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Positive periodic solutions of general nonlinear third-order ordinary differential equations. (Chinese. English summary) Zbl 1449.34119

Summary: Using the fixed point index theory of cones, we consider the existence of positive \(2\pi\)-periodic solutions for general third-order ordinary differential equation \[Lu (t) = f (t, u (t), u' (t), u'' (t)) (t\in\mathbb{R}),\] where \[Lu (t) = u''' (t) + a_2 u'' (t) + {a_1}u' (t) + {a_0}u (t)\] is a third-order ordinary differential operator, \(f:\mathbb{R} \times [0,\infty) \times {\mathbb{R}^2} \to [0,\infty)\) is a continuous function and \(f (t, x, y, z)\) is \(2\pi\)-periodic with respect to \(t\). Under the conditions that the nonlinear term \(f\) satisfies some easily verifiable inequalities, some existence results for positive \(2\pi\)-periodic solutions of the equation are obtained that allow \(f (t, x, y, z)\) satisfies superlinear or sublinear growth with respect to \(x\), \(y\), \(z\).

MSC:

34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
47N20 Applications of operator theory to differential and integral equations
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