Abstract and Applied Analysis

Solvability of Three-Point Boundary Value Problems at Resonance with a p -Laplacian on Finite and Infinite Intervals

Hairong Lian, Patricia J. Y. Wong, and Shu Yang

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Three-point boundary value problems of second-order differential equation with a p-Laplacian on finite and infinite intervals are investigated in this paper. By using a new continuation theorem, sufficient conditions are given, under the resonance conditions, to guarantee the existence of solutions to such boundary value problems with the nonlinear term involving in the first-order derivative explicitly.

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Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 658010, 16 pages.

First available in Project Euclid: 5 April 2013

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Lian, Hairong; Wong, Patricia J. Y.; Yang, Shu. Solvability of Three-Point Boundary Value Problems at Resonance with a $p$ -Laplacian on Finite and Infinite Intervals. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 658010, 16 pages. doi:10.1155/2012/658010. https://projecteuclid.org/euclid.aaa/1365168321

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  • A. Cabada and R. L. Pouso, “Existence results for the problem ${(\varphi ({u}^{'}))}^{'}=f(t,u,{u}^{'})$ with nonlinear boundary conditions,” Nonlinear Analysis, vol. 35, no. 2, pp. 221–231, 1999.
  • M. García-Huidobro, R. Manásevich, P. Yan, and M. Zhang, “A $p$-Laplacian problem with a multi-point boundary condition,” Nonlinear Analysis, vol. 59, no. 3, pp. 319–333, 2004.
  • W. Ge and J. Ren, “An extension of Mawhin's continuation theorem and its application to boundary value problems with a $p$-Laplacian,” Nonlinear Analysis, vol. 58, no. 3-4, pp. 477–488, 2004.
  • W. Ge, Boundary Value Problems for Nonlinear Ordinnary Differential Equaitons, Science Press, Beijing, China, 2007.
  • D. Jiang, “Upper and lower solutions method and a singular superlinear boundary value problem for the one-dimensional $p$-Laplacian,” Computers & Mathematics with Applications, vol. 42, no. 6-7, pp. 927–940, 2001.
  • H. Lü and C. Zhong, “A note on singular nonlinear boundary value problems for the one-dimensional $p$-Laplacian,” Applied Mathematics Letters, vol. 14, no. 2, pp. 189–194, 2001.
  • J. Mawhin, “Some boundary value problems for Hartman-type perturbations of the ordinary vector $p$-Laplacian,” Nonlinear Analysis, vol. 40, pp. 497–503, 2000.
  • M. del Pino, P. Drábek, and R. Manásevich, “The Fredholm alternative at the first eigenvalue for the one-dimensional $p$-Laplacian,” Journal of Differential Equations, vol. 151, no. 2, pp. 386–419, 1999.
  • D. O'Regan, “Some general existence principles and results for ${(\phi ({y}^{\prime }))}^{\prime }=qf(t,y,{y}^{\prime })$, $0<t<1$,” SIAM Journal on Mathematical Analysis, vol. 24, no. 3, pp. 648–668, 1993.
  • M. Zhang, “Nonuniform nonresonance at the first eigenvalue of the $p$-Laplacian,” Nonlinear Analysis, vol. 29, no. 1, pp. 41–51, 1997.
  • Z. Du, X. Lin, and W. Ge, “On a third-order multi-point boundary value problem at resonance,” Journal of Mathematical Analysis and Applications, vol. 302, no. 1, pp. 217–229, 2005.
  • W. Feng and J. R. L. Webb, “Solvability of three point boundary value problems at resonance,” Nonlinear Analysis, vol. 30, no. 6, pp. 3227–3238, 1997.
  • C. P. Gupta, “A second order $m$-point boundary value problem at resonance,” Nonlinear Analysis, vol. 24, no. 10, pp. 1483–1489, 1995.
  • B. Liu, “Solvability of multi-point boundary value problem at resonance. IV,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 275–299, 2003.
  • R. Ma, Nonlocal BVP for Nonlinear Ordinary Differential Equations, Academic Press, 2004.
  • R. P. Agarwal and D. O'Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, 2001.
  • H. Lian and W. Ge, “Solvability for second-order three-point boundary value problems on a half-line,” Applied Mathematics Letters, vol. 19, no. 10, pp. 1000–1006, 2006.