The method of lower and upper solutions for mixed fractional four-point boundary value problem with \(p\)-Laplacian operator. (English) Zbl 1357.34018
In this paper, the authors consider a mixed fractional four-point boundary value problem with \(p\)-Laplacian operator. By using the monotone iterative technique and lower and upper solutions method, some new results on the existence of positive solutions for the four-point boundary value problem are established. At last, an example is given to illustrate the potential applications of the main results. This paper extends single fractional derivative to mixed fractional derivatives.
Reviewer: Wengui Yang (Sanmenxia)
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
Keywords:
fractional differential equation; mixed fractional derivatives; four-point boundary value problem; positive solution; \(p\)-Laplacian operator; lower and upper solutionsReferences:
| [1] | Diethelm, K., The analysis of fractional differential equations, (Lectures Notes in Mathematics (2010), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1215.34001 |
| [2] | Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier B.V: Elsevier B.V Amsterdam, The Netherlands · Zbl 1092.45003 |
| [3] | Zhou, Y., Basic Theory of Fractional Differential Equations (2014), World Scientific: World Scientific Singapore · Zbl 1336.34001 |
| [4] | Bai, Z.; Sun, W., Existence and multiplicity of positive solutions for singular fractional boundary value problems, Comput. Math. Appl., 63, 1369-1381 (2012) · Zbl 1247.34006 |
| [5] | Bai, Z.; Zhang, Y., Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput., 218, 1719-1725 (2011) · Zbl 1235.34007 |
| [6] | Ma, D., Positive solutions of multi-point boundary value problem of fractional differential equation, Arab. J. Math. Sci., 21, 225-236 (2015) · Zbl 1317.34015 |
| [7] | Ge, F.; Kou, C., Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations, Appl. Math. Comput., 257, 308-316 (2015) · Zbl 1338.34103 |
| [8] | Feng, W., Topological methods on solvability, multiplicity and eigenvalues of a nonlinear fractional boundary value problem, Electron. J. Qual. Theory Differ. Equ., 2015, 70, 1-16 (2015) · Zbl 1349.47139 |
| [9] | Wang, J.; Zhou, Y.; Wei, W., A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun. Nonlinear Sci. Numer. Simul., 16, 4049-4059 (2011) · Zbl 1223.45007 |
| [10] | Yang, L.; Shen, C.; Xie, D., Multiple positive solutions for nonlinear boundary value problem of fractional order differential equation with the Riemann-Liouville derivative, Adv. Differential Equations, 284, 1-12 (2014) · Zbl 1353.34011 |
| [11] | Liu, X.; Jia, M., Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations, Math. Methods Appl. Sci., 39, 475-487 (2016) · Zbl 1333.34010 |
| [12] | Ntouyas, S. K.; Etemad, S., On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, Appl. Math. Comput., 266, 235-243 (2015) · Zbl 1410.34028 |
| [13] | Liu, X.; Lin, L.; Fang, H., Existence of positive solutions for nonlocal boundary value problem of fractional differential equation, Cent. Eur. J. Phys., 11, 1423-1432 (2013) |
| [14] | Jia, M.; Liu, X., Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions, Appl. Math. Comput., 232, 313-323 (2014) · Zbl 1410.34019 |
| [15] | Leibenson, L. S., General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk Kirgizskoi SSSR, 9, 7-10 (1983), (in Russian) |
| [16] | Liu, Y.; Yang, X., Resonant boundary value problems for singular multi-term fractional differential equations, J. Differ. Equ. Appl., 5, 409-472 (2013) · Zbl 1322.34010 |
| [17] | Liu, X.; Jia, M.; Xiang, X., On the solvability of a fractional differential equation model involving the \(p\)-Laplacian operator, Comput. Math. Appl., 64, 3267-3275 (2012) · Zbl 1268.34020 |
| [18] | Jafari, H.; Baleanu, D.; Khan, H., Existence criterion for the solutions of fractional order \(p\)-Laplacian boundary value problems, Bound. Value Probl. (1), 1-10 (2015) · Zbl 1381.34016 |
| [19] | Han, Z.; Lu, H.; Zhang, C., Positive solutions for eigenvalue problems of fractional differential equation with generalized \(p\)-Laplacian, Appl. Math. Comput., 257, 526-536 (2015) · Zbl 1338.34015 |
| [20] | Liu, X.; Jia, M.; Ge, W., Multiple solutions of a \(p\)-Laplacian model involving a fractional derivative, Adv. Differential Equations, 126, 1-12 (2013) · Zbl 1390.34059 |
| [21] | Tang, X.; Yan, C.; Liu, Q., Existence of solutions of two-point boundary value problems for fractional \(p\)-Laplace differential equations at resonance, J. Appl. Math. Comput., 41, 119-131 (2013) · Zbl 1296.34029 |
| [22] | Liu, Z.; Lu, L., A class of BVPs for nonlinear fractional differential equations with \(p\)-Laplacian operator, Electron. J. Qual. Theory Differ. Equ., 70, 1-16 (2012) · Zbl 1340.34299 |
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