New w-convergence Conditions for the Newton-Kantorovich Method

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1 Punjab University Journal of Mathematics (ISSN ) Vol. 46(1) (214) pp New w-convergence Conditions for the Newton-Kantorovich Method Ioannis K. Argyros Department of Mathematicsal Sciences, Cameron University, Lawton, OK 7355, USA, Hongmin Ren College of Information Engineering, Hangzhou Polytechnic, Hangzhou 31142, Zhejiang, PR China Abstract.We present new sufficient semilocal convergence conditions for the Newton-Kantorovich method in order to approximate a locally unique solution of onlinear equation in a Banach space setting. Examples are given to show that our results apply but earlier ones do not apply to solve nonlinear equations. AMS (MOS) Subject Classification Codes: 65J15, 65H1, 65G99, 64H25, 47J1, 49M15 Key Words: Newton-Kantorovich method; Banach space; majorizing series, telescopic series. 1. INTRODUCTION In this study, we are concerned with the problem of approximating a locally unique solution x of equation F (x) =, (1.1) where F is a Fréchet-differentiable operator defined on an open convex subset D of a Banach space X with values in a Banach space Y. A large number of problems in applied mathematics also in engineering are solved by finding the solutions of certain equations [3,7,11]. These solutions can rarely be found in closed form. That is why numerical methods are used to solve such equations. The most popular method for generating a sequence {x n } approximating x is undoubtedly Newton-Kantorovich method x n+1 = x n F (x n ) 1 F (x n ) (n ) (x D). (1.2) Here, F (x) L(X, Y ) the space of bounded linear operators from X into Y denotes the Fréchet derivative of operator F. Local as well as semilocal convergence results for Newton-Kantorovich method (1.2) under various Lipschitz-type assumptions have been given by many authors [1-14]. 77

2 78 Ioannis K. Argyros, Hongmin Ren The various semilocal convergence conditions for Newton-Kantorovich method (1.2) are only sufficient but not necessary. Hence, it is possible using the same information as before to find weaker sufficient convergence conditions. As an example (see also Section 3) we showed [3-7] that the famous Newton-Kantorovich hypothesis (see (3.1)) for solving nonlinear equations can always be replaced by an at least as weak condition (see (3.2)). Similar results for Newton-type methods have been given in [1,2,8-14]. Note that the applicability of these methods is extended, whenever weaker sufficient convergence conditions become available. Hence, such studies results are extremely important in computational mathematics. The affine invariant condition F (x ) 1 (F (x) F (y)) q( x y ) for all x, y D, where, q : [, + ) [, + ) is continuous, non-decreasing has been used by many authors in the study of the Newton-Kantorovich method [5,6,7,8,9,12,13,14]. Here, we present new sufficient convergence conditions. Our results extend to solve equations F (x) + G(x) =, (1.3) using x n+1 = x n F (x n ) 1 (F (x n ) + G(x n )) (n ) (x D), (1.4) where F is as above G : D Y is a continuous operator. The paper is organized as follows: Section 2 contains the semilocal convergence of methods (1.2) (1.4), whereas in Section 3 we provide special cases, numerical examples. 2. SEMILOCAL CONVERGENCE We present the semi-local convergence analysis of methods (1.2) (1.4) in this section. The following auxiliary result is used repeatedly in this paper. Lemma 1. (Banach lemma on invertible operators [11]) Let T L(X). Then, T 1 exists if only if there is a bounded linear operator P in X such that P 1 exists If T 1 exists, then T 1 I P T < 1. P 1 I P T. Next, in Lemma 2, 3 Theorem 5 we use method (1.2) to approximate a solution x of equation (1.1). Let x D. Suppose that the following conditions hold: (C 1 ) F (x ) 1 L(Y, X) F (x ) 1 β, (C 2 ) < F (x ) 1 F (x ) η, (C 3 ) there exists a continuous strictly increasing function w : [, + ) [, + ) with w 1 : [, + ) [, + ) continuous such that for all s, t x, y D w 1 (s) + w 1 (t) w 1 (s + t)

3 New w-convergence conditions for the Newton-Kantorovich method 79 F (x) F (y) w( x y ). It is convenient for us to define for a = b = 1, scalar sequences +1 = 1 β w(b n η), (2.1) c n = w(tb n η)dtb n, (2.2) b n+1 = β+1 c n. (2.3) We provide a connection between Newton-Kantorovich method {x n } scalar sequences { }, {b n }, {c n }. Lemma 2. Under the (C 1 ) (C 3 ) conditions further suppose: (C 4 ) x n D (C 5 ) β w(b n η) < 1. Then, the following estimates hold: (I n ) F (x n ) 1 β, (II n ) x n+1 x n = F (x n ) 1 F (x n ) b n η, (III n ) F (x n+1 ) c n η. Proof. We shall use induction to show items (I n ) (III n ). (I ) (II ) follow immediately from the initial conditions. To show (III ), we use (1.2) for n =, (II ) (C 3 ) to obtain in turn F (x 1 ) = F (x 1 ) F (x ) F (x )(x 1 x ) = [F (x + t(x 1 x )) F (x )](x 1 x )dt. So, we get that F (x 1 ) = [F (x + t(x 1 x )) F (x )](x 1 x )dt w(t x 1 x )dt x 1 x w(tb η)dtb η = c η. (2.4) (2.5) If x k+1 D (k n), then it follows from (C 3 ) (C 5 ) the induction hypotheses that: F (x k ) 1 F (x k+1 ) F (x k ) a k βw( x k+1 x k ) βa k w(b k η) < 1. It follows from (2.6) the Banach Lemma 1 that F (x k+1 ) 1 L(Y, X), (2.6) F (x k+1 ) 1 which shows (I n ) for all n. As in (2.4), we also have: F (x k ) 1 1 F (x k ) 1 F (x k+1 ) F (x k ) a k β 1 βa k w(b k η) = a k+1β, F (x k+1 ) = F (x k+1 ) F (x k ) F (x k )(x k+1 x k ) = [F (x k + t(x k+1 x k )) F (x k )](x k+1 x k )dt. (2.7) (2.8)

4 8 Ioannis K. Argyros, Hongmin Ren Consequently, we get that F (x k+1 ) w(t x k+1 x k )dt x k+1 x k w(tb kη)dtb k η = c k η, which shows (III n ) for all n. Moreover, by (1.2), (2.7) (2.9) we have that F (x k+1 ) 1 F (x k+1 ) F (x k+1 ) 1 F (x k+1 ) βa k+1 c k η = b k+1 η. (2.9) (2.1) That completes the induction for (II n ). Next, we shall show the convergence of sequence {x n }, which is equivalent to proving that {b n } is a Cauchy sequence. To this effect we need the following result: Lemma 3. Suppose: Condition (C 5 ) holds. Then, the following assertions hold: (a) Scalar sequence { } increases, (b) lim n b n =, (c) r = k= b k <, b k = 1 η w 1 ( 1 β ( 1 a k 1 a k+1 )) (d) If U(x, rη) = {x X x x rη} D, then (C 4 ) holds. Proof.. (a) We shall show using induction that { }, {b n }, {c n }, 1 β w(b n η) are positive sequences. In view of the initial conditions, a, b, c 1 βa w(b η) are positive. Assume a k, b k, c k 1 βa k w(b k η) are positive for k n. It follows from hypothesis c k >, (2.3) that a k+1 b k+1 >. Moreover, by (2.1), a k+1 >, consequently w(b k+1 η) >. Furthermore, 1 βa k+1 w(b k+1 η) > by (C 5 ). The induction is completed. Solving (2.1) for w(b n η), we obtain By telescopic sum, we have: w(b n η) = 1 β ( ). (2.11) n 1 w(b k η) = 1 β (1 1 ) (2.12) k= = 1 1 β n 1 k= w(b kη). (2.13) But 1 β n 1 k= w(b kη) decreases, so { } given by (2.13) increases. Note also that a = 1. (b) By (a), { } increases < 1 1. (2.14) Therefore, { 1 } is monotonic on the compact set [, 1] as such it converges to some limit denoted by a. By letting n in (2.11), we get lim n b n = lim n 1 η w 1 ( 1 β ( )) = 1 η w 1 ( lim n 1 β ( )) = 1 η w 1 () =. (2.15)

5 New w-convergence conditions for the Newton-Kantorovich method 81 (c) r is finite since by the first hypothesis in (C 3 ) (2.15): 1 r = η w 1 ( 1 β ( 1 1 )) 1 1 a k a k+1 η w 1 ( β ( 1 1 )) = 1 a k a k+1 η w 1 ( 1 (1 a)). β k= (d) We have x 1 x b η = η x 1 U(x, rη). Assume x k U(x, rη) D for all k n. Then, we have by Lemma 1 in turn that k= x k+1 x x k+1 x k + + x 1 x (b k + + b )η rη (2.16) x k+1 U(x, rη) D. We can show the semilocal convergence result for Newton-Kantorovich method (1.2). Theorem 4. Under conditions (C 1 ) (C 3 ), (C 5 ), further suppose (C 6 ) U(x, rη) D. Then, sequence {x n } generated by Newton-Kantorovich method (1.2) is well defined, remains in U(x, rη) for all n, converges to a solution x U(x, rη) of equation F (x) =. Moreover, the following estimates hold: x n x b k η < rη. (2.17) k=n Furthermore, x is the only solution of equation F (x) = in where provided that r rη is the maximum number satisfying β D 1 = D D, (2.18) D = U(x, r ) (2.19) w((1 t)rη + tr )dt = 1. (2.2) Proof. It follows from Lemmas 2 3 (see also (II n )) that {x n } is a Cauchy sequence in a Banach space X as such it converges to some x U(x, rη) (since U(x, rη) is a closed set). We have lim n w(b n η) =, which implies by (2.2), the continuity of function w the assertion (b) in Lemma 3 that lim n c n =. By letting k in (2.9) using the continuity of operator F, we obtain F (x ) =. By (C 6 ), we get n n x n+1 x x k+1 x k b k η < rη, k= x n+1 U(x, rη) x = lim n x n U(x, rη). Let m > n. Then, we have x n x m m 1 k=n x k x k+1 By letting m in (2.21), we obtain (2.17). k= m 1 k=n b k η < rη. (2.21)

6 82 Ioannis K. Argyros, Hongmin Ren Finally, to show uniqueness, let y D be a solution of equation F (x) =. Define linear operator M = By (C 1 ), (C 3 ) (2.2), we obtain in turn: F (x + t(y x ))dt. (2.22) F (x ) 1 M F (x ) β w((1 t) x x + t y x )dt < β w((1 t)rη + tr (2.23) )dt = 1. It follows from (2.23), the Banach lemma 2.1 that M 1 exists. Using the identity = F (y ) F (x ) = M(y x ), we deduce x = y. Remark 5. It follows from (C 3 ) that (C 3 ) there always exists a continuous non-decreasing function w : [, + ) [, + ) with w () = such that for all x D F (x) F (x ) w ( x x ). Note that w w holds in general, w w can be arbitrarily large [3-7]. Hence (C 3 ) is not an additional hypothesis. In view of (C 3 ), (2.5) c, a 1 can be defined in a tighter way by a a 1 = 1 βa w (b η) c = w (tb η)dtb. The new { }, {b n } {c n } sequences are tighter majorizing (for {x n }) than before under the same computational cost. Moreover, the uniqueness ball is extended, since w can replace w (see (2.23)) in condition (2.2). The results obtained here can be extended to hold for equations containing ot necessarily differentiable term. In the remaining results we use method (1.4) to approximate a solution x of equation (1.3). Let us suppose: (C 7 ) there exists a continuous, non-decreasing function v : [, + ) [, + ) with v() = such that for all x, y D: Define sequence {c n } by G(x) G(y) v( x y ) x y. c n = [ w(tb n η)dt + v(b n η)]b n, where as { } {b n } are given by (2.1) (2.3), respectively. Then, using the identity: F (x n+1 ) + G(x n+1 ) = [F (x n + t(x n+1 x n )) F (x n )](x n+1 x n )dt +G(x n+1 ) G(x n ), (2.24)

7 New w-convergence conditions for the Newton-Kantorovich method 83 (instead of (2.4) (for n = ), (2.8)) following the rest of the proof of Theorem 4 (excluding the uniqueness part) we arrive at: Theorem 6. Under the conditions (C 1 ) (C 3 ), (C 5 ) (C 7 ) the following hold: Sequence {x n } generated by Newton-Kantoroovich method (1.4), is well defined, remains in U(x, rη) for all n, converges to a solution x U(x, rη) of equation F (x) + G(x) =. Moreover, the following estimates hold: x n x b k η < rη. (2.25) k=n We can show a uniqueness result but we use a condition other than (2.2). Proposition 7. Under the hypotheses of Theorem 2.6, further suppose: there exists r 1 rη such that β( w(tr 1 )dt + v(r 1 )) aq < 1 for some q (, 1), (2.26) then x is the unique solution of equation F (x) + G(x) = in D 3 = D D 2, where D 2 = U(x, r 1 ), a is given in Lemma 2.3. Proof. Let y D 3 be a solution of equation F (x) + G(x) =. Using (1.4), we get the identity x n+1 y = F (x n ) 1 [F (x n ) F (y ) F (x n )(x n y + G(x n ) G(y )] = F (x n ) 1 [ ( F (y + t(x n y )) F (x n ) ) dt(x n y ) +G(x n ) G(y )], (2.27) so, x n+1 y β[ w(t x n y )dt + v( x n y )] x n y β[ w(tr 1)dt + v(r 1 )] x n y q x n y. (2.28) Hence, we get x n+1 y q n x y q n r 1, which implies lim n x n = y. But we know that lim n x n = x. Hence, we deduce x = y. It turns out that condition (C 5 ) can be replaced by the at least as weak (C 5 ) βw (d n η) < 1. Indeed, introduce scalar sequences {p n }, {d n } by We get p = β, d = 1, n d n = p n+1 = k= b k p 1 βw (d n η). d n = 1 η w 1 ( 1 p 1 p n+1 ).

8 84 Ioannis K. Argyros, Hongmin Ren Then, in view of the observation F (x ) 1 F (x k+1 ) F (x ) βw ( x k+1 x ) βw ( x k+1 x k + x k x k x 1 x ) βw ((b k + b k b )η) = βw (d k η), (2.29) estimate (2.7) can be replaced by the at least as precise F (x k+1 ) 1 p 1 βw ( x k+1 x k + x k x k x 1 x ) p 1 βw (d k η). (2.3) With these changes, we arrive at the following analogs of Lemmas 2, 3, Theorem 4, 6 Proposition 7. Lemma 8. Suppose (C 1 ) (C 4 ), (C 3 ) (C 5 ) hold. Then, the following estimates hold F (x n ) 1 p n, x n+1 x n b n η F (x n+1 ) c n η. Lemma 9. Suppose (C 5 ) holds. Then, sequence (a) {p n } increases, (b) {d n } is increasingly convergent, (c) lim n b n =, (d) r = k= b k < (e) If U(x, rη) D, then (C 4 ) holds. Similarly, we obtain analogs of Theorem 4,6 Proposition 7 (simply replace (C 5 ) by (C 5 ) )). Remark 1. The results obtained here can further be refined, if we further assume: (C 3 ) there exists a function p : [, 1] [, + ) such that w(st) p(s)w(t) for all s [, 1] t [, + ). This condition has been successfully used to sharpen the error bounds for particular expressions [5,6,7,8,9,12,13,14]. Note that such a function p always exists. Indeed, if w is a nonzero function on R +, then one can define p : [, 1] [, + ) by p(s) = sup{ w(st) w(t) : t [, + ), with w(t) > }. Note that in this case the results obtained in this study hold with w(tb nη)dt replaced by P w(b n η), where P = p(s)ds. Finally, note that the results obtained here can be provided in affine invariant form, if we replace operator F by F (x ) 1 F [5-7].

9 New w-convergence conditions for the Newton-Kantorovich method SPECIAL CASES AND APPLICATIONS Condition (C 5 ) is difficult to verify in general. However, (C 5 ) holds in some very interesting cases. Let us consider the Lipschitz case, i.e., w(s) = Ls, w (s) = L s, G =. Then, the famous for its simplicity clarity Newton-Kantorovich hypothesis h K = βlη 1 2 (3.1) implies condition (C 5 ) r K = Moreover, our condition given in [3], [11] by where, h K [11]. h AH = βlη 1 2, (3.2) L = 1 8 (L + 4L + L 2 + 8L L) also implies (C 5 ) where, Note that α = r AH = 2 2 αβ, 4L L + L 2 + 8L L. h K 1 2 h AH 1 2 but not necessarily vise versa unless if L = L. In the first example we show that the Kantorovich hypothesis (see (3.1)) is satisfied with the bigger uniqueness ball of solution than before [2], [7]. Example 11. Let X = Y = R be equipped with the max-norm. Let x = 1, D = U(x, 1 q), q [, 1) define function F on D by F (x) = x 3 q. (3.3) Then, we obtain that β = 1 3, L = 6(2 q), L = 3(3 q) η = 1 3 (1 q). Then, the famous for its simplicity clarity Kantorovich hypothesis for solving equations using (NKM) [1,2,7] is satisfied, say for q =.6, since h K = βlη = 2 3 (2 q)(1 q) = < 1 2. (3.4) Hence, (NKM) converges starting at x = 1. We also have that r = , η = , rη = , L = 7.2 < L = 8.4 r = 2 βl rη = That is our Theorem 2.4 guarantees the convergence of (NKM) to x = 3.6 = the uniqueness ball is better than the one given in (KT). In the second example we apply Theorem 4 to onlinear integral equation of Chrasekhar-type.

10 86 Ioannis K. Argyros, Hongmin Ren Example 12. Let us consider the equation x(s) = 1 + s 4 x(s) x(t) dt, s [, 1]. (3.5) s + t Note that solving (3.5) is equivalent to solving F (x) =, where F : C[, 1] C[, 1] defined by [F (x)](s) = x(s) 1 s 4 x(s) x(t) dt, s [, 1]. (3.6) s + t Using (3.6), we obtain that the Fréchet-derivative of F is given by [F (x)y](s) = y(s) s 4 y(s) x(t) s + t dt s 4 x(s) y(t) dt, s [, 1]. (3.7) s + t Let us choose the initial point x (s) = 1 for each s [, 1]. Then, we have that β = , η = , L = L = ln 2 = , h = r = (see also [1,2,3,7]). Then, hypotheses of Theorem 2.4 are satisfied. In consequence, equation F (x) = has a solution x in U(1, ρ), where ρ = rη = Acknowledgement This work was supported by National Natural Science Foundation of China (Grant No ). REFERENCES [1] S. Amat, S. Busquier M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim. 25, (24), [2] S. Amat S. Busquier, Third-order iterative methods under Kantorovich conditions, J. Math. Anal. Appl. 336 (1), (27), [3] I.K. Argyros, On the Newton-Kantorovich hypotheses for solving equations, J. Comput. Appl. Math. 169, (24), [4] I.K. Argyros, A semilocal convergence analysis for directional Newton methods, Math. Comput. AMS, 8, (211), [5] I.K. Argyros S. Hilout, Improved generalized differentiability conditions for Newton-like methods, J. Complexity, 26, (21), [6] I.K. Argyros S. Hilout, On the convergence of Newton s method under w condition second derivative, Appl. Math. 38 (3), (211), [7] I.K. Argyros, Y. Cho S. Hilout, Numerical Methods for Equations Its Applications, CRC Press/Taylor Frnnws Publ. Gr. New York, 21. [8] J.A. Ezquerro M.A. Hernández, Generalized differentiability conditions for Newton s method, IMA J. Numer. Anal. 22 (2), (22), [9] J.A. Ezquerro M.A. Hernández, On an application of Newton s method to nonlinear equations with w condition second derivative, BIT 42 (3), (22), [1] J.M. Gutierrez, A new semilocal convergence theorem for Newton s method, J. Comput. Appl. Math. 79, (1997), [11] L.V. Kantorovich G.P. Akilov, Functional Analysis in Normed Spaces, Prgamon Press, Oxford, [12] F.A. Potra V. Ptak, Sharp error bounds for Newton process, Numer. Math. 34, (198), [13] P.D. Proinov, New general convergence theory for iterative processes its applications to Newton- Kantorovich type theorem, J. Complexity, 26, (21), [14] P.P. Zabrejko D.F.Nquen, The majorant method in the theory of Newton-Kantorovich approximations the Ptak error estimations, Numer. Funct. Anal. Optim. 9, (1987),