Research Article
BibTex RIS Cite

Existence of three solutions for Kirchhoff-type three-point boundary value problems

Year 2021, Volume: 50 Issue: 2, 304 - 317, 11.04.2021

Shapour Heidarkhani Amjad Salari

https://doi.org/10.15672/hujms.912780
Cited By: 2

Abstract

The present paper is an attempt to investigate the multiplicity results of solutions for a three-point boundary value problem of Kirchhoff-type. Indeed, we will use variational methods for smooth functionals, defined on the reflexive Banach spaces in order to achieve the existence of at least three solutions for the equation. Finally, by presenting one example, we will ensure the applicability of our results.

Keywords

three solutions three-point boundary value problem second-order Kirchhoff-type equation Lipschitz condition variational methods critical point theory

References

  • [1] D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling 27, 49-57, 1998.
  • [2] G. Autuori, F. Colasuonno and P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Var. Elliptic Eqs. 57, 379-395, 2012.
  • [3] G. Autuori, F. Colasuonno and P. Pucci, Lifespan estimates for solutions of polyharmonic Kirchhoff systems, Math. Mod. Meth. Appl. Sci. 22 (2), 1150009, 36 pages, 2012.
  • [4] G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math. 16 (5), 1450002, 43 pages, 2014.
  • [5] J.R. Cannon, The One-dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23, Addison-Wesley, Menlo Park, California, USA, 1984.
  • [6] J.R. Cannon, E.P. Esteva and J. Van der hock, A Galerkin procedure for the diffusion equation subject to specification of mass, SIAM J. Numer. Anal. 24, 499-515, 1987.
  • [7] F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (17), 5962-5974, 2011.
  • [8] Z. Du and L. Kong, Existence of three solutions for systems of multi-point boundary value problems, Electron. J. Qual. Theory Diff. Equ. 10 (1), 1-17, 2009.
  • [9] Z. Du, C. Xue and W. Ge, Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative, Arch. Math. 84 (4), 341-349, 2005.
  • [10] W. Feng and J.R.L. Webb, Solvability of m-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212 (2), 467-480, 1997.
  • [11] G.M. Figueiredo, G. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii’s genus, Asymptot. Anal. 94 (3-4), 347-361, 2015.
  • [12] J.R. Graef, S. Heidarkhani and L. Kong, Infinitely many solutions for systems of multi-point boundary value problems, Topol. Methods Nonlinear Anal. 42 (1), 105- 118, 2013.
  • [13] J.R. Graef and L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonl. Anal. 14 (1), 39-60, 2007.
  • [14] J.R. Graef, L. Kong and Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (1), 39–52, 2011.
  • [15] C.P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl. 168 (2), 540-551, 1992.
  • [16] X. He and W. Ge, Triple solutions for second order three-point boundary value problems, J. Math. Anal. Appl. 268 (1), 256-265, 2002.
  • [17] S. Heidarkhani, Multiple solutions for a class of multipoint boundary value systems driven by a one dimensional $(p_1, \ldots , p_n)$-Laplacian operator, Abstr. Appl. Anal. 2012, Article ID 389530, 15 pages, 2012.
  • [18] S. Heidarkhani, Infinitely many solutions for systems of n two-point Kirchhoff-type boundary value problems, Ann. Polon. Math. 107, 133-152, 2013.
  • [19] S. Heidarkhani, G.A. Afrouzi and D. O’Regan, Existence of three solutions for a Kirchhoff-type boundary-value problem, Electronic J. Differ. Equ. 2011, No. 91, 1-11, 2011.
  • [20] S. Heidarkhani and A. Salari, Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type, Stud. Sci. Math. Hungarica, 54 (1), 119140, 2017.
  • [21] J. Henderson, Solutions of multi-point boundary value problems for second order equations, Dynam. Syst. Appl. 15 (1), 111-117, 2006.
  • [22] J. Henderson, B. Karna and C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133(5), 1365-1369, 2005.
  • [23] J. Henderson and S.K. Ntouyas, Positive solutions for systems of nth order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Diff. Equ. 2007, No. 18, 1-12, 2007.
  • [24] V.A. Il’in and E.I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm–Liouville operator, Differ. Equ. 23 (7), 979-987, 1987.
  • [25] G. Infante, Positive solutions of some three-point boundary value problems via fixed point index for weakly inward A-proper maps, Fixed Point Theory Appl. 2005 (2), 177-184, 2005.
  • [26] N.I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Diff. Uravn. 13 (2), 294-304, 1977.
  • [27] N.I. Kamyuin, A boundary value problem in the theory of the heat conduction with nonclassical boundary condition, USSR Comput. Math. Phy. 4 (6), 33-59, 1964.
  • [28] G. Kirchhoff, Vorlesungen über mathematische Physik, Mechanik, Teubner, Leipzig, 1883.
  • [29] X. Lin, Existence of three solutions for a three-point boundary value problem via a three-critical-point theorem, Carpathian J. Math. 31, 213-220, 2015.
  • [30] J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Stud. 30, 284-346. North-Holland, Amsterdam, 1978.
  • [31] R. Ma, Positive solutions for second order three-point boundary value problems, Appl. Math. Lett. 14 (1), 1-5, 2001.
  • [32] R. Ma, Multiplicity results for a three-point boundary value problems at resonance, Nonlinear Anal. 53 (6), 777-789, 2003.
  • [33] R. Ma and H. Wang, Positive solutions of nonlinear three-point boundary-value problems, J. Math. Anal. Appl. 279 (1), 216-227, 2003.
  • [34] X. Mingqi, G. Molica Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity 29 (2), 357-374, 2016.
  • [35] G. Molica Bisci and P. Pizzimenti, Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition, Proc. Edinb. Math. Soc. 57 (3), 779-809, 2014.
  • [36] G. Molica Bisci and V. Rădulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math. 17 (1), 1450001, 17 pages, 2014.
  • [37] G. Molica Bisci and V. Rădulescu, Mountain pass solutions for nonlocal equations, Annales AcademiæScientiarum FennicæMathematica 39, 579-59, 2014.
  • [38] G. Molica Bisci, and D. Repovš, On doubly nonlocal fractional elliptic equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26, 161-176, 2015.
  • [39] G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math. 19 (1), 1550088, 23 pp, 2017.
  • [40] M. Moshinsky, Sobre los problemas de condiciones a la frontier en una dimension de caracteristicas discontinues, Bol. Soc. Mat. Mexicana 7, 1-25, 1950.
  • [41] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221 (1), 246-255, 2006.
  • [42] B. Ricceri, A further three critical points theorem, Nonlinear Anal. TMA 71 (9), 4151-4157, 2009.
  • [43] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (4), 543-549, 2010.
  • [44] Y. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2), 589-603, 2005.
  • [45] J. Sun, H. Chen, J. Nieto and M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal. 72 (12), 4575-4586, 2010.
  • [46] Y. Sun, L. Liu, J. Zhang and R.P. Agarwal, Positive solutions of singular three-point boundary value problems for second-order differential equations, J. Comput. Appl. Math. 230 (2), 738–750, 2009.
  • [47] S. Timoshenko, Theory of elastic stability, McGraw Hill, New York, 1961.
  • [48] X. Xu, Multiplicity results for positive solutions of some semi-positone three-point boundary value problems, J. Math. Anal. Appl. 291 (2), 673-689, 2004.
  • [49] Q. Yao, Positive solutions of singular third-order three-point boundary value problems, J. Math. Anal. Appl. 354 (1), 207-212, 2009.
  • [50] E. Zeidler, Nonlinear functional analysis and its applications, II/B, Springer-Verlag, New York, 1990.

Year 2021, Volume: 50 Issue: 2, 304 - 317, 11.04.2021

Shapour Heidarkhani Amjad Salari

https://doi.org/10.15672/hujms.912780
Cited By: 2

Abstract

References

  • [1] D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling 27, 49-57, 1998.
  • [2] G. Autuori, F. Colasuonno and P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Var. Elliptic Eqs. 57, 379-395, 2012.
  • [3] G. Autuori, F. Colasuonno and P. Pucci, Lifespan estimates for solutions of polyharmonic Kirchhoff systems, Math. Mod. Meth. Appl. Sci. 22 (2), 1150009, 36 pages, 2012.
  • [4] G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Commun. Contemp. Math. 16 (5), 1450002, 43 pages, 2014.
  • [5] J.R. Cannon, The One-dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23, Addison-Wesley, Menlo Park, California, USA, 1984.
  • [6] J.R. Cannon, E.P. Esteva and J. Van der hock, A Galerkin procedure for the diffusion equation subject to specification of mass, SIAM J. Numer. Anal. 24, 499-515, 1987.
  • [7] F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (17), 5962-5974, 2011.
  • [8] Z. Du and L. Kong, Existence of three solutions for systems of multi-point boundary value problems, Electron. J. Qual. Theory Diff. Equ. 10 (1), 1-17, 2009.
  • [9] Z. Du, C. Xue and W. Ge, Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative, Arch. Math. 84 (4), 341-349, 2005.
  • [10] W. Feng and J.R.L. Webb, Solvability of m-point boundary value problems with nonlinear growth, J. Math. Anal. Appl. 212 (2), 467-480, 1997.
  • [11] G.M. Figueiredo, G. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii’s genus, Asymptot. Anal. 94 (3-4), 347-361, 2015.
  • [12] J.R. Graef, S. Heidarkhani and L. Kong, Infinitely many solutions for systems of multi-point boundary value problems, Topol. Methods Nonlinear Anal. 42 (1), 105- 118, 2013.
  • [13] J.R. Graef and L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonl. Anal. 14 (1), 39-60, 2007.
  • [14] J.R. Graef, L. Kong and Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (1), 39–52, 2011.
  • [15] C.P. Gupta, Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl. 168 (2), 540-551, 1992.
  • [16] X. He and W. Ge, Triple solutions for second order three-point boundary value problems, J. Math. Anal. Appl. 268 (1), 256-265, 2002.
  • [17] S. Heidarkhani, Multiple solutions for a class of multipoint boundary value systems driven by a one dimensional $(p_1, \ldots , p_n)$-Laplacian operator, Abstr. Appl. Anal. 2012, Article ID 389530, 15 pages, 2012.
  • [18] S. Heidarkhani, Infinitely many solutions for systems of n two-point Kirchhoff-type boundary value problems, Ann. Polon. Math. 107, 133-152, 2013.
  • [19] S. Heidarkhani, G.A. Afrouzi and D. O’Regan, Existence of three solutions for a Kirchhoff-type boundary-value problem, Electronic J. Differ. Equ. 2011, No. 91, 1-11, 2011.
  • [20] S. Heidarkhani and A. Salari, Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type, Stud. Sci. Math. Hungarica, 54 (1), 119140, 2017.
  • [21] J. Henderson, Solutions of multi-point boundary value problems for second order equations, Dynam. Syst. Appl. 15 (1), 111-117, 2006.
  • [22] J. Henderson, B. Karna and C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations, Proc. Amer. Math. Soc. 133(5), 1365-1369, 2005.
  • [23] J. Henderson and S.K. Ntouyas, Positive solutions for systems of nth order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Diff. Equ. 2007, No. 18, 1-12, 2007.
  • [24] V.A. Il’in and E.I. Moiseev, Nonlocal boundary value problem of the second kind for a Sturm–Liouville operator, Differ. Equ. 23 (7), 979-987, 1987.
  • [25] G. Infante, Positive solutions of some three-point boundary value problems via fixed point index for weakly inward A-proper maps, Fixed Point Theory Appl. 2005 (2), 177-184, 2005.
  • [26] N.I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Diff. Uravn. 13 (2), 294-304, 1977.
  • [27] N.I. Kamyuin, A boundary value problem in the theory of the heat conduction with nonclassical boundary condition, USSR Comput. Math. Phy. 4 (6), 33-59, 1964.
  • [28] G. Kirchhoff, Vorlesungen über mathematische Physik, Mechanik, Teubner, Leipzig, 1883.
  • [29] X. Lin, Existence of three solutions for a three-point boundary value problem via a three-critical-point theorem, Carpathian J. Math. 31, 213-220, 2015.
  • [30] J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), North-Holland Math. Stud. 30, 284-346. North-Holland, Amsterdam, 1978.
  • [31] R. Ma, Positive solutions for second order three-point boundary value problems, Appl. Math. Lett. 14 (1), 1-5, 2001.
  • [32] R. Ma, Multiplicity results for a three-point boundary value problems at resonance, Nonlinear Anal. 53 (6), 777-789, 2003.
  • [33] R. Ma and H. Wang, Positive solutions of nonlinear three-point boundary-value problems, J. Math. Anal. Appl. 279 (1), 216-227, 2003.
  • [34] X. Mingqi, G. Molica Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity 29 (2), 357-374, 2016.
  • [35] G. Molica Bisci and P. Pizzimenti, Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition, Proc. Edinb. Math. Soc. 57 (3), 779-809, 2014.
  • [36] G. Molica Bisci and V. Rădulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math. 17 (1), 1450001, 17 pages, 2014.
  • [37] G. Molica Bisci and V. Rădulescu, Mountain pass solutions for nonlocal equations, Annales AcademiæScientiarum FennicæMathematica 39, 579-59, 2014.
  • [38] G. Molica Bisci, and D. Repovš, On doubly nonlocal fractional elliptic equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26, 161-176, 2015.
  • [39] G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math. 19 (1), 1550088, 23 pp, 2017.
  • [40] M. Moshinsky, Sobre los problemas de condiciones a la frontier en una dimension de caracteristicas discontinues, Bol. Soc. Mat. Mexicana 7, 1-25, 1950.
  • [41] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221 (1), 246-255, 2006.
  • [42] B. Ricceri, A further three critical points theorem, Nonlinear Anal. TMA 71 (9), 4151-4157, 2009.
  • [43] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (4), 543-549, 2010.
  • [44] Y. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2), 589-603, 2005.
  • [45] J. Sun, H. Chen, J. Nieto and M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal. 72 (12), 4575-4586, 2010.
  • [46] Y. Sun, L. Liu, J. Zhang and R.P. Agarwal, Positive solutions of singular three-point boundary value problems for second-order differential equations, J. Comput. Appl. Math. 230 (2), 738–750, 2009.
  • [47] S. Timoshenko, Theory of elastic stability, McGraw Hill, New York, 1961.
  • [48] X. Xu, Multiplicity results for positive solutions of some semi-positone three-point boundary value problems, J. Math. Anal. Appl. 291 (2), 673-689, 2004.
  • [49] Q. Yao, Positive solutions of singular third-order three-point boundary value problems, J. Math. Anal. Appl. 354 (1), 207-212, 2009.
  • [50] E. Zeidler, Nonlinear functional analysis and its applications, II/B, Springer-Verlag, New York, 1990.
There are 50 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Shapour Heidarkhani This is me 0000-0002-7908-8388

Amjad Salari This is me 0000-0001-9170-9856

Publication Date April 11, 2021
Published in Issue Year 2021 Volume: 50 Issue: 2

Cite

APA Heidarkhani, S., & Salari, A. (2021). Existence of three solutions for Kirchhoff-type three-point boundary value problems. Hacettepe Journal of Mathematics and Statistics, 50(2), 304-317. https://doi.org/10.15672/hujms.912780
AMA Heidarkhani S, Salari A. Existence of three solutions for Kirchhoff-type three-point boundary value problems. Hacettepe Journal of Mathematics and Statistics. April 2021;50(2):304-317. doi:10.15672/hujms.912780
Chicago Heidarkhani, Shapour, and Amjad Salari. “Existence of Three Solutions for Kirchhoff-Type Three-Point Boundary Value Problems”. Hacettepe Journal of Mathematics and Statistics 50, no. 2 (April 2021): 304-17. https://doi.org/10.15672/hujms.912780.
EndNote Heidarkhani S, Salari A (April 1, 2021) Existence of three solutions for Kirchhoff-type three-point boundary value problems. Hacettepe Journal of Mathematics and Statistics 50 2 304–317.
IEEE S. Heidarkhani and A. Salari, “Existence of three solutions for Kirchhoff-type three-point boundary value problems”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 2, pp. 304–317, 2021, doi: 10.15672/hujms.912780.
ISNAD Heidarkhani, Shapour - Salari, Amjad. “Existence of Three Solutions for Kirchhoff-Type Three-Point Boundary Value Problems”. Hacettepe Journal of Mathematics and Statistics 50/2 (April 2021), 304-317. https://doi.org/10.15672/hujms.912780.
JAMA Heidarkhani S, Salari A. Existence of three solutions for Kirchhoff-type three-point boundary value problems. Hacettepe Journal of Mathematics and Statistics. 2021;50:304–317.
MLA Heidarkhani, Shapour and Amjad Salari. “Existence of Three Solutions for Kirchhoff-Type Three-Point Boundary Value Problems”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 2, 2021, pp. 304-17, doi:10.15672/hujms.912780.
Vancouver Heidarkhani S, Salari A. Existence of three solutions for Kirchhoff-type three-point boundary value problems. Hacettepe Journal of Mathematics and Statistics. 2021;50(2):304-17.

Cited By

Multiple solutions to a quasilinear periodic boundary value problem with impulsive effects
Electronic Journal of Qualitative Theory of Differential Equations
https://doi.org/10.14232/ejqtde.2024.1.12
Three solutions for discrete anisotropic Kirchhoff-type problems
Demonstratio Mathematica
https://doi.org/10.1515/dema-2022-0209

https://dergipark.org.tr/en/pub/hujms