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A Hartman-Nagumo inequality for the vector ordinary $$p$$-Laplacian and applications to nonlinear boundary value problems. (English) Zbl 1041.34011
The authors prove some existence results on solutions of the system of differential equations $$(\phi_{p}(x'))' = f(t,x,x'), \;0 \leq t \leq 1,$$ verifying either the periodic boundary conditions $$x(0) = x(1), \;x'(0) = x'(1),$$ or the Dirichlet boundary conditions $$x(0) = x_{0}, \;x(1) = x_{1}.$$ Here, $$p > 1,$$ $$x \in \mathbb{R}^{N}$$ and $$\phi_{p}(x) = \| x \|^{p-2}x,$$ if $$x \neq 0$$, $$\phi_{p}(0) = 0.$$ On the other hand, $$f: [0,1]\times \mathbb{R}^{N}\times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$$ is continuous and $$x_{0}, x_{1}$$ are some given points of $$\mathbb{R}^{N}.$$ The proofs are based on a generalization of the Hartman-Nagumo inequality to the case of the vector ordinary p-Laplacian and classical degree theory.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 47H11 Degree theory for nonlinear operators
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