Nonlinear Three Point Boundary Value Problem
DOI:
https://doi.org/10.5644/SJM.08.1.07Keywords:
Fixed point theorem, three point boundary value problem, Leray Schauder nonlinear alternativeAbstract
In this work, we establish sufficient conditions for the existence of solutions for a three point boundary value problem generated by a third order differential equation. We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. Then by using the Leray Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we give two examples to illustrate our results.
2000 Mathematics Subject Classification. 34B10, 34B15, 34B18, 34G20
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R.P. Agarwal and D. O’Regan, Infinite interval problems modelling phenomena which arise in the theory of plasma and electrical theory, Studies. Appl. Math., 111 (2003), 339–358.
D. R. Anderson and J. M. Davis, Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. Math. Anal. Appl., 267 (2002), 135–157.
K. Deimling, Nonlinear Functional Analysis. Springer, Berlin, 1985.
G. Infante and J. R. L. Webb, Three point boundary value problems with solutions that change sign, J. Integ. Eqns Appl., 15 (2003), 37–57.
G. Infante and J. R. L. Webb, Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc., 49 (2006), 637–656.
V. A. Il’in and E. I. Moiseev, Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations., 23 (7) (1987), 803–810.
W. Feng and J. R. L. Webb, Solvability of three point nonlinear boundary value problems at resonance, Nonlinear Analysis TMA., 30 (6) (1997), 3227–3238.
John R. Graef and Bo Yang, Positive solutions for a third order nonlocal boundary value problem, Discrete Contin. Dyn. Syst., Ser. S., 1 (2008), 89–97.
John R. Graef, L.J. Kong and Bo Yang, Positive solutions to a nonlinear third order three point boundary value problem, Electron. J. Differential Equations, Conf., 19 (2010), 151–159.
John R. Graef, L.J. Kong and Bo Yang, Positive solutions for third order multipoint singular boundary value problems, Czechoslovak Mathematical Journal., 60 (1) (2010), 173–182.
John R. Graef and J. R. L. Webb, Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal., 71 (2009), no. 5-6, 1542–1551.
A. Guezane-Lakoud and R. Khaldi, Study of a third-order three-point boundary value problem, AIP Conf. Proc., November 11, Volume 1309 (2010), 329–335.
A. Guezane-Lakoud and S. Kelaiaia, Solvability of a three-point nonlinear boundary value problem, Electron. J. Differential Equations., 2010 (139), 1–9.
A. Guezane-Lakoud, S. Kelaiaia and A. M. Eid, A positive solution for a nonlocal boundary value problem, Int. J. Open Problems Compt. Math., 4 (1) (2011), 36–43.
P. Guidotti and S. Merino, Gradual loss of positivity and hidden invariant cones in a scalar heat equation, Differential Integral Equations., 13 (2000), 1551–1568.
L.J. Kong and J.S.W. Wong, Positive solutions for multi-point boundary value problems with nonhomogeneous boundary conditions, J. Math. Anal. Appl., 367 (2010), 588–611.
L.J. Kong and Q.K. Kong, Higher order boundary value problems with nonhomogeneous boundary conditions, Nonlinear Anal., 72 (1) (2010), 240–261.
R. Ma, A Survey On nonlocal boundary value problems, Applied Mathematics ENotes., 7 (2007), 257–279.
S. A. Krasnoselskii, A remark on a second order three-point boundary value problem, J. Math. Anal. Appl., 183 (1994), 581–592.
P. K. Palamides, G. Infante, P. Pietramala, Nontrivial solutions of a nonlinear heat flow problem via Sperner’s Lemma., Appl. Math. Lett., 22 (2009), no. 9, 1444–1450.
Alex K. Palamides and Nikolaos M. Stavrakakis, Existence and uniqueness of a positive solution for a third-order three-point boundary-value problem, Electron. J. Differential Equations., 2010 No. 155, pp 12.
Y. Sun, Positive solutions for third-order three-point nonhomogeneous boundary value problems., Appl. Math. Lett., 22 (1) (2009) 45–51.
J. R. L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal., 63 (2005), 672–685.
J. R. L. Webb and Gennaro Infante, Non-local boundary value problems of arbitrary order. J. Lond. Math. Soc., (2) 79 (2009), no. 1, 238–258.
