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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.

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