On the Local Convergence of Regula-falsi-type Method for Generalized Equations

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1 Journal of Advances in Applied Mathematics, Vol., No. 3, July On the Local Convergence of Regula-falsi-type Method for Generalized Equations Farhana Alam 1*, M. H. Rashid M. A. Alom 3 1 Department of Mathematics, University of Rajshahi, Rajshahi-605, Bangladesh Department of Computer Science Engineering, North Bengal International University, Rajshahi, Bangladesh chaiti_math@yahoo.com Department of Mathematics, University of Rajshahi, Rajshahi-605, Bangladesh harun_math@ru.ac.bd 3 Department of Mathematics, University of Rajshahi, Rajshahi-605, Bangladesh Department of Mathematics, Khulna University of Engineering & Technology, Khulna-903, Bangladesh asraf.math.kuet@gmail.com Abstract Let X Y be Banach spaces. We study a Regula-falsi-type method for solving the generalized equations 0 fx + gx + F x, where f : X Y is Fréchet differentiable in a neighborhood of a solution, g : X Y is Fréchet differentiable at F : X Y is a set valued mapping with closed graph. Under some suitable assumptions, we prove the existence of any sequence generated by the Regula-falsi-type method establish the local convergence results of the sequence generated by this method. Indeed, we will show that the sequence generated by this method converges linearly super-linearly. Keywords: Generalized equations, Set-valued mapping, Pseudo-Lipschitz continuity, Super-linear convergence, Divided difference, Local convergence. 1 Introduction Let X Y be Banach spaces. We deal with the problem of seeking a point x X satisfying 0 fx + gx + F x, 1 where f : X Y is Fréchet differentiable in a neighborhood of a solution of 1, g : X Y is Fréchet differentiable at but may be not differentiable in a neighborhood of F : X Y is a set valued mapping with closed graph. It is clarified that when F = 0, 1 is reduced to the classical problem of solving systems of nonlinear equations: fx + gx = 0. Cătinas [1] proposed the following method for solving by using the combination of Newton s method with the secants method when f is differentiable g is a continuous function admitting first second order divided differences: 0 fx k + gx k + fx k + [x k 1, x k ; g]x k+1 x k, k = 1,,..., 3 where fx k denotes the Fréchet derivative of f at x [x, y; g] the first order divided difference of g on the points x y. To solve the problem 1, Geoffroy Piétrus [] extended the method 3 proposed the following method: 0 fx k + gx k + fx k + [x k 1, x k ; g]x k+1 x k + F x k+1. 4 Under some assumptions, they proved that the sequence generated by this method converges superlinearly to the solution of 1. Next time for solving 1, Jean-Alexis Piétrus [3] presented the following method: 0 fx k + gx k + fx k + [x k+1 x k, x k ; g]x k+1 x k + F x k+1. 5

2 116 Journal of Advances in Applied Mathematics, Vol., No. 3, July 017 They proved that the sequence generated by 5 converges superlinearly by considering that f the first order divided difference of g are p-hȯlder continuous around a solution that f + g + F 1 is pseudo-lipschitz around 0, with F having closed graph. Rashid et al. [4] improved extended the result, which has been given by Jean-Alexis Piétrus [3] show that if f the first order divided difference of g are p-hȯlder continuous at a solution, then the method 5 converges superlinearly. It is remarkable that when g = 0, 1 is reduced to the following variational inclusion: 0 fx + F x. 6 Here 6 is a very general framework many problems from applied mathematical areas, such as variational inequality problems including linear nonlinear complementary problems, systems of nonlinear equations, abstract inequality systems etc, can be cast as problem 6 see [5,6,7,8,10,11] the references therein. In the case g = 0, Geoffroy et al. [1] considered a second degree Taylor polynomial expansion of f under suitable first second order differentiability assumptions showed that the existence of a sequence cubically converges to the solution of 1. When the single valued functions involved in 1 are differentiable, Newton-like methods can be considered to solve this problem, such an approach has been used in many contributions to this subject see, e.g., [4,8,11,13,14,15]. Let x X. The subset of X, denoted by Dx, is defined by D x0 x = d X : 0 fx + gx + fx + [x 0, x; g]d + F x + d. To solve 1, Geoffroy Piétrus [, Theorem 3.1] considered two starting points x 0 x 1 in a suitable neighborhood of x provided super-linear convergence result. In view of computational computations, this convergence is very slow because in their result, the terms x k x k 1 are involved in the upper bound of the relation x k+1 x C x k x max x k x, x k 1 x. This drawback motivates us to consider the following Regula-falsi-type method: Our aim is to prove the existence of a sequence Algorithm 1 The Regula-falsi-type Method Step 1. Select x 0 X put k := 0. Step. If 0 D x0 x k, then stop; otherwise, go to Step 3. Step 3. If 0 / D x0 x k, choose d k such that d k D x0 x k. Step 4. Set x k+1 := x k + d k. Step 5. Replace k by k + 1 go to Step. generated by Algorithm 1 which is locally linearly superlinearly convergent to the solution of 1. Geoffroy Piétrus [] considered two starting points x 0 x 1 in a neighborhood of the solution to obtain their result, where as in our study we will consider one argument as a starting point x 0 in a neighborhood of the solution of 1. The content of this paper is organized as follows: In section, we recall some necessary notations, notions preliminary results. In section 3, we consider the Regula-falsi-type method to show the existence convergence of the sequence generated by Algorithm 1. In the last section, we give a summary of the results obtained in this study. Notations Preliminary Results Let X Y be Banach spaces let F : X Y be a set valued mapping. The graph of F is defined by the set gphf := x, y X Y : y F x the inverse of F is defined by F 1 y := x X : y F x. Let x X r > 0. By Br, x, we denote the closed ball centered at x with radius r, while LX, Y sts for the set of all bounded linear operators from X to Y. All the norms are denoted by. Let B X E X. The distance from a point x to a set B is defined by distx, B := inf x a : a B for each x X, while the excess from a set E to the set B is defined by ee, B := supdistx, B : x E. We take the definitions of the first second divided difference operators from []:

3 Journal of Advances in Applied Mathematics, Vol., No. 3, July Definition.1. An operator belonging to the space LX, Y denoted by [x 0, y 0 ; g] is called the first order divided difference of the operator g : X Y on the points x 0, y 0 X if both of the following properties hold: a [x 0, y 0 ; g]y 0 x 0 = gy 0 gx 0 for x 0 y 0 ; b If g is Fréchet differentiable at x 0 X then [x 0, x 0 ; g] = g x 0. Definition.. An operator belonging to the space LX, LX, Y denoted by [x 0, y 0, z 0 ; g] is called the second order divided difference of the operator g : X Y on the points x 0, y 0, z 0 X if both of the following properties hold: a [x 0, y 0, z 0 ; g]z 0 x 0 = [y 0, z 0 ; g] [x 0, y 0 ; g] for the distinct points x 0, y 0 z 0 ; b If g is twice differentiable at x 0 X then [x 0, x 0, x 0 ; g] = g x 0. We recall the definition of pseudo-lipschitz continuity for set-valued mappings from [11] whose notion was introduced by Aubin [16] has been studied extensively; see examples [4,8,13,14] the references therein. Definition.3. Let Γ : Y X be a set-valued mapping let ȳ, gphγ. Then Γ is said to be pseudo-lipschitz around ȳ, if there exist constants r > 0, rȳ > 0 M > 0 such that the following inequality holds: eγ y 1 Br,, Γ y M y 1 y for any y 1, y Brȳ, ȳ. 7 We finish this section with the following lemma. This lemma is known as Banach fixed point lemma which has been proved by Dontchev Hagger in [17]. This fixed-point lemma is the vital mechanism to prove the existence of any sequence generated by Algorithm 1. Lemma.1. Let Φ : X X be a set-valued mapping. Let η 0 X, r 0, λ 0, 1 be such that distη 0, Φη 0 < r1 λ 8 eφx 1 Br, η 0, Φx λ x 1 x for any x 1, x Br, η 0. 9 Then Φ has a fixed point in Br, η 0, that is, there exists x Br, η 0 such that x Φx. If Φ is single-valued, then there exists x Br, η 0 such that x = Φx. 3 Convergence Analysis Suppose X Y are Banach spaces. Let be a solution of 1. This section is devoted to study the local convergence of the sequence generated by Algorithm 1. Let r > 0. We suppose that F has closed graph, f is Fréchet differentiable its derivative is continuous in a neighborhood of with a constant ε > 0, g is differentiable at admits a second order divided difference satisfying the following condition: there exists κ > 0 such that for all x, y, z Br,, [x, y, z; g] κ. To this end, let x X let us define the mapping Q x by Q x := fx + fx x + g + F. The following result establishes the equivalence relation between f + g + F 1. This result is the refinement of the result in [9] or [8]. Lemma 3.1. Let f : X Y be a function let, ȳ gph f + g + F. Assume that f is Fréchet differentiable its derivative is continuous in an open neighborhood Ω of g admits a second order divided difference on Ω. Then the followings are equivalent: i The mapping f + g + F 1 is pseudo-lipschitz at ȳ, ; ii The mapping is pseudo-lipschitz at ȳ,.

4 118 Journal of Advances in Applied Mathematics, Vol., No. 3, July 017 Proof. Define a function h : X Y by hx := fx + f + fx. The proof is similar to that of [9], because the proof does not depend on the property of g. Define a single valued mapping Z x : X Y by f fx + gy gx + fy Z x y := fx + [x 0, x; g]y x, for y x, f fx + fy, for y = x. 10 Define a set valued mapping Φ x : X X by Φ x = [Z x ]. 11 For every x, x X, we have that Z x x Z x x = f fx x x + gx gx [x 0, x; g]x x f fx x x + [x, x ; g] [x 0, x; g] x x = f fx + [x, x ; g] [x 0, x; g] x x Linear Convergence The following lemma plays an important role to prove our first main theorem. Lemma 3.. Let be a solution of 1. Suppose that is pseudo-lipschitz at 0, with constant M. Assume that f is continuous in the neighborhood of with constant ε > 0 g admits second order divided difference in the neighborhood of. Then, there exists δ > 0 such that for each x Bδ,, there is ˆx Bδ, satisfying 0 fx + gx + fx + [x 0, x; g]ˆx x + F ˆx 13 ˆx 1 x. 14 Proof. The M-pseudo-Lipschitz continuity of r 0 > 0 such that around 0, implies that there exist r > 0, e y 1 Br,, y M y 1 y for all y 1, y Br 0, Let ε > 0 κ > 0 be such that Mε + κ < The continuity property of f the second order divided difference of g imply that there exists r > 0 such that fy fx ε for all x, y B r, 17 Let δ > 0 be such that Fix x X with x, define [x, y, z; g] κ for all x, y, z B r,. 18 δ min r, r 0 3ε + 4κ, 3Mε 1Mκ, r x := 3M ε + κ x. 0

5 Journal of Advances in Applied Mathematics, Vol., No. 3, July For x Bδ, using 16, we have r x 3Mε + κδ < δ. 1 We will apply Lemma.1 to the map Φ x with η 0 := r := r x λ := to conclude that there 3 exists a fixed point ˆx Br x, such that ˆx Φ x ˆx, that is, Z x ˆx Q ˆx, which implies that 0 fx + gx + fx + [x 0, x; g]ˆx x + F ˆx, i.e. 13 holds. Furthermore, ˆx Br x, Bδ, so ˆx r x 3Mε + κ 1 x. i.e. 14 holds. Thus, to complete the proof, it is sufficient to show that Lemma.1 is applicable for the map Φ x with η 0 := r := r x λ :=. To do this, it remains to prove that both assertions of Lemma.1 hold. It is obvious that 0 Br x,. According to the definition of the excess e, we have dist, Φ x 0 Br x,, Φ x For all u Br x, Bδ, with u x, we have from 10 that 0 Bδ,, [Z x ] 0 Br,, [Z x ]. Z x u = f fx + gu gx fx + [x 0, x; g] u x + fu f fx f x + f fx u x + [x, u; g] [x 0, x; g] u x. 3 Since f fx f x = 1 [ f + t x f] xdt, by using the continuity 0 property of f with constant ε, we have that f fx f x ε x. 4 Using 4 the second order divided difference concept with a constant κ, we have from 3 that Z x u ε x + ε u x + [x 0, x, u; g] u x 0 u x ε x + u x + κ u x 0 u x 3εδ + 4κδ < 3εδ + 4κδ 5 = 3ε + 4κδ < r 0. Therefore Z x u Br 0, 0 for all u Br x, with u x. Furthermore, in the case when u = x, we have from 3 that Z x x = f fx fx. 6 Using 4 the third assumption from 19 in 6, we obtain Z x x ε x = εδ < r 0. Thus we obtain Z x u Br 0, 0 for all u Br x,. 7

6 10 Journal of Advances in Applied Mathematics, Vol., No. 3, July 017 On the other h, letting u = in 3 for δ 1 by 19, we obtain Z x f fx fxx + [x, ; g] [x 0, x; g] x f fx fxx + [x, ; g] [x 0, x; g] x ε x + [x 0, x, ; g] x 0 x ε + κ x 0 x ε + κδ x ε + κ x. This together with 15 with y 1 = 0 y = Z x implies that dist, Φ x M Z x M ε + κ x = 1 r x = 1 λr. 3 This shows that the assertion 8 of Lemma.1 is satisfied. Now, we show that assertion 9 of Lemma.1 is also satisfied. Let x, x Br x,. Then by 1, we have that x, x Br x, Bδ, hence Z x x, Z x x Br 0, 0 by 7. This together with 15 with y 1 = Z x x y = Z x x implies that e Φ x x Br x,, Φ x x Φ x x Bδ,, Φ x x [Z x x ] Br,, [Z x x ] M Z x x Z x x. 8 In the case x x, since g admits second order divided difference, 18 is applicable to concluding that [x, x ; g] [x 0, x; g] x x = [x, x ; g] [x, x ; g] + [x, x ; g] [x 0, x; g] x x = [x, x, x ; g]x x + [x 0, x, x ; g]x x 0 x x [x, x, x ; g] x x + [x 0, x, x ; g] x x 0 x x 4κδ x x. This, together with 1 17, gives that Z x x Z x x f fx + [x, x ; g] [x 0, x; g] x x ε + 4δκ x x. Combining this with 8 using the relation 1Mκδ 3Mε by 19, we obtain that e Φ x x Br x,, Φ x x Mε + 4δκ x x 3 x x = λ x x. This implies that assertion 9 of Lemma.1 is satisfied. This completes the proof of the Lemma. The following theorem shows that the sequence x n generated by Algorithm 1 converges linearly.

7 Journal of Advances in Applied Mathematics, Vol., No. 3, July Theorem 3.1. Let be a solution of 1. Suppose that is pseudo-lipschitz at 0, with constant M. Assume that f is continuous in the neighborhood of with constant ε > 0 g admits second order divided difference in the neighborhood of. Then, there exists δ > 0 such that for every starting point x 0 Bδ,, there is a sequence x n generated by Algorithm 1 converges to which satisfies that Proof. By Lemma 3., there exists δ > 0 such that x k+1 1 x k for each k = 0, 1,, for each x Bδ, = there is ˆx Bδ, such that hold. 30 Let x 0 Bδ,. Then, it follows from 30 that there exists ˆx Bδ, such that ˆx Φ x0 ˆx. This implies that Z x0 ˆx Q ˆx, which translates to 0 fx 0 + gx 0 + fx 0 + [x 0, x 0 ; g]ˆx x 0 + F ˆx 31 ˆx 1 x 0. 3 Thus, 31 indicates that D x0 x 0. Consequently, we can choose d 0 D x0 x 0. By Algorithm 1, x 1 := x 0 + d 0 is defined. Thus, we obtain from 31 3 that 0 fx 0 + gx 0 + fx 0 + [x 0, x 0 ; g]x 1 x 0 + F x 1 x 1 1 x 0 so 9 holds for k = 0. We will proceed by induction on k. Now we assume that x 0, x 1,..., x k are generated by Algorithm 1 satisfying 9. Then by 30, we have that D xk x k. Then by Algorithm 1, we can select d k D x0 x k such that x k+1 = x k + d k. Then 31 3 give that 0 fx k + gx k + fx k + [x 0, x k ; g]x k+1 x k + F x k+1 x k+1 1 x k, so 9 holds for all k. This completes the proof of the Theorem. 3. Superlinear Convergence This section is devoted to study the superlinear convergence of the sequence generated by Algorithm 1. Let r > 0 we assume that f is Lipschitz continuous in a neighborhood Br, of the solution of 1, that is, there exists L > 0 such that fy fx L x y, for all x, y B r, 33 g admits second order divided difference, that is, there exists κ > 0 such that We assume that L κ are related in such a way that [x, y, z; g] κ for all x, y, z B r,. 34 Set L < 1 4Mκ 7M. 35 γ := 7ML + 4κ It follows from 35 that γ < 1 4 < 1. The following lemma is an analogue of Lemma 3.. However, the proof technique is slightly different.

8 1 Journal of Advances in Applied Mathematics, Vol., No. 3, July 017 Lemma 3.3. Let be a solution of 1. Assume that is pseudo-lipschitz at 0, with constant M. Suppose that f is Lipschitz continuous in the neighborhood of with constant L > 0 g admits second order divided difference in the neighborhood of with constant κ > 0. Let γ be defined by 36. Then, there exists δ > 0 such that for each x Bδ,, there is ˆx Bδ, satisfying 0 fx + gx + fx + [x 0, x; g]ˆx x + F ˆx 37 ˆx γ x max x, x Proof. By the assumed pseudo-lipschitz property of that there exist r > 0, r 0 > 0 such that with constant M > 0 around 0, implies Let δ > 0 be such that e y 1 Br,, y M y 1 y for all y 1, y Br 0, Fix x X with x, define It follows for x Bδ, that r0 1 δ min r,, 5L + 8κ 1 7ML + 4κ, r x := γ x max x, x r x 7ML + 4κ 4 δ max δ, δ = 7ML + 4κδ. 4 This, together with the relation 7ML + 4κδ 1 in 40, implies that r x 7ML + 4κδ δ δ. 4 4 The proof will be completed if we can apply Lemma.1 to the map Φ x with η 0 := r := r x λ := 1 7 to show that there exists a fixed point ˆx Br x, such that ˆx Φ x ˆx, that is, Z x ˆx Q ˆx. This implies that 0 fx + gx + fx + [x 0, x; g]ˆx x + F ˆx, i.e. D x0 x hence 37 holds. Furthermore, ˆx Br x, Bδ, so ˆx r x = γ x max x, x 0. i.e. 38 holds. Thus, to finish the proof, it is sufficient to show that Lemma.1 is applicable for the map Φ x with η 0 := r := r x λ := 1. To do this, it remains to prove that both assertions of Lemma.1 hold. It is obvious that 0 Br x,. According to the definition of the excess e, we have dist, Φ x 0 Br x,, Φ x 0 Bδ,, [Z x ] 0 Br,, [Z x ]. 43

9 Journal of Advances in Applied Mathematics, Vol., No. 3, July For all u Br x, Bδ, with u x, we have from 10 that Note by 33 that Z x u = f fx + gu gx fx + [x 0, x; g] u x + fu f fx f x + f fx u x + [x, u; g] [x 0, x; g] u x f fx f x + f fx u x + [x, u; g] [x 0, x; g] u x. 44 = f fx f x L [ f + t x f] xdt f + t x f x dt t x dt = L x. 45 This together with the relation 5L + 8κδ r 0 by 40 notion of second order divided difference with a constant κ, we have from 44 that Z x u L x + L x u x + [x 0, x, u; g] u x 0 u x L + L + 4κ δ 5L + 8κ = δ < r 0. It follows that for all u Br x, with u x, Z x u Br 0, 0. Furthermore, in the case when u = x, we have from 44 that Z x x = f fx fx. 47 Using 45 the third relation from 40 in 47, we obtain Z x x L x = Lδ < r 0. This implies that Z x u Br 0, 0, for all u Br x,. On the other h, letting u = in 44 for δ 1 by 40, we obtain Z x f fx f x + f fx x + [x, ; g] [x 0, x; g] x L x + L x + [x 0, x, ; g] x 0 x 3L x + κ x 0 x 3L + κ x max x, x 0 3L + 4κ < x max x, x 0. This together with with y 1 = 0 y = Z x implies that dist, Φ x M Z x 3ML + 4κ = x max x, x 0 r x = 1 λr. 46

10 14 Journal of Advances in Applied Mathematics, Vol., No. 3, July 017 This shows that the assertion 8 of Lemma.1 is satisfied. Now, we verify that assertion 9 of Lemma.1 is also satisfied. Let x, x Br x,. Then by 4, we have that x, x Br x, Bδ, hence Z x x, Z x x Br 0, 0 by 46. This together with 39 with y 1 = Z x x y = Z x x implies that e Φ x x Br x,, Φ x x Φ x x Bδ,, Φ x x [Z x x ] Br,, [Z x x ] M Z x x Z x x. 48 In the case x x, since g admits second order divided difference, 34 is applicable to concluding that [x, x ; g] [x 0, x; g] x x = [x, x ; g] [x, x ; g] + [x, x ; g] [x 0, x; g] x x = [x, x, x ; g]x x + [x 0, x, x ; g]x x 0 x x [x, x, x ; g] x x + [x 0, x, x ; g] x x 0 x x 4κδ x x. This, together with 1 33, gives that Z x x Z x x f fx + [x, x ; g] [x 0, x; g] x x L x + 4κδ x x L + 4κδ x x. Combining this with 48 using the relation 7ML + 4κδ 1 by 40, we obtain that e Φ x x Br x,, Φ x x ML + 4κδ x x 1 7 x x = λ x x. This implies that assertion 9 of Lemma.1 is satisfied. This completes the proof of the Lemma. Theorem 3.. Let be a solution of 1. Assume that is pseudo-lipschitz at 0, with constant M. Suppose that f is Lipschitz continuous in the neighborhood of with constant L > 0 g admits second order divided difference in the neighborhood of. Let γ be defined by 36. Then, there exists δ > 0 such that for every starting point x 0 Bδ,, there is a sequence x n generated by Algorithm 1 converges to which satisfies that x k+1 γ x k max Proof. By Lemma 3.3, there exists δ > 0 such that x k, x 0 for each k = 0, 1,, for each x Bδ, = there is ˆx Bδ, such that hold. 50 Let x 0 Bδ,. Then, it follows from 50 that there exists ˆx Bδ, such that ˆx Φ x0 ˆx. This implies that Z x0 ˆx Q ˆx, which translates to 0 fx 0 + gx 0 + fx 0 + [x 0, x 0 ; g]ˆx x 0 + F ˆx 51 ˆx γ x 0 max x 0, x 0. 5

11 Journal of Advances in Applied Mathematics, Vol., No. 3, July Thus, 51 indicates that D x0 x 0. Consequently, we can choose d 0 D x0 x 0. By Algorithm 1, x 1 := x 0 + d 0 is defined. Thus, we obtain from 51 5 that 0 fx 0 + gx 0 + fx 0 + [x 0, x 0 ; g]x 1 x 0 + F x 1 x 1 γ x 0 max x 0, x 0. so 49 holds for k = 0. We will proceed by induction on k. Now we assume that x 0, x 1,..., x k are generated by Algorithm 1 satisfying 49. Then by 50, we have that D xk x k. Then by Algorithm 1, we can select d k D x0 x k such that x k+1 = x k + d k. Then 51 5 give that 0 fx k + gx k + fx k + [x 0, x k ; g]x k+1 x k + F x k+1 x k+1 γ x k max x k, x 0. so 49 holds for all k. This completes the proof of the Theorem. 4 Concluding Remarks We have established local convergence results for Regula-falsi-type method for solving the generalized equation 1 under some suitable assumptions. When is pseudo-lipschitz continuous, f is continuous, g admits second order divided difference, we have shown that the sequence generated by Algorithm 1 converges linearly. However, when is pseudo-lipschitz continuous, f is Lipschitz continuous, g admits second order divided difference, we have presented super-linear convergence results of the Regula-falsi-type method defined by Algorithm 1. This study extends improves the results corresponding to []. Acknowledgments This work is supported financially by the Ministry of Education, Bangladesh, whose grant number is References 1. E. Cătinas, On some iterative methods for solving nonlinear equations, Rev. Anal. Numér. Théor. Approx., , M.H. Geoffroy, A. Piétrus, Local convergence of some iterative methods for solving generalized equations, J. Math. Anal. Appl., 90, 004, C. Jean-Alexis, A. Pietrus, On the convergence of some methods for variational inclusions, Rev. R. Acad. Cien. serie A. Mat., 10, 008, M.H. Rashid, J.H. Wang C. Li, Convergence Analysis of a method for variational inclusions, Applicable Analysis, 9110, 01, A.L. Dontchev R.T. Rockafellar, Regularity conditioning of solution mappings in variational analysis, Set-valued Anal., 11, 004, M.C. Ferris J.S. Pang, Engineering economic applications of complimentarity problems, SIAM Rev., 39, 1997, J.S. He, J.H. Wang C. Li, Newton s method for undetermined systems of equations under the modified γ-condition, Numer. Funct. Anal. Optim., 8, 007, M.H. Rashid, On the convergence of extended Newton-type method for solving variational inclusions, Journal of Cogent Mathematics, 11, 014, M.H. Rashid, A. Sardar, Convergence of the Newton-type method for generalized equations, GANIT J. Bangladesh Math. Soc., , M.H. Rashid, Convergence analysis of gauss-type proximal point method for variational inequalities, Open Science Journal of Mathematics Application, 1, 014, 5-14.

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