Existence of Positive Solutions for a Coupled System of Nonlinear Fractional Differential Equations

Ghanmi, Abdeljabbar; Horrigue, Samah

doi:10.1007/s11253-019-01623-w

Cited by 11 publications

(5 citation statements)

References 12 publications

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“…Now, since ϕ λ,α (u k ) is uniformly bounded and using (12), u k is uniformly bounded in E α,ψ p and hence x k is uniformly bounded in E α,ψ p . Therefore, we get…”

Section: Preliminaries and Variational Settingmentioning

confidence: 97%

“…Since fractional differential operators are very important, several authors have concentrated on the development of such operators, for example, the well-known Riemann-Liouville operators, Caputo derivative, and Hadamard derivative. Moreover, many papers have been published in this direction we cite for example the papers of Ben Ali et al [4], Chamekh et al [6], Ghanmi and Horrigue [13,12], Horrigue [20], Torres [41], and Wang et al [42].…”

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confidence: 99%

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Positive, negative, and sign-changing solutions for Kirchhoff problem involving Riemann fractional derivative with respect to a function

Alyami

2024

MFC

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In this work, we establish the existence of three solutions (one of them is positive, one is negative and the other one has non-constant sing) for some fractional boundary value problems involving the ψ-Riemann-Liouville derivative and the p-Laplace operator. More precisely, we use the variational method to prove that the associated functional energy satisfies the Palais-smale condition, and has a critical point in three-manifold sets. In my opinion, this is the first time that the signs of three existing solutions have been studied for a problem involving the ψ-Riemann-Liouville derivative.

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“…Now, since ϕ λ,α (u k ) is uniformly bounded and using (12), u k is uniformly bounded in E α,ψ p and hence x k is uniformly bounded in E α,ψ p . Therefore, we get…”

Section: Preliminaries and Variational Settingmentioning

confidence: 97%

mentioning

confidence: 99%

Positive, negative, and sign-changing solutions for Kirchhoff problem involving Riemann fractional derivative with respect to a function

Alyami

2024

MFC

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“…Because fractional differential operators are very important, several papers involving different derivatives, we cite for instance the papers of Ben Ali et al [5], Chamekh et al [29], Ghanmi and Horrigue [31,32], Horrigue [34], Torres [25], and Wang et al [43].…”

Section: Introductionmentioning

confidence: 99%

Existences Results for Some Nonsingular $p$-Kirchhoff Problems with $\psi$-Hilfer Fractional Derivative

Alzumi,

Shammakh,

Ghanmi

2024

Preprint

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In this paper, we prove the existence of three solutions for a $p$-Kirchhoff problem with $\psi$-Hilfer fractional derivative. To be more precise, we use the variational method and we prove that the associated functional energy admits a critical point in each of the three constructed sets, these critical points are weak solutions for a studied problem. Moreover, by definition of these sets, one of these solutions is positive, the second is negative, and the third one change sign. At the end of this work, we present an example to validate our main results.

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“…Due to its importance and their various applications, several researchers are concentrated on the study of problems involving different fractional operators. For details and examples, one can see the monographs [8][9][10][11][12][13][14][15] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Variational approach for the Kirchhoff problem involving the p$$ p $$‐Laplace operator and the ψ$$ \psi $$‐Hilfer derivative

Alsaedi

Ghanmi

2023

Math Methods in App Sciences

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This work aims to develop the variational framework for some Kirchhoff problems involving both the p$$ p $$‐Laplace operator and the ψ$$ \psi $$‐Hilfer derivative. Precisely, we use the mountain pass theorem to prove the existence of nontrivial solutions. Moreover, the multiplicity of solutions is proved by the use of the Z2$$ {Z}_2 $$‐symmetry mountain pass theorem. Our main results generalize the paper of Torres (J Fract Calculus Appli. 2014;5(1):1‐10) and the work of Sousa et al. (Comp Appl Math. 2019;38:4).

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Existence of Positive Solutions for a Coupled System of Nonlinear Fractional Differential Equations

Cited by 11 publications

References 12 publications

Positive, negative, and sign-changing solutions for Kirchhoff problem involving Riemann fractional derivative with respect to a function

Positive, negative, and sign-changing solutions for Kirchhoff problem involving Riemann fractional derivative with respect to a function

Existences Results for Some Nonsingular $p$-Kirchhoff Problems with $\psi$-Hilfer Fractional Derivative

Variational approach for the Kirchhoff problem involving the p$$ p $$‐Laplace operator and the ψ$$ \psi $$‐Hilfer derivative

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