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Stable periodic solutions for Nazarenko's equation
Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
In this work, we apply the 'disconjugacy theory' and Elias's spectrum theory to study the disconjugacy $ u^{(4)} + \beta u''-\alpha u = 0 $ with two parameters $ \alpha,\beta\in\mathbb{R} $ and the spectrum structure of the linear operator $ u^{(4)} + \beta u''-\alpha u $ coupled with the clamped beam conditions $ u(0) = u'(0) = u(1) = u'(1) = 0 $. As the application of our results, we obtain the global structure of nodal solutions of the corresponding nonlinear analogue based on the bifurcation theory.
References:
[1] |
R. P. Agarwal and Y. M. Chow,
Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math., 10 (1984), 203-217.
doi: 10.1016/0377-0427(84)90058-X. |
[2] |
A. Cabada, J. Á. Cid and L. Sanchez,
Positivity and lower and upper solutions for fourth order boundary value problems, Nonlinear Anal., 67 (2007), 1599-1612.
doi: 10.1016/j.na.2006.08.002. |
[3] |
A. Cabada and R. R. Enguiça,
Positive solutions of fourth order problems with clamped beam boundary conditions, Nonlinear Anal., 74 (2011), 3112-3122.
doi: 10.1016/j.na.2011.01.027. |
[4] |
A. Cabada and L. Saavedra,
Disconjugacy characterization by means of spectral $(k, n-k)$ problems, Appl. Math. Lett., 52 (2016), 21-29.
doi: 10.1016/j.aml.2015.08.007. |
[5] |
A. Cabada and L. Saavedra, The eigenvalue characterization for the constant sign Green's functions of $(k, n – k)$ problems, Bound. Value Probl., 44 (2016), 35pp.
doi: 10.1186/s13661-016-0547-1. |
[6] |
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. |
[7] |
U. Elias,
Eigenvalue problems for the equations $Ly + p(x)y = 0$, J. Differ. Equ., 29 (1978), 28-57.
doi: 10.1016/0022-0396(78)90039-6. |
[8] |
U. Elias, Oscillation Theory of Two-Term Differential Equations, Mathematics and Its Applications, Vol. 396, Kluwer Academic Publishers Group, Dordrecht, 1997, viii+217 pp.
doi: 10.1007%2F978-94-017-2517-0. |
[9] |
C. P. Gupta,
Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. Anal., 26 (1988), 289-304.
doi: 10.1080/00036818808839715. |
[10] |
P. Habets and L. Sanchez,
A monotone method for fourth order boundary value problems involving a factorizable linear operator, Port. Math., 64 (2007), 255-279.
doi: 10.4171/PM/1786. |
[11] |
J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators, Oper. Theory Adv. Appl., vol.177, Birkhäuser/Springer, Basel, Boston/Berlin, 2007. |
[12] |
R. Ma,
Nodal solutions for a fourth-order two-point boundary value problem, J. Math. Anal. Appl., 314 (2006), 254-265.
doi: 10.1016/j.jmaa.2005.03.078. |
[13] |
R. Ma, H. Wang and M. Elsanosi,
Spectrum of a linear fourth-order differential operator and its applications, Math. Nachr., 286 (2013), 1805-1819.
doi: 10.1002/mana.201200288. |
[14] |
R. Ma, J. Wang and Y. Long, Lower and upper solution method for the problem of elastic beam with hinged ends, J. Fixed Point Theory Appl., 20 (2018), 13 pp.
doi: 10.1007/s11784-018-0530-9. |
[15] |
R. Ma, J. Zhang and S. Fu,
The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl., 215 (1997), 415-422.
doi: 10.1006/jmaa.1997.5639. |
[16] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[17] |
B. P. Rynne,
Infinitely many solutions of superlinear fourth order boundary value problems, Topol. Meth. Nonlinear Anal., 19 (2002), 303-312.
doi: 10.12775/TMNA.2002.016. |
[18] |
B. P. Rynne,
Global bifurcation for $2m$th-order boundary value problems and infinitely many solutions of superlinear problems, J. Differ. Equ., 188 (2003), 461-472.
doi: 10.1016/S0022-0396(02)00146-8. |
[19] |
J. R. L. Webb, G. Infante and D. Franco,
Positive solutions of nonlinear fourth-order boundary value problems with local and non-local boundary conditions, Proc. R. Soc. Edinb. Sect. A, 138 (2008), 427-446.
doi: 10.1017/S0308210506001041. |
[20] |
Y. Wei, Q. Song and Z. Bai,
Existence and iterative method for some fourth order nonlinear boundary value problems, Appl. Math. Lett., 87 (2019), 101-107.
doi: 10.1016/j.aml.2018.07.032. |
[21] |
J. Xu and X. Han, Nodal solutions for a class of fourth-order two-point boundary value problems, Bound. Value Probl., (2010), Art. ID 570932, 11 pp.
doi: 10.1155/2010/570932. |
show all references
References:
[1] |
R. P. Agarwal and Y. M. Chow,
Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math., 10 (1984), 203-217.
doi: 10.1016/0377-0427(84)90058-X. |
[2] |
A. Cabada, J. Á. Cid and L. Sanchez,
Positivity and lower and upper solutions for fourth order boundary value problems, Nonlinear Anal., 67 (2007), 1599-1612.
doi: 10.1016/j.na.2006.08.002. |
[3] |
A. Cabada and R. R. Enguiça,
Positive solutions of fourth order problems with clamped beam boundary conditions, Nonlinear Anal., 74 (2011), 3112-3122.
doi: 10.1016/j.na.2011.01.027. |
[4] |
A. Cabada and L. Saavedra,
Disconjugacy characterization by means of spectral $(k, n-k)$ problems, Appl. Math. Lett., 52 (2016), 21-29.
doi: 10.1016/j.aml.2015.08.007. |
[5] |
A. Cabada and L. Saavedra, The eigenvalue characterization for the constant sign Green's functions of $(k, n – k)$ problems, Bound. Value Probl., 44 (2016), 35pp.
doi: 10.1186/s13661-016-0547-1. |
[6] |
W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. |
[7] |
U. Elias,
Eigenvalue problems for the equations $Ly + p(x)y = 0$, J. Differ. Equ., 29 (1978), 28-57.
doi: 10.1016/0022-0396(78)90039-6. |
[8] |
U. Elias, Oscillation Theory of Two-Term Differential Equations, Mathematics and Its Applications, Vol. 396, Kluwer Academic Publishers Group, Dordrecht, 1997, viii+217 pp.
doi: 10.1007%2F978-94-017-2517-0. |
[9] |
C. P. Gupta,
Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. Anal., 26 (1988), 289-304.
doi: 10.1080/00036818808839715. |
[10] |
P. Habets and L. Sanchez,
A monotone method for fourth order boundary value problems involving a factorizable linear operator, Port. Math., 64 (2007), 255-279.
doi: 10.4171/PM/1786. |
[11] |
J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators, Oper. Theory Adv. Appl., vol.177, Birkhäuser/Springer, Basel, Boston/Berlin, 2007. |
[12] |
R. Ma,
Nodal solutions for a fourth-order two-point boundary value problem, J. Math. Anal. Appl., 314 (2006), 254-265.
doi: 10.1016/j.jmaa.2005.03.078. |
[13] |
R. Ma, H. Wang and M. Elsanosi,
Spectrum of a linear fourth-order differential operator and its applications, Math. Nachr., 286 (2013), 1805-1819.
doi: 10.1002/mana.201200288. |
[14] |
R. Ma, J. Wang and Y. Long, Lower and upper solution method for the problem of elastic beam with hinged ends, J. Fixed Point Theory Appl., 20 (2018), 13 pp.
doi: 10.1007/s11784-018-0530-9. |
[15] |
R. Ma, J. Zhang and S. Fu,
The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl., 215 (1997), 415-422.
doi: 10.1006/jmaa.1997.5639. |
[16] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[17] |
B. P. Rynne,
Infinitely many solutions of superlinear fourth order boundary value problems, Topol. Meth. Nonlinear Anal., 19 (2002), 303-312.
doi: 10.12775/TMNA.2002.016. |
[18] |
B. P. Rynne,
Global bifurcation for $2m$th-order boundary value problems and infinitely many solutions of superlinear problems, J. Differ. Equ., 188 (2003), 461-472.
doi: 10.1016/S0022-0396(02)00146-8. |
[19] |
J. R. L. Webb, G. Infante and D. Franco,
Positive solutions of nonlinear fourth-order boundary value problems with local and non-local boundary conditions, Proc. R. Soc. Edinb. Sect. A, 138 (2008), 427-446.
doi: 10.1017/S0308210506001041. |
[20] |
Y. Wei, Q. Song and Z. Bai,
Existence and iterative method for some fourth order nonlinear boundary value problems, Appl. Math. Lett., 87 (2019), 101-107.
doi: 10.1016/j.aml.2018.07.032. |
[21] |
J. Xu and X. Han, Nodal solutions for a class of fourth-order two-point boundary value problems, Bound. Value Probl., (2010), Art. ID 570932, 11 pp.
doi: 10.1155/2010/570932. |
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