Fractional calculus extends standard diferentiation and integration to noninteger orders, which creates a wider mathematical framework. This enables for the simulation of process with memory and mutations, which are common in many science and engineering systems. Additional studies support both the theory and applications of fractional calculus [1][2][3][4][5][6][7]. It remains an efective framework for understanding and solving complex problems. Fractional derivatives such as Caputo fractional derivative, Riemann-Liouville fractional derivative and ψ-Fractional derivatives have specific benefits for modeling complex operations.

Almeida [8] expanded the standard definition for Caputo fractional derivatives by introducing a flexible formulation that includes more general integral operators and diferent types of kernels. The ψ-Caputo derivatives enriches a function ψ(t), resulting in more flexibility in modeling complex systems with variable scaling attributes (refer [9]. The ψ-Caputo derivative uses a kernel function ψ to model diferent systems behaviors, making it a useful tool for describing complex phenomena. Sharma et al. [10] discussed the existence and controllability results for fractional integro-diferential stochastic delayed system with impulsive efects. A function that may generate numerous outputs from the single input is known as the multi-valued function. A single-valued function is the specific case of the multivalued function. It is defined that there is exactly one output for every input. A single-valued function is a part multi-valued functions that it has a single element in their output set.

In [11], Hariharan et al. investigated the existence and uniqueness of Hilfer-Kkatugampola fractional diferential equations with almost sectorial operators. Zhou and Zhang in [12] combined the probability density function and the Laplace transform to construct a framework of moderate solutions. Moreover, Peng and Jia [14] investigated the existence and uniqueness of the solutions for fractional diferential equation whose derivative has an impact on an arbitrary function. In [15] Yang discussed the existence uniqueness of mild solution for a class of ψ-Caputo fractional stochastic evolution equation with varying-time delay driven by fractional Brownian motion. In [16], authors investigated on ψ-fractional stochastic system with Rosenblatt process. In [17], authors were studied long term behaviours of the mild solution to a Hilfer time-fractional stochastic diferential equation. Sharma et al. [18] investigated the existence of the mild solution and approximate controllability of a class of Caputo conformable fractional neutral-type stochastic system with the instantaneous impulsive efects and nonlocal conditions.

In [19], Sharma et al. discussed the existence and controllability results for fractional Sobolev-type stochastic system involving delays and impulses. El-Sayed and Ibrahim [20] demonstrated the Cauchy problem of the multi-valued fractional diferential equation. Nandhaprasadh and Udhayakumar [21] examined the existence of mild solutions for fractional stochastic evolution equations with infinite delay on an infinite interval. Hammami and Horrigue [22] investigated the existence and uniqueness of solutions to the ψ-Riemann-Liouville fractional stochastic diferential equation is determined using Banach’s fixed point theorem. Omaba et al. [23] used Banach fixed point theorem to prove existence and uniqueness of mild solution.

Motivated by the above articles, the following system is investigated by the fractional stochastic diferential inclusion expressed in the Caputo sense in the presence of stochastic noise,

here we take is a single valued function. Then the above inclusion, we consider the equation defined by

where are the two well-defined maps, is a set valued and B is a single valued map and B is a Banach space.

The main contribution of this work is constructing and analysing the generalized fractional-order system with the ψ-Caputo derivative. Using single-valued nonlinear functions and Banach’s fixed point theorem, we proved the existence and uniqueness of the solutions. Along with the asymptotic behavior of these solutions, we are verified their stability under the suitable conditions. An energy growth bound was derived to characterize the boundedness of the system. An example is presented to illustrate the significance of the existence and uniqueness results.

This work is organized as follows: Section 2 to recalls some fundamental concepts, results and properties related to fractional calculus and stochastic diferential equations. Section 3 constructs the existence and uniqueness of the solution to the non-linear stochastic ψ-Caputo fractional diferential equation. Section 4 demonstrates the upper growth bound. Section 5 establishes an exponential growth bound on the energy solution at time . Finally, Section 6 provides an example for verifying our methodology.

In this section, we provide some basic concepts of stochastic diferential equations and ψ−fractional operator. Let us take , where , 0 < and let ψ be a non−negative function on P that is continuous and whose derivative is also non−negative. Simply, a given , we represent the function defined by

Definition 2.1.[24][25]The ψ-Caputo fractional derivative of the function 0 \end{document} ]]> , then the order γ is defined by

where, and

by

The fractional integral of the function f with respect to a function ψ is defined

provided the integral exists.

Lemma 2.2.[26]Let and p , f \in C \gamma , \psi [ P , \mathbb { R } ] \end{document} ]]> , then we have,

In particular, any non-negative integer 4

Lemma 2.3.[27][28] 0 \end{document} ]]>and 0 \end{document} ]]> , then we obtain

Lemma 2.4.[29][30]If and , then we obtain

In particular for , we get

Definition 2.5.[31]A one dimensional Brownian motion is a real valued stochastic process that satisfies:

I.

II. Independent increment: When , the increments and are independent.

III. Normal increment : The diference follows a normal distribution for all , where denotes a normal distribution with the mean λ and the variance

IV. Brownian motion has almost continuous, then the function is continuous.

Lemma 2.6.[22][31] (Ito isometryb ) Let be a stochastic process(means that - measurable for each , we have

Definition 2.7. [22][31]We delegate a group of square-integrable functions on Λ to , with the following norm defined as

Theorem 2.8.[32] (Banach’s Fixed Point Theorem) Let T be a complete metric space and let be a contraction on T. Then M has unique fixed point such that

Lemma 2.9.[33] (Gronwall Inequality) Let 0 \end{document} ]]> and be two continuous function satisfies

Then, the regular Gronwall inequality is given by,

Around this article, referring the filtered probability space which satisfies the standard requirements and by . Then the following notations must be used.

is a m - dimensional Brownian motion.

R and are well-defined maps.

For easier use, to indicate the components of throughout the text using the superscript . The following stochastic diferential equation is taken into consideration.

where . Using the Lemma (2.4), the problem can change in its equivalent integral form:

The existence and uniqueness of solutions to the nonlinear stochastic Caputo fractional diferential equation will be demonstrated in this section. For this, we assume that all or and

(1) Lipschitz continuity: For any and , there exist a positive constant J, we have

(2) Linear growth: For any and , there exist a positive constant G, we have

Define an operator A : by:

To prove the solutions of the existence and uniqueness, we first prove the following lemma.

Lemma 3.1. Let U and V be the random field solutions, which satisfies

Proof. By using the conditions such that, algebraic inequality , the Itob isometry, the Lipschitz continuity and the linear growth we get the following evaluation

This completes the proof.

The proof of the following theorem is completed by applying Theorem (2.8) together with the lemma (3.1).

Theorem3.2.Given L be a constant in inequality(11)which satisfies

Thus, the equation (14) has unique solution.

The upper growth moment bound was proved in this section. It is essential to demonstrate the succeeding lemma.

Lemma 4.1.Suppose that U is the random field solution which resolves the problem and fulfills

Given that the Linear growth(12)holds, we obtain:

Proof. By using the following conditions such that,the linear growth condition algebraic inequality , and the Cauchy-Schwartz inequality, the Itob isometry equation (6), we get

Take,

Which follows that

Therefore, we conclude it by

Now, we can simply demonstrate an upper growth bound for a random solu tion is bounded.

Then the following theorem, we combine the Gronwall inequality with Lemma (4.1). □

Theorem 4.2.Given that the condition of linear growth stated in(12), we get

From the previous result, the energy solution is satisfies an exponential growth bound at a given time . Therefore, behavior of an energy solution and finding a growth of time is very large. Then the growth rate of the solution has the finite upper bound.

Corollary 5.1. Let the condition of Theorem (4.2) and are exist, thus we can get

Proof. Recall from Theorem (4.2) that

Taking log on both sides then the equation becomes,

Then, dividing both sides by and taking lim sup, we get

Hence proved.

In this section, we demonstrate the fractional stochastic diferential equation and verifying that it is unique and bounded.

Example 6.1. Given and $P = [1]$ to define the following fractional stochastic diferential equation:

Here we place and

It can be observed that the maps g and meet the conditions (11) and (12) with

Further,

By verifying Theorem (4.2) and the existence and uniqueness of the solution (15), we obtain

Additionally, solution achieves

Therefore, the system (15) is unique and bounded.

Example 6.2.Given and $P =[0,1] to define the following fractional stochastic diferential equation:

Here we place g and

Then, it can be observed that the maps and meets the conditions (11) and (12) with

Further,

verifying Theorem (4.2) and the existence and uniqueness of the solution (16), we obtain

Additionally, solution achieves

Therefore, the system (16) is unique and bounded.

In this study, the ψ-Caputo fractional derivative involving the single valued function was discussed and a general analytical framework ensuring the existence and uniqueness of the solutions. The existence and uniqueness of solution was derived by using the Banach’s fixed point theorem. The obtained energy growthbound and asymptotic results describe the bounded and long-term behavior of the system. An example is provided to demonstrate our findings. This methodology provides a basis for future investigations of nonlinear and coupled systems. Practical applications will be improved by the development of the numerical techniques.