Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. This includes Dirichlet, mixed, Robin, fractional, Sturm-Liouville, integral, nonlocal, multi-point, anti-periodic, conjugate, right-focal and impulsive conditions. Furthermore, our study includes Riemann-Liouville, Caputo, Hadamard, Prabhakar, Hilfer and conformable type fractional derivatives. Results for boundary value problems involving fractional p-Laplacian, fractional operators with nonsingular Mittag-Leffler kernels, q-difference, discrete, and impulsive equations, are also taken into account.Definition 2. (Riemann-Liouville fractional derivative) The Riemann-Liouville fractional derivative of order α ≥ 0 is defined by (D 0 f )(t) = f (t) andwhere m is the smallest integer greater than or equal to α.Definition 3. (Caputo fractional derivative) The Caputo fractional derivative of order α ≥ 0 is defined by ( C D 0 f )(t) = f (t) and

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“…. (27) Later, Jleli and Samet [21] obtained a Lyapunov-type inequality for a boundary value problems with Sturm-Liouville boundary conditions…”

Section: Fractional Calculusmentioning

confidence: 99%

“…In 2016, Dhar et al [27] derived Lyapunov-type inequalities for the following boundary value problem…”

Section: Fractional Calculusmentioning

confidence: 99%

Recent Developments of Lyapunov–Type Inequalities for Fractional Differential Equations

Ntouyas

Ahmad

Horikis

2019

Differential and Integral Inequalities

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“…In 2016, Dhar et al [16] derived Lyapunov-type inequalities for two-point Riemann-Liouville fractional boundary value problems associated with fractional integral boundary conditions. This article stresses the importance of choosing well-posed boundary conditions for Riemann-Liouville fractional boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

“…This article shows a gap in the literature on Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with fractional boundary conditions. In 2016, Dhar et al [3] derived Lyapunov-type inequalities for two-point Riemann-Liouville type fractional boundary value problems associated with fractional integral boundary conditions. This article stresses the importance of choosing well-posed boundary conditions for Riemann-Liouville type fractional boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov-Type Inequalities for Riemann-Liouville Type Fractional Boundary Value Problems with Fractional Boundary Conditions

Jonnalagadda

Satpathi

Basua

2019

Advances in the Theory of Nonlinear Analysis and Its Application

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“…In this paper, we investigate existence and nonexistence of positive solutions for a class of Riemann-Stieltjes integral boundary value problems of fractional differential equations with parameters In recent decades, with the wide applications of fractional differential equations in physics, engineering, biology, chemistry, and many other fields, researchers have been paying more and more attention to them, see [1][2][3][4][5][6][7][8][9][10][11][12] and the references therein. At the same time, many problems of fluid mechanics, bioengineering, chemical engineering, and so on could be attributed to the integral boundary value problems, which are nonlocal problems.…”

Section: Introductionmentioning

confidence: 99%

A class of nonlocal problems of fractional differential equations with composition of derivative and parameters

Jia

Liu

et al. 2019

Adv Differ Equ

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In this paper, we study existence and nonexistence of positive solutions for a class of Riemann-Stieltjes integral boundary value problems of fractional differential equations with parameters. By using the fixed point index theory, some new sufficient conditions for the existence of at least one, two and the nonexistence of positive solutions are obtained. The results we obtain show the influence of parameter λ and parameter a on the existence of positive solutions. Finally, some examples are given to illustrate our main results.

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Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions

Cited by 16 publications

References 24 publications

Recent Developments of Lyapunov–Type Inequalities for Fractional Differential Equations

Recent Developments of Lyapunov–Type Inequalities for Fractional Differential Equations

Lyapunov-Type Inequalities for Riemann-Liouville Type Fractional Boundary Value Problems with Fractional Boundary Conditions

A class of nonlocal problems of fractional differential equations with composition of derivative and parameters

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