A constrained approximation theorem for integral functionals on $L^p$

Let $(T,\mathcal{F},\mu )$ be a $\sigma$-finite measure space, $E$ a separable real Banach space and $p\ge 1$. Given a sequence of functions $f,f_1,f_2,\mathrm{.}\mathrm{.}\mathrm{.}$ from $T\times E$ to $\mathbf{R}$, under general assumptions, we prove that, for each closed hyperplane $V$ of $L^p(T,E)$, for each $u\in V$, and for each sequence $\{{\lambda }_n\}$ converging to ${\int }_{T}f(t,u(t))d\mu$, there exists a sequence $\{u_n\}$ in $V$ converging to $u$ and such that ${\int }_T{f}_n(t,u_n(t))d\mu ={\lambda }_n$ for all $n$ large enough.