In a linear programming model we have two decision variables $x_i$ and $x_{ij}$, and two parameters (i.e., non-decision variables) $a_i^k$ and $b_{ij}^k$.
We would like to confirm whether the following constraints are considered linear and thus does not invalidate the linearity of the model:
Since $a_i^k$ and $b_i^k$ are constants for the model for all $i\in \mathcal{I},$ and for all $k \in \mathcal{K}$, then they do not affect the constraints in any way that makes them non-linear.
What makes something non-linear in linear programming is if the decision variable is dependent another decision variable, or the degree of the problem is greater than one. For example, the following constraints would be non-linear in your model if they would appear:
$$x_i^2 +x_ix_{ij} -x_{ij}^2\le0\quad\text{ or }\quad \frac{x_i}{x_{ij}}\le |x_i|\quad\text{ and so on.}$$
What will significantly affect your model is the strict inequality constraint, as it may make your feasible region open and an optimal solution (if that's the goal of the problem) may not be able to be found. However, the constraints are still linear.
