This paper focuses on investigating the existence of positive solutions to boundary value problems (BVPs) of higher-order nonlinear fractional differential equations (FDES) involving fractional boundary conditions. We examine the characteristics of Green's functions. By utilizing the powerful methodologies of Schauder's fixed-point theorem, upper and lower solutions, and cone theory techniques, we obtain significant existence results for nonsingular boundary value problems. For singular problems, we employ cone theory techniques in conjunction with the Leray-Schauder nonlinear alternative to obtain positive solutions. To further clarify and validate the primary findings, several illustrative examples are provided.
Citation: Yue Du, Chunyu Liang, Yujun Cui, Yumei Zou. Existence of positive solutions for nonlinear boundary value problems involving fractional boundary conditions[J]. AIMS Mathematics, 2026, 11(5): 13999-14024. doi: 10.3934/math.2026576
This paper focuses on investigating the existence of positive solutions to boundary value problems (BVPs) of higher-order nonlinear fractional differential equations (FDES) involving fractional boundary conditions. We examine the characteristics of Green's functions. By utilizing the powerful methodologies of Schauder's fixed-point theorem, upper and lower solutions, and cone theory techniques, we obtain significant existence results for nonsingular boundary value problems. For singular problems, we employ cone theory techniques in conjunction with the Leray-Schauder nonlinear alternative to obtain positive solutions. To further clarify and validate the primary findings, several illustrative examples are provided.
| [1] | I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Vol. 198, New York/London/Toronto: Academic Press, 1999. http://dx.doi.org/10.1016/s0076-5392(99)x8001-5 |
| [2] |
F. B. Tatom, The relationship between fractional calculus and fractals, Fractals, 3 (1995), 217–229. http://dx.doi.org/10.1142/S0218348X95000175 doi: 10.1142/S0218348X95000175
|
| [3] |
A. M. A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., 33 (1998), 181–186. http://dx.doi.org/10.1016/S0362-546X(97)00525-7 doi: 10.1016/S0362-546X(97)00525-7
|
| [4] |
A. A. Kilbas, J. J. Trujillo, Differential equations of fractional order: methods, results and problems, Appl. Anal., 78 (2001), 153–192. http://dx.doi.org/10.1080/00036810108840931 doi: 10.1080/00036810108840931
|
| [5] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 2006. |
| [6] | H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, 1985. http://dx.doi.org/10.1016/s0304-0208(08)x7157-9 |
| [7] |
X. Hao, L. Zhang, L. Liu, Positive solutions of higher order fractional integral boundary value problem with a parameter, Nonlinear Anal.: Model. Control, 24 (2019), 210–223. http://dx.doi.org/10.15388/NA.2019.2.4 doi: 10.15388/NA.2019.2.4
|
| [8] |
Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495–505. http://dx.doi.org/10.1063/1.3142922 doi: 10.1063/1.3142922
|
| [9] |
X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal. Theory Methods Appl., 71 (2009), 4676–4688. http://dx.doi.org/10.1016/j.na.2009.03.030 doi: 10.1016/j.na.2009.03.030
|
| [10] |
C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050–1055. http://dx.doi.org/10.1016/j.aml.2010.04.035 doi: 10.1016/j.aml.2010.04.035
|
| [11] |
Y. Zhao, S. Sun, Z. Han, Q. Li, The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2086–2097. http://dx.doi.org/10.1016/j.cnsns.2010.08.017 doi: 10.1016/j.cnsns.2010.08.017
|
| [12] |
P. Chen, X. Zhang, Y. Wang, Y. Wu, The uniqueness and iterative properties of positive solution for a coupled singulart tempered fractional system with different characteristics, Fractal Fract., 8 (2024), 636. http://dx.doi.org/10.3390/fractalfract8110636 doi: 10.3390/fractalfract8110636
|
| [13] |
X. Zhang, P. Chen, H. Tian, Y. Wu, The iterative properties for positive solutions of a tempered fractional equation, Fractal Fract., 7 (2023), 761. http://dx.doi.org/10.3390/fractalfract7100761 doi: 10.3390/fractalfract7100761
|
| [14] |
C. Zhai, M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems, Nonlinear Anal. Theory Methods Appl., 75 (2012), 2542–2551. http://dx.doi.org/10.1016/j.na.2011.10.048 doi: 10.1016/j.na.2011.10.048
|
| [15] |
S. F. Aljurbua, Extended existence results of solutions for FDEs of order $1 < \gamma\leq 2$, AIMS Math., 9 (2024), 13077–13086. http://dx.doi.org/10.3934/math.2024638 doi: 10.3934/math.2024638
|
| [16] |
N. Kosmatov, A singular boundary value problem for nonlinear differential equations of fractional order, J. Appl. Math. Comput., 29 (2009), 125–135. http://dx.doi.org/10.1007/s12190-008-0104-x doi: 10.1007/s12190-008-0104-x
|
| [17] |
X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64–69. http://dx.doi.org/10.1016/j.aml.2008.03.001 doi: 10.1016/j.aml.2008.03.001
|
| [18] |
J. R. Graef, L. Kong, B. Yang, Positive solutions for a fractional boundary value problem, Appl. Math. Lett., 56 (2016), 49–55. http://dx.doi.org/10.1016/j.aml.2015.12.006 doi: 10.1016/j.aml.2015.12.006
|
| [19] |
S. F. Aljurbua, Exploring solutions to a speciffc class of fractional differential equations of order $3 < \hat{u}\leq 4$, Bound. Value Probl., 2024 (2024), 71. http://dx.doi.org/10.1186/s13661-024-01878-8 doi: 10.1186/s13661-024-01878-8
|
| [20] |
S. F. Aljurbua, Generalized existence results for solutions of nonlinear fractional differential equations with nonlocal boundary conditions, Ain Shams Eng. J., 15 (2024), 103035. http://dx.doi.org/10.1016/j.asej.2024.103035 doi: 10.1016/j.asej.2024.103035
|
| [21] |
I. Bachar, H. Maagli, H. Eltayeb, Existence and iterative method for some Riemann fractional nonlinear boundary value problems, Mathematics, 7 (2019), 961. http://dx.doi.org/10.3390/math7100961 doi: 10.3390/math7100961
|
| [22] |
Y. Cui, C. Liang, Y. Zou, Existence and uniqueness of solutions for a class of fractional differential equation with lower-order derivative dependence, AIMS Math., 10 (2025), 3797–3818. http://dx.doi.org/10.3934/math.2025176 doi: 10.3934/math.2025176
|
| [23] |
I. Bachar, H. Maagli, H. Eltayeb, Existence and uniqueness of solutions for fractional boundary value problems under mild Lipschitz condition, J. Funct. Spaces, 2021 (2021), 1–6. http://dx.doi.org/10.1155/2021/6666015 doi: 10.1155/2021/6666015
|
| [24] |
H. A. Hammad, S. F. Aljurbua, A. S. M. Al Luhayb, Advanced fixed point methods for analyzing coupled Caputo Q-fractional boundary value problems with supportive examples, J. Funct. Spaces, 2025 (2025), 6620500. http://dx.doi.org/10.1155/jofs/6620500 doi: 10.1155/jofs/6620500
|
| [25] |
Z. Bai, The method of lower and upper solutions for a bending of an elastic beam equation, J. Math. Anal. Appl., 248 (2000), 195–202. http://dx.doi.org/10.1006/jmaa.2000.6887 doi: 10.1006/jmaa.2000.6887
|
| [26] |
Q. A. Dang, T. K. Q. Ngo, Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term, Nonlinear Anal.: Real World Appl., 36 (2017), 56–68. http://dx.doi.org/10.1016/j.nonrwa.2017.01.001 doi: 10.1016/j.nonrwa.2017.01.001
|
| [27] |
S. S. Almuthaybiri, C. C. Tisdell, Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis, Open Math., 18 (2020), 1006–1024. http://dx.doi.org/10.1515/math-2020-0056 doi: 10.1515/math-2020-0056
|
| [28] | Z. Bai, S. Zhang, S. Sun, C. Yin, Monotone iterative method for fractional differential equations, Electron. J. Differ. Equ., 2016 (2016), 1–8. |
| [29] | S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ. Equ., 2006 (2006), 1–12. |
| [30] |
X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 37 (2014), 26–33. http://dx.doi.org/10.1016/j.aml.2014.05.002 doi: 10.1016/j.aml.2014.05.002
|
| [31] | K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, Berlin: Springer, 2010. http://dx.doi.org/10.1007/978-3-642-14574-2 |
| [32] | K. Deimling, Nonlinear functional analysis, Berlin: Springer-Verlag Heidelberg, 1985. http://dx.doi.org/10.1007/978-3-662-00547-7 |
| [33] | D. Guo, V. Lakshmikantham, Nonlinear problems in abstract cones, Orlando: Academic press, 1988. http://dx.doi.org/10.1016/c2013-0-10750-7 |
| [34] | A. Granas, J. Dugundji, Fixed point theory, Springer, 2003. http://dx.doi.org/10.1007/978-0-387-21593-8 |
| [35] | R. Precup, Methods in nonlinear integral equations, Kluwer Academic Publishers, 2002. http://dx.doi.org/10.1007/978-94-015-9986-3 |
Yue Du, Chunyu Liang, Yujun Cui, Yumei Zou. Existence of positive solutions for nonlinear boundary value problems involving fractional boundary conditions[J]. AIMS Mathematics, 2026, 11(5): 13999-14024. doi: 10.3934/math.2026576
DownLoad: