Existence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy

In this study, we introduce conditions for the existence of solutions for an iterative functional differential equation of fractional order. We prove that the solutions of the above class of fractional differential equations are bounded by Tsallis entropy. The method depends on the concept of Hyers-Ulam stability. The arbitrary order is suggested in the sense of Riemann-Liouville calculus.


Introduction
In 1940, Ulam [1,2] established the following question on the stability of the Cauchy equation: if a function φ approximately satisfies functional equation E, when does an exact solution of E which φ approximates exist ?
In 1941, D. H. Hyers introduced a solution to the problem [3].In 1978, Th.M. Rassias [4] modified Hyers' theorem by establishing the existence of unique linear mappings near approximate additive mappings.The issue has been assumed for various types of spaces (see [5][6][7]).Li and Hua [8] investigated the Hyers-Ulam stability of finite polynomial equations.Bidkham, Mezerji, and Gordji [9] introduced the Hyers-Ulam stability of the generalized finite polynomial equation.Recently, Cimpean and Popa [10] proved the Hyers-Ulam stability for Euler's equation in R n .Popa and Rasa [11] obtained the infimum of the Hyers-Ulam stability constants for different operators, such as the Stancu, Bernstein, and Kantorovich operators.
The Ulam stability of fractional differential equations was introduced for the first time by Wang and Zhou using the Caputo derivative [12].Ibrahim investigated this type of stability for different classes of fractional differential equations by utilizing the fractional derivatives of the Srivastava-Owa differential operator (a type of Riemann-Liouville operator) in a complex domain [13][14][15].The Ulam-Hyers stability for the Cauchy fractional differential equation in the unit disk was also investigated.Various studies are reported in [16,17].Wang and Lin also imposed stability by utilizing the Hadamard-type fractional integral equations [18].Recently, the stability of the sequential fractional differential equation with respect to the Miller-Ross formula was investigated on the basis of the Banach contraction mapping theorem [19].
In this study, we impose conditions for the existence of solutions for an iterative functional differential equation of fractional order Equation (1).We prove that the solutions of the aforementioned class of fractional differential equations are bounded by the Tsallis entropy (fractional entropy).The strategy is based on the concept of the Hyers-Ulam stability.The arbitrary order is derived with respect to the Riemann-Liouville calculus.

Preliminaries
We consider the following concepts: In view of the Riemann-Liouville operators (differential and integral), the fractional calculus can be defined and the fractional integral operator can be formulated as follows: Moreover, for the continuous function φ, the fractional derivative of order ρ > 0 is defined by Consequently, when a = 0 we have The function φ : R n + → χ, n ∈ N is called homogeneous of degree γ with respect to λ ∈ R + such that φ λy [1] (t), λy [2] (t), ..., λy [n] (t) = λ γ φ y [1] (t), ..., y [n] (t) where R + = (0, ∞), and (χ, .) is a Banach space over R. Definition 2. The function φ : R n+1 [1] (t), ty [2] (t), ..., ty [n] (t) = t γ φ y [0] (t), y [1] (t), ..., y [n] (t) where Definition 3. Let be a nonnegative number.Then, Equation ( 1) is considered stable in the Hyers-Ulam sense if δ > 0 such that for every φ ∈ C 1 ∈ (R n+1 + , χ), we derive Equation (2) and for all y ∈ R + the function η ∈ R + exists with the following property: An entropy of the scalar variable was imposed by Mathai, and the properties and its links to the Tsallis non-extensive statistical mechanics and the Mathai pathway pattern were introduced and generalized by Mathai and Haubold [24].In the present work, we deal with the following measure of entropy, which was derived by Tsallis [25] T The discreet form is as follows: Various fractional entropies have been proposed in the literature [26][27][28][29][30].We show that a solution of Equation ( 1) is bounded by the Tsallis entropy under some conditions.This solution satisfies the Hyers-Ulam stability.

Main results
Our first main result is explained in the following theorem: γ then every solution y of Equation ( 1) is bounded by the Tsallis entropy.
Proof.We let y be a solution of Equation ( 1) satisfying Equation (3).Putting the following equation: is absolutely convergent.We consider the following function: Thus, in view of Theorem 1, we derive the following equation: Consequently, we obtain the following inequality: The existence of stability is proven.As such, we assume two solutions to Equation (1), namely, y and u such that y = u with the property in Equation ( 6).Thus, there exists t 0 ∈ (0, T ] such that y(t 0 ) = u(t 0 ) .For all t > 0 the following inequality holds: Since is arbitrary, which leads to a contradiction.Thus, we prove the uniqueness of the solution.
Next, we discuss the stability of the following case: subjected to the initial value We formulate the following theorem: for some λ < then Equation ( 7) has Hyers-Ulam stability.
Proof.The solution to Equation (7) has the following form: Moreover, we determine that where |y(t)| = max y i (t).Thus, we obtain the following inequality: We define the function G : L 1 (0, T ] −→ L 1 (0, T ] as follows: Evidently, G(y) is absolutely convergent because Moreover, G(y) satisfies the following inequality: Thus for m → ∞ we derive the following conclusion: Therefore, G is a µ-contraction mapping in L 1 (0, T ].On the basis of the Banach contraction mapping theorem, we find a unique v ∈ L 1 (0, T ] such that Then, we verify the stability of Equation (7).For m → ∞ we get    Therefore, Equation ( 7) has Ulam-Hyers stability.

Conclusions
In this study, we investigated the generalized Ulam-Hyers stability and Ulam-Hyers-Rassias stability for iterative fractional differential equations with respect to the Riemann-Liouville derivative.These stabilities depend on fractional entropy.We demonstrated that the solutions of the aforementioned class of fractional differential equations are bounded by the Tsallis entropy.We also utilized the Banach contraction fixed-point theorem.

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