Extended Newton-type Method for Nonsmooth Generalized Equation under ( n, α ) -point-based Approximation

Let X and Y be Banach spaces and Ω ⊆ X . Let f : Ω ⟶ Y be a single valued function which is nonsmooth. Suppose that F : X ⇉ 2 Y is a set-valued mapping which has closed graph. In the present paper, we study the extended Newton-type method for solving the nonsmooth generalized equation 0 ∈ f ( x ) + F ( x ) and analyze its semilocal and local convergence under the conditions that ( f + F ) − 1 is Lipschitz-like and f admits a certain type of approximation which generalizes the concept of point-based approximation so-called ( n, α ) -point-based approximation. Applications of ( n, α ) -point-based approximation are provided for smooth functions in the cases n � 1 and n � 2 as well as for normal maps. In particular, when 0 < α < 1 and the derivative of f , denoted ∇ f , is ( ℓ , α ) -H¨older continuous, we have shown that f admits ( 1 , α ) -point-based approximation for n � 1 while f admits ( 2 , α ) -point-based approximation for n � 2, when 0 < α < 1 and the second derivative of f , denoted ∇ 2 f , is ( K, α ) -H¨older. Moreover, we have constructed an ( n, α ) -point-based approximation for the normal maps f C + F when f has an ( n, α ) -point-based approximation. Finally, a numerical experiment is provided to validate the theoretical result of this study.


Introduction
Robinson [1,2] introduced the generalized equation as a general tool for describing, analyzing and solving different problems in a unified manner and it has been studied extensively. Typical examples are systems of inequalities, variational inequalities, linear and nonlinear complementary problems, system of nonlinear equations, equilibrium problems, first-order necessary conditions for nonlinear programming etc. (ey also have plenty of applications in engineering and economics. For more details on these applications and many other ones that we did not mention here, one can refer to [1][2][3]. In this study, let X and Y be Banach spaces, F: X⇉2 Y be a set-valued mapping with closed graph and f: Ω⊆X ⟶ Y be a nonsmooth single-valued function that admits (n, α)-point-based approximation A on Ω with a constant L > 0. We are concerned with the problem of approximating the point x∈ Ω (which is called the solution of (1)) of the following nonsmooth generalized equation: (e classical Newton method is very well known and extensively used to find solutions of (1) when F � 0 { }, where f has Lipschitz continuous Fréchet derivatives. A survey of local and semilocal convergence results for Newton method's are found and mentioned in [4][5][6][7]. When f is nonsmooth, such a classical linearization is no longer available and we need to seek a replacement. In other words, if f doesn't possess Fréchet derivatives, it is not so clear how a Newton algorithm should be designed. (ere are many investigators have worked on this question and the applicants have presented different methods for a few things that are important in certain cases and have proved their justification; see for example [4,[8][9][10][11][12][13][14]. Several papers have worked on the Newton-type methods for nonsmooth equations and variational inequalities; see for example [8,9,15] for inspiration and advanced works on these topics.
In the case when F � 0 { } and f is a nonsmooth function, Robinson [9, (eorem 3.2] considered point-based approximation with Lipschitzian property to show the convergence of Newtons method under the Newton-Kantorovich-type hypothesis. Argyros [10] presented a semilocal convergence analysis of Newtons method based on a suitable point-based approximation. More explicitly, he has taken weaker assumptions in point-based approximation by considering Hölderian property instead of Lipschitzian property in order to cover a wider range of problems than those discussed in [9] and hence showed the convergence result for Newton's method. In addition, Kummer [16] presented a necessary and adequate condition for superlinear convergence of the Newton method, which was originally designed for derivative-type approximations of a nonsmooth function around an isolated zero. Relevant results, for solving the nonsmooth generalized (1) are given in [8,17,18].
To solve the nonsmooth generalized (1), Geoffroy and Piétrus in [19] considered the following method 0 ∈ A x k , x k+1 + F x k+1 for each k � 0, 1, 2, . . . , (2) where A: X × X ⟶ Y is an approximation of f, and presented a local convergence result by using the assumptions that f admits an (n, α)-point-based approximation A and the set-valued map (A(x * , ·) + F(·)) − 1 is M-pseudo-Lipschitz around (0, x * ). For the first time, Dontchev [11] introduced the iterative procedure (2) for solving (1) and presented the nonsmooth analogue of the Kantorovich-type theorem for this procedure by assuming the Aubin continuity of the map (A(x 0 , ·) + F(·)) − 1 at (0, x 1 ) (or, equivalently, (f + F) − 1 is Aubin continuous at (f(x 1 ) − A(x 0 , x 1 ), x 1 ) ), where x 1 is the first iterate of (2). Let x ∈ Ω⊆X. (e subset of Ω, denoted by M(x), is defined by Although the method (2) guarantees the existence of a convergent sequence x k for solving (1), the constructed points x 1 , x 2 , . . . , x k are not unique and therefore, for a starting point near to a solution, the sequences generated by the method (2) are not uniquely defined. For example, the convergence result established in [19, (eorem 3.3], guarantees the existence of a convergent sequence. Hence, in view of numerical computation, this kind of Newton-type methods are not convenient in practical application. Based on these ideas, Rashid [8] introduced and studied the following algorithm and presented semilocal and local convergence results under the assumptions that f has a pointbased approximation and (f + F) − 1 is Lipsctiz-like mapping: It is noted that, in the case when A is replaced by the classical linearization of f, the Algorithm 1 is reduced to the Gauss-Newton-type method introduced by Rashid et al. [20].
Moreover, when the single-valued function involved in (1) is smooth, there has been increased amount of interest on semilocal and local analysis (see, for example, [8,[20][21][22][23] and the references therein).
Our approach is somewhat different. In this study, we give a more general approach, namely (n, α)-point-based approximation, which is an extension of the concept of point-based approximation introduced by Robinson [9] and it can apply to a wide range of particular problems. Because of the presence of Step 3 in Algorithm 1, we have shown in the main proof ((eorem 2) that each of the constructed points x 1 , x 2 , . . . , x k has limit. (erefore, in numerical computational view point, Algorithm 1 gives the more accurate result than the result given by the method (2).
In the present paper, we present semilocal and local convergence of Algorithm 1 under some mild conditions for the function f and the set-valued mapping f + F. In fact, the main motivation of this research is to analyze the semilocal and local convergence of the sequence generated by Algorithm 1 for solving the nonsmooth generalized (1) using the notion of (n, α)-point-based approximation introduced by Geoffroy and Piétrus [19] and Lipschitz-like property. Based on the information around the initial point, the main result is the convergence criterion, developed in the section 3, which provides some sufficient conditions, for a starting point near to the solution, ensuring the convergence to the solution of any sequence generated by Algorithm 1. As a result, local convergence result for the extended Newtontype method is obtained.
(is paper is organized as follows: In section 2, we recall some definitions, notations and preliminarily results that will need afterwards. In Section 3, we show the existence of the sequence generated by Algorithm 1 and then establish the convergence of the extended Newton-type method by using the concept of (n, α)-point-based approximation as well as Lipchitz-like property. In Section 4, we have given some applications of (n, α)-point-based approximation for smooth functions in the case when n � 1, n � 2 and 0 < α < 1 and for normal maps f C + F which is reformulated by Rashid [8]. In the last section, a numerical experiment is provided to justify the theoretical result of this study.

Preliminaries
(roughout this paper, we assume that X and Y are two real or complex Banach spaces and N is the set of all Natural numbers and N * � N − 0 { }. Suppose that f: X ⟶ Y is a Fréchet differentiable function and F: X⇉2 Y is a set-valued mapping with closed graph. Let x ∈ X and r > 0. (e closed ball centered at x with radius r is denoted by B r (x).
All the norms are denoted by ‖ · ‖. (e domain dom F and the inverse F − 1 are respectively defined by Let D⊆X. (e distance from a point x to a set D is defined by 2 International Journal of Mathematics and Mathematical Sciences while the excess from the set D to the set C⊆X is defined by Definition 1. Consider the set-valued mapping F: X⇉2 Y .
(en the graph of F is defined by Definition 2. A set-valued function F: X⇉2 Y is said to be a closed graph if the set (x, y): y ∈ F(x) is a closed subset of X × Y in the product topology i.e. for all sequences x k k∈N and y k k∈N such that x k ⟶ x and y k ⟶ y and y k ∈ F(x k ) for all n, we have y ∈ F(x).
(e notions of pseudo-Lipschitz and Lipchitz-like setvalued mappings were introduced by Aubin in [24] and have been studied extensively; see for example [25,26]. We recall these notions from [20]. Definition 3. Let Γ: y⇉2 X be a set-valued mapping and let (y, x) ∈ gphΓ. Let r x > 0, r y > 0 and M > 0. Γ is said to be (a) Lipchitz-like on B r y (y) relative to B r x (x) with constant M if the following inequality holds: (b) pseudo-Lipschitz around (y, x) if there exist constants r y ′ > 0, r x ′ > 0 and M ′ > 0 such that Γ is Lipchitzlike on B r y ′ (y) relative to B r x ′ (x) with constant M ′ .

Remark 1.
(e pseudo-Lipschitz property of a set-valued mapping Γ is equivalent to the openness with linear rate of Γ − 1 (the covering property) and to the metric regularity of Γ − 1 (a basic well-posedness property in optimization) (see [23,24,27,28] for more details).
Definition 4. Let f: Ω⊆X ⟶ Y be a function and n ∈ N * , α > 0. (en a function A: Ω × Ω ⟶ Y is said to be a (n, α)-point-based approximation ((n, α)-PBA in brief ) on Ω for f with modulus κ if there exists a scalar κ such that, for each u, v ∈ Ω, both of the following assertions hold: It is clear that when n � 1 and α � 1, Definition 4 agrees with Robinson's definition of point-based approximation introduced in [9].
Recall the following definition of strict differentiability, which has been taken from [11].
(e following result is a version of [11, Lemma 2]. (is result establishes the connection between the strict differentiability of f and (n, α)-PBA of a function f.

Lemma 2. Let A be a (n, α)-point-based approximation of a function f in Ω with a constant κ and let
is strictly differentiable at the point x * and its strict derivative at x * is zero.
(e following lemma is taken from [25, Corollary 2]. Lemma 3. Let F: X⇉2 Y be a set-valued mapping with closed graph and let f, g: 3en the following are equivalent:

Remark 3. Combining Lemma 2 and Lemma 3 we conclude that if
International Journal of Mathematics and Mathematical Sciences is Lipschitz-like at (y * , x * ) if and only if the map (A(x * , ·) + F(·)) − 1 possesses the same property. (e following theorem on the convergence of the nonsmooth function using (n, α)-point-based approximation is due to Geoffroy and Piétrus; see [19, (eorem 3.3]: 3en for every c > Mk/π n,α , one can find δ > 0 such that for every starting point x 0 ∈ B δ (x * ), there exists a sequence x k generated by (2), which satisfies Dontchev and Hager [25] proved Banach fixed point theorem, which has been employing the standard iterative concept for contracting mapping. To prove the existence of the sequence generated by Algorithm 1, the following lemma will be played an important rule in this study.
Then Φ has a fixed point in B(x * , r), that is, there exists

Convergence Analysis
(roughout the whole study we assume that X and Y are real or complex Banach spaces. Let n ∈ N * , α > 0 and F: X⇉2 Y be a set-valued mapping with closed graph. (en Furthermore, the following equivalence is clear: In particular, Let (x, y) ∈ gph(f + F) and let r x > 0, r y > 0. Furthermore, throughout in this section we assume that Let us recall that (1) is an abstract model for various problems. From now on, we make the following assumptions.
(i) F has closed graph; (ii) f admits an (n, α)-point-based approximation with modulus L, denoted by A, on some open neigh- (e following lemma plays an important role to the convergence analysis of the extended Newton-type method which is defined by Algorithm 1. (e proof is a refinement of the one for [11, Lemma 1]. (10), so that (11)

Lemma 5. Suppose the assumptions (i)-(iii) hold and let r be defined in
Proof. Since f has a (n, α)-point-based approximation A on an open neighbourhood of x∈ (f + F) − 1 (y) with a constant L and the map (f + F) − 1 is Lipschitz-like around (y, x) with a constant M, then by Remark 3 we have that R − 1 x (·) is Lipschitz-like around (y, x) with a constant M < L, that is, there exist constants r x > 0, r y > 0 and M such that Note, by (20 and 21), that r > 0. Now let It is sufficent to show that there exist x ″ ∈ R − 1 x (y 2 ) such that 4 International Journal of Mathematics and Mathematical Sciences To this end, we shall verify that there exists a sequence hold for each k � 2, 3, 4, . . .. We proceed by mathematical induction. Denote Note by (24) that It follows, from (13) and the relation r ≤ r y − Lr n+α (is implies that z i ∈ B r y (y) for each i � 1, 2. Letting x (y 1 ) by (13) and it follows from (18) that which can be rewritten as (is, by the definition of z 1 , means that (18). (is together with (24) implies that According to the concept of Lipschitz-like property of R − 1 x (·) and noting that z 1 , z 2 ∈ B r y (y), it follows from (23) that there exists Moreover, by the definition of z 2 and noting x 1 � x ′ , we have which together with (18) implies that (is shows that 26 and 27 are true with constructed points x 1 and x 2 . Suppose that the points x 1 , x 2 , . . . , x m have constructed so that 26 and 27 are true for k � 2, 3, . . . , m. We need to construct x m+1 such that (26 and 27) are also true for k � m + 1. To do this, setting (en, by the inductional assumption together with the concept of (n, α)-point-based approximation of A, we obtain that We have ‖x 1 − x‖ ≤ r x /2 and ‖y 1 − y 2 ‖ ≤ 2r from (24) and using (27) we get MLr α By (20), we have 4.2 α Mr ≤ r x (2 α − MLr α x ) and then (39) becomes Consequently, Furthermore, using 13 and 20, we get that, for each i � 0, 1 International Journal of Mathematics and Mathematical Sciences It follows that z m i ∈ B r y (y) for each i � 0, 1. Since assumption (14) holds for k � m, we have which can be written as (is, together with 18 and 40, yields that Using (23) again, inasmuch as z m 0 , z m 1 ∈ B r y (y), there exists an element where the last inequality holds by (38). By the definition of z m 1 , we have which together with (18) implies (is together with (46) completes the induction step and the existence of sequence x k satisfying (14) and (15).
Since MLr α x /2 α < 1, we see from (27) that x k is a Cauchy sequence. Define x ″ : � lim k⟶∞ x k . Note that F has closed graph. (en, taking limit in (26), we get MLr α (is completes the proof of the Lemma 5.
Before going to state the main theorem in this study, for our convenience, we define the map Z x : X ⟶ Y, for each x ∈ X, by and the set-valued map Φ x : X⇉2 X by (en we have that (e main result of this study read as follows, which provides some sufficient conditions ensuring the convergence of the extended Newton-type method for nonsmooth generalized (1) from starting point x 0 .

Theorem 2. Suppose that η > 1. Let x∈ X, Ω be an open and convex subset of X containing x and let f be a function which has (n, α)-point-based approximation A on Ω with a constant L > 0. Suppose that the map F has closed graph and the map
Let r be defined by (10) so that (11) holds. Let δ > 0 be such that (a) δ ≤ min r x /4, r.π n,α /4 n+α , (r y π n,α /L(3 n+α + 2 n+α + 1)) 1/n+α , 1}, 3en there exists some δ > 0 such that any sequence x m generated by Algorithm 1 with starting point x 0 ∈ B δ (x) converges to a solution x * of nonsmooth generalized (1), that is, x * satisfies 0 ∈ f(x * ) + F(x * ).

Proof.
By assumption (b), it can be easily written that It follows from (54) that Since π n,α ‖y‖ < Lδ n+α by assumption (c) and (26) holds, there exists 0 < δ ≤ δ be such that 6 International Journal of Mathematics and Mathematical Sciences Let x 0 ∈ B δ (x). We will proceed by mathematical induction. We will show that Algorithm 1 generates at least one sequence and any sequence x m generated by Algorithm 1 for (1) satisfies the following assertions: for each m � 0, 1, 2, . . .. For this purpose we define Owing to the fact 4δ ≤ r x in assumption (a) and η > 1, by assumption (b) we can write as follows (e above inequality gives either By the facts π n,α ‖y‖ < Lδ n+α from condition (c)and (34), the inequality (33) reduces to, for each x ∈ B 2δ (x) Since δ n ≤ δ, we get that, ≤ 3 2 ML π n,α δ α 2 n+α + 1 .δ It is trivial that (58) is true for m � 0. To show, (32) holds for m � 0, firstly we need to verify that x 1 exists, that is, we need to show that M(x 0 ) ≠ ∅. To do this, we consider the mapping Φ x 0 defined by (24) and apply Lemma 4 to the map Φ x 0 with η 0 � x. Let us check that both assumptions (5) and (6) of Lemma 4, with r: � r x 0 and λ: � 1/3 hold. Noting that (3) and according to the definition of the excess e and the map Φ x 0 , we obtain From the notion of (n, α)-point-based approximation A of f with L, we obtain that Note that Lδ n+α (2 n+α + 3 n+α + 1) ≤ π n,α r y because of assumption (a), π n,α ‖y‖ < Lδ n+α by assumption (c) and ‖x 0 − x‖ ≤ δ ≤ δ. It follows from (65), for each International Journal of Mathematics and Mathematical Sciences (is implies that In particular, letting x � x in (65). (en we have that and hence Hence, by the assumed Lipschitz-like property of R − 1 x and (68), we have from (64) that that is, the assumption (5) of Lemma 4 is satisfied.
Below, we will show that the assumption (6) of Lemma 4 holds. To do this, let x ′ , x ″ ∈ B r x 0 (x). (en from assumption (a) and (35), we have that Applying (52), we get that With the help of first relation in (62) and combining the above two inequalities we get, (74) (is means that the assumption (6) of Lemma 4 is also satisfied. Since both assumptions (5) and (6) of Lemma 4 are satisfied, we can say that Lemma 4 is applicable and therefore, we conclude that there exists By Algorithm 1, x 1 : � x 0 + d 0 is defined. Hence x 1 is generated for (1).
Furthermore, by the definition of M(x 0 ), we can write Now we are ready to show that (59) is hold for m � 0. Note that r > 0 by assumption (a). (en (21) is satisfied by (20). Lemma 5 states us that the mapping by assumption (a) and the choice of δ.
Furthermore, assumptions (a), (c) and the 2nd relation of the inequality (62) imply that Noting that x 0 ∈ B r x /2 (x) as mentioned earlier and by (78)) we have that 0 ∈ B r/3 (y) .
(us, by applying Lemma 1, we obtain that

International Journal of Mathematics and Mathematical Sciences
According to Algorithm 1 and using (77 and 80) we have From (56 and 81) we get, (is shows that (59) is hold for m � 0. Suppose that the points x 1 , x 2 , . . . , x k have obtained by Algorithm 1 satisfying (2) such that (31 and 32) are hold for m � 0, 1, 2, . . . , k − 1. We show that assertions (31) and (32) are also hold for m � k. Since (31) and (32) are true for each m ≤ k − 1, we have the following inequality and so x k ∈ B 2δ (x). (is shows that (58) holds for m � k.
Next we show that the assertion (59) is also hold for m � k. Let x k ∈ B r x k (x). If we apply Lemma 4 to the map Φ x k with η � x, r: � r x k and λ: � 1/3, then by the analogue argument as we did for the case k � 0 one can find that , using the idea of (n, α)-point-based approximation of f, the inequality 4 n+α δ ≤ rπ n,α from assumption (a), we obtain that It is noted earlier that x k ∈ B r x /2 (x). Moreover, (78) implies that 0 ∈ B r/3 (y). (is, together with (84), implies that Lemma 1 is applicable for the map R − 1 x k (·) and hence we have that Since M(x k ) ≠ ∅, Algorithm 1 ensure us the existence of a point x k+1 which satisfy the following inequality International Journal of Mathematics and Mathematical Sciences (is shows that (59) holds for m � k. (us, we can see from (59) that x m is a Cauchy sequence and hence convergent to some x * . Since the graph of F is closed, we can pass to the limit in x k+1 ∈ R − 1 x k (0) obtaining that x * is a solution of (1). (erefore, the proof is completed. In particular, in the case when x is a solution of (1), that is, y � 0, (eorem 2 is reduced to the following corollary, which gives the local convergent result of the extended Newton-type method for solving nonsmooth generalized (1). □ Corollary 1. Suppose that η > 1 and x be a solution of (1). Let Ω be an open and convex subset of X containing x and r > 0 be such that B r (x) is an open and convex set. Suppose that the function f is continuous which has an (n, α)-point-based approximation A on B r (x) with a constant L > 0, the map F has closed graph. Assume that the map (en there exists some δ > 0 such that any sequence x m generated by Algorithm 1 starting from x 0 ∈ B δ (x) converges to a solution x * of nonsmooth generalized (1), that is, Proof. By hypothesis R − 1 x (·) is pseudo-Lipschitz around (0, x). (en there exists constants r 0 , r x and M such that that is, the map R − 1 x (·) is Lipschitz-like on B r 0 (0) relative to B r (x) with constant M.

Application of (n, α)-pointbased approximation
(is section is devoted to present applications of (n, α)-point-based approximation. In particular, when the Fréchet derivative of f is (ℓ, α)-Hölder, the function A is an (1, α)-point-based approximation for f . Moreover, when f is a twice Fréchet differentiable function such that ∇ 2 f is (K, α)-Hölder, then the function A is an (2, α)-point-based approximation for f. In addition, application of (n, α)-point-based approximation is provided for normal maps.
(1) Suppose that the Fréchet derivative of f is (ℓ, α)-Hölder continuous. We show that the function is an (1, α)-point-based approximation for f. In this case, by using the Algorithm 1 we can infer that there exists a sequence x k which converges superlinearly and this result recovers the convergence result of Geoffroy and Piétrus in [19]. In this regards, define the function Λ(u, v) by It follows that (is yields that A satisfies the first property of (1, α)-point-based approximation on Ω. To proof the second property of (1, α)-point-based approximation, we assume that y, z ∈ Ω. (en, we have that (97) (is shows that the second property of (1, α)-PBA for f also holds. (erefore, we say that when the Fréchet derivative of f is (ℓ, α)-Hölder with exponent α ∈ (0, 1), the function A: (2) Let r x > 0 be such that B r x /2 (x)⊆X. Suppose that f is a twice Fréchet differentiable function on B r x /2 (x) such that ∇ 2 f is (K, α)-Hölder on B r x /2 (x) and with exponent α ∈ (0, 1). Choose ℓ > 0 and L > 0 be such that Let p, q ∈ B r x /2 (x) and define the function (en, (eorem 2 ensures the existence of a sequence x k which converges super-quadratically and the result of (eorem 2 coincides with the result of [22,29].
For the proof of second property, we assume that a, b be any elements of B r x /2 (x), (en, we have that (is also can be written as △ ′ (p, q, a, b) � [∇f(p) − ∇f(q)](a − b) + Since there exist an open subset B r x /2 (x)⊆X and a positive number K such that ‖∇ 2 f‖ ≤ K on B r x /2 (x). Let a, b ∈ B r x /2 (x). (en, ‖a − b‖ ≤ r x . (en, by applying the notion of (ℓ, α)-Hölder continuity property of ∇f and (K, α)-Hölder continuity property of ∇ 2 f , we get 12 International Journal of Mathematics and Mathematical Sciences Step 1 Select η ∈ [1, t∞), x 0 ∈ Ω, and put i: � 0.
Step 2 If 0 ∈ M(x i ), then stop; otherwise, go to Step 3.
Step 3 If 0 ∉ M(x i ), choose d i such that d i ∈ M(x i ) and ‖d i ‖ ≤ ηdist(0, M(x i )).
Step 4 Set x i+1 : � x i + d i .
Step 5 Replace i by i + 1 and go to Step 2.
ALGORITHM 1: ((e Extended Newton-type Method)(ENM).   International Journal of Mathematics and Mathematical Sciences