  An interesting observation is that most pairs of weakly homogeneous mappings
have no strongly monotonic property, which is one of the key conditions to
ensure the unique solvability of the generalized variational inequality. This
paper focuses on studying the unique solvability of the generalized variational
inequality with a pair of weakly homogeneous mappings. By using a weaker
condition than the strong monotonicity and some additional conditions, we
achieve several results on the unique solvability of the underlying problem.
These results are exported by making use of the exceptional family of elements
or derived from new obtained Karamardian-type theorems or established under the
exceptional regularity condition. They are new even when the problem comes down
to its important subclasses studied in recent years.
