of neutral functional differential equations, impulsive systems, 

controllability theories, and interdisciplinary interactions bridging mathematics with physics. N. U. 

Ahmed’s seminal work, "Semigroup Theory with Applications to Systems and Control" [1], serves as 

a cornerstone in understanding the fundamental principles of semigroup theory and its versatile 

applications in systems and control theory. Ahmed’s text provides a comprehensive exploration of 

semigroups, offering insights into their algebraic structures and their pivotal role in analyzing dynamic 

mailto:tguna84@gmail.coma


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systems and control processes. 

The works by Baghli and Benchohra [2] delve into the uniqueness and existence results for partial and 

neutral functional differential equations in Frechet spaces, shedding light on the intricate dynamics of 

these equations with infinite delay. Additionally, Lupulescu and Lungan [5] contribute to the field by 

studying random integral equations on time scales, offering novel perspectives on the interplay 

between randomness and differential equations. Gunasekar et all [12,23,24] explore the existence 

results for nonlocal impulsive neutral functional integro-differential equations, unraveling the 

complexities of impulsive systems with nonlocal interactions. Furthermore, Baleanu et al. investigate 

the approximate controllability of second-order nonlocal impulsive functional integro-differential 

systems in Banach spaces, providing valuable insights into the controllability properties of such 

systems. 

Recent research has also focused on exploring the synergies between physics, mathematics, and 

computer science. Hazra et al. [21] present a modeling framework that elucidates the interdisciplinary 

interactions among these fields, fostering a deeper understanding of complex phenomena. Similarly, 

Han et al. [22] delve into the formation of trade networks, highlighting the role of economies of scale 

and product differentiation in shaping global economic dynamics. 

In mathematical and control theory research, various studies delve into the analysis and controllability 

of complex dynamical systems, aiming to understand their behavior and design effective control 

strategies. The research by Baleanu et al. focuses on the approximate controllability of second-order 

nonlocal impulsive functional integro-differential systems in Banach spaces. Their study investigates 

the ability to steer such systems arbitrarily close to desired states using control inputs. This research 

contributes to understanding the controllability properties of systems with impulsive and nonlocal 

behaviors. 

Anguraj et al. explore the existence results for an impulsive neutral functional differential equation 

with state-dependent delay. By analyzing the existence of solutions to this equation, the study provides 

theoretical insights into the behavior of impulsive systems with state-dependent delays. 

These references collectively contribute to advancing our understanding of the controllability 

properties of complex dynamical systems, particularly those involving impulses, delays, and nonlocal 

interactions. They provide valuable theoretical insights and mathematical techniques for analyzing and 

designing control strategies for such systems, with implications for various scientific and engineering 

applications. 

The Second order impulsive neutral functional differential equation with delay and random effects is 

of the form.  

𝑑

𝑑ℎ
[𝜑′(ℎ, ℵ) + 𝜌(ℎ, 𝜑ℎ(. , ℵ), ℵ)] = 𝐴𝜑(ℎ, ℵ) + Υ(ℎ,𝜛, 𝜑ℎ(. , ℵ), ℵ);   ℎ ∈ 𝐽 = (0, 𝜚], ℎ ≠ ℎ𝜉 ,

                                              𝜑(0, ℵ) = 𝜙0(ℵ), 𝜉 = 1,2,3, . . . , 𝑚  

                                             𝜑′(0, ℵ) = 𝜙′
0
(ℵ),

                                          Δ𝜑(ℎ𝜉 , ℵ) = 𝐼𝜉 (𝜑(ℎ𝜉 , ℵ)) ,

                                         Δ′𝜑(ℎ𝜉 , ℵ) = 𝐼𝜉′ (𝜑(ℎ𝜉 , ℵ)) .

(1)                                       



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The approximate controllability of random impulsive neutral functional differential equation with 

finite delay. 

 

𝑑

𝑑ℎ
[𝜑′(ℎ, ℵ) + 𝜌(ℎ, 𝜑ℎ(. , ℵ), ℵ)] = 𝐴𝜑(ℎ, ℵ) + Υ(h, 𝜑ℎ(. , ℵ), ℵ) + 𝐵𝑦(ℎ, ℵ);    ℎ ∈ 𝐽

                                    𝜑(0, ℵ)          = 𝜙0(ℵ),

                                    𝜑′(0, ℵ)         = 𝜙′0(ℵ),

                                 Δ𝜑(ℎ𝜉 , ℵ)        = 𝐼𝜉(𝜑(ℎ𝜉 , ℵ)),

                                Δ′𝜑(ℎ𝜉 , ℵ)        = 𝐼𝜉′(𝜑(ℎ𝜉 , ℵ)).
  

 (2) 

𝐴 symbolizes the infinitesimal source of a constantly evolving set of cosine transformations denoted 

by {𝑇1(ℎ): ℎ ∈ ℝ}, where these transformations are bounded linear operations occurring within a 

Banach Space 𝒮 , defined with the norm ||. ||  and Υ: 𝐽 × 𝐽 × 𝒟 × Ω → 𝒮, 𝜌: 𝐽 ∗ 𝒟 × Ω → 𝒮  are 

continuous functions and 𝐵: ℰ → ℝ+ where ℰ is the banach space and Ω is a random operator in a 

stochastic domain. 

2  Preliminaries 

In this section, we’ll review fundamental concepts and terminology that are essential for understanding 

the key findings of our study. Lately, there’s been more interest in studying a specific type of problem 

involving how things change over time, even when the speed of change isn’t fixed. 

 𝜑′′(ℎ, ℵ) = 𝐴𝜑(ℎ, ℵ) + Υ(ℎ, ℵ),    0 ≤ ℎ ≤ 𝜚    𝜑(0, ℵ) = 𝑥0(ℵ),    𝜑′(0, ℵ) = 𝑦0(ℵ) (3) 

Here, 𝐴:𝐷(𝐴) ⊆ 𝒮 → 𝒮, where ℎ ∈ 𝐽 = [0, 𝜚], denotes a closed operator that is densely defined. 

Furthermore, let Υ: 𝐽 × Ω → 𝒮  denote an appropriate function. Numerous studies have examined 

equations of this nature. Typically, the solutions of the problem is linked to the presence of an 

evolution operator 𝑇2(ℎ,𝜛) for the corresponding homogeneous equation.  

 𝜑′′(ℎ, ℵ) = 𝐴𝜑(ℎ, ℵ),    0 ≤ 𝜛, ℎ ≤ 𝜚, (4) 

Definition 1 Let (𝒟, ∥⋅∥𝒟) be a seminormed linear space of functions defined on (−δ, 0] and taking 

values in a Banach space 𝒮. The space 𝒟 satisfies the following axioms: 

(A)For any continuous function φ: (−δ, 0] → 𝒮 and ϕ0 ∈ 𝒟, the following conditions hold for all 

h ∈ J   

    1.  The function φh ∈ 𝒟.  

    2.  There exists a positive constant K such that  

 |φ(h, ℵ)| ≤ K ∥ φh(⋅, ℵ) ∥𝒟. 

Furthermore, there exist functions U, ϑ, ϑ′:ℝ+ → ℝ+, where U is continuous and bounded, and ϑ, ϑ′ 

are locally bounded and independent of φ, such that  

 ∥ φh(⋅, ℵ) ∥X≤ U(h)sup{|φ(m, ℵ)|:−δ ≤ m ≤ 0} + ϑ ∥ ϕ0(ℵ) ∥𝒟+ ϑ′ ∥ ϕ0′(ℵ) ∥𝒟. 

(B)The function φh is 𝒟-valued and continuous on J for the functions φ described in (A). 

(C)The space 𝒟 is complete.  



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Definition 2 A collection of bounded linear maps {T1(h): h ∈ 𝒥}  in the Banach space 𝒮  is 

considered a strongly continuous cosine function when it meets these criteria   

    1.  Addition Condition: T1(ϖ + h) + T1(ϖ − h) = 2T1(ϖ)T1(h) for all ϖ, h ∈ 𝒥.  

    2.  Identity Property: T1(0) = I, where I denotes the identity operator.  

    3.  Continuity Requirement: T1(h)φ remains continuously dependent on h over 𝒥 for any fixed 

φ ∈ 𝒮.  

In this scenario, A serves as the fundamental element behind a continuously evolving set of operations 

known as the strongly continuous cosine function, denoted by {T1(h): h ∈ 𝒥}. These operations 

involve bounded linear maps defined within the Banach space 𝒮, where distances are measured using 

the norm ∥⋅∥ . The associated sine function with {T1(h): h ∈ ℝ} , denoted as {T2(h): h ∈ 𝒥} , is 

expressed as  

 T2(h)φ = ∫
h

0
T1(ϖ)φ dϖ    forφ ∈ 𝒟andh ∈ 𝒥. 

Additionally, ϑ and ϑa represent positive constants ensuring ∥ T1(h) ∥≤ ϑ and ∥ T2(h) ∥≤ ϑa for 

every h ∈ J. 

Definition 3 Approximate controllability, an essential concept in control theory, addresses the 

capability to roughly guide a system from one state to another utilizing control inputs within a 

designated timeframe. 

Formally, system represented by a state space 𝒟 , where we can influence its behavior through 

admissible control inputs from the space U. The evolution of this system is described by an equation: 

 φ′(h, ℵ) = Aφ(h, ℵ) + By(h, ℵ), 

where φ(h, ℵ) ∈ 𝒟 denotes the system’s state at time h, y(h, ℵ) ∈ U denotes the control input, A is 

the system’s operator or matrix, and B is the control operator or matrix. 

Approximate controllability is achieved if, given any starting state x0 and any desired terminal state 

xf, there is a sequence of control inputs {φξ(h, ℵ)} such that the system’s solution ζ(h, ℵ) of the 

dynamical system satisfies x(0, ℵ) = ϕ0(ℵ) and limξ→∞xξ(ϱ, ℵ) = ϕ′0(ℵ) for some finite time ϱ, 

where xξ(h, ℵ) is the system"s solution resulting from the control input φξ(h, ℵ). 

Lemma 1 (Leray-Schauder Nonlinear Alternative) 

Let us denote a Banach space 𝒮. Inside 𝒮, there’s a closed and convex subset Z. Within Z, there’s a 

relatively open subset U containing the point 0. Then, there’s a mapping Υ: U → Z that’s compact, 

meaning it preserves the "closeness" of points when mapping from U to Z. In that case, either   

    1.  Υ possesses a fixed point in U, or  

    2.  A point ζ ∈ ∂U satisfies ζ ∈ λΥ(ζ) for some λ ∈ (0,1).  

Lemma 2 A set 𝒟 ⊂ 𝒮 is relatively compact in 𝒮 if and only 𝒟ξ is relatively compact in 

C[(hξ, hξ+1]; 𝒮) for each ξ = 0,1, … , n. 



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Now, let’s discuss how we can determine if the equations are approximately controllable within their 

interior, without involving impulses, delays, or nonlocal conditions. To do this, we consider the 

following scenario: For any starting point φ0 within our space 𝒮 and any function y belonging to the 

L2 space over the interval (0, ϱ] with values in U, we examine the initial-value problem:  

 φ′(h, ℵ) = Aφ(h, ℵ) + By(h, ℵ),    φ ∈ 𝒮,        φ(0, ℵ)) = φ0(ℵ), (5) 

where the control function φ belongs to L2(0, ϱ; U), has precisely one mild solution represented by  

 φ(h, ℵ) = T(h)φ0(ℵ) + ∫
h

0
T(h − ϖ)By(ϖ, ℵ) dϖ,    h ∈ (0, ϱ]. 

Definition 4 For the above system , the controllability mapping G: L2((0, ϱ]; U) → 𝒮 is defined for 

h > 0 as follows:  

 Gu = ∫
h

0
T(h − ϖ)By(ϖ, ℵ) dϖ. 

The corresponding adjoint operator G∗: 𝒮 → L2((0, ϱ]; 𝒮) is determined by the rule  

 (G∗φ)(ϖ) = B∗T∗(ϱ − ϖ)φ    ∀ϖ ∈ [0, ϱ], ∀z ∈ 𝒮. 

Consequently, the Grammian operator W:𝒮 → 𝒮 is  

 kφ = GG∗φ = ∫
τ

0
T(ϱ − ϖ)BB∗T∗(ϱ − ϖ) dϖ. 

Remark 1 The series of linear operators (Γ(ℵ))α: 𝒮 → L2((0, ϱ]; U) , where 0 < α ≤ 1 , can be 

defined as follows:  

 (Γ(ℵ))αφ = B
∗T∗(⋅)(αI + GG∗)−1φ = G∗(αI + GG∗)−1φ,    (3.6) 

This set of operators fulfills the condition:  

 lim
α→0

G(Γ(ℵ))α = I, 

in the strong topology. 

3  Existence Results 

In this section, we show that there are solutions to the problem described by equations (1.1). To do this, 

we list some conditions we’ll need to consider. 

Definition 5  If φ0 = ∅  and the continuous function x: (0, ϱ] × Ω → 𝒮, T > 0  and 𝒟 =

C[(−δ, ϱ], 𝒮] solves the integral equation then it is considered a mild solution to equation (1.1). 

 

ζ(h, ℵ) = T1(h)ϕ0(ℵ) + T2(h)[ϕ′0(ℵ) + ρ(0, ϕ0(ℵ), ℵ)] − ∫
h

0
T1(h − ϖ)ρ(ϖ,φϖ(. , ℵ), ℵ)dϖ

 +∫
h

0
T2(h − ϖ)Υ(ϖ,φϖ(. , ℵ), ℵ)dϖ + ∑0<hξ<h T1(h − hξ)Iξ(φ(hξ, ℵ))

 +∑0<hξ<h T2(h − hξ)I′ξ(φ(hξ, ℵ)).

 

For your convenience, we have listed the hypotheses that will be discussed in the following section. 

(G1)There exist a continuous function a0, b0, c0, d0: J × Ω → ℝ such that  

 ||Υ(h, x, ℵ)|| ≤ a0(ℵ)||x, ℵ||𝒟
α0 + b0(ℵ) 



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 ||ρ(h, x, ℵ)|| ≤ c0(ℵ)||x, ℵ||𝒟
β0 + d0(ℵ) 

for all x ∈ 𝒮, ℵ ∈ Ω 

(G2)(i)For all h,ϖ ∈ J, the function Υ(h, . , . ): 𝒟 × Ω → 𝒮 is continuous and for all (x, ℵ) ∈ 𝒟 × Ω 

the function Υ(. , x, ℵ): J → 𝒮 is strongly measurable. 

(ii) For all h ∈ J, the function Υ(h, . , . ): 𝒟 × Ω → 𝒮  is continuous and for all (x, ℵ) ∈ 𝒟 × Ω the 

function Υ(. , x, ℵ): J → 𝒮 is strongly measurable. 

(G4)Let Iξ, I′ξ ∈ C(𝒮, 𝒮), ξ = 1,2,3, . . . , m are all compact operator 

 ||Iξ(h, x, ℵ)|| ≤ aξ(ℵ)||x, ℵ||ℝ
αξ

 

 ||I′ξ(h, x, ℵ)|| ≤ a′ξ(ℵ)||x, ℵ||ℝ
αξ

 

(G5)The function Υ: J × J × 𝒟 × Ω → 𝒮, ρ: J × 𝒟 × Ω → 𝒮 is continuous, and a constant ℒ exists for 

which 

 ||Υ(h, x1, ℵ) − Υ(h, x2, ℵ)|| ≤ ℒ(ℵ)(||(x1, ℵ) − (x2, ℵ)||𝒟
α0) 

 ||ρ(h, x1, ℵ) − ρ(h, x2, ℵ)|| ≤ ℒ1(ℵ)(||(x1, ℵ) − (x2, ℵ)||𝒟
β0) 

(G6) There is a random funtion R:Ω → ℝ+ such that  

 ϑ||ϕ0|| + ϑa||ϕ′0(ℵ) + ρ(0, ϕ0(ℵ), ℵ)|| + ϑahb0(ℵ) + ϑhd0(ℵ) +

ϑ∑0<hξ<h aξ(ℵ)||ζ(hξ, ℵ)||ℝ
αξ + ϑa∑0<hξ<h a′ξ(ℵ)||ζ(hξ, ℵ)||ℝ

αξ +

ϑa ∫
h

0
||[ sup

ϖ∈(0.ϱ]
||ζϖ(. , ℵ), ℵ||𝒟

α0dϖ + ϑ∫
h

0
||[ sup

ϖ∈(0.ϱ]
||ζϖ(. , ℵ), ℵ||𝒟

β0dϖ ≤ R(ℵ). 

Theorem 3.1  Assuming that conditions (G1)-(G6) are met, the problem described in (1.1) will have a 

mild random solution on the interval (0, ϱ].  

 Proof: Let us consider a map (Γ(ℵ)): Ω × PCδ = [(−δ, ϱ], 𝒮] → PCδ  be a random operator is 

defined by (Γ(ℵ))ζ(h) where h ∈ (−δ, ϱ]  

 (Γ(ℵ))ζ(h, ℵ) =

{
 
 

 
 
Υ(h, ℵ), h ∈ (−δ, ϱ]

T1(h)ϕ0(ℵ) + T2(h)[ϕ′0(ℵ) + ρ(0, ϕ0(ℵ), ℵ)]  

− ∫
h

0
T1(h − ϖ)ρ(ϖ, ζϖ(. , ℵ), ℵ)dϖ + ∫

h

0
T2(h − ϖ)Υ(ϖ, ζϖ(. , ℵ), ℵ)dϖ  

+∑0<hξ<h T1(h − hξ)Iξ(ζ(hξ, ℵ)) + ∑0<hξ<h T2(h − hξ)I′ξ(ζ(hξ, ℵ)), h, ϖ ∈ J

 (6) 

We aim to illustrate that (Γ(ℵ)) satisfies all conditions outlined in Lemma 2.1. To enhance clarity, the 

proof will be broken down into many stages. 

Step 1: The mapping (Γ(ℵ)) takes sets that are bounded and maps them into other bounded sets. 

To demonstrate this, it’s sufficient to establish that there is a +ve constant r(ℵ) so that for every ζ ∈

ℬr(δ), where δ is defined as follows: 



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 ℬr(δ): = {ζ ∈ PCδ: sup
δ≤h≤ϱ

∥ ζ(h, ℵ) ∥≤ r(ℵ)} 

one has ∥ (Γ(ℵ))ζ ∥PC≤ R(ℵ). 

||(Γ(ℵ))𝜁(ℎ)|| ≤ ||𝑇1(ℎ)𝜙0(ℵ)|| + ||𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]||

 +∫
ℎ

0

||𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛|| + ||∫
ℎ

0

𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)||𝑑𝜛

 +|| ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))|| + || ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))||𝑑𝜛

 ≤ 𝜗||𝜙0(ℵ)|| + 𝜗𝑎||𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)|| + 𝜗∫
ℎ

0

||𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛||

 +𝜗𝑎∫
ℎ

0

||Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)||𝑑𝜛 + 𝜗 ∑

0<ℎ𝜉<ℎ

||𝐼𝜉(𝜁(ℎ𝜉 , ℵ))|| + 𝜗𝑎 ∑

0<ℎ𝜉<ℎ

||𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))||

 ≤ 𝜗||𝜙0|| + 𝜗𝑎||𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)|| + 𝜗∫
ℎ

0

[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛽0 + 𝑑0(ℵ)]𝑑𝜛

 +𝜗𝑎∫
ℎ

0

[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛼0 + 𝑏0(ℵ)]𝑑𝜛 + 𝜗 ∑

0<ℎ𝜉<ℎ

𝑎𝜉(ℵ)||𝜁(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉

 +𝜗𝑎 ∑

0<ℎ𝜉<ℎ

𝑎′𝜉(ℵ)||𝜁(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉

  

 

 ≤ 𝜗||𝜙0|| + 𝜗𝑎||𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)|| + 𝜗𝑎ℎ𝑏0(ℵ) + 𝜗ℎ𝑑0(ℵ) + 𝜗 ∑

0<ℎ𝜉<ℎ

𝑎𝜉(ℵ)||𝜁(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉

 +𝜗𝑎 ∑

0<ℎ𝜉<ℎ

𝑎′𝜉(ℵ)||𝜁(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉 + 𝜗𝑎∫

ℎ

0

||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛼0𝑑𝜛 +

 𝜗∫
ℎ

0

||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛽0𝑑𝜛

 ≤ 𝑅(ℵ)

 

 Hence (Γ(ℵ)) is bounded set in 𝑃𝐶𝛿. 

Step 2:We now show that (Γ(ℵ)) is continuous on ℬ𝑟(𝛿). 

Let us consider that for 𝜁1, 𝜁2 ∈ ℬ𝑟(𝛿), ℎ ∈ 𝐽, 



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||(Γ(ℵ))𝜁1(ℎ) − (Γ(ℵ))𝜁2(ℎ)|| ≤ ||∫
ℎ

0

𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜁1,𝜛(. , ℵ), ℵ) − 𝜌(𝜛, 𝜁2,𝜛(. , ℵ), ℵ)||𝑑𝜛

 +||∫
ℎ

0

𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁1,𝜛(. , ℵ), ℵ) − Υ(𝜛, 𝜁2,𝜛(. , ℵ), ℵ)||𝑑𝜛

 +|| ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)[𝐼𝜉(𝜁1(ℎ𝜉 , ℵ)) − 𝐼𝜉(𝜁2(ℎ𝜉 , ℵ))]||

 +|| ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)[𝐼′𝜉(𝜁1(ℎ𝜉 , ℵ)) − 𝐼′𝜉(𝜁2(ℎ𝜉 , ℵ))]||𝑑𝜛

 ≤ 𝜗∫
ℎ

0

||𝜌(𝜛, 𝜁1,𝜛(. , ℵ), ℵ) − 𝜌(𝜛, 𝜁2,𝜛(. , ℵ), ℵ)||𝑑𝜛

 +𝜗𝑎∫
ℎ

0

||Υ(𝜛, 𝜁1,𝜛(. , ℵ), ℵ) − Υ(𝜛, 𝜁2,𝜛(. , ℵ), ℵ)||𝑑𝜛

 +𝜗 ∑

0<ℎ𝜉<ℎ

||𝐼𝜉(𝜁1(ℎ𝜉 , ℵ)) − 𝐼𝜉(𝜁2(ℎ𝜉 , ℵ))|| +

 𝜗𝑎 ∑

0<ℎ𝜉<ℎ

||𝐼′𝜉(𝜁1(ℎ𝜉 , ℵ)) − 𝐼′𝜉(𝜁2(ℎ𝜉 , ℵ))||

 ≤ 𝜗∫
ℎ

0

sup
𝜛∈(0.𝜚]

||(𝜁1,𝜛(. , ℵ), ℵ) − (𝜁2,𝜛(. , ℵ), ℵ)||𝒟
𝛽0𝑑𝜛

 +𝜗𝑎∫
ℎ

0

sup
𝜛∈(0.𝜚]

||(𝜁1,𝜛(. , ℵ), ℵ) − (𝜁2,𝜛(. , ℵ), ℵ)||𝒟
𝛼0𝑑𝜛

 +𝜗 ∑

0<ℎ𝜉<ℎ

𝑎𝜉(ℵ)||𝜁1(ℎ𝜉 , ℵ) − 𝜁2(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉

 +𝜗𝑎 ∑

0<ℎ𝜉<ℎ

𝑎′𝜉(ℵ)||𝜁1(ℎ𝜉 , ℵ) − 𝜁2(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉

  

 

  

||(Γ(ℵ))𝜁1(ℎ) − (Γ(ℵ))𝜁2(ℎ)|| ≤ 𝜗 ∑0<ℎ𝜉<ℎ 𝑎𝜉(ℵ)||𝜁1(ℎ𝜉 , ℵ) − 𝜁2(ℎ𝜉 , ℵ)||ℝ
𝛽𝜉

 +𝜗 ∫
ℎ

0
sup

𝜛∈(0.𝜚]
||(𝜁1,𝜛(. , ℵ), ℵ) − (𝜁2,𝜛(. , ℵ), ℵ)||𝒟

𝛽0𝑑𝜛

 +𝜗𝑎 ∑0<ℎ𝜉<ℎ 𝑎′𝜉(ℵ)||𝜁1(ℎ𝜉 , ℵ) − 𝜁2(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉

 +𝜗𝑎 ∫
ℎ

0
sup

𝜛∈(0.𝜚]
||(𝜁1,𝜛(. , ℵ), ℵ) − (𝜁2,𝜛(. , ℵ), ℵ)||ℝ

𝛼0𝑑𝜛

 

for all ℎ ∈ (−𝛿, 𝜚] , and their compactness for ℎ > 0  proves the uniform operator topology is 

continuous. Since 𝜁1, 𝜁2 ∈ ℬ𝑟(𝛿) , the righthand side of the above inequalities are 

independent.Therefore ||((Γ(ℵ))𝜁1)(ℎ) − ((Γ(ℵ))𝜁2)(ℎ)|| → 0 as (𝜁1 − 𝜁2) → 0. Hence (Γ(ℵ)) is 

continuous. 

Step 3: The operator (Γ(ℵ)) is compact. 

To establish this, we decompose (Γ(ℵ)) into (Γ1(ℵ)) + (Γ(ℵ))2, where both (Γ1(ℵ)) and (Γ2(ℵ)) 



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are operators acting on ℬ𝑟(𝛿). Specifically, they are characterized as follows 

(Γ1(ℵ))𝜁(ℎ) = 𝑇1(ℎ)𝜙0(ℵ) + 𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] − ∫
ℎ

0

𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 +∫
ℎ

0

𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛   and

(Γ2(ℵ))𝜁(ℎ) = ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ)) + ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ)),   forall   ℎ ∈ (−𝛿, 𝜚].

 

We will begin by demonstrating that (Γ1(ℵ)) is a compact operator. 

(i)The set (Γ1(ℵ))(ℬ𝑟(𝛿)) exhibits equicontinuity. 

Now, consider 𝛿 ≤ ℎ1 < ℎ2 ≤ 𝜚 and let 𝜖 > 0 be small. then 



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||(Γ1(ℵ))𝜁(ℎ2) − (Γ1(ℵ))𝜁(ℎ1)|| ≤ ||[𝑇1(ℎ2) − 𝑇1(ℎ2)]𝜙0(ℵ)|| + ||[𝑇2(ℎ2) − 𝑇2(ℎ1)][, ℵ)]||

 +||∫
ℎ1

0

𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛 −∫
ℎ2

0

𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛||

 +||∫
ℎ1

0

𝑇2(ℎ2 −𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ) − ∫
ℎ2

0

𝑇2(ℎ1 −𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)||𝑑𝜛

 ≤ ||[𝑇1(ℎ2) − 𝑇1(ℎ2)]𝜙0(ℵ)|| + ||[𝑇2(ℎ2) − 𝑇2(ℎ1)][𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]||

 +||∫
ℎ1−𝜖

0

(𝑇1(ℎ2 −𝜛) − 𝑇1(ℎ1 −𝜛))𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛||

 +||∫
ℎ1−𝜖

0

(𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛))Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛||

 +||∫
ℎ1

ℎ1−𝜖

𝑇1(ℎ2 −𝜛) − 𝑇1(ℎ1 −𝜛))𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛||

 +||∫
ℎ1

ℎ1−𝜖

𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛))Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛||

 +||∫
ℎ2

ℎ1

𝑇1(ℎ2 −𝜛) − 𝑇1(ℎ1 −𝜛))𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)||𝑑𝜛

 +||∫
ℎ2

ℎ1

𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛))Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)||𝑑𝜛

 ≤ ||[𝑇1(ℎ2) − 𝑇1(ℎ2)]𝜙0(ℵ)|| + ||[𝑇2(ℎ2) − 𝑇2(ℎ1)][𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]||

 +∫
ℎ1−𝜖

0

||𝑇1(ℎ2 −𝜛) − 𝑇1(ℎ1 −𝜛)||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛽0 + 𝑑0(ℵ)]𝑑𝜛

 +∫
ℎ1−𝜖

0

||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛼0 + 𝑏0(ℵ)]𝑑𝜛

 +∫
ℎ1

ℎ1−𝜖

||𝑇1(ℎ2 −𝜛) − 𝑇1(ℎ1 −𝜛)||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛽0 + 𝑑0(ℵ)]𝑑𝜛

 +∫
ℎ1

ℎ1−𝜖

||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛼0 + 𝑏0(ℵ)]𝑑𝜛

 +∫
ℎ2

ℎ1

||𝑇1(ℎ2 −𝜛) − 𝑇1(ℎ1 −𝜛)||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛽0 + 𝑑0(ℵ)]𝑑𝜛

 +∫
ℎ2

ℎ1

||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)||[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛼0 + 𝑏0(ℵ)]𝑑𝜛

 

We observe that as ℎ2 − ℎ1  approaches zero, | |(Γ1(ℵ))𝜁(ℎ2) − (Γ(ℵ))𝜁(ℎ1)| |  tends to zero 

regardless of 𝜁 ∈ ℬ𝑟(𝛿). Because the operator 𝑇2(ℎ) is compact for ℎ > 0, it ensures continuity in 

the uniform operator topology. As a result, (Γ1(ℵ)) transforms ℬ𝑟(𝛿) into a family of functions that 

are equicontinuous. 

Next, we need to demonstrate that the set (Γ1(ℵ))(ℬ𝑟(𝛿))(ℎ) is precompact within 𝒮. 

Consider fixed values 𝛿 < ℎ ≤ 𝜛 ≤ 𝜚, and let 𝜖 be a real number such that 0 < 𝜖 < ℎ. For 𝜁 ∈



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𝐵𝑟(𝛿), ((Γ1(ℵ)), 𝜖)(ℎ) is given by 

 𝑇1(ℎ)𝜙0(ℵ) + 𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] − ∫
ℎ−𝜖

0

𝑇2(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 +∫
ℎ−𝜖

0

𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 

Utilizing the compactness property of 𝑇2(ℎ)  for ℎ > 0 , we establish that the set 

{((Γ(ℵ))1,𝜖𝜁)(ℎ): 𝜁 ∈ 𝐵𝑟(𝛿)} is precompact for 𝜁 ∈ 𝐵𝑟(𝛿) and 0 < 𝜖 < ℎ. Additionally, for each 

𝜁 ∈ 𝐵𝑟(𝛿), we ensure that  

 

||((Γ1(ℵ))𝜁)(ℎ) − ((Γ(ℵ))1,𝜖𝜁)(ℎ)|| ≤ ∫
ℎ

ℎ−𝜖
||𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜂, 𝜁𝜛(. , ℵ), ℵ)||

 + ∫
ℎ

ℎ−𝜖
||𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)||

                                                                     ≤ 𝜗 ∫
ℎ

ℎ−𝜖
[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛽0 + 𝑑0(ℵ)]𝑑𝜛

 +𝜗𝑎 ∫
ℎ

ℎ−𝜖
[ sup
𝜛∈(0.𝜚]

||𝜁𝜛(. , ℵ), ℵ||𝒟
𝛼0 + 𝑏0(ℵ)]𝑑𝜛

 

Thus, there is a precompact sets that can be made arbitrarily close to the set {((Γ1(ℵ))𝜁): 𝜁 ∈ 𝐵𝑟(𝛿)}. 

Therefore, the set {((Γ1(ℵ))𝜁): 𝜁 ∈ 𝐵𝑟(𝛿)}  is relatively compact in 𝒮 . It’s evident that 

(Γ1(ℵ))(𝐵𝑟(𝛿))  is bounded uniformly.Since we have established that (Γ1(ℵ))(𝐵𝑟(𝛿))  forms an 

equicontinuous family, the Arzelà-Ascoli theorem indicates that it is sufficient to show that (Γ1(ℵ)) 

maps 𝐵𝑟(𝛿) into a relatively compact set in 𝒮. 

Now, we need to confirm that (Γ(ℵ))2 is a compact operator as well. By applying Lemma 2.1, we 

establish its complete continuity. The property of (Γ(ℵ))2 being continuous can be demonstrated by 

considering the state space. Conversely, for 𝑟 > 0, ℎ ∈ (ℎ𝜉 , ℎ𝜉+1] ∩ (0, 𝜚], 𝑖 ≥ 1, and 𝜁 ∈ ℬ𝑟 =

ℬ𝑟(0, ℬ𝑟(𝛿)), we observe that 

 (Γ(ℵ))𝜁(ℎ) ∈

{
 
 

 
 ∑

𝜉
𝑗=1 𝑇(ℎ − ℎ𝑗)𝐼𝑗(ℬ𝑟∗(0, 𝒮)), ℎ ∈ (ℎ𝜉 , ℎ𝜉+1),

∑𝜉𝑗=0 𝑇(ℎ𝜉+1 − ℎ𝑗)𝐼𝑗(ℬ𝑟∗(0, 𝒮)), ℎ = ℎ𝜉+1,

∑𝜉𝑗=0 𝑇(ℎ𝜉 − ℎ𝑗)𝐼𝑗(ℬ𝑟∗(0, 𝒮)) + 𝐼𝜉(ℬ𝑟∗(0, 𝒮)), ℎ = ℎ𝜉

 (7) 

This demonstrates that [(Γ(ℵ))2(𝐵𝑟)]𝜉(ℎ) is relatively compact in 𝒮 for each ℎ ∈ [ℎ𝜉 , ℎ𝜉+1], as the 

maps 𝐼𝑗 are completely continuous. Additionally, by leveraging the compactness of the operators 𝐼𝜉  

along with the strong continuity of (𝑇(ℎ))𝑡0 , we can show that [(Γ(ℵ))2(𝐵𝑟)]𝜉  is uniformly 

continuous at ℎ for every ℎ ∈ [ℎ𝜉 , ℎ𝜉+1] and for each 𝜉 = 1,2, … , 𝑛. Therefore, according to Lemma 

2.2, (Γ(ℵ))2 is completely continuous. 

Step 4:  Certainly, Our goal is to identify an open set 𝑈 ⊆ 𝑃𝐶𝛿 such that for any point 𝜁 lying on the 

boundary of 𝑈, it won’h be in the set 𝜆(Γ(ℵ))(𝜁) for 𝜆 ∈ (0,1). Therefore, for every ℎ ∈ (0, 𝜚],  

 



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(Γ(ℵ))𝜁(ℎ) = 𝜆𝜁(ℎ, ℵ)

                      = 𝜆𝑇1(ℎ)𝜙0(ℵ) + 𝜆𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] − 𝜆∫
ℎ−𝜖

0

𝑇2(ℎ)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 +𝜆∫
ℎ

0

𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛 + 𝜆 ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))

 +𝜆 ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))

 

for each ℎ ∈ (0, 𝜚], we have ||𝜁(ℎ, ℵ)|| ≤ ||(Γ(ℵ))𝜁(ℎ)|| and 

||(Γ(ℵ))𝜁(ℎ)|| ≤ ||𝑇1(ℎ)𝜙0(ℵ)|| + ||𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]||

 +∫
ℎ

0

||𝑇2(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)||𝑑𝜛 +∫
ℎ

0

||𝑇2(ℎ − 𝜛)Υ(Π, 𝜁𝜛(. , ℵ), ℵ)||𝑑𝜛

 +|| ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))|| + || ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))||

 

By step 1, ||(Γ(ℵ))𝜁(ℎ)|| ≤ 𝑅(ℵ) We can find a constant 𝑅(ℵ) such that ∥ 𝜁 ∥𝑃𝐶≠ 𝑅(ℵ). Set  

 𝑈 = {𝜁 ∈ 𝑃𝐶([𝛿, 𝜚], 𝒮)  |  sup
𝛿≤ℎ≤𝜚

∥ 𝜁(ℎ, ℵ) ∥< 𝑅(ℵ)} 

The results obtained from Steps 1-3 in Theorem 3.1 imply that it’s enough to show that (Γ(ℵ)):𝑈 →

𝑃𝐶𝛿 is a compact mapping. With the selection of 𝑈, no 𝜑 ∈ 𝜕𝑈 exists for which 𝜁 ∈ 𝜆(Γ(ℵ))(𝜁) for 

𝜆 ∈ (0,1). Based on Lemma 2.1, we assume that the operator (Γ(ℵ)) has a fixed point 𝜁∗ ∈ 𝑈.. Thus, 

we obtain  

𝜁∗(ℎ, ℵ) = 𝑇1(ℎ)𝜙0(ℵ) + 𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] + ∫
ℎ

0
𝑇2(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛

∗ (. , ℵ), ℵ)𝑑𝜛

 +∫
ℎ

0
𝑇2(ℎ − 𝜛)Υ(Π, 𝜁𝜛

∗ (. , ℵ), ℵ)𝑑𝜛 + ∑0<ℎ𝜉<ℎ 𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))

 +∑0<ℎ𝜉<ℎ 𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁
∗(ℎ𝜉 , ℵ))

(8) 

 This suggests that 𝜁∗(ℎ, ℵ) possesses a fixed point and serves as a mild solution to problem (1.1). 

This concludes the proof of the theorem. 

4  Approximate Contollability of Random Neutral Functional Differential Equation 

Definition 6 The problem (1.2) is controllable on the interval (0, ϱ] if, for any given final state ζ1(ℵ), 

there is a control y(h, ℵ) in L2(J, Ω) such that the solution ζ(h, ℵ) of (1.2) reaches ζ1(ℵ) at time ϱ.  

 We now present our primary existence result regarding problem (1.2). The definition of a mild 

random solution comes first. 

If ζ0 = ∅ and the continuous function ζ: PC(J, 𝒮) × Ω → PC(J, 𝒮) and 𝒟 = [(−δ, ϱ], 𝒮] solves the 

integral equation then it is referred to as a mild solution to equation (1.1).  

Definition 7  A function ζ(⋅, ℵ) ∈ PC(J, 𝒮) is considered a mild solution of problem (1.2) with initial 

conditions if it satisfies the following integral equation. 



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Vol 32 No. 1 (2025) 

 

47 https://internationalpubls.com 

 

ζ(h, ℵ) = T1(h)ϕ0(ℵ) + T2(h)[ϕ′0(ℵ) + ρ(0, ϕ0(ℵ), ℵ)] − ∫
h

0
T2(h − ϖ)ρ(ϖ, ζϖ(. , ℵ), ℵ)dϖ

 +∫
h

0
T2(h − ϖ)[Υ(ϖ, ζϖ(. , ℵ), ℵ) + By(h, ℵ)]dϖ

 +∑0<hξ<h T1(h − hξ)Iξ(ζ(hξ, ℵ)) + ∑0<hξ<h T2(h − hξ)I′ξ(ζ(hξ, ℵ))

 

 For your convenience, we have listed the additional hypotheses that will be discussed in the following 

section. 

Let 

( G7 )The linear operator k: L2(J, 𝒮) → 𝒮  given by ky = ∫
ϱ

0
T2(ϱ − ϖ)By(ϖ, ℵ)dϖ  has a 

pseudo-inverse operator k−1 in L2(J, S)/kerk  

(G8) There exist a random function Q:Ω → ℝ+ where  

 

 ϑaBk
−1 ∫

h

0
[||ζ1(ℵ)|| + ϑ||ϕ0(ℵ)|| + ϑa||ϕ′0(ℵ) + ρ(0, ϕ0(ℵ), ℵ)||

 +ϑ∫
ϱ

0
[ sup
η∈(0.ϱ]

||ζη(. , ℵ), ℵ||𝒟
β0 + d0(ℵ)]dη + ϑa ∫

ϱ

0
[ sup
η∈(0.ϱ]

||ζη(. , ℵ), ℵ||𝒟
α0 + b0(ℵ)]dη

 +ϑ∑0<hξ<ϱ aξ(ℵ)||ζ(hξ, ℵ)||ℝ
αξ + ϑa∑0<hξ<ϱ a′ξ(ℵ)||ζ(hξ, ℵ)||ℝ

αξ]dϖ ≤ Q(ℵ)

 

 Theorem 4.1 If (𝐺1) - (𝐺8) are fulfilled, then the problem (1.2) is approximately controllable on 𝐽. 

Proof: Let us specify the control:  

𝑦(ℎ, ℵ) = 𝑘−1(𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] + ∫
𝜚

0

𝑇1(𝜚 − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 −∫
𝜚

0

𝑇2(𝜚 − 𝜛)Υ(Π, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛 − ∑

0<ℎ𝜉<𝜚

𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))

 − ∑

0<ℎ𝜉<𝜚

𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))

 

 We define the operator (Γ(ℵ))′: 𝑃𝐶𝛿 = Ω × 𝑃𝐶([𝛿, 𝜚], 𝒮) → 𝑃𝐶𝛿  be a random operator and is 

defined by  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

48 https://internationalpubls.com 

((Γ(ℵ))′𝜁)(ℎ) = 𝑇1(ℎ)𝜙0(ℵ) + 𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] − ∫
ℎ

0

𝑇1(ℎ − 𝜛)[𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 +∫
ℎ

0

𝑇2(ℎ − 𝜛)[Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 +∫
ℎ

0

𝑇2(ℎ − 𝜛)𝐵𝑘
−1[(𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0

𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∫
𝜚

0

𝑇2(𝜚 − 𝜂)Υ(Η, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂

 − ∑

0<ℎ𝜉<𝜚

𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ)))] − ∑

0<ℎ𝜉<𝜚

𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))]𝑑𝜛

 + ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ)) + ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))]   ℎ ∈ (−𝛿, 𝜚].

 

(Γ(ℵ))′ = (Γ(ℵ))′1 + (Γ(ℵ))′2  

(Γ(ℵ))′1𝜁(ℎ) = 𝑇1(ℎ)𝜙0(ℵ) + 𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] − ∫
ℎ

0

𝑇2(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 +∫
ℎ

0

𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛 + ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))

 + ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))

 

 

(Γ(ℵ))′2𝜁(ℎ) = ∫
ℎ

0
𝑇2(ℎ − 𝜛)𝐵𝑘

−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 + ∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂

 −∑0<ℎ𝜉<𝜚 𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ)) − ∑0<ℎ𝜉<𝜚 𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))]𝑑𝜛

 

 We have already outlined four scenarios for (Γ1(ℵ)) in theorem(3.1). Hence, it suffices to validate 

the outcome for (Γ(ℵ))2. 

Step 1: (Γ(ℵ))2 takes bounded sets and maps them to bounded sets. 

Specifically, it is sufficient to establish that we can find a +ve constant 𝑞(ℵ) such that for every 𝜁 ∈

ℬ𝑞(𝛿), defined as: 

 ℬ𝑟(𝛿):= {𝜁 ∈ 𝑃𝐶𝛿: sup
𝛿≤ℎ≤𝜚

∥ 𝜁(ℎ, ℵ) ∥≤ 𝑞(ℵ)} 

one has ∥ (Γ(ℵ))2𝜁 ∥𝑃𝐶≤ 𝑄(ℵ). 

 



Communications on Applied Nonlinear Analysis 

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||(Γ(ℵ))′2𝜁(ℎ)|| ≤ ∫
ℎ

0
||𝑇2(ℎ − 𝜛)𝐵𝑘

−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂

 −∑0<ℎ𝜉<𝜚 𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ)) − ∑0<ℎ𝜉<𝜚 𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))]||𝑑𝜛

 ≤ ∫
ℎ

0
||𝑇2(ℎ − 𝜛)||𝐵𝑘

−1[||𝜁1(ℵ)|| + ||𝑇1(𝜚)𝜙0(ℵ)|| + ||𝑇2(𝜚)[𝜙
′
0
(ℵ]||

 +|| ∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂|| + || ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂||

 +∑0<ℎ𝜉<𝜚 ||𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))|| + || ∑0<ℎ𝜉<𝜚 𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))||]𝑑𝜛

 ≤ 𝜗𝑎𝐵𝑘
−1 ∫

ℎ

0
[||𝜁1(ℵ)|| + 𝜗||𝜙0(ℵ)|| + 𝜗𝑎||𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)||

 +𝜗 ∫
𝜚

0
[ sup
𝜂∈(0.𝜚]

||𝜁𝜂(. , ℵ), ℵ||𝒟
𝛽0 + 𝑑0(ℵ)]𝑑𝜂 + 𝜗𝑎 ∫

𝜚

0
[ sup
𝜂∈(0.𝜚]

||𝜁𝜂(. , ℵ), ℵ||𝒟
𝛼0

 +𝜗∑0<ℎ𝜉<𝜚 𝑎𝜉(ℵ)||𝜁(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉 + 𝜗𝑎 ∑0<ℎ𝜉<𝜚 𝑎′𝜉(ℵ)||𝜁(ℎ𝜉 , ℵ)||ℝ

𝛼𝜉]𝑑𝜛

 ≤ 𝑄(ℵ)

 (9) 

 Hence (Γ(ℵ))′2 is bounded in 𝑃𝐶𝛿 

Step 2: We now demonstrate that (Γ(ℵ))′2 is continuous on ℬ𝑟(𝛿). Let us consider 𝜁1, 𝜁2 ∈ ℬ𝑟(𝛿) 

and ℎ ∈ 𝐽.  

||(Γ(ℵ))′2𝜁1(ℎ) − (Γ(ℵ))′2𝜁2(ℎ)|| ≤ ∫
ℎ

0

||𝑇1(ℎ − 𝜛)𝐵𝑘
−1[∫

𝜚

0

𝑇1(𝜚 − 𝜂)[𝜌(𝜂, (𝜁1,𝜂(. , ℵ), ℵ), ℵ)

 −𝜌(𝜂, (𝜁2,𝜂(. , ℵ), ℵ), ℵ)]𝑑𝜂

 −∫
𝜚

0

𝑇2(𝜚 − 𝜂)[Υ(𝜂, (𝜁1,𝜂(. , ℵ), ℵ), ℵ) − Υ(𝜂, (𝜁2,𝜂(. , ℵ), ℵ), ℵ)]𝑑𝜂

 − ∑

0<ℎ𝜉<𝜚

𝑇1(𝜚 − ℎ𝜉)[𝐼𝜉(𝜁1(ℎ𝜉 , ℵ)) − 𝐼𝜉(𝜁2(ℎ𝜉 , ℵ))]

 − ∑

0<ℎ𝜉<𝜚

𝑇2(𝜚 − ℎ𝜉)[𝐼′𝜉(𝜁1(ℎ𝜉 , ℵ)) − 𝐼′𝜉(𝜁2(ℎ𝜉 , ℵ))]||]𝑑𝜛

 ≤ ∫
ℎ

0

𝜗𝑎𝐵𝑘
−1[∫

𝜚

0

𝜗 sup
𝜂∈(0.𝜚]

||(𝜁1,𝜂(. , ℵ), ℵ) − (𝜁2,𝜂(. , ℵ), ℵ)||𝒟
𝛽0𝑑𝜂

 +∫
𝜚

0

𝜗𝑎 sup
𝜂∈(0.𝜚]

||(𝜁1,𝜂(. , ℵ), ℵ) − (𝜁2,𝜂(. , ℵ), ℵ)||𝒟
𝛼0𝑑𝜂

 +𝜗 ∑

0<ℎ𝜉<𝜚

𝑎𝜉(ℵ)||𝜁1(ℎ𝜉 , ℵ) − 𝜁2(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉

 +𝜗𝑎 ∑

0<ℎ𝜉<𝜚

𝑎′𝜉(ℵ)||𝜁1(ℎ𝜉 , ℵ) − 𝜁2(ℎ𝜉 , ℵ)||ℝ
𝛼𝜉]𝑑𝜛

 

For all ℎ ∈ (−𝛿, 𝜚], and due to their compactness for ℎ > 0, the uniform operator topology is 

continuous. Given 𝜁1, 𝜁2 ∈ ℬ𝑟(𝛿), the independence of the right side of the inequalities above is 

evident. Consequently, as (𝜁1 − 𝜁2) → 0, we have ∥ ((Γ(ℵ))′2𝜁1)(ℎ) − ((Γ(ℵ))′2𝜁2)(ℎ) ∥→ 0. This 

implies that (Γ(ℵ)) is continuous. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

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Step 3: (Γ(ℵ))′2 is a compact operator. 

To establish this, we analyze the decomposition (Γ(ℵ))′2 = (Γ𝑎(ℵ))′2 + (Γ𝑏(ℵ))′2, where (Γ1(ℵ)) 

and (Γ(ℵ))2 denote operators on ℬ𝑟(𝛿). They are defined as follows:  

(Γ𝑎(ℵ))′2 = ∫
ℎ

0

||𝑇2(ℎ − 𝜛)𝐵𝑘
−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0

𝑇1(𝜚 − 𝜂)[𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)]𝑑𝜂 − ∫
𝜚

0

𝑇2(𝜚 − 𝜂)[Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)]𝑑𝜂]𝑑𝜛

(Γ𝑏(ℵ))′2 = ∫
ℎ

0

||𝑇2(ℎ − 𝜛)𝐵𝑘
−1[ ∑

0<ℎ𝜉<𝜚

𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ)) − ∑

0<ℎ𝜉<𝜚

𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))]𝑑𝜛

 

We first prove that (Γ(ℵ))2,𝑎(ℬ𝑟(𝛿)) is equicontinuous. 

Let 𝛿 ≤ ℎ1 < ℎ2 ≤ 𝜚 and 𝜖 > 0 be small, then 

 

||(Γ𝑎 (ℵ))′2𝜁(ℎ2) − (Γ𝑎(ℵ))′2𝜁(ℎ1)|| ≤ ∫
ℎ1
0
||𝑇2(ℎ2 −𝜛)𝐵𝑘

−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)

 [𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] + ∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂]𝑑𝜛

 −∫
ℎ2
0
𝑇2(ℎ1 −𝜛)𝐵𝑘

−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂]||𝑑𝜛

 ≤ ∫
ℎ1−𝜃

0
||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)𝐵𝑘

−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂]||𝑑𝜛

 +|| ∫
ℎ1
ℎ1−𝜃

||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)𝐵𝑘
−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂]||𝑑𝜛

 +∫
ℎ1
ℎ1−𝜃

||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)𝐵𝑘
−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 + ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂]||𝑑𝜛

 ≤ ∫
ℎ1−𝜃

0
||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)𝐵𝑘

−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)[ sup

𝜂∈(0.𝜚]
||𝜁𝜂(. , ℵ), ℵ||𝒟

𝛽0 + 𝑑0(ℵ)]𝑑𝜂 + ∫
𝜚

0
𝑇2(𝜚 − 𝜂)[ sup

𝜂∈(0.𝜚]
||𝜁𝜂(. , ℵ), ℵ||𝒟

𝛼0 + 𝑏0(ℵ)]𝑑𝜂]||𝑑𝜛

 +|| ∫
ℎ1
ℎ1−𝜃

||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)𝐵𝑘
−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)[ sup

𝜂∈(0.𝜚]
||𝜁𝜂(. , ℵ), ℵ||𝒟

𝛽0 + 𝑑0(ℵ)]𝑑𝜂 + ∫
𝜚

0
𝑇2(𝜚 − 𝜂)[ sup

𝜂∈(0.𝜚]
||𝜁𝜂(. , ℵ), ℵ||𝒟

𝛼0 + 𝑏0(ℵ)]𝑑𝜂]||𝑑𝜛

 +∫
ℎ1
ℎ1−𝜃

||𝑇2(ℎ2 −𝜛) − 𝑇2(ℎ1 −𝜛)𝐵𝑘
−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 +∫
𝜚

0
𝑇1(𝜚 − 𝜂)[ sup

𝜂∈(0.𝜚]
||𝜁𝜂(. , ℵ), ℵ||𝒟

𝛽0 + 𝑑0(ℵ) + ∫
𝜚

0
𝑇2(𝜚 − 𝜂)[ sup

𝜂∈(0.𝜚]
||𝜁𝜂(. , ℵ), ℵ||𝒟

𝛼0 + 𝑏0(ℵ)]𝑑𝜂]||𝑑𝜛

 

 As ℎ2 − ℎ1  approaches zero, | |(Γ𝑎(ℵ))′2𝜁(ℎ2) − (Γ𝑎(ℵ))′2𝜁(ℎ1)| |  tends to zero for any 𝜁 ∈

ℬ𝑟(𝛿). This convergence is due to the operator’s compactness 𝑇2(ℎ) for ℎ > 0, ensuring continuity 

in the uniform operator norm. Consequently, (Γ1(ℵ)) maps ℬ𝑟(𝛿) into an uniformly continuous 



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family of functions. 

We now aim to demonstrate that the set (Γ𝑎(ℵ))′2(ℬ𝑟(𝛿))(ℎ) is precompact in 𝒮. 

Given 𝛿 < ℎ ≤ 𝜛 ≤ 𝜚 , let 𝜖  be a real number where 0 < 𝜖 < ℎ . For 𝜁 ∈ ℬ𝑟(𝛿) ,we specify 

((Γ(ℵ))′2,𝑎,𝜖𝜁)(ℎ) as 

 
 ∫

ℎ−𝜖

0
𝑇2(ℎ2 −𝜛)𝐵𝑘

−1[𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ) − 𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)]

 −∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ) + ∫

𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)]𝑑𝜂

 

 By leveraging the compactness of 𝑇2(ℎ)  for ℎ > 0 , we conclude that the set 

{((Γ(ℵ))′2,𝑎,𝜖𝜁)(ℎ): 𝜁 ∈ ℬ𝑟(𝛿)} is precompact for 𝜁 ∈ ℬ𝑟(𝛿) and 0 < 𝜖 < ℎ. Moreover, for every 

𝜁 ∈ ℬ𝑟(𝛿), we assert  

 

||((Γ𝑎(ℵ))2𝜁)(ℎ) − ((Γ(ℵ))2,𝑎,𝜖𝜁)(ℎ)|| ≤ ∫
ℎ

ℎ−𝜖
||𝑇2(ℎ2 −𝜛)𝐵𝑘

−1[∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)||𝑑𝜂

 +∫
𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)||𝑑𝜂]𝑑𝜛

              ≤ ∫
ℎ

ℎ−𝜖
𝜗𝑎𝐵𝑘

−1[∫
𝜚

0
𝜗[ sup
𝜂∈(0.𝜚]

||𝜁𝜂(. , ℵ), ℵ||𝒟
𝛽0 + 𝑑0(ℵ)]𝑑𝜂

 +∫
𝜚

0
𝜗𝑎[ sup

𝜂∈(0.𝜚]
||𝜁𝜂(. , ℵ), ℵ||𝒟

𝛼0 + 𝑏0(ℵ)]𝑑𝜂]𝑑𝜛

 

Thus, we can find sets that are precompact and close to {((Γ𝑎(ℵ))′2𝜁): 𝜁 ∈ 𝐵𝑟(𝛿)}. As a result, 

{((Γ𝑎(ℵ))′2𝜁): 𝜁 ∈ 𝐵𝑟(𝛿)} itself becomes precompact within 𝒮. It’s clear that (Γ𝑎(ℵ))′2(𝐵𝑟(𝛿)) is 

uniformly bounded. Given that we have demonstrated (Γ𝑎(ℵ))′2(𝐵𝑟(𝛿))  constitutes an 

equicontinuous family, the Arzelà-Ascoli theorem implies that it is sufficient to show that (Γ𝑎(ℵ))′2 

maps 𝐵𝑟(𝛿) into a precompact set in 𝒮. 

Next, it is necessary to confirm that (Γ𝑏(ℵ))′2 is also a compact operator. 

By step 3 of theorem 3.1(above theorem) we prove that (Γ𝑏(ℵ))′2 is compact. 

Step 4: Next, we establish the existence of an open set 𝑈 ⊆ 𝑃𝐶𝛿  such that 𝜁 ∉ 𝜆(Γ(ℵ))′(𝜁) for 𝜆 ∈

(0,1) and 𝜁 ∈ 𝜕𝑈. Consider 𝜆 ∈ (0,1) and let 𝜁 ∈ 𝑃𝐶𝛿  be a potential solution of 𝜁 = 𝜆(Γ(ℵ))′(𝜁) 

for some 0 < 𝜆 < 1. Consequently, for every ℎ ∈ (0, 𝜚], we have 



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𝜁(ℎ, ℵ) = 𝜆𝑇1(ℎ)𝜙0(ℵ) + 𝜆𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] − 𝜆∫
ℎ

0

𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛

 +𝜆∫
ℎ

0

𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛(. , ℵ), ℵ)𝑑𝜛 + 𝜆∫
ℎ

0

𝑇2(ℎ − 𝜛)𝐵𝑘
−1[(𝜁1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ)

 −𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] + ∫
𝜚

0

𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂

 −∫
𝜚

0

𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂(. , ℵ), ℵ)𝑑𝜂 − ∑

0<ℎ𝜉<𝜚

𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ)))

 − ∑

0<ℎ𝜉<𝜚

𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))]𝑑𝜛 + 𝜆 ∑

0<ℎ𝜉<ℎ

𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁(ℎ𝜉 , ℵ))

 +𝜆 ∑

0<ℎ𝜉<ℎ

𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁(ℎ𝜉 , ℵ))

 

 

By step 1 of theorem 3.1 and 3.2, ||(Γ(ℵ))′𝜁(ℎ)|| ≤ 𝑅(ℵ) + 𝑄(ℵ) We can find a constant 𝑅(ℵ) +

𝑄(ℵ) such that ∥ 𝜁 ∥𝑃𝐶≠ 𝑅(ℵ) + 𝑄(ℵ). Set  

 𝑈 = {𝜁 ∈ 𝑃𝐶([𝛿, 𝜚], 𝒮)  |  sup
𝛿≤ℎ≤𝜚

∥ 𝜁(ℎ) ∥< 𝑅(ℵ) + 𝑄(ℵ)} 

Based on Steps 1-3 of Theorem 3.2, it is sufficient to show that (Γ(ℵ))′: 𝑈 → 𝑃𝐶𝛿 is a compact map. 

Given the choice of 𝑈, there is no 𝜑 ∈ 𝜕𝑈 for which 𝜁 ∈ 𝜆(Γ(ℵ))(𝜁) with 𝜆 ∈ (0,1). According to 

Lemma 2.1, we assume that the operator (Γ(ℵ)) has a fixed point 𝜁∗ ∈ 𝑈. Thus, we derive  

𝜁∗(ℎ, ℵ) = 𝑇1(ℎ)𝜙0(ℵ) + 𝑇2(ℎ)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] − 𝜆 ∫
ℎ

0
𝑇1(ℎ − 𝜛)𝜌(𝜛, 𝜁𝜛

∗ (. , ℵ), ℵ)𝑑𝜛

 +𝜆 ∫
ℎ

0
𝑇2(ℎ − 𝜛)Υ(𝜛, 𝜁𝜛

∗ (. , ℵ), ℵ)𝑑𝜛 + ∫
ℎ

0
𝑇2(ℎ − 𝜛)𝐵𝑘

−1[(𝜁∗,1(ℵ) − 𝑇1(𝜚)𝜙0(ℵ)

 −𝑇2(𝜚)[𝜙′0(ℵ) + 𝜌(0, 𝜙0(ℵ), ℵ)] + ∫
𝜚

0
𝑇1(𝜚 − 𝜂)𝜌(𝜂, 𝜁𝜂

∗(. , ℵ), ℵ)𝑑𝜂

 − ∫
𝜚

0
𝑇2(𝜚 − 𝜂)Υ(𝜂, 𝜁𝜂

∗(. , ℵ), ℵ)𝑑𝜂 − ∑0<ℎ𝜉<𝜚 𝑇1(𝜚 − ℎ𝜉)𝐼𝜉(𝜁
∗(ℎ𝜉 , ℵ)))

 −∑0<ℎ𝜉<𝜚 𝑇2(𝜚 − ℎ𝜉)𝐼′𝜉(𝜁
∗(ℎ𝜉 , ℵ))]𝑑𝜛 + 𝜆∑0<ℎ𝜉<ℎ 𝑇1(ℎ − ℎ𝜉)𝐼𝜉(𝜁

∗(ℎ𝜉 , ℵ))

 +𝜆∑0<ℎ𝜉<ℎ 𝑇2(ℎ − ℎ𝜉)𝐼′𝜉(𝜁
∗(ℎ𝜉 , ℵ))

 

 This implies, that 𝜁∗(ℎ, ℵ) has a fixed point and 𝜁∗(ℎ, ℵ) is a mild solution of problem (1.2). This 

completes the proof of this theorem. 

5  Example 

 This section introduces an example to illustrate our findings. Before delving into the application of 

our abstract results, we must first establish some technical prerequisites. In what follows, let 𝒮 =

𝐿2([0, 𝜋]) , 𝐷(𝐴) = {𝜑 ∈ 𝒮: 𝑥′′ ∈ 𝒮, 𝜑(0) = 𝜑(𝜋) = 0} , and 𝐴:𝐷(𝐴) ⊆ 𝒮 → 𝒮  denote the linear 

operator defined by 𝐴𝜑 = 𝜑′′.It’s widely recognized that 𝐴 acts as the infinitesimal generator of a 

strongly continuous cosine family (𝑇1(ℎ))ℎ∈ℝ  on 𝒮 . Moreover, 𝐴 has a discrete spectrum, with 

eigenvalues −𝑛2 for 𝑛 ∈ 𝜗, each corresponding to the eigenvectors 𝑧𝑛(𝜚) = (
2

𝜋
)1/2. 



Communications on Applied Nonlinear Analysis 

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Consider the following impulsive partial neutral functional integro-differential equation of the form:  

 
𝜕

𝜕
[
𝜕

𝜕ℎ
𝑧(ℎ, 𝑥, ℵ) − 𝜌(ℎ, 𝑧(cosℎ, 𝑥, ℵ), ℵ) =

𝜕2

𝜕𝑥2
𝑧(ℎ, 𝑥, ℵ) + Υ(h, 𝑧(sinℎ, 𝑥, ℵ), ℵ),   ℵ ∈ (−∞, 0]  (11) 

  Δ𝑧(ℎ𝜉 , 𝑥, ℵ) = ∫
𝜋

0
𝑞𝜉(𝑥, 𝑦)𝑧(ℎ𝜉 , 𝑦, ℵ)𝑑𝑦    and    Δ′𝑧(ℎ𝜉 , 𝑥, ℵ) = ∫

𝜋

0
𝑞′𝜉(𝑥, 𝑦)𝑧(ℎ𝜉 , 𝑦, ℵ)𝑑𝑦,    𝜉 =

1, … ,𝑚, (12) 

  𝑧(ℎ, 0, ℵ) = 𝑧(ℎ, 𝜋, ℵ) = 0;     𝑧(0, 𝑥, ℵ) = 𝑧0(𝑥, ℵ); 𝑧ℎ(0, 𝑥, ℵ) = 𝑧1(𝑥, ℵ),    ℎ ∈ 𝐽 = [0,1],    0 ≤

𝑥 ≤ 𝜋, (13) 

  𝑧(0, 𝑥, ℵ) = 𝑧0(𝑥, ℵ),    and    𝑧ℎ(0, 𝑥, ℵ) = 𝑧1(𝑥, ℵ),    0 ≤ 𝑥 ≤ 𝜋. (14) 

where we assume the following conditions: 

The functions Υ(⋅, ℵ) and are continuous on [0,1] with 𝑛 = sup0≤𝜛≤1|Υ(𝜛, ℵ)| < 1. 

The functions 𝑞𝜉 , 𝑞′𝜉: [0, 𝜋] × [0, 𝜋] → ℝ, 𝑘 = 1,1, … ,𝑚, are continuously differentiable, and  

𝜓𝜉 = (∫
𝜋

0

∫
𝜋

0

(
𝜕

𝜕𝑥
𝑞𝜉(𝑥, 𝑦))

2

𝑑𝑥𝑑𝑦)

1
2

< ∞

𝜓′𝜉 = (∫
𝜋

0

∫
𝜋

0

(
𝜕

𝜕𝑥
𝑞′𝜉(𝑥, 𝑦))

2

𝑑𝑥𝑑𝑦)

1
2

< ∞,

 

for every 𝜉 = 1,2, … ,𝑚. 

To address this system, we introduce the operators in the following manner Υ: 𝐽 × 𝐽 × 𝒟 × Ω → 𝒮, 

and 𝜌: 𝐽 × 𝒟 × Ω → 𝒮, 

 𝜌(ℎ, 𝑧ℎ(. , ℵ), ℵ)(𝑥) = 𝜌(ℎ, 𝑧(cosℎ, 𝑥, ℵ), ℵ) 

 Υ(h, 𝑧ℎ(. , ℵ), ℵ)(𝑥) = Υ(h, 𝑧(sinℎ, 𝑥, ℵ), ℵ) 

 𝐼𝜉(𝑧, ℵ)(𝑥) = ∫
𝜋

0
𝑞𝜉(𝑥, 𝑦)𝑧(ℎ𝜉 , 𝑦, ℵ)𝑑𝑦    𝜉 = 1,2, . . . , 𝑚 

 𝐼′𝜉(𝑧, ℵ)(𝑥) = ∫
𝜋

0
𝑞′𝜉(𝑥, 𝑦)𝑧(ℎ𝜉 , 𝑦, ℵ)𝑑𝑦    𝜉 = 1,2, . . . , 𝑚. 

Sure, here’s a simplified version: 

The equations (5.13-5.16) can be transformed into a more general form, denoted as (1.1). By using the 

functions mentioned earlier, we meet the requirements stated in Theorem 3.1. Therefore, according to 

Theorem 3.1, we can conclude that the given nonlocal impulsive Cauchy problem (5.13-5.16) has a 

mild solution over the interval 𝐽. 

6  Conclusion 

This study delves into a specific class of mathematical problems concerning second-order equations 

with delays, a topic widespread in scientific and engineering disciplines. By situating these equations 

within the realm of Banach spaces, distinct challenges and pathways for analysis and control are 

uncovered. Through rigorous examination of the existence and approximate controllability of 

solutions, this research significantly contributes to understanding dynamical systems with delayed 



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feedback. Mathematical tools such as cosine family theory and the Leray-Schauder theorem are 

leveraged to establish stringent conditions for solution existence, with implications for theoretical 

advancements and practical applications. Moreover, empirical validation through a practical example 

provides invaluable insights into the behavior of these equations in real-world scenarios, effectively 

bridging the gap between theory and application. This comprehensive investigation advances 

understanding of complex dynamical systems with delayed feedback and offers practical insights for 

developing robust control strategies and engineering solutions across various domains. 

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