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A Knobloch-type result for \(p(t\))-Laplacian systems. (English) Zbl 1033.34023
The authors study the periodic boundary value problem \[ (| u'|^{p(t)- 2} u')'= f(t,u),\quad u(0)= u(T),\quad u'(0)= u'(T), \] where the function \(f\in C(\mathbb{R}\times \mathbb{R}^n, \mathbb{R}^N)\) is \(T\)-periodic with respect to \(t\), and \(p\in C(\mathbb{R}, \mathbb{R})\) is a \(T\)-periodic function such that \(p(t)> 1\) for \(t\in \mathbb{R}\).
The authors prove that, if there exists some \(r> 0\) such that \(\langle f(t,u), u\rangle\geq 0\) for \(t\in\mathbb{R}\) and \(u\in\mathbb{R}^N\) with \(| u|= r\), then the problem has a solution \(u\) satisfying \(| u(t)|\leq r\), \(t\in\mathbb{R}\).

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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