Existence results for singular boundary value problem of nonlinear fractional differential equation. (English) Zbl 1217.34034

Cui, Yujun

Abstr. Appl. Anal. 2011, Article ID 605614, 9 p. (2011).

Summary: By applying a fixed point theorem for mappings that are decreasing with respect to a cone, this paper investigates the existence of positive solutions for the nonlinear fractional boundary value problem:
\[ D^\alpha_{0^+}u(t)+f(t,u(t))=0, \quad 0<t<1, \]
\[ u(0)=u'(0)=u'(1)=0, \]
where \(2<\alpha<3\), \(D^\alpha_{0^+}\) is the Riemann-Liouville fractional derivative.

MSC:

34B16	Singular nonlinear boundary value problems for ordinary differential equations
34A08	Fractional ordinary differential equations
47N20	Applications of operator theory to differential and integral equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

singular boundary value problem; nonlinear fractional differential equation; Riemann-Liouville fractional derivative

References:

[1]	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 Science, Amsterdam, The Netherlands, 2006. · Zbl 1206.26007 · doi:10.1016/S0304-0208(06)80001-0
[2]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0943.82582 · doi:10.1007/BF01048101
[3]	I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[4]	S. Zhang, “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
[5]	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
[6]	X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4676-4688, 2009. · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030
[7]	J. A. Gatica, V. Oliker, and P. Waltman, “Singular nonlinear boundary value problems for second-order ordinary differential equations,” Journal of Differential Equations, vol. 79, no. 1, pp. 62-78, 1989. · Zbl 0685.34017 · doi:10.1016/0022-0396(89)90113-7
[8]	J. Henderson and W. Yin, “Singular (k,n - k) boundary value problems between conjugate and right focal,” Journal of Computational and Applied Mathematics, vol. 88, no. 1, pp. 57-69, 1998. · Zbl 0901.34026 · doi:10.1016/S0377-0427(97)00207-0
[9]	P. W. Eloe and J. Henderson, “Singular nonlinear (k,n - k) conjugate boundary value problems,” Journal of Differential Equations, vol. 133, no. 1, pp. 136-151, 1997. · Zbl 0882.34029 · doi:10.1023/A:1022983130486
[10]	P. W. Eloe and J. Henderson, “Singular nonlinear multipoint conjugate boundary value problems,” Communications in Applied Analysis, vol. 2, no. 4, pp. 497-511, 1998. · Zbl 0903.34016
[11]	J. J. DaCunha, J. M. Davis, and P. K. Singh, “Existence results for singular three point boundary value problems on time scales,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 378-391, 2004. · Zbl 1069.34012 · doi:10.1016/j.jmaa.2004.02.049
[12]	Y. Feng and S. Liu, “Solvability of a third-order two-point boundary value problem,” Applied Mathematics Letters, vol. 18, no. 9, pp. 1034-1040, 2005. · Zbl 1094.34506 · doi:10.1016/j.aml.2004.04.016
[13]	Q. Yao and Y. Feng, “The existence of solution for a third-order two-point boundary value problem,” Applied Mathematics Letters, vol. 15, no. 2, pp. 227-232, 2002. · Zbl 1008.34010 · doi:10.1016/S0893-9659(01)00122-7
[14]	K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. · Zbl 0559.47040
[15]	D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988. · Zbl 0661.47045