| Title:
|
Infinitely many solutions for boundary value problems arising from the fractional advection dispersion equation (English) |
| Author:
|
Chen, Jing |
| Author:
|
Tang, Xian Hua |
| Language:
|
English |
| Journal:
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Applications of Mathematics |
| ISSN:
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0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
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60 |
| Issue:
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6 |
| Year:
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2015 |
| Pages:
|
703-724 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the existence of infinitely many solutions to the boundary value problem \begin {gather} \frac {{\rm d}}{{\rm d} t}\Big (\frac {1}{2} _{0}D_{t}^{-\beta }(u'(t)) +\frac {1}{2} _{t}D_{T}^{-\beta }(u'(t))\Big )+\nabla F(t,u(t))=0 \quad \text {\rm a.e.}\ t\in [0,T],\nonumber \\ u(0)=u(T)=0.\nonumber \end {gather} Under more general assumptions on the nonlinearity, we obtain new criteria to guarantee that this boundary value problem has infinitely many solutions in the superquadratic, subquadratic and asymptotically quadratic cases by using the critical point theory. (English) |
| Keyword:
|
fractional boundary value problem |
| Keyword:
|
critical point theory |
| Keyword:
|
variational methods |
| MSC:
|
26A33 |
| MSC:
|
35G60 |
| idZBL:
|
Zbl 06537669 |
| idMR:
|
MR3436569 |
| DOI:
|
10.1007/s10492-015-0118-2 |
| . |
| Date available:
|
2015-11-17T20:37:48Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144454 |
| . |
| Reference:
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[1] Ambrosetti, A., Rabinowitz, P. H.: Dual variational methods in critical point theory and applications.J. Funct. Anal. 14 (1973), 349-381. Zbl 0273.49063, MR 0370183, 10.1016/0022-1236(73)90051-7 |
| Reference:
|
[2] Bai, C.: Existence of solutions for a nonlinear fractional boundary value problem via a local minimum theorem.Electron. J. Differ. Equ. (electronic only) 2012 (2012), Article No. 176, 9 pages. Zbl 1254.34009, MR 2991410 |
| Reference:
|
[3] Bai, C.: Existence of three solutions for a nonlinear fractional boundary value problem via a critical points theorem.Abstr. Appl. Anal. 2012 (2012), Article ID 963105, 13 pages. Zbl 1253.34008, MR 2969986 |
| Reference:
|
[4] Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with ``strong'' resonance at infinity.Nonlinear Anal., Theory Methods Appl. 7 (1983), 981-1012. Zbl 0522.58012, MR 0713209 |
| Reference:
|
[5] Benson, D. A., Wheatcraft, S. W., Meerschaert, M. M.: Application of a fractional advection-dispersion equation.Water Resour. Res. 36 (2000), 1403-1412, DOI:10.1029/\allowbreak2000WR900031. 10.1029/2000WR900031 |
| Reference:
|
[6] Bin, G.: Multiple solutions for a class of fractional boundary value problems.Abstr. Appl. Anal. 2012 (2012), Article ID 468980, 16 pages. Zbl 1253.34009, MR 2991017 |
| Reference:
|
[7] Chen, J., Tang, X. H.: Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory.Abstr. Appl. Anal. 2012 (2012), Article No. 648635, 21 pages. Zbl 1235.34011, MR 2872321 |
| Reference:
|
[8] Chen, J., Tang, X. H.: Infinitely many solutions for a class of fractional boundary value problem.Bull. Malays. Math. Sci. Soc. (2) 36 (2013), 1083-1097. Zbl 1280.26011, MR 3108797 |
| Reference:
|
[9] Clark, D. C.: A variant of the Lusternik-Schnirelman theory.Indiana Univ. Math. J. 22 (1972), 65-74. Zbl 0228.58006, MR 0296777, 10.1512/iumj.1973.22.22008 |
| Reference:
|
[10] Ervin, V. J., Roop, J. P.: Variational formulation for the stationary fractional advection dispersion equation.Numer. Methods Partial Differ. Equations 22 (2006), 558-576. Zbl 1095.65118, MR 2212226, 10.1002/num.20112 |
| Reference:
|
[11] Fix, G. J., Roop, J. P.: Least squares finite-element solution of a fractional order two-point boundary value problem.Comput. Math. Appl. 48 (2004), 1017-1033. MR 2107380, 10.1016/j.camwa.2004.10.003 |
| Reference:
|
[12] Graef, J. R., Kong, L., Kong, Q.: Multiple solutions of systems of fractional boundary value problems.Appl. Anal. 94 (2015), 1288-1304. Zbl 1323.34007, MR 3325346, 10.1080/00036811.2014.930822 |
| Reference:
|
[13] Heidarkhani, S.: Infinitely many solutions for nonlinear perturbed fractional boundary value problems.An. Univ. Craiova, Ser. Mat. Inf. 41 (2014), 88-103. Zbl 1324.58005, MR 3234477 |
| Reference:
|
[14] Izydorek, M., Janczewska, J.: Homoclinic solutions for a class of the second order Hamiltonian systems.J. Differ. Equations 219 (2005), 375-389. Zbl 1080.37067, MR 2183265, 10.1016/j.jde.2005.06.029 |
| Reference:
|
[15] Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory.Comput. Math. Appl. 62 (2011), 1181-1199. Zbl 1235.34017, MR 2824707, 10.1016/j.camwa.2011.03.086 |
| Reference:
|
[16] Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory.Int. J. Bifurcation Chaos Appl. Sci. Eng. 22 (2012), Article ID 1250086, 17 pages. Zbl 1258.34015, MR 2926062, 10.1142/S0218127412500861 |
| Reference:
|
[17] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations.North-Holland Mathematics Studies 204 Elsevier, Amsterdam (2006). Zbl 1092.45003, MR 2218073 |
| Reference:
|
[18] Kong, L.: Existence of solutions to boundary value problems arising from the fractional advection dispersion equation.Electron. J. Differ. Equ. (electronic only) 2013 (2013), Article No. 106, 15 pages. Zbl 1291.34016, MR 3065059 |
| Reference:
|
[19] Li, F., Liang, Z., Zhang, Q.: Existence of solutions to a class of nonlinear second order two-point boundary value problems.J. Math. Anal. Appl. 312 (2005), 357-373. Zbl 1088.34012, MR 2175224, 10.1016/j.jmaa.2005.03.043 |
| Reference:
|
[20] Li, Y.-N., Sun, H.-R., Zhang, Q.-G.: Existence of solutions to fractional boundary-value problems with a parameter.Electron. J. Differ. Equ. (electronic only) 2013 (2013), Article No. 141, 12 pages. Zbl 1294.34006, MR 3084621 |
| Reference:
|
[21] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems.Applied Mathematical Sciences 74 Springer, New York (1989). Zbl 0676.58017, MR 0982267, 10.1007/978-1-4757-2061-7 |
| Reference:
|
[22] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations.A Wiley-Interscience Publication John Wiley & Sons, New York (1993). Zbl 0789.26002, MR 1219954 |
| Reference:
|
[23] Nyamoradi, N.: Infinitely many solutions for a class of fractional boundary value problems with Dirichlet boundary conditions.Mediterr. J. Math. 11 (2014), 75-87. MR 3160613, 10.1007/s00009-013-0307-8 |
| Reference:
|
[24] Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications.Mathematics in Science and Engineering 198 Academic Press, San Diego (1999). Zbl 0924.34008, MR 1658022 |
| Reference:
|
[25] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations.Reg. Conf. Ser. Math. 65. American Mathematical Society Providence (1986). Zbl 0609.58002, MR 0845785, 10.1090/cbms/065 |
| Reference:
|
[26] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications.Gordon and Breach, New York (1993). Zbl 0818.26003, MR 1347689 |
| Reference:
|
[27] Sun, H.-R., Zhang, Q.-G.: Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique.Comput. Math. Appl. 64 (2012), 3436-3443. Zbl 1268.34027, MR 2989371, 10.1016/j.camwa.2012.02.023 |
| Reference:
|
[28] Tang, X. H.: Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity.J. Math. Anal. Appl. 401 (2013), 407-415. MR 3011282, 10.1016/j.jmaa.2012.12.035 |
| Reference:
|
[29] Willem, M.: Minimax Theorems.Progress in Nonlinear Differential Equations and Their Applications 24 Birkhäuser, Boston (1996). Zbl 0856.49001, MR 1400007 |
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