Positive solutions for nonlinear fractional differential equations. (English) Zbl 1377.26006

Boulares, Hamid; Ardjouni, Abdelouaheb; Laskri, Yamina

Positivity 21, No. 3, 1201-1212 (2017).

Summary: We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation \[ \begin{cases} ^{C}D^{\alpha}x\left( t\right) =f(t,x(t))+^{C}D^{\alpha -1}g\left( t,x\left( t\right) \right),\;0<t\leq T,\\ x\left( 0\right) =\theta_{1}>0,\;x'\left( 0\right) =\theta_{2}>0, \end{cases} \]
where \(1<\alpha \leq 2\). In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mapping and employ Schauder fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. The results obtained here extend the work of M. M. Matar [Acta Math. Univ. Comen., New Ser. 84, No. 1, 51–57 (2015; Zbl 1340.34031)]. Finally, an example is given to illustrate our results.

MSC:

26A33	Fractional derivatives and integrals
34A12	Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G20	Nonlinear differential equations in abstract spaces

Keywords:

fractional differential equations; positive solutions; upper and lower solutions; existence; uniqueness; fixed point theorems

Citations:

Zbl 1340.34031

References:

[1]	Bai, Z., Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) · Zbl 1079.34048
[2]	Bai, Z.B., Qiu, T.T.: Existence of positive solution for singular fractional differential equation. Appl. Math. Comput. 215, 2761-2767 (2009) · Zbl 1185.34004
[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]	Kaufmann, E., Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 3, 1-11 (2008) · Zbl 1183.34007
[5]	Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[6]	Kou, C., Zhou, H., Yan, Y.: Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. Nonlinear Anal. 74, 5975-5986 (2011) · Zbl 1235.34022
[7]	Matar, M.: On existence of positive solution for initial value problem of nonlinear fractional differential equations of order \[1 < \alpha \le 21\]<α≤2. Acta Math. Univ. Comen. 84(1), 51-57 (2015) · Zbl 1340.34031
[8]	Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993) · Zbl 0789.26002
[9]	Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[10]	Smart, D.R.: Fixed point theorems. Cambridge University Press, Cambridge (1980) · Zbl 0427.47036
[11]	Wang, C., Wang, R., Wang, S., Yang, C.: Positive solution of singular boundary value problem for a nonlinear fractional differential equation. Bound. Value Probl. 2011, 1-12 (2011) (Art ID 297026) · Zbl 1079.34048
[12]	Wang, C., Zhang, H., Wang, S.: Positive solution of a nonlinear fractional differential equation involving Caputo derivative. Discret. Dyn. Nat. Soc. 2012, 1-16 (2012) (Art ID425408) · Zbl 1248.34006
[13]	Zhang, S.: Existence results of positive solutions to boundary value problem for fractional differential equation. Positivity 13(3), 583-599 (2009) · Zbl 1202.26018
[14]	Zhang, S.: The existence of a positive solution for a fractional differential equation. J. Math. Anal. Appl. 252, 804-812 (2000) · Zbl 0972.34004