Electron. J. Diff. Equ., Vol. 2016 (2016), No. 206, pp. 1-17.

Existence of solutions for p-Laplacian-like differential equation with multi-point nonlinear Neumann boundary conditions at resonance

Le Xuan Truong, Le Cong Nhan

Abstract:
This work concerns the multi-point nonlinear Neumann boundary-value problem involving a p-Laplacian-like operator
$$\displaylines{ (\phi( u'))' = f(t, u, u'),\quad t\in (0,1), \cr u'(0) = u'(\eta), \quad \phi(u'(1)) = \sum_{i=1}^m{\alpha_i \phi(u'(\xi_i))}, }$$
where $\phi:\mathbb{R} \to \mathbb{R}$ is an odd increasing homeomorphism with $\phi(\pm \infty) = \pm \infty$ such that
$$ 0<\alpha(A):=\limsup_{s\to +\infty}\frac{\phi(A + s)}{\phi(s)} <\infty, \quad \text{for } A >0. $$
By using an extension of Mawhin's continuation theorem, we establish sufficient conditions for the existence of at least one solution.

Submitted December 24, 2014. Published July 29, 2016.
Math Subject Classifications: 34B10, 34B15.
Key Words: Continuation theorem; p-Laplacian differential equation; resonance.

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Le Xuan Truong
Division of Computational Mathematics and Engineering
Institute for Computational Science
Ton Duc Thang University
Ho Chi Minh City, Vietnam
email: lexuantruong@tdt.edu.vn
Le Cong Nhan
Mathematic Department
An Giang University
18 Ung Van Khiem Str, An Giang, Vietnam
email: lcnhanmathagu@gmail.com

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