Geodesic
On two-order fractional boundary value problem with generalized Riemann-Liouville derivative
Ufa mathematical journal, Tome 15 (2023) no. 2, pp. 135-156 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper we focus our study on the existence, uniqueness and Hyers-Ulam stability for the following problem involving generalized Riemann-Liouville operators: \begin{equation*} \mathcal{D}_{0^+}^{\rho_1,\Psi} \Big(\mathcal{D}_{0^+}^{\rho_2,\Psi} + \nu \Big) \mathrm{u}(\mathfrak{t}) = \mathfrak{f}(\mathfrak{t}, \mathrm{u}(\mathfrak{t})). \end{equation*} It is well known that the existence of solutions to the fractional boundary value problem is equivalent to the existence of solutions to some integral equation. Then it is sufficient to show that the integral equation has only one fixed point. To prove the uniqueness result, we use Banach fixed point Theorem, while for the existence result, we apply two classical fixed point theorems due to Krasnoselskii and Leray-Scauder. Then we continue by studying the Hyers-Ulam stability of solutions which is a very important aspect and attracted the attention of many authors. We adapt some sufficient conditions to obtain stability results of the Hyers-Ulam type.
Keywords: fractional derivatives, generalized Riemann-Liouville derivative, fixed point theorem, fractional Boundary value problem, Hyers-Ulam stability. 
@article{UFA_2023_15_2_a11,
     author = {H. Serrai and B. Tellab and Kh. Zennir},
     title = {On two-order fractional boundary value problem with generalized {Riemann-Liouville} derivative},
     journal = {Ufa mathematical journal},
     pages = {135--156},
     year = {2023},
     volume = {15},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2023_15_2_a11/}
}
TY  - JOUR
AU  - H. Serrai
AU  - B. Tellab
AU  - Kh. Zennir
TI  - On two-order fractional boundary value problem with generalized Riemann-Liouville derivative
JO  - Ufa mathematical journal
PY  - 2023
SP  - 135
EP  - 156
VL  - 15
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UFA_2023_15_2_a11/
LA  - en
ID  - UFA_2023_15_2_a11
ER  - 
%0 Journal Article
%A H. Serrai
%A B. Tellab
%A Kh. Zennir
%T On two-order fractional boundary value problem with generalized Riemann-Liouville derivative
%J Ufa mathematical journal
%D 2023
%P 135-156
%V 15
%N 2
%U http://geodesic.mathdoc.fr/item/UFA_2023_15_2_a11/
%G en
%F UFA_2023_15_2_a11
H. Serrai; B. Tellab; Kh. Zennir. On two-order fractional boundary value problem with generalized Riemann-Liouville derivative. Ufa mathematical journal, Tome 15 (2023) no. 2, pp. 135-156. http://geodesic.mathdoc.fr/item/UFA_2023_15_2_a11/

[1] M. I. Abbas, M. A. Ragusa, “Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag-Leffler functions”, Appl. Anal., 101:9 (2021), 3231–3245 | DOI | MR

[2] M. S. Abdo, K. Shah, H. A. Wahash, S. K. Panchal, “On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative”, Chaos Solitons Fractal, 135 (2020), 109867 | DOI | MR | Zbl

[3] B. Ahmad, S. K. Ntouyas, A. Alsaedi, “A study of nonlocal integro-multi-point boundary value problems of sequential fractional integro-differential inclusions”, Ukr. Math. J., 73:6 (2021), 888–907 | DOI | MR | Zbl

[4] B. Alqahtani, A. Fulga, E. Karapinar, “Fixed point results on $\delta$-symmetric quasi-metric space via simulation function with an application to Ulam stability”, Mathematics, 6:10 (2018), 208 | DOI | MR | Zbl

[5] J. Alzabut, G. M. Selvam, R.A. El-Nabulsi, D. Vignesh, M. E. Samei, “Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions”, Symmetry, 13:3 (2021), 473 | DOI

[6] D. Baleanu, A. Jajarmi, M. Hajipour, “A new formulation of the fractional optimal control problems involving Mittag-Leffler kernel”, J. Optim. Theory Appl., 175:3 (2017), 718–737 | DOI | MR | Zbl

[7] D. Baleanu, S. Etemad, S. Rezapour, “A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions”, Bound. Value Probl., 2020 (2020), 64 | DOI | MR | Zbl

[8] D. Baleanu, S. Etemad, S. Rezapour, “On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators”, Alex. Eng. J., 59:5 (2020), 3019–3027 | DOI | MR

[9] D. Baleanu, S. Rezapour, S. Etemad, A. Alsaedi, “On a time-fractional integro-differential equation via three-point boundary value conditions”, Math. Probl. Eng., 2015 (2015), 785738 | DOI | MR | Zbl

[10] D. Boucenna, A. Boulfoul, A. Chidouh, A. Ben Makhlouf, B. Tellab, “Some results for initial value problem of nonlinear fractional equation in Sobolev space”, J. Appl. Math. Comput., 67:1-2 (2021), 605–621 | DOI | MR | Zbl

[11] A. Boulfoul, B. Tellab, N. Abdellouahab, Kh. Zennir, “Existence and uniqueness results for initial value problem of nonlinear fractional integro-differential equation on an unbounded domain in a weighted Banach space”, Math. Methods Appl. Sci., 44:5 (2021), 3509–3520 | DOI | MR | Zbl

[12] M. A. Dokuyucu, E. Celik, H. Bulut, H. M. Baskonus, “Cancer treatment model with the Caputo-Fabrizio fractional derivative”, Eur. Phys. J. Plus., 133:3 (2018), 92 | DOI

[13] S. Etemad, S. Rezapour, M. E. Samei, “On a fractional Caputo-Hadamard inclusion problem with sum boundary value conditions by using approximate endpoint property”, Math. Methods Appl. Sci., 43:17 (2020), 9719–9734 | DOI | MR | Zbl

[14] S. Etemad, B. Tellab, C. T. Deressa, J. Alzabut, Y. Li, S. Rezapour, “On a generalized fractional boundary value problem based on the thermostat model and its numerical solutions via Bernstein polynomials”, Adv. Diff. Equ, 2021 (2021), 458 | DOI | MR | Zbl

[15] S. Etemad, B. Tellab, J. Alzabut, S. Rezapour, M. I. Abbas, “Approximate solutions and Hyers-Ulam stability for a system of the coupled fractional thermostat control model via the generalized differential transform”, Adv. Diff. Equ, 2021 (2021), 428 | DOI | MR | Zbl

[16] A. Granas, J. Dugundji, Fixed point theory, Springer, New York, 2003 | MR | Zbl

[17] M. Hajipour, A. Jajarmi, D. Baleanu, “An efficient nonstandard finite difference scheme for a class of fractional chaotic systems”, J. Comput. Nonl. Dyn., 13:2 (2017), 021013 | DOI | MR

[18] D. H. Hyers, “On the stability of the linear functional equation”, Proc. Natl. Acad. Sci. USA, 27:4 (1941), 222–224 | DOI | MR | Zbl

[19] A. Jajarmi, M. Hajipour, D. Baleanu, “New aspects of the adaptive synchronization and hyperchaos suppression of a financial model”, Chaos Solitons Fractals, 99 (2017), 285–296 | DOI | MR | Zbl

[20] F. Jarad, T. Abdeljawad, “Generalized fractional derivatives and Laplace transform”, Discrete Contin, Dyn. Syst. Ser. S, 13:3 (2020), 709–722 | MR | Zbl

[21] S. A. Khan, K. Shah, G. Zaman, F. Jarad, “Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative”, Chaos Interdiscip. J. Nonl. Sci., 29:1 (2019), 013128 | DOI | MR | Zbl

[22] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of the fractional differential equations, Elsevier, Amsterdam, 2006 | MR

[23] N. Kosmatov, W. Jiang, “Resonant functional problems of fractional order”, Chaos Solitons Fractals, 91 (2016), 573–579 | DOI | MR | Zbl

[24] C. Lu, C. Fu, H. Yang, “Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions”, Appl. Math. Comput., 327 (2018), 104–116 | MR | Zbl

[25] H. Mohammadi, S. Kumar, S. Rezapour, S. Etemad, “A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control”, Chaos Solitons Fractals, 144 (2021), 110668 | DOI | MR

[26] S. Rezapour, B. Tellab, C.T. Deressa, S. Etemad, K. Nonlaopon, “H-U-type stability and numerical solutions for a nonlinear model of the coupled systems of navier bvps via the generalized differential transform method”, Fractal and Fractional, 5:4 (2021), 166 | DOI | MR

[27] S. Rezapour, S. Etemad, B. Tellab, P. Agarwal, J. L. G. Guirao, “Numerical solutions caused by $DGJIM$ and $ADM$ methods for multi-term fractional bvp involving the generalized $\psi$-RL-operators”, Symmetry, 16:4 (2021), 532 | DOI

[28] I.A. Rus, “Ulam stabilities of ordinary differential equations in a Banach space”, Carpath. J. Math., 26:1 (2010), 103–107 | MR | Zbl

[29] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, Yverdon, 1993 | MR | Zbl

[30] J. Singh, D. Kumar, Z. Hammouch, A. Atangana, “A fractional epidemiological model for computer viruses pertaining to a new fractional derivative”, Appl. Math. Comput., 316:1 (2018), 504–515 | MR | Zbl

[31] S. Sitho, S. Etemad, B. Tellab, S. Rezapour, S. K. Ntouyas, J. Tariboon, “Approximate solutions of an extended multi-order boundary value problem by implementing two numerical algorithms”, Symmetry, 13:8 (2021), 1341 | DOI

[32] W. Sudsutad, J. Alzabut, S. Nontasawatsri, C. Thaiprayoon, “Stability analysis for a generalized proportional fractional Langevin equation with variable coefficient and mixed integro-differential boundary conditions”, J. Nonl. Funct. Anal., 2020 (2020), 23

[33] S. T. M. Thabet, S. Etemad, S. Rezapour, “On a new structure of the pantograph inclusion problem in the Caputo conformable setting”, Bound. Value Probl., 2020 (2020), 171 | DOI | MR | Zbl

[34] S. T. M. Thabet, S. Etemad, S. Rezapour, “On a coupled Caputo conformable system of Pantograph problems”, Turk. J. Math., 45:1 (2021), 496–519 | DOI | MR | Zbl

[35] S. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964 | MR | Zbl

[36] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999 | MR | Zbl