Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. (English) Zbl 1255.34010

Zhang, Xinguang; Liu, Lishan; Wu, Yonghong

Math. Comput. Modelling 55, No. 3-4, 1263-1274 (2012).

Summary: Let \(D^\alpha_{0+}\) be the standard Riemann-Liouville derivative. We discuss the existence of multiple positive solutions for the following fractional differential equation with a negatively perturbed term \[ \begin{cases}-D^\alpha_{0+}u(t)=p(t)f(t,u(t))-q(t),& 0<t<1,\\ u(0)=u'(0)=u(1)=0,&{}\end{cases} \] where \(2<\alpha\leq 3\) is a real number, the perturbed term \(q:(0,1)\to[0,+\infty)\) is Lebesgue integrable and may be singular at some zero measures set of [0,1], which implies the nonlinear term may change sign.

MSC:

34A08	Fractional ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
45J05	Integro-ordinary differential equations

Keywords:

fractional differential equation; multiple positive solutions; Green function; perturbed term

References:

[1]	Agrawal, O., Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272, 368-379 (2002) · Zbl 1070.49013
[2]	Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Appl., 204, 609-625 (1996) · Zbl 0881.34005
[3]	Leggett, R.; Williams, L., Multiple positive solutions of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033
[4]	Podlubny, I., (Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198 (1999), Academic Press: Academic Press New Tork, London, Toronto) · Zbl 0918.34010
[5]	Samko, S.; Kilbas, A.; Marichev, O., (Fractional Integral and Derivatives. Fractional Integral and Derivatives, Theory and Applications (1993), Gordon and Breach: Gordon and Breach Switzerland) · Zbl 0818.26003
[6]	Bai, Z.; Lv, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048
[7]	Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252, 804-812 (2000) · Zbl 0972.34004
[8]	Zhang, S., Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 1, 136-148 (2003) · Zbl 1026.34008
[9]	Miller, K.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[10]	Kilbas, A.; Srivastava, H.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier B.V.: Elsevier B.V. Netherlands · Zbl 1092.45003
[11]	Kilbas, A.; Trujillo, J., Differential equations of fractional order: methods, results and problems-I, Appl. Anal., 78, 153-192 (2001) · Zbl 1031.34002
[12]	Kilbas, A.; Trujillo, J., Differential equations of fractional order: methods, results and problems-II, Appl. Anal., 81, 435-493 (2002) · Zbl 1033.34007
[13]	Babakhani, A.; Gejji, V., Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 434-442 (2003) · Zbl 1027.34003
[14]	Yu, C.; Gao, G., On the solution of nonlinear fractional order differential equation, Nonlinear Anal. TMA, 63, 971-976 (1998)
[15]	Jafari, H.; Gejji, V., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput., 180, 700-706 (2006) · Zbl 1102.65136
[16]	Zhang, S., Positive solutions for boundary value problems of nonlinear fractional differential equations, Electron. J. Differential Equations, 2006, 1-12 (2006)
[17]	Jiang, D.; Yuan, C., The positive properties of the green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Anal. TMA, 72, 710-719 (2010) · Zbl 1192.34008
[18]	Bai, C., Positive solutions for nonlinear fractional differential equations with coefficient that changes sign, Nonlinear Anal. TMA, 64, 677-685 (2006) · Zbl 1152.34304
[19]	Zhang, S., Nonnegative solution for singular nonlinear fractional differential equation with coefficient that changes sign, Positivity, 12, 711-724 (2006) · Zbl 1172.26306
[20]	Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer Berlin · Zbl 0559.47040