Existence of Solutions for a Hadamard Fractional Boundary Value Problem at Resonance
Abstract
1. Introduction
2. Preliminaries
- (i)
- for every ;
- (ii)
- for every ;
- (iii)
- , with a continuous projector such that .
- (a)
- ;
- (b)
- ; .
3. Main Result
- (A1)
- The function is a Carathéodory function, satisfying the conditions
- (a)
- is measurable for each ;
- (b)
- is continuous (in the last n variables) for a.e. ;
- (c)
- For each there exists such that
- (A2)
- The functions have bounded variations, and satisfy the resonant conditions (that is, ), and (that is, ); in addition, , where is given by (10).
- (i)
- , ;
- (ii)
- , .
- (a)
- the operator is continuous;
- (b)
- the set is bounded;
- (c)
- the operator is continuous;
- (d)
- the set is relatively compact in .
- (A3)
- There exist the nonnegative functions , a.e. , such that
- (A4)
- There exist the constants such that if for all and , if or , then either or .
- (A5)
- There exists a constant such that for all , and , if or , then either
- (1)
- For , then . So, we obtain for all . Then we find and , orBecause the determinant of the above system is , we deduce that . Hence, by , we obtain and .
- (2)
- For , then . So, we find the system
- (3)
- For , by the relationship , we obtain
- (i)
- , for any ;
- (ii)
- , for any ,
4. An Example
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Mawhin, J. Topological Degree Methods in Nonlinear Boundary Value Problems; Regional Conference Series in Mathematics No. 40; American Mathematical Society: Providence, RI, USA, 1979. [Google Scholar]
- Mawhin, J. Topological Degree and Boundary Value Problems for Nonlinear Differential Equations. In Topological Methods for Ordinary Differential Equations, Proceedings of the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.), Montecatini Terme, Italy, 24 June–2 July 1991; Springer: Berlin/Heidelberg, Germany, 1993; Volume 1537, pp. 74–142, Lecture Notes in Mathematics. [Google Scholar]
- Bohner, M.; Domoshnitsky, A.; Padhi, S.; Srivastava, S.N. Existence of solutions by coincidence degree theory for Hadamard fractional differential equations at resonance. Turk. J. Math. 2024, 48, 296–306. [Google Scholar] [CrossRef]
- Tudorache, A.; Luca, R. Positive solutions to a system of coupled Hadamard fractional boundary value problems. Fractal Fract. 2024, 8, 543. [Google Scholar] [CrossRef]
- Luca, R.; Tudorache, A. On a system of Hadamard fractional differential equations with nonlocal boundary conditions on an infinite interval. Fractal Fract. 2023, 7, 458. [Google Scholar] [CrossRef]
- Tudorache, A.; Luca, R. Positive solutions for a system of Hadamard fractional boundary value problems on an infinite interval. Axioms 2023, 12, 793. [Google Scholar] [CrossRef]
- Domoshnitsky, A.; Srivastava, S.N.; Padhi, S. Existence of solutions for a higher order Riemann-Liouville fractional differential equation by Mawhin’s coincidence degree theory. Math. Methods Appl. Sci. 2023, 46, 12018–12034. [Google Scholar] [CrossRef]
- Wang, Y.; Huang, Y.; Li, X. Positive solutions for fractional differential equation at resonance under integral boundary conditions. Demonstr. Math. 2022, 55, 238–253. [Google Scholar] [CrossRef]
- Zhang, W.; Liu, W. Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance. Bound. Value Prob. 2018, 2018, 36. [Google Scholar] [CrossRef]
- Zhang, W.; Liu, W. Existence of solutions for fractional multi-point boundary value problems on an infinite interval at resonance. Mathematics 2020, 8, 126. [Google Scholar] [CrossRef]
- Baitiche, Z.; Guerbati, K.; Hammouche, H.; Benchohra, M.; Graef, J. Sequential fractional differential equations at resonance. Funct. Differ. Equ. 2020, 26, 167–184. [Google Scholar] [CrossRef]
- Chen, T.; Liu, W.; Hu, Z. A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. 2012, 75, 3210–3217. [Google Scholar] [CrossRef]
- Chen, T.; Liu, W.; Zhang, H. Some existence results on boundary value problems for fractional p-Laplacian equation at resonance. Bound. Value Prob. 2016, 2016, 51. [Google Scholar] [CrossRef]
- Cheng, L.; Liu, W.; Ye, Q. Boundary value problem for a coupled system of fractional differential equations with p-Laplacian operator at resonance. Electr. J. Differ. Equ. 2014, 2014, 1–12. [Google Scholar]
- Djebali, S.; Aoun, A.G. Resonant fractional differential equations with multi-point boundary conditions on (0,+∞). J. Nonlinear Funct. Anal. 2019, 2019, 21. [Google Scholar]
- Ge, F.D.; Zhou, H.C.; Kou, C.H. Existence of solutions for a coupled fractional differential equations with infinitely many points boundary conditions at resonance on an unbounded domain. Differ. Equ. Dyn. Syst. 2019, 27, 395–411. [Google Scholar] [CrossRef]
- Hu, L. Existence of solutions to a coupled system of fractional differential equations with infinite-point boundary value conditions at resonance. Adv. Differ. Equ. 2016, 2016, 200. [Google Scholar] [CrossRef]
- Hu, Z.; Liu, W.; Liu, J. Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance. Adv. Differ. Equ. 2013, 2013, 312. [Google Scholar] [CrossRef]
- Jiang, W. Solvability of fractional differential equations with p-Laplacian at resonance. Appl. Math. Comput. 2015, 260, 48–56. [Google Scholar] [CrossRef]
- Jiang, W.; Huang, X.; Wang, B. Boundary value problems of fractional differential equations at resonance. Phys. Procedia 2012, 25, 965–972. [Google Scholar] [CrossRef]
- Kosmatov, N.; Jiang, W. Resonant functional problems of fractional order. Chaos Solitons Fractals 2016, 91, 573–579. [Google Scholar] [CrossRef]
- Ma, W.; Meng, S.; Cui, Y. Resonant integral boundary value problems for Caputo fractional differential equations. Math. Prob. Engin. 2018, 5438592. [Google Scholar] [CrossRef]
- Wang, Y. Necessary conditions for the existence of positive solutions to fractional boundary value problems at resonance. Appl. Math. Lett. 2019, 97, 34–40. [Google Scholar] [CrossRef]
- Wang, Y.; Liu, L. Positive solutions for a class of fractional 3-point boundary value problems at resonance. Adv. Differ. Equ. 2017, 2017, 7. [Google Scholar] [CrossRef]
- Wang, Y.; Wu, Y. Positive solutions of fractional differential equation boundary value problems at resonance. J. Appl. Anal. Comput. 2020, 10, 2459–2475. [Google Scholar]
- Zhang, W.; Liu, W. Existence of solutions for fractional multi-point boundary value problems at resonance with three-dimensional kernels. Adv. Differ. Equ. 2018, 2018, 15. [Google Scholar] [CrossRef]
- Zhang, W.; Liu, W.; Chen, T. Solvability for a fractional p-Laplacian multipoint boundary value problem at resonance on infinite interval. Adv. Differ. Equ. 2016, 2016, 183. [Google Scholar] [CrossRef]
- Zou, Y.; He, G. The existence of solutions to integral boundary value problems of fractional differential equations at resonance. J. Funct. Spaces 2017, 2785937. [Google Scholar] [CrossRef]
- Garra, R.; Mainardi, F.; Spada, G. A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Solitons Fractals 2017, 102, 333–338. [Google Scholar] [CrossRef]
- Garra, R.; Orsingher, E.; Polito, F. A note on Hadamard fractional differential equations with varying coefficients and their applications in probability. Mathematics 2018, 6, 4. [Google Scholar] [CrossRef]
- Ahmad, B.; Alsaedi, A.; Ntouyas, S.K.; Tariboon, J. Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies Series; Elsevier Science B.V.: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Luca, R.; Tudorache, A. Existence of Solutions for a Hadamard Fractional Boundary Value Problem at Resonance. Fractal Fract. 2025, 9, 119. https://doi.org/10.3390/fractalfract9020119
Luca R, Tudorache A. Existence of Solutions for a Hadamard Fractional Boundary Value Problem at Resonance. Fractal and Fractional. 2025; 9(2):119. https://doi.org/10.3390/fractalfract9020119
Chicago/Turabian StyleLuca, Rodica, and Alexandru Tudorache. 2025. "Existence of Solutions for a Hadamard Fractional Boundary Value Problem at Resonance" Fractal and Fractional 9, no. 2: 119. https://doi.org/10.3390/fractalfract9020119
APA StyleLuca, R., & Tudorache, A. (2025). Existence of Solutions for a Hadamard Fractional Boundary Value Problem at Resonance. Fractal and Fractional, 9(2), 119. https://doi.org/10.3390/fractalfract9020119