This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $ 0\leq c < d $ and fixed points at the end of the interval $ [0, d] $. For $ i = 0, 1 $, the boundary conditions are as follows: $ a_{i}, b_{i} > 0 $, $ a_{0} p(c) = -b_{0} p(d), \ a_{1} p^{'}(c) = -b_{1} p^{'}(d) $. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.
Citation: Saleh Fahad Aljurbua. Extended existence results for FDEs with nonlocal conditions[J]. AIMS Mathematics, 2024, 9(4): 9049-9058. doi: 10.3934/math.2024440
This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $ 0\leq c < d $ and fixed points at the end of the interval $ [0, d] $. For $ i = 0, 1 $, the boundary conditions are as follows: $ a_{i}, b_{i} > 0 $, $ a_{0} p(c) = -b_{0} p(d), \ a_{1} p^{'}(c) = -b_{1} p^{'}(d) $. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.
| [1] | I. Podlubny, Fractional differential equations, In: Mathematics in science and engineering, Elsevier, 198 (1999). |
| [2] | S. Abbas, M. Benchohra, G. M. N'Guerekata, Topics in fractional differential equations, New York: Springer, 2012. https://doi.org/10.1007/978-1-4614-4036-9 |
| [3] | D. R. Smart, Fixed point theorems, Cambridge University Press, 1980. |
| [4] | Y. Zhou, Fractional evolution equations and inclusions: Analysis and control, Academic Press, 2016. https://doi.org/10.1016/C2015-0-00813-9 |
| [5] |
A. Ali, I. Suwan, T. Abdeljawad, Abdullah, Numerical simulation of time partial fractional diffusion model by Laplace transform, AIMS Mathematics, 7 (2022), 2878–2890. https://doi.org/10.3934/math.2022159 doi: 10.3934/math.2022159
|
| [6] |
M. M. Al-Sawalha, O. Y. Ababneh, R. Shah, N. A. Shah, K. Nonlaopon, Combination of Laplace transform and residual power series techniques of special fractional-order non-linear partial differential equations, AIMS Mathematics, 8 (2023), 5266–5280. https://doi.org/10.3934/math.2023264 doi: 10.3934/math.2023264
|
| [7] |
X. Zuo, W. Wang, Existence of solutions for fractional differential equation with periodic boundary condition, AIMS Mathematics, 7 (2022), 6619–6633. https://doi.org/10.3934/math.2022369 doi: 10.3934/math.2022369
|
| [8] | B. Ahmad, J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via leray-Schauder degree theory, Topol. Methods Nonlinear Anal., 35 (2010), 295–304. |
| [9] |
B. Ahmad, V. Otero-Espinar, Existence of solutions for fractional differential inclusions with antiperiodic boundary conditions, Bound. Value Probl., 2009 (2009), 625347. https://doi.org/10.1155/2009/625347 doi: 10.1155/2009/625347
|
| [10] | R. P. Agarwal, B. Ahmad, J. J. Nieto, Fractional differential equations with nonlocal (parametric type) anti-periodic boundary conditions, Filomat, 31 (2017), 1207–1214. |
| [11] |
A. Hyder, M. A. Barakat, D. Rizk, R. Shah, K. Nonlaopon, Study of HIV model via recent improved fractional differential and integral operators, AIMS Mathematics, 8 (2023), 1656–1671. https://doi.org/10.3934/math.2023084 doi: 10.3934/math.2023084
|
| [12] |
M. D. Johansyah, A. K. Supriatna, E. Rusyaman, J. Saputra, Application of fractional differential equation in economic growth model: A systematic review approach, AIMS Mathematics, 6 (2021), 10266–10280. https://doi.org/10.3934/math.2021594 doi: 10.3934/math.2021594
|
| [13] |
C. Derbazi, H. Hammouche, Caputo-Hadamard fractional differential equations with nonlocal fractional integro-differential boundary conditions via topological degree theory, AIMS Mathematics, 5 (2020), 2694–2709. https://doi.org/10.3934/math.2020174 doi: 10.3934/math.2020174
|
| [14] |
M. Manjula, K. Kaliraj, T. Botmart, K. S. Nisar, C. Ravichandran, Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses, AIMS Mathematics, 8 (2023), 4645–4665. https://doi.org/10.3934/math.2023229 doi: 10.3934/math.2023229
|
Saleh Fahad Aljurbua. Extended existence results for FDEs with nonlocal conditions[J]. AIMS Mathematics, 2024, 9(4): 9049-9058. doi: 10.3934/math.2024440
DownLoad: