Min-max method for some classes of Kirchhoff problems involving the $ \psi $-Hilfer fractional derivative

Hadeel Zaki Mohammed Azumi; Wafa Mohammed Ahmed Shammakh; Abdeljabbar Ghanmi; Hadeel Zaki Mohammed Azumi; Wafa Mohammed Ahmed Shammakh; Abdeljabbar Ghanmi

doi:10.3934/math.2023835

AIMS Mathematics

2023, Volume 8, Issue 7: 16308-16319. doi: 10.3934/math.2023835

Previous Article Next Article

Research article Special Issues

Min-max method for some classes of Kirchhoff problems involving the $ \psi $-Hilfer fractional derivative

1.
Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah 21493, Saudi Arabia
2.
Faculté des Sciences de Tunis, LR10ES09 Modélisation mathématique, analyse harmonique et théorie du potentiel, Université de Tunis El Manar, Tunis 2092, Tunisie

Received: 09 March 2023 Revised: 19 April 2023 Accepted: 23 April 2023 Published: 08 May 2023
MSC : 31B30, 35J35, 35J60

In this work, we develop some variational settings related to some singular $ p $-Kirchhoff problems involving the $ \psi $-Hilfer fractional derivative. More precisely, we combine the variational method with the min-max method in order to prove the existence of nontrivial solutions for the given problem. Our main result generalizes previous ones in the literature.
- variational methods,
- min-max method,
- $ p $-Laplacian operator,
- $ \psi $-Hilfer operators
Citation: Hadeel Zaki Mohammed Azumi, Wafa Mohammed Ahmed Shammakh, Abdeljabbar Ghanmi. Min-max method for some classes of Kirchhoff problems involving the $ \psi $-Hilfer fractional derivative[J]. AIMS Mathematics, 2023, 8(7): 16308-16319. doi: 10.3934/math.2023835

Related Papers:

Abstract

In this work, we develop some variational settings related to some singular $ p $-Kirchhoff problems involving the $ \psi $-Hilfer fractional derivative. More precisely, we combine the variational method with the min-max method in order to prove the existence of nontrivial solutions for the given problem. Our main result generalizes previous ones in the literature.

References

[1]	R. P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional differential equations, Georgian Math. J., 16 (2009), 401–411. https://doi.org/10.1515/GMJ.2009.401 doi: 10.1515/GMJ.2009.401
[2]	K. B. Ali, A. Ghanmi, K. Kefi, Existence of solutions for fractional differential equations with Dirichlet boundary conditions, Electron. J. Differ. Eq., 2016 (2016), 116.
[3]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[4]	R. Alsaedi, A. Ghanmi, Variational approach for the Kirchhoff problem involving the $p$-Laplace operator and the $\psi$-Hilfer derivative, Math. Method. Appl. Sci., 2023. https://doi.org/10.1002/mma.9053 doi: 10.1002/mma.9053
[5]	J. K. Brooks, Equicontinuous sets of measures and applications to Vitali's integral convergence theorem and control measures, Adv. Math., 10 (1973), 165–171. https://doi.org/10.1016/0001-8708(73)90104-7 doi: 10.1016/0001-8708(73)90104-7
[6]	W. Chen, G. Pang, A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction, J. Comput. Phys., 309 (2016), 350–367. https://doi.org/10.1016/j.jcp.2016.01.003 doi: 10.1016/j.jcp.2016.01.003
[7]	Y. Cho, I. Kim, D. Sheen, A fractional-order model for MINMOD millennium, Math. Biosci., 262 (2015), 36–45. https://doi.org/10.1016/j.mbs.2014.11.008 doi: 10.1016/j.mbs.2014.11.008
[8]	J. V. da C. Sousa, E. C. de Oliveira, On the $\psi$-fractional integral and applications, Comput. Appl. Math., 38 (2019), 4. https://doi.org/10.1007/s40314-019-0774-z doi: 10.1007/s40314-019-0774-z
[9]	J. V. da C. Sousa, J. Zuo, D. O'Regand, The Nehari manifold for a $\psi$-Hilfer fractional p-Laplacian, Appl. Anal., 101 (2022), 5076–5106. https://doi.org/10.1080/00036811.2021.1880569 doi: 10.1080/00036811.2021.1880569
[10]	J. V. da C. Sousa, E. C. de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
[11]	J. V. da C. Sousa, L. S. Tavares, C. Torres, A variational approach for a problem involving a $\psi$-Hilfer fractional operator, J. Appl. Anal. Comput., 11 (2021), 1610–1630. https://doi.org/10.11948/20200343 doi: 10.11948/20200343
[12]	R. Ezati, N. Nyamoradi, Existence of solutions to a Kirchhoff $\psi$-Hilfer fractional $p$-Laplacian equations, Math. Method. Appl. Sci., 44 (2021), 12909–12920. https://doi.org/10.1002/mma.7593 doi: 10.1002/mma.7593
[13]	A. Ghanmi, M. Althobaiti, Existence results involving fractional Liouville derivative, Bol. Soc. Paran. Mat., 39 (2021), 93–102. https://doi.org/10.5269/bspm.42010 doi: 10.5269/bspm.42010
[14]	A. Ghanmi, M. Kratou, K. Saoudi, A multiplicity results for a singular problem involving a Riemann-Liouville fractional derivative, Filomat, 32 (2018), 653–669. https://doi.org/10.2298/FIL1802653G doi: 10.2298/FIL1802653G
[15]	A. Ghanmi, S. Horrigue, Existence of positive solutions for a coupled system of nonlinear fractional differential equations, Ukr. Math. J., 71 (2019), 39–49. https://doi.org/10.1007/s11253-019-01623-w doi: 10.1007/s11253-019-01623-w
[16]	A. Ghanmi, Z. Zhang, Nehari manifold and multiplicity results for a class of fractional boundary value problems with $p$-Laplacian, Bull. Korean Math. Soc., 56 (2019), 1297–1314. https://doi.org/10.4134/BKMS.b181172 doi: 10.4134/BKMS.b181172
[17]	N. M. Grahovac, M. M. $\grave{{\rm{Z}}}$igi$\grave{{\rm{c}}}$, Modelling of the hamstring muscle group by use of fractional derivatives, Comput. Math. Appl., 59 (2010), 1695–1700. https://doi.org/10.1016/j.camwa.2009.08.011 doi: 10.1016/j.camwa.2009.08.011
[18]	R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing Company, 2000.
[19]	F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Int. J. Bifurcat. Chaos, 22 (2012), 1250086. https://doi.org/10.1142/S0218127412500861 doi: 10.1142/S0218127412500861
[20]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, elsevier, 2006.
[21]	G. Kirchhoff, Vorlesungen über mechanik, Teubner, 1897.
[22]	R. L. Magin, M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, J. Vib. Control, 14 (2008), 1431–1442. https://doi.org/10.1177/1077546307087439 doi: 10.1177/1077546307087439
[23]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
[24]	Y. A. Rossikhin, M. Shitikova, Analysis of two colliding fractionally damped spherical shells in modeling blunt human head impacts, Open Phys., 11 (2013), 760–778. https://doi.org/10.2478/s11534-013-0194-4 doi: 10.2478/s11534-013-0194-4
[25]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, theory and functions, Switzerland: Gordon and breach science publishers, 1993.
[26]	S. Saravanakumar, P. Balasubramaniam, Non-instantaneous impulsive Hilfer fractional stochastic differential equations driven by fractional Brownian motion, Stoch. Anal. Appl., 39 (2021), 549–566. https://doi.org/10.1080/07362994.2020.1815545 doi: 10.1080/07362994.2020.1815545
[27]	C. Torres, Mountain pass solution for a fractional boundary value problem, J. Fract. Calc. Appl., 5 (2014), 1–10.
[28]	C. Torres Ledesma, Boundary value problem with fractional $p$-Laplacian operator, Adv. Nonlinear Anal., 5 (2016), 133–146. https://doi.org/10.1515/anona-2015-0076 doi: 10.1515/anona-2015-0076
[29]	C. Torres Ledesma, Existence and concentration of solutions for a nonlinear fractional Schödinger equation with steep potential well, Commun. Pure Appl. Anal., 15 (2016), 535–547. https://doi.org/10.3934/cpaa.2016.15.535 doi: 10.3934/cpaa.2016.15.535
[30]	C. Torres Ledesma, Existence of solution for a general fractional advection-dispersion equation, Anal. Math. Phys., 9 (2019), 1303–1318. https://doi.org/10.1007/s13324-018-0234-8 doi: 10.1007/s13324-018-0234-8
[31]	N. H. Tuan, A. T. Nguyen, N. H. Can, Existence and continuity results for Kirchhoff parabolic equation with Caputo–Fabrizio operator, Chaos Soliton. Fract., 167 (2023), 113028. https://doi.org/10.1016/j.chaos.2022.113028 doi: 10.1016/j.chaos.2022.113028

Reader Comments

Your name:*

Email:*
© 2023 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)