On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces

Abstract

We study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. (2009).

1. Introduction and Preliminaries

Let 𝐸 be a real Banach space with dual 𝐸^∗. The symbol 𝐷(𝑇) stands for the domain of 𝑇.

Let 𝑇∶𝐷(𝑇)→𝐸 be a mapping.

Definition 1. The mapping 𝑇 is said to be Lipschitzian if there exists a constant 𝐿>0 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦 (1) for all 𝑥,𝑦∈𝐷(𝑇).

Definition 2. The mapping 𝑇 is called strongly pseudocontractive if there exists 𝑡>1 such that 𝑥−𝑦≤(1+𝑟)𝑥−𝑦−𝑟𝑡𝑇𝑥−𝑇𝑦 (2) for all 𝑥,𝑦∈𝐷(𝑇) and 𝑟>0. If 𝑡=1 in inequality (2), then 𝑇 is called pseudocontractive.

We will denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽(𝑥)=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖²=𝑓^∗2, (3) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It follows from inequality (2) that 𝑇 is strongly pseudocontractive if and only if there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that (𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦,𝑗𝑥−𝑦≥𝑘𝑥−𝑦² (4) for all 𝑥,𝑦∈𝐷(𝑇), where 𝑘=(𝑡−1)/𝑡∈(0,1). Consequently, from inequality (4) it follows easily that 𝑇 is strongly pseudocontractive if and only if 𝑥−𝑦≤𝑥−𝑦+𝑠(𝐼−𝑇−𝑘𝐼)𝑥−(𝐼−𝑇−𝑘𝐼)𝑦 (5) for all 𝑥,𝑦∈𝐷(𝑇) and 𝑠>0.

Closely related to the class of pseudocontractive maps is the class of accretive operators.

Let 𝐴∶𝐷(𝐴)→𝐸 be an operator.

Definition 3. The operator 𝐴 is called accretive if 𝑥−𝑦≤𝑥−𝑦+𝑠𝐴𝑥−𝐴𝑦 (6) for all 𝑥,𝑦∈𝐷(𝐴) and 𝑠>0.

Also, as a consequence of Kato [1], this accretive condition can be expressed in terms of the duality mapping as follows.

For each 𝑥,𝑦∈𝐷(𝐴), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0. (7) Consequently, inequality (2) with 𝑡=1 yields that 𝐴 is accretive if and only if 𝑇∶=(𝐼−𝐴) is pseudocontractive. Furthermore, from setting 𝐴∶=(𝐼−𝑇), it follows from inequality (5) that 𝑇 is strongly pseudocontractive if and only if (𝐴−𝑘𝐼) is accretive, and, using (7), this implies that 𝑇=(𝐼−𝐴) is strongly pseudocontractive if and only if there exists 𝑘∈(0,1) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝑘𝑥−𝑦² (8) for all 𝑥,𝑦∈𝐷(𝐴). The operator 𝐴 satisfying inequality (8) is called strongly accretive. It is then clear that 𝐴 is strongly accretive if and only if 𝑇=(𝐼−𝐴) is strongly pseudocontractive. Thus, the mapping theory for strongly accretive operators is closely related to the fixed point theory of strongly pseudocontractive mappings. We will exploit this connection in the sequel.

The notion of accretive operators was introduced independently in 1967 by Kato [1] and Browder [2]. An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem 𝑑𝑢(0)𝑑𝑡+𝐴𝑢=0,𝑢=𝑢₀ (9) is solvable if 𝐴 is locally Lipschitzian and accretive on 𝐸. If 𝑢 is independent of 𝑡, then 𝐴𝑢=0 and the solution of this equation corresponds to the equilibrium points of the system (9). Consequently, considerable research efforts have been devoted, especially within the past 15 years or so, to developing constructive techniques for the determination of the kernels of accretive operators in Banach spaces (see, e.g., [3–19]. Two well-known iterative schemes, the Mann iterative method (see, e.g., [20]) and the Ishikawa iterative scheme (see, e.g., [21]), have successfully been employed.

The Mann and Ishikawa iterative schemes are global and their rate of convergence is generally of the order 𝑂(𝑛^−1/2). It is clear that if, for an operator 𝑈, the classical iterative sequence of the form, 𝑥_𝑛+1=𝑈𝑥_𝑛, 𝑥₀∈𝐷(𝑈) (the so-called Picard iterative sequence) converges, then it is certainly superior and preferred to either the Mann or the Ishikawa sequence since it requires less computations and, moreover, its rate of convergence is always at least as fast as that of a geometric progression.

In [22, 23], Chidume proved the following results.

Theorem 4. Let 𝐸 be an arbitrary real Banach space and 𝐴∶𝐸→𝐸 Lipschitz (with constant 𝐿>0) and strongly accretive with a strong accretive constant 𝑘∈(0,1). Let 𝑥^∗ denote a solution of the equation 𝐴𝑥=0. Set 𝜖∶=(1/2)(𝑘/(1+𝐿(3+𝐿−𝑘))) and define 𝐴_𝜖∶𝐸→𝐸 by 𝐴_𝜖𝑥∶=𝑥−𝜖𝐴𝑥 for each 𝑥∈𝐸.
For arbitrary 𝑥₀∈𝐸, define the sequence {𝑥_𝑛}^∞_𝑛=0 in 𝐸 by 𝑥_𝑛+1=𝐴_𝜖𝑥_𝑛,𝑛≥0. (10) Then {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑥^∗ with 𝑥_𝑛+1−𝑥^∗≤𝛿^𝑛𝑥₀−𝑥^∗, (11) where 𝛿=(1−(1/2)𝑘𝜖)∈(0,1). Moreover, 𝑥^∗ is unique.

Corollary 5. Let 𝐸 be an arbitrary real Banach space and 𝐾 a nonempty convex subset of 𝐸. Let 𝑇∶𝐾→𝐾 be Lipschitz (with constant 𝐿>0) and strongly pseudocontractive (i.e., 𝑇 satisfies inequality (5) for all 𝑥,𝑦∈𝐾). Assume that 𝑇 has a fixed point 𝑥^∗∈𝐾. Set 𝜖₀∶=(1/2)(𝑘/(1+𝐿(3+𝐿−𝑘))) and define 𝑇_𝜖0∶𝐾→𝐾 by 𝑇_𝜖0𝑥=(1−𝜖₀)𝑥+𝜖₀𝑇𝑥 for each 𝑥∈𝐾. For arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛}^∞_𝑛=0 in 𝐾 by 𝑥_𝑛+1=𝑇_𝜖0𝑥_𝑛,𝑛≥0. (12)
Then {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑥^∗ with 𝑥_𝑛+1−𝑥^∗≤𝛿^𝑛𝑥₀−𝑥^∗, (13) where 𝛿=(1−(1/2)𝑘𝜖₀)∈(0,1). Moreover, 𝑥^∗ is unique.

Recently, Ćirić et al. [24] improved the results of Chidume [22, 23], Liu [14], and Sastry and Babu [18] as in the following results.

Theorem 6. Let 𝐸 be an arbitrary real Banach space and 𝐴∶𝐸→𝐸 a Lipschitz (with constant 𝐿>0) and strongly accretive with a strong accretive constant 𝑘∈(0,1). Let 𝑥^∗ denote a solution of the equation 𝐴𝑥=0. Set 𝜀∶=(𝑘−𝜂)/𝐿(2+𝐿), 𝜂∈(0,𝑘) and define 𝐴_𝜀∶𝐸→𝐸 by 𝐴_𝜀𝑥∶=𝑥−𝜀𝐴𝑥 for each 𝑥∈𝐸. For arbitrary 𝑥₀∈𝐸, define the sequence {𝑥_𝑛}^∞_𝑛=0 in 𝐸 by 𝑥_𝑛+1=𝐴_𝜀𝑥_𝑛,𝑛≥0. (14) Then {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑥^∗ with 𝑥_𝑛+1−𝑥^∗≤𝜃^𝑛𝑥₀−𝑥^∗, (15) where 𝜃=(1−((𝑘−𝜂)/(𝑘(𝑘−𝜂)+𝐿(2+𝐿)))𝜂)∈(0,1). Thus the choice 𝜂=𝑘/2 yields 𝜃=1−𝑘²/2[𝑘+2𝐿(2+𝐿)]. Moreover, 𝑥^∗ is unique.

Corollary 7. Let 𝐸 be an arbitrary real Banach space and 𝐾 a nonempty convex subset of 𝐸. Let 𝑇∶𝐾→𝐾 be Lipschitz (with constant 𝐿>0) and strongly pseudocontractive (i.e., 𝑇 satisfies inequality (5) for all 𝑥,𝑦∈𝐾). Assume that 𝑇 has a fixed point 𝑥^∗∈𝐾. Set 𝜀₀∶=(𝑘−𝜂)/𝐿(2+𝐿), 𝜂∈(0,𝑘) and define 𝑇_𝜀0∶𝐾→𝐾 by 𝑇_𝜀0𝑥=(1−𝜀₀)𝑥+𝜀₀𝑇𝑥 for each 𝑥∈𝐾. For arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛}^∞_𝑛=0 in 𝐾 by 𝑥_𝑛+1=𝑇_𝜀0𝑥_𝑛,𝑛≥0. (16) Then {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑥^∗ with 𝑥_𝑛+1−𝑥^∗≤𝜃^𝑛𝑥₀−𝑥^∗, (17) where 𝜃=(1−((𝑘−𝜂)/(𝑘(𝑘−𝜂)+𝐿(2+𝐿)))𝜂)∈(0,1). Moreover, 𝑥^∗ is unique.

In this paper, we study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. [24].

2. Main Results

In the following theorems, 𝐿>1 will denote the Lipschitz constant of the operator 𝐴 and 𝑘 will denote the strong accretive constant of the operator 𝐴 as in inequality (8). Furthermore, 𝜖>0 is defined by 𝜖∶=𝑘−𝜂(().𝐿+1+𝐿)𝑘−𝜂,𝜂∈0,𝑘 (18) With these notations, we prove the following theorem.

Theorem 8. Let 𝐸 be an arbitrary real Banach space and 𝐴∶𝐸→𝐸 Lipschitz and strongly accretive with a strong accretive constant 𝑘∈(0,1). Let 𝑥^∗ denote a solution of the equation 𝐴𝑥=0. Define 𝐴_𝜖∶𝐸→𝐸 by 𝐴_𝜖𝑥_𝑛=(1−𝜖)𝑥_𝑛−1+𝜖𝑥_𝑛−𝜖𝐴𝑥_𝑛 for each 𝑥_𝑛∈𝐸. For arbitrary 𝑥₀∈𝐸, define the sequence {𝑥_𝑛}^∞_𝑛=0 in 𝐸 by 𝑥_𝑛=𝐴_𝜖𝑥_𝑛,𝑛≥1. (19) Then {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑥^∗ with 𝑥_𝑛+1−𝑥^∗≤𝜌^𝑛𝑥₀−𝑥^∗, (20) where 𝜌=(1−((𝑘−𝜂)/(𝐿+((𝑘−𝜂)(1+𝐿+𝑘)))𝜂)∈(0,1). Thus the choice 𝜂=𝑘/2 yields 𝜌=1−𝑘²/2[2𝐿+𝑘(1+𝐿+𝑘)]. Moreover, 𝑥^∗ is unique.

Proof. Existence of 𝑥^∗ follows from [5, Theorem 13.1]. Define 𝑇=(𝐼−𝐴) where 𝐼 denotes the identity mapping on 𝐸. Observe that 𝐴𝑥^∗=0 if and only if 𝑥^∗ is a fixed point of 𝑇. Moreover, 𝑇 is strongly pseudocontractive since 𝐴 is strongly accretive, and so 𝑇 also satisfies inequality (5) for all 𝑥,𝑦∈𝐸 and 𝑠>0. Furthermore, the recursion formula 𝑥_𝑛=𝐴_𝜖𝑥_𝑛 becomes 𝑥_𝑛=(1−𝜖)𝑥_𝑛−1+𝜖𝑇𝑥_𝑛,𝑛≥1. (21) Observe that 𝑥^∗=(1+𝜖)𝑥^∗+𝜖(𝐼−𝑇−𝑘𝐼)𝑥^∗−(1−𝑘)𝜖𝑥^∗, (22) and from the recursion formula (21) 𝑥_𝑛−1=(1+𝜖)𝑥_𝑛+𝜖(𝐼−𝑇−𝑘𝐼)𝑥_𝑛−(1−𝑘)𝜖𝑥_𝑛+𝜖²𝑥_𝑛−1−𝑇𝑥_𝑛, (23) which implies that 𝑥_𝑛−1−𝑥^∗=(1+𝜖)𝑥_𝑛−𝑥^∗+𝜖(𝐼−𝑇−𝑘𝐼)𝑥_𝑛−(𝐼−𝑇−𝑘𝐼)𝑥^∗−(1−𝑘)𝜖𝑥_𝑛−𝑥^∗+𝜖²𝑥_𝑛−1−𝑇𝑥_𝑛. (24) This implies using inequality (5) with 𝑠=𝜖/(1+𝜖) and 𝑦=𝑥^∗ that 𝑥_𝑛−1−𝑥^∗≥(1+𝜖)𝑥_𝑛−𝑥^∗+𝜖×)𝑥1+𝜖(𝐼−𝑇−𝑘𝐼_𝑛−(𝐼−𝑇−𝑘𝐼)𝑥^∗−(1−𝑘)𝜖𝑥_𝑛−𝑥^∗−𝜖²𝑥_𝑛−1−𝑇𝑥_𝑛≥(1+𝜖)𝑥_𝑛−𝑥^∗−(1−𝑘)𝜖𝑥_𝑛−𝑥^∗−𝜖²𝑥_𝑛−1−𝑇𝑥_𝑛=(1+𝑘𝜖)𝑥_𝑛−𝑥^∗−𝜖²𝑥_𝑛−1−𝑇𝑥_𝑛. (25) Observe that 𝑥_𝑛−1−𝑇𝑥_𝑛≤𝑥_𝑛−1−𝑇𝑥_𝑛−1+𝑇𝑥_𝑛−1−𝑇𝑥_𝑛≤𝐴𝑥_𝑛−1+𝑥_𝑛−1−𝑥_𝑛+𝐴𝑥_𝑛−1−𝐴𝑥_𝑛≤𝐿𝑥_𝑛−1−𝑥^∗+(1+𝐿)𝑥_𝑛−1−𝑥_𝑛=𝐿𝑥_𝑛−1−𝑥^∗+(1+𝐿)𝜖𝑥_𝑛−1−𝑇𝑥_𝑛, (26) and so 𝑥_𝑛−1−𝑇𝑥_𝑛≤𝐿()𝜖𝑥1−1+𝐿^𝑛−1−𝑥^∗, (27) so that from (25) we obtain 𝑥_𝑛−1−𝑥^∗≥(1+𝑘𝜖)𝑥_𝑛−𝑥^∗−𝐿𝜖²()𝜖𝑥1−1+𝐿^𝑛−1−𝑥^∗. (28) Therefore 𝑥_𝑛−𝑥^∗≤1+𝐿𝜖²/(1−(1+𝐿)𝜖)𝑥1+𝑘𝜖^𝑛−1−𝑥^∗, (29) where 𝜌=1+𝐿𝜖²/(1−(1+𝐿)𝜖)𝜖1+𝑘𝜖=1−1+𝑘𝜖𝑘−𝐿𝜖()𝜖𝜖1−1+𝐿=1−𝜂1+𝑘𝜖=1−𝑘−𝜂𝐿+𝑘−𝜂(1+𝐿+𝑘)𝜂. (30) From (29) and (30), we get 𝑥_𝑛−𝑥^∗≤𝜌𝑥_𝑛−1−𝑥^∗≤⋯≤𝜌^𝑛𝑥₀−𝑥^∗⟶0 (31) as 𝑛→∞. Hence 𝑥_𝑛→𝑥^∗ as 𝑛→∞. Uniqueness follows from the strong accretivity property of 𝐴.

The following is an immediate corollary of Theorem 8.

Corollary 9. Let 𝐸 be an arbitrary real Banach space and 𝐾 a nonempty convex subset of 𝐸. Let 𝑇∶𝐾→𝐾 be Lipschitz (with constant 𝐿>1) and strongly pseudocontractive (i.e., 𝑇 satisfies inequality (5) for all 𝑥,𝑦∈𝐾). Assume that 𝑇 has a fixed point 𝑥^∗∈𝐾. Set 𝜖₀∶=(𝑘−𝜂)/(𝐿+(1+𝐿)(𝑘−𝜂)), 𝜂∈(0,𝑘) and define 𝐴_𝜖0∶𝐾→𝐾 by 𝐴_𝜖0𝑥_𝑛=(1−𝜖₀)𝑥_𝑛−1+𝜖₀𝑥_𝑛−𝜖₀𝐴𝑥_𝑛 for each 𝑥_𝑛∈𝐾. For arbitrary 𝑥₀∈𝐾, define the sequence {𝑥_𝑛}^∞_𝑛=0 in 𝐾 by 𝑥_𝑛=𝐴_𝜖0𝑥_𝑛,𝑛≥1. (32) Then {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑥^∗ with 𝑥_𝑛+1−𝑥^∗≤𝜌^𝑛₀𝑥₀−𝑥^∗, (33) where 𝜌₀=(1−((𝑘−𝜂)/(𝐿+(𝑘−𝜂)(1+𝐿+𝑘)))𝜂)∈(0,1). Thus the choice 𝜂=𝑘/2 yields 𝜌₀=1−𝑘²/2[2𝐿+𝑘(1+𝐿+𝑘)]. Moreover, 𝑥^∗ is unique.

Proof. Observe that 𝑥^∗ is a fixed point of 𝑇 if and only if it is a fixed point of 𝑇_𝜖0. Furthermore, the recursion formula (32) is simplified to the formula 𝑥_𝑛=1−𝜖₀𝑥_𝑛−1+𝜖₀𝑇𝑥_𝑛, (34) which is similar to (21). Following the method of computations as in the proof of the Theorem 8, we obtain 𝑥_𝑛−𝑥^∗≤1+𝐿𝜖²₀/1−(1+𝐿)𝜖₀1+𝑘𝜖₀𝑥_𝑛−1−𝑥^∗=1−𝑘−𝜂()𝜂𝑥𝐿+𝑘−𝜂1+𝐿+𝑘^𝑛−1−𝑥^∗. (35) Set 𝜌₀=1−((𝑘−𝜂)/(𝐿+(𝑘−𝜂)(1+𝐿+𝑘)))𝜂. Then from (35) we obtain 𝑥_𝑛−𝑥^∗≤𝜌₀𝑥_𝑛−1−𝑥^∗≤⋯≤𝜌^𝑛₀𝑥₀−𝑥^∗⟶0 (36) as 𝑛→∞. This completes the proof.

Remark 10. Since 𝐿>1 and 𝑘<𝐿, we have 𝐿>𝑘−𝜂. (37) So we can easily obtain 1()>1𝐿+𝑘−𝜂1+𝐿+𝑘𝑘().𝑘−𝜂+𝐿2+𝐿 (38)
Now 𝜌=1−𝑘−𝜂()𝜂𝐿+𝑘−𝜂1+𝐿+𝑘<1−𝑘−𝜂𝑘()𝜂𝑘−𝜂+𝐿2+𝐿=𝜃. (39) Thus the relation between Ćirić et al. [24] and our parameter of convergence, that is, between 𝜃 and 𝜌, respectively, is the following: 𝜌<𝜃. (40)

Our convergence parameter 𝜌 shows the overall improvement for 𝜃, and consequently the results of Ćirić et al. [24] are improved.

Acknowledgment

The authors would like to thank the referees for useful comments and suggestions.