On the Kantorovich Theory for Nonsingular and Singular Equations

: We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods

Method (2) specializes to the Gauss-Newton method (GNM) for solving nonlinear least squares problems, the generalized NTM for undetermined systems, and an NTM for ill-posed Equations .
As an example of ( 1) and ( 2), let T 1 and T 2 stand for Hilbert spaces.Then, consider the task of finding a local minimum ũ of min H(u), where H(u) = 1 2 ∥Υ(u)∥ 2 .Then, the GNM is defined by to solve Υ ′ (u) * Υ(u) = 0, (5) where Υ ′ (x n ) † is the Moore-Penrose inverse [5,15,31], and Υ ′ (u) * is the adjoint of the linear operator Υ ′ (u) (see also the Remark 1).Ben-Israel [5,17] utilized the conditions for all u, ũ in a neighborhood of x 0 ∈ Ω.He also used these conditions with Υ ′ (u) # [5,17].These results are not semilocal since they require information about Υ ′ ( ũ) + or Υ ′ ( ũ) # .Moreover, if Υ ′ (x 0 ) −1 ∈ B(T 2 , T 1 ), they require conditions not required in the Kantorovich theory [1,16,33,[37][38][39][40][41].Later Deuflhard and Heindl [39], Haussler [29], and Yamamoto [1] gave Kantorovich-type theorems for the GNM like (4) using convergence conditions involving either OI of Moore-Penrose inverses: for each u, ũ ∈ Ω.This condition is strong and does not hold in concrete examples (see Section 4 in [31]).A Kantorovich-like result with generalized inverses can be found in [42] without (6).However, it was assumed that T 1 and T 2 are finite-dimensional and T 2 = R(E(x 0 ), where R(D) denotes the range of a linear operator D. Other drawbacks of the earlier works are that only properties of OI are used.That is BEB = B and the projectional properties of EB and BE.However, the stability and perturbation bounds for OI are not given for method (1).However, this was accomplished through the elegant work of Nashed and Chen [31].This work reduces to the Kantorovich theory for (2) when E(u) # is replaced by E(u) −1 without additional conditions.Later works on the convergence analysis using Lipchit-type conditions, particularly for the Newton-Gauss method (5), can be found in [10,24] and the references therein.Next, we address the problems with the implementation of method (1) which constitutes the motivation for this paper.Let ∆ ∈ B(T 1 , T 2 ) and let But the main problem with the implementation of (2) of (7) still remains.This problem requires the invertibility of . This inversion can be avoided.Let m be a fixed natural number.Define the operators Then, we can consider the replacement of (7) given as By letting m → +∞ in the definition of L, we have that Thus, it is worth studying the convergence of (8) instead of (2), since we avoid the inversion of the operator ∆ 2 .Let us provide examples of possible choices for the operator ∆.First, consider the case when the operator is a positive integer, and J denotes the Jacobian of the operator F.Then, choose ∆ = J(x 0 ) in the semi-local convergence case of ∆ = J(x * ) in the local case, where x * ∈ Ω is assumed to be a solution of the equation Υ(u) = 0.The selection ∆ = J( ū) has been used in [43,44], for ū ∈ Ω.In the setting of a Banach space for E = Υ ′ (u), the operator ∆ can be chosen to be Numerous selections for ∆ connected to OI or generalized inverses (GI) can also be found in [5,11,13,31] and the references therein.Other selections for ∆ are also possible provided that they satisfy the convergence conditions (C 5 ) and (C 6 ) of Section 3. The convergence analysis relies on the relaxed generalized continuity used to control the derivative Υ ′ and majorizing sequences for the iterates {x n } (see also Section 2).The results in this article specialize immediately to solve nonsingular equations if E(u) # is replaced by The rest of the article provides the preliminaries in Section 2; the convergence of ( 8) is in Section 3; and the applications are in Section 4. The article's concluding remarks appear in Section 5.

Preliminaries
We reproduce standard results on OI and GI to make the article as self-contained as possible.More properties can be found in [5,[11][12][13]31].Let ∆ ∈ B(T 1 , T 2 ).An operator B ∈ B(T 2 , T 1 ) is said to be an inner inverse (II) of ∆ if ∆B∆ = ∆, and an OI of It is well known that II and bounded OI always exist.The zero is always an OI.So, we consider only nonzero outer inverses.Suppose the operator B is either an inverse or an OI of ∆.Then, ∆B and B∆ are linear idempotents (algebraic projectors).Suppose that B is an inverse of E, then for N( Ē), N(B∆) = N(∆) and R(∆) = R(∆B).Consequently, the following decompositions hold If B is an inner and an outer inverse of ∆, then B is called a GI of ∆.Moreover, there exists a unique GI B = ∆ + P,Q satisfying ∆B∆ = ∆, B∆B = B, B∆ = I − P and ∆B = Q, where P is a given projection of T 1 into N(∆) and Q a given projection of T 2 into R(∆).In the special case when T 1 and T 2 are Hilbert spaces, and P, Q are orthogonal projections.Then, ∆ + P,Q is the Moore-Penrose inverse of ∆.We need the following six auxiliary Lemmas, the proof of which can be found in Section 2 in [31].
Define the parameter We need some estimates.
. Moreover, the following estimates hold.
the operator L −1 ∈ B(T 2 , T 1 ) and where ∆ # 1 , L are as defined in the introduction.
Proof.The operator ∆ # 1 is a bounded OI of E(x) by Lemma 2 for B = E(x).Moreover, we have in turn by the definition of L: by the choice of r.It is followed by the Lemma 2 that

Semi-Local Convergence
The convergence of the method ( 8) is shown using scalar majorizing sequences.
Therefore, the convergence of the sequence {x n } relates to that of {p n }.
By this condition and (12) that 0 ≤ s n ≤ s n+1 ≤ r and there exists s * ∈ [κ, r] such that lim n→+∞ s n = s * .
The functions φ 0 , φ and φ 1 connect to the operators on the method (8).
(ii) Under the conditions (C 1 ) − (C 8 ), further suppose that the operator I + ∆ + ∆ − E(x) −1 E(x) sends N(A) to R(A) provided that for x ∈ Ω, the inverse of Then, by Lemma 4, E(x n ∆ + is a GI.Thus, the proof of Theorem 1 establishes the convergence of method (8) for GI.(iii) By the Lemma 6, the condition (20) can be exchanged by rank (E(x)) ≤ rank(E(x 0 )) for ∆ = E(x 0 ) and if T 1 and T 2 are finite dimensional.In general Banach spaces, the condition (20) can be switched by the stronger R(E(x)) ⊂ R(E(x 0 )) (for ∆ = E(x 0 )) (see Lemma 5) or by the conditions of the Lemma 6.
Example 6. Method (8 ′ ): Set m = 1, 5 and ∆ = Υ ′ (x 0 ).It follows that Definition 2. Let {x n } be a sequence.Then, the computational order of convergence (COC) is for Let {x n } be a sequence.Then, the approximate computational order of convergence (ACOC) is for θ The Tables 1-5 demonstrate that the cheaper-to-implement method ( 8) is behaving the same as Newton's method for a large enough m.

Conclusions
We developed a semi-local Kantorovich-like analysis of Newton-type methods for solving singular nonlinear operator equations using outer or generalized inverses.These methods do not use inverses as in earlier studies but a sum of operators.This sum converges to the inverse and makes the implementation of these methods easier than the ones using inverses.The analysis of the methods relies on the concept of generalized continuity for the operators involved and majorizing sequences.Examples complement the theory.Due to its generality, this article's technique can be applied on other method with inverses along the same lines [6,14,19,32,39,43,[45][46][47][48][49].It is worth noting that the method (8) should be used for sufficiently small m.Otherwise, if m is very large, it may be as expensive to implement as method (4).

Definition 1 .
Let {x n } be a sequence in T 1 .Then, real sequence {p n } satisfying∥x n+1 − x n ∥ ≤ p n+1 − p n , ∀ n ≥ 0 is calleda majorizing sequence for {x n }.If the sequence {p n } converges, then also {x n } converges, and for x * = lim n→∞ x n and p * = lim n→∞ p n , we have