Fractional difference equations are a generalization of integer-order difference equations and differential equations, where the order of the difference operator is a non-integer value. An equilibrium point of a fractional difference equation is a solution that remains constant over time.

To find the equilibrium point of a fractional difference equation, we first need to express it in a form that is amenable to analysis. This involves rewriting the equation as a fixed point iteration, which can be written as:

x_{n+1} = f(x_n)

where x_n is the value of the solution at time n, x_{n+1} is the value of the solution at time n+1, and f(x_n) is a function that describes the update rule for the solution.

Once we have the fixed point iteration form, we can find the equilibrium point by solving for x such that:

x = f(x)

In other words, the equilibrium point is the value of x that satisfies this equation. This can be done analytically, or through numerical methods such as iterative methods like Newton-Raphson method.

Note that fractional difference equations can have multiple equilibrium points, which can be stable or unstable depending on the behavior of the solution near them. Understanding the stability properties of equilibrium points is crucial for predicting the long-term behavior of solutions to fractional difference equations.