MULTIPLE SOLUTIONS FOR FRACTIONAL THREE-POINT BOUNDARY VALUE PROBLEMS: A VARIATIONAL APPROACH

Wei Zhang

doi:10.11948/20240587

2026 Volume 16 Issue 3

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Wei Zhang. MULTIPLE SOLUTIONS FOR FRACTIONAL THREE-POINT BOUNDARY VALUE PROBLEMS: A VARIATIONAL APPROACH[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1608-1620. doi: 10.11948/20240587

Citation:

Wei Zhang. MULTIPLE SOLUTIONS FOR FRACTIONAL THREE-POINT BOUNDARY VALUE PROBLEMS: A VARIATIONAL APPROACH[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1608-1620. doi: 10.11948/20240587

MULTIPLE SOLUTIONS FOR FRACTIONAL THREE-POINT BOUNDARY VALUE PROBLEMS: A VARIATIONAL APPROACH

Wei Zhang^1, ,

1.
School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan, Anhui 232001, China

Corresponding author: Email: zhangwei_azyw@163.com(W. Zhang)
Fund Project: The author was supported by the Key Project of Graduate Education and Teaching Reform of Anhui Province (2024jyjxggyjY180) and the Anhui Provincial Natural Science Foundation (2208085QA05)

Abstract

Abstract

We are concerned with the following fractional boundary value problem:
$\begin{cases} {}_tD_T^\alpha ({}_0^CD_t^\alpha u(t))=\lambda f(u(t)), \quad t \in (0, T), \hfill \\ u(0) = 0, \quad u(T) = \beta u(\eta), \end{cases}$
where $ \lambda{>}0$ is a parameter, $ \beta\in \mathbb{R}$, $ \eta \in (0, T)$. By proving the appropriate fractional derivative space, we establish the variational structure of the proposed problem and derive the existence result for multiple weak solutions using a variant of the mountain pass theorem. To demonstrate the applicability of the main results, we present an example. This paper explores the application of the variational approach in solving fractional three-point BVPs. Our findings are novel and distinct from the conclusions of existing literature, holding significant theoretical value.
- Fractional differential equation /
- three-point boundary conditions /
- multiple weak solutions /
- critical point theorem
MSC: 34A08, 34B15, 34B10

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References

[1]	R. P. Agarwal and D. O'Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 2001. Google Scholar
[2]	B. Ahmad, M. Alnahdi, S. K. Ntouyas and A. Alsaedi, On a mixed nonlinear fractional boundary value problem with a new class of closed integral boundary conditions, Qual. Theory Dyn. Syst., 2023, 22(3), 96. https://doi.org/10.1007/s12346-023-00781-4. doi: 10.1007/s12346-023-00781-4 CrossRef Google Scholar
[3]	P. B. Bailey, L. F. Shampine and P. E. Waltman, Nonlinear Two Point Boundary Value Problems, Academic Press, New York, 1968. Google Scholar
[4]	G. Bonanno and G. D'Aguì, A variant of the mountain-pass theorem, Differential Integral Equations, 2013, 26(9–10), 1149–1156. Google Scholar
[5]	W. Chen, An intuitive study of fractional derivative modeling and fractional quantum in soft matter, J. Vib. Control, 2008, 14(9–10), 1651–1657. Google Scholar
[6]	W. Chen, H. G. Sun and X. C. Li. Fractional Derivative Modeling for Mechanical and Engineering Problems, Science Press, Beijing, 2010. Google Scholar
[7]	D. Craiem, F. J. Rojo, J. M. Atienza, R. L. Armentano and G. V. Guinea, Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries, Phys. Med. Biol., 2008, 53, 4543–4554. doi: 10.1088/0031-9155/53/17/006 CrossRef Google Scholar
[8]	S. Dhar and L. Kong, A critical point approach to multiplicity results for a fractional boundary value problem, Bull. Malays. Math. Sci. Soc., 2020, 43(5), 3617–3633. doi: 10.1007/s40840-020-00886-y CrossRef Google Scholar
[9]	K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Math., Springer-Verlag, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2. Google Scholar
[10]	R. Guefaifia, S. Boulaaras and F. Kamache, Existence of weak solutions for a new class of fractional boundary value impulsive systems with Riemann-Liouville derivatives, J. Integral Equations Appl., 2021, 33(3), 301–313. Google Scholar
[11]	H. A. Hammad, R. A. Rashwan, A. Nafea, M. E. Samei and S. Noeiaghdam, Stability analysis for a tripled system of fractional pantograph differential equations with nonlocal conditions, J. Vib. Control., 2024, 30(3–4), 632–647. Google Scholar
[12]	V. Hutson and J. S. Pym, Applications of Functional Analysis and Operator Theory. Mathematics in Science and Engineering, Academic Press, Inc., New York-London, 1980. Google Scholar
[13]	K. Iatime, L. Guedda and S. Djebali, System of fractional boundary value problems at resonance, Fract. Calc. Appl. Anal., 2023, 26(3), 1359–1383. doi: 10.1007/s13540-023-00157-0 CrossRef Google Scholar
[14]	F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 2011, 62(3), 1181–1199. doi: 10.1016/j.camwa.2011.03.086 CrossRef Google Scholar
[15]	F. Jiao and Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos, 2012, 22(4), 1250086. DOI: 10.1142/S0218127412500861. CrossRef Google Scholar
[16]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. Google Scholar
[17]	Y. H. Lan and Y. Zhou, Non-fragile observer-based robust control for a class of fractional-order nonlinear systems, Systems Control Lett., 2013, 62(12), 1143–1150. doi: 10.1016/j.sysconle.2013.09.007 CrossRef Google Scholar
[18]	C. E. T. Ledesma and N. Nyamoradi, ($k$,$\psi$)-Hilfer impulsive variational problem, Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A-Mat., 2023, 117(1), 42. DOI: org/10.1007/s13398-022-01377-4. CrossRef $k$,$\psi$)-Hilfer impulsive variational problem" target="_blank">Google Scholar
[19]	D. Li, Y. Li, F. Chen and Y. An, Variational formulation for the Sturm-Liouville problem of fractional differential equation with generalized ($p$,$q$)-Laplacian operator, J. Appl. Anal. Comput., 2023, 13(3), 1225–1238. $p$,$q$)-Laplacian operator" target="_blank">Google Scholar
[20]	W. Lian, Z. Bai and Z. Du, Existence of solution of a three-point boundary value problem via variational approach, Appl. Math. Lett., 2020, 104, 106283. DOI: org/10.1016/j.aml.2020.106283. CrossRef Google Scholar
[21]	X. Lin, Existence of three solutions for a three-point boundary value problem via a three-critical-point theorem, Carpathian J. Math., 2015, 31(2), 213–220. doi: 10.37193/CJM.2015.02.09 CrossRef Google Scholar
[22]	D. Min and F. Chen, Variational methods to the $p$-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem, Fract. Calc. Appl. Anal., 2021, 24(4), 1069–1093. doi: 10.1515/fca-2021-0046 CrossRef $p$-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem" target="_blank">Google Scholar
[23]	J. V. C. Sousa, L. S. Tavares and C. E. T. Ledesma, A variational approach for a problem involving a $\psi$-Hilfer fractional operator, J. Appl. Anal. Comput., 2021, 11(3), 1610–1630. $\psi$-Hilfer fractional operator" target="_blank">Google Scholar
[24]	Y. Tian and J. J. Nieto, The applications of critical-point theory to discontinuous fractional-order differential equations, Proc. Edinb. Math. Soc., 2017, 60(4), 1021–1051. doi: 10.1017/S001309151600050X CrossRef Google Scholar
[25]	Y. Wang, C. Li, H. Wu and H. Deng, Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value, Discrete Contin. Dyn. Syst. Ser. S, 2022, 15(7), 1767–1776. doi: 10.3934/dcdss.2022005 CrossRef Google Scholar
[26]	F. Yang and K. Q. Zhu, A note on the definition of fractional derivatives applied in rheology, Acta Mech. Sin., 2011, 27(6), 866–876. doi: 10.1007/s10409-011-0526-9 CrossRef Google Scholar
[27]	Q. Yang, D. Chen, T. Zhao and Y. Chen, Fractional calculus in image processing: A review, Fract. Calc. Appl. Anal., 2016, 19(5), 1222–1249. doi: 10.1515/fca-2016-0063 CrossRef Google Scholar
[28]	E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, New York, Springer, 1990. Google Scholar
[29]	W. Zhang and J. Ni, New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval, Appl. Math. Lett., 2021, 118, 107165. https://doi.org/10.1016/j.aml.2021.107165. doi: 10.1016/j.aml.2021.107165 CrossRef Google Scholar
[30]	W. Zhang, Z. Wang and J. Ni, Variational method to the fractional impulsive equation with Neumann boundary conditions, J. Appl. Anal. Comput., 2024, 14(5), 2890–2902. Google Scholar
[31]	X. Zhang, Z. Shao and Q. Zhong, Multiple positive solutions for higher-order fractional integral boundary value problems with singularity on space variable, Fract. Calc. Appl. Anal., 2022, 25(4), 1507–1526. doi: 10.1007/s13540-022-00073-9 CrossRef Google Scholar
[32]	Y. Zhang, Y. Cui and Y. Zou, Existence and uniqueness of solutions for fractional differential system with four-point coupled boundary conditions, J. Appl. Math. Comput., 2023, 69(3), 2263–2276. doi: 10.1007/s12190-022-01834-8 CrossRef Google Scholar