Multiple-set split feasibility problems for total asymptotically strict pseudocontractions mappings

The purpose of this article is to propose and investigate an algorithm for solving the multiple-set split feasibility problems for total asymptotically strict pseu-docontractions mappings in infinite-dimensional Hilbert spaces. The results presented in this article improve and extend some recent results of A. Moudafi, H. K. Xu, Y. Censor, A. Segal, T. Elfving, N. Kopf, T. Bortfeld, X. A. Motova, Q. Yang, A. Gibali, S. Reich and others.

2000 AMS Subject Classification: 47J05; 47H09; 49J25.

Throughout this article, we always assume that H₁, H₂ are real Hilbert spaces, "→", "⇀" are denoted by strong and weak convergence, respectively, and F(T) is the fixed point set of a mapping T.

Let G be a nonempty closed convex subset of H₁ and T : G → G a mapping.

T is said to be a contraction if there exists a constant α ∈ (0,1) such that

Banach contraction principle guarantees that every contractive mapping defined on complete metric spaces has a unique fixed point.

T is said to be a weak contraction if

where ψ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that ψ is positive on (0, ∞), ψ(0) = 0, and lim_t→∞ ψ(t) = ∞. We remark that the class of weak contractions was introduced by Alber and Guerre-Delabriere [1]. In 2001, Rhoades [2] showed that every weak contraction defined on complete metric spaces has a unique fixed point.

T is said to be nonexpansive if

T is said to be asymptotically nonexpansive if there exists a sequence {k_n} ⊂ [1, ∞) with k_n → 1 as n → ∞ such that

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [3] as a generalization of the class of nonexpansive mappings. They proved that if G is a nonempty closed convex bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on G, then T has a fixed point.

T is said to be total asymptotically nonexpansive if

where ϕ : [0, ∞) → [0, ∞) is a continuous and strictly increasing function with ϕ(0) = 0, and {μ_n} and {ξ_n} are nonnegative real sequences such that μ_n → 0 and ξ_n → 0 as n → ∞. The class of mapping was introduced by Alber et al. [4]. From the definition, we see that the class of total asymptotically nonexpansive mappings includes the class of asymptotically nonexpansive mappings as special cases, see [5,6] for more details.

T is said to be strictly pseudocontractive if there exists a constant κ ∈ [0, 1) such that

The class of strict pseudocontractions was introduced by Browder and Petryshyn [7] in a real Hilbert space. In 2007, Marino and Xu [8] obtained a weak convergence theorem for the class of strictly pseudocontractive mappings, see [8] for more details.

T is said to be an asymptotically strict pseudocontraction if there exist a constant κ ∈ [0, 1) and a sequence {k_n} ⊂ [1, ∞) with k_n → 1 as n → ∞ such that

The class of asymptotically strict pseudocontractions was introduced by Qihou [9] in 1996. Kim and Xu [10] proved that the class of asymptotically strict pseudocontractions is demiclosed at the origin and also obtained a weak convergence theorem for the class of mappings; see [10] for more details.

In this article, we introduce the following mapping.

Definition 1.1 Let H be a real Hilbert space, and G be a nonempty closed convex subset of H. A mapping T : G → G is said to be (κ, {μ_n}, {ξ_n}, ϕ)-total asymptotically strict pseudocontractive, if there exists a constant κ ∈ [0, 1) and sequences {μ_n} ⊂ [0, ∞), {ξ_n} ⊂ [0, ∞) with μ_n → 0 and ξ_n → 0 as n → ∞, and a continuous and strictly increasing function ϕ : [0, ∞) → [0, ∞) with ϕ(0) = 0 such that

Now, we give an example of total asymptotically strict pseudocontractive mapping.

Let C be a unit ball in a real Hilbert space l² and let T : C → C be a mapping defined by

where {a_i} is a sequence in (0, 1) such that .

It is proven in Goebal and Kirk [3] that

(i) ;

(ii) .

Denote by , then

Letting and {ξ_n} be a nonnegative real sequence with ξ_n → 0, then , we have

Remark 1.2 If ϕ(λ) = λ² and ξ_n = 0, then total asymptotically strict pseudocontractive mapping is asymptotically strict pseudocontraction mapping.

It is easy to see the following proposition holds.

Proposition 1.3 Let T : G → G be a (κ, {μ_n}, {ξ_n}, ϕ)-total asymptotically strict pseudocontractive mapping. If , then for each q ∈ F(T) and for each x ∈ G, the following inequalities hold and are equivalent:

The split feasibility problem (SFP) in finite-dimensional spaces was first introduced by Censor and Elfving [11] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [12]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [13-15].

The SFP in an infinite-dimensional Hilbert space can be found in [12,14,16-18].

The purpose of this article is to introduce and study the following multiple-set SFP(MSSFP) for total asymptotically strict pseudocontraction in the framework of infinite-dimensional Hilbert spaces:

where A : H₁ → H₂ is a bounded linear operator, S_i : H₁ → H₁ and T_i : H₂ → H₂, i = 1, 2, ..., N are mappings, and . In the sequel, we use Γ to denote the set of solutions of (MSSFP)--(1.12), i.e.,

To prove our main results, we first recall some definitions, notations, and conclusions.

Let E be a Banach space. A mapping T : E → E is said to be demi-closed at origin, if for any sequence {x_n} ⊂ E with x_n ⇀ x* and ||(I - T)x_n|| → 0, then x* = Tx*.

A Banach space E is said to have the Opial property, if for any sequence {x_n} with x_n ⇀ x*, then

Remark 1.4 It is well known that each Hilbert space possesses the Opial property.

Definition 1.5 Let H bea real Hilbert space.

(1) A mapping T : H → H is said to be uniformly L-Lipschitzian, if there exists a constant L > 0, such that

(2) A mapping T : H → H is said to be semi-compact, if for any bounded sequence {x_n} ⊂ H with lim_n→∞ ||x_n - Tx_n|| = 0, then there exists a subsequence such that converges strongly to some point x* ∈ H.

Lemma 1.6 [10] Let H be a real Hilbert space. If {x_n} is a sequence in H weakly convergent to z, then

Proposition 1.7 Assume that G is a closed convex subset of a real Hilbert space H and let T : G → G be a (κ, {μ_n}, {ξ_n}, ϕ)-total asymptotically strict pseudocon-traction mapping and uniformly L-Lipschitzian. Then the demiclosedness principle holds for I - T in the sense that if {x_n} is a sequence in G such that x_n ⇀ x*, and lim sup_m→∞ lim sup_n→∞ ||x_n - T^mx_n|| = 0 then (I - T)x* = 0. In particular, x_n ⇀ x*, and (I - T)x_n → 0 ⇒ (I - T)x* = 0, i.e., T is demiclosed at origin.

Proof Since {x_n} is bounded, we can define a function f on H by

By Lemma 1.6, the weak convergence x_n ⇀ x* implies that

In particular, for each m ≥ 1,

On the other hand, since T is a (κ, {μ_n}, {ξ_n})-total asymptotically strict pseudo-contraction mapping, by (1.8), we get

Taking lim sup_m→∞ on both sides and observing the facts that lim_m→∞ μ_m = 0, lim_m→∞ ξ_m = 0 and lim sup_m→∞ lim sup_n→∞ ||x_n - T^mx_n|| = 0, we derive that

Since lim sup_m→∞ f(T^mx*) = f(x*)+lim sup_m→∞ ||T^mx* - x*||², and f(x*) = lim sup_n→∞ ||x_n - x*||², it follows from (1.15) that lim sup_m→∞ ||x* - T^mx*||² = 0. That is, T^mx* → x*; hence Tx* = x*.

Lemma 1.8 [19] Let {a_n}, {b_n} and {δ_n} be sequences of nonnegative real numbers satisfying

If and , then the limit lim_n→∞ a_n exists.

For solving the multiple-set split feasibility problem (1.12), let us assume that the following conditions are satisfied:

1. H₁ and H₂ are two real Hilbert spaces, A : H₁ → H₂ is a bounded linear operator;

2. Let G, be a nonempty closed convex subset of H₁ and H₂ respectively, S_i : G → G, i = 1, 2,...,N, is a uniformly L_i-Lipschitzian and (β_i, {μ_i,n}, {ξ_i,n}, ϕ_i)-total asymptotically strictly pseudocontractive mapping and , is a uniformly -Lipschitzian and -total asymptotically strictly pseudocontractive mapping which satisfy the following conditions:

(i) ;

(ii) ;

(iii) ;

(iv) and .

(v)

We are now in a position to give the following result:

Theorem 2.1 Let and ϕ be the same as above. In addition, there exist positive constants M and M* such that ϕ(λ) ≤ M*λ² for all λ ≥ M. Let {x_n} be the sequence generated by:

where is a sequence in [0, 1] and γ > 0 is a constant satisfying the following conditions:

(vi) and , where δ ∈ (0, 1 - β) is a positive constant.

(I) If (where Γ is the set of solutions to (MSSFP)--(1.12)), then {x_n} converges weakly to a point x* ∈ Γ.

(II) In addition, if there exists a positive integer j such that S_j is semi-compact, then {x_n} and {u_n} both converge strongly to x* ∈ Γ.

The proof of conclusion (I)

(1) First we prove that for each p ∈ Γ, the following limits exist

In fact, since ϕ is an increasing function, it results that ϕ(λ) ≤ ϕ(M), if λ ≤ M and ϕ(λ) ≤ M*λ², if λ ≥ M. In either case, we can obtain that

Since p ∈ Γ, then and . From (2.1) and (1.10) we have

On the other hand, since

and

It follows from (1.11) we have

Substituting (2.6) and (2.7) into (2.5) and simplifying it, we have

Substituting (2.8) into (2.4) and after simplifying we have

where

By condition (vi) we have

By condition (iv), and . Hence, from Lemma 1.8 we know that the following limit exists

Consequently, from (2.9) and (2.10) we have that

This together with the condition (vi) implies that

and

It follows from (2.5), (2.10) and (2.12) that the limit ||u_n - p|| exists.

The conclusion (1) is proved.

(2) Next we prove that

In fact, it follows from (2.1) that

In view of (2.11) and (2.12) we have that

Similarly, it follows from (2.1), (2.12), and (2.14) that

The conclusion (2.13) is proved.

(3) Next we prove that for each j = 1, 2,..., N - 1,

In fact, from (2.11) we have

Since S_j is uniformly L_j-Lipschitzian continuous, it follows from (2.13) and (2.17) that

Similarly, for each j = 1, 2,..., N - 1, from (2.13) we have

Since T_j is uniformly -Lipschitzian continuous, by the same way as above, from (2.13) and (2.18), we can also prove that

(4) Finally we prove that x_n ⇀ x* and u_n ⇀ x* which is a solution of (MSSFP)--(1.12).

Since {u_n} is bounded. There exists a subsequence such that (some point in H₁). Hence, for any positive integer j = 1, 2,..., N, there exists a subsequence {n_i(j)} ⊂ {n_i} with n_i(j)(modN) = j such that . Again from (2.16) we have

Since S_j is demiclosed at zero (see Proposition 1.7), it gets that x* ∈ F(S_j). By the arbitrariness of j = 1, 2,..., N, we have .

Moreover, from (2.1) and (2.12) we have

Since A is a linear bounded operator, it gets . For any positive integer k = 1, 2,..., N, there exists a subsequence {n_i(k)} ⊂ {n_i} with n_i(k)(modN) = k such that . In view of (2.16) we have

Since T_k is demiclosed at zero, we have Ax* ∈ F(T_k). By the arbitrariness of k = 1, 2,..., N, it yields . This together with x* ∈ C shows that x* ∈ Γ, i.e., x* is a solution to the (MSSFP)--(1.12).

Now we prove that x_n ⇀ x* and u_n ⇀ x*.

In fact, if there exists another subsequence such that with y* ≠ x*. Consequently, by virtue of (2.2) and the Opial property of Hilbert space, we have

This is a contradiction. Therefore, u_n ⇀ x*. By using (2.1) and (2.12), we have

The proof of conclusion (II).

Without loss of generality, we can assume that S₁ is semi-compact. It follows from (2.20) that

Therefore, there exists a subsequence of (for the sake of convenience we still denote it by such that (some point in H₁). Since . This implies that x* = u*, and so . By virtue of (2.2) we know that lim_n→∞ ||u_n - x*|| = 0 and lim_n→∞ ||x_n - x*|| = 0, i.e., {u_n} and {x_n} both converge strongly to x* ∈ Γ.

This completes the proof of Theorem 2.1.

Remark 2.2 Since the class of total asymptotically strict pseudocontractive mappings includes the class of asymptotically strict pseudocontractions mappings and the class of strict pseudocontractions mappings as special cases, Theorem 2.1 improves and extend the corresponding results of Censor et al. [14,15], Yang [17], Moudafi [20], Xu [21], Censor and Segal [22], Censor et al. [23] and others.

The authors declare that they have no competing interests.

All authors have read and approved the final manuscript.

The authors would like to thank the referees for useful comments and suggestions. This study was supported by the Natural Science Foundation of Sichuan Province (No. 08ZA008).

1. Alber, YaI, Guerre-Delabriere, S: On the projection methods for fixed point problems. Analysis. 21, 17–39 (2001)

2. Rhoades, BE: Some theorems on weakly contractive maps. Nonlinear Anal: Theory Methods Appl. 47, 2683–2693 (2001). Publisher Full Text

3. Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Am Math Soc. 35, 171–174 (1972). Publisher Full Text

4. Alber, YaI, Chidume, CE, Zegeye, H: Approximating fixed points of total asymptotically nonex-pansive mappings. Fixed Point Theory Appl. 10673, 20 (2006)

5. Chidume, CE, Ofoedu, EU: A new iteration process for approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. Int J Math Math Sci. 615107, 17 (2009)

6. Chidume, CE, Ofoedu, EU: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J Math Anal Appl. 333, 128–141 (2007). Publisher Full Text

7. Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl. 20, 197–228 (1967). Publisher Full Text

8. Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl. 329, 336–346 (2007). Publisher Full Text

9. Qihou, L: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal: Theory Methods Appl. 26, 1835–1842 (1996). Publisher Full Text

10. Kim, TH, Xu, HK: Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions. Nonlinear Anal: Theory Methods Appl. 68, 2828–2836 (2008). Publisher Full Text

11. Censor, Y, Elfving, T: A multi-projection algorithm using Bregman projections in a product space. Numer Algorithms. 8, 221–239 (1994). Publisher Full Text

12. Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Problem. 18, 441–453 (2002). Publisher Full Text

13. Censor, Y, Bortfeld, T, Martin, B, Trofimov, A: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys Med Biol. 51, 2353–2365 (2006). PubMed Abstract | Publisher Full Text

14. Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications. Inverse Problem. 21, 2071–2084 (2005). Publisher Full Text

15. Censor, Y, Motova, XA, Segal, A: Pertured projections and subgradient projections for the multiple-setssplit feasibility problem. J Math Anal Appl. 327, 1244–1256 (2007). Publisher Full Text

16. Xu, HK: A variable Krasnosel'skii-Mann algorithm and the multiple-sets split feasibility problem. Inverse Problem. 22, 2021–2034 (2006). Publisher Full Text

17. Yang, Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Problem. 20, 1261–1266 (2004). Publisher Full Text

18. Zhao, J, Yang, Q: Several solution methods for the split feasibility problem. Inverse Problem. 21, 1791–1799 (2005). Publisher Full Text

19. Aoyama, K, Kimura, W, Takahashi, W, Toyoda, M: Approximation of common fixed points of acountable family of nonexpansive mappings on a Banach space. Nonlinear Anal: Theory Methods Appl. 67(8), 2350–2360 (2007). Publisher Full Text

20. Moudafi, A: The split common fixed point problem for demi-contractive mappings. Inverse Prob-lem. 26(055007), 6 (2010)

21. Xu, HK: Iterative methods for split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problem. 26(105018), 17 (2010)

22. Censor, Y, Segal, A: The split common fixed point problem for directed operators. J Convex Anal. 16, 587–600 (2009)

23. Censor, Y, Gibali, A, Reich, S: Algorithms for the split variational inequality problem. In: Numer Algorithm (accepted)