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Positive solutions to nonlinear higher-order nonlocal boundary value problems for fractional differential equations. (English) Zbl 1208.34002

Summary: We study the existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equations of the type
\[ ^c{\mathcal D}^\delta_{0+}u(t)+f(t,u(t))=0,\quad 0<t<1, \]
\[ u(1)=\beta u(\eta)+\lambda_2,\;u'(0)=\alpha u'(\eta)-\lambda_1,\quad u''(0)=0,\;u'''(0)=0,\dots, u^{(n-1)}(0)=0, \]
where, \(n-1<\delta<n\), \(n(\geq 3)\in\mathbb N\), \(0<\eta,\alpha,\beta<1\), the boundary parameters \(\lambda_1,\lambda_2\in\mathbb R^+\) and \(^cD^\delta_{0+}\) is the Caputo fractional derivative. We use classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.

MSC:

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

[1] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. · Zbl 1197.01038 · doi:10.1142/9789812817747
[2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1155.35396 · doi:10.1134/S1064562406010029
[3] J. Sabatier, O. P. Agrawal, J. A. Tenreiro, and Machado, Advances in Fractional Calculus, Springer, Dordrecht, The Netherlands, 2007. · Zbl 1197.68021 · doi:10.1007/978-3-540-72504-6_33
[4] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011
[5] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[6] B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 494720, 9 pages, 2009. · Zbl 1186.34009 · doi:10.1155/2009/494720
[7] R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp. 2859-2862, 2010. · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029
[8] M. Belmekki, J. J. Nieto, and R. Rodríguez-López, “Existence of periodic solution for a nonlinear fractional differential equation,” Boundary Value Problems, vol. 2009, Article ID 324561, 18 pages, 2009. · Zbl 1181.34006 · doi:10.1155/2009/324561
[9] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677-2682, 2008. · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[10] Z. Shuqin, “Existence of solution for a boundary value problem of fractional order,” Acta Mathematica Scientia, vol. 26, no. 2, pp. 220-228, 2006. · Zbl 1106.34010 · doi:10.1016/S0252-9602(06)60044-1
[11] Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495-505, 2005. · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[12] C. Bai and J. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 150, no. 3, pp. 611-621, 2004. · Zbl 1061.34001 · doi:10.1016/S0096-3003(03)00294-7
[13] M. El-Shahed, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007. · Zbl 1149.26012 · doi:10.1155/2007/10368
[14] S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2006, pp. 1-12, 2006. · Zbl 1096.34016
[15] C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050-1055, 2010. · Zbl 1204.34007 · doi:10.1016/j.aml.2010.04.035
[16] D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 710-719, 2010. · Zbl 1192.34008 · doi:10.1016/j.na.2009.07.012
[17] S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5545-5550, 2009. · Zbl 1185.26011 · doi:10.1016/j.na.2009.04.045
[18] S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300-1309, 2010. · Zbl 1189.34050 · doi:10.1016/j.camwa.2009.06.034
[19] C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363-1375, 2010. · Zbl 1189.34014 · doi:10.1016/j.camwa.2009.06.029
[20] Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916-924, 2010. · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033
[21] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherland, 1964. · Zbl 0121.10604
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