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

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
Full Text:

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., () [5] Samko, S.; Kilbas, A.; Marichev, O., () [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 New York · Zbl 0789.26002 [10] Kilbas, A.; Srivastava, H.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), 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 Berlin · Zbl 0559.47040
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.