Convex functions occurring in variational problems and the absolute minimum problem

For the minimum problem of the functional $\int_{(a,\,x^0)}^{(b,\,x^1)}f(t,x(t),\dot x(t))\,dt$ (where $f(t,x,u)\colon T\times R^n\times R^n\to(-\infty,\infty)$, and the case $f=\infty$ corresponds to some constraints imposed on $x$ and $u$) we consider the problem of the existence of a function $\varphi(t,x)\colon T\times\ R^n\to R$ which has the following property: if $x_m(t)$ is a minimizing sequence, then, for any $\alpha$ and $\beta$ wich $a\leqslant\alpha<\beta\leqslant b$, and for any $x(t)$,
\begin{multline*} \widetilde\varphi(\beta,x(\beta))-\varphi(\alpha,x(\alpha))-\int_\alpha^\beta f(t,x(t),\dot x(t))\,dt\\ \leqslant\varliminf\biggl[\varphi(\beta,x_m(\beta))-\varphi(\alpha,x_m(\alpha))-\int_\alpha^\beta f(t,x_m(t),\dot x_m(t))\,dt\biggr] \end{multline*}
(every function $\varphi$ which has this property yields a necessary condition for the absolute minimum). We prove existence criterions for an arbitrary and continuous function $\varphi$.
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