Volume 2012, Issue 1 250538

Research Article

Open Access

Hybrid Method with Perturbation for Lipschitzian Pseudocontractions

Lu-Chuan Ceng

Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China shnu.edu.cn

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Ching-Feng Wen,

Corresponding Author

Ching-Feng Wen

[email protected]

Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan kmu.edu.tw

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Lu-Chuan Ceng,

Lu-Chuan Ceng

Department of Mathematics, Shanghai Normal University and Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China shnu.edu.cn

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Ching-Feng Wen,

Corresponding Author

Ching-Feng Wen

[email protected]

Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan kmu.edu.tw

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First published: 17 July 2012

https://doi.org/10.1155/2012/250538

Citations: 1

Academic Editor: Jen-Chih Yao

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Abstract

Assume that F is a nonlinear operator which is Lipschitzian and strongly monotone on a nonempty closed convex subset C of a real Hilbert space H. Assume also that Ω is the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings on C. By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterative algorithm with perturbation F, which generates two sequences from an arbitrary initial point x₀ ∈ H. These two sequences are shown to converge in norm to the same point P_Ωx₀ under very mild assumptions.

1. Introduction and Preliminaries

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ||·|| and C a nonempty closed convex subset of H. Let T : C → C be a self-mapping of C. Recall that T is said to be a pseudocontractive mapping if

(1.1)

and T is said to be a strictly pseudo-contractive mapping if there exists a constant k ∈ [0,1) such that

(1.2)

For such cases, we also say that T is a k-strict pseudo-contractive mapping. We use F(T) to denote the set of fixed points of T.

It is well known that the class of strictly pseudo-contractive mappings strictly includes the class of nonexpansive mappings which are the mappings T on C such that

(1.3)

Iterative methods for nonexpansive mappings have been extensively investigated; see [1–16] and the references therein.

However, iterative methods for strictly pseudo-contractive mappings are far less developed than those for nonexpansive mappings though Browder and Petryshyn initiated their work in 1967; the reason is probably that the second term appearing on the right-hand side of (1.2) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strictly pseudo-contractive mapping T. However, on the other hand, strictly pseudo-contractive mappings have more powerful applications than nonexpansive mappings do in solving inverse problems; see Scherzer [17]. Therefore, it is interesting to develop iterative methods for strictly pseudo-contractive mappings. As a matter of fact, Browder and Petryshyn [18] showed that if a k-strict pseudo-contractive mapping T has a fixed point in C, then starting with an initial x₀ ∈ C, the sequence {x_n} generated by the recursive formula:

(1.4)

where α is a constant such that k < α < 1 converges weakly to a fixed point of T.

Recently, Marino and Xu [19] have extended Browder and Petryshyn′s result by proving that the sequence {x_n} generated by the following Mann′s algorithm:

(1.5)

converges weakly to a fixed point of T, provided that the control sequence {α_n} satisfies the condition that k < α_n < 1 for all n and

. However, this convergence is in general not strong. It is well known that if C is a bounded and closed convex subset of H, and T : C → C is a demicontinuous pseudocontraction, then T has a fixed point in C (Theorem 2.3 in [20]). However, all efforts to approximate such a fixed point by virtue of the normal Mann′s iteration algorithm failed.

In 1974, Ishikawa [21] introduced a new iteration algorithm and proved the following convergence theorem.

Theorem I (see [21].)If C is a compact convex subset of a Hilbert space H, T : C → C is a Lipschitzian pseudocontraction and x₀ ∈ C is chosen arbitrarily, then the sequence {x_n} _n≥0 converges strongly to a fixed point of T, where {x_n} is defined iteratively for each positive integer n ≥ 0 by

(1.6)

where {α_n} and {β_n} are sequences of real numbers satisfying the conditions (i) 0 ≤ α_n ≤ β_n < 1; (ii) β_n → 0 as n → ∞; (iii)

Since its publication in 1974, it remains an open question whether or not Mann′s iteration algorithm converges under the setting of Theorem I to a fixed point of T if the mapping T is Lipschitzian pseudo-contractive. In [22], Chidume and Mutangadura gave an example of a Lipschitzian pseudocontraction with a unique fixed point for which Mann′s iteration algorithm fails to converge.

In an infinite-dimensional Hilbert space, Mann and Ishikawa′s iteration algorithms have only weak convergence, in general, even for nonexpansive mapping. So, in order to get strong convergence for strictly pseudo-contractive mappings, several attempts have been made based on the CQ method (see, e.g., [19, 23, 24]). The last scheme, in such a direction, seems for us to be the following due to Zhou [25]:

(1.7)

He proved, under suitable choice of the parameters α_n and β_n, that the sequence {x_n} generated by (1.7) strongly converges to P_F(T)x₀.

Among classes of nonlinear mappings, the class of pseudocontractions is one of the most important. This is due to the relation between the class of pseudocontractions and the class of monotone mappings (we recall that a mapping A is monotone if 〈Ax − Ay, x − y〉 ≥ 0 for all x, y ∈ H). A mapping A is monotone if and only if (I − A) is pseudo-contractive. It is well known (see, e.g., [26]) that if S is monotone, then the solutions of the equation Sx = 0 correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts, especially within the past 30 years or so, have been devoted to iterative methods for approximating fixed points of a pseudo-contractive mapping T (see e.g., [27–32] and the references therein).

Very recently, motivated by the work in [19, 25, 33] and the related work in the literature, Yao et al. [34] suggested and analyzed a hybrid algorithm for pseudo-contractive mappings in Hilbert spaces. Further, they proved the strong convergence of the proposed iterative algorithm for Lipschitzian pseudo-contractive mappings.

Theorem YLM (see [34].)Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a L-Lipschitzian pseudo-contractive mapping such that F(T) ≠ ∅. Assume that the sequence α_n ∈ [a, b] for some a, b ∈ (0, 1/(L + 1)). Let x₀ ∈ H. For C₁ = C and , let {x_n} be the sequence in C generated iteratively by

(1.8)

Then {x_n} converges strongly to P_F(T)x₀.

Inspired by the above research work of Yao et al. [34], in this paper we will continue this direction of research. Let C be a nonempty closed convex subset of a real Hilbert space H. We will propose a new hybrid iterative scheme with perturbed mapping for approximating fixed points of a Lipschitzian pseudo-contractive self-mapping on C. We will establish a strong convergence theorem for this hybrid iterative scheme. To be more specific, let T : C → C be a L-Lipschitzian pseudo-contractive mapping and F : C → H a mapping such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strong monotone. Let {α_n}⊂(0,1), {λ_n}⊂[0,1) and take a fixed number μ ∈ (0, 2η/κ²). We introduce the following hybrid iterative process with perturbed mapping F. Let x₀ ∈ H. For C₁ = C and

, two sequences {x_n}, {y_n} are generated as follows:

(1.9)

It is clear that if λ_n = 0, for all n ≥ 1, then the hybrid iterative scheme (1.9) reduces to the hybrid iterative process (1.8). Under very mild assumptions, we obtain a strong convergence theorem for the sequences {x_n} and {y_n} generated by the introduced method. Our proposed hybrid method with perturbation is quite general and flexible and includes the hybrid method considered in [34] and several other iterative methods as special cases. Our results represent the modification, supplement, extension, and improvement of [34, Algorithm 3.1 and Theorem 3.1]. Further, we consider the more general case, where

are N L-Lipschitzian pseudo-contractive self-mappings on C with N ≥ 1 an integer. In this case, we propose another hybrid iterative process with perturbed mapping F for approximating a common fixed point of

. Let x₀ ∈ H. For C₁ = C and

, two sequences {x_n} and {y_n} are generated as follows:

(1.10)

where T_n : = T_nmod N, for integer n ≥ 1, with the mod function taking values in the set {1,2, …, N} (i.e., if n = jN + q for some integers j ≥ 0 and 0 ≤ q < N, then T_n = T_N if q = 0 and T_n = T_q if 1 < q < N). It is clear that if N = 1, then the hybrid iterative scheme (1.10) reduces to the hybrid iterative process (1.9). Under quite appropriate conditions, we derive a strong convergence theorem for the sequences {x_n} and {y_n} generated by the proposed method.

We now give some preliminaries and results which will be used in the rest of this paper. A Banach space X is said to satisfy Opial′s condition if whenever {x_n} is a sequence in X which converges weakly to x, then

(1.11)

It is well known that every Hilbert space H satisfies Opial′s condition (see, e.g., [35]). Throughout this paper, we shall use the notations: “⇀” and “→” standing for the weak convergence and strong convergence, respectively. Moreover, we shall use the following notation: for a given sequence {x_n} ⊂ X, ω_w(x_n) denotes the weak ω-limit set of {x_n}, that is,

(1.12)

In addition, for each point x ∈ H, there exists a unique nearest point in C, denoted by P_Cx, such that

(1.13)

where P_C is called the metric projection of H onto C. It is known that P_C is a nonexpansive mapping.

Now we collect some lemmas which will be used in the proof of the main result in the next section. We note that Lemmas 1.1 and 1.2 are well known.

Lemma 1.1. Let H be a real Hilbert space. There holds the following identity:

(1.14)

Lemma 1.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Given x ∈ H and z ∈ C. Then z = P_Cx if and only if there holds the relation:

(1.15)

Lemma 1.3 (see [23].)Let C be a nonempty closed convex subset of H. Let {x_n} be a sequence in H and u ∈ H. Let q = P_Cu. If {x_n} is such that ω_w(x_n) ⊂ C and satisfies the condition:

(1.16)

Then x_n → q.

Lemma 1.4 (see [27].)Let X be a real reflexive Banach space which satisfies Opial′s condition. Let C be a nonempty closed convex subset of X, and T : C → C be a continuous pseudo-contractive mapping. Then, I − T is demiclosed at zero.

Let T : C → C be a nonexpansive mapping and F : C → H be a mapping such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strongly monotone, that is, F satisfies the following conditions:

(1.17)

respectively. For any given numbers λ ∈ [0,1) and μ ∈ (0, 2η/κ²), we define the mapping T^λ : C → H:

(1.18)

Lemma 1.5 (see [36].)If 0 ≤ λ < 1 and 0 < μ < 2η/κ², then there holds for T^λ : C → H:

(1.19)

where

In particular, whenever T = I the identity operator of H, we have

(1.20)

2. Main Result

In this section, we introduce a hybrid iterative algorithm with perturbed mapping for pseudo-contractive mappings in a real Hilbert space H.

Algorithm 2.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a pseudo-contractive mapping and F : C → H be a mapping such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strong monotone. Let {α_n} ⊂ (0,1), {λ_n}⊂[0,1) and take a fixed number μ ∈ (0,2η/κ²). Let x₀ ∈ H. For C₁ = C and , define two sequences: {x_n} and {y_n} of C as follows:

(2.1)

Now we prove the strong convergence of the above iterative algorithm for Lipschitzian pseudo-contractive mappings.

Theorem 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a L-Lipschitzian pseudo-contractive mapping such that F(T) ≠ ∅, and let F : C → H be a mapping such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strong monotone. Assume that {α_n}⊂[a, b] for some a, b ∈ (0, 1/(L + 1)) and {λ_n}⊂[0,1) such that lim _n→∞λ_n = 0. Take a fixed number μ ∈ (0, 2η/κ²). Then the sequences {x_n} and {y_n} generated by (2.1) converge strongly to the same point P_F(T)x₀.

Proof. Firstly, we observe that P_F(T) and {x_n} are well defined. From [19, 27], we note that F(T) is closed and convex. Indeed, by [27], we can define a mapping g : C → C by g(x) = (2I − T) ⁻¹ for every x ∈ C. It is clear that g is a nonexpansive self-mapping such that F(T) = F(g). Hence, by [23, Proposition 2.1 (iii)], we conclude that F(g) = F(T) is a closed convex set. This implies that the projection P_F(T) is well defined. It is obvious that {C_n} is closed and convex. Thus, {x_n} is also well defined.

Now, we show that F(T) ⊂ C_n for all n ≥ 0. Indeed, taking p ∈ F(T), we note that (I − T)p = 0, and (1.1) is equivalent to

(2.2)

Using Lemma 1.1 and (2.2), we obtain

(2.3)

Since T is L-Lipschitzian, utilizing Lemma 1.5 we derive

(2.4)

From (2.1), we observe that x_n − y_n = α_n(I − P_C(I − λ_nμF)T)x_n. Hence, utilizing Lemma 1.5 and (2.4) we obtain

(2.5)

Combining (2.3) and (2.5), we get

(2.6)

At the same time, we observe that

(2.7)

Therefore, from (2.6) and (2.7) we have

(2.8)

which implies that

(2.9)

that is,

(2.10)

From , we have

(2.11)

Utilizing F(T) ⊂ C_n, we also have

(2.12)

So, for all u ∈ F(T) we have

(2.13)

which hence implies that

(2.14)

Thus, {x_n} is bounded and so are {y_n} and {Ty_n}.

From and , we have

(2.15)

Hence,

(2.16)

and therefore

(2.17)

This implies that lim _n→∞| | x_n − x₀|| exists.

From Lemma 1.1 and (2.15), we obtain

(2.18)

Since x_n+1 ∈ C_n+1 ⊂ C_n, from | | x_n − x_n+1|| → 0 and λ_n → 0 it follows that

(2.19)

Noticing that α_n ∈ [a, b] for some a, b ∈ (0, 1/(L + 1)), thus, we obtain

(2.20)

Also, we note that | | Ty_n − P_C(I − λ_nμF)Ty_n|| ≤ λ_nμ| | F(Ty_n)|| → 0. Therefore, we get

(2.21)

On the other hand, utilizing Lemma 1.5 we deduce that

(2.22)

that is,

(2.23)

Meantime, it is clear that

(2.24)

Consequently,

(2.25)

Now (2.25) and Lemma 1.4 guarantee that every weak limit point of {x_n} is a fixed point of T, that is, ω_w(x_n) ⊂ F(T). In fact, the inequality (2.14) and Lemma 1.3 ensure the strong convergence of {x_n} to P_F(T)x₀. Since | | x_n − y_n|| = | | α_n(I − P_C(I − λ_nμF)T)x_n|| → 0, it is immediately known that {y_n} converges strongly to P_F(T)x₀. This completes the proof.

Corollary 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a nonexpansive mapping such that F(T) ≠ ∅, and let F : C → H be a mapping such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strong monotone. Assume that {α_n}⊂[a, b] for some a, b ∈ (0, 1/2) and {λ_n}⊂[0,1) such that lim _n→∞λ_n = 0. Take a fixed number μ ∈ (0, 2η/κ²). Then the sequences {x_n} and {y_n} generated by (2.1) converge strongly to the same point P_F(T)x₀.

Corollary 2.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a L-Lipschitzian pseudo-contractive mapping such that F(T) ≠ ∅. Assume that {α_n}⊂[a, b] for some a, b ∈ (0, 1/(L + 1)) and {λ_n}⊂[0,1) such that lim _n→∞λ_n = 0. Then the sequences {x_n} and {y_n} generated by the scheme

(2.26)

converge strongly to the same point P_F(T)x₀.

Proof. Put μ = 2 and F = (1/2)I in Theorem 2.2. Then, in this case we have κ = η = 1/2, and hence

(2.27)

This implies that μ = 2 ∈ (0, 2η/κ²) = (0,4). Meantime, it is easy to see that the scheme (2.1) reduces to (2.26). Therefore, by Theorem 2.2, we obtain the desired result.

Corollary 2.5 ([34, Corollary 3.2]). Let A : H → H be a L-Lipschitzian monotone mapping for which A⁻¹(0) ≠ ∅. Assume that the sequence {α_n}⊂[a, b] for some a, b ∈ (0, 1/(L + 2)). Then the sequence {x_n} generated by the scheme

(2.28)

strongly converges to

Proof. Put λ_n = 0 and T = I − A in Corollary 2.4. Then, it is easy to see that the scheme (2.26) reduces to (2.28). Therefore, by Corollary 2.4, we derive the desired result.

Next, consider the more general case where Ω is expressed as the intersection of the fixed-point sets of N pseudo-contractive mappings T_i : C → C with N ≥ 1 an integer, that is,

(2.29)

In this section, we propose another hybrid iterative algorithm with perturbed mapping for a finite family of pseudo-contractive mappings in a real Hilbert space H.

Algorithm 2.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let be N pseudo-contractive self-mappings on C with N ≥ 1 an integer, and let F : C → H be a mapping such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strong monotone. Let {α_n} ⊂ (0,1), {λ_n} ⊂ [0,1), and take a fixed number μ ∈ (0 , 2η/κ²). Let x₀ ∈ H. For C₁ = C and , define two sequences {x_n}, {y_n} of C as follows:

(2.30)

where

(2.31)

for integer n ≥ 1, with the mod function taking values in the set {1,2, …, N} (i.e., if n = jN + q for some integers j ≥ 0 and 0 ≤ q < N, then T_n = T_N if q = 0 and T_n = T_q if 1 < q < N).

Theorem 2.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let be N L-Lipschitzian pseudo-contractive self-mappings on C such that , and let F : C → H be a mapping such that for some constants κ, η > 0, F is κ-Lipschitzian and η-strong monotone. Assume that {α_n}⊂[a, b] for some a, b ∈ (0, 1/(L + 1)) and {λ_n}⊂[0,1) such that lim _n→∞λ_n = 0. Take a fixed number μ ∈ (0, 2η/κ²). Then the sequences {x_n}, {y_n} generated by (2.30) converge strongly to the same point P_Ωx₀.

Proof. Firstly, as stated in the proof of Theorem 2.2, we can readily see that each F(T_i) is closed and convex for i = 1,2, …, N. Hence, Ω is closed and convex. This implies that the projection P_Ω is well defined. It is clear that the sequence {C_n} is closed and convex. Thus, {x_n} is also well defined.

Now let us show that Ω ⊂ C_n for all n ≥ 0. Indeed, taking p ∈ Ω, we note that (I − T_n)p = 0 and

(2.32)

Using Lemma 1.1 and (2.32), we obtain

(2.33)

Since each T_i is L-Lipschitzian for i = 1,2, …, N, utilizing Lemma 1.5 we derive

(2.34)

From (2.30), we observe that x_n − y_n = α_n(I − P_C(I − λ_nμF)T_n)x_n. Hence, utilizing Lemma 1.5 and (2.34) we obtain

(2.35)

Combining (2.33) and (2.35), we get

(2.36)

Meantime, we observe that

(2.37)

Therefore, from (2.36) and (2.37) we have

(2.38)

which implies that

(2.39)

that is,

(2.40)

From , we have

(2.41)

Utilizing Ω ⊂ C_n, we also have

(2.42)

So, for all u ∈ Ω we have

(2.43)

which hence implies that

(2.44)

Thus {x_n} is bounded and so are {y_n} and {T_ny_n}.

From and , we have

(2.45)

Hence,

(2.46)

and therefore

(2.47)

This implies that lim _n→∞| | x_n − x₀|| exists.

From Lemma 1.1 and (2.45), we obtain

(2.48)

Thus,

(2.49)

Obviously, it is easy to see that lim _n→∞| | x_n − x_n+i|| = 0 for each i = 1,2, …, N. Since x_n+1 ∈ C_n+1 ⊂ C_n, from | | x_n − x_n+1|| → 0 and λ_n → 0 it follows that

(2.50)

Noticing that α_n ∈ [a, b] for some a, b ∈ (0, 1/(L + 1)), thus, we obtain

(2.51)

Also, we note that | | T_ny_n − P_C(I − λ_nμF)T_ny_n|| ≤ λ_nμ| | F(T_ny_n)|| → 0. Therefore, we get

(2.52)

On the other hand, utilizing Lemma 1.5 we deduce that

(2.53)

that is,

(2.54)

Furthermore, it is clear that

(2.55)

Consequently,

(2.56)

and hence for each i = 1,2, …, N:

(2.57)

So, we obtain lim _n→∞| | x_n − T_n+ix_n|| = 0 for each i = 1,2, …, N. This implies that

(2.58)

Now (2.58) and Lemma 1.4 guarantee that every weak limit point of {x_n} is a fixed point of T_l. Since l is an arbitrary element in the finite set {1,2, …, N}, it is known that every weak limit point of {x_n} lies in Ω, that is, ω_w(x_n) ⊂ Ω. This fact, the inequality (2.44) and Lemma 1.3 ensure the strong convergence of {x_n} to P_Ωx₀. Since | | x_n − y_n|| = | | α_n(I − P_C(I − λ_nμF)T_n)x_n|| → 0, it follows immediately that {y_n} converges strongly to P_Ωx₀. This completes the proof.

Remark 2.8. Algorithm 3.1 in [34] for a Lipschitzian pseudocontraction is extended to develop our hybrid iterative algorithm with perturbation for N-Lipschitzian pseudocontractions; that is, Algorithm 2.6. Theorem 2.7 is more general and more flexible than Theorem 3.1 in [34]. Also, the proof of Theorem 2.7 is very different from that of Theorem 3.1 in [34] because our technique of argument depends on Lemma 1.5. Finally, we observe that several recent results for pseudocontractive and related mappings can be found in [37–42].

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Hybrid Method with Perturbation for Lipschitzian Pseudocontractions

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Hybrid Method with Perturbation for Lipschitzian Pseudocontractions

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1. Introduction and Preliminaries

2. Main Result

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