## Existence of three solutions for Kirchhoff-type three-point boundary value problems.(English)Zbl 07426781

Summary: 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.

### MSC:

 35J20 Variational methods for second-order elliptic equations 34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text:

### References:

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Van der hock, A Galerkin procedure for the diffusion equation subject to specification of mass, SIAM J. Numer. Anal. 24, 499-515, 1987. · Zbl 0677.65108 [7] [7] F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (17), 5962-5974, 2011. · Zbl 1232.35052 [8] [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. · Zbl 1201.34030 [9] [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. · Zbl 1074.34011 [10] [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. · Zbl 0883.34020 [11] [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. · Zbl 1330.35500 [12] [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. · Zbl 1292.34018 [13] [13] J.R. Graef and L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonl. Anal. 14 (1), 39-60, 2007. · Zbl 1145.34007 [14] [14] J.R. Graef, L. Kong and Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (1), 39-52, 2011. · Zbl 1222.34019 [15] [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. · Zbl 0763.34009 [16] [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. · Zbl 1043.34015 [17] [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. · Zbl 1241.35098 [18] [18] S. Heidarkhani, Infinitely many solutions for systems of n two-point Kirchhoff-type boundary value problems, Ann. Polon. Math. 107, 133-152, 2013. · Zbl 1291.34044 [19] [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] [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. · Zbl 1399.34091 [21] [21] J. Henderson, Solutions of multi-point boundary value problems for second order equations, Dynam. Syst. Appl. 15 (1), 111-117, 2006. · Zbl 1104.34310 [22] [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. · Zbl 1061.34009 [23] [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. · Zbl 1182.34029 [24] [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. · Zbl 0668.34024 [25] [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. · Zbl 1107.34007 [26] [26] N.I. Ionkin, The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Diff. 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Ma, Multiplicity results for a three-point boundary value problems at resonance, Nonlinear Anal. 53 (6), 777-789, 2003. · Zbl 1037.34011 [33] [33] R. Ma and H. Wang, Positive solutions of nonlinear three-point boundary-value problems, J. Math. Anal. Appl. 279 (1), 216-227, 2003. · Zbl 1028.34014 [34] [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. · Zbl 1334.35406 [35] [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. · Zbl 1333.35081 [36] [36] G. Molica Bisci and V. Rădulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math. 17 (1), 1450001, 17 pages, 2014. · Zbl 1318.35019 [37] [37] G. Molica Bisci and V. Rădulescu, Mountain pass solutions for nonlocal equations, Annales AcademiæScientiarum FennicæMathematica 39, 579-59, 2014. · Zbl 1309.35017 [38] [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. · Zbl 1329.49018 [39] [39] G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math. 19 (1), 1550088, 23 pp, 2017. · Zbl 1352.35005 [40] [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] [41] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221 (1), 246-255, 2006. · Zbl 1357.35131 [42] [42] B. Ricceri, A further three critical points theorem, Nonlinear Anal. TMA 71 (9), 4151-4157, 2009. · Zbl 1187.47057 [43] [43] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (4), 543-549, 2010. · Zbl 1192.49007 [44] [44] Y. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2), 589-603, 2005. · Zbl 1074.34028 [45] [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. · Zbl 1198.34036 [46] [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. · Zbl 1173.34016 [47] [47] S. Timoshenko, Theory of elastic stability, McGraw Hill, New York, 1961. [48] [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. · Zbl 1056.34035 [49] [49] Q. Yao, Positive solutions of singular third-order three-point boundary value problems, J. Math. Anal. Appl. 354 (1), 207-212, 2009. · Zbl 1169.34314 [50] [50] E. Zeidler, Nonlinear functional analysis and its applications, II/B, Springer-Verlag, New York, 1990. · Zbl 0684.47029 [51] [1] D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling 27, 49-57, 1998. · Zbl 0906.34014 [52] [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. · Zbl 1246.35044 [53] [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. · Zbl 1320.35092 [54] [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. · Zbl 1325.35129 [55] [5] J.R. Cannon, The One-dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications, 23, Addison-Wesley, Menlo Park, California, USA, 1984. [56] [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. · Zbl 0677.65108 [57] [7] F. Colasuonno and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (17), 5962-5974, 2011. · Zbl 1232.35052 [58] [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. · Zbl 1201.34030 [59] [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. · Zbl 1074.34011 [60] [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. · Zbl 0883.34020 [61] [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. · Zbl 1330.35500 [62] [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. · Zbl 1292.34018 [63] [13] J.R. Graef and L. Kong, Existence of solutions for nonlinear boundary value problems, Comm. Appl. Nonl. Anal. 14 (1), 39-60, 2007. · Zbl 1145.34007 [64] [14] J.R. Graef, L. Kong and Q. Kong, Higher order multi-point boundary value problems, Math. Nachr. 284 (1), 39-52, 2011. · Zbl 1222.34019 [65] [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. · Zbl 0763.34009 [66] [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. · Zbl 1043.34015 [67] [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. · Zbl 1241.35098 [68] [18] S. Heidarkhani, Infinitely many solutions for systems of n two-point Kirchhoff-type boundary value problems, Ann. Polon. Math. 107, 133-152, 2013. · Zbl 1291.34044 [69] [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. [70] [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. · Zbl 1399.34091 [71] [21] J. Henderson, Solutions of multi-point boundary value problems for second order equations, Dynam. Syst. Appl. 15 (1), 111-117, 2006. · Zbl 1104.34310 [72] [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. · Zbl 1061.34009 [73] [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. · Zbl 1182.34029 [74] [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. · Zbl 0668.34024 [75] [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. · Zbl 1107.34007 [76] [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. [77] [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. [78] [28] G. Kirchhoff, Vorlesungen über mathematische Physik, Mechanik, Teubner, Leipzig, 1883. · JFM 08.0542.01 [79] [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. · Zbl 1349.34063 [80] [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. [81] [31] R. Ma, Positive solutions for second order three-point boundary value problems, Appl. Math. Lett. 14 (1), 1-5, 2001. · Zbl 0989.34009 [82] [32] R. Ma, Multiplicity results for a three-point boundary value problems at resonance, Nonlinear Anal. 53 (6), 777-789, 2003. · Zbl 1037.34011 [83] [33] R. Ma and H. Wang, Positive solutions of nonlinear three-point boundary-value problems, J. Math. Anal. Appl. 279 (1), 216-227, 2003. · Zbl 1028.34014 [84] [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. · Zbl 1334.35406 [85] [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. · Zbl 1333.35081 [86] [36] G. Molica Bisci and V. Rădulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math. 17 (1), 1450001, 17 pages, 2014. · Zbl 1318.35019 [87] [37] G. Molica Bisci and V. Rădulescu, Mountain pass solutions for nonlocal equations, Annales AcademiæScientiarum FennicæMathematica 39, 579-59, 2014. · Zbl 1309.35017 [88] [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. · Zbl 1329.49018 [89] [39] G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math. 19 (1), 1550088, 23 pp, 2017. · Zbl 1352.35005 [90] [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. [91] [41] K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221 (1), 246-255, 2006. · Zbl 1357.35131 [92] [42] B. Ricceri, A further three critical points theorem, Nonlinear Anal. TMA 71 (9), 4151-4157, 2009. · Zbl 1187.47057 [93] [43] B. Ricceri, On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (4), 543-549, 2010. · Zbl 1192.49007 [94] [44] Y. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2), 589-603, 2005. · Zbl 1074.34028 [95] [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. · Zbl 1198.34036 [96] [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. · Zbl 1173.34016 [97] [47] S. Timoshenko, Theory of elastic stability, McGraw Hill, New York, 1961. [98] [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. · Zbl 1056.34035 [99] [49] Q. Yao, Positive solutions of singular third-order three-point boundary value problems, J. Math. Anal. Appl. 354 (1), 207-212, 2009. · Zbl 1169.34314 [100] [50] E. Zeidler, Nonlinear functional analysis and its applications, II/B, Springer-Verlag, New York, 1990. · Zbl 0684.47029
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