# American Institute of Mathematical Sciences

March  2020, 13(3): 695-708. doi: 10.3934/dcdss.2020038

## Variational principles in the frame of certain generalized fractional derivatives

 1 Department of Mathematics, Çankaya University 06790, Ankara, Turkey 2 Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia 3 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 4 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

* Corresponding author

Received  August 2018 Revised  September 2018 Published  March 2019

Fund Project: The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under discussion.

Citation: Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038
##### References:

show all references

##### References:
 [1] Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039 [2] Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak, Hussam Alrabaiah. Lyapunov type inequality in the frame of generalized Caputo derivatives. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2335-2355. doi: 10.3934/dcdss.2020212 [3] Shakir Sh. Yusubov, Elimhan N. Mahmudov. Optimality conditions of singular controls for systems with Caputo fractional derivatives. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021182 [4] Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001 [5] Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021021 [6] Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131 [7] Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020 [8] Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial & Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723 [9] Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 [10] Giovanni Bonfanti, Arrigo Cellina. The validity of the Euler-Lagrange equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 511-517. doi: 10.3934/dcds.2010.28.511 [11] Menita Carozza, Jan Kristensen, Antonia Passarelli di Napoli. On the validity of the Euler-Lagrange system. Communications on Pure & Applied Analysis, 2015, 14 (1) : 51-62. doi: 10.3934/cpaa.2015.14.51 [12] Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449 [13] Zbigniew Gomolka, Boguslaw Twarog, Jacek Bartman. Improvement of image processing by using homogeneous neural networks with fractional derivatives theorem. Conference Publications, 2011, 2011 (Special) : 505-514. doi: 10.3934/proc.2011.2011.505 [14] Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $C^{1}$ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [15] Hayat Zouiten, Ali Boutoulout, Delfim F. M. Torres. Regional enlarged observability of Caputo fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 1017-1029. doi: 10.3934/dcdss.2020060 [16] Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021 [17] G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 485-501. doi: 10.3934/dcdss.2020027 [18] Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. doi: 10.3934/dcdss.2020040 [19] Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127 [20] Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282

2020 Impact Factor: 2.425