Existence and Uniqueness Theorems for a Variable-Order Fractional Differential Equation with Delay

Telli, Benoumran; Souid, Mohammed Said; Alzabut, Jehad; Khan, Hasib

doi:10.3390/axioms12040339

Cited by 20 publications

(6 citation statements)

References 30 publications

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“…Remark 1. In some papers, such as [23,24], the definitions of variable-order fractional differentials/integrals differ, incorrectly citing the definitions of [20,21].…”

Section: Variable-order Generalized Proportional Integrals and Caputo...mentioning

confidence: 99%

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

O’Regan,

Agarwal,

Hristova

et al. 2024

Mathematics

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An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

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“…Remark 1. In some papers, such as [23,24], the definitions of variable-order fractional differentials/integrals differ, incorrectly citing the definitions of [20,21].…”

Section: Variable-order Generalized Proportional Integrals and Caputo...mentioning

confidence: 99%

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

O’Regan,

Agarwal,

Hristova

et al. 2024

Mathematics

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“…, m} and an ϵ-approximation α(t) of the fractional order δ(t), defined by (4). Let the function η k ∈ AC ψ ([0, T k ], R) be a solution of inequality (15), k = 1, 2, . .…”

Section: Ulam-type Stability Of Approximate Solutionsmentioning

confidence: 99%

“…For the Riemann-Liouville fractional derivative of variable order for differential equations, existence is studied in [7], and the concept of approximate solution of the given problem is introduced. This concept is based on the application of piecewise constant orders of fractional integrals and derivatives used for various problems in [15,16].…”

Section: Introductionmentioning

confidence: 99%

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

O’Regan,

Hristova,

Agarwal

2023

Fractal Fract

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An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

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“…In this regard, various boundary and initial condition problems have been researched using modern functional analysis tools and nonlinear analysis techniques. For instance, in [9,35,37,40], the theory of hybrid fixed points has been used to the investigation of various hybrid situations. Additionally, the authors have created several results pertaining to existence theory for various BVPs of HFDEs in publications we refer to [2,8,10,11,34].…”

Section: Introductionmentioning

confidence: 99%

On boundary values system of sequential hybrid fractional differential equations

Shah,

Gul,

Khan

et al. 2024

J. Math. Computer Sci.

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This work is devoted to study a broad system of sequential hybrid fractional differential equations (S-HFDEs). Sufficient conditions related to the existence theory for the aforementioned system are investigated by utilizing the coincidence degree theory of topology. The mentioned degree theory is a powerful instrument which can be utilized to examine a wide range of nonlinear problems for qualitative theory. In addition, for the system of boundary value problems (BVPs) of S-HFDEs under consideration, a result concerning the Ullam-Hyers (U-H) stability is also developed. A pertinent example is also given to verify our theoretical results.

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Existence and Uniqueness Theorems for a Variable-Order Fractional Differential Equation with Delay

Cited by 20 publications

References 30 publications

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

On boundary values system of sequential hybrid fractional differential equations

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