Lyapunov-type inequalities for sequential fractional boundary value problems using Hilfer’s fractional derivative

Zhang, Wei; Liu, Wenbin

doi:10.1186/s13660-019-2050-6

Cited by 5 publications

(1 citation statement)

References 28 publications

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“…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]. The first paper considering integral boundary conditions is also duo to Ferreira [22].…”

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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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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Nonlinear sequential fractional boundary value problems involving generalized ψ-Caputo fractional derivatives

Dien

2022

Filomat

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This paper is devoted to study the nonlinear sequential fractional boundary value problems involving generalized ?-Caputo fractional derivatives with nonlocal boundary conditions. We investigate the Green function and some of its properties, from which we obtain a new Lyapunov-type inequality for our problem. A lower bound for the possible eigenvalues of our problem is derived. Furthermore, we apply some properties of the Green function to obtain some existence results for our problem. It is worth mentioning that our results still work with some source functions including singularities.

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Lyapunov-type inequalities for sequential fractional boundary value problems using Hilfer’s fractional derivative

Cited by 5 publications

References 28 publications

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

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

Nonlinear sequential fractional boundary value problems involving generalized ψ-Caputo fractional derivatives

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