Solvability of Semilinear Operator Equations with Growing Nonlinearity

Let U and V be Hilbert spaces, L: UªV be a bounded linear operator, and N: UªV be a nonlinear operator. Consider the semilinear operator equation LuqNus¨ and the corresponding linear equation Lus¨ where ¨ gV. In this paper we prove appropriate solvability of the semilinear system under the following assumptions: i. the linear equation is approximately solvable; ii. N is compact; iii.5Nu5FAqB5u5for A)0, B)0, and B is such that