## Existence of positive solutions for a kind of fractional boundary value problems.(English)Zbl 1474.34175

Summary: We are concerned with the following nonlinear three-point fractional boundary value problem: $$D_{0 +}^\alpha u (t) + \lambda a (t) f (t, u (t)) = 0$$, $$0 < t < 1$$, $$u (0) = 0$$, and $$u (1) = \beta u (\eta)$$, where $$1 < \alpha \leq 2$$, $$0 < \beta < 1$$, $$0 < \eta < 1$$, $$D_{0 +}^\alpha$$ is the standard Riemann-Liouville fractional derivative, $$a (t) > 0$$ is continuous for $$0 \leq t \leq 1$$, and $$f \geq 0$$ is continuous on $$[0,1] \times [0, \infty)$$. By using Krasnoesel’skii’s fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34A08 Fractional ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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### References:

 [1] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311, 2, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052 [2] Agarwal, R. P.; O’Regan, D.; Staněk, S., Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 371, 1, 57-68 (2010) · Zbl 1206.34009 · doi:10.1016/j.jmaa.2010.04.034 [3] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 204, 2, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456 [4] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Analysis: Theory, Methods & Applications, 72, 2, 916-924 (2010) · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033 [5] Agarwal, R. P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Applicandae Mathematicae, 109, 3, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6 [6] El-Shahed, M., Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstract and Applied Analysis, 2007 (2007) · Zbl 1149.26012 · doi:10.1155/2007/10368 [7] Xu, X.; Jiang, D.; Yuan, C., Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Analysis: Theory, Methods & Applications, 71, 10, 4676-4688 (2009) · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030 [8] Ahmad, B., Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Applied Mathematics Letters, 23, 4, 390-394 (2010) · Zbl 1198.34007 · doi:10.1016/j.aml.2009.11.004 [9] Liang, S.; Zhang, J., Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Analysis: Theory, Methods & Applications, 71, 11, 5545-5550 (2009) · Zbl 1185.26011 · doi:10.1016/j.na.2009.04.045 [10] Jaradat, O. K.; Al-Omari, A.; Momani, S., Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Analysis: Theory, Methods & Applications, 69, 9, 3153-3159 (2008) · Zbl 1160.34300 · doi:10.1016/j.na.2007.09.008
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