Mild Solutions to the Cauchy Problem for Some Fractional Differential Equations with Delay

Liang, Jin; Mu, Yunyi

doi:10.3390/axioms6040030

Cited by 3 publications

(1 citation statement)

References 20 publications

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“…Fractional differential equation is an interesting research field. The reason is that it can be used to solve practical problems from the fields of science and engineering, such as physics, chemistry, electrodynamics of complex media, polymer rheology and so on ( [1][2][3][4][5][6]). Recently the research of fractional differential equations has made great progress.…”

Section: Introductionmentioning

confidence: 99%

Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

Wang

2021

Fractal Fract

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The aim of this paper is to establish the existence and uniqueness results for differential equations of Hilfer-type fractional order with variable coefficient. Firstly, we establish the equivalent Volterra integral equation to an initial value problem for a class of nonlinear fractional differential equations involving Hilfer fractional derivative. Secondly, we obtain the existence and uniqueness results for a class of Hilfer fractional differential equations with variable coefficient. We verify our results by providing two examples.

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Section: Introductionmentioning

confidence: 99%

Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

Wang

2021

Fractal Fract

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Initial-value / Nonlocal Cauchy Problems for Fractional Differential Equations Involving ψ-Hilfer Multivariable Operators

Liang

Xiao

2020

Fract Calc Appl Anal

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In this paper, we investigate two types of problems (the initial-value problem and nonlocal Cauchy problem) for fractional differential equations involving ψ-Hilfer derivative in multivariable case (ψ-m-Hilfer derivative). First we propose and discuss ψ-fractional integral, ψ-fractional derivative and ψ-Hilfer type fractional derivative of a multivariable function f : ℝm → ℝ (m is a positive integer). Then, using the properties of the ψ-m-Hilfer fractional derivative with m = 1 (the ψ-Hilfer derivative), we derive an equivalent relationship between solutions to the initial-value (Cauchy) problem and solutions to some integral equations, and also present an existence and uniqueness theorem. Based on the equivalency relationship, we establish new and general existence results for the nonlocal Cauchy problem of fractional differential equations involving ψ-Hilfer multivariable operators in the space of weighted continuous functions. Moreover, we obtain a new Gronwall-type inequality with singular kernel, and derive the dependence of the solution on the order and the initial condition for the fractional Cauchy problem with the help of this Gronwall-type inequality. Finally, some examples are given to illustrate our results. Compared with the recent paper [2] and other previous works, the novelties in this paper are in treating the multivariable case of operators (f : ℝm → ℝ, m is a positive integer).

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Properties for ψ-Fractional Integrals Involving a General Function ψ and Applications

Liang¹,

Mu²

2019

Mathematics

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In this paper, we are concerned with the ψ-fractional integrals, which is a generalization of the well-known Riemann–Liouville fractional integrals and the Hadamard fractional integrals, and are useful in the study of various fractional integral equations, fractional differential equations, and fractional integrodifferential equations. Our main goal is to present some new properties for ψ-fractional integrals involving a general function ψ by establishing several new equalities for the ψ-fractional integrals. We also give two applications of our new equalities.

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Mild Solutions to the Cauchy Problem for Some Fractional Differential Equations with Delay

Cited by 3 publications

References 20 publications

Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

Initial-value / Nonlocal Cauchy Problems for Fractional Differential Equations Involving ψ-Hilfer Multivariable Operators

Properties for ψ-Fractional Integrals Involving a General Function ψ and Applications

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