Hyers–Ulam Stability and Existence of Solutions to the Generalized Liouville–Caputo Fractional Differential Equations

In this research article, a novel Φ -fractional Bielecki-type norm introduced by Sousa and Oliveira [23] is used to obtain results on uniqueness and Ulam stability of solutions for a new class of multi-terms fractional differential equations in the framework of generalized Caputo fractional derivative. The uniqueness results are obtained by employing Banach' and Perov's fixed point theorems. While the Φ -fractional Gronwall type inequality and the concept of the matrices converging to zero are implemented to examine different types of stabilities in the sense of Ulam-Hyers (UH) of the given problems. Finally, two illustrative examples are provided to demonstrate the validity of our theoretical findings.

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“…It is worth noting that the results obtained in this paper are generalizations and partial continuation of some results obtained in [11,12].…”

Section: Introductionsupporting

confidence: 78%

Some new uniqueness and Ulam stability results for a class of multiterms fractional differential equations in the framework of generalized Caputo fractional derivative using the Φ-fractional Bielecki-type norm

Derbazi¹,

Baitiche²,

Fečkan³

2021

Turk J Math

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“…It is worth noting that the results obtained in this paper are generalizations and partial continuation of some results obtained in [3,12,15,16,29,30].…”

Section: Introductionsupporting

confidence: 77%

Uniqueness and Ulam-Hyers-Mittag-Leffler stability results for the delayed fractional multi-terms differential equation involving the $Φ$--Caputo fractional Derivative

Derbazi,

Baitiche

2020

Preprint

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The principal aim of the present paper is to establish the uniqueness and Ulam-Hyers Mittag-Leffler (UHML) stability of solutions for a new class of multi-terms fractional time-delay differential equations in the context of the Φ-Caputo fractional derivative. To achieve this purpose, the generalized Laplace transform method alongside facet with properties of the Mittag-Leffler functions (M-LFs), are utilized to give a new representation formula of the solutions for the aforementioned problem. Besides that, the uniqueness of the solutions of the considered problem is also proved by applying the well-known Banach contraction principle coupled with the Φ-fractional Bielecki-type norm. While the Φ-fractional Gronwall type inequality and the Picard operator (PO) technique combined with abstract Gronwall lemma are used to prove the UHML stability results for the proposed problem. Lastly, an example is offered to assure the validity of the obtained theoretical results.

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“…In the continuation, by introducing the Gauss hypergeometric series, we define the Gauss hypergeometric stability of equation ( 1) [15][16][17]. Also, we consider the Picard operator and Henry-Gronwall inequality, which we use in the next section [18,19]. Definition 1.…”

Section: Preliminariesmentioning

confidence: 99%

Picard Method for Existence, Uniqueness, and Gauss Hypergeomatric Stability of the Fractional-Order Differential Equations

Eidinejad

Saadati

Sen

2021

Mathematical Problems in Engineering

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In this paper, we consider a class of fractional-order differential equations and investigate two aspects of these equations. First, we consider the existence of a unique solution, and then, using a new class of control functions, we investigate the Gauss hypergeometric stability. We use Chebyshev and Bielecki norms in order to prove these aspects by the Picard method. Finally, we give some examples to illustrate our results.

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Hyers–Ulam Stability and Existence of Solutions to the Generalized Liouville–Caputo Fractional Differential Equations

Cited by 17 publications

References 29 publications

Some new uniqueness and Ulam stability results for a class of multiterms fractional differential equations in the framework of generalized Caputo fractional derivative using the Φ-fractional Bielecki-type norm

Some new uniqueness and Ulam stability results for a class of multiterms fractional differential equations in the framework of generalized Caputo fractional derivative using the Φ-fractional Bielecki-type norm

Uniqueness and Ulam-Hyers-Mittag-Leffler stability results for the delayed fractional multi-terms differential equation involving the $Φ$--Caputo fractional Derivative

Picard Method for Existence, Uniqueness, and Gauss Hypergeomatric Stability of the Fractional-Order Differential Equations

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