American Institute of Mathematical Sciences

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

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

 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.

Citation: Yanqiong Lu, Ruyun Ma. Disconjugacy conditions and spectrum structure of clamped beam equations with two parameters. Communications on Pure & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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