Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

Jleli, Mohamed; Kirane, Mokhtar; Samet, Bessem

doi:10.1155/2017/5123240

Cited by 16 publications

(12 citation statements)

References 33 publications

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“…The generalization of the above Lyapunov inequality to fractional boundary value problems have been the interest of some researchers in the last few years. For examples, we refer the reader to [ 16 – 22 ]. For discrete fractional counterparts of Lyapunov inequalities we refer to [ 23 ] and for the q -fractional types we refer to [ 24 ].…”

Section: Introductionmentioning

confidence: 99%

A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

Abdeljawad

2017

J Inequal Appl

128

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In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order α ∈ [0, 1] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo (ABC) and Riemann (ABR) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order 2 < α ≤ 3 in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

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Section: Introductionmentioning

confidence: 99%

A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

Abdeljawad

2017

J Inequal Appl

128

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“…The result, as proved by Lyapunov in [L93], asserts that if q ∈ C ([a, b]; R) , then a necessary condition for the boundary value problem (1.1) Looking for a generalization for fractional differential equations, in [F13], Ferreira investigated a Lyapunov-type inequality for the Riemann-Liouville fractional boundary value problem Recently, some Hartman-Wintner-type inequalities were obtained for different fractional boundary value problems. In this direction, we refer to Cabrera, Sadarangania, Samet [CSS17] and Jleli, Kirane, Samet [JKS17].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative

Kirane¹,

Torebek²

2018

Journal of Mathematical Inequalities

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In this work, we obtain a Lyapunov-type and a Hartman-Wintner-type inequalities for a linear and a nonlinear fractional differential equation with generalized Hilfer operator subject to Dirichlet-type boundary conditions. We prove existence of positive solutions to a nonlinear fractional boundary value problem. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.

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“…Jleli et al 8 studied the following fractional differential equation:

𝔻_{a, g}^{α} u false(t false) + q false(t false) u false(t false) = 0, t \in false[a, b false],

supplemented with the boundary conditions

u false(a false) = u false(b false) = 0,

where

𝔻_{a, g}^{α}

is the Riemman fractional derivative of order 1 < α < 2 with respect to a certain nondecreasing function

g \in C^{1} false(false[a, b false], ℝ false)

such that g ′ ( x ) > 0, for all x ∈ ( a , b ), and a continuous function

q : ℝ \to ℝ

. They derived A Hartman–Wintner‐type inequality,

\int_{a}^{b} {[(g (b) - g (s)) (g (s) - g (a))]}^{α - 1} | q (s) | g^{'} (s) d s \geq Γ (α) {(g (b) - g (a))}^{α - 1};

several Lyapunov‐type inequalities. …”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

Lyapunov‐ and Hartman–Wintner‐type inequalities for a nonlinear fractional BVP with generalized ψ‐Hilfer derivative

Zohra

Hedia

Kirane

2020

Math Methods in App Sciences

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Hartman-Wintner-Type Inequality for a Fractional Boundary Value Problem via a Fractional Derivative with respect to Another Function

Cited by 16 publications

References 33 publications

A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

A Lyapunov-type inequality for a fractional boundary value problem with Caputo-Fabrizio derivative

Lyapunov‐ and Hartman–Wintner‐type inequalities for a nonlinear fractional BVP with generalized ψ‐Hilfer derivative

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