Some properties of $q$-ary bent functions

Let $F$ be a function from a finite field $Q$ to a finite field $P$. Here, both fields are of characteristic 2, $|P|=q\geq2$ and $Q$ is the expansion of the field $P$. The period of $F$ is defined as the period of the sequence $u(i)= F(\theta^i)$ ($\theta$ – primitive element of $Q$, $i\in\mathbb N_0$). Besides, let $N_a(F)$ be a number of solutions in $Q$ of equation $F(x)=a$, $a\in P$.
Consider $F$ to be a bent function. In this case, it is shown that if the period of $F$ is not maximal one, then exact values of $N_a(F)$, $a\in P$, can be derived. Moreover, if values of $N_a(F)$, $a\in P$, are of a special form, then the value of the period of $F$ is divisible by some exact value.