On a Lyapunov-type inequality and the zeros of a certain Mittag–Leffler function

In this paper, by variational methods, some Lyapunov-type inequalities are established for fractional quasilinear problems involving left and right Riemann-Liouville fractional derivative operators. To the authors' knowledge, this is the first work, where Lyapunov-type inequalities for fractional boundary value problems are investigated by using variational methods. As an application of the obtained inequalities, we extend the notion of generalized eigenvalues to a fractional quasilinear system, and we derive some geometric properties of the fractional generalized spectrum.

Section: Introductionmentioning

confidence: 91%

Positive solutions of a weakly singular periodic eco-economic system with changing-sign perturbation

Jleli¹,

Kirane²,

Samet³

2017

“…We prove that problem (1.1) has a nontrivial solution for α ∈ (3,4] provided that the real and continuous function q satisfies…”

Section: Introductionmentioning

confidence: 98%

Nonexistence of solutions to a fractional differential boundary value problem

Al-Qurashi¹,

Ragoub²

2016

We investigate new results about Lyapunov-type inequality by considering a fractional boundary value problem subject to mixed boundary conditions. We give a necessary condition for nonexistence of solutions for a class of boundary value problems involving Riemann-Liouville fractional order. The order considered here is 3 < α ≤ 4. The investigation is based on a construction of Green's function and on finding its corresponding maximum value. In order to illustrate the result, we provide an application of Lyapunov-type inequality for an eigenvalue problem and we show how the necessary condition of existence can be employed to determine intervals for the real zeros of the Mittag-Leffler function.

“…Explicitly, the author showed that if the above problem (1.4) has a nontrivial solution, then 5) which yields the standard Lyapunov inequality (1.2) if we take α = 2 in (1.5), where Γ is the gamma function. From then on, some Lyapunov-type inequalities for other fractional boundary value problems were established, see, for example, [7,10,11,14,16] and the references listed therein. On the other hand, there are some papers on Lyapunov-type inequalities for partial differential equations, we refer the reader to [2,3,9] for related results.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov inequality for a class of fractional differential equations with Dirichlet boundary conditions

Chen¹,

Liou²,

Lo³

2017