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Periodic solutions for a class of nonlinear 2\(n\)th-order differential equations. (English) Zbl 1207.34052

The paper deals with a \(2n^{th}\)-order nonlinear differential equation of the type
\[ u^{(2n)}(t)+a(u^{(n)}(t)) u^{(n+1)}(t)- P(u'(t))u''(t)+f(u(t))u'(t)-g(t,u(t))=e(t), \]
where \(n\geq 2\) is an even integer, \(g:\mathbb R^2\rightarrow \mathbb R\) is continuous, \(T\)-periodic in \(t\), \(a,P,f:\mathbb R\rightarrow R\) are continuous, and \(e:\mathbb R\rightarrow \mathbb R\) is continuous, \(T\)-periodic with \(\int_0^T e(t)\,dt=0\). Applying Mawhin’s continuation theorem of coincidence degree theory, under appropriate hypotheses, the existence of \(T\)-periodic solutions is established. Applications to fourth-order equations are given. Such equations find applications in fluid mechanics and nonlinear elasticity mechanics.

MSC:

34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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