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Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory. (English) Zbl 1235.34011
Summary: We study the existence and multiplicity of solutions for the following fractional boundary value problem: $(d/dt)((1/2)_0 D^{-\beta}_t (u'(t)) + (1/2)_t D^{-\beta}_T (u'(t))) + \nabla F(t, u(t)) = 0$, a. e. $t \in [0, T], u(0) = u(T) = 0$, where $F(t, \cdot)$ are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results.

MSC:
34A08Fractional differential equations
34B99Boundary value problems for ODE
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Full Text: DOI
References:
[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, The Netherlands, 2006. · Zbl 1206.26007 · doi:10.1016/S0304-0208(06)80001-0
[2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0943.82582 · doi:10.1007/BF01048101
[3] 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
[4] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, Pa, USA, 1993. · Zbl 0924.44003 · doi:10.1080/10652469308819017
[5] 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 A, vol. 71, no. 7-8, pp. 2391-2396, 2009. · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[6] R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973-1033, 2010. · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[7] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 69, no. 8, pp. 2677-2682, 2008. · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[8] J. Vasundhara Devi and V. Lakshmikantham, “Nonsmooth analysis and fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 70, no. 12, pp. 4151-4157, 2009. · Zbl 1237.49022 · doi:10.1016/j.na.2008.09.003
[9] B. Ahmad, “Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 4, pp. 390-394, 2010. · Zbl 1198.34007 · doi:10.1016/j.aml.2009.11.004
[10] Y. Zhou, F. Jiao, and J. Li, “Existence and uniqueness for fractional neutral differential equations with infinite delay,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 7-8, pp. 3249-3256, 2009. · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[11] J. Rong Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 262-272, 2011. · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[12] 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 · eudml:130751
[13] 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. · Zbl 1189.34050 · doi:10.1016/j.camwa.2009.06.034
[14] X.-K. Zhao and W. Ge, “Unbounded solutions for a fractional boundary value problems on the infinite interval,” Acta Applicandae Mathematicae, vol. 109, no. 2, pp. 495-505, 2010. · Zbl 1193.34008 · doi:10.1007/s10440-008-9329-9
[15] Y. Zhang and Z. Bai, “Existence of solutions for nonlinear fractional three-point boundary value problems at resonance,” Journal of Applied Mathematics and Computing, vol. 36, no. 1-2, pp. 417-440, 2011. · Zbl 1225.34013 · doi:10.1007/s12190-010-0411-x
[16] W. Jiang, “The existence of solutions to boundary value problems of fractional differential equations at resonance,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 74, no. 5, pp. 1987-1994, 2011. · doi:10.1016/j.na.2010.11.005
[17] S. Zhang, “Existence of a solution for the fractional differential equation with nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1202-1208, 2011. · Zbl 1217.34011 · doi:10.1016/j.camwa.2010.12.071
[18] S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 11, pp. 5545-5550, 2009. · Zbl 1185.26011 · doi:10.1016/j.na.2009.04.045 · eudml:231936
[19] Z. Wei, W. Dong, and J. Che, “Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 10, pp. 3232-3238, 2010. · Zbl 1202.26017 · doi:10.1016/j.na.2010.07.003
[20] 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
[21] G. Cerami, “An existence criterion for the critical points on unbounded manifolds,” Istituto Lombardo. Accademia di Scienze e Lettere: Rendiconti A, vol. 112, no. 2, pp. 332-336, 1978 (Italian). · Zbl 0436.58006
[22] P. H. Rabinowitz, “Periodic solutions of Hamiltonian systems,” Communications on Pure and Applied Mathematics, vol. 31, no. 2, pp. 157-184, 1978. · Zbl 0358.70014 · doi:10.1002/cpa.3160310203
[23] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. · Zbl 0676.58017
[24] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986. · Zbl 0609.58002
[25] G. Fei, “On periodic solutions of superquadratic Hamiltonian systems,” Electronic Journal of Differential Equations, vol. 2002, no. 8, pp. 1-12, 2002. · Zbl 0999.37039 · emis:journals/EJDE/Volumes/2002/08/abstr.html · eudml:122201
[26] Y. H. Ding and S. X. Luan, “Multiple solutions for a class of nonlinear Schrödinger equations,” Journal of Differential Equations, vol. 207, no. 2, pp. 423-457, 2004. · Zbl 1072.35166 · doi:10.1016/j.jde.2004.07.030
[27] M. J. Esteban and E. Séré, “Stationary states of the nonlinear Dirac equation: a variational approach,” Communications in Mathematical Physics, vol. 171, no. 2, pp. 323-350, 1995. · Zbl 0843.35114 · doi:10.1007/BF02099273
[28] C.-L. Tang and X.-P. Wu, “Subharmonic solutions for nonautonomous sublinear second order Hamiltonian systems,” Journal of Mathematical Analysis and Applications, vol. 304, no. 1, pp. 383-393, 2005. · Zbl 1076.34049 · doi:10.1016/j.jmaa.2004.09.032
[29] C.-L. Tang and X.-P. Wu, “Periodic solutions for second order systems with not uniformly coercive potential,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 386-397, 2001. · Zbl 0999.34039 · doi:10.1006/jmaa.2000.7401
[30] C. Troestler and M. Willem, “Nontrivial solution of a semilinear Schrödinger equation,” Communications in Partial Differential Equations, vol. 21, no. 9-10, pp. 1431-1449, 1996. · Zbl 0864.35036 · doi:10.1080/03605309608821233
[31] X. Fan and X. Han, “Existence and multiplicity of solutions for p(x)-Laplacian equations in \Bbb RN,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 59, no. 1-2, pp. 173-188, 2004. · Zbl 1134.35333 · doi:10.1016/j.na.2004.07.009
[32] F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problems via critical point theory,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1181-1199, 2011. · Zbl 1235.34017 · doi:10.1016/j.camwa.2011.03.086
[33] S. Ma and Y. Zhang, “Existence of infinitely many periodic solutions for ordinary p-Laplacian systems,” Journal of Mathematical Analysis and Applications, vol. 351, no. 1, pp. 469-479, 2009. · Zbl 1153.37009 · doi:10.1016/j.jmaa.2008.10.027
[34] Q. Zhang and C. Liu, “Infinitely many periodic solutions for second order Hamiltonian systems,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 816-833, 2011. · Zbl 1230.37081 · doi:10.1016/j.jde.2011.05.021