×

zbMATH — the first resource for mathematics

Positive solutions for coupled nonlinear fractional differential equations. (English) Zbl 1442.34021
Summary: We consider the existence of positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary values. Assume the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones \(K_1, K_2\) and computing the fixed point index in product cone \(K_1 \times K_2\), we obtain that the system has a pair of positive solutions. It is remarkable that it is established on the Cartesian product of two cones, in which the feature of two equations can be opposite.
MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agrawal, O. P., Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272, 1, 368-379 (2002) · Zbl 1070.49013
[2] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311, 2, 495-505 (2005) · Zbl 1079.34048
[3] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 204, 2, 609-625 (1996) · Zbl 0881.34005
[4] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives. Fractional Integrals and Derivatives, Theorey and Applications (1993), Yverdon, Switzerland: Gordon and Breach Science, Yverdon, Switzerland · Zbl 0818.26003
[5] Zhang, S., Existence of positive solution for some class of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 278, 1, 136-148 (2003) · Zbl 1026.34008
[6] Machado, J. T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1140-1153 (2011) · Zbl 1221.26002
[7] Jafari, H.; Daftardar-Gejji, V., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Applied Mathematics and Computation, 180, 2, 700-706 (2006) · Zbl 1102.65136
[8] Zhang, S., The existence of a positive solution for a nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 252, 2, 804-812 (2000) · Zbl 0972.34004
[9] Kosmatov, N., A singular boundary value problem for nonlinear differential equations of fractional order, Journal of Applied Mathematics and Computing, 29, 1-2, 125-135 (2009) · Zbl 1191.34006
[10] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, 204 (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003
[11] Krasnoselskii, M. A., Positive Solutions of Operator Equations, 381 (1964), Groningen, The Netherlands: P. Noordhoff, Groningen, The Netherlands
[12] Qiu, T.; Bai, Z., Existence of positive solutions for singular fractional differential equations, Electronic Journal of Differential Equations, 2008, article 146 (2008)
[13] Xu, X.; Jiang, D.; Yuan, C., Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Analysis. Theory, Methods & Applications, 71, 10, 4676-4688 (2009) · Zbl 1178.34006
[14] Zhang, S., Nonnegative solution for singular nonlinear fractional differential equation with coefficient that changes sign, Positivity, 12, 4, 711-724 (2008) · Zbl 1172.26306
[15] Feng, M.; Zhang, X.; Ge, W., New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, Boundary Value Problems, 2011 (2011)
[16] Guo, D.; Cho, Y. J.; Zhu, J., Partial Ordering Methods in Nonlinear Problems (2004), New York, NY, USA: Nova Science Publishers, New York, NY, USA
[17] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Analysis: Theory, Methods & Applications, 72, 2, 916-924 (2010) · Zbl 1187.34026
[18] Ahmad, B.; Nieto, J. J., Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstract and Applied Analysis, 2009 (2009) · Zbl 1186.34009
[19] Li, C. F.; Luo, X. N.; Zhou, Y., Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Computers & Mathematics with Applications, 59, 3, 1363-1375 (2010) · Zbl 1189.34014
[20] Zhang, S., Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Computers & Mathematics with Applications, 59, 3, 1300-1309 (2010) · Zbl 1189.34050
[21] Zhao, Y.; Sun, S.; Han, Z.; Zhang, M., Positive solutions for boundary value problems of nonlinear fractional differential equations, Applied Mathematics and Computation, 217, 16, 6950-6958 (2011) · Zbl 1227.34011
[22] Wang, J.; Xiang, H.; Liu, Z., Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, International Journal of Differential Equations, 2012 (2010) · Zbl 1207.34012
[23] Zhao, J.; Wang, P.; Ge, W., Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces, Communications in Nonlinear Science and Numerical Simulation, 16, 1, 402-413 (2011) · Zbl 1221.34053
[24] Yang, W., Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Computers & Mathematics with Applications, 63, 1, 288-297 (2012) · Zbl 1238.34047
[25] Cheng, X.; Lü, H., Multiplicity of positive solutions for a (p1, p2) -Laplacian system and its applications, Nonlinear Analysis: Real World Applications, 13, 5, 2375-2390 (2012) · Zbl 1272.34031
[26] Cheng, X.; Zhong, C., Existence of positive solutions for a second-order ordinary differential system, Journal of Mathematical Analysis and Applications, 312, 1, 14-23 (2005) · Zbl 1088.34016
[27] Cheng, X.; Zhang, Z., Existence of positive solutions to systems of nonlinear integral or differential equations, Topological Methods in Nonlinear Analysis, 34, 2, 267-277 (2009) · Zbl 1195.45018
[28] Deimling, K., Nonlinear Functional Analysis (1985), Berlin, Germany: Springer, Berlin, Germany · Zbl 0559.47040
[29] Podlubny, I., Fractional Differential Equations (1993), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0918.34010
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.