On the solvability in Hilbert space of certain nonlinear operator equations depending on parameters

The equation (*) Au − λTu + μCu = f is studied in a real separable Hilbert space H. Here, λ, μ > 0 are fixed constants and f ε H is fixed. The operators A: D H → H, C: D H → H are monotone and compact, respectively, where D denotes a closed ball in H. The operator T: H → H is linear, compact, self-adjoint and positive-definite. Degree-theoretic arguments are used for the existence of solutions of (*) and extensions of recent results of Kesavan are established. For example, it is shown, under additional assumptions, that there exists a constant μ₀ > 0 such that (*) is solvable for all μ μ₀ and all λ ε R.