×

zbMATH — the first resource for mathematics

Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. (English) Zbl 1189.34014
Summary: We are concerned with the nonlinear differential equation of fractional order \[ D^{\alpha}_{0+}u(t)+f(t,u(t))=0,~0<t<1,~1<\alpha\leq 2, \] where \(D^{\alpha}_{0+}\) is the standard Riemann-Liouville fractional derivative, subject to the boundary conditions \(u(0)=0\), \(D^{\beta}_{0+}u(1)=aD^{\beta}_{0+}u(\xi)\). We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems.

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[2] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[3] Podlubny, I., Fractional differential equations, mathematics in science and engineering, (1999), Academic Press New York
[4] Bakakhani, A.; Gejji, V.D., Existence of positive solutions of nonlinear fractional differential equations, J. math. anal. appl., 278, 434-442, (2003) · Zbl 1027.34003
[5] Delbosco, D.; Rodina, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005
[6] El-Sayed, A.M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear anal., 33, 181-186, (1998) · Zbl 0934.34055
[7] Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear anal., 69, 3337-3343, (2008) · Zbl 1162.34344
[8] 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
[9] Zhou, Yong, Existence and uniqueness of solutions for a system of fractional differential equations, J. frac. calc. appl. anal., 12, 195-204, (2009) · Zbl 1396.34003
[10] Zhou, Yong, Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. dyn. syst. differ. equ., 1, 4, 239-244, (2008) · Zbl 1175.34081
[11] Bai, Z.B.; Lü, H.S., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. anal. appl., 311, 495-505, (2005) · Zbl 1079.34048
[12] Kosmatov, N., A singular boundary value problem for nonlinear differential equations of fractional order, J. appl. math. comput., (2008)
[13] Kaufmann, E.R.; Mboumi, E., Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. qual. theory diff. equ., 2008, 3, 1-11, (2008) · Zbl 1183.34007
[14] Bai, C.Z., Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electron. J. qual. theory diff. equ., 2008, 24, 1-10, (2008)
[15] Krasnosel’skii, M.A., Topological methods in the theory of nonlinear integral equations, (1964), Pergamon Elmsford, (A.H. Armstrong, Trans.) · Zbl 0111.30303
[16] Agarwal, R.P.; Meehan, M.; O’Regan, D., Fixed point theory and applications, (2001), Cambridge University Press Cambridge · Zbl 0960.54027
[17] Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach space, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033
[18] Granas, A.; Guenther, R.B.; Lee, J.W., Some general existence principle in the caratheodory theory of nonlinear systems, J. math. pures appl., 70, 153-196, (1991) · Zbl 0687.34009
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.