Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters

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June 2020, 19(6): 3283-3302. doi: 10.3934/cpaa.2020145

Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters

Yanqiong Lu ^, and Ruyun Ma

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

^* Corresponding author

Received August 2019 Revised January 2020 Published March 2020

Fund Project: The first author is supported by National Natural Science Foundation of China (No.11901464, No.11671322, No.11801453), Gansu provincial National Science Foundation of China (No.1606RJYA232) and NWNU-LKQN-15-16

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.

Keywords: Clamped beam equations, disconjugacy, spectrum, positive solution, bifurcation.

Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18.

Citation: Yanqiong Lu, Ruyun Ma. Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3283-3302. doi: 10.3934/cpaa.2020145

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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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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