Convergence Properties of Extended Newton-type Iteration Method for Generalized Equations

In this paper, we introduce and study the extended Newton-type method for solving generalized equation 0 ∈ f (x)+g(x)+ F (x), where f : Ω ⊆ X → Y is Fréchet differentiable in a neighborhood Ω of a point x̄ in X, g : Ω ⊆ X → Y is linear and differentiable at a point x̄, and F is a set-valued mapping with closed graph acting in Banach spaces X and Y. Semilocal and local convergence of the extended Newton-type method are analyzed.


Introduction
In this study we are concerned with the problem of approximating a solution of a generalized equations.Let X and Y be Banach spaces and Ω ⊆ X.Let f : Ω → Y be a Fréchet differentiable function and its Fréchet derivative is denoted by ∇ f , g : Ω → Y be a linear and differentiable function at x but may not differentiable in a neighborhood Ω of x and its first order divided difference on the points x and y is denoted by [x, y; g] and F : X ⇒ 2 Y be a set-valued mapping with closed graph.We consider here a generalized equation problem to approximate a point x ∈ Ω satisfying 0 ∈ f (x) + g(x) + F (x). (1) For solving (1), Alexis & Pietrus (2008) introduced the following Newton-like method: and obtained local convergence of this method.In particular, the authors obtained superlinear and quadratic convergence of the method (2) when ∇ f is Lipschitz continuous.To solve (1), Rashid, Wang & Li (2012) established local convergence results for the method (2) under the weaker conditions than Alexis & Pietrus (2008).Specifically, Rashid, Wang & Li (2012) extended the results by fixing a gap in the proof of Theorem 1 in Alexis & Pietrus (2008).
Moreover, for solving (1), Hilout, Alexis, & Piétrus (2006) considered the following sequence          x 0 and x 1 are given starting points and they proved the convergence of this method is superlinear when f is only continuous and differentiable at x * .Furthermore, it should be mentioned that Argyros (2004) has studied local as well as semilocal convergence analysis for two-point Newton-like methods in a Banach space setting under very general Lipschitz type conditions for solving (1) in the case when F = {0}.When g = 0, this study has been extended by Rashid (2017aRashid ( , 2017bRashid ( , 2018)).
Let x ∈ X and the subset of X, denoted by N(x), is defined by Algorithm 1 (The Newton-like Method) Step 1. Select x 0 ∈ X, and put k := 0.
Step 3. If 0 N(x k ), choose d k such that d k ∈ N(x k ).
Step 5. Replace k by k + 1 and go to Step 2. Argyros & Hilout (2008) obtained the quadratic convergence of the sequence generated by Algorithm 1 when ∇ f is Lipschitz continuous.
Under some suitable conditions around a solution x * of the generalized equation ( 1), Argyros & Hilout (2008) showed in their Theorem 4.1 that there exists a neighborhood U of x * such that, for any point in U, there exists a sequence generated by Algorithm 1 which is quadratically convergent to the solution x * .This reflects that the convergence result, established in Argyros & Hilout (2008), guarantees the existence of a convergent sequence.Therefore, for any initial point near to a solution, the sequences generated by Algorithm 1 are not uniquely defined and not every generated sequence is convergent.Hence, in view of numerical computation, this kind of methods are not convenient in practical application.This difficulties inspired us to introduce a method "so-called" extended Newton-type (EN-type) method.Thus, we propose the following EN-type method: Algorithm 2 (The EN-type Method)) Step 1. Select η ∈ [1, ∞), x 0 ∈ X, and put k := 0.
Step 5. Replace k by k + 1 and go to Step 2.
The difference between Algorithms 1 and 2 is that Algorithm 2 generates at least one sequence and every generated sequence is convergent but this does not happen for Algorithm 1.Since the sequences generated by Algorithm 1 are not uniquely defined, in comparison with Algorithms 1 and 2, we can infer that Algorithm 2 is more precise than Algorithm 1 in numerical computation.
If the set N(x) is replaced by the set then the Algorithm 2 reduces to the same algorithm corresponding one given by Rashid (2014).
In the case when g = 0, Rashid, Yu, Li & Wu (2013) introduced Gauss-Newton-type method to solve the generalized equation ( 1) and established its semilocal convergence.Moreover, in the same case, Rashid introduced different kinds of methods for solving (1) and obtained their semilocal and local convergence; see for examples (Rashid (2016); Rashid & Sardar (2015); Rashid (2015)).However, in our best knowledge, there is no other study on semilocal convergence analysis discovered for the Algorithm 1.
The purpose of this study is to analyze the semilocal convergence of the extended Newton-type method defined by Algorithm 2. The main tool is the Lipschitz-like property of set-valued mappings, which was introduced by Aubin (1984). in the context of nonsmooth analysis and studied by many mathematicians (see for example, Alexis & Piétrus (2008); Argyros & Hilout (2008); Dontchev (1996a); Hilout, Alexis,& Piétrus (2006); Piétrus (2000b)) and the references therein.The main results are the convergence criteria, established in Section 3, which, based on the attraction region around the initial point, provide some sufficient conditions ensuring the convergence to a solution of any sequence generated by Algorithm 2. As a result, local convergence results for the extended Newton-type method are obtained.
This paper is organized as follows: In section 2, we recall a few necessary preliminary results and also recall a fixed-point theorem which has been proved by Dontchev & Hager (1994).This fixed-point theorem is the main tool to prove the existence of the sequence generated by Algorithm 2. In section 3, we consider the extended Newton-type method as well as the concept of Lipschitz-like property to show the existence and the convergence of the sequence generated by Algorithm 2. In the last section, a summary of the major results of this study are given.

Preliminaries
In this section we give some notations and collect some results that will be helpful to prove our main results.Throughout the whole study, suppose that X and Y are two real or complex Banach spaces.Let x ∈ X.Let B(x, r) = {u ∈ X : ∥u − x∥ ≤ r} be denote the closed ball centered at x with radius r > 0. Let F : X ⇒ 2 Y be a set-valued mapping with closed graph.The domain of F , denoted by domF , is defined by domF := {x ∈ X : F (x) ∅}.
The inverse of F , denoted by F −1 , is defined by and the graph of F , denoted by gphF , is defined by Moreover, the excess from the set A to the set B is defined by The space of linear operators from X to Y is denoted by L(X, Y) and the norms are denoted by ∥ • ∥.Now, we recall some definitions, results and then state the Banach fixed point theorem.We begin with the definition of the first order divided difference operators.The notion of divided differences of nonlinear operators is given by Argyros (2007), which is given below: Definition 2.1.Let g ∈ L(X, Y).Then g is said to have the first order divided difference on the points x and y in X (x y) if the following properties hold: Recall from Rashid, Yu, Li & Wu (2013), the notions of pseudo-Lipschitz and Lipchitz-like set-valued mappings.These notions were introduced by Aubin (see, Aubin (1984); Aubin & Frankowska (1990), for more details) and have been studied extensively.
Remark 2.1.The set-valued mapping Γ is Lipschitz-like on B(ȳ, r ȳ) relative to B( x, r x) with constant µ > 0, which is equivalent to the following statement: if for every y 1 , y 2 ∈ B(ȳ, r ȳ) and for every x 1 ∈ Γ(y 1 ) ∩ B( x, r x), there exists The following lemma is due to Lemma 2.1 of Rashid, Yu, Li & Wu (2013).This lemma is useful and its proof is a little bit similar to that for Theorem 1.49(i) of Mordukhovich (2006).
Lemma 2.1.Let Γ : Y ⇒ 2 X be a set-valued mapping and let (ȳ, x) ∈ gph Γ. Assume that Γ is Lipschitz-like on B(ȳ, r ȳ) relative to B( x, r x) with constant µ.Then dist (x, Γ(y)) ≤ µ dist(y, Γ −1 (x)) holds for every x ∈ B( x, r x) and y ∈ B(ȳ, We close this section with the following lemma.This lemma is a fixed point statement which has been proved by Dontchev & Hager (1994) and employing the standard iterative concept for contracting mapping.This lemma will be used to prove the existence of the sequence generated by Algorithm 2.
Lemma 2.2.Let Φ : X ⇒ 2 X be a set-valued mapping.Let x * ∈ X, r > 0 and 0 < λ < 1 be such that The previous lemma is a generalization of a fixed point theorem which has been given by Ioffe & Tikhomirov (1979), where in assertion (b) the excess e is replaced by Hausdorff distance.

Convergence analysis of EN-type Method
Let Ω be a subset of X. Suppose that f : Ω → Y is a Fréchet differentiable function on a neighborhood Ω of x with its derivative denoted by ∇ f , g : Ω → Y is linear and differentiable at x and let F : X ⇒ 2 Y be a set-valued mapping with closed graph.This section is devoted to prove the existence and convergence of the sequences generated by extended Newton-type method, defined by the Algorithm 2, on a neighborhood Ω of a point x.
Fix x ∈ X.Then for every x ∈ X, we have that (5) Therefore, we define the mapping G x by It follows, from the construction of N(x), that Moreover, for any z ∈ X and y ∈ Y, we have the following equivalence: In particular, let ( x, ȳ) ∈ gphG x.Then, the closed graphness of G x imply that The following result establishes the equivalence between ( f + g + F ) −1 and G −1 x .This result is the modification of Rashid & Sardar (2015).
Assume that g is Fréchet differentiable at x and admits first order divided difference.Then the following are equivalent: The proof is similar to that of Rashid & Sardar (2015), because the proof does not depend on the property of g.
For our convenience, let r x > 0, r ȳ > 0 and B( x, r x) ⊆ Ω ∩ dom F .Assume that the function g is Fréchet differentiable at x and admits a first order divided difference, that is, there exist ν > 0 such that for all x, y, u, v and the mapping Moreover, the closed graph property of G x implies that f + g + F is continuous at x for ȳ i.e. the following condition is hold: Let ε > 0 and write The following lemma plays a crucial role for convergence analysis of the extended Newton-type method.The proof is a refinement of Lemma 3.1 in Rashid, Yu, Li & Wu (2013).
Lemma 3.2.Suppose that G −1 x is Lipschitz-like on B(ȳ, r ȳ) relative to B( x, r x) with constant M. Let ε be defined by (11) and x ∈ B( x, r x 2 ).Assume that ∇ f is continuous on B(( x, r x 2 ).Let r be defined by (10) such that (11 It is enough to show that there exist To this finish , we will verify that there exists a sequence {x n } ⊂ B( x, r x) such that hold for each n = 2, 3, 4, . ... We proceed by mathematical induction on n.Letting Since ∇ f is continuous around x with the constant ε, it gives that It follows, from ( 12) and the relation r ≤ r ȳ − 2εr x by (10), that The above inequality implies that x (y 1 ) by ( 12) and it follows from (6) that The alternative form of the above inclusion is as follows: By the definition of u 1 , this yields that Hence x 1 ∈ G −1 x (u 1 ) by ( 6).This gives, for (12), that x is Lipschitz-like on B(ȳ, r ȳ) relative to B( x, r x), then for every u 1 , u 2 ∈ B(ȳ, r ȳ), we have through (8) that there exists In addition, by the construction of u 2 and x 1 = x ′ , we obtain that This, together with (6), gives that This implies that ( 13) and ( 14) are true with the constructed points x 1 and x 2 .
Suppose that the points x 1 , x 2 , . . ., x k are constructed so that ( 13) and ( 14) are true for n = 2, 3, . . ., k.We have to construct the point x k+1 such that ( 13) and ( 14) are also true for n = k + 1.For showing this, let, for each i = 0, 1, Then, for the above inductional assumption, we get We have from (12) that ∥x 1 − x∥ ≤ r x 2 and ∥y 1 − y 2 ∥ ≤ 2r.This, together with (14), implies that Note by (10) that 4Mr ≤ r x(1 − Mε).Therefore, we have from the above inequality that Moreover, we obtain that Furthermore, using ( 12) and ( 17), one has that, for each i = 0, 1, By the relation r ≤ r ȳ − 2εr x in (10), it follows that ∥u k i − ȳ∥ ≤ r ȳ.This shows that u k i ∈ B(ȳ, r ȳ) for each i = 0, 1.By our assumption ( 13) is true for n = k.Thus, we have that The above inequality can be written as follows: Then by the construction of u k 0 , we have that . This together with (6) implies that x , there exists an element Then by (15), it follows that By the construction of u k 1 , we get that ).This, together with (6), implies that The inequality (18) together with the above inclusion completes the induction step and confirming the existence of a sequence {x k } which satisfies ( 13) and ( 14).
Since Mε < 1, we see from ( 14) that {x k } is a Cauchy sequence and hence it is convergent, to say x ′′ , that is, x ′′ := lim k→∞ x k .Note that F has closed graph.Then, taking limit in (13), we get x (y 2 ).Therefore, we obtain That is, This completes the proof of the Lemma 3.2.
Before going to prove our main results, we would like to introduce some notations.For our convenience, first define the mapping J x : X → Y, for each x ∈ X, by and the set-valued mapping Φ Then for any x ′ , x ′′ ∈ X, we have

Linear Convergence
The first main theorem of this study read as follows, which gives some sufficient conditions confirming the convergence of the extended Newton-type method with starting point x 0 .
Theorem 3.1.Suppose that η > 1 and that G −1 x is Lipschitz-like on B(ȳ, r ȳ) relative to B( x, r x) with constant M. Let r be defined in (10) and let x ∈ B( x, r x 2 ).Suppose that ε > 0 be such that (11) is hold and ∇ f is continuous on B( x, r x 2 ) with constant ε.Let ν > 0 and δ > 0 be such that Suppose that f + g + F is continuous at x for ȳ i.e. (9) is hold.Then there exists some δ > 0 such that any sequence {x n } generated by Algorithm 2 with initial point in B( x, δ) converges to a solution x * of (1), that is, x * satisfies 0 ∈ f (x * ) + g(x * ) + F (x * ).
Now, we show that (23) holds also for n = 0.The continuity property of ∇ f implies that and note that r > 0 by assumption (a).Therefore, (11) satisfies (10).Since G −1 x is Lipschitz-like, it follows from Lemma 3.2 that the mapping G −1 x is Lipschitz-like on B(ȳ, r) relative to B( x, r x 2 ) with constant by assumption (a) and by the choice of δ.Furthermore, by the relation 3(ε + 3ν)δ ≤ r in assumption (a) and assumption (c) imply that and hence (21 It is noted earlier that x 0 ∈ B( x, r x 2 ) and 0 ∈ B(ȳ, r 3 ) by (32).Thus, applying Lemma 2.1 it can be shown that The above relation together with (31) yields that According to Algorithm 2 and using ( 33) and (34), we obtain This implies that ∥x 1 − x 0 ∥ = ∥d 0 ∥ ≤ qδ and therefore, ( 23) is hold for n = 0.
Assume that x 1 , x 2 , . . ., x k are constructed so that ( 22) and ( 23) are hold for n = 0, 1, 2, . . ., k − 1.We will show that there exists x k+1 such that ( 22) and ( 23) are also hold for n = k.Since ( 22) and ( 23) are true for each n ≤ k − 1, we have the following inequality This shows that ( 22) holds for n = k.Now with almost the same argument as we did for the case when n = 0, it can be shown that ( 23) hold for n = k.The proof is complete.
When ȳ = 0, that is, x is a solution of (1), Theorem 3.1 is reduced to the following corollary, which gives the local convergent result for the extended Newton-type method.
Corollary 3.1.Suppose that η > 1 and x is a solution of ( 1).Let G −1 x be pseudo-Lipschitz around (0, x).Let r > 0, ν > 0 and suppose that ∇ f is continuous on B( x, r) and that Then there exists some δ such that any sequence {x n } generated by Algorithm 2 with initial point in B( x, δ) converges to a solution x * of (1).

Proof. Let G −1
x be pseudo-Lipschitz around (0, x).Then there exist constants r 0 , rx and M satisfy the following condition: Thus, by the definition of Lipschitz-like property we can say that Q −1 x is Lipschitz-like on B(0, r 0 ) relative to B( x, rx ) with constant M which satisfy (35).Then, for each 0 < r ≤ rx , one has that x is Lipschitz-like on B(0, r 0 ) relative to B( x, r) with constant M. Let ε ∈ (0, 1) be such that M((6η+1)ε+3ν) ≤ 1.By the continuity of ∇ f we can choose r x ∈ (0, rx ) such that r x 2 ≤ r, r 0 − 2εr x > 0 and By (36), we can choose 0 < δ ≤ 1 such that Thus it is routine to check that inequalities (a)-(c) of Theorem 3.1 are satisfied.Therefore, Theorem 3.1 is applicable to complete the proof.
This gives Hence by 3δ ≤ 5r * in assumption (a) together with second inequality of (45), we get Thanks to assumption (c).Utilizing the first inequality from (45) together with assumption (c), we obtain from (44) that Note that ( 42) is trivial for n = 0.In order to show that ( 43) is hold for n = 0, first we need to prove N(x 0 ) ∅.The nonemptyness of N(x 0 ) will ensure us to deduce the existence of the point x 1 .To complete this, we will apply Lemma 2.2 to the map Φ x 0 with η 0 = x.Let us check that both assertions (3) and ( 4) of Lemma 2.2 hold with r := r x 0 and λ := 1 5 .
Here we note by ( 7) that x ∈ G −1 x (ȳ) ∩ B( x, 2δ).Then, according to the definition of the excess e and the mapping Φ x 0 defined by ( 19), we have that For each x ∈ B( x, 2δ) ⊆ B( x, r x 2 ) and Lipschitz continuous property of ∇ f , we obtain It follows, from the facts 3(L + 4ν)δ 2 ≤ r ȳ and 2∥ȳ∥ < (L + 4ν)δ 2 respectively in assumptions (a) and (c), that This shows that J x 0 (x) ∈ B(ȳ, r ȳ).In particular, let x = x in (49).Then it is easily shown that Using the Lipschitz-like property of G −1 x and ( 51) in ( 48), we have that is, the assertion (3) of Lemma 2.2 is satisfied.Now, we show that assertion (4) of Lemma 2.2 holds.To end this, let x ′ , x ′′ ∈ B( x, r x 0 ).Then we have that x ′ , x ′′ ∈ B( x, r x 0 ) ⊆ B( x, 2δ) ⊆ B( x, r x) by ( 47) and J x 0 (x ′ ), J x 0 (x ′′ ) ∈ B(ȳ, r ȳ) by ( 50).This together with Lipschitz-like property of G −1 x implies that Combining above two inequalities and first inequality from (45), we obtain that It seems that the assertion (4) of Lemma 2.2 is also satisfied.Thus, we have seen that both assertions (3) and (4) of Lemma 2.2 are fulfilled.So, we can conclude that Lemma 2.2 is applicable to deduce the existence of a point x1 ∈ By Algorithm 2, x 1 := x 0 + d 0 is defined.Furthermore, by the construction of N(x 0 ) and ( 5), we have that and so dist(0, N(x 0 )) = dist(x 0 , G −1 x 0 (0)).( 52) Now we are ready to show that ( 43) is hold for n = 0.
Note by assumption (a) that r * > 0.Then, from (37) we conclude that This shows that Lemma 3.2 is applicable with ε := Lr x.According to our assumption G −1 x is Lipschitz-like on B(ȳ, r * ) relative to B( x, r x).Then, it follows from Lemma 3.2 that for each x ∈ B( x, r x 2 ), the mapping by assumption (a).On the other hand, (41) implies that dist(0, We have shown by ( 46) that 0 ∈ B(ȳ, r * 3 ) and it is noted earlier that x 0 ∈ B( x, r x 2 ).Thus by appying Lemma 2.1, we get the following inequality: But, by (52), we can obtain According to Algorithm 2 and using ( 39), ( 41) and ( 53), we have This means that and therefore, ( 43) is true for n = 0.We assume that x 1 , x 2 , . . ., x k are constructed and (42), and (43) are true for n = 0, 1, 2, . . ., k − 1.We show that there exists x k+1 such that (42) and ( 43) are also hold for n = k.Since ( 42) and ( 43) are true for each n ≤ k − 1, we have the following inequality: 2 This shows that (42) holds for n = k.
Finally, we will show that the assertion (43) holds for n = k.For doing this, we will apply again the contraction mapping Then by Algorithm 2, set x k+1 := x k + d k .Moreover, applying Lemma 3.2 we infer that G −1 x k is Lipschitz-like on B(ȳ, r * ) relative to B( x, r x 2 ) with constant M 1−MLr x .Therefore, we can obtain the following inequality: 2 This implies that (43) holds for n = k and therefore the proof is complete.
Consider the special case when x is a solution of (1)(that is, ȳ = 0) in Theorem 3.2.Then the following corollary, which gives the local quadratic convergence result for the extended Newton-type method.The proof of this corollary is similar to that we did for Corollary 3.1.
Then there exist some δ > 0 such that any sequence {x n } generated by Algorithm 2 with initial point in B( x, δ) converges quadratically to a solution x * of (1).

Concluding Remarks
The semilocal and local convergence results for the extended Newton-type method are established when η > 1, ∇ f is continuous and Lipschitz continuous, g admits first order divided difference as well as G −1 x is Lipschitz-like.This work extends and improves the result corresponding to (Argyros & Hilout (2008); Rashid (2016)).