Existence of solutions for a third-order boundary value problem with $p$-Laplacian operator and nonlinear boundary conditions

De-Xiang Ma; Shu-Zhen Sun

doi:10.4171/pm/1828

JournalspmVol. 66, No. 1pp. 13–27

Existence of solutions for a third-order boundary value problem with $p$ -Laplacian operator and nonlinear boundary conditions

De-Xiang Ma
North China Electric Power University, Beijing, China
Shu-Zhen Sun
North China Electric Power University, Beijing, China

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Abstract

In this paper we study the third-order nonlinear boundary value problem

(Φ (u^{''}))^{'} (t) + f (t, u (t), u^{'} (t), u^{''} (t)) = 0 a.e. t \in [0, 1], u (0) = 0, g (u^{'} (0), u^{''} (0)) = A, h (u^{'} (1), u^{''} (1)) = B,

where $A, B \in R$ , $f : [0, 1] \times R^{3} \to R$ is a Carathéodory function, $g, h \in C^{0} (R^{2}, R)$ and $Φ \in C^{0} (R, R)$ . Using apriori estimates, the Nagumo condition, upper and lower solutions and the Schauder fixed point theorem, we are able to prove existence of solutions of this problem.

Cite this article

De-Xiang Ma, Shu-Zhen Sun, Existence of solutions for a third-order boundary value problem with $p$ -Laplacian operator and nonlinear boundary conditions. Port. Math. 66 (2009), no. 1, pp. 13–27

DOI 10.4171/PM/1828