# American Institute of Mathematical Sciences

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

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

 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.
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. [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. [3] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), 2992-3007. [4] G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations 244 (2008), 3031-3059. [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). [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. [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. [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. [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. [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 [11] J. R. Graef, L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonlinear Anal. 14 (2007), 39-60. [12] J. R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (2011), 39-52. [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 [14] J. Henderson, Solutions of multipoint boundary value problems for second order equations, Dynam. Syst. Appl. 15 (2006), 111-117. [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. [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). [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. [18] R. Ma, Existence of positive solutions for superlinear $m$-point boundary value problems, Proc. Edinburgh Math. Soc. 46 (2003), 279-292. [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. [20] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410. [21] E. Zeidler, Nonlinear functional analysis and its applications, Vol. II., New York 1985.

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##### 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. [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. [3] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), 2992-3007. [4] G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations 244 (2008), 3031-3059. [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). [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. [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. [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. [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. [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 [11] J. R. Graef, L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonlinear Anal. 14 (2007), 39-60. [12] J. R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (2011), 39-52. [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 [14] J. Henderson, Solutions of multipoint boundary value problems for second order equations, Dynam. Syst. Appl. 15 (2006), 111-117. [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. [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). [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. [18] R. Ma, Existence of positive solutions for superlinear $m$-point boundary value problems, Proc. Edinburgh Math. Soc. 46 (2003), 279-292. [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. [20] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. 113 (2000), 401-410. [21] E. Zeidler, Nonlinear functional analysis and its applications, Vol. II., New York 1985.
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