No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Lyapunov Type Inequality for a Nonlinear Fractional Hybrid Boundary Value Problem
Abstract
In this paper, we present a Lyapunov type inequality for a nonlinear fractional hybrid boundary value problem. We illustrate the main result through a series of examples.
... The next theorem deals with the Lyapunov-type inequality for the ψ-Hilfer boundary value problem (15). ...
... In 2019, Lopez et al. [15] considered the fractional hybrid boundary value problem: ...
This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann–Liouville, ψ-Caputo, ψ-Hilfer, hybrid, Caputo–Fabrizio, Hadamard, Katugampola, Hilfer–Katugampola, p-Laplacian, and proportional fractional derivative operators.
... Motivated by the contributions [7,8] and following the same approach, several authors continued the investigation of Lyapunov-type inequalities for different types of fractional boundary value problems. For instance, we refer the reader to the series of papers [9,11,12,13,16,17]. ...
... For fractional boundary value problems, Lyapunov-type inequalities were considered for the first time by Ferreira in [4,5]. In addition, there is a work which is generalises the Lyapunov-type inequalities [6,7,10,11,17] and for nonlinear fractional differential equations in [1,2,12,16,[18][19][20]. ...
In this paper, we extend our exploration of a fractional boundary value problem involving left Riemann–Liouville and right Caputo fractional derivatives with order as in [13]. Firstly, we investigate the properties of associated Green’s function. Secondly, we prove the existence of positive solutions, where the Guo-Krasnoselskii fixed-point theorem and properties of the Green function play key roles in the proof. Also, we show the generalised Lyapunov and Hartman–Wintner-type inequalities for the associated fractional boundary value problem. Finally, we present some examples for fractional boundary value problems.
... which is known as Hartman inequality in the literature and it is the best Lyapunovtype inequality for being stronger than both inequalities (1.2) and (1.3). Although Lyapunov-type inequalities have been obtained by some authors [12,14,15] for continuous fractional boundary value problems, there are few relevant results, see for example the paper by Ferreira [5] for the right-focal discrete boundary value problem of the form ...
By employing Green's function, we obtain new Lyapunov type and Hartman type inequalities for higher-order discrete fractional boundary value problems. Reported results essentially generalize some theorems existing in the literature. As an application, we discuss the corresponding eigenvalue problems.
... has a nontrivial solution, where C a D α is the Caputo fractional derivative of order α, 1 < α ≤ Based on the above-mentioned two studies, the subject of fractional Lyapunov-type inequalities has received significant research attention, and a variety of interesting results have been established. For some recent works on the topic, we refer the reader to the works [13][14][15][16][17][18][19][20][21][22][23][24][25][26], the survey paper [27] and the references cited therein. For example, according to the literature report [13], the authors generalized Lyapunov-type inequality (1.2) to the p-Laplacian problem: ...
In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms Da+βHC(HCDa+α+p(t))x(t)+q(t)x(t)=0,01. 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.
In this paper, we obtain a Lyapunov-type and a Hartman–Wintner-type inequalities for a nonlinear fractional hybrid equation with left Riemann–Liouville and right Caputo fractional derivatives of order subject to Dirichlet boundary conditions. It is also shown that failure of the Lyapunov-type and Hartman–Wintner-type inequalities, corresponding nonlinear boundary value problem has only trivial solutions. In the case , our results coincide with the classical Lyapunov and Hartman–Wintner inequalities, respectively.
Abstract This paper is devoted to studying the Lyapunov-type inequality for sequential Hilfer fractional boundary value problems. We first provide some properties of Hilfer fractional derivative, and then establish Lyapunov-type inequalities for a sequential Hilfer fractional differential equation with two types of multi-point boundary conditions. Our results generalize and compliment the existing results in the literature.
ResearchGate has not been able to resolve any references for this publication.