Periodic Solutions for a Third Order Differential Equation Under Conditions on the Potential

Feliz M. Minhós

Abstract: We prove an existence result to the nonlinear periodic problem $$ \left\{\eqalign{&\hskip1.4cm x^{\prime \prime \prime}+a\,x^{\prime \prime}+g(x^{\prime})+c\,x=p(t),\cr &x(0)=x(2\pi )\,, x^{\prime}(0)=x^{\prime}(2\pi )\,, x^{\prime\prime}(0)=x^{\prime\prime}(2\pi ),\cr}\right. $$ where $g:\Bbb{R}\mapsto\Bbb{R}$ is continuous, $p:[0,2\pi]\mapsto \Bbb{R}$ belongs to $\Bbb{L}^1(0,2\pi)$, $a\in\Bbb{R}$, $c\in\Bbb{R}\backslash\{0\}$, under conditions on the asymptotic behaviour of the primitive of the nonlinearity $g$. This work uses the Leray-Schauder degree theory and improves a result contained in $[\text{EO}]$, weakening the condition on the oscillation of $g$. The arguments used were suggested by $[\text{GO}]$, $[\text{HOZ}]$ and $[\text{SO}]$.

Classification (MSC2000): 34B15

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