On the convergence of extended Newton-type method for solving variational inclusions

Abstract: In this paper, we introduce and study the extended Newton-type method for solving the variational inclusion 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 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 variational inclusions. Let X and Y be Banach spaces. We consider here a variational inclusions problem to find a point x ∈ Ω satisfying

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This study is concerned with the problem of approximating a point which satisfies the following variational inclusion where X and Y be Banach spaces, f :X → Y is Frechet differentiable, g: X→Y admits first order divided difference and F:X ⇉ 2 Y is a set-valued mapping with closed graph.
To solve (1), for an initial point near to a solution, the sequences generated by the Newton-type method are not uniquely defined and not every generated sequence is convergent. The convergence result, obtained by Alexis and Pietrus (2008) or Rashid, Wang and Li (2012), guarantees the existence of convergent sequence. Therefore, from the viewpoint of practical computations, these kind of Newton-type methods are not convenient in practical applications. To overcome this drawback, we propose an extended Newtontype method (ENM) for solving (1) and analyze its semilocal and local convergence results. I believe this finding would appeal to the readership of Cogent Mathematics.
(1) f (x)+g(x)+F(x) ∋ 0 where [y k ,x k ;f ] is the first-order divided difference of f on the points y k and x k . This operator will be defined in Section 2. 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 (Argyros, 2004) in the case when F=0.
Alexis and Piétrus (2008) introduced a method for solving the variational inclusions (1.1), which can be defined as follows: where ∇f (x) denotes the Fréchet derivative of f at x and [x,y;g], the first order divided difference of g on the points x and y; and proved the convergence is superlinear and quadratic when ∇f is Lipschitz continuous. Rashid, Wang, and Li (2012) established local convergence results for the method (1.3) under the weaker conditions than Alexis and Pietrus (2008). In particular, Rashid et al. (2012) extended the results by fixing a gap in the proof of corresponding ones (Alexis & Piétrus, 2008, Theorem 1).
Although the method (1.3) guarantees the existence of a convergent sequence {x k }, the points x 1 ,x 2 , … of the sequence {x k } are not converges separately. Therefore, for a starting point near to a solution, the sequences generated by the method (1.3) are not uniquely defined. For instance, the convergence result, obtained by Alexis and Piétrus (2008) or Rashid et al. (2012), guarantees the existence of a convergent sequence. Hence, in view of numerical computation, these kind of methods are not convenient in practical application. This drawback motivates us to introduce a method 'so-called' extended Newton-type method. The difference between the method (1.3) and our proposed method is that the extended Newton-type method generates a convergent sequence {x k } whose each point x 1 ,x 2 , … converge individually but this does not happen for the method (1.3).
Thus, we propose the following extended Newton-type method: x 0 and x 1 are given starting points Page 3 of 19 Rashid, Cogent Mathematics (2014) Remark 1.1 If g = 0, then the set D will be replaced by the set Then the Algorithm 1.1 reduces to the same algorithm corresponding one given by Rashid, Yu, Li, and Wu (2013).
There have been studied many fruitful works on semilocal convergence analysis for the Gauss-Newton method in the case when F = 0 and g = 0 [see Dedieu and Kim (2002), Dedieu and Shub (2000), Xu and Li (2008), for more details] or when F = C and g = 0 [see Li and Ng (2007), for details]. Rashid et al. (2013) have studied semilocal convergence analysis for the Gauss-Newton-type method to solve the generalized Equation (1.2). However, in our best knowledge, there is no study on semilocal convergence analysis discovered for the general case (1.1), even for the method (1.3).
Our purpose here is to analyze the semilocal convergence of the extended Newton-type method defined by Algorithm 1.1. 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 and Piétrus (2008), Argyros and Hilout (2008), Dontchev (1996a), Hilout et al. (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 1.1. 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 some necessary notations, notions, some preliminary results and also recall a fixed-point theorem which has been proved by Dontchev and Hager (1994). This fixed-point theorem is the main tool to prove the existence of the sequence generated by Algorithm 1.1. 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 1.1. In the last section, we give a summary of the major results to close our paper.

Notations and preliminaries
In this section, we give some notations and collect some results that will be helpful to prove our main results. Throughout this paper, we suppose that X and Y are two real or complex Banach spaces. Let x ∈ X and r > 0. The closed ball centered at x with radius r is denoted by r (x). Let F:X ⇉ 2 Y be a set-valued mapping with dom F ≠ ∅. The domain dom F, the inverse F −1 and the graph gphF of F are, respectively, defined by The distance function of A is defined by while the excess from the set A to the set C ⊆ X is defined by All the norms are denoted by ‖ ⋅ ‖ and the space of linear operators from X to Y is denoted by (X,Y). Now, we recall a few definitions, some results and then state the Banach fixed point theorem. We begin with the definition of the first-order divided difference operators. The following definition, is given by Argyros (2007), introduces the notion of divided differences of nonlinear operators.
Definition 2.1 An operator belonging to the space (X,Y) is called the first order divided difference of the operator g:X → Y on the points x and y in X (x ≠ y) if the following properties hold: Recall from Rashid et al. (2013) the notions of pseudo-Lipschitz and Lipschitz-like set-valued mappings. These notions were introduced by Aubin [see, Aubin (1984), Aubin and Frankowska (1990), for more details] and have been studied extensively.
Definition 2.2 Let Γ:Y ⇉ 2 X be a set-valued mapping and let (ȳ,x) ∈ gphΓ. Let r̄x > 0, r̄y > 0 and M > 0. Then Γ is said to be (a) Lipschitz-like on r̄y (ȳ) relative to r̄x (x) with constant M if the following inequality holds: Remark 2.1 Γ is Lipschitz-like on r̄y (ȳ) relative to r̄x (x) with constant M is equivalent to the following statement: if for every y 1 ,y 2 ∈ r̄y (ȳ) and for every The following lemma is useful and it has been taken from [Rashid et al., 2013, Lemma 2.1].
Lemma 2.1 Let Γ:Y ⇉ 2 X be a set-valued mapping and let (ȳ,x) ∈ gph Γ. Assume that Γ is Lipschitzlike on r̄y (ȳ) relative to r̄x (x) with constant M. Then holds for every x ∈ r̄x (x) and y ∈ rȳ e(Γ(y 1 )∩ r̄x (x),Γ(y 2 )) ≤ M‖y 1 − y 2 ‖ for any y 1 ,y 2 ∈ r̄y (ȳ) We end this section with the following lemma. This lemma is a fixed-point statement which has been proved by Dontchev and Hager (1994) and employing the standard iterative concept for contracting mapping. This lemma is used to prove the existence of the sequence generated by Algorithm 1.1.
Lemma 2.2 Let Φ:X ⇉ 2 X be a set-valued mapping. Let 0 ∈ X, r > 0 and 0 < < 1 be such that and Then Φ has a fixed point in The previous lemma is a generalization of a fixed-point theorem which has been given by Ioffe and Tikhomirov (1979), where in assertion (b) the excess e is replaced by Hausdorff distance.

Convergence analysis of extended Newton-type method
Throughout this section, we suppose that f : Ω ⊆ X → Y is a Fréchet differentiable function on a neighborhood Ω of x with its derivative denoted by ∇f , g : Ω ⊆ X → Y is linear and differentiable at x, and let F : X ⇉ 2 Y be set-valued mapping with closed graph. We prove the existence and convergence of the sequences generated by extended Newton-type method, defined by the Algorithm 1.1, on a neighborhood Ω of a point x. Let x ∈ X and define the mapping Q x by Then We remark that Furthermore, we have the following equivalence In particular, Let (x,ȳ) ∈ gph (f + g + F) and let r̄x > 0, r̄y > 0. Throughout the whole paper, we assume that r̄x (x) ⊆ Ω∩dom F, the function g is Fréchet differentiable at x and admits a first-order divided difference satisfying the following condition: The following lemma plays a crucial role for convergence analysis of the extended Newton-type method. The proof is a refinement of the one for (Rashid et al., 2013, Lemma 3.1).
Proof Noted that (3.5) and (3.6) imply r > 0. Now let It suffices to show that there exist x �� ∈ Q −1 x (y 2 ) such that To this end , we shall verify that there exists a sequence {x k } ⊂ r̄x (x) such that and hold for each k = 2,3,4, …. We proceed by mathematical induction on k. Write Note by (3.7) that It follows, from (3.7) and the relation r ≤ r̄y − 2 r̄x by (3.4), that ‖y 1 − y 2 ‖ for any y 1 , y 2 ∈ r (ȳ).
and so Consequently, Furthermore, using (3.7) and (3.12), one has that, for each i = 0,1, It follows from the definition of r in (3.4) that z n i ∈ r̄y (ȳ) for each i = 0,1. Since assumption (3.8) holds for k = n, we have which can be written as (3.11) ‖x n −x‖ ≤ r̄x. (3.12) , Since M < 1, we see from (3.9) 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 (3.8), we get y 2 ∈ f (x)+g(x �� )+∇f (x)(x �� − x)+F(x �� ), that is, Therefore, we obtain That is, This completes the proof of the Lemma 3.1.

□
For our convenience, we define the mapping Z x :X → Y, for each x ∈ X, by and the set-valued mapping Φ x :X ⇉ 2 X by Then for any x � , x �� ∈ X, we have Our first main theorem is as follows, which provides some sufficient conditions ensuring the convergence of the extended Newton-type method with initial point x 0 .

Suppose that
Then there exists some ̂> 0 such that any sequence {x n } generated by Algorithm 1.1 with initial point in ̂(x ) converges to a solution x * of (1.1), that is, x * satisfies 0 ∈ f (x * )+g(x * )+F(x * ).
Hence, by (3.21), (3.23), (3.26) and the assumed Lipschitz-like property, we have that is, the assertion (2.1) of Lemma 2.2 is satisfied. Now, we show that the assertion (2.2) of Lemma 2.2 holds. To end this, let x � ,x �� ∈ r x 0 (x). Then we have that x � ,x �� ∈ r x 0 (x) ⊆ 2 (x) ⊆ r̄x (x) by (3.22) and assumption (a), and Z x 0 (x � ), Z x 0 (x �� ) ∈ r̄y (ȳ) by (3.25). This together with the assumed Lipschitz-like property implies that Using (3.15) and the choice of x 0 , we have (3.24) as in assumption (a) together with (3.27) that This yields that the assertion (2.2) of Lemma 2.2 is satisfied. Since both assertions of Lemma 2.2 are fulfilled, we can say that the Lemma 2.2 is applicable and hence we can By Algorithm 1.1, x 1 : = x 0 + d 0 is defined. Furthermore, by the definition of D(x 0 ), we can write and so Now, we show that (3.20) holds also for n = 0. Note by (3.16) that and note also that r > 0 by assumption (a). Therefore, (3.5) satisfies (3.6). Hence, by the assumed Lipschitz-like property of Q −1 x (⋅), it follows from Lemma 3.1 that the mapping Q −1 According to Algorithm 1.1 and using (3.10) and (3.31), we have This implies that and therefore, (3.20) is hold for n = 0.
We assume that x 1 ,x 2 , … ,x k are constructed so that (3.19) and (3.20) are hold for n = 0,1,2, … ,k − 1. We will show that there exists x k+1 such that (3.19) and (3.20) are also hold for n = k. Since (3.19) and (3.20) are true for each n ≤ k − 1, we have the following inequality This shows that (3.19) holds for n = k. Now with almost the same argument as we did for the case when n = 0, we can show that (3.20) hold for n = k. The proof is complete. □ In particular, in the case when x is a solution of (1.2), i.e. ȳ = 0, 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 > 1 and x satisfies 0 ∈ f (x)+g(x)+F(x). Let Q −1 x (⋅) be pseudo-Lipschitz around (0,x). Let r > 0, > 0 and suppose that ∇f is continuous on r (x) and that Then there exists some ̂ such that any sequence {x n } generated by Algorithm 1.1 with initial point in ̂(x ) converges to a solution x * of (1.1), that is, x * satisfies that 0 ∈ f (x * )+g(x * )+F(x * ).
By (3.33), we can choose 0 < ≤ 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.

□
In the following theorem, we show that if ∇f is Lipschitz continuous around x, then the sequence generated by Algorithm 1.1 converges quadratically.
Theorem 3.2 Let > 1 and suppose that Q −1 x (⋅) is Lipschitz-like on r (ȳ) relative to r̄x (x) with constant M and that ∇f is Lipschitz continuous on rx

Suppose that
Then there exist some ̂> 0 such that any sequence {x n } generated by Algorithm 1.1 with initial point in ̂(x ) converges quadratically to a solution x * of (1.1).

Proof Setting
Then, thanks to the assumption (b) for allowing us to write the fact that It follows from (3.35) that Taking 0 <̂≤ such that It is noting that such ̂ exists by (3.34) and assumption (c). Let x 0 ∈ ̂(x ). To complete the proof of this theorem, we use almost similar argument that we used for completing the proof of Theorem 3.1.    dist(0,f (x 0 )+g(x 0 )+F(x 0 )) ≤ (L + 6 ) 2 4 for each x 0 ∈ ̂(x ).
We show that Algorithm 1.1 generates at least one sequence and such sequence {x n } generated by Algorithm 1.1 satisfies the following assertions: and hold for each n = 0,1,2, …. Let Owing to the fact ≤ r̄x 4 in assumption (a) and > 1, by assumption (b) we can write as follows This gives and hence by ≤ 6r in assumption (a) together with second inequality of (3.41), we get thanks to assumption (c). Utilizing the first inequality from (3.41) and assumption (c) together, we obtain from (3.40) that Note that (3.38) is trivial for n = 0. In order to show that (3.39) is hold for n = 0, we need to prove D(x 0 ) ≠ �. The nonemptyness of D(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 0 =x. Let us check that both assertions (2.1) and (2.2) of Lemma 2.2 hold with r: = r x 0 and : = 2 3 . Noting that x ∈ Q −1 x (ȳ)∩ 2 (x) by (3.2) and according to the definition of the excess e and the mapping Φ x 0 by (3.14), we obtain By the assumed Lipschitz continuity of ∇f and for each x ∈ 2 (x) ⊆ rx 2 (x), we obtain that  and (L + 6 ) ≤ 2 9 (3.42) ‖ȳ‖ < (L + 6 ) 2 4 ≤ 2 9 ⋅ 4 ⋅ 6r =r 3 ; (3.43) r x < 3 2 4M(L + 6 ) 2 + M(L + 6 ) 2 2 = 27 4 M(L + 6 ) 2 ≤ 2 for each x ∈ 2 (x). (3.44)