Let $$W^{1,p}_T = \{u \in W^{1,p}([0,T];\mathbb{R}^N) \mid u(0) = u(T)\},$$ where $W^{1,p}([0,T],\mathbb{R}^n)$ is the usual Sobolev space of functions from $[0,T]$ to $\mathbb{R}^N$. Let $$F:[0,T] \times \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$$ satisfying that there exists functions $a \in C(\mathbb{R}^+,\mathbb{R}^+)$ and $b \in L^1([0,T])$ such that $$|F(t,x,y)| \leq a(|x|)(|b(t)| + |y|^p)$$ for almost every $t \in [0,T]$ and for all $x,y \in \mathbb{R}^N$. Then the function $$u \in W^{1,p}_T \mapsto F(\cdot,u(\cdot),\dot{u}(\cdot)) \in L^1([0,T],\mathbb{R})$$ is continuous, where $\dot{u}$ is the weak derivative of $u$.
This is a result used in Mawhin and Willen's book, "Critical point theory and Hamiltonian systems", (Theorem 1.14) which the authors claim that follows from a theorem of Krasnosel'skii, though they don't say which theorem nor do they say where to find it. The bibliography lists two works from Krasnosel'skii that I wasn't able to get access to.
I tried to prove it by contradiction in the following way:
Let $(u_k)$ be a sequence in $W^{1,p}_T$ converging to $u \in W^{1,p}_T$. Assume that $F(\cdot,u_k(\cdot),\dot{u}_k(\cdot))$ does not converge to $F(\cdot,u(\cdot),\dot{u}(\cdot))$. Passing to a subsequence, if necessary, we can assume that there is $\varepsilon > 0$ such that $$||F(\cdot,u_k(\cdot),\dot{u}_k(\cdot)) - F(\cdot,u(\cdot),\dot{u}(\cdot))||_1 \geq \varepsilon$$ for all $k$. Since $u_k$ converges uniformly to $u$ and $\dot{u}_k$ converges in $L^p$ to $\dot{u}$, passing again to a subsequence, we have that $$(t,u_k(t),\dot{u}_k(t)) \to (t,u(t),\dot{u}(t))$$ for almost every $t$. Then I tried to use the condition on $F$ to show that $F(\cdot,u_k(\cdot),\dot{u}_k(\cdot))$ is dominated by a $L^1$ function, in order to apply the dominated convergence theorem and reach a contradiction, but couldn't do it.
Any help would be much appreciated.