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

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

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.

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. CabadaJ. 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. CabadaJ. 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. MaJ. 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. CabadaJ. 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. CabadaJ. 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. MaJ. 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) $
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