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The existence of solutions to boundary value problems of fractional differential equations at resonance. (English) Zbl 1236.34006

The author obtains a solution of the Riemann-Liouville fractional differential equation

D 0+ α u(t)=f(t,u(t),D 0+ α-1 u(t))a·e·t(0,1)

satisfying the non-local conditions

u(0)=0,D 0+ α-1 u(0)= i=1 m a i D 0+ α-1 u(ξ i ),D 0+ α-2 u(1)= i=1 n b i D 0+ α-2 u(η i )·

It is assumed that 2<α<3, 0<ξ 1 <<ξ m <1, 0<η 1 <<η n <1, i=1 m a i =1, and i=1 n b i η i =1. The existence of a solution at resonance follows from the coincidence degree theorem of Mawhin.

34A08Fractional differential equations
34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
[1]Xue, C.; Ge, W.: The existence of solutions for multi-point boundary value problem at resonance, Acta math. Sin. 48, 281-290 (2005) · Zbl 1124.34310
[2]Feng, W.; Webb, J. R. L.: Solvability of m-point boundary value problems with nonlinear growth, J. math. Anal. appl. 212, 467-480 (1997) · Zbl 0883.34020 · doi:10.1006/jmaa.1997.5520
[3]Liu, B.: Solvability of multi-point boundary value problem at resonance (II), Appl. math. Comput. 136, 353-377 (2003) · Zbl 1053.34016 · doi:10.1016/S0096-3003(02)00050-4
[4]Gupta, C. P.: Solvability of multi-point boundary value problem at resonance, Results math. 28, 270-276 (1995) · Zbl 0843.34023
[5]Prezeradzki, B.; Stanczy, R.: Solvability of a multi-point boundary value problem at resonance, J. math. Anal. appl. 264, 253-261 (2001) · Zbl 1043.34016 · doi:10.1006/jmaa.2001.7616
[6]Ma, R.: Existence results of a m-point boundary value problem at resonance, J. math. Anal. appl. 294, 147-157 (2004) · Zbl 1070.34028 · doi:10.1016/j.jmaa.2004.02.005
[7]Nagle, R. K.; Pothoven, K. L.: On a third-order nonlinear boundary value problem at resonance, J. math. Anal. appl. 195, 148-159 (1995) · Zbl 0847.34026 · doi:10.1006/jmaa.1995.1348
[8]Karakostas, G. L.; Tsamatos, P. Ch.: On a nonlocal boundary value problem at resonance, J. math. Anal. appl. 259, 209-218 (2001) · Zbl 1002.34057 · doi:10.1006/jmaa.2000.7411
[9]Lu, S.; Ge, W.: On the existence of m-point boundary value problem at resonance for higher order differential equation, J. math. Anal. appl. 287, 522-539 (2003) · Zbl 1046.34029 · doi:10.1016/S0022-247X(03)00567-5
[10]Liu, Y.; Ge, W.: Solvability of nonlocal boundary value problems for ordinary differential equations of higher order, Nonlinear anal. 57, 435-458 (2004) · Zbl 1052.34024 · doi:10.1016/j.na.2004.02.023
[11]Du, Z.; Lin, X.; Ge, W.: Some higher-order multi-point boundary value problem at resonance, J. comput. Appl. math. 177, 55-65 (2005) · Zbl 1059.34010 · doi:10.1016/j.cam.2004.08.003
[12]Du, B.; Hu, X.: A new continuation theorem for the existence of solutions to P-Laplacian BVP at resonance, Appl. math. Comput. 208, 172-176 (2009) · Zbl 1169.34307 · doi:10.1016/j.amc.2008.11.041
[13]Kosmatov, N.: Multi-point boundary value problems on an unbounded domain at resonance, Nonlinear anal. 68, 2158-2171 (2008) · Zbl 1138.34006 · doi:10.1016/j.na.2007.01.038
[14]Lian, H.; Pang, H.; Ge, W.: Solvability for second-order three-point boundary value problems at resonance on a half-line, J. math. Anal. appl. 337, 1171-1181 (2008) · Zbl 1136.34034 · doi:10.1016/j.jmaa.2007.04.038
[15]Zhang, X.; Feng, M.; Ge, W.: Existence result of second-order differential equations with integral boundary conditions at resonance, J. math. Anal. appl. 353, 311-319 (2009) · Zbl 1180.34016 · doi:10.1016/j.jmaa.2008.11.082
[16]Agarwal, R. P.; O’regan, D.; Stanek, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. math. Anal. appl. 371, 57-68 (2010) · Zbl 1206.34009 · doi:10.1016/j.jmaa.2010.04.034
[17]X. Su, Boundary value problem for a coupled system of nonlinear differential equations, Appl. Math. Lett. 22 64–69. · Zbl 1163.34321 · doi:10.1016/j.aml.2008.03.001
[18]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[19]Podlubny, Igor: Fractional differential equations, (1999)
[20]Bai, Z.; Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[21]Jiang, D.; Yuan, C.: The positive properties of the Green function for Dirichlet-type of nonlinear fractional differential equations and its application, Nonlinear anal. 72, 710-719 (2010) · Zbl 1192.34008 · doi:10.1016/j.na.2009.07.012
[22]Bai, Z.; Qiu, T.: Existence of positive solution for singular fractional differential equation, Appl. math. Comput. 215, 2761-2767 (2009) · Zbl 1185.34004 · doi:10.1016/j.amc.2009.09.017
[23]Kaufmann, E. R.; Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. theory differ. Equ. 3, 1-11 (2008) · Zbl 1183.34007 · doi:emis:journals/EJQTDE/2008/200803.html
[24]Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Differ. equ. 36, 1-12 (2006(2006)) · Zbl 1096.34016 · doi:emis:journals/EJDE/Volumes/2006/36/abstr.html
[25]Jafari, H.; Gejji, V. D.: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. math. Comput. 180, 700-706 (2006) · Zbl 1102.65136 · doi:10.1016/j.amc.2006.01.007
[26]Benchohra, M.; Hamani, S.; Ntouyas, S. K.: Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear anal. 71, 2391-2396 (2009) · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[27]Xu, X.; Jiang, D.; Yuan, C.: Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear anal. 71, 4676-4688 (2009) · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030
[28]Liang, S.; Zhang, J.: Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear anal. 71, 5545-5550 (2009) · Zbl 1185.26011 · doi:10.1016/j.na.2009.04.045
[29]Yang, A.; Ge, W.: Positive solutions for boundary value problems of N-dimension nonlinear fractional differential system, Bound. value probl. (2008) · Zbl 1167.34314 · doi:10.1155/2008/437453
[30]Kosmatov, N.: A boundary value problem of fractional order at resonance, Electron. J. Differ. equ. 2010, No. 135, 1-10 (2010) · Zbl 1210.34007 · doi:emis:journals/EJDE/Volumes/2010/135/abstr.html
[31]Mawhin, J.: NSFCBMS regional conference series in mathematics, (1979)