Generalized degree for classes of finite dimensional upper hemi-continuous mappings in separable Banach spaces

Let E be a real separable Banach space, E∗ the dual space of E, and Ω⊂E an open bounded subset, and let T:D(T)⊆E→2E∗ be a finite dimensional upper hemi-continuous mapping with D(T)∩Ω≠∅. A generalized degree theory is constructed for such a mapping. This degree is then applied to study the existence of approximate weak solutions to the equation 0∈Tx.