The equation
$$x[n]=y[n-1]-\frac52y[n]+y[n+1]\tag{1}$$
is just a way of representing the input-output relation of a linear time-invariant discrete-time system. It does not mean that the input depends on the output. The input is always given and the system behaves in such a way that the output satisfies $(1)$.
Note that from $(1)$ we cannot determine whether or not the corresponding system is causal. You could rewrite $(1)$ as
$$y[n+1]=x[n]-y[n-1]+\frac52y[n]\tag{2}$$
which would suggest that the system is causal, because it seems that the output at time $n+1$ only depends on past values of the input and output. However, we could also rewrite $(1)$ as
$$y[n-1]=x[n]+\frac52y[n]-y[n+1]\tag{3}$$
which would suggest that the output only depends on future values of the input and output. Finally, another way of rewriting $(1)$ is
$$y[n]=\frac25\big(y[n-1]+y[n+1]-x[n]\big)\tag{4}$$
which seems to imply that the output depends on current, past and future values.
From the difference equation $(1)$ we simply cannot tell if the system is causal or not. Eq. $(1)$ represents three different systems, as expressed by Eqs $(2)$, $(3)$ and $(4)$: one causal, one acausal (left-sided), and one non-causal (two-sided). In order to completely describe the system, we generally need more information than just the difference equation.
