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second order nonlinear ordinary differential equation; periodic problem; lower and upper functions
The paper deals with the boundary value problem \[ u^{\prime \prime }+k\,u=g(u)+e(t),\quad u(0)=u(2\pi ),\,\,u^{\prime }(0)=u^{\prime }(2\pi ), \] where $k\in \mathbb{R}$, $g\:I\mapsto \mathbb{R}$ is continuous, $e\in \mathbb{L}J$ and $\lim _{x\rightarrow 0+}\int _x^1g(s)\,\hspace{0.56905pt}\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.
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