CARPATHIAN 31 (2015), No. 3,
J.
MATH. 349 - 357
Online version available at http://carpathian.ubm.ro Print Edition: ISSN 1584 - 2851 Online Edition: ISSN 1843 - 4401
A general convergence theorem for multiple-set split feasibility problem in Hilbert spaces A BDUL R AHIM K HAN1 , M UJAHID A BBAS 2 and Y EKINI S HEHU3 and Q AMRUL H ASSAN A NSARI4 A BSTRACT. We establish strong convergence result of split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinite dimensional Hilbert spaces. 1. I NTRODUCTION Let C and Q be nonempty, closed and convex subsets of real Hilbert spaces H1 and H2 , respectively. The split feasibility problem (SFP) is to find a point x ∈ C such that Ax ∈ Q,
(1.1)
where A : H1 → H2 is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [5] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. The SFP has attracted the attention of many authors due to its application in signal processing (see, for example, [14, 16]). Note that the split feasibility problem ( 1.1 ) can be formulated as a fixed-point equation: (1.2)
PC (I − γA∗ (I − PQ )A)x∗ = x∗ ;
that is, x∗ solves the SFP (1.1 ) if and only if x∗ solves the fixed point equation (1.2), where A∗ is the adjoint of A, PC and PQ are metric projections onto C and Q, respectively. This implies that we can use fixed-point algorithms ( [4]) to solve SFP. A popular algorithm that solves the SFP (1.1) is CQ algorithm due to Byrne [2]. Byrne [3] applied Krasnosel’skiiMann (KM) iteration to the CQ algorithm, and Zhao and Yang [17] applied KM iteration to the perturbed CQ algorithm to solve the SFP. It is well known that the CQ algorithm and the KM algorithm for a split feasibility problem do not necessarily converge strongly in the infinite-dimensional Hilbert spaces. Let us recall some definitions: Definition 1.1. Let T : H → H be a mapping and F (T ) be its set of fixed points. T is said to be (a) nonexpansive if ||T x − T y|| ≤ ||x − y||, ∀x, y ∈ H. (b) quasi-nonexpansive if ||T x − T p|| ≤ ||x − p||, ∀x ∈ H, p ∈ F (T ). (c) firmly nonexpansive mapping if ||T x − T y||2 ≤ ||x − y||2 − Received: 01.09.2014; In revised form: 13.01.2015; Accepted: 15.01.2015 2010 Mathematics Subject Classification. 47H06, 47H09, 47J05, 47J25. Key words and phrases. Total asymptotically strict pseudocontractive mapping, single-valued (multi-valued) quasi-nonexpansive mapping, split common fixed-point problems, strong convergence, Hilbert space. Corresponding author: Abdul Rahim Khan;
[email protected] 349
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A. R. Khan, M. Abbas, Y. Shehu and Qamrul Hassan Ansari 2
||(x − y) − (T x − T y)|| , ∀x, y ∈ H. (d) quasi-firmly nonexpansive mapping if ||T x − T p||2 ≤ ||x−p||2 −||x−T x||2 , ∀x ∈ H, p ∈ F (T ). (e) strictly pseudocontractive mapping if there exists a constant k ∈ [0, 1) such that ||T x − T y||2 ≤ ||x − y||2 + k||(x − y) − (T x − T y)||2 , ∀x, y ∈ H. (f ) total asymptotically strict pseudocontractive mapping, if there exist a constant k ∈ [0, 1) and sequences {µn } ⊂ [0, ∞), {ξn } ⊂ [0, ∞) with µn → 0 and ξn → 0 as n → ∞, and a continuous and strictly increasing function φ : [0, ∞) → [0, ∞) with φ(0) = 0 such that for all n ≥ 1, x, y ∈ H ||T x − T y||2 ≤ ||x − y||2 + k||(x − y) − (T x − T y)||2 + µn φ(||x − y||) + ξn , ∀x, y ∈ H. The class of firmly-nonexpansive mappings is well known to include resolvents and projection operators, while the class of quasi-firmly nonexpansive mappings contains subgradient projection operators (see, e.g., [11] and the reference therein). In the sequel, we denote by CB(H) the collection of all nonempty, closed and bounded ˜ on CB(H) is defined by subsets of H. ThenHausdorff metric H o ˜ H(A, B) := max sup d(x, B), sup d(y, A) , ∀A, B ∈ CB(H), where D(x, K) := inf d(x, y). x∈A
y∈B
y∈K
Definition 1.2. Let S : H → CB(H) be a multi-valued mapping. An element p ∈ H is said to be a fixed point of S, if p ∈ Sp. The mapping S is said to be: (i) nonexpansive, if ˜ ˜ H(Sx, Sy) ≤ ||x − y||, ∀x, y ∈ H; (ii) quasi-nonexpansive, if F (S) 6= ∅ and H(Sx, Sp) ≤ ||x − p||, ∀x ∈ H, p ∈ F (S). We assume that H1 and H2 are two real Hilbert spaces, A : H1 → H2 bounded linear operator and A∗ : H2 → H1 the adjoint of A. We investigate the following multiple-set split feasibility problem (MSSFP) for a family of multi-valued quasi-nonexpansive mappings and a total asymptotically pseudo-contractive mapping in infinite dimensional Hilbert spaces, i.e., find x ∈ C such that Ax ∈ Q,
(1.3) {Si }∞ i=1
where, : H1 → CB(H1 ) is a family of multi- valued quasi-nonexpansive mappings and T : H2 → H2 is a total asymptotically strict pseudo-contractive mapping with nonempty fixed-point sets C := ∩∞ i=1 F (Si ); and Q := F (T ), and denote the solution set of (MSSFP) by Γ := {y ∈ C : Ay ∈ Q} = C ∩ A−1 (Q).
(1.4)
Recall that ∩∞ i=1 F (Si ) and F (T ) are nonempty, closed and convex subsets of H1 and H2 , respectively ([18]). If Γ 6= ∅, we have that Γ is closed and convex subset of H1 . Theorem 1.1. [9] Let {Si }∞ i=1 : H1 → CB(H1 ) be a family of multi-valued quasi-nonexpansive mappings and for each i ≥ 1, Si is demi-closed at 0. Let T : H2 → H2 be a uniformly L-Lipschitzan continuous and total asymptotically strict pseudocontractive mapping satisfying ∞ ∞ P P µn < ∞ and ξn < ∞. Suppose there exist constants M0 > 0, M1 > 0 such that n=1
n=1
φ(λ) ≤ M0 λ2 , ∀λ > M1 . Let C := ∩∞ i=1 F (Si ) 6= ∅; and Q := F (T ). Assume that for each p ∈ C; Si p = {p}, for each i ≥ 1. Let {xn } be the sequence generated by: x1 ∈ H1 chosen arbitrarily, ∞ P x αi,n wi,n , wi,n ∈ Si yn n+1 = α0,n yn + (1.5) i=1 yn = xn + γA∗ (T n − I)Axn , n ≥ 1,
Multiple-set split feasibility problem in Hilbert spaces
351
where {αi,n } ⊂ (0, 1) and γ > 0 satisfy the following conditions: ∞ P 1−k αi,n = 1, for each n ≥ 1; (b) for each i ≥ 1, lim inf α0,n αi,n > 0 (c) γ ∈ 0, ||A|| . (a) 2 n→∞
i=0
If Ω (the set of solutions of multiple-set split feasibility problem 1.3) is nonempty, then both ∞ {xn }∞ n=1 and {yn }n=1 converge weakly to some point x ∈ Ω. If in addition, there exists some positive integer m such that Sm is semi-compact, then ∞ both {xn }∞ n=1 and {yn }n=1 converge strongly to x ∈ Ω. Question: Can we modify the iterative scheme (1.5) so that the strong convergence is guaranteed without compactness of the operator? In this paper, we modify the iterative scheme (1.5) and prove strong convergence result for multiple-set split feasibility problem (1.3) for a family of multi-valued quasinonexpansive mappings and a total asymptotically pseudo-contractive mapping in infinitely dimensional Hilbert spaces without compactness condition on the operator. Our results improve the corresponding results of Chang et al. [9], Censor et al. [6, 7], Yang [16], Moudafi [13], Xu [14], Censor and Segal [8], Masad and Reich [12] and others. 2. P RELIMINARIES Definition 2.3. A mapping T : H → H is called (1) demiclosed at the origin if for any sequence {xn } ⊂ H with xn * x ( weak convergence ) and ||xn − T xn || → 0, we have T x = x (2) uniformly L-Lipschitzian continuous if there exists a constant L > 0 such that ||T n x − T n y|| ≤ L||x − y||, ∀x, y ∈ H, n ≥ 1 (3) A multi-valued mapping S : H → CB(H) is said to be demiclosed at origin if for any sequence {xn } ⊂ H with xn * x and d(xn , Sxn ) → 0, we have x ∈ Sx. Next, we state some well-known lemmas which will be used in the sequel. Lemma 2.1. Let H be a real Hilbert space. Then ∀x, y ∈ H, the following results hold: (i) ||x + y||2 = ||x||2 + 2hx, yi + ||y||2 ; (ii) ||x + y||2 ≤ ||x||2 + 2hy, x + yi. Lemma 2.2. ( Yang et al., [18]) Let T : H → H be a uniformly L-Lipschitzian continuous and total asymptotically strict pseudocontractive mapping, then T is demi-closed at origin. Lemma 2.3. ( Chang et al., [10]) Let E be a uniformly convex Banach space, Br (0) := {x ∈ E : ||x|| ≤ r} be a closed ball with center 0 and radius r > 0. For any given sequence {x1 , x2 , . . . , xn , . . .} ⊂ Br (0) and any given number sequence {λ1 , λ2 , . . . , λn , . . .} with λi ≥ ∞ P 0, λi = 1, there exists a continuous strictly increasing and convex function g : [0, 2r) → i=1
[0, ∞) with g(0) = 0 such that for any i, j ∈ N, i < j the following holds: ∞ ∞ X 2 X λ n xn ≤ λn ||xn ||2 − λi λj g(||xi − xj ||). n=1
n=1
Lemma 2.4. ( [1]) Let {λn } and {γn } be nonnegative, {αn } be positive real numbers such that λn+1 ≤ λn − αn λn + γn , n ≥ 1. Let for all n > 1, αγnn ≤ c1 and αn ≤ α. Then λn ≤ max{λ1 , K∗ }, where K∗ = (1 + α)c1 . Lemma 2.5. ( [15]) Let {an } be a sequence of nonnegative real P numbers satisfying the relation: an+1 ≤ (1−αn )an +αn σn +γn , n ≥ 0, where (i) {an } ⊂ [0, 1], αn = ∞; (ii) lim sup σn ≤ 0; (iii) γn ≥ 0; (n ≥ 1), Σγn < ∞. Then, an → 0 as n → ∞.
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3. M AIN RESULTS We assume that the following conditions are satisfied: (1) {Si }∞ i=1 : H1 → H1 is a family of multi-valued quasi-nonexpansive mappings and for each i ≥ 1, Si is demiclosed at origin; (2) T : H2 → H2 is a uniformly L-Lipschitzan continuous and total asymptotically strict pseudocontractive mapping satisfying the fol∞ ∞ P P lowing conditions: (i) µn < ∞; ξn < ∞; (ii) {αn } is a real sequence in (0,1) n=1
n=1
such that µn = o(αn ); ξn = o(αn ); lim αn = 0; n→∞
∞ P
αn = ∞; (iii) there exist constants
n=1
M0 > 0, M1 > 0 such that φ(λ) ≤ M0 λ , ∀λ > M1 ; (3) C := ∩∞ i=1 F (Si ) 6= ∅; and Q := F (T ) 6= ∅; (4) for each p ∈ C; Si p = {p}, for each i ≥ 1. A modification of the algorithm (1.5) is given in the following result to have the desired strong convergence. 2
Theorem 3.2. Let {Si }∞ i=1 , T, C, Q, k, {µn }, {ξn }, φ and L satisfy the above conditions (1)-(5). Let {xn } be the sequence generated by x1 ∈ H1 , un = (1 − αn )xn , yn = un + γA∗ (T n − I)Aun , ∞ P (3.1) βi,n wi,n , wi,n ∈ Si yn , n ≥ 1. xn+1 = β0,n yn + i=1
where {βi,n } ⊂ (0, 1) and γ > 0 satisfy the following conditions: ∞ P 1−k (a) βi,n = 1, for each n ≥ 1; (b) for each i ≥ 1, lim inf β0,n βi,n > 0 (c) γ ∈ 0, ||A|| . If Ω 2 n→∞
i=0
(the set of solutions of multiple-set split feasibility problem in (3.1)) is nonempty, then the sequence {xn }∞ n=1 converges strongly to an element of Ω. Proof. Note that φ is continuous, it follows that φ attains maximum (say M ) in [0, M1 ] and by our assumption, φ(λ) ≤ M0 λ2 , ∀λ > M1 . In either case, we have that φ(λ) ≤ M + M0 λ2 , ∀λ ∈ [0, ∞). Let x∗ ∈ Ω. Then from the convexity of ||.||2 , we obtain ||un − x∗ ||2
= ||(1 − αn )xn − x∗ ||2 = ||(1 − αn )(xn − x∗ ) + αn (−x∗ )||2 ≤ (1 − αn )||xn − x∗ ||2 + αn ||x∗ ||2 .
(3.2)
From (3.2) and Lemma 2.1 (i), we obtain that ||yn − x∗ ||2 = ||un − x∗ + γA∗ (T n − I)Aun ||2 (3.3)
= ||un − x∗ ||2 + 2γhun − x∗ , A∗ (T n − I)Aun i + γ 2 ||A∗ (T n − I)Aun ||2 .
Since γ 2 ||A∗ (T n − I)Aun ||2 (3.4)
=
γ 2 hA∗ (T n − I)Aun , A∗ (T n − I)Aun i
=
γ 2 hAA∗ (T n − I)Aun , (T n − I)Aun i
≤
γ 2 ||A||2 ||(T n − I)Aun ||2 ,
As Ax∗ ∈ Q = F (T ) and T is a total asymptotically strict pseudocontractive mapping, so we obtain
Multiple-set split feasibility problem in Hilbert spaces
353
hun − x∗ , A∗ (T n − I)Aun i = hA(un − x∗ ), (T n − I)Aun i = hA(un − x∗ ) + (T n − I)Aun − (T n − I)Aun , (T n − I)Aun i = hT n Aun − Ax∗ , (T n − I)Aun i − ||(T n − I)Aun ||2 i 1h = ||T n Aun − Ax∗ ||2 + ||(T n − I)Aun ||2 − ||Aun − Ax∗ ||2 2 − ||(T n − I)Aun ||2 i 1h ≤ ||Aun −Ax∗ ||2 +k||(T −I)Axn ||2 +µnφ(||Aun −Ax∗ ||)+ξn 2 i 1h + ||(T n − I)Aun ||2 − ||Aun − Ax∗ ||2 2 − ||(T n − I)Aun ||2 µn ξn k−1 (3.5) ||(T n − I)Aun ||2 + M + M0 ||Aun − Ax∗ ||2 ) + . ≤ 2 2 2 Substituting (3.5) and (3.4) into (3.3), we have ||yn − x∗ ||2
≤
||un − x∗ ||2 − γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 +µn γ(M + M0 ||Aun − Ax∗ ||2 + γξn .
(3.6)
Using (3.6) and (3.2) in (3.1), we obtain ||xn+1 − x∗ ||2
≤ β0,n ||yn − x∗ ||2 +
∞ X
βi,n ||wi,n − x∗ ||2
i=1
= β0,n ||yn − x∗ ||2 +
∞ X
βi,n (d(wi,n , Si (x∗ ))2
i=1
≤ β0,n ||yn − x∗ ||2 +
∞ X
βi,n (H(Si yn , Si (x∗ ))2
i=1
= ||yn − x∗ ||2 ≤ ||un − x∗ ||2 − γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 +µn γ(M + M0 ||Aun − Ax∗ ||2 + γξn ≤ (1 − αn )||xn − x∗ ||2 + αn ||x∗ ||2 − γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 +µn γ M + M0 ||A||2 ((1 − αn )||xn − x∗ ||2 + αn ||x∗ ||2 ) + γξn =
||xn − x∗ ||2 − (αn − µn γM0 ||A||2 (1 − αn ))||xn − x∗ ||2 + αn ||x∗ ||2 +µn γM + γαn M0 ||A||2 ||x∗ ||2 + γξn −γ(1 − k − γ||A||2 )||(T n − I)Aun ||2
≤
||xn − x∗ ||2 − (αn − µn γM0 ||A||2 (1 − αn ))||xn − x∗ ||2 + αn ||x∗ ||2 +µn γM + γαn M0 ||A||2 ||x∗ ||2 + γξn
(3.7)
= ||xn − x∗ ||2 − (αn − µn γM0 ||A||2 (1 − αn ))||xn − x∗ ||2 + σn ,
where σn = αn ||x∗ ||2 µn γM + γαn M0 ||A||2 ||x∗ ||2 + γξn . Since µn = o(αn ) and ξn = o(αn ), we may assume without loss of generality that there exist constants k0 ∈ (0, 1) and M2 > 0 µn 1−k0 σn ≤ M0 γ||A|| ≤ M2 . Thus, we obsuch that for all n ≥ 1, we have α 2 (1−α ) and α n n n ∗ 2 ∗ 2 ∗ 2 tain ||xn+1 − x || ≤ ||xn − x || − αn k0 ||xn − x || + σn . By Lemma 2.4, we have that
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A. R. Khan, M. Abbas, Y. Shehu and Qamrul Hassan Ansari ∗ 2
||xn − x || ≤ max{||xn − x∗ ||2 , (1 + k0 )M2 }. Therefore, {xn } is bounded. Furthermore, the sequences {yn } and {un } are bounded. The rest of the proof will be divided into two parts. Case 1. Suppose that there exists n0 ∈ N such that {||xn − x∗ ||}∞ n=n0 is nonincreasing. ∗ 2 ∗ 2 Then {||xn − x∗ ||}∞ n=1 converges and ||xn − x || − ||xn+1 − x || → 0, n → ∞. From (3.6) and (3.2), we have that ||xn+1 − x∗ ||2
≤
||yn − x∗ ||2 ≤ ||un − x∗ ||2 − γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 +µn γ(M + M0 ||Aun − Ax∗ ||2 + γξn
≤
||xn − x∗ ||2 − γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 +µn γ(M + M0 ||Aun − Ax∗ ||2 ) + γξn + αn ||x∗ ||2 .
So we get γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 ≤ ||xn − x∗ ||2 − ||xn+1 − x∗ ||2 + µn γ(M + M0 ||Aun − Ax∗ ||2 ) + γξn + αn ||x∗ ||2 . This implies that γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 → 0, n → ∞. Hence, we obtain ||(T n − I)Aun || → 0, n → ∞.
(3.8)
Also, we observe that ||yn − un || = γA∗ ||(T n − I)Aun || → 0, n → ∞ and ||un − xn || = αn ||xn || → 0, n → ∞. Furthermore, ||yn − xn || ≤ ||yn − un || + ||un − xn || → 0, n → ∞. Using (3.6) and Lemma 2.3 in (3.1), we have ||xn+1 − x∗ ||2
≤ β0,n ||yn − x∗ ||2 +
∞ X
βi,n ||wi,n − x∗ ||2 − β0,n βi,n g(||yn − wi,n ||)
i=1
≤
||un − x∗ ||2 − γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 +µn γ(M + M0 ||Aun − Ax∗ ||2 ) + γξn − β0,n βi,n g(||yn − wi,n ||).
This implies by (3.2) and condition (c) that β0,n βi,n g(||yn − wi,n ||) ≤
||un − x∗ ||2 − ||xn+1 − x∗ ||2 +µn γ(M + M0 ||Aun − Ax∗ ||2 ) + γξn
≤
||xn − x∗ ||2 − ||xn+1 − x∗ ||2 + αn ||x∗ ||2 +µn γ(M + M0 ||Aun − Ax∗ ||2 ) + γξn .
This implies by condition (b) that lim g(||yn − wi,n ||) = 0. As g is continuous and strictly n→∞
increasing with g(0) = 0, so we have lim ||yn − wi,n || = 0 ∀wi,n ∈ Si yn . n→∞
Hence, we have lim d(yn , Si yn ) ≤ lim ||yn − wi,n || = 0.
(3.9)
n→∞
n→∞
Now obtain from (3.1) that ||xn+1 − xn ||2
≤ β0,n ||yn − xn ||2 +
∞ X
βi,n ||wi,n − xn ||2
i=1
≤ β0,n ||yn − xn ||2 +
∞ X
2 βi,n ||wi,n − yn || + ||yn − xn || .
i=1
Since lim ||yn − xn || = 0 and lim ||yn − wi,n || = 0, we have lim ||xn+1 − xn || = 0. n→∞
Consequently,
n→∞
n→∞
Multiple-set split feasibility problem in Hilbert spaces
||un+1 − un || =
||(1 − αn+1 )xn+1 − (1 − αn )xn ||
≤
(3.10)
355
|αn+1 − αn |||xn+1 || + (1 − αn )||xn+1 − xn || → 0, n → ∞.
Using the fact that T is uniformly L-Lipschitzian, we have ||T Aun − Aun || ≤ ||T Aun − T n+1 Aun || + ||T n+1 Aun − T n+1 Aun+1 || + ||T n+1 Aun+1 − Aun+1 || + ||Aun+1 − Aun || ≤ L||Aun −T n Aun ||+(L + 1)||Aun+1 −Aun ||+||T n+1 Aun+1 −Aun+1 || ≤ L||Aun −T n Aun ||+(L + 1)||A||||un+1 −un ||+||T n+1 Aun+1 −Aun+1 ||. By (3.8) and (3.10), we obtain ||(T − I)Aun || → 0, n → ∞.
(3.11)
Since {xn } is bounded, there exists {xnj } of {xn } such that xnj * z ∈ H1 . Using the fact that xnj * z ∈ H1 and ||yn − xn || → 0, n → ∞, we have that ynj * z ∈ H1 . Similarly, unj * z ∈ H1 since ||un − xn || → 0, n → ∞. By the demiclosedness of Si at origin and (3.9), we have that z ∈ F (Si ). Hence, z ∈ ∩∞ i=1 F (Si ) = C. On the other hand, since A is a linear bounded operator, it follows from unj * z ∈ H1 that Aunj * Az ∈ H2 . Hence, from (3.11), we have that ||T Aunj − Aunj || = ||T Aunj − Aunj || → 0, j → ∞. Since T is demiclosed at zero, therefore we have that Az ∈ F (T ) = Q. Hence, z ∈ Ω. Next, we prove that {xn } converges strongly to z. From (3.6) and Lemma 2.1 (ii), we have ||xn+1 − z||2
≤
||yn − z||2 ≤ ||un − z||2 − γ(1 − k − γ||A||2 )||(T n − I)Aun ||2 +µn γ(M + M0 ||Aun − Az||2 ) + γξn
≤
||un − z||2 + µn γ(M + M0 ||Aun − Az||2 ) + γξn
≤
||un − z||2 + M ∗ µn + γξn
=
||(1 − αn )(xn − z) − αn z||2 + M ∗ µn + γξn
≤ (1 − αn )2 ||xn − z||2 − 2αn hun − z, zi + M ∗ µn + γξn ≤ (1 − αn )||xn − z||2 − 2αn hun − z, zi + M ∗ µn + γξn ,
(3.12)
where M ∗ > γsup(M + M0 ||Aun − Az||2 ) > 0. It is clear that −2hun − z, zi → 0, n → n≥1
∞ and
∞ P n=1
M ∗ µn < ∞;
∞ P
γξn < ∞. Now, applying Lemma 2.5 to (3.12), we have
n=1
||xn − z|| → 0. That is, xn → z, n → ∞. Case 2: Assume that {||xn − x∗ ||} is not monotonically decreasing sequence. Set Γn = ||xn − x∗ ||2 and let τ : N → N be a mapping for all n ≥ n0 (for some n0 large enough) by τ (n) := max{k ∈ N : k ≥ n, Γn ≤ Γn+1 }. Clearly, τ is a non decreasing sequence such that τ (n) → ∞ as n → ∞ and Γτ (n) − Γτ (n) ∀n ≥ n0 . By (3.9), it is easy to see that lim d(yτ (n) , Si yτ (n) ) ≤ lim ||yτ (n) − wi,τ (n) || = 0. Furthermore, we can show that n→∞
n→∞
||(T − I)Auτ (n) || → 0, n → ∞. By an argument similar to the one in case 1, we conclude immediately that xτ (n) , yτ (n) and uτ (n) weakly converge to z as τ (n) → ∞. At the same
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A. R. Khan, M. Abbas, Y. Shehu and Qamrul Hassan Ansari
time, from (3.12), we note that, for all n ≥ n0 , 0
≤
||xτ (n)+1 − z||2 − ||xτ (n) − z||2
≤ ατ (n) [−2huτ (n) − z, zi − ||xτ (n) − z||2 ] + M ∗ µτ (n) + γξτ (n) , which implies ||xτ (n) − z||2 ≤ −2huτ (n) − z, zi + M ∗ µτ (n) + γξτ (n) . Hence, we deduce that limn→∞ ||xτ (n) − z|| = 0. Therefore, limn→∞ Γτ (n) = limn→∞ Γτ (n)+1 = 0. Furthermore, for n ≥ n0 , it is easy to see that Γτ (n) ≤ Γτ (n)+1 if n 6= τ (n) (that is τ (n) < n),because Γj ≥ Γj+1 for τ (n) + 1 ≤ j ≤ n. As a consequence, we obtain for all n ≥ n0 , 0 ≤ Γn ≤ max{Γτ (n) , Γτ (n)+1 } = Γτ (n)+1 . Hence lim Γn = 0, that is, {xn } converges strongly to z. Corollary 3.3. Let {Si } : H1 → H1 be a family of single-valued quasi-nonexpansive mappings, and for each i ≥ 1, I − Si is demi-closed at 0. Let A, A∗ , T, C, Q, k, {µn }, {ξn }, L, φ, {βi,n } ∞ and γ > 0 be the same as in Theorem 3.2. Let {yn }∞ n=1 and {xn }n=1 be sequences generated by x1 ∈ H1 , un = (1 − αn )xn , yn = un + γA∗ (T n − I)Aun , ∞ P βi,n Si yn , n ≥ 1. xn+1 = β0,n yn + i=1
Then the conclusions in Theorem 3.2 still hold. Remark 3.1. The asymptotic behaviour of the sequence (3.1) is like that of (1.5). Note that ∞ P {αn } in (3.1) satisfies the conditions: lim αn = 0; αn = ∞, which show that {αn } is a n→∞
n=1
slowly decreasing sequence and that explains the good behaviour of our iterative scheme (3.1). Acknowledgement. The authors A. R. Khan and M. Abbas are grateful to King Fahd University of Petroleum and Minerals for supporting research project IN 121037. R EFERENCES [1] Alber, Y., Espinola, R. and Lorenzo, P., Strongly convergent approximations to fixed points of total asymptotically nonexpansive mappings, Acta Math. Sin. (Engl. Ser), 24 (2008), No. 6, 1005–1022 [2] Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems, 18 (2002), No. 2, 441–453 [3] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), No. 1, 103–120 [4] Ceng, L. C., Ansari, Q. H., and Yao, J. C., An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633–642 [5] Censor, Y. and Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221–239 [6] Censor, Y., Elfving, T., Kopf, N. and Bortfeld, T., The multiple-sets split feasibility problem and its applications, Inverse Problem, 21 (2005), 2071–2084 [7] Censor, Y., Motova, X. A. and Segal, A., Pertured projections and subgradient projections for the multiple-setssplit feasibility problem, J. Math. Anal. Appl., 327 (2007), 1244–1256 [8] Censor, Y. and Segal, A., The split common fixed point problem for directed operators, J. Convex Anal., 16 (2009), 587–600 [9] Chang, S. S., Joseph Lee, H. W., Chan, C. K., Wang, L. and Qin, L. J., Split feasibility problem for quasinonexpansive multi-valued mappings and total asymptotically strict pseudo-contractive mapping, Appl. Math. Comput., 219 (2013), 10416–10424 [10] Chang, S. S., Kim, J. K. and Wang, X. R., Modified block iterative algorithm for solving convex feasibility problems in Banach spaces, J. Inequal. Appl., 14 (2010), doi:10.1155/2010/86968 [11] Maruster, S. and Popirlan, C., On the Mann-type iteration and the convex feasibility problem, J. Comput. Appl. Math., 212, (2008), 390–396
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[12] Masad, E. and Reich, S., A note on the multiple-set split feasibility problem in Hilbert spaces, J. Nonlinear Convex Anal., 8 (2007), 367–371 [13] Moudafi, A., The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 587–600 [14] Xu, H. K., A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22, (2006), 2021–2034 [15] Xu, H. K., Iterative algorithm for nonlinear operators, J. London Math. Soc, 66 (2002), 1–17 [16] Yang, Q., The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, 20 (2004), 1261–1266 [17] Yang, Q. and Zhao, J., Generalized KM theorems and their applications, Inverse Problems, 22 (2006), 83–844 [18] Yang, L., Chang, S. S., Cho, Y. J. and Kim, J. K., Multiple-set split feasibility problems for total asymptotically strict pseudocontractions mappings, Fixed Point Theory Appl., 2011 (77) (2011), http://dx.doi.org/10 1186/11871812-2011-77 1 D EPARTMENT OF
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