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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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##### References:
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