LOCAL CONVERGENCE OF NEWTON-LIKE METHODS FOR GENERALIZED EQUATIONS

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1 East Asian Mathematical Journal Vol. 25 (2009), No. 4, pp OCA CONVERGENCE OF NEWTON-IKE METHODS FOR GENERAIZED EQUATIONS Ioannis K. Argyros Abstract. We provide a local convergence analysis for Newton-like methods for the solution of generalized equations in a Banach space setting. Using some ideas of ours introduced in [2] for nonlinear equations we show that under weaker hypotheses computational cost than in [7] a larger convergence radius finer error bounds on the distances involved can be obtained. 1. Introduction In this study we are concerned with the problem of approximating a solution x of the generalized equation o f(x) + g(x) + F (x), (1) where X, Y are Banach spaces, f : X Y is a Fréchet-differentiable operator in a neighborhood U of x, g : X Y is continuous at x F denotes a set-valued map from X into the subsets of Y. If F = {0} g = 0 equation (1) reduces to a regular nonlinear equation. If F = {0} g 0 equation is again a regular nonlinear equation studied in [2] the references there. Here we are interested in generating a sequence {x n } (n 0) approximating x in cases when F = {0} g = 0 or not. The most popular method for approximating x is undoubtedly Newton-like method of the form o f(x n ) + g(x n ) + ( f (x n ) + [x n 1, x n, g] ) (x n+1 x n ) + F (x n+1 ) (2) where f (x) denotes the Fréchet-derivative of operator f [x, y; g] simply denoted by [x, y] is the first order divided difference of g at the points x, y satisfying [x, y] (X, Y ), [x, y](y x) = g(y) g(x) for x y. (3) If g is Fréchet-differentiable at x X then [x, x] = g (x). Received October 9, 2008; Accepted April 8, Mathematics Subject Classification. 65H10, 65G99, 47H17, 49M15. Key words phrases. Newton-like methods, Banach space, local convergence, radius of convergence, generalized equations, Fréchet derivative, ipschitz/center-ipschitz condition. 425 c 2009 The Youngnam Mathematical Society

2 426 IOANNIS K. ARGYROS Geoffroy Pietrus provided a local convergence analysis for method (2) in [7]. Here we are motivated by this paper, our work in [2] optimization considerations. Using more precise error estimates a combination of ipschitz as well as center ipschitz conditions on f g we provide a finer convergence analysis than before [5] [7] with the advantages already stated in the abstract of this paper. 2. ocal Convergence Analysis of Method (2) We need the definition of a divided difference of order 2 [9], the definition Aubin continuity of a set-valued map [1] a generalization of the Ioffe Tikhomirov theorem on fixed points of operators [6], [8]. Definition 1. We say that an operator in (X, (X, Y )) denoted by [x, y, z; g] or simply [x, y, z] is called a divided difference of order two of the operator y : X Y at the points x, y, z X if [x, y, z](z x) = [y, z] [x, y] If g is twice Fréchet-differentiable at x X then for all distinct points x, y z from X. (4) [x, x, x] = g (x) 2 Definition 2. A set-valued map ΓX Y is said to be M-pseudo-ipschitz about (x 0, y 0 ) Graph Γ = {(x, y) X Y y Γ(x)} if there exist neighborhoods V of y 0 U of x 0 such that e ( Γ(v) U, Γ(w) ) M v w for all v, w V. (5) From now on we set for x X, r > 0 U(x, r) = {z X z x r}. emma 3. et (X, ρ) be a Banach space, let T map X to the closed subsets of X, let q 0 X, let r > 0, λ [0, 1) be such that the following hold true: dist(q 0, T (q 0 )) < r(1 λ), (6) e ( T (v) U(q 0, r), T (w) ) λρ(v, w) for all v, w U(q 0, r). (7) Then T has a fixed point in U(q 0, r). If T is single-valued, then x is the unique fixed point of T in U(q 0, r). We will make the following assumptions: (A 1 ) F has a closed graph; (A 2 ) f is Fréchet differentiable in some neighborhood V of x ; (A 3 ) g is differentiable at x ;.

3 OCA CONVERGENCE OF NEWTON-IKE METHODS 427 (A 4 ) f is -ipschitz on V 0 -center ipschitz on V. That is there exist positive constants 0 such that F (y 1 ) F (y 2 ) y 1 y (8) F (y) F (x ) 0 y x for all y, y 1, y 2 V ; (9) (A 5 ) there exists a positive constant K such that for all x, y, z V, (A 6 ) the set-valued map [x, y, z] K; (10) G(x) 1 = [f(x ) + f (x )(x x ) + g(x) + F (x)] 1 (11) is M-pseudo-ipschitz around (0, x ). We can state the main local convergence result for method (2): Theorem 4. Under assumptions (A 1 ) (A 6 ) the following hold true: for every c > M ( 2 + K) = c 0 there exists δ > 0 such that for any distinct initial guesses x 0, x 1 U(x, δ), there exists a sequence {x n } (n 0) generated by Newton-like method (2) such that x n+1 x c x n x max{ x n x, x n 1 x } (n 1). (12) Before starting the proof it is convenient to define operators R n T n by R n (x) = f(x ) + g(x) + f (x )(x x ) f(x n ) g(x n ) (f (x n ) + [x n 1, x n ])(x x n ) (n 1)(13) T n (x) = G 1 [R n (x)] (n 1). (14) Note that x k+1 is a fixed point of T k if only if R k (x k+1 ) G(x k+1 ), i.e., if only if o f(x k ) + g(x k ) + ( f (x k ) + [x k 1, x k ] ) (x k+1 x k ) + F (x k+1 ). (15) We also need the auxiliary result: Proposition 5. Under the hypotheses of Theorem 4, there exists δ > 0 such that for all x 0, x 1 U(x, δ) (x 0, x 1, x distinct), the map T 1 has a fixed point x 2 in U(x, δ) satisfying x 2 x c x 1 x max{ x 1 x k, x 0 x }. (16) Proof. In view of (A 6 ) there exist positive constants a b such that e ( G 1 (y 1 ) U(x, a), G 1 (y 2 ) ) M y 1 y 2 for all y 1, y 2 U(0, b). (17) Choose a fixed δ (0, δ 0 ) where δ 0 = min {a, 1c ( ) } 1/2, 2b. (18) K

4 428 IOANNIS K. ARGYROS We shall show conditions (6) (7) of emma 3 hold true where q 0 = x T = T 1, for some constants r λ to be determined. We first note that dist(x, T 1 (x )) e ( G 1 (0) U(x, δ), T 1 (x ) ). (19) et x 0, x 1 U(x, δ) such that x 0, x 1 x are distinct, then we obtain in turn by (3), (4), (8) (10) (18) R 1 (x ) f(x ) + g(x ) f(x 1 ) g(x 1 ) (f (x 1 ) + [x 0, x 1 ])(x x 1 )) f(x ) f(x 1 ) f (x 1 )(x x 1 ) + g(x ) g(x 1 ) [x 0, x 1 ](x x 1 ) = f(x ) f(x 1 ) f (x 1 )(x x 1 ) + [x 0, x 1, x ](x x 0 )(x x 1 ) 2 x x K x x 0 x x 1 2 x x 1 + K x x 0 x x K δ x x K δ 2 b, (20) by the choice of δ. In view of (17) we get e ( G 1 (0) U(x, δ), T 1 (x ) ) = e ( G 1 (0) U(x, δ), G 1 [R 1 (x )] ) M 2 x x 1 + K x x 0 x x 1. (21) Using (19) we obtain in turn [ ] dist(x, T 1 (x )) M 2 x x 1 + K x x 0 x x 1 M 2 + K x x 1 max{ x x 0, x x 1 }. (22) Choose c fixed c > M ( 2 + K). Then there exist λ (0, 1) such that M ( 2 + K) c(1 λ). That is dist(x, T 0 (x )) c(1 λ) x x 1 max{ x x 0, x x 1 }. (23) etting q 0 = x, r = r 1 = c x x 1 max{ x x 0, x x 1 } condition (6) holds true.

5 OCA CONVERGENCE OF NEWTON-IKE METHODS 429 We shall show condition (7) also holds true. By δc < 1 x 0, x 1 U(x, δ) we have r 1 δ a. et x U(x, δ), then we get in turn R 1 (x) f(x ) f(x) f (x )(x x) + f(x) f(x 1 ) f (x 1 )(x x 1 ) + g(x) g(x 1 ) [x 0, x 1 ](x x 1 ) K δ 2, (24) 2 which implies z 1 (x) U(0, b) for x U(x, δ) by the choice of δ. et w, z U(x, r 1 ) then by (17) e ( T 1 (w) U(x, r 1 ), T 1 (z) ) e ( T 1 (w) U(x, δ), T 1 (z) ) M R 1 (w) R 1 (z) M (F (x ) F (x 1 ))(w z) + M g(w) g(z) [x 0, x 1 ](w z) M (F (x ) F (x 1 ))(w z) + M ( [x 1, z, w](w x 1 ) + [x 0, x 1, w](w x 0 ) ) (z w) Mδ( 0 + 4K) z w. (25) Without loss of generality we may assume λ δ < M( 0 + 4K) = δ 1, (26) which implies condition (7). By emma 3 there exists a fixed point x 2 U(x, r 1 ) for the map T 1. That completes the proof of Proposition 5. Proof of Theorem 4. Using induction on k 1 setting q 0 = x, r k = c x k x max{ x k 1 x, x k x } we conclude by Proposition 5 that the map T k has a fixed point x k+1 in U(x, r k ). It follows that x k+1 x c x k x max{ x k x, x k 1 x } (k 1). That completes the proof of Theorem 4. As in [7] we consider two modifications of method (2): Remark 6. (a) If (2) is replaced by o f(x n ) + g(x n ) + ( f (x n ) + [x 0, x n ] ) (x n+1 x n ) + F (x n+1 ) (27) then under hypotheses (A 1 ) (A 6 ) the conclusions of Theorem 4 hold with (12) replaced by x n+1 x c x n x max{ x n x, x 0 x }. (28) Note that regular-false method (27) [3] is slower than method (2). (b) If (2) is replaced by o f(x n ) + y(x n ) + ( f (x n ) + [x n+1, x n ] ) (x n+1 x n ) (29)

6 430 IOANNIS K. ARGYROS or o f(x n ) + f (x n )(x n+1 x n ) + g(x n+1 ) + F (x n+1 ) (30) then if c > c 0 is replaced by c > c 1 = M 2 (H 5 ) is dropped under hypotheses (A 1 ) (A 4 ) (A 6 ) the conclusions of Theorem 4 hold true with (12) replaced by the faster (quadratic convergence): Remark 7. In general x n+1 x c x n x 2. (31) 0 (32) holds 0 can be arbitrarily large [2] [4]. If equality holds in (32), then our results reduce to the corresponding ones in [7]. Otherwise they constitute an improvement. Indeed denote by δ0 0 δ1 1 used in [7] given by δ0 0 = min {a, 1c ( ) } 1/2, 2b (33) K δ1 1 λ = M( + 4K). (34) It follows from (18), (26), (33) (34) that δ 0 0 δ 0 (35) δ 1 1 δ 1. (36) Note also that the choice of δ influences the choice of c. In view of (35) (36) we conclude that under the same computational cost (since in practice the computation of constant requires the computation of 0 ) hypotheses a larger convergence radius δ a smaller ratio c can be obtained. These observations are very important in computational mathematics [2] [11]. References [1] Aubin, J. -P., ipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), [2] Argyros, I. K., A unifying local-semilocal convergence analysis applications for twopoint Newton-like methods in Banach space, J. Math. Anal. Applic. 298 (2004), [3] Argyros, I. K., Newton Methods, Nova Science Publ. Inc., New York, [4] Argyros, I. K., On the secant method for solving nonsmooth equations, J. Math. Anal. Appl. (to appear, 2006). [5] Dontchev, A.. Hager, W. W., An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), [6] Dontchev, A.., ocal convergence of the Newton method for generalized equations, C. R. Acad. Sci. Paris, t. 332, Série I (1996), , Mathematical Analysis. [7] Geoffory, M. H. Pietrus, A., A local convergence of some iterative methods for generalized equations, J. Math. Anal. Applic. 290 (2004), [8] Ioffe, A. D. Tikhomirov, V. M., Theory of Extremal Problems, North Holl, Amsterdam, 1979.

7 OCA CONVERGENCE OF NEWTON-IKE METHODS 431 [9] Kantorovich,. V. Akilov, G. P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, [10] Rockafellar, R. T., ipschitz properties of multifunctions, Nonlinear Analysis, 9 (1985), [11] Xiao, B. Harker, P. T., A nonsmooth Newton method for variational inequalities I: Theory, Math. Programming, 65 (1994), Ioannis K. Argyros Cameron University Department of Mathematical Sciences awton, OK 73505, USA address: iargyros@cameron.edu