American Institute of Mathematical Sciences

Advanced
February  2020, 25(2): 489-505. doi: 10.3934/dcdsb.2019250

Multiplicity results for fourth order problems related to the theory of deformations beams

Alberto Cabada 1,, and  Rochdi Jebari 2,

1. 

Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain
2. 

Faculté des Sciences de Tunis, Université de Tunis El-Manar, Campus Universitaire 2092 - El Manar, Tunisie

* Corresponding author: Alberto Cabada

Received  November 2018 Revised  March 2019 Published  November 2019

The main purpose of this paper is to establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem with integral condition. By using a new technique of construct a positive cone, we apply the Krasnoselskii compression/expansion and Leggett-Williams fixed point theorems in cones to show our multiplicity results. Finally, a particular case is studied, and the existence of multiple solutions is proved for two different particular functions.

Keywords: Green functions, fourth-order boundary value problem, multiplicity of positive solutions, fixed-point index theory, Leggett-Williams fixed point theorem, cone.
Mathematics Subject Classification: Primary: 34B15, 34B27; Secondary: 34B09, 34B10.
Citation: Alberto Cabada, Rochdi Jebari. Multiplicity results for fourth order problems related to the theory of deformations beams. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 489-505. doi: 10.3934/dcdsb.2019250
References:
[1]

A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415–426. doi: 10.1016/S0022-247X(86)80006-3.  Google Scholar

[2]

Z. B. Bai and H. Y. Wang, On the positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357-368.  doi: 10.1016/S0022-247X(02)00071-9.  Google Scholar

[3]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[4]

A. Cabada and C. Fernández-Gómez, Constant sign solutions of two-point fourth order problems, Appl. Math. Comput., 263 (2015), 122-133.  doi: 10.1016/j.amc.2015.03.112.  Google Scholar

[5]

A. Cabada, J. A. Cid and B. Máquez-Villamarín, Computation of Green's functions for boundary value problems with Mathematica, Appl. Math. Comput., 219 (2012), 1919-1936.  doi: 10.1016/j.amc.2012.08.035.  Google Scholar

[6]

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

[7]

A. Cabada, Green's Functions in the Theory of Ordinary Differential Equations, SpringerBriefs in Mathematics, Springer, New York, 2014. doi: 10.1007/978-1-4614-9506-2.  Google Scholar

[8]

A. Cabada and L. Saavedra, The eigenvalue characterization for the constant sign Green's functions of (k, n-k) problems, Bound. Value Probl., 2016 (2016), 35 pp. doi: 10.1186/s13661-016-0547-1.  Google Scholar

[9]

A. Cabada and L. Saavedra, Characterization of constant sign Green's function for a two-point boundary-value problem by means of spectral theory, Electron. J. Differential Equations, 2017 (2017), 96 pp.  Google Scholar

[10]

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

[11]

J. M. Davis and J. Henderson, Uniqueness implies existence for fourth-order Lidstone boundaryvalue problems, Panamer. Math. J., 8 (1998), 23–35.  Google Scholar

[12]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[13]

J. R. Graef, J. Henderson and B. Yang, Positive solutions to a fourth-order three point boundary value problem, Discrete Contin. Dyn. Syst., (2009), 269–275.  Google Scholar

[14]

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

[15]

R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673-688.  doi: 10.1512/iumj.1979.28.28046.  Google Scholar

[16]

R. Y. Ma, J. H. Zhang and F. M. Shengmao, 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

[17]

E. L. Reiss, A. J. Callegari and D. S. Ahluwalia, Ordinary Differential Equations with Applications, Holt, Rhinehart and Winston, New York, 1976. Google Scholar

[18]

S. Timoshenko, Strength of Materials, Van Nostrand, 1955. Google Scholar

[19]

S. P. Timoshenko and S. W. Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959. Google Scholar

[20]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc., 74 (2006), 673-693.  doi: 10.1112/S0024610706023179.  Google Scholar

[21]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear Differ. Equ. Appl., 15 (2008), 45-67.  doi: 10.1007/s00030-007-4067-7.  Google Scholar

show all references

References:
[1]

A. R. Aftabizadeh, Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116 (1986), 415–426. doi: 10.1016/S0022-247X(86)80006-3.  Google Scholar

[2]

Z. B. Bai and H. Y. Wang, On the positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357-368.  doi: 10.1016/S0022-247X(02)00071-9.  Google Scholar

[3]

G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176.  doi: 10.1016/j.jmaa.2008.01.049.  Google Scholar

[4]

A. Cabada and C. Fernández-Gómez, Constant sign solutions of two-point fourth order problems, Appl. Math. Comput., 263 (2015), 122-133.  doi: 10.1016/j.amc.2015.03.112.  Google Scholar

[5]

A. Cabada, J. A. Cid and B. Máquez-Villamarín, Computation of Green's functions for boundary value problems with Mathematica, Appl. Math. Comput., 219 (2012), 1919-1936.  doi: 10.1016/j.amc.2012.08.035.  Google Scholar

[6]

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

[7]

A. Cabada, Green's Functions in the Theory of Ordinary Differential Equations, SpringerBriefs in Mathematics, Springer, New York, 2014. doi: 10.1007/978-1-4614-9506-2.  Google Scholar

[8]

A. Cabada and L. Saavedra, The eigenvalue characterization for the constant sign Green's functions of (k, n-k) problems, Bound. Value Probl., 2016 (2016), 35 pp. doi: 10.1186/s13661-016-0547-1.  Google Scholar

[9]

A. Cabada and L. Saavedra, Characterization of constant sign Green's function for a two-point boundary-value problem by means of spectral theory, Electron. J. Differential Equations, 2017 (2017), 96 pp.  Google Scholar

[10]

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

[11]

J. M. Davis and J. Henderson, Uniqueness implies existence for fourth-order Lidstone boundaryvalue problems, Panamer. Math. J., 8 (1998), 23–35.  Google Scholar

[12]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[13]

J. R. Graef, J. Henderson and B. Yang, Positive solutions to a fourth-order three point boundary value problem, Discrete Contin. Dyn. Syst., (2009), 269–275.  Google Scholar

[14]

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

[15]

R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673-688.  doi: 10.1512/iumj.1979.28.28046.  Google Scholar

[16]

R. Y. Ma, J. H. Zhang and F. M. Shengmao, 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

[17]

E. L. Reiss, A. J. Callegari and D. S. Ahluwalia, Ordinary Differential Equations with Applications, Holt, Rhinehart and Winston, New York, 1976. Google Scholar

[18]

S. Timoshenko, Strength of Materials, Van Nostrand, 1955. Google Scholar

[19]

S. P. Timoshenko and S. W. Krieger, Theory of Plates and Shells, McGraw-Hill, New York, 1959. Google Scholar

[20]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc., 74 (2006), 673-693.  doi: 10.1112/S0024610706023179.  Google Scholar

[21]

J. R. L. Webb and G. Infante, Positive solutions of nonlocal boundary value problems involving integral conditions, Nonlinear Differ. Equ. Appl., 15 (2008), 45-67.  doi: 10.1007/s00030-007-4067-7.  Google Scholar

Figure 1.  Graph of $ \frac{z_M'''(1)-z_M'''(0)-1}{M} $ on $ [-m_0^4, m_1^4) $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
[2]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

[3]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151
[4]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[5]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[6]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[7]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[8]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[9]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[10]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[11]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[12]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[13]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[14]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[15]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349
[16]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[17]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[18]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[19]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[20]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

2019 Impact Factor: 1.27

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences