American Institute of Mathematical Sciences

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2013, 2013(special): 273-281. doi: 10.3934/proc.2013.2013.273

Existence of nontrivial solutions to systems of multi-point boundary value problems

John R. Graef 1, , Shapour Heidarkhani 2, and  Lingju Kong 1,

1. 

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States
2. 

Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149

Received  July 2012 Revised  December 2012 Published  November 2013

In this paper, sufficient conditions are established for the existence of at least one nontrivial solution of the multi-point boundary value system $$ \left\{\begin{array}{ll} -(\phi_{p_i}(u'_{i}))'=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}),\ x\in(0,1),\\ u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\ u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j), \end{array} \right. i=1,\ldots,n. $$ The approach is based on variational methods and critical point theory.
Keywords: Multi-point boundary value problem, critical point theory..
Mathematics Subject Classification: Primary: 34B10, 34B1.
Citation: John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273
References:
[1]

D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math. 39 (2009), 707-727.  Google Scholar

[2]

D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal. 22 (2003), 93-103.  Google Scholar

[3]

G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), 2992-3007.  Google Scholar

[4]

G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations 244 (2008), 3031-3059.  Google Scholar

[5]

Z. Du and L. Kong, Existence of three solutions for systems of multi-point boundary value problems, Electron. J. Qual. Theory Diff. Equ., Spec. Ed. I, (2009), No. 10, 17 pp. (electronic).  Google Scholar

[6]

P. W. Eloe and B. Ahmad, Positive solutions of a nonlinear $n$th order boundary value problem with nonlocal conditions, Appl. Math. Lett. 18 (2005), 521-527.  Google Scholar

[7]

H. Feng and W. Ge, Existence of three positive solutions for $M$-point boundary-value problem with one-dimensional, Taiwanese J. Math. 14 (2010), 647-665.  Google Scholar

[8]

W. Feng and J. R. L. Webb, Solvability of $m$-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212 (1997), 467-480.  Google Scholar

[9]

J. R. Graef, S. Heidarkhani, and L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems, Appl. Anal. 90 (2011), 1909-1925.  Google Scholar

[10]

J. R. Graef, S. Heidarkhani, and L. Kong, Infinitely many solutions for systems of multi-point boundary value problems using variational methods, Topol. Methods Nonlinear Anal. 42 (2013), 105-118 Google Scholar

[11]

J. R. Graef, L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonlinear Anal. 14 (2007), 39-60.  Google Scholar

[12]

J. R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (2011), 39-52.  Google Scholar

[13]

S. Heidarkhani, Multiple solutions for a class of multipoint boundary value systems driven by a one dimensional ($p_1,\ldots, p_n$)-Laplacian operator, Abstract Appl. Anal. 2012 (2012), Article ID 389530, 15 pages, doi:10.1155/2012/389530  Google Scholar

[14]

J. Henderson, Solutions of multipoint boundary value problems for second order equations, Dynam. Syst. Appl. 15 (2006), 111-117.  Google Scholar

[15]

J. Henderson, B. Karna, and C. C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133 (2005), 1365-1369.  Google Scholar

[16]

J. Henderson and S. K. Ntouyas, Positive solutions for systems of $n$th order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Diff. Equ. 2007, No. 18, 12 pp. (electronic).  Google Scholar

[17]

D. Ma and X. Chen, Existence and iteration of positive solutions for a multi-point boundary value problem with a $p$-Laplacian operator, Portugal. Math. (N. S.) 65 (2008), 67-80.  Google Scholar

[18]

R. Ma, Existence of positive solutions for superlinear $m$-point boundary value problems, Proc. Edinburgh Math. Soc. 46 (2003), 279-292.  Google Scholar

[19]

R. Ma and D. O'Regan, Solvability of singular second order $m$-point boundary value problems, J. Math. Anal. Appl. 301 (2005), 124-134.  Google Scholar

[20]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410.  Google Scholar

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E. Zeidler, Nonlinear functional analysis and its applications, Vol. II., New York 1985. Google Scholar

show all references

References:
[1]

D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math. 39 (2009), 707-727.  Google Scholar

[2]

D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal. 22 (2003), 93-103.  Google Scholar

[3]

G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), 2992-3007.  Google Scholar

[4]

G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations 244 (2008), 3031-3059.  Google Scholar

[5]

Z. Du and L. Kong, Existence of three solutions for systems of multi-point boundary value problems, Electron. J. Qual. Theory Diff. Equ., Spec. Ed. I, (2009), No. 10, 17 pp. (electronic).  Google Scholar

[6]

P. W. Eloe and B. Ahmad, Positive solutions of a nonlinear $n$th order boundary value problem with nonlocal conditions, Appl. Math. Lett. 18 (2005), 521-527.  Google Scholar

[7]

H. Feng and W. Ge, Existence of three positive solutions for $M$-point boundary-value problem with one-dimensional, Taiwanese J. Math. 14 (2010), 647-665.  Google Scholar

[8]

W. Feng and J. R. L. Webb, Solvability of $m$-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212 (1997), 467-480.  Google Scholar

[9]

J. R. Graef, S. Heidarkhani, and L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems, Appl. Anal. 90 (2011), 1909-1925.  Google Scholar

[10]

J. R. Graef, S. Heidarkhani, and L. Kong, Infinitely many solutions for systems of multi-point boundary value problems using variational methods, Topol. Methods Nonlinear Anal. 42 (2013), 105-118 Google Scholar

[11]

J. R. Graef, L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonlinear Anal. 14 (2007), 39-60.  Google Scholar

[12]

J. R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (2011), 39-52.  Google Scholar

[13]

S. Heidarkhani, Multiple solutions for a class of multipoint boundary value systems driven by a one dimensional ($p_1,\ldots, p_n$)-Laplacian operator, Abstract Appl. Anal. 2012 (2012), Article ID 389530, 15 pages, doi:10.1155/2012/389530  Google Scholar

[14]

J. Henderson, Solutions of multipoint boundary value problems for second order equations, Dynam. Syst. Appl. 15 (2006), 111-117.  Google Scholar

[15]

J. Henderson, B. Karna, and C. C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133 (2005), 1365-1369.  Google Scholar

[16]

J. Henderson and S. K. Ntouyas, Positive solutions for systems of $n$th order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Diff. Equ. 2007, No. 18, 12 pp. (electronic).  Google Scholar

[17]

D. Ma and X. Chen, Existence and iteration of positive solutions for a multi-point boundary value problem with a $p$-Laplacian operator, Portugal. Math. (N. S.) 65 (2008), 67-80.  Google Scholar

[18]

R. Ma, Existence of positive solutions for superlinear $m$-point boundary value problems, Proc. Edinburgh Math. Soc. 46 (2003), 279-292.  Google Scholar

[19]

R. Ma and D. O'Regan, Solvability of singular second order $m$-point boundary value problems, J. Math. Anal. Appl. 301 (2005), 124-134.  Google Scholar

[20]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410.  Google Scholar

[21]

E. Zeidler, Nonlinear functional analysis and its applications, Vol. II., New York 1985. Google Scholar

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