Int. Journal of Math. Analysis, Vol. 2, 28, no. 15, 713-72 The Approximation of Some Invertible Operators Using Newton s Method in Banach Spaces Alexandru V. Blaga National College Mihai Eminescu 5 Mihai Eminescu Street Satu Mare 4414, Romania alblaga25@yahoo.com Abstract. Under the hypothesis that the derivative of an operator satisfies Lipschitz conditions, it is known that Newton s method is convergent. These conditions ensure from Banach theorem of inversability for operators that in balls of a namely radius, these are inversable. The aim of this article is to emphasize an algoritm of approximation for some invertible types of operators using Newton s method. Mathematics Subject Classification: 65H1, 47H15, 49R2 Keywords: Newton s method, convergence ball, Lipschitz condition with L average, bounded linear maps, Inverse Function Theorem, Banach s Theorem 1. Preliminaries Newton s method is generally used to resolve ecuations such as F (x) =, where F is an operator defined from a domain D of real or complex space X, to a Banach space Y. The method ensures the convergence of the sequence defined by (1.1.1) x n+1 = x n f (x n ) 1 f(x n ),n=, 1,.... It is determined in [4] a ball of an optimum radius, in which is ensured the inversability states the existence of a local inverse fx 1, defined on B(f(x ),α) Y and which satisfied the following (1.1.2) f 1 x (f(x )) = x (1.1.3) (1.1.4) f(f 1 x (y)) = y, for any y B(f(x ),α) f 1 x is differentiable.
714 A. V. Blaga The article [4] is dedicated to the determination of the lower exact bound of the ball of α radius, denoted by B(f(x ),α). We will use the same principle to prove that we can approximate an invertible at right operator through a sequence of operators that are not necessarily invertible. 2. Background in linear spaces Next, X and Y are Banach space where the norms will be named the same,, in order to simplify things. If A : X Y is a linear operator, we remind that the norm of the operator A is given by Ax A = sup x X x. Additionally, we will use B(X, Y ) to define the set of linear operators which are bounded A : X Y. Show that the norm makes B(X, Y ) a normed linear space. Definition 2.1. The linear map A B(X, Y ) is invertible if there is a B B(Y,X) so that AB = I Y and BA = I X (where I X is the identity map on X). The map B is called the inverse of A and is denoted by B = A 1. Next we will remind Banach s theorem of inversability. Theorem 2.1. ([3]). Let X be a Banach space and A B(X, Y ) with I X A < 1. Then A is invertible and the inverse is given by (2.2.1) A 1 = (I X A) k k= and satisfies the relation (2.2.2) A 1 1 < 1 I X A. Moreover, if < ρ < 1 then A I X < ρ and B I X < ρ implies A 1 B 1 1 A B. (1 ρ) 2 For operators from the Banach space B(X, Y ) we have the result given by Theorem 2.2. ([3]). Let X, Y be Banach a spaces and A, B B(X, Y ). Assume that A is invertible. If B satisfies (2.2.3) A B < 1 A 1 then B is also invertible and B 1 A 1 1 A 1 A B, B 1 A 1 A 1 2 (2.2.4) B A 1 A 1 A B. The following non-trivial example points out an application between B(X, Y ) and B(Y,X) which is differentiable on any element from B(X, Y ).
Approximation of invertible operators 715 Proposition 2.1. ([3]). Let X and Y be a Banach spaces and let U B(X, Y ) be the set of invertible elements (this is an open set from Theorem 2.2). Define a map f : U B(Y,X) by (2.2.5) f(x) =X 1, for any X U. Then f is differentiable and for any A U the derivative f (A) :B(X, Y ) B(Y,X) is the linear map whose value V B(X, Y ) is (2.2.6) f (A)V = A 1 VA 1. Proof. Let L : B(X, Y ) B(Y,X) be the linear map LV = A 1 VA 1. Then for X U we have f(x) f(a) L(X A) =X 1 (X A)A 1 (X A)A 1, so that f(x) f(a) L(X A) X 1 A 1 2 X A 2. The map X X 1 is continuous (Theorem 2.2), so lim X A X 1 = A 1,thus f(x) f(a) L(X A) lim X A X A lim X A X 1 A 1 2 X A = 3. Main results We consider f B(X, Y ) invertible and the map A : B(X, Y ) B(Y,X) which satisfies following conditions i) A has a continuous derivative in the ball B(f,r) ii) A (f ) 1 exists and A (f ) 1 A satisfies the center Lipschitz condition with the L average (3.3.1) A (f ) 1 A (f) I ρ(f) L(u)du, for any f B(f,r) B(X, Y ), where ρ(f) = f f and L is a positive integrable function in the interval [,r]. By Banach s theorem (Theorem 2.1) when r r, for f B(f,r ), A (f) 1 exists and (3.3.2) A (f) 1 A (f ) r 1 1 ρ(f) L(u)du where r satisfies L(u)du = 1. In these conditions the operator A 1 f does exist and satisfies the relations (1.2), (1.3) and (1.4) on the domain B(A(f ),b/ A (f ) 1 ). The result is ensured by Theorem 3.1. ([4]). Suppose that r r and b = r hypothesis of condition (3.1) we have (3.3.3) B(A(f ),b/ A (f ) 1 ) A(B(f,r )) L(u)udu. Then in the
716 A. V. Blaga and in the open ball on the left A 1 f exists and: (3.3.4) A 1 f (A(f )) = f (3.3.5) A(A 1 f (f)) = f for any f B(A(f ),b/ A (f ) 1 ) (3.3.6) A 1 f is differentiable. Moreover, the radius of the ball is the optimum one. If we extended the inequality (3.1) to B(f,r) that is (3.3.7) A (f ) 1 A (f) I ρ(f) L(u)du, for any f B(f,r) then the relation (3.3) can also be extended to the two closed sets, result given in [4]. Being the set U from Proposition 2.3 is open there exists a R> for which B(f,r) U. For r = R the relations (3.1), (3.2) and (3.3) are verified. The existence of R is ensured by Proposition 3.1. If f is invertible and f B(X, Y ) for which f f R< 1 f 1, then f is in U. Proof. The proof of the above proposition immediately results from Theorem 2.2 and the inclusion B(f,R) is satisfied. Next, we need Proposition 3.2. ([4]). Let the map h :[,T] R defined by (3.3.8) where T satisfies (3.3.9) h(t) =β t + T t L(u)(t u)du, t T L(u)(T u)du = T. Then when <β<b, h is decreasing monotonically on [,r ], while it is increasing monotonically on [r,t] and h(β) >, h(r )=β b<, h(t )= β>. Moreover h has a unique zero in each interval, denoted by r 1 and r 2. They satisfy (3.3.1) β<r 1 < r b β<r <r 2 <T.
Approximation of invertible operators 717 Theorem 3.2. If g B(A(f ),β/ A (f ) 1 ) and we consider the sequences (3.3.11) (3.3.12) f n+1 = f n A (f ) 1 (A(f n ) g), n=, 1,... t n+1 = t n + h(t n ),n=, 1,... then we have i) t n is increases monotonically and converges to r 1 ii) f n+1 f n t n+1 t n. Proof. i) Because h(t)+t is increases monotonically and t =<t 1 = β< <r 1, the monotony are deduced using induction, and so convergence for t n, lim t n = r 1. ii) We have f 1 f A (f ) 1 A(f ) g β = t 1 t. If ii) is true for n 1, we consider θ 1 and f n 1+θ = f n 1 + θ(f n f n 1 ), t n 1+θ = t n 1 + θ(t n t n 1 ) for which we have f n 1+θ f f 1 f + f 2 f 1 +...+ f n 1 f n 2 + θ f n f n 1 (t 1 t )+(t 2 t 1 )+...+ (t n 1 t n 2 )+θ(t n t n 1 )=t n 1+θ <r 1 R = r. We also have the equality f n+1 f n = A (f ) 1 (A(f n ) A(f n 1 ) A (f )(f n f n 1 )) = = 1 and from (3.1) we have (A (f ) 1 A (f n 1+θ ) I)(f n+1 f n 1 )dθ 1 f n+1 f n A (f ) 1 A (f n 1+θ ) I f n f n 1 dθ ( 1 ) ρ(fn 1+θ ) L(u)du f n+1 f n dθ 1 tn 1+θ L(u)du(t n t n 1 )dθ = tn L(u)(t n u)du tn 1 L(u)(t n 1 u)du = t n+1 t n. So ii) is true for any n and from ii) we deduce that the sequence {f n } n=1 is convergent. So we have lim f n = f and from (3.11) lim A(f n )=A(f) =g. On the other hand, f n f r 1 implies f n B(f,r 1 ) which implies f B(f,r 1 ) B(f,r ) B(f,R) U. Remark 3.1. From Theorem 3.1 the radius of the ball of inversability b/ A (f ) 1 is the optimum one. The extension of this ball is determinated by the maximization of b when f is fixed and the function L satisfies the relation (3.1). If we choose two examples of the function L we can compare the values of b and from the sequences (3.1) we obtain:
718 A. V. Blaga (a) L :[,r] [, ), L(u) =2u. From r L(u)du = 1, we deduce r =1 and b = r L(u)du = 2 and the sequence 3 (3.3.13) <β<r 1 < 3 2 β<r =1<r 2 < 3=R (b) L :[,r] [, ), L(u) = 3 r 16 u. From L(u)du = 1, we deduce r =4 and b = r L(u)du = 12 and the sequence 5 (3.3.14) <β<r 1 < 5 3 β<r =4<r 2 < 2 3 5 = R. For instance, the concrete forms of Theorem 3.1 are respectively Theorem 3.3. When L satisfies conditions (a) we have (3.3.15) B(A(f ), 2/3 A (f ) 1 ) A(B(f, 1)). Theorem 3.4. When L satisfies conditions (b) we have (3.3.16) B(A(f ), 12/5 A (f ) 1 ) A(B(f, 4)). From the two examples (a) and (b) the second example leads to a larger value. Next, we have an result that leads to a b max, given by Proposition 3.3. Let L :[,r] [, ) be a function that satisfies the relation (3.1) and l :[,r] R is differentiable with l (u) =L(u) for any u [,r]. That is b max = r r l(u)du, when l() =. Proof. From l (u) =L(u) we deduce that the function l is increasing monotonically. Since r L(u)du = 1, we have l(r ) l() = 1, so l(r ) = 1. In finding b, we have b = r L(u)du = r r l(u)du. Considering the function b :[,r] R, b(t) =t t l(u)du, we have b (t) =1 l(t). For t r, we have l(t) l(r ) = 1, so on [,r ], we have b (t). Similarly on [r,r] we have b (t). The monotony of b(t) is obvious from we have b(t) b(r )=r r l(u)du. The finding of b(r ) in the two cases leads to the same values. The optimum radius in these conditions is r r l(u)du A (f ) 1. Theorem 3.5. For any operator g B(A(f ),β/ A (f ) 1 ) there exists a sequence of operators not necessarily invertibles which approximate it. Proof. We consider the sequence {A(f n )}, n =, 1,... which is correctly defined because for f B(X, Y ) given and invertible, from (3.11) we can determine f 1,soA(f 1 ). Inductively the sequence {A(f n )}, n =, 1,... is correctly defined. From Theorem 3.2, for any x X, we have lim f n (x) =f(x) because B(X, Y ) is a Banach space. From f U we deduce the existence of the operator F B(Y,X) which is the inverse of f. On the other hand,
Approximation of invertible operators 719 g = A f is invertible at right and lim A(f n )=g. The sequence of operators A(f n ) B(Y,X) are not necessarily invertible. It is also ensured the inverse at right of g and we known its inverse because (g F A 1 f )(u) = [(A f) (F A 1 f )](u) =[A (f F ) A 1 f )](u) =A(A 1 f (u)) = u (see (3.5)), for any u B(A(f ),β/ A (f ) 1 ). From Theorem 3.2 we can also write A(f n+1 ) A(f) A (t n+1 r 1 ) and (A 1 f ) (g) =A f (f), with A(f) =g In Proposition 2.1 it is proven the differentiability of the application f : U B(Y,X) given by f(x) =X 1 and f (A)V = A 1 VA 1. Next, we consider Y = X and since any operator g B(A(f ),β/ A (f ) 1 ) allows the determinate of a sequence of operators that approximate it, we will consider the restriction of the identic operator I at this ball, denoted by I 1. We denote the sequence that approximates I 1 with A n, that is lim A n I 1 =, or lim A n (x) =I 1, for any x B(A(f ),β/ A (f ) 1 ). With these notes, we have Proposition 3.4. If f : U B(X, X) B(X, X), f(a) =A 1 and and the sequences (3.3.17) f n (A) =A 1 A n,n=, 1,..., (3.3.18) L n : B(X, X) B(X, X), L n (V )= A 1 A n VA 1 A n,n=, 1,..., then lim f n (A)V = A 1 VA 1, for any V B(X, X). Proof. We have f n (A) f n (B) L n (A B) =A 1 A n B 1 B n + A 1 A n (A B)A 1 A n =[ B 1 (A B)A 1 + A 1 A n (A B)A 1 ]A n = =(A 1 A n B 1 )(A B)A 1 A n. The application f(a) =A 1 is continuous from Theorem 2.2, so we have lim = f(b) =B 1, that is A B f n (A) f n (B) L n (A B) lim A B A B lim A B A 1 A n B 1 A 1 A n = B 1 A n B 1 A 1 A n = A n I 1 B 1 A 1 A n, which leads to ( ) f n (A) f n (B) L n (A B) lim lim A B A B lim B 1 A n I 1 A 1 A n = because A n is bunded. Remark 3.2. Proposition 3.4 points out the sequence of operators L n, which approximate the differentiale of the operator f and f (A)V = A 1 VA 1 starting from the sequence of operators f n. This is the
72 A. V. Blaga product of an invertible and differentiable operator f with a sequence of linear operators A n, which are not necessarily differentiable. Even if lim f n (A) = A 1, the result is not obtained immediately. References [1] A.M. Ostrowski, Solutions of equations in Euclidean and Banach spaces, Academic Press, New York, 1973 [2] F.A. Potra and V. Ptak, Sharp error bounds for Newton s process, Numer. Math. 34(198), 63-72 [3] Ralph Howard, The inverse function theorem for Lipschitz maps, University of South Carolina Columbia [4] Wang Xinghua, Convergences of Newton s method and inverse function theorem in Banach space, Mathematics of Computation, Volume 68, Number 225(1999),169-186 [5] Wang Xinghua, Chong Li, Covergence of Newton s method and uniqueness of the solution of equations in Banach spaces, Acta Mathematica Sinica, English Series, April, 23, Vol. 19, No. 2, 45-412 Received: January 16, 28