American Institute of Mathematical Sciences

Advanced
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 & 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

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.  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

[1]

Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
[2]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[3]

Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
[4]

GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803
[5]

Ruyun Ma, Yanqiong Lu. Disconjugacy and extremal solutions of nonlinear third-order equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1223-1236. doi: 10.3934/cpaa.2014.13.1223
[6]

Jiu Liu, Jia-Feng Liao, Chun-Lei Tang. Positive solution for the Kirchhoff-type equations involving general subcritical growth. Communications on Pure & Applied Analysis, 2016, 15 (2) : 445-455. doi: 10.3934/cpaa.2016.15.445
[7]

Jagmohan Tyagi, Ram Baran Verma. Positive solution to extremal Pucci's equations with singular and gradient nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2637-2659. doi: 10.3934/dcds.2019110
[8]

Gui-Dong Li, Yong-Yong Li, Xiao-Qi Liu, Chun-Lei Tang. A positive solution of asymptotically periodic Choquard equations with locally defined nonlinearities. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1351-1365. doi: 10.3934/cpaa.2020066
[9]

Michael E. Filippakis, Donal O'Regan, Nikolaos S. Papageorgiou. Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507
[10]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875
[11]

T. F. Ma, M. L. Pelicer. Attractors for weakly damped beam equations with $p$-Laplacian. Conference Publications, 2013, 2013 (special) : 525-534. doi: 10.3934/proc.2013.2013.525
[12]

Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537
[13]

Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497
[14]

Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335
[15]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5003-5036. doi: 10.3934/dcds.2015.35.5003
[16]

Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849
[17]

Alexander Krasnosel'skii, Alexei Pokrovskii. On subharmonics bifurcation in equations with homogeneous nonlinearities. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 747-762. doi: 10.3934/dcds.2001.7.747
[18]

Takayuki Niimura. Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2561-2591. doi: 10.3934/dcds.2020141
[19]

V. Mastropietro, Michela Procesi. Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities. Communications on Pure & Applied Analysis, 2006, 5 (1) : 1-28. doi: 10.3934/cpaa.2006.5.1
[20]

Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020321

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (77)
  • HTML views (82)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences