The paper is devoted to the study of the existence of solutions to the problem \[ (\varphi_{p}(u'))' = f(t,u), \quad t \in [0,T], \] under the Dirichlet conditions \(u(0) = u(T) = 0\) or the periodic boundary conditions \(u(0) = u(T), \;u'(0) = u'(T)\). Here, \(\varphi_{p} : \mathbb R^{n} \rightarrow \mathbb R^{n}\) is defined by \(\varphi_{p}(u) = |u |^{p-2}u, \text{ if } \;u \neq 0, \varphi_{p}(0) = 0\), \(p > 1\), i.e., the vector p-Laplacian ordinary operator, and \(f: [0,T] \times \mathbb R^{n} \rightarrow \mathbb R^{n}\) is continuous. Under the additional hypothesis: \[ \exists M > 0: \langle f(t,u),u\rangle \geq 0,\quad \forall t \in [0,T], \;\forall u \in \mathbb R^{n}, \] satisfying \(|u |= M\), the author proves the existence of at least one solution \(u\) to the previous problem such that \(|u(t) |\leq M\), \(\forall t \in [0,T]\).
This result extends some results of P. Hartman [Trans. Am. Math. Soc. 96, 493-509 (1960; Zbl 0098.06101)] and H.-W. Knobloch [J. Differ. Equations 9, 67-85 (1971; Zbl 0211.11801)] for \(p=2\). The proof requires the use of new techniques, for the presence of the p-Laplacian, the application of the Schauder fixed-point theorem to a suitable modification of the original problem and some ideas from the method of upper and lower solutions associated with a second-order equation. The results of this paper have been recently generalized, by the author and A. Ureña, to the case of derivative-depending nonlinearities [A Hartman-Nagumo inequality for the vector ordinary \(p\)-Laplacian and applications to nonlinear boundary value problems, preprint].