The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives. (English) Zbl 1254.34016

Summary: We study the following singular eigenvalue problem for a higher-orer fractional differential equation \[ \begin{aligned} -{\mathcal D}^\alpha x(t)=\lambda f(x(t),{\mathcal D}^{\mu_1} x(t),{\mathcal D}^{\mu_2} x(t),\dots,{\mathcal D}^{\mu_{n-1}}x(t)),\quad & 0< t< 1,\\ x(0)= 0,\quad{\mathcal D}^{\mu_i}x(0)= 0,\quad{\mathcal D}^{\mu}x(1)= \sum^{p-2}_{j=1} a_j{\mathcal D}^\mu x(\xi_j),\quad & 1\leq i\leq n-1,\end{aligned} \] where \(n\geq 3\), \(n\in\mathbb{N}\), \(n-1<\alpha\leq n\), \(n-1-1<\alpha- \mu_l< n-l\), for \(l= 1,2,\dots,n-2\), and \(\mu- \mu_{n-1}> 0\), \(\alpha-\mu_{n-1}\leq 2\), \(\alpha-\mu> 1\), \(a_j\in [0,+\infty)\), \(0<\xi_1< \xi_2<\cdots< \xi_{p-2}< 1\), \[ 0<\sum^{p-2}_{j=1} a_j \xi^{\alpha-\mu-1}_j< 1, \] \({\mathcal D}^\alpha\) is the standard Riemann-Liouville derivative, and\(f: (0,+\infty)^n\to [0,+\infty)\) is continuous. Firstly, we give the Green function and its properties. Then we established an eigenvalue interval for the existence of positive solutions from Schauder’s fixed point theorem and the lower and upper solutions method. The interesting point of this paper is that \(f\) may be singular at \(x_i= 0\), for \(i= 1,2,\dots, n\).


34A08 Fractional ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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[1] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72, 916-924 (2010) · Zbl 1187.34026
[2] Zhang, S., Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl., 59, 1300-1309 (2010) · Zbl 1189.34050
[3] Goodrich, C. S., Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23, 1050-1055 (2010) · Zbl 1204.34007
[4] Goodrich, C. S., Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl., 62, 1251-1268 (2011) · Zbl 1253.34012
[5] Wang, Y.; Liu, L.; Wu, Y., Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 74, 3599-3605 (2011) · Zbl 1220.34006
[6] Ahmad, B.; Alsaedi, A., Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations, Fixed Point Theory Appl., 2010 (2010), (Article ID 364560, 17 pages) · Zbl 1208.34001
[7] Ahmad, B.; Nieto, J., Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal., 2009 (2009), (Article ID 494720, 9 pages) · Zbl 1186.34009
[8] Miller, K.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[9] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering (1999), Academic Press: Academic Press New York, London, Toronto · Zbl 0924.34008
[10] Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and Applications of Fractional Differential Equations (2006), Elsevier B.V.: Elsevier B.V. Netherlands · Zbl 1092.45003
[11] Feng, M.; Zhang, X.; Ge, W., New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl., 2011 (2011), (Article ID 720702, 20 pages) · Zbl 1214.34005
[12] El-Shahed, M.; Nieto, J., Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. Math. Appl., 59, 3438-3443 (2010) · Zbl 1197.34003
[13] Ahmad, B.; Nieto, J., Existence results for higher order fractional differential inclusions with nonlocal boundary conditions, Nonlinear Stud., 17, 131-138 (2010) · Zbl 1206.34010
[14] Hussein Salem, A., On the fractional order \(m\)-point boundary value problem in reflexive Banach spaces and weak topologies, J. Comput. Appl. Math., 224, 565-572 (2009) · Zbl 1176.34070
[15] Rehman, M.; Khan, R., Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett., 23, 1038-1044 (2010) · Zbl 1214.34007
[16] Lakshmikantham, V.; Vatsala, A., General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21, 828-834 (2008) · Zbl 1161.34031
[17] Bhaskar, T.; Lakshmikantham, V.; Leela, S., Fractional differential equations with a Krasnoselskii-Krein type condition, Nonlinear Anal., 3, 734-737 (2009) · Zbl 1181.34008
[18] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048
[19] Zhang, X.; Han, Y., Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett., 25, 555-560 (2012) · Zbl 1244.34009
[20] Ahmad, B.; Alsaedi, A., Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type, Nonlinear Anal. Hybrid Syst., 3, 501-509 (2009) · Zbl 1179.45008
[21] Benchohraa, M.; Hamania, S.; Ntouyas, S., Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal., 71, 2391-2396 (2009) · Zbl 1198.26007
[22] Zhang, X.; Liu, L.; Wu, Y., Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Modelling, 55, 1263-1274 (2012) · Zbl 1255.34010
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