Definition
Following Antman (1983, p. 283), the formal definition of a variational inequality is the following one.
Definition 1. Given a Banach space , a subset of , and a functional from to the dual space of the space , the variational inequality problem is the problem of solving respect to the variable belonging to the following inequality:
where is the duality pairing.
In general, the variational inequality problem can be formulated on any finite – or infinite-dimensional Banach space. The three obvious steps in the study of the problem are the following ones:
- Prove the existence of a solution: this step implies the mathematical correctness of the problem, showing that there is at least a solution.
- Prove the uniqueness of the given solution: this step implies the physical correctness of the problem, showing that the solution can be used to represent a physical phenomenon. It is a particularly important step since most of the problems modeled by variational inequalities are of physical origin.
- Find the solution.
Read more about this topic: Variational Inequality
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