Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a, b]

Laadjal, Zaid; Adjeroud, Nacer; Ma, Qingwei

doi:10.7153/jmi-2019-13-54

Cited by 11 publications

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“…Now we present a lower bound of the eigenvalues for the eigenvalue problem (5). We have the following result about the nonexistence for solutions of the boundary value problem (5).…”

Section: Resultsmentioning

confidence: 97%

“…This section is devoted to prove the main results of the problem (4), and present a lower bound of the eigenvalues for the eigenvalue problem (5).…”

Section: Resultsmentioning

confidence: 99%

“…Motivated by the above mentioned works and the papers [5,6], in this paper, we investigated the sharp estimate for the unique solution of the following fractional differential equation with Hadamard derivative:…”

Section: Introductionmentioning

confidence: 99%

“…If the eigenvalue problem(5) has a non-trivial continuous solution, then|λ| ≥ σ σ+1 Γ(σ) (σ − 1) σ−1 ln b a σ ,(18)Proof. From Lemma 7, the solution of the problem (5) can be written as followsu (x) = b a λG (x, τ ) u (τ ) dτ.which yieldsu ≤ |λ| u max x∈[a,b] b a |G (x, τ )| dτSince u is non-trivial, then u = 0.…”

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confidence: 99%

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Sharp estimates for the unique solution of the Hadamard-type two-point fractional boundary value problems

Laadjal¹,

Adjeroud²

2020

Preprint

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In this short note, we present the sharp estimate for the existence of a unique solution for a Hadamardtype fractional differential equations with two-point boundary value conditions. The method of analysis is obtained by using the integral of the Green's function and the Banach contraction principle. Further, we will also obtain a sharper lower bound of the eigenvalues for an eigenvalue problem. Two examples are presented to clarify the applicability of the essential results.

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“…Now we present a lower bound of the eigenvalues for the eigenvalue problem (5). We have the following result about the nonexistence for solutions of the boundary value problem (5).…”

Section: Resultsmentioning

confidence: 97%

“…This section is devoted to prove the main results of the problem (4), and present a lower bound of the eigenvalues for the eigenvalue problem (5).…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Sharp estimates for the unique solution of the Hadamard-type two-point fractional boundary value problems

Laadjal¹,

Adjeroud²

2020

Preprint

Self Cite

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“…Secondly, the boundary conditions (1.4) or (1.7) are replaced by multipoint boundary conditions or integral boundary conditions. For instance, in [16][17][18], Lyapunov inequalities for Hadamard fractional differential equations are given. Lyapunov-type inequalities regarding sequential fractional differential equations are obtained in [19][20][21].…”

Section: Theorem 12 If the Fractional Boundary Value Problemmentioning

confidence: 99%

Lyapunov-type inequalities for differential equation with Caputo–Hadamard fractional derivative under multipoint boundary conditions

Wang¹,

Wu²,

Chen³

2021

J Inequal Appl

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Lyapunov-type inequalities for fractional Langevin-type equations involving Caputo-Hadamard fractional derivative

Zhang

2022

J Inequal Appl

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In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\beta } \bigl({}_{H}^{C}D_{a + }^{\alpha }+ p(t)\bigr)x(t) + q(t)x(t) = 0,\quad 0 < a < t < b, \end{aligned} $$ D a + β H C ( H C D a + α + p ( t ) ) x ( t ) + q ( t ) x ( t ) = 0 , 0 < a < t < b , and $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\eta }{ \phi _{p}}\bigl[\bigl({}_{H}^{C}D_{a + }^{\gamma }+ u(t)\bigr)x(t)\bigr] + v(t){\phi _{p}}\bigl(x(t)\bigr) = 0,\quad 0 < a < t < b, \end{aligned} $$ D a + η H C ϕ p [ ( H C D a + γ + u ( t ) ) x ( t ) ] + v ( t ) ϕ p ( x ( t ) ) = 0 , 0 < a < t < b , subject to mixed boundary conditions, respectively, where $p(t)$ p ( t ) , $q(t)$ q ( t ) , $u(t)$ u ( t ) , $v(t)$ v ( t ) are real-valued functions and $0 < \beta < 1 < \alpha < 2$ 0 < β < 1 < α < 2 , $1 < \gamma $ 1 < γ , $\eta < 2$ η < 2 , ${\phi _{p}}(s) = |s{|^{p - 2}}s$ ϕ p ( s ) = | s | p − 2 s , $p > 1$ p > 1 . The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic.

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Lyapunov-type inequality for the Hadamard fractional boundary value problem on a general interval [a, b]

Abstract: In this paper, using two different methods, we studied an open problem and obtained several results for Lyapunov-type and Hartman-Wintner-type inequalities for a Hadamard fractional differential equation on a general interval [a,b] , (1 a < b) with the boundary value conditions.

Cited by 11 publications

References 0 publications

Sharp estimates for the unique solution of the Hadamard-type two-point fractional boundary value problems

Sharp estimates for the unique solution of the Hadamard-type two-point fractional boundary value problems

Lyapunov-type inequalities for differential equation with Caputo–Hadamard fractional derivative under multipoint boundary conditions

Lyapunov-type inequalities for fractional Langevin-type equations involving Caputo-Hadamard fractional derivative

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