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Odd periodic solutions of fully second-order ordinary differential equations with superlinear nonlinearities. (English) Zbl 1378.34065

Summary: This paper is concerned with the existence of periodic solutions for the fully second-order ordinary differential equation \[ u^{\prime\prime}(t) = f(t, u(t), u^{\prime}(t)), \quad t \in \mathbb{R}, \] where the nonlinearity \(f : \mathbb{R}^3 \rightarrow \mathbb{R}\) is continuous and \(f(t, x, y)\) is \(2 \pi\)-periodic in \(t\). Under certain inequality conditions such that \(f(t, x, y)\) may have superlinear growth on \((x, y)\), the existence of odd \(2 \pi\)-periodic solutions is obtained via the Leray-Schauder fixed point theorem.

MSC:

34C25 Periodic solutions to ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
47N20 Applications of operator theory to differential and integral equations
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