# Study of Third-Order Three-Point Boundary Value Problem with Dependence on the First-Order Derivative

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 41)

## Abstract

Under certain conditions on the nonlinearity f and by using Leray–Schauder nonlinear alternative and the Banach contraction theorem, we prove the existence and uniqueness of nontrivial solution of the following third-order three-point boundary value problem (BVP1):
$$\displaystyle\begin{array}{rcl} & & \left \{\begin{array}{c} {u}^{{\prime\prime\prime}} + f\left (t,u\left (t\right ),{u}^{{\prime}}\left (t\right )\right ) = 0,\ \ \ t \in \left (0,1\right ) \\ \alpha {u}^{{\prime}}\left (1\right ) =\beta u\left (\eta \right ), u\left (0\right ) = {u}^{{\prime}}\left (0\right ) = 0 \end{array} \right. \\ & & \begin{array}{c} \text{where} \beta, \text{ }\alpha \in \mathbb{R}_{+}^{{\ast}},\text{ }0 <\eta < 1; \end{array} \\ \end{array}$$
then we study the positivity by applying the well-known Guo–Krasnosel’skii fixed-point theorem. The interesting point lies in the fact that the nonlinear term is allowed to depend on the first-order derivative u.

### References

1. 1.
D. R. Anderson, Green’s function for a third-order generalized right focal problem, J. Math. Anal. Appl. 288 (2003), 1–14.
2. 2.
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
3. 3.
A. Guezane-Lakoud and L. Zenkoufi, Positive solution of a three-point nonlinear boundary value problem for second order differential equations,IJAMAS, 20 (2011), 38–46.
4. 4.
A. Guezane-Lakoud, S. Kelaiaia and A. M. Eid, A positive solution for a non-local boundary value problem, Int. J. Open Problems Compt. Math., Vol. 4, No. 1, (2011), 36–43.
5. 5.
A. Guezane-Lakoud and S. Kelaiaia, Solvability of a three-point nonlinear boundary-value problem, EJDE, Vol. 2010, No. 139, (2010), 1–9.Google Scholar
6. 6.
J. R. Graef and Bo Yang, Existence and nonexistence of positive solutions of a nonlinear third order boundary value problem, EJQTDE, 2008, No. 9, 1–13.Google Scholar
7. 7.
J. R. Graef and B. Yang, Positive solutions of a nonlinear third order eigenvalue problem, Dynam. Systems Appl. 15 (2006), 97–110.
8. 8.
D.Guo and V.Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988.
9. 9.
L. J. Guo, J. P. Sun and Y. H. Zhao, Existence of positive solutions for nonlinear third-order three-point boundary value problem, Nonlinear Anal.,Vol 68, 10 (2008), 3151–3158.
10. 10.
B. Hopkins and N. Kosmatov, Third-order boundary value problems with sign-changing solutions, Nonlinear Anal., 67(2007), 126–137S
11. 11.
Li, Positive solutions of nonlinear singular third-order two-point boundary value problem, J. Math. Anal. Appl. 323 (2006), 413–425.Google Scholar
12. 12.
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.Google Scholar
13. 13.
Y. Sun, Positive solutions of singular third-order three-point boundary value problem, J. Math. Anal. Appl. 306 (2005), 589–603.

## Authors and Affiliations

1. 1.Faculty of Sciences, Department of MathematicsUniversity Badji MokhtarAnnabaAlgeria
2. 2.Department of MathematicsUniversity of 8 May 1945 GuelmaGuelmaAlgeria