Multiple Solutions for an Equation of Kirchhoff Type: Theoretical and Numerical Aspects

André Luís Machado Martinez, Emerson Vitor Castelani, Glaucia Maria Bressan, Elenice Weber Stiegelmeier


A nonlinear boundary value problem related to an equation of Kirchhoff type is considered. The existence of multiple positive solutions is proved through Avery-Peterson Fixed Point Theorem. A numerical method based on Levenberg-Marquadt algorithm combined with a heuristic process is present in order to align numerical and theoretical aspects.


Multiple solution, Kirchhoff Equation, numerical solutions

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T. F. Ma, E. S. Miranda, and M. B. de Souza Cortes, “A nonlinear differential equation involving reflection of the argument,” Arch. Math.(Brno), vol. 40, no. 1, pp. 63–68, 2004.

G. R. Kirchhoff, Vorlesungen über mathematische physik: mechanik, vol. 1. Teubner, 1876.

P. Amster and M. C. Mariani, “A fixed point operator for a nonlinear boundary value problem,” Journal of mathematical analysis and applications, vol. 266, no. 1, pp. 160–168, 2002.

A. Arosio and S. Panizzi, “On the well-posedness of the kirchhoff string,” Transactions of the American Mathematical Society, vol. 348, no. 1, pp. 305–330, 1996.

M. Chipot and J. F. Rodrigues, “On a class of nonlocal nonlinear elliptic problems,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 26, no. 3, pp. 447–467, 1992.

T. F. Ma, “Existence results for a model of nonlinear beam on elastic bearings,” Applied Mathematics Letters, vol. 13, no. 5, pp. 11–15, 2000.

G. Caristi, S. Heidarkhani, and A. Salari, “Variational approaches to kirchhofftype second-order impulsive differential equations on the half-line,” Results in Mathematics, vol. 73, no. 1, p. 44, 2018.

A. L. M. Martinez, E. V. Castelani, J. Da Silva, and W. V. I. Shirabayashi, “A note about positive solutions for an equation of kirchhoff type,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 2082–2090, 2011.

R. Avery and A. C. Peterson, “Three positive fixed points of nonlinear operators on ordered banach spaces,” Computers & Mathematics with Applications, vol. 42, no. 3-5, pp. 313–322, 2001.

J. J. Moré, “The levenberg-marquardt algorithm: implementation and theory,” in Numerical analysis, pp. 105–116, Springer, 1978.

M. I. A. Lourakis, “A brief description of the levenberg-marquardt algorithm implemented by levmar,” Foundation of Research and Technology, vol. 4, no. 1, pp. 1–6, 2005.

C. T. Kelley, Iterative methods for optimization, vol. 18. Siam, 1999.


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