On resonant mixed Caputo fractional differential equations

Guezane-Lakoud, A.; Kılıçman, Adem

doi:10.1186/s13661-020-01465-7

Cited by 6 publications

(3 citation statements)

References 19 publications

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“…However, among several types of fractional differential equations found in the literature, it is imperative to mention that the Caputo and Riemann–Liouville derivatives are studied separately in many cases. Moreover, the recent papers on the study of boundary value problems at resonance having mixed type fractional‐order derivatives is not satisfactory, and the topic has not been extensively studied so far (see other studies 18–31 ). The initial attempts in recent years are as follows.…”

Section: Introductionmentioning

confidence: 99%

“…In Song and Cui, 21 the authors investigated, by means of Mawhins coincidence degree, the following integral boundary value problems of the mixed fractional differential equations under resonance:

\begin{matrix} -^{C} D_{1^{-}}^{α} D_{0^{+}}^{β} u false(t false) & = f false(t, u false(t false), D_{0^{+}}^{β} u false(t false), D_{0^{+}}^{β + 1} u false(t false) false), t \in false(0,1 false), \\ u false(0 false) & = u^{'} false(0 false) = 0, u false(1 false) = \int_{0}^{1} u false(t false) d A false(t false), \end{matrix}

where

^{C} D_{1^{-}}^{α}

and

D_{0^{+}}^{β}

are the right Caputo fractional derivative of order α ∈ (1, 2] and the left Riemann–Liouville fractional derivative of order β ∈ (0, 1], respectively,

f \in C false(false[0,1 false] \times ℝ^{3}, ℝ false)

, A ( t ) is a bounded‐variation function,

\int_{0}^{1} u false(t false) d A false(t false)

is the Riemann–Stieltjes integral of u with respect to A . And a very recent study 22 considers the existence of solutions for the following type of equation:

\begin{matrix} D_{1^{-}}^{θ} D_{0^{+}}^{υ} x false(t false) & <...> \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear differential equations involving mixed fractional derivatives with functional boundary data

Zhang

Sun

2022

Math Methods in App Sciences

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Section: Introductionmentioning

confidence: 99%

\begin{matrix} -^{C} D_{1^{-}}^{α} D_{0^{+}}^{β} u false(t false) & = f false(t, u false(t false), D_{0^{+}}^{β} u false(t false), D_{0^{+}}^{β + 1} u false(t false) false), t \in false(0,1 false), \\ u false(0 false) & = u^{'} false(0 false) = 0, u false(1 false) = \int_{0}^{1} u false(t false) d A false(t false), \end{matrix}

where

^{C} D_{1^{-}}^{α}

and

D_{0^{+}}^{β}

are the right Caputo fractional derivative of order α ∈ (1, 2] and the left Riemann–Liouville fractional derivative of order β ∈ (0, 1], respectively,

f \in C false(false[0,1 false] \times ℝ^{3}, ℝ false)

, A ( t ) is a bounded‐variation function,

\int_{0}^{1} u false(t false) d A false(t false)

is the Riemann–Stieltjes integral of u with respect to A . And a very recent study 22 considers the existence of solutions for the following type of equation:

\begin{matrix} D_{1^{-}}^{θ} D_{0^{+}}^{υ} x false(t false) & <...> \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Nonlinear differential equations involving mixed fractional derivatives with functional boundary data

Zhang

Sun

2022

Math Methods in App Sciences

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“…In the last few decades, a special consideration has been paid to fractional differential equations (FDEs) due to their wide range applications into real world phenomena (see [1][2][3][4]). Various attempts have been made in order to present these phenomena in a superior way and to explore new fractional derivatives with different approaches such as Riemann-Liouville, Caputo, Hadamard, Hilfer-Hadamard, and Grünwald-Letnikov [5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

et al. 2021

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In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.

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Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator

Sun

2024

Demonstratio Mathematica

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In this article, a functional boundary value problem involving mixed fractional derivatives with p ( x ) p\left(x) -Laplacian operator is investigated. Based on the fixed point theorems and Mawhin’s coincidence theory’s extension theory, some existence theorems are obtained in the case of non-resonance and the case of resonance. Some examples are supplied to verify our main results.

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On resonant mixed Caputo fractional differential equations

Cited by 6 publications

References 19 publications

Nonlinear differential equations involving mixed fractional derivatives with functional boundary data

Nonlinear differential equations involving mixed fractional derivatives with functional boundary data

A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator

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