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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{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.

_{0}. Recently, Ibrahim et al. [23] investigated the existence and uniqueness of

^{[}

^{j}

^{]}(t) := y(y

^{[}

^{j−}

^{1]}(t)) indicates the j − th iterate of self-mapping y, where j = 1, 2, …, n.

## 2. Preliminaries

**Definition 1.**The function ϕ : ${\mathbb{R}}_{+}^{n+1}\to \chi $, n ∈ ℕ is called homogeneous of degree γ with respect to λ ∈ ℝ

_{+}such that

_{+}= (0, ∞), and (χ, ‖·‖) is a Banach space over ℝ.

**Definition 2.**The function ϕ : ${\mathbb{R}}_{+}^{n+1}\to \chi $, n ∈ ℕ is called homogeneous of degree 0 < γ ≤ 1 with respect to t ∈ ℝ

_{+}if

_{0}(t) = y

^{[0]}(t) := t

^{β}, 0 < β ≤ 1.

**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}∈ ( ${\mathbb{R}}_{+}^{n+1}$, χ), we derive Equation (2) and

_{+}the function η ∈ ℝ

_{+}exists with the following property:

## 3. Main results

**Theorem 1.**Suppose that ϕ ∈ C( ${\mathbb{R}}_{+}^{n+1}$, χ) achieving (2) and

**Proof.**The relationship

**Theorem 2.**Assume that ϕ ∈ C( ${\mathbb{R}}_{+}^{n+1}$, χ) achieving (2) and

**Proof.**We let y be a solution of Equation (1) satisfying Equation (3). Putting the following equation:

^{0}∈ (0,T] such that

**Theorem 3.**Assume that

**Proof.**The solution to Equation (7) has the following form:

_{i}(t). Thus, we obtain the following inequality:

_{1}(0, T]→ L

_{1}(0, T] as follows:

_{1}(0, T]. On the basis of the Banach contraction mapping theorem, we find a unique v ∈ L

_{1}(0, T] such that

## 4. Conclusions

## Acknowledgments

**PACS classifications**: 02.70

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Ibrahim, R.W.; Jalab, H.A.
Existence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy. *Entropy* **2015**, *17*, 3172-3181.
https://doi.org/10.3390/e17053172

**AMA Style**

Ibrahim RW, Jalab HA.
Existence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy. *Entropy*. 2015; 17(5):3172-3181.
https://doi.org/10.3390/e17053172

**Chicago/Turabian Style**

Ibrahim, Rabha W., and Hamid A. Jalab.
2015. "Existence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy" *Entropy* 17, no. 5: 3172-3181.
https://doi.org/10.3390/e17053172