Chebyshev-Secant-type Methods for Non

Chebyshev-Secant-type Methods for Nondifferentiable Operators

I. K. Argyros, J. A. Ezquerro, J. M. Gutiérrez, M. A. Hernández & S. Hilout Milan Journal of Mathematics Issued by the Seminario Matematico e Fisico di Milano ISSN 1424-9286 Volume 81 Number 1 Milan J. Math. (2013) 81:25-35 DOI 10.1007/s00032-012-0189-4

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Author's personal copy Milan J. Math. Vol. 81 (2013) 25–35 DOI 10.1007/s00032-012-0189-4 Published online September 29, 2012 © 2012 Springer Basel

Milan Journal of Mathematics

Chebyshev-Secant-type Methods for Non-differentiable Operators I. K. Argyros, J. A. Ezquerro, J. M. Gutiérrez, M. A. Hernández and S. Hilout Abstract. In this paper we give a semilocal convergence theorem for a family of iterative methods for solving nonlinear equations defined between two Banach spaces. This family is obtained as a combination of the well known Secant method and Chebyshev method. We give a very general convergence result that allow the application of these methods to non-differentiable problems. Mathematics Subject Classification (2010). 47H17, 65J15. Keywords. Banach space, nondifferentiable operator, the secant method, semilocal convergence, divided difference, the Chebyshev method.

1. Introduction A mathematical problem that usually occurs in all branches of science and engineering is to solve a nonlinear equation of the form F (x) = 0.

(1.1)

It is known that the mathematical treatment of this problem consists of considering iterative methods to approximate numerically a solution of equation (1.1). Obviously, the problem becomes more complicated when the operator F is not differentiable. In this case, we have to consider iterative methods that do not use derivatives of the operator F in their definition. Although this situation is not very common, there are some iterative methods with this feature: for instance, the well-known Secant method [20]. In this paper, we consider (1.1), where F : Ω ⊆ X → Y is a nonlinear operator defined on a non-empty open convex domain Ω of a Banach space X with values in a Banach space Y . Let L(X, Y ) denote the space of bounded linear operators from The research of the second, third and fourth authors has been supported in part by the project MTM2011-28636-C02-01 of the Spanish Ministry of Science and Innovation.

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I. K. Argyros, J. A. Ezquerro, J. M. Gutiérrez, M. A. Hernández and S. Hilout

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X to Y . An operator [x, y; F ] ∈ L(X, Y ) is called a divided difference of first order for the operator F on the points x and y (x = y) if the following equality holds [x, y; F ](x − y) = F (x) − F (y). Using this definition, the Secant method is described by the following algorithm: xn+1 = xn − [xn−1 , xn ; F ]−1 F (xn );

x0 , x−1 given.

(1.2)

The convergence of (1.2) to a solution of (1.1) has been studied by different authors ([2], [3], [4], [19], [20], [21]). The basic assumption is that the divided difference of first order for the operator F is Lipschitz or Hölder continuous on some ball around the starting point x0 . But this assumption forces the operator F to be differentiable ([4], [20]). In this paper, we have two main aims. Firstly, we consider a multiparametric family of iterative methods that do not use derivatives and generalize method (1.2). Secondly, we obtain a semilocal convergence result for non-differentiable operators. To do this, we change the conditions normally imposed on divided differences. Thus, we relax the requirements that the first order divided difference operator F is Lipschitz or Hölder continuous, and just assume the following condition: [x, y; F ] − [v, w; F ] ≤ ω(x − v, y − w);

x, y, v, w ∈ Ω,

where ω : R+ ×R+ → R+ is a continuous non-decreasing function in its two arguments. It is clear that this condition generalizes the conditions previously indicated. In fact, when ω(u1 , u2 ) = k(u1 + u2 ), we obtain the Lipschitz continuous case and, when ω(u1 , u2 ) = k(up1 + up2 ), we obtain the the (k, p)-Hölder continuous case. Moreover, in general, this condition does not involve the differentiability of the operator F. The family of iterative methods that we consider in this paper arises from the Chebyshev method, a well known third-order convergence method that has been studied, for instance, in [9] or [1]. Firstly, Hern´ andez [15] and later Ezquerro and Hernández [11], modified this method by avoiding the computation of the second derivative of F and reducing the number of evaluations of the first derivative of F . Actually, these authors have obtained a modification of the Chebyshev method with order of convergence at least three, which only need to evaluate the first derivative of F . This family of iterative methods is written as follows [11]: ⎧ x0 ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎨ yk = xk − F (xk ) F (xk ), zk = xk + p (yk − xk ), p ∈ (0, 1], ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ xk+1 = xk − 2 F (xk )−1 ((p2 + p − 1) F (xk ) + F (zk )), p

k ≥ 0,

So, they obtain a uniparametric family of iterative methods which depends only on the first derivative of the operator F , which is evaluated only at one point. Then,

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Chebyshev-Secant-type Methods

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to construct a family of iterative methods free of derivatives, as the classical Secant method, we consider an approximation of the first derivative of F from a divided difference of first order; that is, F (xn ) [xn−1 , xn , F ], where, [x, y; F ] is a divided difference of first order for the operator F at the points x, y ∈ Ω. So, we consider the family of iterative methods introduced in [8] and called Chebyshev-Secant-type methods. These methods are written as follows: ⎧ x−1 , x0 ∈ Ω, ⎪ ⎪ ⎪ ⎪ ⎨ yn = xn − A−1 F (xn ), An = [xn−1 , xn ; F ], n ⎪ ⎪ zn = xn + a (yn − xn ), ⎪ ⎪ ⎩ xn+1 = xn − A−1 n ≥ 0, n (b F (xn ) + c F (zn )),

(1.3)

where a, b, c are non-negative parameters to be chosen so that the sequence {xn } converges to x∗ . Notice that these methods (1.3) do not use derivatives. In fact they reduce to (1.2) if b = 1 and c = 0. Even more, if ac = 0 and b > 0, the Chebyshev-Secant-type methods defined in (1.3) become the following Secant-type method: xn+1 = xn − τ [xn−1 , xn ; F ]−1 F (xn );

x0 , x−1 given,

(1.4)

where τ is a non-negative parameter. Some authors (see [10] or [18]) refer to Secanttype methods as the methods written in the form xn+1 = xn + sn ,

where

Bn sn = F (xn ) − F (xn−1 ),

for an appropriate operator Bn . In our case, Bn is a divided difference operator. Another usual choice for Bn is the one considered in the well-known Broyden’s method [18], where Bn is updated at each step and its inverse can be calculated by using the Sherman-Morrison formula. The kind of methods defined in (1.4) have been studied, for instance, in [16], [17]. However, if ac = 0, the methods defined in (1.3) are essentially different from the Secant method (1.2) or the Secant-type methods (1.4). Moreover, every method included in (1.3) has superlinear convergence. So, for instance, method (1.3) with a = b = c = 1 has R-order of convergence at least two, as we can see in [12, 13]. For other values of the parameters a, b and c, we can follow a process analogous to those given in [12, 13] and prove the superlinear convergence. Next, in Section 2, we provide a semilocal convergence result for the family of iterative methods (1.3) when they are applied to non-differentiable operators. So, we extend the results given in [8] where only the differentiable case is considered. Finally, in Section 3, we consider two examples, where we analyse two non-differentiable problems. Throughout the paper we denote B(x, ) = {y ∈ X; y − x < } and B(x, ) = {y ∈ X; y − x ≤ }.



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2. Convergence analysis We analyse the semilocal convergence of the family of iterative methods given by (1.3). To do this, we use a technique based on proving a system of recurrence relations. Firstly, we suppose that there exists a first-order divided difference [x, y; F ] ∈ L(X, Y ), for all x, y ∈ Ω. Let us suppose that (1 − b) = (1 − a)c and a ∈ [0, 1], x−1 , x0 ∈ Ω are such that x0 − x−1 ≤ α, −1 the linear operator A0 is invertible, A−1 0 ≤ β and A0 F (x0 ) ≤ η, [x, y; F ] − [u, v; F ] ≤ ω(x − u, y − v), x, y, u, v ∈ Ω, where ω : R+ × R+ −→ R+ is a continuous non-decreasing function in both arguments, (C5) we denote m = max{p, acβω(α, (1 + p)η), acβω((1 + p)η, η), acβω(η, η)}, where p = acβω(α, η), and suppose that the equation 1 (1 + ϕ(t))η = t 1 − ϕ(t) 1 + (1 + ϕ(t)) , (2.1) ac

(C1) (C2) (C3) (C4)

m , has at least one positive root; we denote the smallest where ϕ(t) = 1−βω(α+t,t) positive root equation by R, of this 1 m (1 + M ) < 1 and M = 1−βω(α+R,R) > 0, (C6) (1 + M )M 1 + ac (C7) B(x0 , R) ⊆ Ω.

Next, we give three technical lemmas. In the first one, we give two expressions of the operator F that we need later. Lemma 2.1. Notice that F (zn ) = (1 − a)F (xn ) + a(Bn − An )(yn − zn ),

n ≥ 0,

F (xn+1 ) = (An+1 − An )(xn+1 − xn ) + ac(An − Bn )(yn − xn ),

n ≥ 0,

where An = [xn−1 , xn ; F ], Bn = [xn , zn ; F ] and (1 − b) = (1 − a)c. Lemma 2.2. Let us consider (C1)–(C7). Then i n

1 (1 + M ) η i = R, (1 + M ) M 1 + (1 + M ) η < 1 ac 1 − M 1 + ac (1 + M ) i=0 where M = ϕ(R). Lemma 2.3. Let us consider (C1)–(C7). Then, for n ≥ 1, we have the following recurrence relations: β −1 [I] There exists A−1 n and An ≤ 1−βω(α+R,R) , 1 [II] yn − xn ≤ M 1 + ac (1 + M ) yn−1 − xn−1 n 1 n (1 + M ) y0 − x0 , ≤ M 1 + ac i 1 (1 + M ) y0 − x0 < R, [III] yn − x0 ≤ (1 + M ) ni=0 M i 1 + ac i 1 (1 + M ) y0 − x0 < R, [IV] zn − x0 ≤ (1 + M ) ni=0 M i 1 + ac n 1 (1 + M ) y0 − x0 , [V] xn+1 − xn ≤ (1 + M )yn − xn ≤ (1 + M )M n 1 + ac i 1 (1 + M ) y0 − x0 < R. [VI] xn+1 − x0 ≤ (1 + M ) ni=0 M i 1 + ac



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Proof. Observe that y0 − x0 ≤ η, z0 − x0 ≤ η and x1 − x0 ≤ 1 + acA−1 0 A0 − B0 y0 − x0 ≤ (1 + p)y0 − x0 ≤ (1 + M )η < R. Now, we can prove that [I]–[VI] is true for n = 1 and assume that [I]–[VI] are true for k = 1, 2, . . . , n − 1. Then, [I]: Since I − A−1 0 An ≤ βω(xn−1 − x−1 , xn − x0 ) ≤ βω(α + R, R) < 1, then, by Banach’s lemma, it follows A−1 n ≤

1 β ≤ . 1 − β ω(xn−1 − x−1 , xn − x0 ) 1 − βω(α + R, R)

(2.2)

[II]: yn − xn ≤ A−1 n F (xn ) ≤ A−1 n ω(xn−1 − xn−2 , xn − xn−1 )xn − xn−1 +ac ω(xn−1 − xn−2 , zn−1 − xn−1 )yn−1 − xn−1 1 (1 + M ) yn−1 − xn−1 ≤ M 1 + ac n 1 ≤ M n 1 + ac (1 + M ) y0 − x0 , [III]: yn − x0 ≤ zn − xn + xn − x0 i 1 (1 + M ) y0 − x0 < R, ≤ (1 + M ) ni=0 M i 1 + ac [IV]: zn − x0 ≤ zn − xn + xn − x0 i 1 (1 + M ) y0 − x0 < R, ≤ (1 + M ) ni=0 M i 1 + ac [V]: xn+1 − xn ≤ 1 + acA−1 n An − Bn yn − xn m yn − xn ≤ 1 + 1−βω(α+R,R) ≤ (1 + M )yn − n xn 1 (1 + M ) y0 − x0 , ≤ (1 + M )M n 1 + ac

[VI]: xn+1 − x0 ≤ xn+1 − xn + xn − x0 i 1 ≤ (1 + M ) ni=0 M i 1 + ac (1 + M ) y0 − x0 < R, so that [I]–[VI] are true for all positive integers k by mathematical induction.

We are then ready to prove a semilocal convergence theorem for the iterative method (1.3) when it is applied to nondifferentiable operators that satisfy conditions (C1)–(C7). Theorem 2.4. Let X and Y be two Banach spaces and let F : Ω ⊆ X → Y be a nonlinear operator defined on a non-empty open convex domain Ω. We suppose that there exists [x, y; F ] ∈ L(X, Y ), for all x, y ∈ Ω, and conditions (C1)–(C7) are satisfied. Then, sequence (1.3), starting from x−1 and x0 , converges to a unique solution x∗ of F (x) = 0. Moreover, the solution x∗ and the iterates xn belong to B(x0 , R) and x∗ is unique in B(x0 , R). Proof. From Lemma 2.3, it follows that (1.3) is a Cauchy sequence, since xn+k − xn ≤ xn+k − xn+k−1 + xn+k−1 − xn+k−2 + · · · + xn+1 − xn ≤ (1 + M ) (yn+k−1 − xn+k−1 + yn+k−2 − xn+k−2 + · · · + yn − xn )



i 1 M 1 + (1 + M ) y0 − x0 ≤ (1 + M ) ac i=n k n 1 1 − M 1 + ac (1 + M ) 1 n η. 1 + (1 + M ) < (1 + M )M 1 ac 1 − M 1 + ac (1 + M ) n+k−1

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Consequently, {xn } is convergent and then limn xn = x∗ ∈ B(x0 , R). Observe now that 1 F (xn ) ≤ ac ω(η, η) 1 + (1 + M ) yn−1 − xn−1 ac and yn−1 − xn−1 → 0 as n → ∞, so that F (x∗ ) = 0. To prove the uniqueness of the solution x∗ , we first assume that y ∗ is another solution of F (x) = 0 in B(x0 , R). Next, we consider the operator A = [x∗ , y ∗ ; F ], so that if A is invertible, we have x∗ = y ∗ , since A(y ∗ − x∗ ) = F (y ∗ ) − F (x∗ ). Indeed, −1 −1 ∗ ∗ A−1 0 A − I ≤ A0 A − A0 ≤ A0 [y , x ; F ] − [x−1 , x0 ; F ]

≤ βω(y ∗ − x−1 , x∗ − x0 ) ≤ βω(α + R, R) < 1,

and the operator A−1 exists.

Remark 2.5. Using the same information as in Theorem 2.4, we can provide at least as tight upper bounds on the distances involved, and at least as precise information on the location of the solution x∗ . Indeed, let us assume (C8) [x−1 , x0 ; F ] − [x, y; F ] ≤ ω0 (x−1 − x, x0 − y), x, y ∈ Ω, where ω0 : R+ × R+ −→ R+ is a continuous non-decreasing function in both arguments, (C9) there exists R1 ≥ R such that β ω0 (α + R1 , R) < 1, and (C10) B(x0 , R1 ) ⊆ Ω. Condition (C8) always follows from (C4) (simply, set ω = ω0 ). Hence, (C8) is not an additional (to (C4)) hypothesis. Note that ω0 (s, t) ≤ ω(s, t) ω(s, t) can be arbitrarily large (see [5]–[7]). ω0 (s, t) It follows from (2.3), and the proof of [I] in Lemma 2.3 that 1 A−1 n ≤ 1 − β ω0 (xn−1 − x−1 , xn − x0 )

(2.3)

holds in general, and

(2.4)

which is an at least as tight estimate as (2.2). In particular, if strict inequality holds in (2.3), then (2.4) can replace (2.2), which leads to tighter error bounds on the distances involved. Conditions (C8)–(C9) extend the uniqueness ball for the solution x∗ . Indeed, assuming y ∗ ∈ U (x0 , R1 ), the uniqueness proof of Theorem 2.4 now gives A−1 0 A − I ≤ β ω0 (α + R1 , R) < 1.



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We finish this section with a semilocal convergence theorem for the methods defined in (1.3) in the case ac = 0 or, equivalently, for the Secant-type methods (1.4) with b > 0. We omit the proof because it follows directly from the proof of Theorem 2.1 in [16]. Theorem 2.6. Let X and Y be two Banach spaces and let F : Ω ⊆ X → Y be a nonlinear operator defined on a non-empty open convex domain Ω such that there exists [x, y; F ] ∈ L(X, Y ), for all x, y ∈ Ω. Let us suppose that −1 (c1) The linear operator A0 is invertible, A−1 0 ≤ β and A0 F (x0 ) ≤ η, (c2) x−1 , x0 ∈ Ω are such that x0 − x−1 ≤ α, (c3) [x, y; F ] − [u, v; F ] ≤ ω(x − u, y − v), x, y, u, v ∈ Ω, where ω : R+ × R+ −→ R+ is a continuous non-decreasing function in both arguments, (c4) we denote m = max{βω(α, η), βω(η, η)}, and suppose that the equation bm − η = 0, t 1− 1 − βω(t + α, t)

has at least one positive root; we denote the smallest positive root of this equation by R, bm (c5) ∈ (0, 1), 1 − βω(α + R, R) (c6) B(x0 , R) ⊆ Ω. Then, sequence (1.4), starting from x−1 and x0 , converges to a unique solution x∗ of F (x) = 0. Moreover, the solution x∗ and the iterates xn belong to B(x0 , R). According to Remark 2.5, the results of Theorem 2.6 are improved along the same lines.

3. Applications We provide two examples in this concluding section. In the first example, we solve a system of two equations with two unknowns that is non-differentiable at the starting points. In the second one, we solve a nonlinear integral equation, where the operator involved is non-differentiable at the solution. 3.1. Example 1 In this example we apply the above results to the following nonlinear system: x21 − x2 + 1 + 19 |x1 − 1| = 0, x1 + x22 − 7 + 19 |x2 | = 0.

(3.1)

Observe that system (3.1) is equivalent to F (x) = 0, where F : R2 → R2 , F = (F1 , F2 ), x = (x1 , x2 ), F1 (x1 , x2 ) = x21 − x2 + 1 + 19 |x1 − 1| and F2 (x1 , x2 ) = x1 + x22 − 7 + 19 |x2 |.



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In this paper the following type of divided difference is chosen: for u, v ∈ R2 , [u, v; F ] ∈ L(R2 , R2 ) and [u, v; F ]i1 =

Fi (u1 , v2 ) − Fi (v1 , v2 ) , u1 − v 1

so that

[u, v; F ] =

u21 −v12 u1 −v1

1

−1

[u, v; F ]i2 =

u22 −v22 u2 −v2

1 + 9

Fi (u1 , u2 ) − Fi (u1 , v2 ) , u2 − v 2

|u1 −1|−|v1 −1| u1 −v1

0

0 |u2 |−|v2 | u2 −v2

i = 1, 2,

.

This choice of the divided differences is usually considered by different authors, as we can see in [4] or [16], although there are also other possibilities for choosing the divided difference operator, see for instance [20]. If we take the max-norm as vector norm and the matrix norm subordinated to this vector norm, we obtain 2 [x, y; F ] − [u, v; F ] ≤ x − u + y − v + , 9 and consequently, from (C4), it follows ω(s, t) = s + t + 2/9. Now, we apply an iterative method included in family (1.3) for approximating a solution of (3.1). For example, we choose a = b = c/2 = 1/2. We start the corresponding iterative method with u−1 = (0, 0) and u0 = (1, 1). After three iterations applying the iterative method, we obtain u2 = (1.404149252, 2.249344679)

and

u3 = (1.135872343, 2.358061769).

After that we consider x−1 = u2 and x0 = u3 . With these choices we can apply Theorem 2.4 to guarantee the convergence of the iterative method to a solution of (3.1). So, we have α = 0.2682,

β = 0.4232,

η = 0.0215,

m = 0.1088,

the solution of (2.1) is then R = 0.0469, so that M = 0.1416 and 1 (1 + M )M 1 + (1 + M ) = 0.5445 < 1. ac Therefore the hypotheses of Theorem 2.4 are satisfied and we can guarantee the convergence of the iterative method to the unique solution x∗ = (1.159360850, 2.361824342) of (3.1) in B(x0 , R) after four iterations more and using ten significative figures. Moreover, if we consider the computational order of convergence ρ (see [14]), xn − x∗ ∞ xn+1 − x∗ ∞ / ln , n ∈ N, ρ ≈ ln xn − x∗ ∞ xn−1 − x∗ ∞ we obtain ρ = 1.6328 . . . (i.e. superlineal convergence).



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3.2. Example 2 In this example we present an application of the previous analysis to the following nonlinear integral equation of mixed Hammerstein type B G(s, t) λ(x(t) − f (t))2 + μ|x(t) − f (t)| dt, s ∈ [A, B], x(s) = f (s) + A

where x, f ∈ C[A, B], the kernel G is the Green function (B−s)(t−A) , t ≤ s, B−A G(s, t) = (s−A)(B−t) , s ≤ t, B−A and λ, μ ∈ R. Observe that x(s) = f (s) is a solution of the integral equation, where it is non-differentiable. To simplify the analysis we choose A = 0, B = 1, λ = μ = 1/3 and f (s) = 0. We determine where a solution of this is located and is unique. Then, the solution is approximated by an iterative method of (1.3). To approximate numerically a solution of the equation, we approach the integral by a Gauss-Legendre quadrature formal with eight nodes, 1 8

φ(t) dt wj φ(tj ). 0

j=1

If we denote the approximations of x(ti ) by xi , i = 1, 2, . . . , 8, we obtain the following nonlinear system: 8

xi =

1

aij x2j + |xj | , 3

i = 1, 2, . . . , 8,

(3.2)

j=1

where aij =

wj tj (1 − ti ) if j ≤ i, wj ti (1 − tj ) if j > i.

ˆ ), or x+x System (3.2) can be now written as x = 13 A(¯ 1 ˆ ) = 0, F (x) ≡ x − A(¯ x+x 3 where F : R8 −→ R8 , x = (x1 , x2 , . . . , x8 )T , 1 = (1, 1, . . . , 1)T , A = (aij )8i,j=1 , ¯ = (x21 , x22 , . . . , x28 )T and x ˆ = (|x1 |, |x2 |, . . . , |x8 |)T . x We choose the same kind of divided difference introduced in the previous example. In fact, for u, v ∈ R8 , [u, v; F ] = ([u, v; F ]ij )8i,j=1 ∈ L(R8 , R8 ), where [u, v; F ]ij =

1 (Fi (u1 , . . . , uj , vj+1 , . . . , v8 ) − Fi (u1 , . . . , uj−1 , vj , . . . , v8 )) , uj − v j

u = (u1 , u2 , . . . , u8 )T and v = (v1 , v2 , . . . , v8 )T , so that, for the previous F , we have [u, v; F ] = I − 13 (P − Q), where P = (pij )8i,j=1 with pij = aij (uj + vj ) and |u |−|v |

Q = (qij )8i,j=1 with qij = aij ujj −vjj . We now choose a = b = 1/2 and c = 1 in (1.3) and take as starting points x−1 = (9/10, 9/10, . . . , 9/10)T and x0 = (1, 1, . . . , 1)T . We work again with the max-norm



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to obtain α = 1/10, β = 1.1330 . . ., η = 1.0413 . . . and ω(s, t) = 13 (0.1235 . . .)(2 + s + t). We also obtain m = 0.0970 . . ., the solution of (2.1) is R = 2.1765 . . ., M = 0.1388 . . . and 1 (1 + M )M 1 + (1 + M ) = 0.5183 . . . < 1. ac In consequence, the hypotheses of Theorem 2.4 are satisfied and method (1.3) with a = b = 1/2 and c = 1 converges to the solution x∗ = (0, 0, . . . , 0)T after eight iterations. Note that 32 significative figures are used in the computations. Furthermore, the existence of the solution is guaranteed in the ball B(x0 , 2.1765 . . .) and the unicity in B(x0 , 2.1765 . . .). Finally, if we interpolate the points and taking into account that the solution of the integral equation satisfies x(0) = x(1) = 0, we obtain the solution x∗ (s) = 0, which lies within the existence domain of solution obtained above.

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