View of An improved semilocal convergence analysis for the midpoint method

J. Numer. Anal. Approx. Theory, vol. 45 (2016) no. 2, pp. 109–127 ictp.acad.ro/jnaat

AN IMPROVED SEMILOCAL CONVERGENCE ANALYSIS FOR THE MIDPOINT METHOD

IOANNIS K. ARGYROS^∗and SANJAY K. KHATTRI^†

Abstract. We expand the applicability of the midpoint method for approximat- ing a locally unique solution of nonlinear equations in a Banach space setting.

Our majorizing sequences are finer than the known results in scientific literature [1, 3, 4, 10–16, 24–26, 28] and the convergence criteria can be weaker. Finally, numerical work is reported that compares favorably to the existing approaches in the literature [6, 8–16, 24–26,28].

MSC 2010. 65B05, 65G99, 65J15, 65N30, 65N35, 65H10, 47H17, 49M15.

Keywords. Midpoint method, semilocal convergence, majorization sequence, Banach space, Fr´echet-derivative.

1. INTRODUCTION

In this study, we are concerned with the problem of approximating a locally unique solutionx^? of equation

(1.1) F(x) = 0,

where, F is a twice Fr´echet differentiable operator defined on a convex subset Dof a Banach spaceXwith values in a Banach spaceY. Numerous problems in science and engineering can be reduced to solving the above equation [23, 32]. Consequently, solving these equations is an important scientific field of research. In many situations, finding a closed form solution for the non-linear equation (1.1) is not possible. Therefore, iterative solution techniques are employed for solving these equations.

The study about convergence analysis of iterative methods is usually divided into two categories: semi-local and local convergence analysis. The semilocal convergence analysis is based upon the information around an initial point to give criteria ensuring the convergence of the iterative procedure. While the local convergence analysis is based on the information around a solution to

∗Department of Mathematical Sciences, Cameron University, Lawton, Oklahoma 73505- 6377, USA, e-mail: [email protected].

†Department of Engineering, Stord Haugesund University College, Norway, e-mail:

[email protected].

find estimates of the radii of convergence balls. In this paper, we study the semilocal convergence of the midpoint method defined as

y_n=x_n− F⁰(x_n)⁻¹F(x_n), (1.2)

xn+1 =xn− F^0xn+y₂ ⁿ⁻¹F(x_n), for each n= 0,1,2, . . . ,

where x₀ ∈ D is an initial point. Here, F⁰(x) denotes the first Fr´echet- derivative of the operator F [23, 32]. It is well-known that the Midpoint method is cubically convergent and it has a long history [see 27–32]. Let U(w, R) and U(w, R) stand, respectively, for the open and closed balls in X with center w and radius R > 0. Let the space of bounded linear operators from X into Y be denoted by L(X,Y). The following set of (C) conditions have been used

(1) There exists x₀ ∈Dsuch that F⁰(x₀)⁻¹∈L(Y,X).

(2) F⁰(x₀)⁻¹F(x₀)≤η.

(3) F⁰(x₀)⁻¹F⁰⁰(x)≤Lfor all x∈D.

(4) F⁰(x₀)⁻¹(F⁰⁰(x)− F⁰⁰(y))≤ M kx−yk, for all x, y∈D.

The following sufficient convergence criteria have been given in connection to the (C) conditions

η≤ ^4M+L2−L

√ L²+2M 3M(L+√

L²+2M) [1, 3, 4, 23–26]

(1.3) or

η≤ ₂¹_K [12, 14], (1.4)

where

K=L q

1 +⁷₆^M_L2.

However, simple numerical examples can be used to show that criteria (1.3) and (1.4) are unsatisfied but the midpoint method (1.2) still converges to the solution x^?. As an example, let X =Y =R, x0 = 1 andD = [ζ,2−ζ] for ζ ∈(0,1). Define functionF on D by

(1.5) F(x) =x⁵−ζ.

Then, using conditions (C), we get

η = ^(1−ζ)₅ , L= 4(2−ζ)³, M= 12(2−ζ)².

Figure 1.1 plots the criteria (1.3) and (1.4) for the problem (1.5). The curve (defined by the right hand side of the inequality (1.3)) intersect the lineη(see Figure 1.1) at ζ ≈0.73 while the curve (defined by the right hand side of the criteria (1.4)) intersect the η line at ζ ≈ 0.72. We notice in the Figure 1.1 that for ζ <0.72 the criteria (1.3) and (1.4) are not satisfied. However, one

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

2 4 6 8 10 12 14 16 18 20·10⁻²

ζ

η

1 2K 4M+L²−L√

L²+2M 3M(L+√

L²+2M)

Fig. 1.1. Convergence criteria (1.3) and (1.4) for (1.5).

may see that the method (1.2) is convergent. For additional examples, see the Section 4.

In our work we expand the applicability of the midpoint method (1.2), in cases where (1.3) or (1.4) are not satisfied, using the (C) conditions together with the following center Lipschitz condition

(1) F⁰(x₀)⁻¹(F⁰(x)− F⁰(x₀))≤L₀kx−x₀kfor all x∈D.

We shall refer to (C₁)-(C₅) as the (H) conditions.

As can be inferred from the studies [1–28], several techniques are usually employed for analyzing the convergence of iterative methods. Among these, the most popular technique is based on the concept of majorizing sequences.

In the studies that lead to the convergence conditions (1.3) and (1.4) the computation of the upper bounds on F⁰(x_n)⁻¹F⁰(x₀) was based on (C₃) and the estimate

(1.6) F⁰(x_n)⁻¹F⁰(x₀)≤ _1−Lkx¹

n−x₀k.

Instead of (C₃) we use the more precise and less expensive condition (C₅) which leads to

(1.7) F⁰(xn)⁻¹F⁰(x₀)≤ _1−L ¹

0kxn−x₀k. Note that

(1.8) L₀ ≤L

holds in general andL/L₀ can be arbitrarily large [22, 23]. This change in the study of the semilocal convergence of the midpoint method leads to tighter

error estimates on the distancesky_n−xnk,kx_n+1−ynk,ky_n−x^?k,kx_n−x^?k and weaker convergence criteria.

The rest of the paper is organized as follows. Section 2 develop results on majorizing sequences for the midpoint method (1.2), where as in the Section 3 we present the semilocal convergence of the Midpoint method. Numerical examples are given in the concluding section 4.

2. MAJORIZING SEQUENCES

In this section, we study the convergence of scalar sequences that are ma- jorizing for the Midpoint method (1.2). Let the positive constants beL₀ >0, L > 0,M ≥0 and η >0. It is convenient for us to define functions γ, a, α, h_i,i= 1,2,3 by

γ(t) = L t

2^h1−^L₂^0ti, γ =γ(η), (2.1)

a(t) = ₂₄¹ 12Lγ(t)²+ 12Lγ(t) + 7Mη, a=a(η), (2.2)

α(t) = a(t)t

h

1−^L₂⁰(1 +γ(t))tⁱ, α=α(η), (2.3)

h₁(t) =a(t)t+ ^L₂⁰(1 +γ(t))t−1 (2.4)

h2(t) = ^L₂α(t)t+^γ(t)L₂ ⁰[2(1 +γ(t)) +α(t)]t−γ(t) (2.5)

and

h3(t) =a(t)t+L0(1 +γ(t))(1 +α(t))t−1.

(2.6)

We denote the minimal positive zeros of the functionsh₁,h₂ and h₃ by η₁,η₂ and η3, respectively. Note that α(t) is well defined on (0, η1) by the choice of η₁. Let us set

(2.7) η0= min{η₁, η2, η3}.

Then, for allt∈(0, η₀) we have

α∈(0,1), (2.8)

h₁(t)<0, (2.9)

h₂(t)≤0 (2.10)

and

h₃(t)≤0.

(2.11)

We can show the following result on the convergence of majorizing sequences for the Midpoint method.

Lemma2.1. Let the positive constants be L0>0,L >0,M ≥0andη >0.

Suppose that

(2.12) η

( ≤η₀ if η₀ 6=η₁,

< η₀ if η₀ =η₁. Then, scalar sequence {t_n} generated by

(2.13)

t₀= 0, s₀=η, t_n+1=s_n+ L(s_n−t_n)² 2^h1−^L₂⁰(sn+tn)ⁱ

,

sn+1 =tn+1+

12L(t_n+1−s_n)²+ ^{6L2(sn−tn)3}

1−^L0

2 (tn+tn)(s_n−t_n)³+ 7M(s_n−t_n)³ 24(1−L0tn+1)

is increasing, bounded from above by

(2.14) t^??=_1−α^1+γη

and converges to its unique least upper bound t^? which satisfies

(2.15) 0≤t^? ≤t^??.

Moreover, the following estimates hold for each n= 0,1,2, . . . 0< tn+1−sn≤γ(sn−tn)≤γαⁿη (2.16)

and

0< sn+1−tn+1 ≤α(sn−tn)≤αⁿ⁺¹η.

(2.17)

Proof. We use mathematical induction to prove (2.16) and (2.17). Estimates (2.16) and (2.17) hold for n= 0 by (2.1)-(2.3) and (2.13), since

s₁−t₁ =

12L(t₁−s₀)³+ ^{6L2(s0−t0)3}

1−^L₂⁰(t0+s0) + 7M(s₀−t₀)³ 24(1−L0t1)

≤ 12Lγ²+ 12Lγ+ 7M(s₀−t0)

24^h1−^L₂⁰(1 +γ)ηⁱ (s0−t0)²

≤ a

1−^L₂⁰(1 +γ)(s₀−t₀)(s₀−t₀)≤α(s₀−t₀) =αη.

(2.18)

Let us assume that (2.16) and (2.17) hold for all k≤η. Then, we have t_k+1−s_k≤γ(s_k−t_k)≤γα^kη,

s_k+1−t_k+1 ≤α(s_k−t_k)≤α^k+1η,

t_n+1 ≤s_k+γα^kη≤t_k+α^kη+γα^kη

≤tk−1+α^k−1η+α^kη+γα^k−1η+γα^kη

≤ · · · ≤t₂+ (α²η+α³η+· · ·+α^kη) + (γα²η+· · ·+γα^kη)

≤s₁+γαη+ (α²η+α³η+· · ·+α^kη) + (γα²η+· · ·+γα^kη)

≤t₁+αη+γαη+ (α²η+α³η+· · ·+α^kη) + (γα²η+· · ·+γα^kη)

≤η+γη+αη+γαη+ (α²η+· · ·+α^kη) + (γα²η+· · ·+γα^kη)

≤ ^1−α_1−α^k+1(1 +γ)η < ^1+γ_1−αη=t^??, (2.19)

and

s_k+1 ≤t_k+1+α^k+1η≤^h1−α_1−α^k+1(1 +γ) +α^k+1iη

<^1+γ_1−γ +α^k+1η≤t^??. Evidently, estimates (2.16) and (2.17) are true provided that

L(s_k−t_k) 21−^L₂⁰(sk+tk)

≤γ (2.20)

and

a(s_k−t_k)

2(1−L0tk+1) ≤α.

(2.21)

Inequality (2.20) can be written as

(2.22) ^Lα₂^kη +^γL₂⁰2^1−α_1−α^k(1 +γ) +α^kη−γ ≤0.

Estimate (2.22) motivates us to define recurrent functionsf_k on [0,1) for each k= 1,2, . . .by

(2.23) f_k(t) = ^{L t}₂^kη +^γL₂⁰ 2^1−t_1−t^k(1 +γ) +t^kη−γ.

We need a relationship between two consecutive functions f_k. We have by (2.23)

f_k+1(t) =f_k(t) +^{L tk+1}₂ ^η −^{L t}₂^kη +^γL₂⁰ 2(1 +γ)(t^k−t^k−1) +t^k+1−t^kη

=f_k(t) + (t−1)^hL₂t+γ L₀(1 +γ) + ^γαL₂ ⁰ⁱt^k−1η.

(2.24)

It follows from (2.24) that

(2.25) fk+1(t)≤fk(t)≤ · · · ≤f1(t).

In view of (2.25), estimate (2.22) holds if

(2.26) f₁(α)≤0

which is true by the choice ofη2. Similarly, estimate (2.21) can be written as (2.27) aα^kη+αL₀(1 +γ)^1−α_1−α^k+1η−α≤0.

Define recurrent functions g_k on [0,1) for each k= 1,2, . . . by (2.28) g_k(t) =at^k−1η+L₀(1 +γ)^1−t_1−t^k+1η−1.

Then using (2.28) we get

(2.29) g_k+1(t) =g_k(t) + (t−1) [a+L0(1 +t)t]t^k−1η.

Hence, it follows from (2.29) that

(2.30) g_k+1(t)≤g_k(t)≤ · · · ≤g₁(t).

In view of (2.30), instead of (2.27), we can show that

(2.31) g₁(α)≤0,

which is true by the choice of η3. The induction for (2.16) and (2.17) is complete. Hence, sequence {t_n} is an increasing, bounded from above by t^??

and as such it converges to its unique least upper boundt^?. The proof of the

Lemma is complete.

We have the following useful and obvious extension of Lemma 2.1 Lemma 2.2. Suppose there existsN ≥0 such that

t₀< s₀ < t₁ <· · ·< t_N < s_N < t_N+1 < _L¹

0. (2.32)

and

s_N −t_N

( ≤η₀ if η₀ 6=η₁

< η₀ if η₀ =η₁. (2.33)

Then, the conclusions of the Lemma 2.1 hold for sequence {t_n}. Moreover, the following estimates hold for each n= 0,1,2,3, . . .

0< tN+1+n−sN+n≤γN(sN+n−tN+n) (2.34)

and

0< sN+1+n−tN+1+n≤αN(sN+n−tN+n) (2.35)

where γ_N =γ(s_N −t_N),α_N =α(s_N −t_N) and t^??_N = _1−α^1+γN

N(s_N −t_N).

Remark 2.3.

(1) Note that for N = 0, the Lemma 2.2 reduces to Lemma 2.1 with α₀ =α and γ₀ =γ.

(2) Let us define sequences{r_n} and {v_n} by (2.36)

r0 = 0, q0 =η, r1=q0+ K₀(q0−r0)² 2 (1−L3η/2), q₁=r₁+

12L₁(r₁−q₀)²+^6L_1−L^{2L02(q0−r0)3}

0r0/2 + 7M₀(q₀−r₀)³

24(1−L₃r₁) ,

r_n+1 =q_n+ L(q_n−r_n)² 2^h1−^L₂⁰(q_n+r_n)ⁱ

,

q_n+1=r_n+1+

12L(rn+1−qn)²+_1−L^{6L2(qn−rn)3}

0(rn+qn)/2 + 7M(r_n−qn)³

24(1−L0rn+1) (n≥1)

for someL₀,L₁,L₂,L₃,K₀,M₀ such that

(2.37) L1 ≤L, L2 ≤L, L⁰₂ ≤L, K₀ ≤L, L3 ≤L0 and M₀ ≤ M and

(2.38)

v₀= 0, u₀=η, v_n+1=u_n+ L(u_n−v_n)² 2^h1−^L₂⁰(vn+un)ⁱ

,

un+1 =vn+1+

12L(vn+1−un)²+ ^{6L2(un−vn)3}

1−^L0

2 (vn+un)+ 7M(u_n−vn)³

24(1−Lv_n+1) .

In view of (1.8), (2.13), (2.36), (2.37)–(2.38) a simple inductive argu- ment shows that

rn≤tn≤vn

(2.39)

qn≤sn≤un, (2.40)

r_n+1−q_n≤s_n+1−t_n≤u_n+1−v_n, (2.41)

q_n+1−r_n+1≤s_n+1−t_n+1≤u_n+1−v_n+1 (2.42)

and

r^?= lim

n→∞rn≤t^? ≤v^? = lim

n→∞vn. (2.43)

Moreover, (2.39)-(2.42) hold as strict inequalities for n ≥ 1 if (1.8) and (2.37) hold as strict inequalities. Sequence {v_n} was shown to be majorizing for the Midpoint method (1.2) provided that (1.3) or (1.4) hold [1, 3, 4, 10–16, 24–26, 28]. We shall prove in the next section that tighter sequences{r_n}and{v_n}are also majorizing for the Midpoint method (1.2). Then, certainly these majorizing sequences also converge under (1.3) or (1.4). However, these sequences converge under the new convergence criteria given in the Lemma 2.1 which can be weaker that (1.3) or (1.4) (see Section 4). In the next Section, we

shall provide the connection of L0, L1, L2, L⁰₂, L3, K₀, M₀ to the equation (1.1) and the Midpoint method (1.2) so that estimates (2.37) are satisfied.

3. SEMI-LOCAL CONVERGENCE OF THE MIDPOINT METHOD

We need the following Ostrowski-type representation for the Midpoint me- thod (1.2).

Lemma3.1. Suppose that the Midpoint method (1.2)is well defined for each n= 0,1,2, . . .. Then, the following identities are true for each n= 0,1,2, . . .

F(x_n+1) = (3.1)

= Z 1

0

F⁰⁰(y_n+θ(x_n+1−y_n))(1−θ)dθ(x_n+1−y_n)² +¹₄

Z 1 0

F⁰⁰(^(xn+y₂ ⁿ⁾ +^θ₂(y_n−x_n))(y_n−x_n)dθF⁰(^xn+y₂ ⁿ)⁻¹×

× Z 1

0

F⁰⁰(xn+^θ₂(yn−xn))dθ(yn−xn)² +

Z 1 0

hF⁰⁰(xn+θ(yn−xn))(1−θ)− ¹₂F⁰⁰(xn+^θ₂(yn−xn))ⁱdθ(yn−xn)². and

x_n+1 − y_n = −¹₂F⁰(^xn+y₂ ⁿ)⁻¹ Z ₁

0

F⁰⁰(x_n + ^θ₂(y_n − x_n))dθ(y_n − x_n)². Proof. The proof of (3.1) can be found in [1–4]. Using (1.2), we get in turn that

x_n+1−y_n=

=F⁰(x_n)⁻¹F(x_n)− F⁰(^xn+y₂ ⁿ)⁻¹F(x_n)

=F⁰(xn)⁻¹− F⁰(^xn+y₂ ⁿ)⁻¹F(xn)

=F⁰(^xn+y₂ ⁿ)^−1hF⁰(^xn+y₂ ⁿ)F⁰(xn)⁻¹− Iⁱ

=F⁰(^xn+y₂ ⁿ)^−1hF⁰(^xn+y₂ ⁿ)− F⁰(x_n)ⁱF⁰(x_n)⁻¹F(x_n)

=F⁰(^xn+y₂ ⁿ)⁻¹ Z ₁

0

F⁰⁰(x_n+θ(^xn+y₂ ⁿ −x_n))(^xn+y₂ ⁿ −x_n)[−(y_n−x_n)]dθ

=−¹₂F⁰(^xn+y₂ ⁿ)⁻¹ Z 1

0

F⁰⁰(xn+^θ₂yn−xn)dθ(yn−xn)².

The proof of the Lemma is complete.

We can show the main semi-local convergence result for the Midpoint me- thod (1.2) under the (H) conditions.

Theorem 3.2. Suppose that the (H) conditions and those of the Lemma 2.1 hold. Moreover, suppose that

(3.2) U(x₀, t^?)⊆D.

Then, sequence {x_n} generated by the Midpoint method (1.2) is well defined, remains in U(x0, t^?) for alln ≥0 and converges to a solution x^? ∈U(x0, t^?) of equation F(x) = 0. Moreover, the following estimates hold

ky_n−xnk ≤sn−tn, (3.3)

kx_n+1−ynk ≤tn+1−sn, (3.4)

kx_n−x^?k ≤t^?−t_n (3.5)

and

ky_n−x^?k ≤t^?−s_n. (3.6)

Furthermore, if there exists R≥t^? such that

(3.7) U(x₀, R)⊆D

and

(3.8) L0

2 (t^?+R)<1, then, the solution x^? is unique in U(x₀, R)

Proof. We shall prove that (3.7) and (3.8) hold using mathematical induc- tion. Using (1.2), (C₂) and (2.13), we get thatky₀−x₀k=kF⁰(x₀)⁻¹F⁰(x₀)k ≤ η=s₀−t₀ ≤t^?. That is (3.3) holds forn= 0 andy₀ ∈U(x₀, t^?). We have by (C₅) and the choice ofη1 that

kF⁰(x₀)^−1hF⁰(^x0+y₂ ⁰)− F⁰(x₀)ⁱk ≤ ^L₂⁰ky₀−x₀k

≤ ^L₂⁰(s0−t0) = ^L₂⁰η <1.

(3.9)

It then follows from (3.9) and the Banach Lemma on invertible operators [23, 32] that

(3.10)

F⁰(^x0+y₂ ⁰)⁻¹ ∈L(Y,X),

F⁰(^x0+y₂ ⁰)⁻¹F⁰(x0)≤ ¹

1−^L₂⁰η. Using (1.2), (2.13), Lemma 3.1 and (3.10) we obtain

kx₁−y0k ≤ ¹₂^Lky0−x0k2

1−^L₂⁰η

≤ ¹₂^L(s0−t0)2

1−^L0

2 η

(3.11) and

(3.12) kx₁−y₀k ≤t₁−s₀ ≤γ(s₀−t₀).

Hence, (3.4) holds for n = 0. We also get that kx₁−x0k ≤ kx₁−y0k+ ky₀−x0k ≤ t1 −s0+s0 −t0 = t1 ≤ t^?, which implies x1 ∈ U(x0, t^?). Let us assume that (3.3), (3.4), y^? ∈ U(x₀, t^?) and x_k+1 ∈ U(x₀, t^?) hold for all k≤n. It follows from the proof of the Lemma 2.1 and (C5) that

F⁰(x₀)⁻¹F⁰(^xk+y₂ ^k)− F⁰(x₀)≤ ^L₂⁰ (kx_k−x₀k+ky_k−x₀k)

≤ ^L₂⁰(t_k+s_k)<1 (3.13)

and

F⁰(x0)⁻¹ F⁰(xk+1)− F⁰(x0)≤L0kx_k+1−x0k

≤L0tk+1<1.

(3.14)

It then follows from (3.13) and (3.14) that

F⁰(^xk+y₂ ^k)⁻¹ ∈L(Y,X), F⁰(xk+1)⁻¹∈L(Y,X),

F⁰(^xk+y₂ ^k)⁻¹F⁰(x0)≤ 1

1− ^L₂⁰(tk+sk), (3.15)

F⁰(x_k+1)⁻¹F⁰(x₀)≤ _1−L¹

0tk+1. (3.16)

Then, we have by (1.2), (C₃), Lemma 3.1, (2.13), (3.15) and the induction hypothesis that

kx_k+1−y_kk ≤ ¹₂ Lky_k−x_kk² 1−^L₂⁰(s_k+t_k)

≤ ^L(sk−tk)2

2

1−^L₂⁰(s_k+t_k) =t_k+1−s_k, (3.17)

which shows (3.4). Moreover, using (1.2), (C₃), (C₄), (2.13), Lemma 3.1, we obtain in turn

Z 1 0

F⁰(x0)^−1hF⁰⁰(xk+θ(yk−xk))(1−θ)−¹₂F⁰⁰(xk+ ^θ₂(yk−xk))ⁱdθ

≤

Z 1 0

F⁰(x0)⁻¹F⁰⁰(xk+θ(yk−xk))− F⁰⁰(xk)dθ

+¹₂

Z ₁

0

F⁰(x₀)^−1hF⁰⁰(x_k)− F⁰⁰(x_k+^θ₂(y_k−x_k))ⁱdθ

≤ M Z 1

0

θ(1−θ)dθky_k−x_kk+^M₄ Z 1

0

θ dθky_k−x_kk

= ^7M₂₄ ky_k−x_kk.

(3.18)

Thus,

F⁰(x₀)⁻¹F(x_k+1) (3.19)

≤ ^L₂kx_k+1−y_kk²+^L₄^{2 1}

1−^L₂⁰(sk+tk)ky_k−x_kk³+^7M₂₄ ky_k−x_kk³

≤ ^L₂(tk+1−sk)²+ ^L2(sk−tk)3

4

1−^L₂⁰(sk+tk)+^7M₂₄ (sk−tk)³ and

ky_k+1−x_k+1k ≤F⁰(x_k+1)⁻¹F⁰(x0)F⁰(x0)⁻¹F(x_k+1)

≤

L(tk+1−s_k)²

2 + ^L2(sk−tk)3

4

1−^L₂⁰(tk+sk) +^7M₂₄(s_k−t_k)³ 1−L0t_k+1

=s_k+1−t_k+1, (3.20)

which shows (3.3). We also have that

ky_k+1−x0k ≤ ky_k+1−xk+1k+kx_k+1−x0k

≤s_k+1−t_k+1+t_k+1−t₀ =s_k+1≤t^?. and

kx_k+2−x₀k ≤ kx_k+2−y_k+1k+ky_k+1−x₀k

≤tk+2−sk+1+sk+1−t0=tk+2≤t^?.

Hence,yk+1 and xk+2 belong inU(x0, t^?). It follows from (3.3), (3.4) and the Lemma 2.1 that sequence {x_n}is complete in a Banach spaceX and as such it converges to somex^? ∈U(x0, t^?) (since U(x0, t^?) is a closed set). By letting k→ ∞in (3.19) we obtain F(x^?) = 0. Estimates (3.5) and (3.6) follows from (3.3) by using standard majorization techniques [23, 32]. Finally to show the uniqueness part. Let y^? ∈ U(x0, R) be a solution of the equation F(x) = 0.

LetQ=^R₀¹F⁰(x^?+θ(y^?−x^?))dθ. Using (C₅), (3.7) and (3.8), we get that

F⁰(x₀)⁻¹Q− F⁰(x₀) ≤

Z ₁

0

kF⁰(x₀)⁻¹F⁰(x^?+θ(y^?−x^?))− F⁰(x₀)dθk

≤L0

Z 1 0

[(1−θ)kx^?−x0k+θky^?−x0k]dθ

≤ ^L₂⁰(t^?+R)<1.

(3.21)

It follows from (3.21) and the Banach lemma on invertible operators [23, 32]

that Q⁻¹ ∈L(Y,X). Then, using the identity

0 =F(y^?)− F(x^?) =Q(y^?−x^?)

we deduce thatx^?=y^?. The proof of the Theorem is complete.

Remark 3.3.

(1) The limit point t^? can be replaced by t^?? (given in closed from by (2.14)) in Theorem 3.2.

(2) The conclusions of Theorem 3.2 hold if hypotheses of Lemma 2.1 are replaced by those of Lemma 2.2.

(3) It follows from the (H) conditions that there exists constantsK₀,L1, L₂,L₃,M₀ satisfying

F⁰(x₀)⁻¹F⁰⁰(x₀+^θ₂(y₀−x₀))≤ K₀ (3.22)

F⁰(x₀)⁻¹F⁰⁰(y₀+θ(x₁−y₀))≤L₁ (3.23)

F⁰(x0)⁻¹F⁰⁰(x0+^θ₂(y0−x0))≤L2

(3.24)

F⁰(x₀)⁻¹F⁰⁰(^x0+y₂ ⁰ +^θ₂(y₀−x₀))≤L⁰₂ (3.25)

F⁰(x₀)^−1hF⁰(^x0+y₂ ⁰)− F⁰(x₀)ⁱ≤ ^L₂³ky₀−x₀k (3.26)

F⁰(x₀)^−1hF⁰⁰(x₀+θ(y₀−x₀))− F⁰⁰(x₀)ⁱ≤ M₀θky₀−x₀k (3.27)

θ = θ or θ/2. For all θ ∈ [0,1], where, y₀ = x₀ − F⁰(x₀)⁻¹F(x₀) and x1 = x0 − F⁰(^x0+y₂ ⁰)⁻¹F(x0). Estimates (3.22) -(3.27) are not additional to the (H) conditions, since in practice the verification of (C₂) - (C₅) requires the computation ofK₀,L1,L2,L3 andM₀. Note that finding these constants only involves computations at the initial data. Moreover, these constants satisfy (2.37). Furthermore, according to the proof of Theorem 3.2, {r_n} is a majorizing sequence for {x_n} which is finer than{t_n}and{v_n}(see also (2.39)-(2.43) and the Tables in the next section).

4. NUMERICAL EXAMPLES

Example 4.1. Let X =Y = R be equipped with the max-norm, x0 = 1, D= [ψ,2−ψ]. Let us define F onD by

(4.1) F(x) =x^m−ψ.

Here, a∈(0,1.0). Through some algebraic manipulations, for the conditions (H), we obtain

η= 1−ψ

m , L= (2−ψ)^m−2(m−1), L₀ = ^(2−ψ)

m−1−1 1−ψ

and M= (m−1)(m−2)(2−ψ)^m−3.

Furthermore, we see that for m= 8 andψ= 0.79 the criteria (1.3) and (1.4) yield

0.026≤0.021 and 0.026≤0.020

respectively. Thus we observe that the criteria (1.3) and (1.4) are not satis- fied. Even though the criteria (1.3) and (1.4) fall short but Midpoint method, starting atx0= 1, converges form= 8 anda= 0.79 as reported in Table 4.1.

Moreover from equations (2.4)–(2.6) we obtain

η₁ = 0.038, η₂ = 0.028, η₃ = 0.027.

n kx_n+1−x_nk kF(x_n)k x_n

0 2.879×10⁻⁰² 2.100×10⁻⁰¹ 1.000×10⁺⁰⁰ 1 2.419×10⁻⁰⁴ 1.576×10⁻⁰³ 9.712×10⁻⁰¹ 2 1.576×10⁻¹⁰ 1.026×10⁻⁰⁹ 9.710×10⁻⁰¹ 3 4.357×10⁻²⁹ 2.836×10⁻²⁸ 9.710×10⁻⁰¹ 4 9.209×10⁻⁸⁵ 5.994×10⁻⁸⁴ 9.710×10⁻⁰¹ 5 8.697×10⁻²⁵² 5.661×10⁻²⁵¹ 9.710×10⁻⁰¹ 6 7.327×10⁻⁷⁵³ 4.769×10⁻⁷⁵² 9.710×10⁻⁰¹ 7 0.000×10⁺⁰⁰ 1.198×10^−2,023 9.710×10⁻⁰¹

Table 4.1. Midpoint method applied to (4.1).

From (2.7), we getη0 =η3 = 0.027. We notice that the assumption (2.12), of Lemma 2.1, holds. That is η = 0.026< η₀ = 0.027. From (3.22)-(3.26), we obtain

K₀ = 7, L1= 7^7+ψ₈ ⁶, L2 = 7, L3 = 6

(2/3+1/3ψ−(−1/3+1/3ψ)²)²−1

k^−1/3+1/3ψ−(−1/3+1/3ψ)²k , M₀ = 7, L⁰₂ = 6^5+ψ₃ .

We can verify that the conditions (2.37) are fulfilled. Additionally, for the sequences {t_n} (given by (2.12)), {r_n} (given by (2.36)) and {v_n} (given by (2.38)), we produce the Table 4.2. In Table 4.2, we observe that the sequence

n tn+1−tn rn+1−rn vn+1−vn

0 3.084×10⁻⁰² 2.909×10⁻⁰² 3.542×10⁻⁰² 1 6.199×10⁻⁰³ 6.515×10⁻⁰⁴ 2.818×10⁻⁰² 2 1.022×10⁻⁰⁴ 7.473×10⁻⁰⁸ −4.567×10⁻⁰³ 3 5.692×10⁻¹⁰ 1.162×10⁻¹⁹ 6.400×10⁻⁰⁴ 4 9.876×10⁻²⁶ 4.366×10⁻⁵⁵ −4.798×10⁻⁰⁷ 5 5.158×10⁻⁷³ 2.317×10⁻¹⁶¹ 2.204×10⁻¹⁶ 6 7.350×10⁻²¹⁵ 3.463×10⁻⁴⁸⁰ −2.134×10⁻⁴⁴ 7 2.126×10⁻⁶⁴⁰ 1.156×10^−1,436 1.937×10⁻¹²⁸ 8 5.147×10^−1,917 0.000×10⁺⁰⁰ −1.450×10⁻³⁸⁰ 9 0.000×10⁺⁰⁰ 0.000×10⁺⁰⁰ 6.081×10^−1,137

Table 4.2. Sequences {tn},{rn} and{vn}.

{r_n} provides tighter error bounds than the sequence {t_n}. The convergence of the sequence{v_n}is not expected, since (1.3) or (1.4) are not satisfied. Note also that{v_n} was essentially used as a majorizing sequence for the Midpoint

method in [1, 3, 4, 10–16, 24–26, 28].