Unified approach to nonlinear Caputo fractional derivative boundary value problems: extending the upper and lower solutions method
Authors
I. Talib
- Nonlinear Analysis Group, Department of Mathematics, Virtual University of Pakistan, Pakistan.
A. Batool
- Department of Mathematics, University of Management and Technology, Lahore, Pakistan.
J. V. da C. Sousa
- Aerospace Engineering, PPGEA-UEMA, Department of Mathematics, DEMATI-UEMA, Sao Luis, MA 65054, Brazil.
M. Lamine
- Mathematics Department, Faculty of Sciences of Tunisia, University of Tunis El Manar, Tunisia.
Abstract
The upper and lower solutions approach has been extended in this research to address nonlinear Caputo FDBVPs. This study proposes generalized findings that unify the existence criteria of specific FDBVPs that have previously been handled separately in the literature. This includes both Dirichlet FDBVPs and Neumann FDBVPs, which are treated as special cases. In addition, we extend the results presented in [A. Batool, I. Talib, M. B. Riaz, C. Tunc, Arab J. Basic Appl. Sci., \(\bf 29\) (2022), 249--256], [A. Batool, I. Talib, R. Bourguiba, I. Suwan, T. Abdeljawad, M. B. Riaz, Int. J. Nonlinear Sci. Numer. Simul., \(\bf 24\) (2023), 2145--2154] and [D. Franco, D. O'Regan, Arch. Inequal. Appl., \(\bf 1\) (2003), 413--419]. To assess the validity of the established results, two examples are considered for examination.
Share and Cite
ISRP Style
I. Talib, A. Batool, J. V. da C. Sousa, M. Lamine, Unified approach to nonlinear Caputo fractional derivative boundary value problems: extending the upper and lower solutions method, Journal of Mathematics and Computer Science, 37 (2025), no. 1, 20--31
AMA Style
Talib I., Batool A., Sousa J. V. da C. , Lamine M., Unified approach to nonlinear Caputo fractional derivative boundary value problems: extending the upper and lower solutions method. J Math Comput SCI-JM. (2025); 37(1):20--31
Chicago/Turabian Style
Talib, I., Batool, A., Sousa, J. V. da C. , Lamine, M.. "Unified approach to nonlinear Caputo fractional derivative boundary value problems: extending the upper and lower solutions method." Journal of Mathematics and Computer Science, 37, no. 1 (2025): 20--31
Keywords
- Upper and lower solutions
- fractional derivative differential equations
- Caputo fractional derivative
- Dirichlet boundary conditions
- Neumann boundary conditions
MSC
References
-
[1]
B. Ahmad, M. Alghanmi, S. K. Ntouyas, A. Alsaedi, A study of fractional differential equations and inclusions involving generalized Caputo-type derivative equipped with generalized fractional integral boundary conditions, AIMS Math., 4 (2019), 26–42
-
[2]
A. Al Mosa, P. Eloe, Upper and lower solutions method for boundary value problems at resonance, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 13 pages
-
[3]
M. Al-Refai, On the Fractional Derivative at Extreme Points, Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 5 pages
-
[4]
A. Batool, I. Talib, R. Bourguiba, I. Suwan, T. Abdeljawad, M. B. Riaz, A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems, Int. J. Nonlinear Sci. Numer. Simul., 24 (2023), 2145–2154
-
[5]
A. Batool, I. Talib, M. B. Riaz, C. Tunç, Extension of lower and upper solutions approach for generalized nonlinear fractional boundary value problems, Arab J. Basic Appl. Sci., 29 (2022), 249–256
-
[6]
M. Bohner, O. Tunç, C. Tunç, Qualitative analysis of Caputo fractional integro-differential equations with constant delays, Comput. Appl. Math., 40 (2021), 17 pages
-
[7]
A. Cabada, L. López-Somoza, Lower and upper solutions for even order boundary value problems, Mathematics, 7 (2019), 1–20
-
[8]
C. Chen, M. Bohner, B. Jia, Method of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications, Fract. Calc. Appl. Anal., 22 (2019), 1307–1320
-
[9]
C. De Coster, P. Habets, Two-point boundary value problems: lower and upper solutions, Elsevier, Amsterdam (2006)
-
[10]
K. Diethelm, The analysis of fractional differential equations, In: Lecture Notes in Mathematics, Springer-Verlag, Berlin (2010)
-
[11]
D. Franco, D. Ó Regan, A new upper and lower solutions approach for second order problems with nonlinear boundary conditions, Arch. Inequal. Appl., 1 (2003), 413–419
-
[12]
B. Ghanbari, S. Kumar, R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), 11 pages
-
[13]
Jeelani, A. M. Saeed, M. S. Abdo, K. Shah, Positive solutions for fractional boundary value problems under a generalized fractional operator, Math. Methods Appl. Sci., 2021 (2021), 9524–9540
-
[14]
N. Kadkhoda, A numerical approach for solving variable order differential equations using Bernstein polynomials, Alex. Eng. J., 59 (2020), 3041–3047
-
[15]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam (2006)
-
[16]
S. Kumar, S. Ghosh, B. Samet, E. F. D. Goufo, An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator, Math. Methods Appl. Sci., 43 (2020), 6062–6080
-
[17]
S. Kumar, A. Kumar, S. Abbas, M. Al Qurashi, D. Baleanu, A modified analytical approach with existence and uniqueness for fractional Cauchy reaction-diffusion equations, Adv. Differ. Equ., 2020 (2020), 18 pages
-
[18]
L. Lin, X. Liu, H. Fang, Method of lower and upper solutions for fractional differential equations, Electron. J. Differ. Equ., 2012 (2012), 13 pages
-
[19]
X. Liu, M. Jia, The method of lower and upper solutions for the general boundary value problems of fractional differential equations with p-Laplacian, Adv. Differ. Equ., 2018 (2018), 15 pages
-
[20]
Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218–223
-
[21]
S. Muthaiah, D. Baleanu, Existence of solutions for nonlinear fractional differential equations and inclusions depending on lower-order fractional derivatives, Axioms, 9 (2020), 17 pages
-
[22]
S. K. Ntouyas, S. Etemad, On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, Appl. Math. Comput., 266 (2015), 235–243
-
[23]
A. Shi, S. Zhang, Upper and lower solutions method and a fractional differential equation boundary value problem, Electron. J. Qual. Theory Differ. Equ., 2009 (2009), 13 pages
-
[24]
G. F. Simmons, Introduction to topology and modern analysis, McGraw-Hill Book Co., Inc., New York-San Francisco, Calif.-Toronto-London (1963)
-
[25]
M. Subramanian, M. Manigandan, C. Tunç, T. N. Gopal, J. Alzabut, On system of nonlinear coupled differential equations and inclusions involving Caputo-type sequential derivatives of fractional order, J. Taibah Univ. Sci., 16 (2022), 1–23
-
[26]
I. Talib, N. A. Asif, C. Tunç, Existence of solutions to second-order nonlinear coupled systems with nonlinear coupled boundary conditions, Electron. J. Differ. Equ., 2015 (2015), 11 pages
-
[27]
D. G. Zill, M. R. Cullen, W. S.Wright, Differential equations with boundary-value problems, Cengage Learning, Boston, MA (2013)
