ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 47H15 Šćepanović, Radoje, University of Montenegro, Yugoslavia On Solvability of Some Nonlinear Equations in Hilbert Spaces In this work we considered solvability of nonlinear equations in Hilbert space H. Denote by F mapping from H into H. Theorem. Let the following conditions be fullfiled:

1. A – linear operator acting in H, domain D(A) of A is dense in H and there is a constant α such that

∥(A − λE)x∥ ≥ (λ − α)∥x∥^{2}, λ > α, x ∈ D(A);

2. ∀x, x + h ∈ H : Re(F (x + h) − F (x), h) ≥ m∥h∥^{2}, m > α;

3. ∃M > 0∀x, x + h ∈ H : ∥F (x + h) − F (x)∥ ≤ M∥h∥. Then there exists unique solution z_{0} ∈ H of the equation A(z) = F (z). Further, we gave some sufficient conditions for unique solvability of equations of the form

Ax = Bx + ∑_{i=1}^{n} C_{i}^{∗}F_{i}(C_{1}x, C_{2}x, …, C_{n}x),

where A, B, C_{1}, C_{2}, …, C_{n} are closed linear operators with their domains dense in H and F_{i} : H^{n} → H.