The Dirichlet problem for the vector ordinary $$p$$ -Laplacian.

We introduce the $$p$$ -Mawhin-Ureña-Nagumo and $$p$$ -Hartman-Nagumo conditions and apply them to prove the Dirichlet problem for the vector ordinary $$p$$ -Laplacian, $$(\varPhi _p(x'))'=f(t,x,x')$$ , for $$t\in [0,1],$$ has a solution $$x$$ with $$(t,x(t))\in \varOmega \subset [0,1]\times \mathbb {R}^n$$ where $$(\varOmega ,v,p)$$ is a $$p$$ -admissible bounding set. For $$1<p<2$$ , we turn the vector ordinary $$p$$ -Laplacian into an equivalent system of second-order ordinary differential equations to prove existence. For $$p>2$$ we approximate the $$p$$ -Laplacian and then turn the approximation into a system, proving existence as a limit of solutions to the approximating problems. [ABSTRACT FROM AUTHOR]