Existence and Kummer Stability for a System of Nonlinear φ -Hilfer Fractional Differential Equations with Application

: Using Krasnoselskii’s ﬁxed point theorem and Arzela–Ascoli theorem, we investigate the existence of solutions for a system of nonlinear φ -Hilfer fractional differential equations. Moreover, applying an alternative ﬁxed point theorem due to Diaz and Margolis, we prove the Kummer stability of the system on the compact domains. We also apply our main results to study the existence and Kummer stability of Lotka–Volterra’s equations that are useful to describe and characterize the dynamics of biological systems.

Fractional differential equations (FDEs) are important due to their applications in engineering, economics, control theory, materials sciences, physics, chemistry, and biology (see [1] and the references therein). Scientists have applied various mathematical approaches through diverse research-oriented aspects of fractional differential systems. For instance, existence, stability, and control theory for fractional differential equations were studied [2,3]. For the first time, Alsina and Ger [4] studied the Hyers-Ulam stability for differential equations. Recently, mathematicians have paid more attention to the study of stability for a wide range of differential systems [5][6][7].

Preliminaries
In this section, we recall some fundamental definitions of the φ-Riemann-Liouville fractional integral, φ-Hilfer fractional derivative, and Kummer's functions. For details, please see [1,8] and the references therein. Let [n, m] be a finite and closed interval with 0 ≤ n < m < ∞ and C[n, m] be the space of continuous functions : [n, m] → R equipped with the following norm || || C[n,m] = max η∈ [n,m] | (η)|.
Furthermore, the weighted space C γ,φ (n, m] is defined as The fractional integrals with the above definition have a semi-group property given by Additionally, for ς, σ > 0, we have [9]: Definition 2. Let (n, m), −∞ ≤ n < m ≤ +∞ be a finite or infinite interval of the line R, φ (η) = 0 for all η ∈ (n, m), and ς > 0, n ∈ N. The left-sided Riemann-Liouville derivative of a function with respect to φ of order ς correspondent to the Riemann-Liouville is defined by Definition 3. Let n − 1 < ς < n with n ∈ N, I = [n, m] ( −∞ ≤ n < m ≤ ∞) and , φ ∈ C n ([n, m], R) be two mappings such that φ (x) > 0 for all x ∈ I. The left-and right-sided φ-Hilfer fractional derivatives H D ς,ν;φ 0 + (.) of the arbitrary function of order ς and type 0 ≤ ν < 1 are defined by respectively.
The solution of a hypergeometric differential equation is called a confluent hypergeometric function [10]. There exist different standard forms of confluent hypergeometric functions, such as Kummer's functions, Tricomi's functions, Whittaker's functions, and Coulomb's wave functions. In this paper, we apply the following Kummer (confluent hypergeometric) function to study our stability: which is the solution of the differential equation where z, P 1 ∈ C and P 2 ∈ C \ Z − 0 . Kummer's function was introduced by Kummer in 1837. The series (2) is also known as the confluent hyper-geometric function of the first kind, and is convergent for any z ∈ C. In this article, we apply it on the real line R as our control function. Clearly, for P 1 = P 2 , we have Letting ς, ν ∈ , we consider the following inequality for > 0 where Φ is the Kummer's function (see [10]), to define a new stability concept called Kummer's stability.
Our approach is motivated by the fact that inversion of a perturbed differential operator may result from the sum of a compact operator and a contraction mapping (see [11][12][13] and the references therein). We begin by stating the following Krasnoselskii FPT, which has many applications in studying the existence of solutions to differential equations: Theorem 2. (Krasnoselskii FPT) Let X be a Banach space and M ⊆ X be a closed, convex, and non-empty set. Additionally, let T, S be mappings so that: The operator T is continuous and compact, and • Mapping S is a contraction.
Then, there exists a w ∈ M so that w = Tw + Sw.
In addition, we mention an alternative FPT presented by Diaz and Margolis in 1967, and it plays a crucial role in proving our stability result [14]. Theorem 3. Consider the generalized complete metric space (X, Υ) and let Θ be a self-map operator which is a strictly contraction mapping with the Lipschitz constant κ < 1. Then, we have two options: (i) either for every n ∈ N, Υ(Θ n+1 z, Θ n z) = +∞; or (ii) if there exists n ∈ N so that the operator Θ satisfies Υ(Θ n+1 z, Θ n z) < ∞ for some z ∈ X, then the sequence {Θ n z} tends to a unique fixed point z * of Θ in the set X * = {v ∈ X : Υ(Θ n v, Θ n v) < ∞}. Furthermore, for all z ∈ X: Now, we are ready to prove that Equation (1) is equivalent to an integral equation. Then, by the above theorem, we infer that a fixed point exists for the integral equation, so Equation (1) has at least one solution. Proposition 1. Assume that g : × R → R and h : 2 × R → R are real-valued continuous mappings, and A is a closed operator, then the following integral equation is equivalent to Equation (1): where γ ≥ 0 and we obtain from Proof. Using the properties of the φ-Hilfer fractional derivative outlined in the preliminaries, we have . So, by the above equality, we have Now, applying I ς,φ to both sides of the above equation and using Theorem 1, we obtain and Then, Conversely, assuming that w ∈ C[0, p] satisfies Equation (4), we claim that the fractional differential Equation (1) holds. We apply H D ς,ν;φ to the Equation (4) and imply by Theorem 1 that This completes the proof.
and assume that the hypotheses (H1)-(H3) are satisfied. Then, the operator T maps the closed ball where where B is the beta function. From the formula Applying H3 and condition (5), we have This completes the proof.
The following theorem shows the existence of solutions to the fractional differential Equation (1) using Krasnoselskii's FPT listed above.

Theorem 4.
Assume that hypotheses H1-H4 are satisfied. Then, Equation (1) has a solution. Suppose that r satisfies condition (5) and B r = {w ∈ C([0, p] : ||w|| ≤ r}. Due to Lemma 1, the operator T maps B r into itself. Now, we use Krasnoselskii FPT to show that T has a fixed point.

Claim 1. The operator T 1 is continuous on B r .
Let {w n } be a sequence in B r that converges to w. We need to prove that T 1 w n → T 1 w. For each η ∈ [0, p], we have (φ(s) − φ(0)) 1−γ g(s, w n (s), (w n (s)) − g(s, w(s), (w(s))|ds Since g and h are continuous, and w n → w as n → +∞ in B r , we can conclude that |T 1 w n )(η) − (T 1 w)(η)| → 0 as n → +∞ by Lebesgue dominated convergence theorem.

Claim 2. T 1 is an equicontinuous operator.
To prove our second claim, we let η 1 , η 2 ∈ with η 2 < η 1 and w ∈ B r , Hence, we have regarding Hypothesis 4, the right-hand side of the above inequality tends to zero whenever η 1 → η 2 , so it clearly claims that T 1 is equicontinuous. Furthermore, using the previous lemma, it is uniformly bounded. Therefore, by Arzela-Ascoli Theorem, T 1 is compact on B r .

Stability Analysis
In this section, we present the Kummer stability with respect to Φ(ς, ν; (φ(η) − φ(0)) ς ) for Equation (1) based on Theorem 3. We begin by assuming the following hypotheses: for all η ∈ [0, p]. (K2) h : 2 × R → R is a continuous function which satisfies a Lipschitz condition in the third argument, i.e., there exists L h > 0 such that for all s, η ∈ and w, v ∈ R.
Theorem 5. Suppose that g and h satisfy K1 and K2. Additionally, let If a continuously differentiable function w : → R for ≥ 0 satisfies for all η ∈ , then there exists a unique continuous function v 0 : → R that satisfies Equation (1) and Proof. Let Y := C 1−γ,φ (0, p] be endowed with the following generalized metric, defined by for all w, v ∈ Y. It is not difficult to see that (Y, d * ) is a complete generalized metric space [5]. Define the operator S : for all η ∈ and w ∈ Y. For any w, v ∈ Y, choose a constant K so that d * (w, v) ≤ K, i.e, for all η ∈ . So, using Remark 1, we have Hence, the operator S is a strict contraction. Moreover, for for all η ∈ . In summary, d * (Sv 0 , v 0 ) ≤ 1 and d * (S n v 0 , S n+1 v 0 ) < +∞ for all n ∈ N. According to Theorem 3, there exists a unique continuous function w : → R such that Sw = w, w satisfies Equation (1) for all η ∈ and for every η ∈ . In addition, it follows from the above calculations that which justifies inequality (9).

Application of φ-Hilfer Fractional Derivative
In this section, we propose the proof of the existence of the solution of the Lotka-Volterra model by considering it being ruled by a φ-Hilfer fractional derivative of the model, as an application.
First, we state the Lotka-Volterra model, which was introduced by Lotka and Volterra [15] independently. This model is known as the predator-prey equations or the Lotka-Volterra equations, and it is given by where X and Y are population size or the population density of different species; X n , Y n are the initial conditions; α, β, δ, and σ represent different growth or decay rates; and N 1 , N 2 are the carrying capacities. The above system shows an interaction between the logistic growth and decay of two different species. Based on the definitions in the previous sections, we can restate the model (17) in the sense of the φ-Hilfer fractional derivative. Taking α = ς, β = ν and σ = γ, where γ = ς + ν(1 − ς), we will apply the model where w X , w Y ∈ R + to analyze the existence, uniqueness, and stability of solutions. (18) has a unique solution (X, Y) on the ball with radius r, and L < 1 2

Conclusions
In this paper, we considered a class of fractional differential equations including a closed linear operator. Next, we used the Krasnoselskii fixed-point theorem to investigate the existing result under some mild conditions. Moreover, we introduced and then proved the Kummer stability of φ-Hilfer fractional differential equations on the compact domains.