Geodesic
On adjoint operators for fractional differentiation operators
Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 131, pp. 284-289
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

On a linear manifold of the space of square summable functions on a finite segment vanishing at its ends, we consider the operator of left-sided Caputo fractional differentiation. We prove that the adjoint for it is the operator of right-sided Caputo fractional differentiation. Similar results are established for the Riemann–Liouville fractional differentiation operators. We also demonstrate that the operator, which is represented as the sum of the left-sided and the right-sided fractional differentiation operators is self adjoint. The known properties of the Caputo and Riemann–Liouville fractional derivatives are used to substantiate the results.
Keywords: Caputo fractional derivative, Riemann-Liouville fractional derivative, adjoint operator, square summable function. 
@article{VTAMU_2020_25_131_a3,
     author = {G. Petrosyan},
     title = {On adjoint operators for fractional differentiation operators},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {284--289},
     year = {2020},
     volume = {25},
     number = {131},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2020_25_131_a3/}
}
TY  - JOUR
AU  - G. Petrosyan
TI  - On adjoint operators for fractional differentiation operators
JO  - Vestnik rossijskih universitetov. Matematika
PY  - 2020
SP  - 284
EP  - 289
VL  - 25
IS  - 131
UR  - http://geodesic.mathdoc.fr/item/VTAMU_2020_25_131_a3/
LA  - ru
ID  - VTAMU_2020_25_131_a3
ER  - 
%0 Journal Article
%A G. Petrosyan
%T On adjoint operators for fractional differentiation operators
%J Vestnik rossijskih universitetov. Matematika
%D 2020
%P 284-289
%V 25
%N 131
%U http://geodesic.mathdoc.fr/item/VTAMU_2020_25_131_a3/
%G ru
%F VTAMU_2020_25_131_a3
G. Petrosyan. On adjoint operators for fractional differentiation operators. Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 131, pp. 284-289. http://geodesic.mathdoc.fr/item/VTAMU_2020_25_131_a3/

[1] S. G. Samco, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, 1993 | MR

[2] A. A. Kilbas, H. M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., North-Holland Mathematics Studies, Amsterdam, 2006 | MR | Zbl

[3] F. Mainardi, S. Rionero, T. Ruggeri, “On the initial value problem for the fractional diffusionwave equation”, Waves and Stability in Continuous Media, 1994, 246–251 | MR

[4] M. Afanasova, Y. Ch. Liou, V. Obukhoskii, G. Petrosyan, “On controllability for a system governed by a fractional-order semilinear functional differential inclusion in a Banach space”, Journal of Nonlinear and Convex Analysis, 20:9 (2019), 1919–1935 | MR

[5] J. Appell, B. Lopez, K. Sadarangani, “Existence and uniqueness of solutions for a nonlinear fractional initial value problem involving Caputo derivatives”, J. Nonlinear Var. Anal., 2018, no. 2, 25–33 | Zbl

[6] T. D. Ke, N.V. Loi, V. Obukhovskii, “Decay solutions for a class of fractional differential variational inequalities”, Fract. Calc. Appl. Anal., 2015, no. 18, 531–553 | MR | Zbl

[7] M. Afanasova, G. Petrosyan, “On the boundary value problem for functional differential inclusion of fractional order with general initial condition in a Banach space”, Russian Mathematics, 63:9 (2019), 1–11 | DOI | MR | Zbl

[8] I. Benedetti, V. Obukhovskii, V. Taddei, “On generalized boundary value problems for a class of fractional differential inclusions”, Fract. Calc. Appl. Anal., 2017, no. 20, 1424–1446 | MR | Zbl

[9] M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao, “Boundary value problems for semilinear differential inclusions of fractional order in a Banach space”, Applicable Analysis, 97:4 (2018), 571–591 | DOI | MR

[10] M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao, “On a Periodic Boundary Value Problem for a Fractional–Order Semilinear Functional Differential Inclusions in a Banach Space”, Mathematics, Special Issue “Fixed Point, Optimization, and Applications”, 7:12 (2019), 5–19 | MR

[11] M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao, “Existence and Approximation of Solutions to Nonlocal Boundary Value Problems for Fractional Differential Inclusions”, Fixed Point Theory and Applications, 2019, no. 2 | MR | Zbl

[12] M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao, “On approximate solutions for a class of semilinear fractional-order differential equations in Banach spaces”, Fixed Point Theory and Applications, 28:4 (2017), 1–28 | MR

[13] Ph. Clement, S.-O. Londen, “On the sum of fractional derivatives and m-accretive operators”, Mathematical Research, 64 (1994), 91–100 | MR

[14] P. Egberts, “On the sum of maximal monotone operators and an application to a nonlinear integro-differential equation”, Differential Integral Equations, 6:5 (1993), 1187–1194 | MR | Zbl