Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 683295, 6 pages http://dx.doi.org/10.1155/2014/683295

Research Article

Implicit Approximation Scheme for the Solution of K-Positive Definite Operator Equation

Naseer Shahzad,1 Arif Rafiq,2 and Habtu Zegeye3

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia

2 Department of Mathematics, Lahore Leads University, Lahore 54810, Pakistan

3 Departement of Mathematics, University of Botswana, Private Bag Box 00704, Gaborone, Botswana

Correspondence should be addressed to Naseer Shahzad; nshahzad@kau.edu.sa Received 5 December 2013; Accepted 9 February 2014; Published 23 March 2014 Academic Editor: Ljubomir B. Ciric

Copyright © 2014 Naseer Shahzad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We construct an implicit sequence suitable for the approximation of solutions of K-positive definite operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001).

1. Introduction

Let E be a real Banach space and let J denote the normalized duality mapping from E to 2E defined by

J(x) = [f e E* :(x,f) = \\x\\2, \\f* || = M|, (1)

where E* denotes the dual space of E and {■, ■) denotes the generalized duality pairing. It is well known that if E* is strictly convex, then J is single valued. We will denote the single-valued duality mapping by j.

Let E be a Banach space. The modulus of smoothness of E is the function.

pE :[0, ot) ^ [0, ot) defined by

Re W = sup (\\x + y\\ + \\x - y\\) - 1 : \\x\\ < 1, \\y\\ < t J .

The Banach space E is called uniformly smooth if lim ^ = 0.

A Banach space E is said to be strictly convex if for two elements x,y e E which are linearly independent we have that \\x + y\\ < \\%\\ + \\y\\.

Let E1 be a dense subspace of a Banach space E. An operator T with domain D(T) 2 E1 is called continuously E1 -invertible if the range of T, R(T), with T in E considered as an operator restricted to E1, is dense in E and T has a bounded inverse on R(T).

Let E be a Banach space and let A be a linear unbounded operator defined on a dense domain, D(A), in E. An operator A will be called K positive definite (Kpd) [1] if there exist a continuously D(A)-invertible closed linear operator K with D(A) c D(K) and a constant c > 0 such that j(Kx) e J(Kx),

(Ax, j (Kx)) > c\\Kx\\2, VxeD(A).

Without loss of generality, we assume that c e (0,1).

In [1], Chidume and Aneke established the extension of Kpd operators of Martynjuk [2] and Petryshyn [3, 4] from Hilbert spaces to arbitrary real Banach spaces. They proved the following result.

Theorem 1. Let E be a real separable Banach space with a strictly convex dual E and let A be a Kpd operator with D(A) = D(K). Suppose

(Ax, j (Ky)) = (Kx, j (Ay)) , Vx,y eD (A). (5)

Then, there exists a constant a > 0 such that for all x e D(A)

\\Ax\\<a\\Kx\\. (6)

Furthermore, the operator A is closed, R(A) = E, and the equation Ax = f has a unique solution for any given f e E.

As the special case of Theorem 1 in which E = L „ (lp) spaces, 2 < p < <x>, Chidume and Aneke [1] introduced an iteration process which converges strongly to the unique solution of the equation Ax = f, where A and K are commuting. Recently, Chidume and Osilike [5] extended the results of Chidume and Aneke [1] to the more general real separable ^-uniformly smooth Banach spaces, 1 < q < >x>, by removing the commutativity assumption on A and K. Later on, Chuanzhi [6] proved convergence theorems for the iterative approximation of the solution of the Kpd operator equation Ax = f in more general separable uniformly smooth Banach spaces.

In [7], Osilike and Udomene proved the following result.

Theorem 2. Let E be a real separable Banach space with a strictly convex dual and let A : D(A) c E ^ E be a Kpd operator with D(A) = D(K). Suppose (Ax, j(Ky)) = (Kx,j(Ay)) for all x,y e D(A). Choose any e1 e (0,c2/(1 + oc(1 - c) + a2)) and define Te : D(A) cE^E by

Tex = x + eK-1f - eK-1Ax.

Then the Picard iteration scheme generated from an arbitrary x0 e D(A) by

_ ПТ1 _ rr<n

ХП+1 = 1£ХП = i£ X0

converges strongly to the solution of the equation Ax = f. Moreover, if x* denotes the solution of the equation Ax = f, then

- X II <

< (1-ce(1-c))nß-1 \\Kx0 -Kx*

The most general iterative formula for approximating solutions of nonlinear equation and fixed point of nonlinear mapping is the Mann iterative method [8] which produces a sequence [xn] via the recursive approach xn+l = anxn + (1 -an)Txn, for nonlinear mapping T : C = D(T) ^ C, where the initial guess x0 e C is chosen arbitrarily. For convergence results of this scheme and related iterative schemes, see, for example, [9-15].

In [16], Xu and Ori introduced the implicit iteration process {xn}, which is the modification of Mann, generated by Xo eC,xn = anxn_l +(1- an)Tnxn, for Tt, i=1,2,...,N, nonexpansive mappings, and Tn = Tn (modN) and {an} c (0,1). They proved the weak convergence of this process to a common fixed point of the finite family of nonexpansive mappings in Hilbert spaces. Since then fixed point problems and solving (or approximating) nonlinear equations based on implicit iterative processes have been considered by many authors (see, e.g., [17-21]).

It is our purpose in this paper to introduce implicit scheme which converges strongly to the solution of the Kpd operator equation Ax = f in a separable Banach space. Even though our scheme is implicit, the error estimate obtained indicates that the convergence of the implicit scheme is faster in comparison to the explicit scheme obtained by Osilike and Udomene [7].

2. Main Results

We need the following results.

Lemma 3 (see [10]). IfE* is uniformly convex then there exists a continuous nondecreasing function b : [0, rn) ^ [0, rn) such thatb(0) = 0, b(St) < Sb(t) for all S > 1 and

+ yf < уху2 + 2 (у, j (x)) + max {\\x\\, 1} Ц^Ц b (||y||),

for all x,y e E.

Lemma 4 (see [22]). If there exists a positive integer N such that for alln>N,ne N (the set of all positive integers),

Pn+1 <(1-вп)Рп + К

lim p„ = 0,

where dn e [0,1), dn = rn and bn = o(dn).

Remark 5 (see [6]). Since K is continuously D(A) invertible, there exists a constant p > 0 such that

\\Kx\\> ß\\x\\, VxeD(K)=D(A).

In the continuation c e (0,1), a and p are the constants appearing in (4), (6), and (12), respectively. Furthermore, e > 0 is defined by

*= C(~n v ne(0,c). (13)

With these notations, we now prove our main results.

Theorem 6. Let E be a real separable Banach space with a strictly convex dual and let A : D(A) c E ^ E be a Kpd operator with D(A) = D(K). Suppose (Ax, j(Ky)) = (Kx, ](Ay)) for all x,ye D(A). Let x* denote a solution of the equation Ax = f. For arbitrary x0 e E, define the sequence

{xn)Zoin E by

xn = xn-1 + eK-1f - eK-1Axn, n>0. (14) Then, {xn}™0 converges strongly to x* with

\\xn -x*\\< pnp-1\\Kx0 -Kx*\\, (15)

where p = 1 - ((c - q)l(a(1 - q) + c - e (0,1). Thus, the

choice q = c/2yields p = 1- (c2/(4a(1-cl2) + 2c)). Moreover,

x is unique.

Proof. The existence of the unique solution to the equation Ax = f comes from Theorem 1. From (4) we have

(Ax-cKx,j(Kx))>0, (16)

and from Lemma 1.1 ofKato [23], we obtain that

\\Kx\\ < \\Kx + y(Ax- cKx)\\, (17)

for all x e D(A) and y > 0. Now, from (14), linearity of K and the fact that Ax* = f we obtain that

Kxn = Kxn—1 + ef - eAxn

= Kxn—1 + eAx - eAxn

which implies that

Kxn—1 = Kxn - eAx + eAxn

n— 1

With the help of (14) and Theorem 1, we have the following estimate:

\\Axn - Ax* || = \\A (xn -x*)\\<a ||K (xn - x*)\\ = a \\Kxn - Kx* \\

= a \\Kxn-1 -Kx* -e (Axn - Ax*)|| < a |- Kx* \\ + ae \\Axn - Ax*

which gives

\\AX" -^^T^-eW^"-1 -KX*W■

Furthermore, inequality (20) can be rewritten as Kxn-1 - Kx*

= (1+e)(Kxn -Kx*) + e (Axn - Ax* -c (Kxn - Kx*)) -e(l-c) (Kxn -Kx*)

= (1+e)

Kxn - Kx

+ (Axn - Ax* - c (Kxn - Kx*)) -e(l-c) (Kxn -Kx*)

= (1+e)

Kxn - Kx*

+ (Axn - Ax* - C (Kxn - Kx*)) -e(l-c) (Kxn—1 -Kx*)+e2 (1-c)(Axn -Ax*).

In addition, from (17) and (22), we get that \\Kxn—i - Kx* ||

>(1+e)

Kxn - Kx*

(Axn - Ax* - c (Kxn - Kx*))

-e(l-c) \\Kxn—1 -Kx*\\-e2 (1 - c) \\Axn - Ax* > (1+e)\\Kxn -Kx*\\-e(1-c)\\Kxn—1 -Kx*\\

-e2 (1-c)-^\\Kxn—i -Kx*\\, 1 - ae

which implies that \\Kxn -Kx**\\

1 + e(1-c)+e2 (1 - c) (aj (1 - ae))

\\Kxn—i -Kx*

= P \\Kxn—1 - Kx*

1 + e(1-c) + e2 (1 - c) (a/ (1 - ae)) 1 + e

1--+-(c-e{1-c)-^-) 1 + e\ 1- ae )

1 + e'

a(1 - q) + c - q

4a(1 - c/2) + 2c From (25) and (26), we have that

\\Kxn-Kx*\\<p\\Kxn 1 -Kx*\\<---<pn\\K(x0-x*)\\.

Hence by Remark 5, we get that

< ¡3-1 \\Kxn -Kx*\\<---< p"fT1 \\Kx0 - Kx*

as n ^ >x>. Thus, xn ^ x* as n ^ >x>.

In [6], Chuanzhi provided the following result.

(28) □

Theorem 7. Let E be a real uniformly smooth separable Banach space, and let A : D(A) c E ^ E be a Kpd operator with D(A) = D(K). Suppose (Ax, j(Ky)) = (Kx,j(Ay)) for all x,ye D(A). For arbitrary f e E and x0 e D(A), define the sequence {xn}™0 by

Xn+1 = Xn + tnYn, Yn = K_1f-K_1Axn, 1

0<tn <

!*n = o

lim tn = 0,

b(at„) < —, n>0, v nJ Ba

xn - x

where b(t) is as in (R), a is the constant appearing in inequality (6), c is the constant appearing in inequality (4), and

B = max |||Ky0||,l|.

Then, 0 converges strongly to the unique solution of Ax = f-

However, its implicit version is as follows.

Theorem 8. Let E be a real uniformly smooth separable Banach space, and let A : D(A) c E ^ E be a Kpd operator with D(A) = D(K). Suppose (Ax, j(Ky)) = (Kx, j(Ay)) for all x,ye D(A). For arbitrary f e E and x0 e D(A), define the sequence {xj^0 by

Xn = Xn-1 + tnT* (31)

yn = K-1f-K-1Axn, (32)

=^ rim*»=0, n-°- (33)

Then, {xn}™0 converges strongly to the unique solution of Ax = f-

Proof. The existence of the unique solution to the equation Ax = f comes from Theorem 1. Using (31) and (32) we obtain

KY„ = KYn-i - *„Ayn.

Consider

\\KYn\\2 = (KYn,j(Kyn)) = (Kyn-1 -tnAyn ,j(Kyn))

= (Kyn-l, j (Kyn)) - tn (Ayn, j (Kyn)) (35)

<\\KYn-i\\ \\Kyn\\-ctn\\Kyn\\2, which implies that

\\KYn\\<\\Kyn-i\\-ctn \\Kyn\\. (36)

Hence, {Kyn}^0 is bounded. Let

M1 = sup \\Kyn\\. (37)

n>0 v '

Also from (6) it can be easily seen that {Ayn}'^=0 is also bounded. Let

M2 = sup ||AyJ .

Denote M = Mi + M2; then M < (x>. By using (34) and Lemma 3, we have

HKYnH2 =HKyn-1 -tnAYnf

< HKy^f - 2tn {Ayn,j(Kyn-l))

+ max |||Kyn-1||,l}||tnAyn||b(||tnAyn|| = HKy^f -2tn (Ayn-1,j(Kyn-1)) + 2tn (Ayn-i -Ayn,j(Kyn-i)) + max IHKyn-iH,l}tn HAynHb(tn HAyn

< (1- 2ctn) HKyn-if + 2tn HAyn-i - AyJ ¡Ky^ + max IHKyn-iH, 1} atn\Ky„|| b (atn \KyJ)

< (l-2ctn)HKyn-iH2 + 2Mtntln + max [M,1]a2M2tnb (t„),

nn = \\AYn-i - AYn\. By using (6) and (34) we obtain that \\AYn-i - Ayn\\ = \\A(yn-i - yn)\\ < a (yn-i - yn = atn \\AyJ < Matn —> 0, as n-

Thus, Denote

0 as n

Pn = \\*« - p\\ ,

en = 2ctn, (43)

on = 2Mtnqn + max {M, 1} a2M2tnb (tn).

Condition (33) assures the existence of a rank n0 e N such that dn = 2ctn < 1, for all n > n0. Since b(t) is continuous, so limn^mb(tn) = 0 (by condition (33)). Now with the help of (33), (42), and Lemma 4, we obtain from (39) that

„limo \\KV«\\ = °. (44)

At last by Remark 5, yn ^ 0 as n ^ >x>; that is Axn ^ f as n ^ >x>. Because A has bounded inverse, this implies that xn ^ A-1 f, the unique solution of Axn = f. This completes the proof. □

Remark 9. (1) According to the estimates (6-8) of Martynjuk [2], we have

xn+i - Kx

1 + e, (1 - c) + ae, (1 - c + a) u *u

< -------- HKxn - Kx* || (45)

1 + e,

= e HKxn - Kx*

1 + ei (1 - c) + aef (1 - c + a)

1 + e,

1--i— (c - a (1 - c + a) e,)

1 + ei u

= 1--— n,

1 + e, '

for q = c-a(1 -c + a)ei or ei = (c-q)/a(1-c + a), q e (0,c). Thus,

d=1-----n

a(1-c + a)+c-q

4a (1 - c + a) + 2c

Table 1

n_1_2_3_4_5_6_7_8_

x^_0.0922_0.0851_0.07859_0.07528_0.06947_0.06411_0.05916_0.05459

Table 2

n_1_2_3_4_5_6_7_8_

x^_0.0893_0.0798_0.07136_0.06376_0.05698_0.05092_0.04550_0.04066

Table 3

n 1 2 3 4 5 6 7 8

Xn 0.0098 0.0096 0.00949 0.00933 0.00917 0.00901 0.00885 0.00870

Table 4

n 1 2 3 4 5 6 7 8

Xn 0.0098 0.0096 0.00941 0.00923 0.00905 0.00887 0.00869 0.00852

(2) For a > c/2, we observe that

4a(1 - c/2) +2c

(4a (1 - c/2) + 2c)(4a(1-c + a) + 2c)

Thus, the relation between Martynjuk [2] and our parameter of convergence, that is, between 9 and p, respectively, is the following:

Despite the fact that our scheme is implicit, inequality (49) shows that the results of Osilike and Udomene [7] are improved in the sense that our scheme converges faster.

Example 10. Suppose E = R, D(A) = R+, Ax = x, Kx = 21 (x* = 0 is the solution of Ax = /); then for the explicit iterative scheme due to Osilike and Udomene [7] we have

Kx„+i = Kxn elAxn,

which implies that

and hence

2xn+1 = 2xn e1Xn>

xn+i = (1 i)Xn'

Also for the implicit iterative scheme we have that Kxn = Kxn-1 - eAxn,

which implies that

1 + e/2

It can be easily seen that for c < 1/2 and a > 1/2, (4) and (6) are satisfied. Suppose c = 1/4 and a = 3/5; then q = 0.125, e = (c- n)/a{1 -tf) = 0.23810, e1 = (c - n)/a{1 - c + a) = 0.15432, p = 0.97596, and 0 = 0.983288 and so p < 0. Take x0 = 0.1; then from (52) we have Table 1 and for (54) we get Table 2.

Example 11. Let us take E = R, D(A) = R+, Ax = (1/4)x, Kx = 2x (x* = 0 is the solution of Ax = /); then for the explicit iterative scheme due to Osilike and Udomene [7] we have

Kx„+i = Kxn elAxn,

which implies that

and hence

2xn+1 = 2xn 4 Xn>

Also for the implicit iterative scheme we have that

Kxn = Kxn-1 - eAxn

which implies that

1 + e/8

It can be easily seen that for c < 1/8 and a > 1/8, (4) and (6) are satisfied. Suppose c = 0.0625 and a = 0.2; then ^ = 0.03125, e = (c- n)/a(1 - q) = 0.16129, e1 = (c - t])/a(1 -c + a) = 0.13736, p = 0.99566, and 0 = 0.99623 and so p<6. Take x0 = 0.01; then from (57) we have Table 3 and for (59) we get Table 4.

Even though our scheme is implicit we observe that it converges strongly to the solution of the Kpd operator equation Ax = f with the error estimate which is faster in comparison to the explicit error estimate obtained by Osilike and Udomene [7].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author, therefore, acknowledges his thanks to DSR for the financial support. This paper is dedicated to Professor Miodrag Matel-jevi'c on the occasion of his 65th birthday.

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