Theoretical and Numerical Aspects of a Third-order Three-point Nonhomogeneous Boundary Value Problem

André L. M. Martinez, Marcelo R. A. Ferreira, Emerson V. Castelani

Abstract


In this paper we are considering a third-order three-point equation with nonhomogeneous conditions in the boundary. Using Krasnoselskii's Theorem and Leray-Schauder Alternative we provide existence results of positive solutions for this problem. Nontrivials examples are given and a numerical method is introduced.


Keywords


numerical solutions, third-order, boundary value problem and Krasnoselskii's Theorem

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References


D. Anderson, “Multiple Positive Solutions for a Three-Point Boundary Value Problem,” Mathematical and Computer Modelling, vol. 27, no. 6, pp. 49–57, 1998.

D. R. Anderson and J. M. Davis, “Multiple Solutions and Eigenvalues for Third Order Right Focal Boundary Value Problems,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 135–157, 2002.

D. R. Anderson, “Green’s function for a third-order generalized right focal problem,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 1–14, 2003.

A. Boucherif and N. Al-Malki, “Nonlinear three-point third-order boundary value problems,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1168–1177, 2007.c SBMAC TEMA 11

H. Chen, “Positive solutions for the nonhomogeneous three-point boundary value problem of second-order differential equations,” Math. Comput. Modelling, vol. 45, no. 7-8, pp. 844–852, 2007.

Z. B. Fei and Xiangli, “Existence of triple positive solutions for a third order generalized right focal problem,” Math. Inequal. Appl, vol. 9, no. 3, p. 2006, 2006.

J. Graef and B. Yang, “Multiple Positive Solutions to a Three Point Third Order Boundary Value Problem,” Discrete and Continuous Dynamical Systems, vol. 2005, pp. 337–344, 2005.

Y. Lin and M. Cui, “A numerical solution to nonlinear multi-point boundary value problems in the reproducing kernel space,” Mathematical Methods in the Applied Sciences, vol. 34, no. 1, pp. 44–47, 2011.

Z. Liu and F. Li, “On the existence of positive solutions of an elliptic boundary value problem,” Chinese Ann. Math. Ser. B, vol. 21, no. 4, pp. 499–510, 2000.

R. Ma, “Existence Theorems for a Second Orderm-Point Boundary Value Problem,” Journal of Mathematical Analysis and Applications, vol. 211, pp. 545–555, jul 1997.

Y. H. Ma and R. Y. Ma, “Positive solutions of a singular nonlinear three-point boundary value problem,” Acta Math. Sci. Ser. A Chin. Ed., vol. 23, no. 5, pp. 583–588, 2003.

Q. L. Yao, “The existence and multiplicity of positive solutions for a third-order three-point boundary value problem,” Acta Mathematicae Applicatae Sinica, vol. 19, no. 1, pp. 117–122, 2003.

H. Yu, L. Haiyan, and Y. Liu, “Multiple positive solutions to third-order threepoint singular semipositone boundary value problem,” Proceedings Mathematical Sciences, vol. 114, no. 4, pp. 409–422, 2004.

Y. Sun, “Positive solutions for third-order three-point nonhomogeneous boundary value problems,” Applied Mathematics Letters, vol. 22, pp. 45–51, jan 2009.

R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed point theory and applications, vol. 141. Cambridge university press, 2001.




DOI: https://doi.org/10.5540/tema.2019.020.03.417

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TEMA - Trends in Applied and Computational Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)
ISSN: 1677-1966  (print version),  2179-8451  (online version)

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