Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics, 63-76

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1 Bulletin of the Transilvania University of Braşov Vol 1(59), No Series III: Mathematics, Informatics, Physics, A CONVERGENCE ANALYSIS OF THREE-STEP NEWTON-LIKE METHOD UNDER WEAK CONDITIONS IN BANACH SPACES Nazli KARACA 1 and Isa YILDIRIM 2 Abstract In this paper, we present a local and semi-local convergence analysis for three-step Newton-like method in order to approximate a solution of nonlinear equations in a Banach space. The convergence ball and error estimates are given for these methods. Also, we give some numerical examples for this study. Our results are generalizations of the analogous ones recently proved by Sahu et al. [3]. 2 Mathematics Subject Classification: 47H1, 49M15, 65K1. Key words: Newton-like method, iterative methods, Fréchet differentiable, nonlinear equations. 1 Introduction Throughout this paper, N will denote the set of all positive integers and B(X, Y ) will designate the space of all bounded linear operators from X to Y. Also, for given any x X and r >, B r [x] = {y X : y x r} and B r (x) = {y X : y x < r}. One of the most important problems in analysis is the solution of nonlinear equation F (x) =. (1) in a Banach space X, where F is a Fréchet differentiable operator on some open convex subset C of X with values in another Banach space Y. To solve these equations we can use different iterative methods. One of them is the Newton method defined by x n+1 = x n [ F (x n ) ] 1 F (xn ), n N, (2) 1 Corresponding author, Faculty of Sciences, Ataturk University, TURKEY, nazli.kadioglu@atauni.edu.tr 2 Faculty of Sciences, Ataturk University, TURKEY, isayildirim@atauni.edu.tr

2 64 Nazli Karaca and Isa Yildirim where x is an initial point, F (x) denotes the Fréchet derivative of F at the point x C. Argyros [1], Dennis [2] and Rheinboldt [6] considered the following modified Newton method to avoid the inverse of derivative which is required at each step in Newton method (2), x n+1 = x n [ F (x ) ] 1 F (xn ), n N. (3) There are two types of the convergence analysis of iterative methods, semilocal and local convergence analysis. The semi-local convergence analysis is, based on the information around an initial point while the local one is, based on the information around a solution. In 212, Sahu et al. [3] introduced the S-iteration process of Newton-like type for finding the solution of operator equation (1) as follows: Let x X nad α (, 1). The sequence {x n+1 } is defined by; x n+1 = z n [F (x )] 1 F (z n ) z n = (1 α)x n + αy n, y n = x n [F (x )] 1 F (x n ), n N. Moreover, he showed that the rate of convergence of this iterative process is faster than (3). Let x C be a solution of (1) such that [F (x )] 1 B(Y, X). For some L,, L 2 >, x C and x C, suppose that [F (x )] 1 and F satisfy the following conditions: F (x) F (x ) L x x, (5) [ F (x ) ] 1 (F (x) F (x )) x x, (6) and [ F (x ) ] 1 (F (x) F (x )) L 2 x x. (7) In 212, Sahu et al. [3] proved some theorems on the semi-local and local convergence of the iterative method (4) to solve the operator equation (1). Recently, Karaca and Yildirim [1] introduced iteration process (8) for the class of nonexpansive mapping in Banach spaces and they proved that this iteration process is faster than Picard iteration, S and normal S-iterations [7]. Let X be a normed space, C a nonempty convex subset of X and T : C C an operator. Then, for arbitrary x C x n+1 = T y n, y n = (1 α n )z n + α n T z n, z n = (1 β n )x n + β n T x n, n N where {α n } and {β n } are sequences in (, 1). (4) (8)

3 Local and semi-local convergence of Newton-like method 65 Now, we introduce iteration process (8) of Newton-like type in order to find the solution of operator equation (1). Let C be an open convex subset of a Banach space X. Assume that F : X Y is a Frechet differentiable operator where Y is a Banach space. Let α, β (, 1). Starting with x X and {x n+1 } is defined by: x n+1 = y n [F (x )] 1 F (y n ) y n = z n α [F (x )] 1 F (z n ) z n = x n β [F (x )] 1 F (x n ), n N, which can be written in the following compact form: x n+1 = y n [ [F (x )] 1 F (y n ) ] y n = (1 α)z n α z n [F (x )] 1 F (z n ) [ ] z n = (1 β)x n + β x n [F (x )] 1 F (x n ), n N. (9) The aim of this paper is to prove semi-local and local convergence analysis of iteration process (9). It is shown that the rate of convergence of (9) is faster than (4). Applications to numerical examples are included. Definition 1. Let C be a nonempty subset of normed space X. T : C X is called: A mapping (i) Contraction if there exists a constant L (, 1) such that T (x) T (y) L x y for all x, y C. (ii) Quasi-contraction [8] if there exists a constant L (, 1) and F (T ) = = {x C : T (x) = x} such that T (x) p L x p for all x C and p F (T ). (iii) Lipschitzian if there exists a constant L > such that T (x) T (y) L x y for all x, y C. In the sequel, we also need the following lemmas. Lemma 1. [4] Let T be a bounded linear operator on a Banach space X. Then the following are equivalent: (a) There is a bounded linear operator P on X such that P 1 exists, and P T < 1 P 1. (b) T 1 exists. Moreover, if T 1 exists, then T 1 P P 1 T P 1 1 P 1 P T.

4 66 Nazli Karaca and Isa Yildirim Lemma 2. [5] Let X and Y be Banach spaces, C be a nonempty open convex subset of X and F : X Y be a Fréchet differentiable operator. Assume that x C is a solution of (1) such that [F (x )] 1 B(Y, X) and the operator F satisfies the conditions (7). Suppose that B r (x ) C, where r = 1 L 2. Then, for any x B r (x ), F (x ) is invertible, and the following inequality holds: ( [ F (x ) ] 1 F (x )) L 2 x x. Lemma 3. [9] Let X and Y be Banach spaces, C be a nonempty open convex subset of X and F : X Y be a Fréchet differentiable operator. Then, for all x, y C, F (x) F (y) = F (y + t(x y)) (x y)dt. 2 Main results Before studying convergence analysis of (9), we will give the following theorem. Let the operators T, S α, S β : B r [x ] X be defined as follows: for all x B r [x ] and for fixed α, β (, 1), T (x) = x [ F (x ) ] 1 F (x) (1) S α (x) = x α [ F (x ) ] 1 F (x) S β (x) = x β [ F (x ) ] 1 F (x). Theorem 1. Let F be a Fréchet differentiable operator defined on an open convex subset C of a Banach space X with values in a Banach space Y. Suppose that [F (x )] 1 B(Y, X) for some x C and the operator F satisfies the conditions (5) and [F (x )] 1 F (x ) k and [F (x )] 1 l for some k, l >. Moreover assume that γ = kll < 1 2 and B r[x ] C, where r = 1 1 2γ γ k. Then, (i) The operators (1) are contractions on B r [x ] with constant δ, (1 α (1 δ)) and (1 β (1 δ)) and the operator equation (1) has a uniqe solution in B r [x ]. (ii) The operator G : B r [x ] X, G (x) = T S α S β (x) generated by (1) is a contraction on B r [x ] with constant δ (1 α (1 δ)) (1 β (1 δ)). Proof. (i) Denote by δ := lrl. This implies that δ < 1. For all x, y B r [x ], we obtain that

5 Local and semi-local convergence of Newton-like method 67 S β (x) S β (y) = x y β [ F (x ) ] 1 (F (x) F (y)) = β x y [ F (x ) ] 1 (F (x) F (y)) + (1 β) x y = β x y [ F (x ) ] 1 F (y + t(x y)) (x y)dt + (1 β) x y = β [ F (x ) ] [ 1 1 F (y + t(x y)) (x y)dt F (x ) (x y)dt] + (1 β) x y β [ F (x ) ] 1 1 F (y + t(x y)) (x y)dt F (x ) (x y)dt + (1 β) x y β [ F (x ) ] 1 F (y + t(x y)) (x y) F (x ) (x y) dt + (1 β) x y βl F (y + t(x y)) F (x ) x y dt + (1 β) x y βl L y + t(x y) x x y dt + (1 β) x y βlrl x y + (1 β) x y = (1 β (1 lrl )) x y = (1 β (1 δ)) x y. Since δ < 1, we get that (1 β (1 δ)) < 1 Thus, the operator S β is a contraction. Now we need to show that S β (B r [x ]) B r [x ]. For x B r [x ], we get that = S β (x) x S β (x) S β (x ) + S β (x ) x x x β [ F (x ) ] 1 (F (x) F (x )) + x β [ F (x ) ] 1 F (x ) x β x x [ F (x ) ] 1 (F (x) F (x )) + (1 β) x x + βk β [ F (x ) ] 1 F (x + t(x x )) F (x ) x x dt + (1 β) x x + βk βl L t(x x ) x x dt + (1 β) x x + βk βll x x 2 tdt + (1 β) x x + βk

6 68 Nazli Karaca and Isa Yildirim βlr2 L 2 r, + (1 β) r + βk which means that S β (B r [x ]) B r [x ]. Similarly, we obtain that S α (x) S α (y) (1 α (1 δ)) x y and T (x) T (y) δ x y. (11) Thus S α and T are contractions with constant (1 α (1 δ)) and δ, respectively. Also, for x B r [x ], we get that S α (x) x r and T (x) x r. This implies that S α (B r [x ]) B r [x ] and T (B r [x ]) B r [x ]. Therefore, by Banach contraction principle, S β, S α and T have a uniqe fixed point in B r [x ]. (ii) From (11) and (11), G (x) G (y) = T S α S β (x) T S α S β (y) δ S α S β (x) S α S β (y) δ (1 α (1 δ)) S β (x) S β (y) δ (1 α (1 δ)) (1 β (1 δ)) x y. Thus, the operator G is a contraction with constant δ (1 α (1 δ)) (1 β (1 δ)). Theorem 2. Let F be a Fréchet differentiable operator defined on an open convex subset C of a Banach space X with values in a Banach space Y. Suppose that [F (x )] 1 B(Y, X) for some x C and the operator F satisfies the conditions (5) and [F (x )] 1 F (x ) k and [F (x )] 1 l for some k, l >. Moreover assume that γ = kll < 1 2 and B r[x ] C, where r = 1 1 2γ γ k. Then, (i) The sequence {x n } defined by (9) is in B r [x ] and it converges strongly to x. (ii) The following inequality holds: x n+1 x λ n+1 x x, (12) where λ = δ (1 α (1 δ)) (1 β (1 δ)) and δ = lrl. Proof. (i) From Theorem 1, there is a unique solution of the equation (1). Let

7 Local and semi-local convergence of Newton-like method 69 x B r [x ]. Using (9), we get x n+1 x = G(x n ) G(x ) (13) = T S α S β (x n ) T S α S β (x ) Thus, x n x, as n. (ii) It follows from (13). δ S α S β (x n ) S α S β (x ) δ (1 α (1 δ)) S β (x n ) S β (x ) δ (1 α (1 δ)) (1 β (1 δ)) x n x. [δ (1 α (1 δ)) (1 β (1 δ))] n+1 x x. Remark 1. From the inequality (3.9) in Theorem 3.5 in [3] and (12) λ = δ (1 α (1 δ)) (1 β (1 δ)) < δ (1 α + αδ). (14) The inequality (14) shows that the error estimate in Theorem 2 is sharper than that of Theorem 3.5 in [3]. Our next result is as follows: Theorem 3. Let F be a Fréchet differentiable operator defined on an open convex subset C of a Banach space X with values in a Banach space Y. Let x C be a solution of (1) such that [F (x )] 1 B(Y, X). Suppose that [F (x )] 1 and F satisfy the conditions (6) and (7) with B r (x ) C and x B r (x ), where r 1 = 1 2 L 2 and r = 2L Moreover, assume that the operators S β, S α and T are defined by (1). Then (i) For x B r (x ), T (x) x γ x x x, (15) S α (x) x (αγ x + 1 α) x x, S β (x) x (βγ x + 1 β) x x, L where γ x = 1 2(1 rl 2 ) ( x x + 2 x x ). (ii) T, S α and S β are quasi-contraction operators on B r (x ) with constant γ, 1 (1 γ)α and 1 (1 γ)β, respectively, where γ = sup x Br(x ) {γ x }. Proof. (i) Let x B r (x ) with x x. From (1), T (x) x = x [ F (x ) ] 1 F (x) x (16) = [ F (x ) ] 1 ( (F (x) F (x )) F (x ) (x x ) ) = ([ F (x ) ] 1 F (x )) 1 [ F (x ) ] 1 ( F (tx + (1 t)x ) (x x ) F (x ) (x x ) ) dt

8 7 Nazli Karaca and Isa Yildirim It follows from (6), (16) and Lemma 2, T (x) x 2(1 rl 2 ) ( x x + 2 x x ) x x = γ x x x. Using the operator S α defined by (1), S α (x) x = x α [ F (x ) ] 1 F (x) x = α(x x ) α [ F (x ) ] 1 (F (x) F (x )) + (1 α)(x x ) α [ F (x ) ] 1 (F (x) F (x )) (x x ) + (1 α) x x = α ([ F (x ) ] 1 F (x )) 1 [ F (x ) ] 1 F (tx + (1 t)x ) ( (x x ) F (x ) (x x ) ) dt + (1 α) x x α ( [ F (x ) ] 1 F (x )) 1 [ F (x ) ] 1 ( F (tx + (1 t)x ) F (x ) ) x x dt + (1 α) x x Again using (6) and Lemma 2, we obtain that S α (x) x α 1 L 2 x x t(x x ) + (x x ) x x dt + (1 α) x x α 1 L 2 x x α 2(1 rl 2 ) ( x x + 2 x x ) x x + (1 α) x x = (αγ x + 1 α) x x. (t x x + x x ) x x dt + (1 α) x x Similarly, taking β instead of α, we deduce that S β (x) x (βγ x + 1 β) x x. (ii) The operators T, S α and S β are quasi-contractions. Indeed, ( ) γ = sup {γ x } = sup x x + 2 x x x B r(x ) 2(1 rl 2 ) x B r(x ) 2(1 rl 2 ) (r + 2 x x ) 3r < 2(1 rl 2 ) = 1

9 Local and semi-local convergence of Newton-like method 71 Since γ < 1, we get that 1 (1 γ)α < 1 and 1 (1 γ)β < 1. Finally, we will give the following theorem on the local convergence analysis for iteration process (9). Theorem 4. Let F be a Fréchet differentiable operator defined on an open convex subset C of a Banach space X with values in a Banach space Y. Let x C be a solution of (1) such that [F (x )] 1 B(Y, X). Suppose that [F (x )] 1 and F satisfy the conditions (6) and (7). Also assume that B r1 (x ) C, where r 1 = 1 L 2 and for any x B r (x 2 ), where r = 2L Then (i) The sequence {x n } defined by (9) is in B r (x ) and it converges strongly to the unique solution x in B r1 (x ). (ii) The following inequality holds: x n+1 x (λ ) n+1 x x (17) where λ = (αγ + 1 α)(βγ + 1 β)γ and γ = 3 2(1 rl 2 ) x x. Proof. (i) Firstly, we will show that x is unique solution of (1) in B r1 (x ). On the contrary, suppose that y is another solution of (1) in B r1 (x ). From Lemma 3, we obtain = F (x ) F (y ) = Now, we can define an operator L by L(h) = F (y + t(x y )) (x y )dt. F (y + t(x y )) hdt, for all h X. Thus, we get I [ F (x ) ] 1 1 [ L = F (x ) ] 1 (F (x ) F (y + t(x y )))dt L 2 2 x y < L 2r 1 2 = 1 2. From Lemma 1, we know that the operator L is invertible. Thus, x = y, which is a contradiction. That is, x is the unique solution of (1) in B r1 (x ). Since the S α and S β defined by (1) are operators on B r [x ], from (9), we get z = S β (x ), y = S α S β (x ) B r (x ). Also the operator V defined by (1) is an operator on B r [x ] from Theorem 3. Therefore, (9) can be written as x n+1 = T S α S β (x n ). (18)

10 72 Nazli Karaca and Isa Yildirim From (18), we have x n+1 x = T S α S β (x n ) x γ yn S α S β (x n ) x By using the definition of γ x, we obtain γ yn = Hence, we have γ yn (αγ zn + 1 α) S β (x n ) x γ yn (αγ zn + 1 α)(βγ zn + 1 β) x n x. 2(1 rl 2 ) ( S αs β (x n ) x + 2 x x ) 2(1 rl 2 ) [(αγ z n + 1 α) S β (x n ) x + 2 x x ] 2(1 rl 2 ) [ S β(x n ) x + 2 x x ] 2(1 rl 2 ) [(βγ z n + 1 β) x n x + 2 x x ] 2(1 rl 2 ) [ x x + 2 x x ] 3 2(1 rl 2 ) x x = γ. x n+1 x [(αγ + 1 α)(βγ + 1 β)γ ] n+1 x x. (19) which implies that x n x as n. (ii) From (19), we obtain that desired error estimate. 3 Numerical Examples In this section, we will give two numerical examples in order to support our main results. The first example shows numerically that our method (9) is faster than the modified Newton method (3) and SIP of Newton-like (4). Example 1. Let X = R, C = (1, 4) and F : C R an operator defined by F (x) = x 2 4, x C. Then F is Frechet differentiable and its Frechet derivative F (x) at any point x C is given by F (x) = 2x. For x = 3, we have [F (x )] 1 = 1 6. Set k = , l = and L = 2, we have γ = kll < 1 2 and [ F (x ) ] 1 l, [ F (x ) ] 1 F (x ) k, F (x) F (x ) L x x.

11 Local and semi-local convergence of Newton-like method 73 Because all the conditions of Theorem 2 are satisfied, the sequence {x n } generated by (9) is in B r [x ] and it converges to a unique x B r [x ]. From the following table and figure, we see that the new method (9) is faster than the modified Newton method (3) and SIP of Newton-like (4). n modified Newton method SIP of Newton-like our method The following example shows the convergence of (9) in infinite dimensional space. Example 2. Let X = C[, 1 4 ] be the space of real-valued continuous functions defined on the interval [, 1 4 ] with norm x = max t 1 x(t). Consider the 4 integral equation of Fredholm type F (x) =, where F (x)(s) = cos 2πs π 2 s sin 2πtx 2 (t)dt, with s [, 1 4 ] and x C[, 1 4 ]. It is easy to find the Frèchet derivative of F as F (x) h(s) = h(s) + π Now, we have F (x ) = cos 2πs π π max x 2 8 s [, 1 4 ] s sin 2πtx(t)h(t)dt, h X. s sin 2πtx 2 (t)dt s sin 2πtdt x 2

12 74 Nazli Karaca and Isa Yildirim and Moreover, we get I F (x ) = F (x) F (x ) = π s sin 2πtx (t)dt π max s sin 2πtdt x s [, 1 4 ] x 4. π s sin 2πtx(t)dt π s sin 2πtx (t)dt π s sin 2πt (x(t) x (t)) dt π max s sin 2πtdt x x s [, 1 4 ] x x 4 and if x 4 < 1, then by Lemma 1, we obtain Thus, we get [ F (x ) ] x. [ F (x ) ] 1 F (x ) 8 + x x. For the initial point x (s) = 1 2, we obtain [ F (x ) ] 1 l = [ F (x ) ] 1 F (x ) k = L =.25. Hence, γ = kll = < 1 2. So the conditions of Theorem 2 are satisfied. Therefore, the sequence {x n } generated by (9) is in B r [x ] and it converges to a uniqe solution x B r [x ] of the integral equation. Acknowledgement. We thank the reviewers for their constructive comments, which helped us improve the manuscript. This work was supported by Ataturk University Rectorship under The Scientific and Research Project of Ataturk University, Project No.: 216/153.

13 Local and semi-local convergence of Newton-like method 75 References [1] Argyros, I. K., Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Comp., New York, 27. [2] Dennis, J. E., On the Kantorovich hypothesis for Newton s method, SIAM Journal on Numerical Analysis, 6 (1969), [3] Sahu, D. R., Singh, K. K., and Singh, V. K., Some Newton-like methods with sharper error estimates for solving operator equations in Banach spaces, Fixed Point Theory and Applications, 1 (212), 1-2. [4] Rall, L. B., Computational Solution of Nonlinear Operator Equations, John Wiley and Sons, New York, [5] Ren, H., and Argyros, I. K., On convergence of the modified Newtons method under Hölder continuous Fréchet derivative, Applied Mathematics and Computation, 213 (29), [6] Rheinboldt, W. C., A Unified Convergence Theory for a Class of Iterative Processes, SIAM Journal on Numerical Analysis, 5 (1968), [7] Sahu, D. R., Applications of the S-iteration process to constrained minimization problems and split feasibility problems, Fixed Point Theory, 12 (211), [8] Scherzer, O., Convergence criteria of iterative methos based on Landweber iteration for solving nonlinear problems, Journal of Mathematical Analysis and its Applications, 194 (1995), [9] Suhubi, E. S., Functional Analysis, Kluwer Academic Publishers, London, 23. [1] Karaca, N. and Yildirim, I., Approximating fixed points of nonexpansive mappings by a faster iteration process, Journal of Advanced Mathematical Studies, 8(215),

14 76 Nazli Karaca and Isa Yildirim