Abstract

The dissertation considers a degree theory and the index of a critical point of demi-continuous, everywhere defined mappings of the monotone type.

A topological degree is derived for mappings from a Banach space to its dual space. The mappings satisfy the condition (S₊), and it is shown that the derived degree has the classical properties of a degree function.

A formula for the calculation of the index of a critical point of a mapping A : X→X^* satisfying the condition (S₊) is derived without the separability of X and the boundedness of A. For the calculation of the index, we need an everywhere defined linear mapping A' : X→X^* that approximates A in a certain set. As in the earlier results, A' is quasi-monotone, but our situation differs from the earlier results because A' does not have to be the Frechet or Gateaux derivative of A at the critical point. The theorem for the calculation of the index requires a construction of a compact operator T = (A' + Γ)^-1Γ with the aid of linear mappings Γ : X→X and A'. In earlier results, Γ is compact, but here it need only be quasi-monotone. Two counter-examples show that certain assumptions are essential for the calculation of the index of a critical point.