Boundary Value Problems for Hybrid Caputo Fractional Differential Equations

Baitiche, Zidane; Guerbati, Kaddour; Benchohra, Mouffak; Zhou, Yong

doi:10.3390/math7030282

Cited by 24 publications

(17 citation statements)

References 26 publications

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“…Define two operators A : E → E and B : Ω → E by Au(t) =Ψ(t, u(t)), t ∈ J, Therefore, the equivalent fraction integral Equation (22) to the hybrid fractional problem (3) can be transformed into the following operator equation:…”

Section: Existence Resultsmentioning

confidence: 99%

“…Recently, a new class of mathematical modelings based on hybrid fractional differential equations with hybrid or non-hybrid boundary value conditions have accomplished a large inquisitiveness of many researchers using different techniques (see, for example, [22][23][24][25]). Fractional hybrid differential equations can be employed in modeling and describing non-homogeneous physical phenomena that take place in their form.…”

Section: Introductionmentioning

confidence: 99%

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On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

2021

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This paper deals with a new class of hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain continuously differentiable and increasing function ϑ. By means of a hybrid fixed point theorem for a product of two operators, an existence result is proved. Furthermore, the sufficient conditions of the continuous dependence on the given parameters are investigated. Finally, a simulative example is given to highlight the acquired outcomes.

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Section: Existence Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

2021

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“…In [20], Baitiche et al studied the existence of solutions for the following hybrid sequential FDEs:…”

Section: Introductionmentioning

confidence: 99%

A Study on ψ‐Caputo‐Type Hybrid Multifractional Differential Equations with Hybrid Boundary Conditions

Fredj

Hammouche

Abdo

et al. 2022

Journal of Mathematics

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In this research paper, we investigate the existence, uniqueness, and Ulam–Hyers stability of hybrid sequential fractional differential equations with multiple fractional derivatives of ψ -Caputo with different orders. Using an advantageous generalization of Krasnoselskii’s fixed point theorem, we establish results of at least one solution, whereas the uniqueness of solution is derived via Banach’s fixed point theorem. Besides, the Ulam–Hyers stability for the proposed problem is investigated by applying the techniques of nonlinear functional analysis. In the end, we provide an example to illustrate the applicability of our results.

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“…After that, the Darbo fixed point theorem has been generalized in many different directions; we suggest some works for reference [33][34][35][36]. e reader may also consult [37][38][39][40][41][42][43] and references therein where several applications of the measure of noncompactness can be found.…”

Section: Introductionmentioning

confidence: 99%

Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving $ψ$ -Caputo Derivative in Banach and Fréchet Spaces

Derbazi

Baitiche

Benchohra

et al. 2020

International Journal of Differential Equations

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Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.

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Boundary Value Problems for Hybrid Caputo Fractional Differential Equations

Cited by 24 publications

References 26 publications

On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

A Study on ψ‐Caputo‐Type Hybrid Multifractional Differential Equations with Hybrid Boundary Conditions

Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving $ψ$ -Caputo Derivative in Banach and Fréchet Spaces

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Boundary Value Problems for Hybrid Caputo Fractional Differential Equations

Cited by 24 publications

References 26 publications

On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

A Study on ψ‐Caputo‐Type Hybrid Multifractional Differential Equations with Hybrid Boundary Conditions

Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving ψ -Caputo Derivative in Banach and Fréchet Spaces

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Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving $ψ$ -Caputo Derivative in Banach and Fréchet Spaces