The solvability of some $ p $-Laplace boundary value problems with Caputo fractional derivative are discussed. By using the fixed-point theory and analysis techniques, some existence results of one or three non-negative solutions are obtained. Two examples showed that the conditions used in this paper are somewhat easy to check.
Citation: Xiaoping Li, Dexin Chen. On solvability of some $ p $-Laplacian boundary value problems with Caputo fractional derivative[J]. AIMS Mathematics, 2021, 6(12): 13622-13633. doi: 10.3934/math.2021792
The solvability of some $ p $-Laplace boundary value problems with Caputo fractional derivative are discussed. By using the fixed-point theory and analysis techniques, some existence results of one or three non-negative solutions are obtained. Two examples showed that the conditions used in this paper are somewhat easy to check.
[1] | A. Atangana, D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., 143 (2017), D4016005. |
[2] | D. Baleanu, A. Mousalou, S. Rezapour, A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative, Adv. Diff. Eq., 2017 (2017), 52. doi: 10.1186/s13662-017-1107-4 |
[3] | D. Baleanu, A. Mousalou, S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Bound. Value Probl., 2017 (2017), 145. doi: 10.1186/s13661-017-0867-9 |
[4] | C. Chen, H. Song, H. Yang, Liouville type theorems for stable solutions of p-Laplace equation in Rn, Nonlinear Anal. Theor., 160 (2017), 44–52. doi: 10.1016/j.na.2017.05.004 |
[5] | F. Chen, D. Baleanu, G. Wu, Existence results of fractional differential equations with Riesz-Caputo derivative, Eur. Phys. J. Spec. Top., 226 (2017), 3411–3425. doi: 10.1140/epjst/e2018-00030-6 |
[6] | M. Krasnoselskii, Positive Solutions of Operator Equations, Groningen, 1964. |
[7] | R. Leggett, L. Williams, Multiple positive fixed points of nonliear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673–688. doi: 10.1512/iumj.1979.28.28046 |
[8] | Z. Li, Z. Bai, Existence of solutions for some two-point fractional boundary value problems under barrier strip conditions, Bound. Value Probl., 2019 (2019), 192. doi: 10.1186/s13661-019-01307-1 |
[9] | X. Liu, M. Jia, Solvability and numerical simulations for BVPs of fractional coupled systems involving left and right fractional derivatives, Appl. Math. Comput., 353 (2019), 230–242. |
[10] | X. Li, M. He, Monotone iterative method for fractional $p$-Laplacian differential equations with four-point boundary conditions, Adv. Differ. Equ., 2020 (2020), 686. doi: 10.1186/s13662-020-03066-1 |
[11] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
[12] | H. Salem, On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies, J. Comp. Appl. Math., 224 (2009), 567–572. |
[13] | K. Sheng, W. Zhang, Problems with p-Laplacian on time scales, Bound. Value Probl., 2018. DOI: 10.1186/s13661-018-0990-2. |
[14] | Y. Tian, Z. Bai, Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Comp. Math. Appl., 59 (2010), 2601–2609. doi: 10.1016/j.camwa.2010.01.028 |
[15] | Y. Tian, S. Sun, Z. Bai, Positive solution of fractional differential equations with p-Laplacian, J. Funct. Space, 2017, Article ID 3187492, 9 pages. Available from: https://doi.org/ 10.1155/2017/3187492. |
[16] | Y. Tian, Y. Wei, S. Sun, Multiplicity for fractional differential equations with p-Laplacian, Bound. Value Probl., 127 (2018), 1–14. |
[17] | L. Guo, L. Liu, Y. Feng, Uniqueness of iterative positive solutions for the singular infinite-point p-Laplacian fractional differential system via sequential technique, Nonlinear Anal. Model. 25 (2020), 786–805. |
[18] | L. Guo, L. Liu, Maximal and minimal iterative positive solutions for singular infinite point p-Laplacian fractional differential equations, Nonlinear Anal. Model., 23 (2018), 851–865. doi: 10.15388/NA.2018.6.3 |
[19] | B. Ahmad, J. Henderson, R. Luca, Boundary value problems for fractional differential equations and systems, Trends in Abstract and Applied Analysis, 9. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2021. |
[20] | S. Zhang, Positive solutions for boundary value problem of nonlinear fractional differential equations, Electron. J. Differ. Eq., 2006 (2006), 1–12. |
[21] | F. Yan, M. Zuo, X. Hao, Positive solution for a fractional singular boundary value problem with p-Laplacian operator, Bound. Value Probl., 2018 (2018), 51. doi: 10.1186/s13661-018-0972-4 |
[22] | M. Galewski, G. Bisci, Existence results for one-dimensional fractional equations, Math. Method. Appl. Sci., 39 (2016), 1480–1492. doi: 10.1002/mma.3582 |