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

John R. Graef; Shapour Heidarkhani; Lingju Kong

doi:10.3934/proc.2013.2013.273

Article Contents

2013, Volume 2013: 273-281. Doi: 10.3934/proc.2013.2013.273 open access

This issue Previous Article Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space Next Article Positive solutions of nonlocal fractional boundary value problems

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

1.
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403
2.
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149

Received: June 30, 2012

Revised: November 30, 2012

Published: October 2013

Abstract Related Papers Cited by

Abstract

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, 34B15.

Citation:

Related Papers

Cited by

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.