# Implicit Approximation Scheme for the Solution ofK-Positive Definite Operator Equation

Authors: Naseer Shahzad; Arif Rafiq; Habtu Zegeye
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We construct an implicit sequence suitable for the approximation of solutions ofK-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).


---

## Body

## 1. Introduction

LetE be a real Banach space and let J denote the normalized duality mapping from E to 2E* defined by
(1)J(x)={f*∈E*:〈x,f*〉=∥x∥2,∥f*∥=∥x∥},
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.LetE be a Banach space. Themodulus of smoothness of E is the function.ρ
E
:
[
0
,
∞
)
→
[
0
,
∞
) defined by
(2)ρE(t)=sup⁡{12(∥x+y∥+∥x-y∥)-1:∥x∥≤1,∥y∥≤t}.
The Banach space E is calleduniformly smooth if
(3)lim⁡t→0ρE(t)t=0.
A Banach space E is said to bestrictly convex if for two elements x,y∈E which are linearly independent we have that ∥x+y∥<∥x∥+∥y∥.LetE1 be a dense subspace of a Banach space E. An operator T with domain D(T)⊇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).LetE 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)⊂D(K) and a constant c>0 such that j(Kx)∈J(Kx),
(4)〈Ax,j(Kx)〉≥c∥Kx∥2,∀x∈D(A).
Without loss of generality, we assume that c∈(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.
LetE 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
(5)〈Ax,j(Ky)〉=〈Kx,j(Ay)〉,∀x,y∈D(A).
Then, there exists a constant α>0 such that for all x∈D(A)(6)∥Ax∥≤α∥Kx∥.
Furthermore, the operator A is closed, R(A)=E, and the equation Ax=f has a unique solution for any given f∈E.As the special case of Theorem1 in which E=Lp (lp) spaces, 2≤p<∞, 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 q-uniformly smooth Banach spaces, 1<q<∞, 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.
LetE be a real separable Banach space with a strictly convex dual and let A:D(A)⊆E→E be a Kpd operator with D(A)=D(K). Suppose 〈Ax,j(Ky)〉=〈Kx,j(Ay)〉 for all x,y∈D(A). Choose any ϵ1∈(0,c2/(1+α(1-c)+α2)) and define Tϵ:D(A)⊆E→E by
(7)Tϵx=x+ϵK-1f-ϵK-1Ax.
Then the Picard iteration scheme generated from an arbitrary x0∈D(A) by
(8)xn+1=Tϵxn=Tϵnx0
converges strongly to the solution of the equation Ax=f. Moreover, if x* denotes the solution of the equation Ax=f, then
(9)∥xn+1-x*∥≤(1-cϵ(1-c))nβ-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+1=αnxn+(1-αn)Txn, for nonlinear mapping T:C=D(T)→C, where the initial guess x0∈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 x0∈C, xn=αnxn-1+(1-αn)Tnxn, for Ti,i=1,2,…,N, nonexpansive mappings, and Tn=Tn(mod⁡N) and {αn}⊂(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 theKpd 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,∞)→[0,∞) such that b(0)=0, b(δt)≤δb(t) for all δ≥1 and
(R)∥x+y∥2≤∥x∥2+2〈y,j(x)〉+max⁡{∥x∥,1}∥y∥b(∥y∥),
for all x,y∈E.Lemma 4 (see [22]).
If there exists a positive integerN such that for all n≥N, n∈ℕ (the set of all positive integers),
(10)ρn+1≤(1-θn)ρn+bn,
then
(11)lim⁡n→∞ρn=0,
where θn∈[0,1), ∑n=1∞‍θn=∞ and bn=o(θn).Remark 5 (see [6]).
SinceK is continuously D(A) invertible, there exists a constant β>0 such that
(12)∥Kx∥≥β∥x∥,∀x∈D(K)=D(A).In the continuationc∈(0,1), α and β are the constants appearing in (4), (6), and (12), respectively. Furthermore, ϵ>0 is defined by
(13)ϵ=c-ηα(1-η),η∈(0,c).With these notations, we now prove our main results.Theorem 6.
LetE be a real separable Banach space with a strictly convex dual and let A:D(A)⊆E→E be a Kpd operator with D(A)=D(K). Suppose 〈Ax,j(Ky)〉=〈Kx,j(Ay)〉 for all x,y∈D(A). Let x* denote a solution of the equation Ax=f. For arbitrary x0∈E, define the sequence {xn}n=0∞ in E by
(14)xn=xn-1+ϵK-1f-ϵK-1Axn,n≥0.
Then, {xn}n=0∞ converges strongly to x* with
(15)∥xn-x*∥≤ρnβ-1∥Kx0-Kx*∥,
where ρ=1-((c-η)/(α(1-η)+c-η))η∈(0,1). Thus, the choice η=c/2 yields ρ=1-(c2/(4α(1-c/2)+2c)). Moreover, x* is unique.Proof.
The existence of the unique solution to the equationAx=f comes from Theorem 1. From (4) we have
(16)〈Ax-cKx,j(Kx)〉≥0,
and from Lemma 1.1 of Kato [23], we obtain that
(17)∥Kx∥≤∥Kx+γ(Ax-cKx)∥,
for all x∈D(A) and γ>0. Now, from (14), linearity of K and the fact that Ax*=f we obtain that
(18)Kxn=Kxn-1+ϵf-ϵAxn=Kxn-1+ϵAx*-ϵAxn,
which implies that
(19)Kxn-1=Kxn-ϵAx*+ϵAxn,
so that
(20)Kxn-1-Kx*=Kxn-Kx*-ϵAx*+ϵAxn.
With the help of (14) and Theorem 1, we have the following estimate:
(21)∥Axn-Ax*∥=∥A(xn-x*)∥≤α∥K(xn-x*)∥=α∥Kxn-Kx*∥=α∥Kxn-1-Kx*-ϵ(Axn-Ax*)∥≤α∥Kxn-1-Kx*∥+αϵ∥Axn-Ax*∥,
which gives
(22)∥Axn-Ax*∥≤α1-αϵ∥Kxn-1-Kx*∥.
Furthermore, inequality (20) can be rewritten as
(23)Kxn-1-Kx*=(1+ϵ)(Kxn-Kx*)+ϵ(Axn-Ax*-c(Kxn-Kx*))-ϵ(1-c)(Kxn-Kx*)=(1+ϵ)[ϵ1+ϵKxn-Kx*+ϵ1+ϵ(Axn-Ax*-c(Kxn-Kx*))]-ϵ(1-c)(Kxn-Kx*)=(1+ϵ)[ϵ1+ϵKxn-Kx*+ϵ1+ϵ(Axn-Ax*-c(Kxn-Kx*))]-ϵ(1-c)(Kxn-1-Kx*)+ϵ2(1-c)(Axn-Ax*).
In addition, from (17) and (22), we get that
(24)∥Kxn-1-Kx*∥≥(1+ϵ)∥ϵ1+ϵKxn-Kx*+ϵ1+ϵ(Axn-Ax*-c(Kxn-Kx*))∥-ϵ(1-c)∥Kxn-1-Kx*∥-ϵ2(1-c)∥Axn-Ax*∥≥(1+ϵ)∥Kxn-Kx*∥-ϵ(1-c)∥Kxn-1-Kx*∥-ϵ2(1-c)α1-αϵ∥Kxn-1-Kx*∥,
which implies that
(25)∥Kxn-Kx**∥≤1+ϵ(1-c)+ϵ2(1-c)(α/(1-αϵ))1+ϵ∥Kxn-1-Kx*∥=ρ∥Kxn-1-Kx*∥,
where
(26)ρ=1+ϵ(1-c)+ϵ2(1-c)(α/(1-αϵ))1+ϵ=1-ϵ1+ϵ(c-ϵ(1-c)α1-αϵ)=1-ϵ1+ϵη=1-c-ηα(1-η)+c-ηη=1-c24α(1-c/2)+2c.
From (25) and (26), we have that
(27)∥Kxn-Kx*∥≤ρ∥Kxn-1-Kx*∥≤⋯≤ρn∥K(x0-x*)∥.
Hence by Remark 5, we get that
(28)∥xn-x*∥≤β-1∥Kxn-Kx*∥≤⋯≤ρnβ-1∥Kx0-Kx*∥⟶0,
as n→∞. Thus, xn→x* as n→∞.In [6], Chuanzhi provided the following result.Theorem 7.
LetE be a real uniformly smooth separable Banach space, and let A:D(A)⊆E→E be a Kpd operator with D(A)=D(K). Suppose 〈Ax,j(Ky)〉=〈Kx,j(Ay)〉 for all x,y∈D(A). For arbitrary f∈E and x0∈D(A), define the sequence {xn}n=0∞ by
(29)xn+1=xn+tnγn,γn=K-1f-K-1Axn,0≤tn≤12c,∑tn=0,lim⁡n→∞tn=0,b(αtn)≤2cBα,n≥0,
where b(t) is as in (R), α is the constant appearing in inequality (6), c is the constant appearing in inequality (4), and
(30)B=max⁡{∥Kγ0∥,1}.
Then, {xn}n=0∞ converges strongly to the unique solution of Ax=f.However, its implicit version is as follows.Theorem 8.
LetE be a real uniformly smooth separable Banach space, and let A:D(A)⊆E→E be a Kpd operator with D(A)=D(K). Suppose 〈Ax,j(Ky)〉=〈Kx,j(Ay)〉 for all x,y∈D(A). For arbitrary f∈E and x0∈D(A), define the sequence {xn}n=0∞ by
(31)xn=xn-1+tnγn,(32)γn=K-1f-K-1Axn,(33)∑tn=∞,lim⁡n→∞tn=0,n≥0.
Then, {xn}n=0∞ converges strongly to the unique solution of Ax=f.Proof.
The existence of the unique solution to the equationAx=f comes from Theorem 1. Using (31) and (32) we obtain
(34)Kγn=Kγn-1-tnAγn.
Consider
(35)∥Kγn∥2=〈Kγn,j(Kγn)〉=〈Kγn-1-tnAγn,j(Kγn)〉=〈Kγn-1,j(Kγn)〉-tn〈Aγn,j(Kγn)〉≤∥Kγn-1∥∥Kγn∥-ctn∥Kγn∥2,
which implies that
(36)∥Kγn∥≤∥Kγn-1∥-ctn∥Kγn∥.
Hence, {Kγn}n=0∞ is bounded. Let
(37)M1=sup⁡n≥0∥Kγn∥.
Also from (6) it can be easily seen that {Aγn}n=0∞ is also bounded. Let
(38)M2=sup⁡n≥0∥Aγn∥.
Denote M=M1+M2; then M<∞.
By using (34) and Lemma 3, we have
(39)∥Kγn∥2=∥Kγn-1-tnAγn∥2≤∥Kγn-1∥2-2tn〈Aγn,j(Kγn-1)〉+max⁡{∥Kγn-1∥,1}∥tnAγn∥b(∥tnAγn∥)=∥Kγn-1∥2-2tn〈Aγn-1,j(Kγn-1)〉+2tn〈Aγn-1-Aγn,j(Kγn-1)〉+max⁡⁡{∥Kγn-1∥,1}tn∥Aγn∥b(tn∥Aγn∥)≤(1-2ctn)∥Kγn-1∥2+2tn∥Aγn-1-Aγn∥∥Kγn-1∥+max⁡⁡{∥Kγn-1∥,1}αtn∥Kγn∥b(αtn∥Kγn∥)≤(1-2ctn)∥Kγn-1∥2+2Mtnηn+max⁡⁡{M,1}α2M2tnb(tn),
where
(40)ηn=∥Aγn-1-Aγn∥.
By using (6) and (34) we obtain that
(41)∥Aγn-1-Aγn∥=∥A(γn-1-γn)∥≤α∥K(γn-1-γn)∥=αtn∥Aγn∥≤Mαtn⟶0,asn⟶∞.
Thus,
(42)ηn⟶0asn⟶∞.
Denote
(43)ρn=∥xn-p∥,θn=2ctn,σn=2Mtnηn+max⁡{M,1}α2M2tnb(tn).
Condition (33) assures the existence of a rank n0∈ℕ such that θn=2ctn≤1, for all n≥n0. Since b(t) is continuous, so lim⁡n→∞⁡b(tn)=0 (by condition (33)). Now with the help of (33), (42), and Lemma 4, we obtain from (39) that
(44)lim⁡n→∞∥Kγn∥=0.
At last by Remark 5, γn→0 as n→∞; that is Axn→f as n→∞. Because A has bounded inverse, this implies that xn→A-1f, the unique solution of Axn=f. This completes the proof.Remark 9.
(
1
) According to the estimates (6–8) of Martynjuk [2], we have
(45)∥Kxn+1-Kx*∥≤1+ϵ1(1-c)+αϵ12(1-c+α)1+ϵ1∥Kxn-Kx*∥=θ∥Kxn-Kx*∥,
where
(46)θ=1+ϵ1(1-c)+αϵ12(1-c+α)1+ϵ1=1-ϵ11+ϵ1(c-α(1-c+α)ϵ1)=1-ϵ11+ϵ1η,
for η=c-α(1-c+α)ϵ1 or ϵ1=(c-η)/α(1-c+α), η∈(0,c). Thus,
(47)θ=1-c-ηα(1-c+α)+c-ηη=1-c24α(1-c+α)+2c.(2) For α>c/2, we observe that
(48)ρ=1-c24α(1-c/2)+2c=θ-4αc2(4α(1-c/2)+2c)(4α(1-c+α)+2c)(α-c2).
Thus, the relation between Martynjuk [2] and our parameter of convergence, that is, between θ and ρ, respectively, is the following:
(49)ρ<θ.
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.
SupposeE=ℝ, D(A)=ℝ+, Ax=x, Kx=2I (x*=0 is the solution of Ax=f); then for the explicit iterative scheme due to Osilike and Udomene [7] we have
(50)Kxn+1=Kxn-ϵ1Axn,
which implies that
(51)2xn+1=2xn-ϵ1xn,
and hence
(52)xn+1=(1-ϵ12)xn.
Also for the implicit iterative scheme we have that
(53)Kxn=Kxn-1-ϵAxn,
which implies that
(54)xn=11+ϵ/2xn-1.
It can be easily seen that for c≤1/2 and α≥1/2, (4) and (6) are satisfied. Suppose c=1/4 and α=3/5; then η=0.125, ϵ=(c-η)/α(1-η)=0.23810, ϵ1=(c-η)/α(1-c+α)=0.15432, ρ=0.97596, and θ=0.983288 and so ρ<θ. Take x0=0.1; then from (52) we have Table 1 and for (54) we get Table 2.Table 1
n
1
2
3
4
5
6
7
8
x
n
0.0922
0.0851
0.07859
0.07528
0.06947
0.06411
0.05916
0.05459Table 2
n
1
2
3
4
5
6
7
8
x
n
0.0893
0.0798
0.07136
0.06376
0.05698
0.05092
0.04550
0.04066Example 11.
Let us takeE=ℝ, D(A)=ℝ+, Ax=(1/4)x, Kx=2x (x*=0 is the solution of Ax=f); then for the explicit iterative scheme due to Osilike and Udomene [7] we have
(55)Kxn+1=Kxn-ϵ1Axn,
which implies that
(56)2xn+1=2xn-ϵ14xn,
and hence
(57)xn+1=(1-ϵ18)xn.
Also for the implicit iterative scheme we have that
(58)Kxn=Kxn-1-ϵAxn,
which implies that
(59)xn=11+ϵ/8xn-1.
It can be easily seen that for c≤1/8 and α≥1/8, (4) and (6) are satisfied. Suppose c=0.0625 and α=0.2; then η=0.03125, ϵ=(c-η)/α(1-η)=0.16129, ϵ1=(c-η)/α(1-c+α)=0.13736, ρ=0.99566, and θ=0.99623and so ρ<θ. 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 theKpd operator equation Ax=f with the error estimate which is faster in comparison to the explicit error estimate obtained by Osilike and Udomene [7].Table 3
n
1
2
3
4
5
6
7
8
x
n
0.0098
0.0096
0.00949
0.00933
0.00917
0.00901
0.00885
0.00870Table 4
n
1
2
3
4
5
6
7
8
x
n
0.0098
0.0096
0.00941
0.00923
0.00905
0.00887
0.00869
0.00852



---

*Source: 683295-2014-03-23.xml*