Filippov-type existence theorems for $ q $-fractional differential inclusions with nonlinear $ q $-integral conditions in Banach spaces

Taher S. Hassan; Ali Rezaiguia; Loredana Florentina Iambor; Ismoil Odinaev; Taher S. Hassan; Ali Rezaiguia; Loredana Florentina Iambor; Ismoil Odinaev

doi:10.3934/math.2026182

AIMS Mathematics

2026, Volume 11, Issue 2: 4539-4556. doi: 10.3934/math.2026182

Previous Article Next Article

Research article Special Issues

Filippov-type existence theorems for $ q $-fractional differential inclusions with nonlinear $ q $-integral conditions in Banach spaces

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il, 2440, Saudi Arabia
2.
Jadara University Research Center, Jadara University, Irbid 21110, Jordan
3.
Department of Mathematics, Faculty of Science, Mansoura University Mansoura, 35516, Egypt
4.
Department of Mathematics, Faculty of Science and Technology, Souk Ahras University, 41000 Souk Ahras, Algeria
5.
Department of Mathematics and Computer Science, University of Oradea, Univeritatii nr. 1, 410087 Oradea, Romania
6.
Department of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, 620002 Yekaterinburg, Russia

Received: 23 October 2025 Revised: 07 January 2026 Accepted: 28 January 2026 Published: 13 February 2026
MSC : 34A60, 39A13, 47H04

This study focused on boundary value problems involving fractional $ q $-difference inclusions subject to nonlinear $ q $-integral conditions:
$ \begin{equation*} \begin{array}{c} ^{c}D_{q, \mathcal{\gamma }}\left( \nu \left( \iota \right) -h\left( \iota , \nu \left( \iota \right) \right) \right) \in F\left( \iota , \nu \left( \iota \right) \right) , \text{ }\iota \in \left[ 0, \ell \right] , 1< \mathcal{\gamma }\leq 2, \\ \nu \left( 0\right) -\nu ^{\prime }\left( 0\right) = a\left( \iota \right) \int_{0}^{\ell }\mathcal{G}_{1}\left( \tau , \nu \left( \tau \right) \right) d_{q}\tau , \\ \nu \left( \ell \right) -\nu ^{\prime }\left( \ell \right) = b\left( \iota \right) \int_{0}^{\ell }\mathcal{G}_{2}\left( \tau , \nu \left( \tau \right) \right) d_{q}\tau . \end{array} \end{equation*} $
By applying appropriate fixed point theorems, we established the existence of solutions to the problem considered. In addition, a Filippov-type result was provided. This work continued the study recently presented in [N. Allouch, J. R. Graef, S. Hamani, Boundary value problem for fractional $ q $-difference equations with integral conditions in Banach spaces, Fractal Fract., 6 (2022), 237].
- Caputo fractional $ q $-difference inclusion,
- fixed point theorems,
- integral boundary conditions
Citation: Taher S. Hassan, Ali Rezaiguia, Loredana Florentina Iambor, Ismoil Odinaev. Filippov-type existence theorems for $ q $-fractional differential inclusions with nonlinear $ q $-integral conditions in Banach spaces[J]. AIMS Mathematics, 2026, 11(2): 4539-4556. doi: 10.3934/math.2026182

Related Papers:

Abstract

This study focused on boundary value problems involving fractional $ q $-difference inclusions subject to nonlinear $ q $-integral conditions:

$ \begin{equation*} \begin{array}{c} ^{c}D_{q, \mathcal{\gamma }}\left( \nu \left( \iota \right) -h\left( \iota , \nu \left( \iota \right) \right) \right) \in F\left( \iota , \nu \left( \iota \right) \right) , \text{ }\iota \in \left[ 0, \ell \right] , 1< \mathcal{\gamma }\leq 2, \\ \nu \left( 0\right) -\nu ^{\prime }\left( 0\right) = a\left( \iota \right) \int_{0}^{\ell }\mathcal{G}_{1}\left( \tau , \nu \left( \tau \right) \right) d_{q}\tau , \\ \nu \left( \ell \right) -\nu ^{\prime }\left( \ell \right) = b\left( \iota \right) \int_{0}^{\ell }\mathcal{G}_{2}\left( \tau , \nu \left( \tau \right) \right) d_{q}\tau . \end{array} \end{equation*} $

By applying appropriate fixed point theorems, we established the existence of solutions to the problem considered. In addition, a Filippov-type result was provided. This work continued the study recently presented in [N. Allouch, J. R. Graef, S. Hamani, Boundary value problem for fractional $ q $-difference equations with integral conditions in Banach spaces, Fractal Fract., 6 (2022), 237].

References

[1]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: models and numerical methods, 2 Eds., World Scientific, 2016. https://doi.org/10.1142/10044
[2]	M. Benchohra, J. Henderson, D. Seba, Boundary value problems for fractional differential inclusions in Banach spaces, Fract. Differ. Calc., 2 (2012), 99–108. https://doi.org/10.7153/fdc-02-07 doi: 10.7153/fdc-02-07
[3]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[4]	A. Rezaiguia, S. Kelaiaia, Existence results for third-order differential inclusions with three-point boundary value problems, Acta Math. Univ. Comenian., 85 (2016), 311–318.
[5]	M. H. Annaby, Z. S. Mansour, $q$-fractional calculus and equations, Berlin, Heidelberg: Springer, 2012. https://doi.org/10.1007/978-3-642-30898-7
[6]	D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional dynamics and control, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-0457-6
[7]	A. Alsaedi, B. Alharbi, B. Ahmad, Higher-order nonlocal multipoint $q$-integral boundary value problems for fractional $q$-difference equations with dual hybrid terms, Open Math., 22 (2024), 20240051. https://doi.org/10.1515/math-2024-0051 doi: 10.1515/math-2024-0051
[8]	B. Alharbi, A. Alsaedi, R. P. Agarwal, B. Ahmad, Existence results for a nonlocal $q$-integro multipoint boundary value problem involving a fractional $q$-difference equation with dual hybrid terms, An. Şt. Univ. Ovidius Constanţa, 32 (2024), 5–31. https://doi.org/10.2478/auom-2024-0026
[9]	A. Alsaedi, B. Ahmad, H. Al-Hutami, Nonlinear multi-term impulsive fractional $q$-difference equations with closed boundary conditions, Qual. Theory Dyn. Syst., 23 (2024), 67. https://doi.org/10.1007/s12346-023-00934-5 doi: 10.1007/s12346-023-00934-5
[10]	M. A. A. Ali, S. T. M. Thabet, T. Abdeljawad, I. Kedim, New results on an impulsive differential equation involving a $q$-analogue of the $\psi$-Caputo fractional derivative, Bound. Value Probl., 2025 (2025), 139. https://doi.org/10.1186/s13661-025-02133-4 doi: 10.1186/s13661-025-02133-4
[11]	A. Cernea, Filippov lemma for a class of Hadamard-type fractional differential inclusions, Fract. Calc. Appl. Anal., 18 (2015), 163–171. https://doi.org/10.1515/fca-2015-0011 doi: 10.1515/fca-2015-0011
[12]	A. Cernea, Continuous version of Filippov's theorem for a Sturm-Liouville type differential inclusion, Electron. J. Differ. Equ., 2008 (2008), 1–7.
[13]	A. Cernea, A Filippov-type existence theorem for some nonlinear $q$-difference inclusions, In: Differential and difference equations with applications, Cham: Springer, 2016, 71–77. https://doi.org/10.1007/978-3-319-32857-7_7
[14]	A. Rezaiguia, S. Kelaiaia, Existence of solution, Filippov's theorem and compactness of the set of solutions for a third-order differential inclusion with three-point boundary conditions, Mathematics, 6 (2018), 40. https://doi.org/10.3390/math6030040 doi: 10.3390/math6030040
[15]	A. Rezaiguia, T. S. Hassan, Topological structure and existence of solutions set for $q$-fractional differential inclusion in Banach space, Mathematics, 11 (2023), 683. https://doi.org/10.3390/math11030683 doi: 10.3390/math11030683
[16]	L. P. Castro, On the solutions to a Riemann-Liouville fractional $q$-derivative boundary value problem, Fract. Calc. Appl. Anal., 28 (2025), 2302–2332. https://doi.org/10.1007/s13540-025-00441-1 doi: 10.1007/s13540-025-00441-1
[17]	B. Alqahtani, S. Abbas, M. Benchohra, S. S. Alzaid, Fractional $q$-difference inclusions in Banach spaces, Mathematics, 8 (2020), 91. https://doi.org/10.3390/math8010091 doi: 10.3390/math8010091
[18]	N. Allouch, J. R. Graef, S. Hamani, Boundary value problem for fractional $q$-difference equations with integral conditions in Banach spaces, Fractal Fract., 6 (2022), 237. https://doi.org/10.3390/fractalfract6050237 doi: 10.3390/fractalfract6050237
[19]	M. Kisielewicz, Differential inclusions and optimal control, Dordrecht: Springer, 1991.
[20]	C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Berlin, Heidelberg: Springer, 1977. https://doi.org/10.1007/BFb0087685
[21]	J. P. Aubin, H. Frankowska, Differential inclusions, In: Set-valued analysis, Boston, MA: Birkhäuser, 2009, 1–27. https://doi.org/10.1007/978-0-8176-4848-0_10
[22]	K. Deimling, Multivalued differential equations, Berlin: Walter de Gruyter, 1992.
[23]	S. Hu, N. S. Papageorgiou, Handbook of multivalued analysis. Volume I: Theory, Dordrecht: Kluwer Academic Publishers, 1997.
[24]	R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Math. Proc. Cambridge Philos. Soc., 66 (1969), 365–370. https://doi.org/10.1017/S0305004100045060 doi: 10.1017/S0305004100045060
[25]	P. Rajković, S. Marinković, M. Stanković, On $q$-analogues of Caputo derivative and Mittag-Leffler function, Fract. Calc. Appl. Anal., 10 (2007), 359–373.
[26]	R. P. Agarwal, B. Ahmad, H. Al-Hutami, A. Alsaedi, Existence results for nonlinear multi-term impulsive fractional integro-difference equations with nonlocal boundary conditions, AIMS Math., 8 (2023), 19313–19333. https://doi.org/10.3934/math.2023985 doi: 10.3934/math.2023985

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)