Generalized equations and newton’s and method

Abstract

In [18], G. S. Silva considered the problem of approximating the solution of the generalized equation F(x)+Q(x) ϶ 0,(11.1) where F : D → H is a Fréchet differentiable function, H is a Hilbert space with inner product ⟨., .⟩ and corresponding norm ||.||, D ⊆ H an open set and T : H ⇉ H is set-valued and maximal monotone. It is well known that the system of nonlinear equations and abstract inequality system can be modelled as equation of the form (11.1) [17]. If ψ : H → (−∞,+ ∞) is a proper lower semicontinuous convex function and Q(x) = ∂ψ(x) = {u ∈ H : ψ(y) ≥ ψ(x)+ ⟨u, y − x⟩}, for all y ∈ H (11.2) then (11.1) becomes the variational inequality problem F(x)+∂ψ(x) ϶ 0, including linear and nonlinear complementary problems. Newton’s method for solving (11.1) for an initial guess x0 is defined by F(xk)+F′(xk)(xk+1−xk)+Q(xk+1) ϶, k = 0,1,2… (11.3) has been studied by several authors [1]-[24].