Solvability of a nonlinear boundary value problem
Authors
A. GuezaneLakoud
 Laboratory of Advanced Materials, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, Annaba, Algeria.
S. KELAIAIA
 Laboratory of Advanced Materials, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, Annaba, Algeria.
Abstract
In this paper we consider three point boundary value problems
of second order. We introduce new and sufficient conditions that allow us to
obtain the existence of a nontrivial solution by using Leray Schauder nonlinear
alternative. As an application, we give some examples to illustrate our results.
Keywords
 Fixed point theorem
 Three point boundary value problem
 Non trivial solution.
MSC
References

[1]
R. A. Agarwal, 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.

[2]
K. Boukerrioua, A. GuezaneLakoud, Some nonlinear integral inequalities arising in differential equations, EJDE, 80 (2008), 1–6.

[3]
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin (1985)

[4]
G. Infante, J. R. L. Webb, Three point boundary value problems with solutions that change sign , J. Integ. Eqns Appl., 15 (2003), 37–57.

[5]
G. Infante, J. R. L. Webb, Loss of positivity in a nonlinear scalar heat equation, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 249–261.

[6]
G. Infante, J. R. L. Webb, Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc. Edinb. Math. Soc., 49 (2006), 637–656.

[7]
G. Infante, Positive solutions of nonlocal boundary value problems with singularities, Discrete Contin. Dyn. Syst. Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl., (2009), 377–384.

[8]
V. A. Il’in, 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), 803810.

[9]
H. Fan, R. Ma , Loss of positivity in a nonlinear second order ordinary differential equations, Nonlinear Anal., 71 (2009), 437–444.

[10]
W. Feng, J. R. L. Webb, Solvability of three point nonlinear boundary value problems at resonance, Nonlinear Analysis TMA., 30 (6) (1997), 3227–3238.

[11]
P. Guidotti, S. Merino, Gradual loss of positivity and hidden invariant cones in a scalar heat equation, Differential Integral Equations, 13 (2000), 1551–1568.

[12]
C. P. Gupta, Solvability of a threepoint nonlinear boundary value problem for a second order differential equation, J. Math. Anal. Appl., 168 (1992), 540–551.

[13]
R. A. Khan, N. A. Asif, Positive solutions for a class of singular two point boundary value problems, J. Nonlinear. Sci. Appl., 2 (2009), 126–135

[14]
S. A. Krasnoselskii , A remark on a second order threepoint boundary value problem, J. Math. Anal. Appl., 183 (1994), 581–592.

[15]
R. Ma, Existence theorems for second order threepoint boundary value problems, J. Math. Anal. Appl., 212 (1997), 545–555.

[16]
R. Ma, A Survey On nonlocal boundary value problems, Applied Mathematics ENotes, 7 (2007), 257–279.

[17]
P. K. Palamides, G. Infante, P. Pietramala, Nontrivial solutions of a nonlinear heat flow problem via Sperner’s Lemma, Applied Mathematics Letters, 22 (2009), 1444–1450.

[18]
L. Shuhong, YP. Sun, Nontrivial solution of a nonlinear second order three point boundary value problem, Appl. Math. J., 22 (1) (2007), 3747.

[19]
S. Sivasankaran, M. Mallika Arjunan, V. Vijayakumar, Existence of global solutions for impulsive functional differential equations with nonlocal conditions , J. Nonlinear. Sci. Appl., 4 (2) (2011), 102–114.

[20]
YP. Sun, Nontrivial solution for a threepoint boundaryvalue problem, EJDE, 111 (2004), 1–10.

[21]
J. R. L. Webb, Optimal constants in a nonlocal boundary value problem, Nonlinear Anal., 63 (2005), 672–685.

[22]
F. Zhang, Multiple positive solution for nonlinear singular third order boundary value problem in abstract spaces, J. Nonlinear. Sci. Appl. , 1 (1) (2008), 36–44.