  In this paper, we study the orbit intersection problem for the linear space
and the algebraic group in positive characteristic. Let $K$ be an algebraically
closed field of positive characteristic and let $\Phi_1, \Phi_{2}: K^d
\longrightarrow K^{d}$ be affine maps, $\Phi_i({\bf x}) = A_i ({\bf x}) + {\bf
x_i}$ (where each $A_i$ is a $d\times d$ matrix and ${\bf x}\in K^d$). If each
${\bf a}_i \in K^d$ is not $\Phi_i$-preperiodic, then we prove that the set
$\left \{(n_1, n_2) \in \Z^{2} \mid \Phi_1^{n_1}({\bf a}_1) =
\Phi_{2}^{n_{2}}({\bf a}_{2})\right\}$ is $p$-normal in $\mathbb{Z}^{2}$ of
order at most $d$. Further, for the regular self-maps $\Phi_1, \Phi_{2}:
\mathbb{G}_m^d \longrightarrow \mathbb{G}_m^d$, we show that the set $ \{(n_1,
n_{2}) \in \N_0^{2} \mid \Phi_1^{n_1}({\bf a}_1) = \Phi_{2}^{n_{2}}({\bf
a}_{2})\}$ (where ${\bf a}_1, {\bf a}_2\in \mathbb{G}_m^d(K)$) lie in a finite
number of linear and exponential one-parameter families. To do so, we use
results on linear equations over multiplicative groups in positive
characteristic and some results on systems of polynomial-exponential equations.
