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  February 2020 Early access  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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