On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

Lakhdar Ragoub; J. F. Gómez-Aguilar; Eduardo Pérez-Careta; Dumitru Baleanu; Lakhdar Ragoub; J. F. Gómez-Aguilar; Eduardo Pérez-Careta; Dumitru Baleanu

doi:10.3934/math.2024239

AIMS Mathematics

2024, Volume 9, Issue 2: 4907-4924. doi: 10.3934/math.2024239

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On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

1.
Mathematics Department, Prince Muqrin University, Almadinah Almunawwarah, Saudi Arabia
2.
Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp-IICBA)/UAEM, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001 Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos, Mexico
3.
Universidad de Guanajuato, Dpto. Electrónica, Carretera Salamanca-Valle de Santiago, Km 3+1.8, Salamanca Gto, México
4.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
5.
Institute of Space Sciences, Magurele-Bucharest, Romania

Received: 26 July 2023 Revised: 30 September 2023 Accepted: 25 October 2023 Published: 23 January 2024
MSC : 34A08, 34A40, 26D10, 33E12

In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.
- Lyapunov's inequality,
- fractional integral,
- Green's function,
- Hilfer Hadamard integral
Citation: Lakhdar Ragoub, J. F. Gómez-Aguilar, Eduardo Pérez-Careta, Dumitru Baleanu. On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative[J]. AIMS Mathematics, 2024, 9(2): 4907-4924. doi: 10.3934/math.2024239

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Abstract

In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 < r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.

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