On the nonlinear perturbations of self-adjoint operators

Abstract

Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation Tx=N(x)Tx=N\left(x), where TT is a self-adjoint operator in a real Hilbert space ℋ{\mathcal{ {\mathcal H} }} and NN is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.

Keywords

• strong monotonicity principle
• uniqueness
• self-adjoint operator
• square root operator
• regularity
• 35j91
• 35j25
• 47h05