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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

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