Journal of Applied Mathematics
- J. Appl. Math.
- Volume 2014 (2014), Article ID 790862, 7 pages.
Positive Solutions for Coupled Nonlinear Fractional Differential Equations
Wenning Liu, Xingjie Yan, and Wei Qi
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Abstract
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 , and computing the fixed point index in product cone , 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.
Article information
Source
J. Appl. Math., Volume 2014 (2014), Article ID 790862, 7 pages.
Dates
First available in Project Euclid: 2 March 2015
Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305756
Digital Object Identifier
doi:10.1155/2014/790862
Mathematical Reviews number (MathSciNet)
MR3212514
Citation
Liu, Wenning; Yan, Xingjie; Qi, Wei. Positive Solutions for Coupled Nonlinear Fractional Differential Equations. J. Appl. Math. 2014 (2014), Article ID 790862, 7 pages. doi:10.1155/2014/790862. https://projecteuclid.org/euclid.jam/1425305756
References
- O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002.Mathematical Reviews (MathSciNet): MR1930721
Zentralblatt MATH: 1070.49013
Digital Object Identifier: doi:10.1016/S0022-247X(02)00180-4 - Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.Mathematical Reviews (MathSciNet): MR2168413
Zentralblatt MATH: 1079.34048
Digital Object Identifier: doi:10.1016/j.jmaa.2005.02.052 - D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.Mathematical Reviews (MathSciNet): MR1421467
Zentralblatt MATH: 0881.34005
Digital Object Identifier: doi:10.1006/jmaa.1996.0456 - S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theorey and Applications, Gordon andBreach Science, Yverdon, Switzerland, 1993.Mathematical Reviews (MathSciNet): MR1347689
- S. Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 136–148, 2003.Mathematical Reviews (MathSciNet): MR1963470
Zentralblatt MATH: 1026.34008
Digital Object Identifier: doi:10.1016/S0022-247X(02)00583-8 - J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011.Mathematical Reviews (MathSciNet): MR2736622
Zentralblatt MATH: 1221.26002
Digital Object Identifier: doi:10.1016/j.cnsns.2010.05.027 - H. Jafari and V. Daftardar-Gejji, “Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 700–706, 2006.Mathematical Reviews (MathSciNet): MR2270047
Zentralblatt MATH: 1102.65136
Digital Object Identifier: doi:10.1016/j.amc.2006.01.007 - S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000.Mathematical Reviews (MathSciNet): MR1800180
Zentralblatt MATH: 0972.34004
Digital Object Identifier: doi:10.1006/jmaa.2000.7123 - N. Kosmatov, “A singular boundary value problem for nonlinear differential equations of fractional order,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 125–135, 2009.Mathematical Reviews (MathSciNet): MR2472100
Zentralblatt MATH: 1191.34006
Digital Object Identifier: doi:10.1007/s12190-008-0104-x - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory andApplications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.Mathematical Reviews (MathSciNet): MR2218073
- M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff, Groningen, The Netherlands, 1964.Mathematical Reviews (MathSciNet): MR0181881
- T. Qiu and Z. Bai, “Existence of positive solutions for singular fractional differential equations,” Electronic Journal of Differential Equations, vol. 2008, article 146, 2008.
- X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4676–4688, 2009.
- S. Zhang, “Nonnegative solution for singular nonlinear fractional differential equation with coefficient that changes sign,” Positivity, vol. 12, no. 4, pp. 711–724, 2008.Mathematical Reviews (MathSciNet): MR2448758
Zentralblatt MATH: 1172.26306
Digital Object Identifier: doi:10.1007/s11117-008-2030-4 - M. Feng, X. Zhang, and W. Ge, “New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 720702, 2011.Mathematical Reviews (MathSciNet): MR2679690
Zentralblatt MATH: 1214.34005
Digital Object Identifier: doi:10.1186/1687-2770-2011-720702 - D. Guo, Y. J. Cho, and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers, New York, NY, USA, 2004.Mathematical Reviews (MathSciNet): MR2084490
- Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010.
- B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID 494720, 9 pages, 2009.Mathematical Reviews (MathSciNet): MR2516016
Zentralblatt MATH: 1186.34009
Digital Object Identifier: doi:10.1155/2009/494720 - C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010.
- S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300–1309, 2010.
- Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950–6958, 2011.Mathematical Reviews (MathSciNet): MR2775685
Zentralblatt MATH: 1221.34068
Digital Object Identifier: doi:10.1016/j.amc.2011.01.103 - J. Wang, H. Xiang, and Z. Liu, “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,” International Journal of Differential Equations, vol. 2012, Article ID 186928, 12 pages, 2010.
- J. Zhao, P. Wang, and W. Ge, “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, vol. 16, no. 1, pp. 402–413, 2011.Mathematical Reviews (MathSciNet): MR2679191
Zentralblatt MATH: 1221.34053
Digital Object Identifier: doi:10.1016/j.cnsns.2009.10.011 - W. Yang, “Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 288–297, 2012.
- X. Cheng and H. Lü, “Multiplicity of positive solutions for a (p1, p2) -Laplacian system and its applications,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2375–2390, 2012.Mathematical Reviews (MathSciNet): MR2911922
Digital Object Identifier: doi:10.1016/j.nonrwa.2012.02.004 - X. Cheng and C. Zhong, “Existence of positive solutions for a second-order ordinary differential system,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 14–23,2005.Mathematical Reviews (MathSciNet): MR2175200
Zentralblatt MATH: 1088.34016
Digital Object Identifier: doi:10.1016/j.jmaa.2005.03.016 - X. Cheng and Z. Zhang, “Existence of positive solutions to systems of nonlinear integral or differential equations,” Topological Methods in Nonlinear Analysis, vol. 34, no. 2, pp. 267–277, 2009.
- K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.Mathematical Reviews (MathSciNet): MR787404
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1993. \endinputMathematical Reviews (MathSciNet): MR1658022
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