EXPANDING THE CONVERGENCE DOMAIN FOR CHUN-STANICA-NETA FAMILY OF THIRD ORDER METHODS IN BANACH SPACES

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1 J. Korean Math. Soc. 5 (15), No. 1, pp EXPANDING THE CONVERGENCE DOMAIN FOR CHUN-STANICA-NETA FAMILY OF THIRD ORDER METHODS IN BANACH SPACES Ioannis Konstantinos Argyros, Santhosh George, Ángel Alberto Magreñán Abstract. We present a semilocal convergence analysis of a third order method for approximating a locally unique solution of an equation in a Banach space setting. Recently, this method was studied by Chun, Stanica Neta. These authors extended earlier results by Kou, Li others. Our convergence analysis extends the applicability of these methods under less computational cost weaker convergence criteria. Numerical examples are also presented to show that the earlier results cannot apply to solve these equations. 1. Introduction In this study we are concerned with the problem of approximating a locally unique solution x of the equation (1.1) F(x) =, where F is a Fréchet-differentiable operator defined on a convex subset D of a Banach space X with values in a Banach space Y. Many problems in computational mathematics other disciplines can be brought in a form like (1.1) using mathematical modelling [, 4, 1, 15, 17,, 1]. The solutions of these equations can rarely be found in closed form. That is why most solution methods for these equations are usually iterative. In particular the practice of Numerical Functional Analysis for finding such solutions is essentially connected to Newton-like methods [1,, 4, 11, 1, 15, 16, 17, 19,, 1]. The study about convergence of iterative procedures is normally centered on two types: semi-local local convergence analysis. The semilocal convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of the iterative procedures. While the local analysis is based on the information around a solution, to find estimates Received January 3, 14; Revised April 4, Mathematics Subject Classification. 65H1, 65G99, 65K1, 47H17, 49M15. Key words phrases. family of third order method, Newton-like methods, Banach space, semilocal convergence, majorizing sequences, recurrent relations, recurrent functions. 3 c 15 Korean Mathematical Society

2 4 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN of the radii of convergence balls. There exist many studies which deal with the local the semilocal convergence analysis of Newton-like methods such as [1]-[]. Majorizing sequences in connection to the Kantorovich theorem have been used extensively for studying the convergence of these methods [, 3, 4, 7, 1, 17]. Rall [1] suggested a different approach for the convergence of these methods, based on recurrent relations. Cela Marquina [5, 6], Parida [18], Parida Gupta [19], Magreñán [15], Ezquerro Hernández [8], Gutiérrez Hernández [9, 1]. Argyros [, 3, 4] used this idea for several high-order methods. In particular, Kou Li [13] introduced a third order family of methods for solving equation (1.1), when X = Y = R defined by y n = x n θf (x n ) 1 F(x n ) for each n =,1,,... (1.) x n+1 = x n θ +θ 1 θ F (x n ) 1 F(x n ) 1 θ F (x n ) 1 F(y n ), where x is an initial point θ R {}. This family uses two evaluations of F one evaluation of F. Third order methods requiring one evaluation of F two evaluation of F can be found in [, 4, 13, ]. It is well known that the convergence domain of high order methods is in general very small. This fact limits the applicability of these methods. In the present study we are motivated by this fact recent work by Chun, Stanica Neta [7] who provided a semilocal convergence analysis of the third order method (1.) in a Banach space setting. Their semilocal convergence analysis is based on recurrent relations. In Section we show convergence of the third order method (1.) using more precise recurrent relations under less computational cost weaker convergence criterion. Moreover, the error estimates on the distances x n+1 x n, x n x are more precise the information on the location of the solution at least as precise. In Section 3 using our technique of recurrent functions we present a semilocal convergence analysis using majorizing sequence. The convergence criterion can be weaker than the older convergence criteria or the criteria of Section. Numerical examples are presented in Section 4 that show the advantages of our work over the older works.. Semilocal convergence I Let U(w,ρ), U(w,ρ) st for the open closed ball, respectively, with centerw X ofradiusρ >.LetalsoL(X,Y)denotethespaceofbounded linear operators from X into Y. The semilocal convergence analysis of third order method (1.), given by Chun, Stanica Neta [7] is based on the following conditions. Suppose: (C): (1) There exists F (x) F (y) K x y for each x y D; () F (x) M for each x D; (3) F (x ) 1 β;

3 EXPANDING THE CONVERGENCE DOMAIN 5 (4) F (x ) 1 F(x ) η. They defined certain parameters sequences by We suppose (C ): a = Kβη, α = θ +θ θ θ, γ = M βη, a = b = 1, d = α+γ, b 1 =, a n a n+1 =, 1 aa n d n b n+1 = a n+1 βηc n, k n = 1+θ (θ 1) + 1 θ θ b n + M a nβb nη, c n = M k n +K θ b n k n + M θ 1 b n d n+1 = αb n+1 +γa n+1 b n+1. (1) F (x ) 1 (F (x) F (y)) K x y for each x,y D; () F (x ) 1 (F (x) F (x )) K x x for each x D; (3) F (x ) 1 F(x ) η. Notice that the new conditions are given in affine invariant form the condition on the second Fréchet-derivative has been dropped. The advantages of presenting results in affine invariant form instead of non-affine invariant form are well known [, 4, 1, 17, ]. If operator F is twice Fréchet differentiable, then (1) in (C ) implies () in (C). In order for us to compare the old approach with the new, let us rewrite the conditions (C) in affine invariant form. We shall call these conditions again (C). (C 1 ) F (x ) 1 (F (x) F (y)) K x y for each x y D; (C ) F (x ) 1 F (x) M for each x D; (C 4 ) F (x ) 1 F(x ) η. The parameters sequences are defined as before but β = 1. Then, we can certainly set K = M. Define parameters a = Kη, α = α, γ = K η, a = b = 1, d = α +γ, b 1 =,

4 6 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN We have that a n+1 = 1 1 K (d n +d n 1 + +d ), b n+1 = a n+1 ηc n, c n = K[(k n ) + θ b n k n + θ 1 (b n ) ], kn = θ+1 (θ 1) + 1 θ θ b n + K a n (b n ) η d n+1 = α b n+1 +γ a n+1(b n+1). (.1) K K holds in general K K can be arbitrarily large []-[4]. Notice that the center Lipschitz condition is not an additional condition to the Lipschitz condition, since in practice the computation of K involves the computation of K as a special case. We have by the definition of a n+1 in turn that Hence, we deduce that a n a n+1 = 1 Kηa n d n a n = a 1 Kηd n 1 n 1 Kηa n 1d n 1 = a n(1 Kηa n 1 d n 1 ) 1 Kηa n 1 (d n +d n 1 ) a n 1 1 Kηa = n 1d n 1 (1 Kηa n 1 d n 1 ) 1 Kηa n 1 (d n +d n 1 ) a n 1 = 1 Kηa n 1 (d n +d n 1 ). a = 1 Kηa n 1 (d n +d n 1 + +d ) 1 = 1 Kη(d n +d n 1 + +d ). (.) a n+1 a n+1 for each n =,1,,... Moreover, strict inequality holds in (.) if K < K. Hence, using a simple inductive argument we also have that (.3) (.4) (.5) b n+1 b n+1, c n c n, k n k n

5 EXPANDING THE CONVERGENCE DOMAIN 7 (.6) d n+1 d n+1. Lemma.1. Under the (C ) conditions the following hold F (x n ) 1 F (x ) a n, F (x n ) 1 F(x n ) b nη, x n+1 x n d nη, x n+1 y n (d n +k n 1 +θb n)η. Moreover, under the (C) conditions the following hold F (x n ) 1 F (x ) a n a n, F (x n ) 1 F(x n ) b nη b n η, x n+1 x n d nη d n η, x n+1 y n (d n +k n 1 +θb n)η (d n +k n 1 +θb n )η. Proof. It follows from the proof of Lemma 1 in [7] by simply noticing: the expressions involving (i) the second Fréchet-derivative F (x n +t(y n x n ))(1 t)(y n x n ) dt F (y n +t(x n+1 y n ))(1 t)(x n+1 y n ) dt are not needed can be replaced, respectively, by [F (y n +t(x n y n ) F (x n )](y n x n )dt [F (y n +t(x n+1 y n ) F (y n )](x n+1 y n )dt. Hence, condition () in (C) is not needed can be replaced by condition (1) in (C ) to produce the same bounds as in [7] (for K = M) (see also the proof of Theorem 3. that follows). (ii) The computation of the upper bounds on F (x n ) 1 F (x ) in [7] uses condition (1) in (C) the estimate F (x n ) 1 (F (x n ) F (x n+1 )) F (x n ) 1 F (x ) K x n x n+1 to arrive at (.7) F (x n ) 1 F (x ) a n+1, whereas we use () in (C ) estimate F (x ) 1 (F (x n ) F (x n+1 )) K x n+1 x n

6 8 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN to arrive at the estimate K ( x n+1 x n + + x 1 x ) K (d n +d n 1 + +d ) (.8) F (x n ) 1 F (x ) a n+1, which is more precise (see also (.)). Lemma.. Suppose that (.9) a 1 b 1 < 1. Then, sequence {p n } defined by p n = a n b n is decreasingly convergent to such that p 1 n+1 ξ1 n+1, ξ 1 := a ξ 1b 1 1 d n (α +γ 1 )ξ1 n. ξ 1 Moreover, if (.1) a 1 b 1 < 1, then, sequence {p n } defined by p n = a n b n is also decreasingly convergent to such that p n+1 p n+1 ξ n+11 ξ, ξ = a 1b 1, d n d n (α+γ)ξ n1 ξ, ξ 1 ξ. Proof. It follows from the proof of Lemma 3 in [7] by simply using {p n}, a 1, b 1, ξ 1 instead of {p n }, a 1, b 1, ξ, respectively. Next, we present the main semilocal convergence result for the third order method (1.) under the (C ) conditions, (.9) the convergence criterion (.11) a(α+γ) < 1. The proof follows from the proof of Theorem 5 in [7] (with the exception of the uniqueness of the solution part) by simply replacing the (C) conditions (.1) by the (C ) conditions (.9) respectively. Theorem.3. Suppose that conditions (C ), (.9) (.11) hold. Moreover, suppose that (.1) U = U(x,r η) D, where (.13) r = d n. n=

7 EXPANDING THE CONVERGENCE DOMAIN 9 Then, sequences {x n } generated by the third order method (1.) is well defined, remains in U for each n =,1,,... converges to a unique solution x of equation F(x) = in U(x, K r η) D. Moreover, the following estimates hold (.14) x n+1 x k=n+1 d kη α+γ ξ 1 η k=n+1 ξ k 1. Proof. As already noted above, we only need to show the uniqueness part. Let y U(x, K r η) be such that F(y ) =. Define Q = F (x + t(y x ))dt. Using condition () in (C ) we get in turn that F (x ) 1 (F (x ) Q) K x +t(y x ) x dt K [(1 t) x x +t y x ]dt < K (.15) [r η + r η] = 1. K It follows from (.15) the Banach lemma on invertible operators [, 4, 1, 17, ] that Q 1 L(Y,X). Then, using the identity we deduce that x = y. = F(x ) F(y ) = Q(x y ), Remark.4. If K = K, operator F is twice Fréchet differentiable, then Lemma.1, Lemma. Theorem.3 reduce to Lemma 1, Lemma 3 Theorem 5 in [7], respectively. Otherwise, i.e., if K < K or if the twice Fréchet differentiability of operator F is not assumed, then our results constitute an improvement. It is worth noticing that if K < K, then (.1) implies (.9) (but not necessarily vice versa) ξ 1 < ξ. 3. Semilocal convergence II We need to introduce some scalar sequences that shall be shown to be majorizing for the third order methods (1.) in Theorem 3.. Let K >,K >,η > θ R {}. Set t = s = θ η. Define polynomials f g by (3.1) f(t)( K θ (3.) +K )t 3 + θ Kt +K( θ 1 θ )t K θ 1 θ θ g(t)=k t 4 + K θ [1+ 1 θ (1+ 1 θ )]t 3 + K θ [ 1 θ (1+ 1 θ ) 1]t + K θ 1 θ (1+ 1 θ )( θ 1 θ 1)t K θ 4 1 θ 1 θ (1+ 1 θ ).

8 3 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN We have f() = K θ 1 θ < for θ ±1 f(1) = K > for K. It follows from the intermediate value theorem that polynomial f has roots in (,1). Denote by δ f the smallest root of f in (,1). Similarly, we have g() = K θ 1 θ θ 1 (1+ 1 θ ) < for θ ±1 g(1) = K 4 + K θ >. Denote by δ g the smallest root of g in (,1). Set (3.3) δ = min{δ f,δ g }. Moreover, suppose that δ satisfies (3.4) 1 θ θ 3 (1+ 1 θ )+ Kη θ δ, [ K θ θ 1 (3.5) < 1 K (1+δ)s θ (3.6) < ] + δ +δ (s t ) δ K θ (1 K (1+δ)s ) { 1 θ (1+ 1 θ ) [ θ 1 θ + ]+ δ +δ δ }(s t ) δ. We shall assume from now on that δ satisfies conditions (3.3)-(3.6). These conditions shall be referred to as the ( ) conditions. Moreover, define scalar sequences {t n }, {s n } by t =, s = t +θη, [ 1 θ t 1 = s + θ 3 for each n =,1,,... (3.7) K θ s n+1 = t n K t n+1 (1+ 1 θ )+ (s t )K θ [ 1 θ θ (s n t n ) + (t n+1 s n ) K ] (s t ) ] +(s n t n )(t n+1 s n ), { t n+ = s n θ (1+ 1 θ θ ) (1 K t n+1 ) [ 1 θ θ (s n t n ) + (t n+1 s n ) ] +(s n t n )(t n+1 s n ) (3.8) + 1 (s n+1 t n+1 ) }. Then, we can show the following auxiliary result for majorizing sequences {t n }, {s n } under the ( ) conditions.

9 EXPANDING THE CONVERGENCE DOMAIN 31 Lemma 3.1. Suppose that the ( ) conditions hold. Then, sequence {t n }, {s n } defined by (3.7) (3.8) are increasingly convergent to their unique least upper bound denoted by t which satisfies (3.9) θη t t := θη 1 δ. Moreover, the following estimates hold for each n =,1,,... (3.1) < s n t n δ n θη (3.11) < t n+1 s n δ n+1 θη. Proof. We shall show estimates (3.1) (3.11) using induction. If n =, (3.1) holds by the definition of t s, whereas (3.11) holds by (3.4). We then have that (3.1) t 1 s +δs = (1+δ)s = 1 δ 1 δ s < t. If n = 1, estimates (3.1) (3.11) hold by (3.5), (3.6), (3.1) (3.1), (3.11) for n =. Suppose that (3.1) (3.11) hold for all m n. Then, we have that (3.13) t m+1 s m +δ m+1 (s t ) t m +δ m (s t ) +δ m+1 (s t ) t +(s t )+δ(s t ) + +δ m+1 (s t ) = 1 δm+ (s t ) < t. 1 δ Next, we shall show (3.1) for m + 1 replacing n. We have by the induction hypotheses (3.13) that s m+1 t m+1 K θ 1 δ 1 K m+ 1 δ [ θ 1 θ (δ m (s t )) + (δm (s t )) ] +δ m+1 (s t ) must be smaller or equal to δ m+1 (s t ), or [ K θ θ 1 (3.14) θ δ m + δm+ 1 K 1 δ m+ 1 δ +δ m+1 ](s t ) δ. Estimate (3.14) motivates us to define recurrent polynomials f m on (,1) by [ θ f m (t) = K tm+ + θ t m+1 + θ 1 t ](s m t ) θ (3.15) +K t(1+t+ +t m+1 )(s t ) t.

10 3 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN We need a relationship between two consecutive polynomials f m. Using (3.15) (3.1) by direct algebraic manipulation we get that (3.16) f m+1 (t) = f m (t)+f(t)t m 1 (s t ). Evidently, condition (3.14) is satisfied, if (3.17) f m (δ). We also have from (3.17) that (3.18) f m+1 (δ) f m (δ), since f(δ). It then, follows from (3.17) (3.18) that (3.17) holds, if (3.19) f (δ), which is true by (3.5). Hence, we showed (3.1) for m+1 replacing n. Next, we shall show (3.11) for m+1 replacing n. We have in turn that s m+ s m+1 K θ 1 δ (1 K m+ 1 δ ) { 1 θ (1+ θ 1 ) [ θ 1 θ (δ m (s t )) + (δm+1 (s t )) ] +δ m+1 (s t ) +(δ m+1 (s t )) } must be smaller or equal to δ m+ (s t ). As in the preceding case we are motivated to define polynomials g m on [,1] by { 1 θ (1+ θ [ 1 ) θ ] } 1 g m (t) = K θ θ t m + tm+ +t m+1 + tm+ θ (3.) (s t )+t K (1+t+ +t m+1 )(s t ) t. Using (3.) (3.) by direct algebraic manipulation we get that (3.1) g m+1 (t) = g m (t)+g(t)t m (s t ). Condition (3.11) is satisfied, if (3.) g m (δ). We also have from (3.1) ( ) that (3.3) g m+1 (δ) g m (δ), since g(δ). Hence, (3.) is satisfied, if (3.4) g (δ), which is true by (3.6). The induction for (3.11) is completed. It then, follows that (3.5) t m+ 1 δm+3 s < t. 1 δ Hence, sequences {t n }, {s n } are increasing, bounded above by t as such they converge to their unique least upper bound t which satisfies (3.9).

11 EXPANDING THE CONVERGENCE DOMAIN 33 We can show the main semilocal convergence result for the third order method (1.) under the (C ) ( ) conditions using {t n } {s n } as majorizing sequences. Theorem 3.. Suppose that (3.6) U(x,t ) D, the (C ) ( ) conditions hold. Then, sequences {x n }, {y n } generated by the third order method (1.) are well defined, remain in U(x,t ) for each n =,1,,... converge to a unique solution x of equation F(x) = in U(x,t ) D. Moreover the following estimates hold for each n =,1,,... (3.7) (3.8) (3.9) y n x n s n t n, x n+1 y n t n+1 s n, x n+1 x n t n+1 t n, (3.3) x n x t t n. Furthermore, if there exists R > t such that (3.31) K (t +R) <, then, the point x is the only solution of equation F(x) = in U(x,R). Proof. We shall first show (3.7) (3.8) using induction. We have by (1.) (3.7) that y x = θ F (x ) 1 F(x ) θ η = s = s t. Hence, (3.7) holds for n =. It follows from the first substep of (1.) that (3.3) F(y ) = F(y ) θf(x ) F (x )(y x ) = (1 θ)f(x )+ [F (x +t(y x )) F (x )](y x )dt. Composing (3.3) by F (x ) 1 using (), (3) in (C ) (3.7) (3.33) F (x ) 1 F(y ) 1 θ F (x ) 1 F(x ) + [F (x +t(y x )) F (x )](y x )dt 1 θ (s t )+ K θ y x ( 1 θ θ + K (s t ))(s t ). Subtracting the first from the second substep in (1.) we get that (3.34) x 1 y = (θ +1)(θ 1) θ F (x ) 1 F(x ) 1 θ F (x ) 1 F(y ).

12 34 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN Hence, using (3.33) (3.34), we get that θ+1 θ 1 x 1 y = F (x ) 1 F(x ) + 1 θ F (x ) 1 F(y ) (3.35) θ θ+1 θ 1 θ (s t )+ 1 θ ( 1 θ θ = t 1 s, + K (s t ))(s t ) which shows (3.8) for n =. Then, (3.9) holds for n =, since x 1 x x 1 y + y x t 1 s +s t = t 1 t t. Then, we have x 1 U(x,t ). Notice that K t < 1 from the proof of Lemma 3.1. Let us suppose x U(x,t ). Then, using () in (C ) we have that (3.36) F (x ) 1 (F (x) F (x )) K x x K t < 1. It follows from (3.36) the Banach lemma that F (x) 1 L(Y,X) (3.37) F (x 1 ) 1 F (x ) 1 1 K x 1 x 1. 1 K t 1 Suppose that (3.7)-(3.9) hold for all m n x m U(x,t ). Using the first step in (1.) we get that F(y m ) = F(y m ) θf(x m ) F (x m )(y m x m ) (3.38) = (1 θ)f(x m )+ [F (x m +t(y m x m )) F (x m )](y m x m )dt. Subtracting the first step in (1.) from the second step to obtain (3.39) F (x m )(x m+1 y m ) = θ3 θ θ +1 F(x m ) 1 θ F(y m). We also have by (3.38) that F(x m+1 ) = F (x m )(x m+1 y m )+F(y m )+[F (y m ) F (x m )](x m+1 y m ) (3.4) +F(x m+1 ) F(y m ) F (y m )(x m+1 y m ) = 1 θ θ F(x m ) 1 θ F(y m) + + [F (x m +t(y m x m )) F (x m )](y m x m )dt θ [F (y m +t(x m+1 y m )) F (y m )](x m+1 y m )dt +[F (y m ) F (x m )](x m+1 y m ). Hence, we get by (3.4) that F (x ) 1 F(x m+1 ) K [ θ 1 θ y m x m

13 (3.41) Then, we get that EXPANDING THE CONVERGENCE DOMAIN 35 K + x m+1 y m [ θ 1 θ (s m t m ) + (t m+1 s m ) ] + y m x m x m+1 y m ] +(s m t m )(t m+1 s m ). y m+1 x m+1 F (x m+1 ) 1 F (x ) F (x ) 1 F(x m+1 ) [ K θ 1 1 K t m+1 θ (s m t m ) + (t m+1 s m ) ] +(s m t m )(t m+1 s m ) = s m+1 t m+1, where, we used (3.37) for x = x m+1 x m+1 x x m+1 x m + + x 1 x t m+1 t m + +t 1 t = t m+1. Hence, we showed (3.7). Then, we have by (3.39) that (3.4) x m+1 y m = θ3 θ θ +1 θ F (x m ) 1 F(x m ) 1 θ F (x m ) 1 F(y m ). It follows from (3.4) that x m+ y m+1 K 1+θ (θ 1) [ 1 K t m+1 θ F (x ) 1 F(x m+1 ) + 1 θ F (x ) 1 F(y m+1 ) ] K θ (1 K t m+1 ) + (t m+1 s m ) + 1 θ ( θ 1 θ (s m t m ) [ 1+θ (θ 1) ( θ 1 θ (s m t m ) +(s m t m )(t m+1 s m )) + (t m+1 s m ) +(s m t m )(t m+1 s m ))) + (s m+1 t m+1 ) ] = t m+ s m+1. Hence, we showed (3.8). Then, we have that x m+ x m+1 x m+ y m+1 + y m+1 x m+1 t m+ s m+1 +s m+1 t m+1

14 36 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN = t m+ t m+1, which shows (3.9). We also have that x m+ x x m+ x m+1 + x m+1 x m + + x 1 x t m+ t m+1 +t m+1 t m + +t 1 t = t m+ < t. Hence, we get x m+ U(x,t ). We showed in Lemma 3.1 that sequences {t n }, {s n } are complete. Hence, it follows from (3.7)-(3.9) that sequences {x n }, {y n } are complete in a Banach space X as such they converge to some x U(x,t ) (since U(x,t ) is a closed set.) By letting m in (3.41), we obtain F(x ) =. Estimate (3.3) follows from (3.9) by using stard majorization techniques [, 4, 1, 17,, 1]. Let us show uniqueness, first in U(x,t ) D. Let y U(x,t ) be such that F(y ) =. Set Q = F (x +t(y x ))dt. Then, using () in (C ) we get that F (x ) 1 (F (x ) Q) K [(1 t) x +t(y x ) x dt K [(1 t) x x +t y x ) x dt K t < 1. It follows that Q 1 exists. Then, from the identity = F(x ) F(y ) = Q(x y ) we deduce that x = y. Similarly, if F(y ) = y U(x,R), we have that F (x ) 1 (F (x ) Q) K (R+t ) < 1, by (3.31). Hence, again we deduce that x = y. Remark 3.3. (a) It follows from the proof of Theorem 3. that sequences { t n },{ s n } defined by t =, s = t +θη, t 1 = s +[ 1 θ θ 3 (1+ 1 θ )+ ( s t )K θ ]( s t ), θ s 1 = t 1 + [ K θ 1 1 K t 1 θ ( s t ) K ( t 1 s ) +K ( s t )( t 1 s )], θ s n+1 = t n+1 + [ θ 1 1 K t n+1 θ ( s n t n ) ( t n+1 s n ) +( s n t n )( t n+1 s n )], K t n+ = s n+1 + θ (1 K t n+1 ) { 1 θ (1+ 1 θ )[ θ 1 θ ( s n t n ) ( t n+1 s n ) +( s n t n )( t n+1 s n )] 1 ( s n+1 t n+1 ) } for each n =,1,,...

15 EXPANDING THE CONVERGENCE DOMAIN 37 Then, a simple induction argument shows that s n s n, t n t n, s n t n s n t n, t n+1 s n t n+1 s n, t = lim t n t. n Clearly, { t n }, { s n }, t can replace {t n }, {s n }, t in Theorem 3.. (b) The limit point t can be replaced by t given in closed form by (3.9). (c) Criteria ( ) or (.9) (.11) are sufficient for the convergence of the third order method (1.). However, these criteria are not also necessary. In practice, we shall test to see which of these criteria are satisfied (if any) then use the best possible error bounds uniqueness results (see also the numerical examples in the next section). 4. Numerical examples Example 4.1. Let x D,X = Y = R, x = 1 D = U(1,1). Define function F on D by (4.1) F(x) = x Then, we get that β = 1, η =.17, M = 1. 3 Now choosing θ = 1.15 we obtain that a =.68, α =.68, γ =.34 as a consequence a 1 b 1 = condition (.9) is violated. Hence, there is no guarantee under the conditions given in [7] that sequence {x n } converges to x. Calculating now δ f δ g, the smallest solutions of the polynomials f(t) g(t) given in (3.1) (3.) respectively between 1, we obtain that δ = min{δ f,δ g } = Moreover, we observe that the conditions are satisfied since 1 θ θ 3 (1+ 1 θ )+ Kη =.7861 δ, θ [ K θ θ ] 1 < 1 K (1+δ)s θ + δ +δ (s t ) =.3634 δ [ K θ 1 < θ (1 K (1+δ)s ) { 1 θ (1+ 1 θ ) θ + ]+ δ +δ δ }(s t )

16 38 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN = = δ. Consequently, convergence to the solution is guaranteed by Theorem 3.. Moreover, the computational order of convergence (COC) is shown in Table 1. Here (COC) is defined by ( xn+1 x ) ( xn x ) ρ ln x n x /ln x n 1 x, n N, Table 1 shows the (COC). Table 1. COC for Example 1 using θ = n COC ρ = 3. Example 4.. Let X = Y = C[,1], the space of continuous functions defined in [,1] equipped with the max-norm. Let Ω = {x C[,1]; x R}, such that R > 1 F defined on Ω given by F(x)(s) = x(s) f(s) λ G(s,t)x(t) 3 dt, x C[,1], s [,1], where f C[,1] is a given function, λ is a real constant the kernel G is the Green function { (1 s)t, t s, G(s,t) = s(1 t), s t. In this case, for each x Ω, F (x) is a linear operator defined on Ω by the following expression: [F (x)(v)](s) = v(s) 3λ G(s,t)x(t) v(t) dt, v C[,1], s [,1]. If we choose x (s) = f(s) = 1, it follows I F (x ) 3 λ /8. Thus, if λ < 8/3, F (x ) 1 is defined Moreover, F (x ) λ. F(x ) λ 8, F (x ) 1 F(x ) λ 8 3 λ.

17 EXPANDING THE CONVERGENCE DOMAIN 39 On the other h, for x,y Ω we have [(F (x) F (y))v](s) = 3λ for x Ω we get in turn that Consequently, F (x) 6 λ 8. F (x) F (y) x y 3 λ ( x + y ) 8 F (x) F (1) x λ ( x +1) 8 Choosing λ = 1.5, R = 4.4 θ = 1.1 we have β = , η =.17119, M = 4.95, a =.466, α = , γ = G(s,t)(x(t) y (t))v(t) dt x y 6R λ, 8 x 1 1+3(1+R) λ. 8 So, as a 1 b 1 = , condition (.9) is violated. Hence, there is no guarantee under the conditions given in [7] that sequence {x n } converges to x. Calculating now δ f δ g, the smallest solutions of the polynomials f(t) g(t) given in (3.1) (3.) respectively between 1, we obtain that δ = min{δ f,δ g } = Moreover, we observe that the conditions are satisfied since 1 θ θ 3 (1+ 1 θ )+ Kη = δ, θ [ K θ θ ] 1 < 1 K (1+δ)s θ + δ +δ (s t ) = δ [ K θ 1 < θ (1 K (1+δ)s ) { 1 θ (1+ 1 θ ) θ + ]+ δ +δ δ }(s t ) = = δ. Consequently, convergence to the solution is guaranteed by Theorem 3.. Acknowledgement. The research is partly supported by the grant MTM C-1.

18 4 I. K. ARGYROS, S. GEORGE, AND Á. A. MAGREÑÁN References [1] S. Amat, Á. A. Magreñán, N. Romero, On a two-step relaxed Newton-type method, App. Math. Comput. 19 (13), no. 4, [] I. K. Argyros, Computational theory of iterative methods, Series: Studies in Computational Mathematics, 15, Editors: C. K. Chui L. Wuytack, Elsevier Publ. Co. New York, U.S.A, 7. [3] I. K. Argyros S. Hilout, Weaker conditions for the convergence of Newton s method, J. Complexity 8 (1), no. 3, [4], Computational Methods in Nonlinear Analysis, World Scientific Publ. Comp., New Jersey, 13. [5] V. Cela A. Marquina, Recurrence relations for rational cubic methods I: The Halley method, Computing 44 (199), no., [6], Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (199), no. 4, [7] C. Chun, P. Stanica, B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (11), no. 6, [8] J. A. Ezquerro M. A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (), no., [9] J. M. Gutiérrez M. A. Hernández, Recurrence relations for the super-halley method, Computers Math. Appl. 36 (1998), no. 7, 1 8. [1], Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 8 (1997), no. 1-, [11] J. M. Gutiérrez, Á. A. Magreñán, N. Romero, On the semilocal convergence of Newton-Kantorovich method under center-lipschitz conditions, Appl. Math. Comput. 1 (13), [1] L. V. Kantorovich G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 198. [13] J. S. Kou T. Li, Modified Chebyshev s method free from second derivative for nonlinear equations, Appl. Math. Comput. 187 (7), no., [14] J. S. Kou, T. Li, X. H. Wang, A modification of Newton method with third-order convergence, Appl. Math. comput. 181 (6), no., [15] Á. A. Magreñán, Estudio de la dinámica del método de Newton amortiguado, PhD Thesis, Servicio de Publicaciones, Universidad de La Rioja, 13. [16] Á. A. Magreñán I. K. Argyros, Two-step Newton methods, J. Complexity 3 (14), no. 4, [17] J. M. Ortega W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic press, New York, 197. [18] P. K. Parida, Study of some third order methods for nonlinear equations in Banach spaces, Ph.D. Dessertation, Indian Institute of Technology, Department of Mathematics, Kharagpur, India, 7. [19] P. K. Parida D. K. Gupta, Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces, J. Math. Anal. Appl. 345 (8), [] F. A. Potra V. Ptrǎ, Nondiscrete induction iterative processes, in Research Notes in Mathematics, Vol. 13, Pitman, Boston, [1] L. B. Rall, Computational solution of nonlinear operator equations, Robert E. Krieger, New York, [] Q. Wu Y. Zhao, Third order convergence theorem by using majorizing function for a modified Newton method in Banach space, Appl. Math. Comput. 175 (6), no.,

19 EXPANDING THE CONVERGENCE DOMAIN 41 Ioannis Konstantinos Argyro Department of Mathematical Sciences Cameron University Lawton, OK 7355, USA address: Santhosh George Department of Mathematical Computational Sciences National Institute of Technology Karnataka, India address: Ángel Alberto Magreñán Departamento de TFG/TFM Universidad Internacional de La Rioja (UNIR) Logroño, Spain address: