##
**Positive solutions for boundary value problem of nonlinear fractional differential equation.**
*(English)*
Zbl 1079.34048

Summary: We investigate the existence and multiplicity of positive solutions to the boundary value problem
\[
D^\alpha_{0+}u(t)+f \bigl(t,u(t) \bigr)=0,\;0<t<1, \quad u(0)=u(1)=0,
\]
where \(1<\alpha\leq 2\) is a real number, \(D_{0+}^\alpha\) is the standard Riemann-Liouville differentiation, and \(f:[0,1] \times[0,\infty) \to [0,\infty)\) is continuous. By means of some fixed-point theorems in a cone, existence and multiplicity results positive solutions are obtained. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.

### MSC:

34K05 | General theory of functional-differential equations |

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

26A33 | Fractional derivatives and integrals |

### Keywords:

fractional differential equation; boundary value problem; positive solution; Green’s function; fixed-point theorem
PDF
BibTeX
XML
Cite

\textit{Z. Bai} and \textit{H. Lü}, J. Math. Anal. Appl. 311, No. 2, 495--505 (2005; Zbl 1079.34048)

### References:

[1] | Babakhani, A.; Gejji, V.D., Existence of positive solutions of nonlinear fractional differential equations, J. math. anal. appl., 278, 434-442, (2003) · Zbl 1027.34003 |

[2] | Delbosco, D., Fractional calculus and function spaces, J. fract. calc., 6, 45-53, (1996) · Zbl 0829.46018 |

[3] | Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005 |

[4] | El-Sayed, A.M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear anal., 33, 181-186, (1998) · Zbl 0934.34055 |

[5] | Gejji, V.D.; Babakhani, A., Analysis of a system of fractional differential equations, J. math. anal. appl., 293, 511-522, (2004) · Zbl 1058.34002 |

[6] | Kilbas, A.A.; Marichev, O.I.; Samko, S.G., Fractional integral and derivatives (theory and applications), (1993), Gordon and Breach Switzerland · Zbl 0818.26003 |

[7] | Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional order: methods, results and problems I, Appl. anal., 78, 153-192, (2001) · Zbl 1031.34002 |

[8] | Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional order: methods, results and problems II, Appl. anal., 81, 435-493, (2002) · Zbl 1033.34007 |

[9] | Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604 |

[10] | Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033 |

[11] | Miller, K.S., Fractional differential equations, J. fract. calc., 3, 49-57, (1993) · Zbl 0781.34006 |

[12] | Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002 |

[13] | Nakhushev, A.M., The sturm – liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms, Dokl. akad. nauk SSSR, 234, 308-311, (1977) · Zbl 0376.34015 |

[14] | Podlubny, I., Fractional differential equations, mathematics in science and engineering, (1999), Academic Press New York |

[15] | I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, Inst. Expe. Phys., Slov. Acad. Sci., UEF-02-94, Kosice, 1994 |

[16] | Zhang, S.Q., The existence of a positive solution for a nonlinear fractional differential equation, J. math. anal. appl., 252, 804-812, (2000) · Zbl 0972.34004 |

[17] | Zhang, S.Q., Existence of positive solution for some class of nonlinear fractional differential equations, J. math. anal. appl., 278, 136-148, (2003) · Zbl 1026.34008 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.