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The existence of positive solutions for fractional differential equations with sign changing nonlinearities. (English) Zbl 1246.34008

Summary: We investigate the existence of at least two positive solutions to eigenvalue problems of fractional differential equations with sign changing nonlinearities in more generalized boundary conditions. Our analysis relies on the Avery-Peterson fixed point theorem in a cone. Some examples are given for the illustration of main results.

MSC:

34A08 Fractional ordinary differential equations

References:

[1] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0943.82582 · doi:10.1007/BF01048101
[3] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2677-2682, 2008. · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[4] R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 6, pp. 2859-2862, 2010. · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029
[5] V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 10, pp. 3337-3343, 2008. · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[6] V. Lakshmikantham and S. Leela, “Nagumo-type uniqueness result for fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2886-2889, 2009. · Zbl 1177.34003 · doi:10.1016/j.na.2009.01.169
[7] T. G. Bhaskar, V. Lakshmikantham, and S. Leela, “Fractional differential equations with a Krasnoselskii-Krein type condition,” Nonlinear Analysis. Hybrid Systems, vol. 3, no. 4, pp. 734-737, 2009. · Zbl 1181.34008 · doi:10.1016/j.nahs.2009.06.010
[8] V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 8, pp. 828-834, 2008. · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[9] C. Yuan, D. Jiang, and X. Xu, “Singular positone and semipositone boundary value problems of nonlinear fractional differential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 535209, 17 pages, 2009. · Zbl 1185.34008 · doi:10.1155/2009/535209
[10] Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6434-6441, 2011. · Zbl 1235.34027 · doi:10.1016/j.na.2011.06.026
[11] C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1251-1268, 2011. · Zbl 1253.34012 · doi:10.1016/j.camwa.2011.02.039
[12] 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. · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030
[13] X. Li, S. Liu, and W. Jiang, “Positive solutions for boundary value problem of nonlinear fractional functional differential equations,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9278-9285, 2011. · Zbl 1223.34107 · doi:10.1016/j.amc.2011.04.006
[14] S. Liang and J. Zhang, “Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1333-1340, 2011. · Zbl 1235.34024 · doi:10.1016/j.camwa.2011.03.073
[15] 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. · Zbl 1102.65136 · doi:10.1016/j.amc.2006.01.007
[16] Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3599-3605, 2011. · Zbl 1220.34006 · doi:10.1016/j.na.2011.02.043
[17] S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5545-5550, 2009. · Zbl 1185.26011 · doi:10.1016/j.na.2009.04.045
[18] 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. · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[19] J. Caballero, J. Harjani, and K. Sadarangani, “Positive solutions for a class of singular fractional boundary value problems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1325-1332, 2011. · Zbl 1235.34010 · doi:10.1016/j.camwa.2011.04.013
[20] R. P. Agarwal, D. O’Regan, and S. Stan\vek, “Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 1, pp. 57-68, 2010. · Zbl 1206.34009 · doi:10.1016/j.jmaa.2010.04.034
[21] D. Jiang and C. Yuan, “The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 2, pp. 710-719, 2010. · Zbl 1192.34008 · doi:10.1016/j.na.2009.07.012
[22] E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlinear fractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1-11, 2008. · Zbl 1183.34007
[23] S. Stan\vek, “The existence of positive solutions of singular fractional boundary value problems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1379-1388, 2011. · Zbl 1228.34020 · doi:10.1016/j.camwa.2011.04.048
[24] T. Jankowski, “Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3775-3785, 2011. · Zbl 1221.34071 · doi:10.1016/j.na.2011.03.022
[25] R. P. Agarwal, B. de Andrade, and G. Siracusa, “On fractional integro-differential equations with state-dependent delay,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1143-1149, 2011. · Zbl 1228.35262 · doi:10.1016/j.camwa.2011.02.033
[26] Z.-W. Lv, J. Liang, and T.-J. Xiao, “Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1303-1311, 2011. · Zbl 1228.65136 · doi:10.1016/j.camwa.2011.04.027
[27] C. Kou, H. Zhou, and Y. Yan, “Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 5975-5986, 2011. · Zbl 1235.34022 · doi:10.1016/j.na.2011.05.074
[28] M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2391-2396, 2009. · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[29] Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3249-3256, 2009. · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[30] G. M. N’Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 5, pp. 1873-1876, 2009. · Zbl 1166.34320 · doi:10.1016/j.na.2008.02.087
[31] 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. · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033
[32] R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1095-1100, 2010. · Zbl 1189.34152 · doi:10.1016/j.camwa.2009.05.010
[33] S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 2006, pp. 1-12, 2006. · Zbl 1096.34016
[34] H. A. H. Salem, “On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 565-572, 2009. · Zbl 1176.34070 · doi:10.1016/j.cam.2008.05.033
[35] W. Zhong and W. Lin, “Nonlocal and multiple-point boundary value problem for fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1345-1351, 2010. · Zbl 1189.34036 · doi:10.1016/j.camwa.2009.06.032
[36] 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. · Zbl 1189.34014 · doi:10.1016/j.camwa.2009.06.029
[37] M. Al-Refai and M. Ali Hajji, “Monotone iterative sequences for nonlinear boundary value problems of fractional order,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3531-3539, 2011. · Zbl 1219.34005 · doi:10.1016/j.na.2011.03.006
[38] X. Su, “Solutions to boundary value problem of fractional order on unbounded domains in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 8, pp. 2844-2852, 2011. · Zbl 1250.34007 · doi:10.1016/j.na.2011.01.006
[39] H. A. H. Salem, “Fractional order boundary value problem with integral boundary conditions involving Pettis integral,” Acta Mathematica Scientia B, vol. 31, no. 2, pp. 661-672, 2011. · Zbl 1240.26009 · doi:10.1016/S0252-9602(11)60266-X
[40] Z. Bai, “Solvability for a class of fractional m-point boundary value problem at resonance,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1292-1302, 2011. · Zbl 1235.34006 · doi:10.1016/j.camwa.2011.03.003
[41] M. U. Rehman, R. A. Khan, and N. A. Asif, “Three point boundary value problems for nonlinear fractional differential equations,” Acta Mathematica Scientia B, vol. 31, no. 4, pp. 1337-1346, 2011. · Zbl 1249.34012 · doi:10.1016/S0252-9602(11)60320-2
[42] B. Ahmad, J. J. Nieto, and A. Alsaedi, “Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions,” Acta Mathematica Scientia B, vol. 31, pp. 2122-2130, 2011. · Zbl 1265.34009
[43] G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 3, pp. 792-804, 2011. · Zbl 1214.34009 · doi:10.1016/j.na.2010.09.030
[44] B. Ahmad and S. K. Ntouyas, “A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order,” Electronic Journal of Qualitative Theory of Differential Equations, no. 22, pp. 1-15, 2011. · Zbl 1340.34063
[45] B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838-1843, 2009. · Zbl 1205.34003 · doi:10.1016/j.camwa.2009.07.091
[46] M. El-Shahed, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007. · Zbl 1149.26012 · doi:10.1155/2007/10368
[47] R. I. Avery and A. C. Peterson, “Three positive fixed points of nonlinear operators on ordered Banach spaces,” Computers & Mathematics with Applications, vol. 42, no. 3-5, pp. 313-322, 2001. · Zbl 1005.47051 · doi:10.1016/S0898-1221(01)00156-0
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