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

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

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

Abstract
Related Papers

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,, {\it Rocky Mountain J. Math.} {\bf 39} (2009), 39 (2009), 707. Google Scholar
[2]	D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem,, {\it Topol. Methods Nonlinear Anal.} {\bf 22} (2003), 22 (2003), 93. Google Scholar
[3]	G. Bonanno, A critical point theorem via the Ekeland variational principle,, {\it Nonlinear Anal.} {\bf 75} (2012), 75 (2012), 2992. Google Scholar
[4]	G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,, {\it J. Differential Equations} {\bf 244} (2008), 244 (2008), 3031. Google Scholar
[5]	Z. Du and L. Kong, Existence of three solutions for systems of multi-point boundary value problems,, {\it Electron. J. Qual. Theory Diff. Equ.}, (2009). Google Scholar
[6]	P. W. Eloe and B. Ahmad, Positive solutions of a nonlinear $n$th order boundary value problem with nonlocal conditions,, {\it Appl. Math. Lett.} {\bf 18} (2005), 18 (2005), 521. Google Scholar
[7]	H. Feng and W. Ge, Existence of three positive solutions for $M$-point boundary-value problem with one-dimensional,, {\it Taiwanese J. Math.} {\bf 14} (2010), 14 (2010), 647. Google Scholar
[8]	W. Feng and J. R. L. Webb, Solvability of $m$-point boundary value problems with nonlinear growth,, {\it J. Math. Anal. Appl.} {\bf 212} (1997), 212 (1997), 467. Google Scholar
[9]	J. R. Graef, S. Heidarkhani, and L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems,, {\it Appl. Anal.} {\bf 90} (2011), 90 (2011), 1909. 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,, {\it Topol. Methods Nonlinear Anal.} {\bf 42} (2013), 42 (2013), 105. Google Scholar
[11]	J. R. Graef, L. Kong, Existence of solutions for nonlinear boundary value problems,, {\it Comm. Appl. Nonlinear Anal.} {\bf 14} (2007), 14 (2007), 39. Google Scholar
[12]	J. R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems,, {\it Math. Nachr.} {\bf 284} (2011), 284 (2011), 39. 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,, {\it Abstract Appl. Anal.} {\bf 2012} (2012), 2012 (2012). Google Scholar
[14]	J. Henderson, Solutions of multipoint boundary value problems for second order equations,, {\it Dynam. Syst. Appl.} {\bf 15} (2006), 15 (2006), 111. Google Scholar
[15]	J. Henderson, B. Karna, and C. C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations,, {\it Proc. Amer. Math. Soc.} {\bf 133} (2005), 133 (2005), 1365. Google Scholar
[16]	J. Henderson and S. K. Ntouyas, Positive solutions for systems of $n$th order three-point nonlocal boundary value problems,, {\it Electron. J. Qual. Theory Diff. Equ.} 2007, (2007). 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,, {\it Portugal. Math. (N. S.)} {\bf 65} (2008), 65 (2008), 67. Google Scholar
[18]	R. Ma, Existence of positive solutions for superlinear $m$-point boundary value problems,, {\it Proc. Edinburgh Math. Soc.} {\bf 46} (2003), 46 (2003), 279. Google Scholar
[19]	R. Ma and D. O'Regan, Solvability of singular second order $m$-point boundary value problems,, {\it J. Math. Anal. Appl.} {\bf 301} (2005), 301 (2005), 124. Google Scholar
[20]	B. Ricceri, A general variational principle and some of its applications,, {\it J. Comput. Appl. Math.} {\bf 113} (2000), 113 (2000), 401. Google Scholar
[21]	E. Zeidler, Nonlinear functional analysis and its applications,, Vol. II., (1985). Google Scholar

show all references

References:

[1]	D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions,, {\it Rocky Mountain J. Math.} {\bf 39} (2009), 39 (2009), 707. Google Scholar
[2]	D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem,, {\it Topol. Methods Nonlinear Anal.} {\bf 22} (2003), 22 (2003), 93. Google Scholar
[3]	G. Bonanno, A critical point theorem via the Ekeland variational principle,, {\it Nonlinear Anal.} {\bf 75} (2012), 75 (2012), 2992. Google Scholar
[4]	G. Bonanno and P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,, {\it J. Differential Equations} {\bf 244} (2008), 244 (2008), 3031. Google Scholar
[5]	Z. Du and L. Kong, Existence of three solutions for systems of multi-point boundary value problems,, {\it Electron. J. Qual. Theory Diff. Equ.}, (2009). Google Scholar
[6]	P. W. Eloe and B. Ahmad, Positive solutions of a nonlinear $n$th order boundary value problem with nonlocal conditions,, {\it Appl. Math. Lett.} {\bf 18} (2005), 18 (2005), 521. Google Scholar
[7]	H. Feng and W. Ge, Existence of three positive solutions for $M$-point boundary-value problem with one-dimensional,, {\it Taiwanese J. Math.} {\bf 14} (2010), 14 (2010), 647. Google Scholar
[8]	W. Feng and J. R. L. Webb, Solvability of $m$-point boundary value problems with nonlinear growth,, {\it J. Math. Anal. Appl.} {\bf 212} (1997), 212 (1997), 467. Google Scholar
[9]	J. R. Graef, S. Heidarkhani, and L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems,, {\it Appl. Anal.} {\bf 90} (2011), 90 (2011), 1909. 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,, {\it Topol. Methods Nonlinear Anal.} {\bf 42} (2013), 42 (2013), 105. Google Scholar
[11]	J. R. Graef, L. Kong, Existence of solutions for nonlinear boundary value problems,, {\it Comm. Appl. Nonlinear Anal.} {\bf 14} (2007), 14 (2007), 39. Google Scholar
[12]	J. R. Graef, L. Kong, Q. Kong, Higher order multi-point boundary value problems,, {\it Math. Nachr.} {\bf 284} (2011), 284 (2011), 39. 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,, {\it Abstract Appl. Anal.} {\bf 2012} (2012), 2012 (2012). Google Scholar
[14]	J. Henderson, Solutions of multipoint boundary value problems for second order equations,, {\it Dynam. Syst. Appl.} {\bf 15} (2006), 15 (2006), 111. Google Scholar
[15]	J. Henderson, B. Karna, and C. C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations,, {\it Proc. Amer. Math. Soc.} {\bf 133} (2005), 133 (2005), 1365. Google Scholar
[16]	J. Henderson and S. K. Ntouyas, Positive solutions for systems of $n$th order three-point nonlocal boundary value problems,, {\it Electron. J. Qual. Theory Diff. Equ.} 2007, (2007). 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,, {\it Portugal. Math. (N. S.)} {\bf 65} (2008), 65 (2008), 67. Google Scholar
[18]	R. Ma, Existence of positive solutions for superlinear $m$-point boundary value problems,, {\it Proc. Edinburgh Math. Soc.} {\bf 46} (2003), 46 (2003), 279. Google Scholar
[19]	R. Ma and D. O'Regan, Solvability of singular second order $m$-point boundary value problems,, {\it J. Math. Anal. Appl.} {\bf 301} (2005), 301 (2005), 124. Google Scholar
[20]	B. Ricceri, A general variational principle and some of its applications,, {\it J. Comput. Appl. Math.} {\bf 113} (2000), 113 (2000), 401. Google Scholar
[21]	E. Zeidler, Nonlinear functional analysis and its applications,, Vol. II., (1985). Google Scholar

[1]	Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457
[2]	Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759
[3]	Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313
[4]	Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[5]	Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118
[6]	Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[7]	John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337
[8]	Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485
[9]	John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269
[10]	Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483
[11]	Antonio Ambrosetti, Massimiliano Berti. Applications of critical point theory to homoclinics and complex dynamics. Conference Publications, 1998, 1998 (Special) : 72-78. doi: 10.3934/proc.1998.1998.72
[12]	Martin Oberlack, Andreas Rosteck. New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 451-471. doi: 10.3934/dcdss.2010.3.451
[13]	K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624
[14]	Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775
[15]	Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[16]	J. R. L. Webb. Remarks on positive solutions of some three point boundary value problems. Conference Publications, 2003, 2003 (Special) : 905-915. doi: 10.3934/proc.2003.2003.905
[17]	K. Q. Lan. Properties of kernels and eigenvalues for three point boundary value problems. Conference Publications, 2005, 2005 (Special) : 546-555. doi: 10.3934/proc.2005.2005.546
[18]	Wenying Feng. Solutions and positive solutions for some three-point boundary value problems. Conference Publications, 2003, 2003 (Special) : 263-272. doi: 10.3934/proc.2003.2003.263
[19]	Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127
[20]	Marek Rychlik. The Equichordal Point Problem. Electronic Research Announcements, 1996, 2: 108-123.

Impact Factor:

American Institute of Mathematical Sciences