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

Abstract

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 2^𝐸∗ defined by 𝐽(𝑥)=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖²,𝑓^∗=‖𝑥‖,(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.

𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝑡)1=sup2𝑥+𝑦+𝑥−𝑦−1∶‖𝑥‖≤1,𝑦≤𝑡.(2) The Banach space 𝐸 is called uniformly smooth if lim_𝑡→0𝜌_𝐸(𝑡)𝑡=0.(3) A Banach space 𝐸 is said to be strictly convex if for two elements 𝑥,𝑦∈𝐸 which are linearly independent we have that ‖𝑥+𝑦‖<‖𝑥‖+‖𝑦‖.

Let 𝐸₁ 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).

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)⊂𝐷(𝐾) and a constant 𝑐>0 such that 𝑗(𝐾𝑥)∈𝐽(𝐾𝑥), 𝐴𝑥,𝑗(𝐾𝑥)≥𝑐‖𝐾𝑥‖²,∀𝑥∈𝐷(𝐴).(4) 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. 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,𝑗𝐾𝑦=𝐾𝑥,𝑗𝐴𝑦,∀𝑥,𝑦∈𝐷(𝐴).(5) Then, there exists a constant 𝛼>0 such that for all 𝑥∈𝐷(𝐴)‖𝐴𝑥‖≤𝛼‖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.

As the special case of Theorem 1 in which 𝐸=𝐿_𝑝 (𝑙_𝑝) 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. Let 𝐸 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 𝑇_𝜖∶𝐷(𝐴)⊆𝐸→𝐸 by 𝑇_𝜖𝑥=𝑥+𝜖𝐾⁻¹𝑓−𝜖𝐾⁻¹𝐴𝑥.(7) Then the Picard iteration scheme generated from an arbitrary 𝑥₀∈𝐷(𝐴) by 𝑥_𝑛+1=𝑇_𝜖𝑥_𝑛=𝑇^𝑛_𝜖𝑥₀(8) converges strongly to the solution of the equation 𝐴𝑥=𝑓. Moreover, if 𝑥^∗ denotes the solution of the equation 𝐴𝑥=𝑓, then 𝑥_𝑛+1−𝑥^∗≤(1−𝑐𝜖(1−𝑐))^𝑛𝛽⁻¹𝐾𝑥₀−𝐾𝑥^∗.(9)

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 {𝑥_𝑛} via the recursive approach 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇𝑥_𝑛, 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 {𝑥_𝑛}, which is the modification of Mann, generated by 𝑥₀∈𝐶, 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+(1−𝛼_𝑛)𝑇_𝑛𝑥_𝑛, for 𝑇_𝑖,𝑖=1,2,…,𝑁, nonexpansive mappings, and 𝑇_𝑛=𝑇_𝑛(mod𝑁) and {𝛼_𝑛}⊂(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]). If 𝐸^∗ 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 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗(𝑥)+max{‖𝑥‖,1}𝑦𝑏𝑦,(R) for all 𝑥,𝑦∈𝐸.

Lemma 4 (see [22]). If there exists a positive integer 𝑁 such that for all 𝑛≥𝑁, 𝑛∈ℕ (the set of all positive integers), 𝜌_𝑛+1≤1−𝜃_𝑛𝜌_𝑛+𝑏_𝑛,(10) then lim_𝑛→∞𝜌_𝑛=0,(11) where 𝜃_𝑛∈[0,1), ∑^∞_𝑛=1𝜃_𝑛=∞ and 𝑏_𝑛=𝑜(𝜃_𝑛).

Remark 5 (see [6]). Since 𝐾 is continuously 𝐷(𝐴) invertible, there exists a constant 𝛽>0 such that ‖𝐾𝑥‖≥𝛽‖𝑥‖,∀𝑥∈𝐷(𝐾)=𝐷(𝐴).(12)

In the continuation 𝑐∈(0,1), 𝛼 and 𝛽 are the constants appearing in (4), (6), and (12), respectively. Furthermore, 𝜖>0 is defined by 𝜖=𝑐−𝜂𝛼().1−𝜂,𝜂∈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)⊆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 {𝑥_𝑛}^∞_𝑛=0 in 𝐸 by 𝑥_𝑛=𝑥_𝑛−1+𝜖𝐾⁻¹𝑓−𝜖𝐾⁻¹𝐴𝑥_𝑛,𝑛≥0.(14) Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑥^∗ with 𝑥_𝑛−𝑥^∗≤𝜌^𝑛𝛽⁻¹𝐾𝑥₀−𝐾𝑥^∗,(15) 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 equation 𝐴𝑥=𝑓 comes from Theorem 1. From (4) we have 𝐴𝑥−𝑐𝐾𝑥,𝑗(𝐾𝑥)≥0,(16) and from Lemma 1.1 of Kato [23], we obtain that ‖𝐾𝑥‖≤𝐾𝑥+𝛾(𝐴𝑥−𝑐𝐾𝑥),(17) for all 𝑥∈𝐷(𝐴) and 𝛾>0. Now, from (14), linearity of 𝐾 and the fact that 𝐴𝑥^∗=𝑓 we obtain that 𝐾𝑥_𝑛=𝐾𝑥_𝑛−1+𝜖𝑓−𝜖𝐴𝑥_𝑛=𝐾𝑥_𝑛−1+𝜖𝐴𝑥^∗−𝜖𝐴𝑥_𝑛,(18) which implies that 𝐾𝑥_𝑛−1=𝐾𝑥_𝑛−𝜖𝐴𝑥^∗+𝜖𝐴𝑥_𝑛,(19) so that 𝐾𝑥_𝑛−1−𝐾𝑥^∗=𝐾𝑥_𝑛−𝐾𝑥^∗−𝜖𝐴𝑥^∗+𝜖𝐴𝑥_𝑛.(20) With the help of (14) and Theorem 1, we have the following estimate: 𝐴𝑥_𝑛−𝐴𝑥^∗=𝐴𝑥_𝑛−𝑥^∗≤𝛼𝐾𝑥_𝑛−𝑥^∗=𝛼𝐾𝑥_𝑛−𝐾𝑥^∗=𝛼𝐾𝑥_𝑛−1−𝐾𝑥^∗−𝜖𝐴𝑥_𝑛−𝐴𝑥^∗≤𝛼𝐾𝑥_𝑛−1−𝐾𝑥^∗+𝛼𝜖𝐴𝑥_𝑛−𝐴𝑥^∗,(21) which gives 𝐴𝑥_𝑛−𝐴𝑥^∗≤𝛼1−𝛼𝜖𝐾𝑥_𝑛−1−𝐾𝑥^∗.(22) Furthermore, inequality (20) can be rewritten as 𝐾𝑥_𝑛−1−𝐾𝑥^∗=(1+𝜖)𝐾𝑥_𝑛−𝐾𝑥^∗+𝜖𝐴𝑥_𝑛−𝐴𝑥^∗−𝑐𝐾𝑥_𝑛−𝐾𝑥^∗−𝜖(1−𝑐)𝐾𝑥_𝑛−𝐾𝑥^∗=(1+𝜖)𝐾𝑥_𝑛−𝐾𝑥^∗+𝜖1+𝜖𝐴𝑥_𝑛−𝐴𝑥^∗−𝑐𝐾𝑥_𝑛−𝐾𝑥^∗−𝜖(1−𝑐)𝐾𝑥_𝑛−𝐾𝑥^∗=(1+𝜖)𝐾𝑥_𝑛−𝐾𝑥^∗+𝜖1+𝜖𝐴𝑥_𝑛−𝐴𝑥^∗−𝑐𝐾𝑥_𝑛−𝐾𝑥^∗−𝜖(1−𝑐)𝐾𝑥_𝑛−1−𝐾𝑥^∗+𝜖²(1−𝑐)𝐴𝑥_𝑛−𝐴𝑥^∗.(23) In addition, from (17) and (22), we get that 𝐾𝑥_𝑛−1−𝐾𝑥^∗≥(1+𝜖)𝐾𝑥_𝑛−𝐾𝑥^∗+𝜖1+𝜖𝐴𝑥_𝑛−𝐴𝑥^∗−𝑐𝐾𝑥_𝑛−𝐾𝑥^∗−𝜖(1−𝑐)𝐾𝑥_𝑛−1−𝐾𝑥^∗−𝜖²(1−𝑐)𝐴𝑥_𝑛−𝐴𝑥^∗≥(1+𝜖)𝐾𝑥_𝑛−𝐾𝑥^∗−𝜖(1−𝑐)𝐾𝑥_𝑛−1−𝐾𝑥^∗−𝜖²()𝛼1−𝑐1−𝛼𝜖𝐾𝑥_𝑛−1−𝐾𝑥^∗,(24) which implies that 𝐾𝑥_𝑛−𝐾𝑥^∗∗≤1+𝜖(1−𝑐)+𝜖²(1−𝑐)(𝛼/(1−𝛼𝜖))1+𝜖𝐾𝑥_𝑛−1−𝐾𝑥^∗=𝜌𝐾𝑥_𝑛−1−𝐾𝑥^∗,(25) where ()𝜌=1+𝜖1−𝑐+𝜖²(1−𝑐)(𝛼/(1−𝛼𝜖))𝜖1+𝜖=1−()𝛼1+𝜖𝑐−𝜖1−𝑐𝜖1−𝛼𝜖=1−𝜂1+𝜖=1−𝑐−𝜂𝛼𝜂𝑐1−𝜂+𝑐−𝜂=1−²().4𝛼1−𝑐/2+2𝑐(26) From (25) and (26), we have that 𝐾𝑥_𝑛−𝐾𝑥^∗≤𝜌𝐾𝑥_𝑛−1−𝐾𝑥^∗≤⋯≤𝜌^𝑛𝐾𝑥₀−𝑥^∗.(27) Hence by Remark 5, we get that 𝑥_𝑛−𝑥^∗≤𝛽⁻¹𝐾𝑥_𝑛−𝐾𝑥^∗≤⋯≤𝜌^𝑛𝛽⁻¹𝐾𝑥₀−𝐾𝑥^∗⟶0,(28) as 𝑛→∞. Thus, 𝑥_𝑛→𝑥^∗ as 𝑛→∞.

In [6], Chuanzhi provided the following result.

Theorem 7. Let 𝐸 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 {𝑥_𝑛}^∞_𝑛=0 by 𝑥_𝑛+1=𝑥_𝑛+𝑡_𝑛𝛾_𝑛,𝛾_𝑛=𝐾⁻¹𝑓−𝐾⁻¹𝐴𝑥_𝑛,0≤𝑡_𝑛≤1,2𝑐𝑡_𝑛=0,lim_𝑛→∞𝑡_𝑛𝑏=0,𝛼𝑡_𝑛≤2𝑐𝐵𝛼,𝑛≥0,(29) where 𝑏(𝑡) is as in (R), 𝛼 is the constant appearing in inequality (6), 𝑐 is the constant appearing in inequality (4), and 𝐵=max𝐾𝛾₀,1.(30) Then, {𝑥_𝑛}^∞_𝑛=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)⊆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 {𝑥_𝑛}^∞_𝑛=0 by 𝑥_𝑛=𝑥_𝑛−1+𝑡_𝑛𝛾_𝑛,(31)𝛾_𝑛=𝐾⁻¹𝑓−𝐾⁻¹𝐴𝑥_𝑛,(32)𝑡_𝑛=∞,lim_𝑛→∞𝑡_𝑛=0,𝑛≥0.(33) Then, {𝑥_𝑛}^∞_𝑛=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 𝐾𝛾_𝑛=𝐾𝛾_𝑛−1−𝑡_𝑛𝐴𝛾_𝑛.(34) Consider 𝐾𝛾_𝑛²=𝐾𝛾_𝑛,𝑗𝐾𝛾_𝑛=𝐾𝛾_𝑛−1−𝑡_𝑛𝐴𝛾_𝑛,𝑗𝐾𝛾_𝑛=𝐾𝛾_𝑛−1,𝑗𝐾𝛾_𝑛−𝑡_𝑛𝐴𝛾_𝑛,𝑗𝐾𝛾_𝑛≤𝐾𝛾_𝑛−1𝐾𝛾_𝑛−𝑐𝑡_𝑛𝐾𝛾_𝑛²,(35) which implies that 𝐾𝛾_𝑛≤𝐾𝛾_𝑛−1−𝑐𝑡_𝑛𝐾𝛾_𝑛.(36) Hence, {𝐾𝛾_𝑛}^∞_𝑛=0 is bounded. Let 𝑀₁=sup_𝑛≥0𝐾𝛾_𝑛.(37) Also from (6) it can be easily seen that {𝐴𝛾_𝑛}^∞_𝑛=0 is also bounded. Let 𝑀₂=sup_𝑛≥0𝐴𝛾_𝑛.(38) Denote 𝑀=𝑀₁+𝑀₂; then 𝑀<∞.
By using (34) and Lemma 3, we have 𝐾𝛾_𝑛²=𝐾𝛾_𝑛−1−𝑡_𝑛𝐴𝛾_𝑛²≤𝐾𝛾_𝑛−1²−2𝑡_𝑛𝐴𝛾_𝑛,𝑗𝐾𝛾_𝑛−1+max𝐾𝛾_𝑛−1,1𝑡_𝑛𝐴𝛾_𝑛𝑏𝑡_𝑛𝐴𝛾_𝑛=𝐾𝛾_𝑛−1²−2𝑡_𝑛𝐴𝛾_𝑛−1,𝑗𝐾𝛾_𝑛−1+2𝑡_𝑛𝐴𝛾_𝑛−1−𝐴𝛾_𝑛,𝑗𝐾𝛾_𝑛−1+max𝐾𝛾_𝑛−1,1𝑡_𝑛𝐴𝛾_𝑛𝑏𝑡_𝑛𝐴𝛾_𝑛≤1−2𝑐𝑡_𝑛𝐾𝛾_𝑛−1²+2𝑡_𝑛𝐴𝛾_𝑛−1−𝐴𝛾_𝑛𝐾𝛾_𝑛−1+max𝐾𝛾_𝑛−1,1𝛼𝑡_𝑛𝐾𝛾_𝑛𝑏𝛼𝑡_𝑛𝐾𝛾_𝑛≤1−2𝑐𝑡_𝑛𝐾𝛾_𝑛−1²+2𝑀𝑡_𝑛𝜂_𝑛+max{𝑀,1}𝛼²𝑀²𝑡_𝑛𝑏𝑡_𝑛,(39) where 𝜂_𝑛=𝐴𝛾_𝑛−1−𝐴𝛾_𝑛.(40) By using (6) and (34) we obtain that 𝐴𝛾_𝑛−1−𝐴𝛾_𝑛=𝐴𝛾_𝑛−1−𝛾_𝑛≤𝛼𝐾𝛾_𝑛−1−𝛾_𝑛=𝛼𝑡_𝑛𝐴𝛾_𝑛≤𝑀𝛼𝑡_𝑛⟶0,as𝑛⟶∞.(41) Thus, 𝜂_𝑛⟶0as𝑛⟶∞.(42) Denote 𝜌_𝑛=𝑥_𝑛,𝜃−𝑝_𝑛=2𝑐𝑡_𝑛,𝜎_𝑛=2𝑀𝑡_𝑛𝜂_𝑛+max{𝑀,1}𝛼²𝑀²𝑡_𝑛𝑏𝑡_𝑛.(43) Condition (33) assures the existence of a rank 𝑛₀∈ℕ such that 𝜃_𝑛=2𝑐𝑡_𝑛≤1, for all 𝑛≥𝑛₀. Since 𝑏(𝑡) is continuous, so lim_𝑛→∞𝑏(𝑡_𝑛)=0 (by condition (33)). Now with the help of (33), (42), and Lemma 4, we obtain from (39) that lim_𝑛→∞𝐾𝛾_𝑛=0.(44) At last by Remark 5, 𝛾_𝑛→0 as 𝑛→∞; that is 𝐴𝑥_𝑛→𝑓 as 𝑛→∞. Because 𝐴 has bounded inverse, this implies that 𝑥_𝑛→𝐴⁻¹𝑓, the unique solution of 𝐴𝑥_𝑛=𝑓. This completes the proof.

Remark 9. (1) According to the estimates (6–8) of Martynjuk [2], we have 𝐾𝑥_𝑛+1−𝐾𝑥^∗≤1+𝜖₁(1−𝑐)+𝛼𝜖²₁(1−𝑐+𝛼)1+𝜖₁𝐾𝑥_𝑛−𝐾𝑥^∗=𝜃𝐾𝑥_𝑛−𝐾𝑥^∗,(45) where 𝜃=1+𝜖₁(1−𝑐)+𝛼𝜖²₁(1−𝑐+𝛼)1+𝜖₁𝜖=1−¹1+𝜖₁𝑐−𝛼(1−𝑐+𝛼)𝜖₁𝜖=1−¹1+𝜖₁𝜂,(46) for 𝜂=𝑐−𝛼(1−𝑐+𝛼)𝜖₁ or 𝜖₁=(𝑐−𝜂)/𝛼(1−𝑐+𝛼), 𝜂∈(0,𝑐). Thus, 𝜃=1−𝑐−𝜂𝛼()𝜂𝑐1−𝑐+𝛼+𝑐−𝜂=1−²().4𝛼1−𝑐+𝛼+2𝑐(47)(2) For 𝛼>𝑐/2, we observe that 𝑐𝜌=1−²4𝛼(1−𝑐/2)+2𝑐=𝜃−4𝛼𝑐²(()())𝑐4𝛼1−𝑐/2+2𝑐)(4𝛼1−𝑐+𝛼+2𝑐𝛼−2.(48) Thus, the relation between Martynjuk [2] and our parameter of convergence, that is, between 𝜃 and 𝜌, respectively, is the following: 𝜌<𝜃.(49)
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=2𝐼 (𝑥^∗=0 is the solution of 𝐴𝑥=𝑓); then for the explicit iterative scheme due to Osilike and Udomene [7] we have 𝐾𝑥_𝑛+1=𝐾𝑥_𝑛−𝜖₁𝐴𝑥_𝑛,(50) which implies that 2𝑥_𝑛+1=2𝑥_𝑛−𝜖₁𝑥_𝑛,(51) and hence 𝑥_𝑛+1=𝜖1−¹2𝑥_𝑛.(52) Also for the implicit iterative scheme we have that 𝐾𝑥_𝑛=𝐾𝑥_𝑛−1−𝜖𝐴𝑥_𝑛,(53) which implies that 𝑥_𝑛=1𝑥1+𝜖/2^𝑛−1.(54) 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 

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=𝑓); then for the explicit iterative scheme due to Osilike and Udomene [7] we have 𝐾𝑥_𝑛+1=𝐾𝑥_𝑛−𝜖₁𝐴𝑥_𝑛,(55) which implies that 2𝑥_𝑛+1=2𝑥_𝑛−𝜖₁4𝑥_𝑛,(56) and hence 𝑥_𝑛+1=𝜖1−¹8𝑥_𝑛.(57) Also for the implicit iterative scheme we have that 𝐾𝑥_𝑛=𝐾𝑥_𝑛−1−𝜖𝐴𝑥_𝑛,(58) which implies that 𝑥_𝑛=1𝑥1+𝜖/8^𝑛−1.(59) 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 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].

Table 3 

Table 4 

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 Mateljevi’c on the occasion of his 65th birthday.