Expanding kantorovich’s theorem for solving generalized equations

Abstract

In [18], G. S. Silva considered the problem of approximating the solution of the generalized equation F(x) + Q(x) ϶ 0, (22.1) 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 modeled as equation of the form (22.1) [17]. If ψ : H → (−∞, + ∞] is a proper lower semi continuous convex function and Q(x) = ∂ψ(x) = {u ∈ H : ψ(y) ≥ ψ(x) + ⟨u, y − x⟩}, for all y ∈ H (22.2) then (22.1) becomes the variational inequality problem F(x) + ∂ψ(x) ∋ 0, including linear and nonlinear complementary problems. Newton’s method for solving (22.1) for an initial guess x0 is defined by F(xk)+F′(xk)(xk+1 − xk)+Q(xk+1) ∋ 0, k = 0,1,2… (22.3) has been studied by several authors [1]-[24]. In [13], Kantorovich obtained a convergence result for Newton’s method for solving the equation F(x) = 0 under some assumptions on the derivative F′(x0) and ||F′(x0)−1F(x0)||. Kantorovich, used the majorization principle to prove his results. Later in [16], Robinson considered generalization of the Kantorovich theorem of the type F(x) ∈ K, where K is a nonempty closed and convex cone, and obtained convergence results and error bounds for this method. Josephy [12], considered a semilocal Newton’s method of the kind (22.3) in order to solving (22.1) with F = NC the normal cone mapping of a convex set C ⊂ R2.