Positive solutions of a system of non-autonomous fractional differential equations. (English) Zbl 1064.34004

Daftardar-Gejji, Varsha

J. Math. Anal. Appl. 302, No. 1, 56-64 (2005).

The paper is about systems of nonautonomous, nonlinear fractional differential equations and the existence of positive solutions. The aim of the paper is to give theorems that provide sufficient conditions for the existence of positive solutions. The basic idea and methodology adopted in the paper is to impose bounds on the (positive) derivatives and to deduce the existence of positive solutions. The analytical tools employed include the Banach and Schauder fixed point theorems.

Reviewer: Neville Ford (Chester)

MSC:

34A34	Nonlinear ordinary differential equations and systems, general theory
26A33	Fractional derivatives and integrals
34C99	Qualitative theory for ordinary differential equations
34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.

Keywords:

fractional differential equations; positive solutions

References:

[1]	Atanackovic, T.M.; Stankovic, B., On a system of differential equations with fractional derivatives arising in rod theory, J. phys. A, 37, 1241-1250, (2004) · Zbl 1059.35011
[2]	Babakhani, A.; Daftardar-Gejji, V., Existence of positive solutions of nonlinear fractional differential equations, J. math. anal. appl., 278, 434-442, (2003) · Zbl 1027.34003
[3]	Daftardar-Gejji, V.; Babakhani, A., Analysis of a system of fractional differential equations, J. math. anal. appl., 293, 511-522, (2004) · Zbl 1058.34002
[4]	Edwards, J.T.; Ford, N.J.; Simpson, A.C., The numerical solution of linear multi-term fractional differential equations: systems of equations, J. comput. appl. math., 148, 401-418, (2002) · Zbl 1019.65048
[5]	Joshi, M.C.; Bose, R.K., Some topics in nonlinear functional analysis, (1985), Wiley Eastern New Delhi · Zbl 0596.47038
[6]	Krasnoselskii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen
[7]	Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[8]	Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004
[9]	Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[10]	Samko, G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[11]	Torvik, P.J.; Bagley, R.L., On the appearance of the fractional derivative in the behaviour of real materials, J. appl. mech., 51, 294-298, (1984) · Zbl 1203.74022
[12]	()
[13]	Zhang, S., The existence of positive solution for a nonlinear fractional differential equation, J. math. anal. appl., 252, 804-812, (2000) · Zbl 0972.34004