Newton’s method for generalized equations using restricted domains

Abstract

In this chapter we are concerned with the study of the generalized equation F(x)+Q(x) ϶ 0, where F : D → H is a nonlinear Fréchet differentiable defined on the open subset D of the Hilbert space H, and Q : H ⇉ H is set-valued and maximal monotone. Many problems from Applied Sciences can be solved finding the solutions of equations in a form like (12.1) [17-22,27,28,30,39]. If ψ : H → (−∞,+∞] is a strict lower semicontinuous convex function and Q(x) = ∂ψ(x) = {u ∈ H : ψ(y) ≥ ψ(x)+⟨u, y−x⟩}, for all y ∈ H, then (12.1) becomes the variational inequality problem F(x)+∂ψ(x) ϶ 0, including linear and nonlinear complementary problems, additional comments about such problems can be found in [1-32, 36-40].