An alternative approach for the analysis and the numerical approximation of ODEs, using a variational framework, is presented. It is based on the natural and elementary idea of minimizing the residual of the differential equation measured in a usual Lp norm. Typical existence results for Cauchy problems can thus be recovered, and finer sets of assumptions for existence are made explicit. We treat, in particular, the cases of an explicit ODE and a differential inclusion. This approach also allows...

In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type $$\begin{array}{c}f(x,\eta ,\xi )\ge a\left(x\right)\frac{{\left|\xi \right|}^p}{{(1+|\eta \left|\right)}^{\alpha }}-b_1\left(x\right){\left|\eta \right|}^{{\beta }_1}-g_1\left(x\right),\\ f(x,\eta ,0)\le b_2\left(x\right){\left|\eta \right|}^{{\beta }_2}+g_2\left(x\right),\end{array}$$ where$0\le \alpha <p$, $1\le {\beta }_1<p$, $0\le {\beta }_2<p$, $\alpha +{\beta }_i\le p$, $a\left(x\right)$, $b_i\left(x\right)$, $g_i\left(x\right)$ ($i=1$, $2$) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^{1,p}\left(a\right)$ , which assume a boundary datum $u_0\in W^{1,p}\left(a\right)\cap L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$.

We consider the following classical autonomous variational problem$$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$where the Lagrangianf is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

We consider the following classical autonomous variational problem$$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$ where the Lagrangian f is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

In this paper we deepen the study of the nonlinear principal components introduced by Salinelli in 1998, referring to a real random variable. New insights on their probabilistic and statistical meaning are given with some properties. An estimation procedure based on spline functions, adapting to a statistical framework the classical Rayleigh–Ritz method, is introduced. Asymptotic properties of the estimator are proved, providing an upper bound for the rate of convergence under suitable mild conditions....