Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications
Abstract
1. Introduction
2. Preliminaries
- (i)
- ;
- (ii)
- ;
- (iii)
- .
- (i)
- ;
- (ii)
- .
3. Main Results
- (i)
- (ii)
- (iii)
- (i)
- (ii)
- (iii)
4. Applications
4.1. Singular Problems
4.2. Semipositone Problem
- (i)
- (ii)
- (iii)
- (1)
- ;
- (2)
- ,
5. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
GF | Green’s function |
FDEs | fractional differential equations |
BC | boundary condition |
R-L | Riemann–Liouville |
FBVPs | fractional boundary value problems |
BVPs | boundary value problems |
References
- Kilbas, A.A.; Srivastava, H.M.; Nieto, J.J. Theory and Applicational Differential Equations; Elsevier Science B.V.: Amsterdam, The Netherlands, 2006; pp. 69–71. [Google Scholar]
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
- Area, I.; Losada, J.; Ndaïrou, F.; Nieto, J.J.; Tcheutia, D.D. Mathematical modeling of 2014 Ebola outbreak. Math. Methods Appl. Sci. 2017, 40, 6114–6122. [Google Scholar] [CrossRef]
- Ndaïrou, F.; Khalighi, M.; Lahti, L. Ebola epidemic model with dynamic population and memory. Chaos Solitons Fractals 2023, 170, 113361. [Google Scholar] [CrossRef]
- Elshehawey, E.; Elbarbary, E.; Afifi, N.; El-Shahed, M. On the solution of the endolymph equation using fractional calculus. Appl. Math. Comput. 2001, 124, 337–341. [Google Scholar] [CrossRef]
- Cui, Y. Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 2016, 51, 48–54. [Google Scholar]
- Zhang, X.; Zhong, Q. Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 2018, 80, 12–19. [Google Scholar] [CrossRef]
- Henderson, J.; Luca, R. Existence of positive solutions for a singular fractional boundary value problem. Nonlinear Anal-Model. 2017, 22, 99–114. [Google Scholar] [CrossRef]
- Xu, J.; Liu, J.; O’Regan, D. Solvability for a Hadamard-type fractional integral boundary value problem. Nonlinear Anal-Model. 2023, 28, 672–696. [Google Scholar] [CrossRef]
- Wang, F.; Liu, L.; Wu, Y. Existence and uniqueness of solutions for a class of higher-order fractional boundary value problems with the nonlinear term satisfying some inequalities. J. Inequalities Appl. 2020, 2020, 196. [Google Scholar] [CrossRef]
- Li, C.; Guo, L. Positive solutions and their existence of a nonlinear hadamard Fractional-order differential equation with a singular source Item using spectral analysis. Fractal Fract. 2024, 8, 377. [Google Scholar] [CrossRef]
- Jiang, D.; Yuan, C. The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. Theory Methods Appl. 2010, 72, 710–719. [Google Scholar] [CrossRef]
- Graef, J.R.; Kong, L.; Wang, M. Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition. Electron. J. Qual. Theory Differ. Equ. 2013, 55, 1–11. [Google Scholar] [CrossRef]
- Graef, J.R.; Kong, L.; Kong, Q.; Wang, M. On a fractional boundary value problem with a perturbation term. J. Appl. Anal. Comput. 2017, 7, 57–66. [Google Scholar]
- Zou, Y. Positive Solutions for a Fractional Boundary Value Problem with a Perturbation Term. J. Funct. Space 2018, 2018, 9070247. [Google Scholar] [CrossRef]
- Gao, Z.; Wang, M. Existence of solutions for a nonlocal fractional boundary value problem. Rocky Mt. J. Math. 2018, 48, 831–843. [Google Scholar] [CrossRef]
- Wanassi, O.K.; Toumi, F. Existence results for perturbed boundary value problem with fractional order. Ric. Mat. 2024, 73, 1367–1383. [Google Scholar] [CrossRef]
- Wang, Y. A New Result Regarding Positive Solutions for Semipositone Boundary Value Problems of Fractional Differential Equations. Fractal. Fract. 2025, 9, 110. [Google Scholar] [CrossRef]
- Wang, Y.; Liu, L. Positive properties of the Green function for two-term fractional differential equations and its application. J. Nonlinear Sci. Appl. 2017, 10, 2094–2102. [Google Scholar] [CrossRef]
- Cabada, A.; Dimitrov, N. Nontrivial solutions of non-autonomous dirichlet fractional discrete problems. Fract. Calc. Appl. Anal. 2020, 23, 980–995. [Google Scholar] [CrossRef]
- Cabada, A.; Dimitrov, N.; Jonnalagadda, J.M. Green’s functions for fractional difference equations with Dirichlet boundary conditions. Chaos Solitons Fractals 2021, 153, 111455. [Google Scholar] [CrossRef]
- Guo, D. Nonlinear Functional Analysis; Shandong Science and Technology Press: Jinan, China, 1985. [Google Scholar]
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Wang, Y. Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications. Fractal Fract. 2025, 9, 261. https://doi.org/10.3390/fractalfract9040261
Wang Y. Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications. Fractal and Fractional. 2025; 9(4):261. https://doi.org/10.3390/fractalfract9040261
Chicago/Turabian StyleWang, Yongqing. 2025. "Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications" Fractal and Fractional 9, no. 4: 261. https://doi.org/10.3390/fractalfract9040261
APA StyleWang, Y. (2025). Positive Properties of Green’s Function for Fractional Dirichlet Boundary Value Problem with a Perturbation Term and Its Applications. Fractal and Fractional, 9(4), 261. https://doi.org/10.3390/fractalfract9040261