Electron. J. Differential Equations, Vol. 2018 (2018), No. 35, pp. 1-16.

Dirichlet boundary value problem for a system of n second order asymptotically asymmetric differential equations

Armands Gritsans, Felix Sadyrbaev, Inara Yermachenko

Abstract:
We consider systems of the form
$$\displaylines{ x_1''+ g_1(x_1) = h_1(x_1,x_2,\ldots,x_n),\cr x_2''+ g_2(x_2) = h_2(x_1,x_2,\ldots,x_n),\cr \cdots \cr x_n''+ g_n(x_n) = h_n(x_1,x_2,\ldots,x_n) }$$
along with the boundary conditions
$$ x_1(0)=x_2(0)=\dots=x_n(0)=0=x_1(1)=x_2(1)=\dots=x_n(1)\,. $$
We assume that right sides are bounded continuous functions, and satisfy $h_i(0,0,\ldots,0)=0$. Also we assume that $g_i(x_i)$ are asymptotically asymmetric functions. By using vector field rotation theory, we provide the existence of solutions.

Submitted April 5, 2017. Published January 24, 2018.
Math Subject Classifications: 34B08, 34B15.
Key Words: Dirichlet boundary value problem; rotation of vector field; asymptotically asymmetric nonlinearities; index of isolated singular point; Fucik spectrum.

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Armands Gritsans
Institute of Life Sciences and Technologies
Daugavpils University
Parades iela 1a, Daugavpils LV 5400, Latvia
email: arminge@inbox.lv
Felix Sadyrbaev
Institute of Life Sciences and Technologies
Daugavpils University
Parades iela 1a, Daugavpils LV 5400, Latvia
email: felix@latnet.lv
Inara Yermachenko
Institute of Life Sciences and Technologies
Daugavpils University
Parades iela 1a, Daugavpils LV 5400, Latvia
email: inara.jermacenko@du.lv

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