The phrase "when there exist" is correct and usable in written English.
You can use it in situations where something is the case only when certain conditions exist. For example, "The project will only move forward when there exist enough resources to do so."
The use of auxiliary information is most useful when the percentage of missing data is high, and when there exist certain combinations of missing data that show little correlation with the data that are present.
As a matter of fact, there are many problems in practice that can be casted as uncertain variational inequality problems when there exist uncertain variables (vector) such as in electricity supply network [1], user equilibrium [19] and wireless networks [20].
Compared to Lyapunov's stability theory, it may be a more powerful tool in cases when searching a Lyapunov functional is a difficult task or when there exist multiple equilibrium points, [1, 12].
The results obtained from this theory are compared with those of analytical results existing in the literature or with those obtained by the finite element method when there exist no available results.
IPR represents a radical form of statist intervention (patent office, patent enforcement, research and development subsidies, tax breaks, monopolistic pricing, and the inevitable colossal subsidy programs, etc) when there exist more efficient models for financing research and development that adopt free market outcomes.
So, when, there exist some values for which there is no solution.
The proposed MST-based method can also deal with the case when there exist interference targets.
Below, we consider the cases when there exist simple roots or one multiple root.
Due to and being equivalent when there exist constants,,, and such that,, and (2.10).
When there exist more sequences, the best one is chosen with respect to a prescribed criterion.
Now, we consider the dynamical scenario when there exist three minimal period-two solutions.
