Strong Convergence Algorithms of the Split Common Fixed Point Problem for Total Quasi-Asymptotically Pseudocontractive Operators

Abstract

We present a new algorithm for solving the two-set split common fixed point problem with total quasi-asymptotically pseudocontractive operators and consider the case of quasi-pseudocontractive operators. Under some appropriate conditions, we prove that the proposed algorithms have strong convergence. The results presented in this paper improve and extend the previous algorithms and results of Censor and Segal (2009), Moudafi (2011 and 2010), Mohammed (2013), Yang et al. (2011), Chang et al. (2012), and others.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Jaan Janno

Department of Mathematics, Ordnance Engineering College, Shijiazhuang, Hebei 050003, China

Received 19 January 2014; Revised 2 March 2014; Accepted 16 March 2014; 3 April 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let C and Q be nonempty closed convex subsets of real Hilbert spaces _{H 1} and _{H 2} , respectively. Let A : _{H 1} [arrow right] _{H 2} be a bounded linear operator. To allow for constraints both in the domain and range of A , Censor and Elfving [1] originally formulated the split feasibility problem (SFP), which is to find a member of set [figure omitted; refer to PDF] A recent generalization, due to Censor and Segal in [2], is called the split common fixed point problem (SCFPP), which is to find a point ^{x *} satisfying [figure omitted; refer to PDF] where _{U i} : _{H 1} [arrow right] _{H 1} ( i = 1,2 , ... t ) and _{T j} : _{H 2} [arrow right] _{H 2} ( j = 1,2 , ... , r ) are some nonlinear operators and A : _{H 1} [arrow right] _{H 2} is also a bounded linear operator. Denote the solution set of SCFPP by [figure omitted; refer to PDF]

In particular, if t = r = 1 , problem (2) is reduced to the two-set SCFPP, where C ... = Fix ... ( U ) and Q ... = Fix ... ( T ) , and the SFP can be retrieved by picking as operators U and T orthogonal projections.

Censor and Segal [2] invented the following CQ-algorithm with directed operators to solve the two-set SCFPP: [figure omitted; refer to PDF] where _{x 0} ∈ H and γ ∈ ( 0,2 / L ) ; L is the largest eigenvalue of the matrix ^{A *} A .

Inspired by the work of Censor and Segal, for _{α n} ∈ ( 0,1 ) , Moudafi presented the following iteration with the demicontractive mappings and quasi-nonexpansive operators in papers [3] and [4], respectively: [figure omitted; refer to PDF] Moudafi's results are weak convergence. In [5, 6], Mohammed utilized the strongly quasi-nonexpansive operators and quasi-nonexpansive operators to solve recursion (5) and obtain weak and strong convergence, respectively. Strong convergence of (5) with pseudo-demicontractive and firmly pseudo-demicontractive mappings can be found in [7, 8]. Furthermore, for several different strong convergence recursions with nonexpansive operators for solving the SCFPP see [9, 10]. For the purpose of generalization, papers [11-13] discussed the total asymptotically strictly pseudocontractive mappings and asymptotically strict pseudocontractive mappings for solving (2) and multiple-set fixed point problem (MSSFP) by the following iteration: [figure omitted; refer to PDF] which is of weak convergence; when U is semicompact, strong convergence of (6) can be deduced. Obviously, (5) is the particular case of (6). On the other hand, papers [14, 15] presented cyclic algorithms of the SCFPP for directed operators and demicontractive mappings, and the results converge weakly.

However, we found that the strong convergence of (6) needs the condition of U to be semicompact. In order to obtain strong algorithm for the two-set SCFPP without more constraints on U or T and continue to generalize the operators, in this paper, we propose a different iteration, which can ensure the strong convergence with more general case when the operators are total quasi-asymptotically pseudocontractive, demiclosed at the origin. We can choose an initial data _{x 1} ∈ _{H 1} arbitrarily and define the sequence { _{x n} } by the recursion: [figure omitted; refer to PDF] where ψ : _{H 1} [arrow right] _{H 1} is a δ -contraction with δ ∈ ( 0,1 ) , T and U are total quasi-asymptotically pseudocontractive mappings, and { _{α n} } , { _{β n} } , and { _{γ n} } are three real sequences satisfying appropriate conditions. Under some mild conditions, we prove that the sequence { _{x n} } generated by (7) converges strongly to the solution of the two-set SCFPP.

2. Preliminaries

In order to reach the main results, we first recall the following facts.

Let C be a nonempty closed and convex subset of a real Hilbert space H with the inner product Y9; · , · YA; and norm || · || . Denote by Fix ... ( T ) the set of fixed points of a mapping T ; that is, Fix ... ( T ) = { x ∈ C : T x = x } .

Definition 1 (see [2, 3, 16, 17]).

(i) Recalled that T : C [arrow right] C is said to be a directed or firmly quasi-nonexpansive operator; if p ∈ Fix ... ( T ) , then [figure omitted; refer to PDF]

(ii) Let D be a closed convex nonempty set of C ; T : C [arrow right] C is nonexpansive; we say that T is attracting with respect to D , if, for every x ∈ C \ D , p ∈ D , [figure omitted; refer to PDF]

(iii) A mapping T : C [arrow right] C is said to be paracontracting or quasi-nonexpansive; if p ∈ Fix ... ( T ) , then [figure omitted; refer to PDF]

(iv) A mapping T : C [arrow right] C is said to be demicontractive or strictly quasi-pseudocontractive; for p ∈ Fix ... ( T ) , there exists a constant β ∈ [ 0,1 ) such that [figure omitted; refer to PDF]

Definition 2 (see [11, 18]).

(i) Let T : C [arrow right] C be a total quasi-asymptotically strictly pseudocontractive if F ( T ) ...0; ∅ , and there exist a constant β ∈ [ 0,1 ] , sequences { _{μ n} } ⊂ [ 0 , ∞ ) , and { _{ξ n} } ⊂ [ 0 , ∞ ) with _{μ n} [arrow right] 0 and _{ξ n} [arrow right] 0 as n [arrow right] ∞ such that [figure omitted; refer to PDF] where [varphi] : [ 0 , ∞ ) [arrow right] [ 0 , ∞ ) is a continuous and strictly increasing function with [varphi] ( 0 ) = 0 .

(ii) A mapping T : C [arrow right] C is said to be total quasi-asymptotically pseudocontractive if F ( T ) ...0; ∅ , and there exist sequences { _{μ n} } ⊂ [ 0 , ∞ ) and { _{ξ n} } ⊂ [ 0 , ∞ ) with _{μ n} [arrow right] 0 and _{ξ n} [arrow right] 0 as n [arrow right] ∞ such that [figure omitted; refer to PDF] where [varphi] : [ 0 , ∞ ) [arrow right] [ 0 , ∞ ) is a continuous and strictly increasing function with [varphi] ( 0 ) = 0 .

(iii) A mapping T : C [arrow right] C is said to be quasi-pseudocontractive if Fix ... ( T ) ...0; ∅ , such that [figure omitted; refer to PDF]

(iv) A mapping T : C [arrow right] C is said to be uniformly k -Lipschitzian if there is a constant k > 0 , such that [figure omitted; refer to PDF]

Remark 3.

Note that the classes of directed operators and attracting operators belong to the class of paracontracting operators. The class of paracontracting operators belongs to the class of demicontractive operators, while the class of quasi-pseudocontractive operators includes the class of demicontractive operators. Further, the class of total quasi-asymptotically pseudocontractive operators, with quasi-pseudocontractive operators as a special case, includes the class of total quasi-asymptotically strictly pseudocontractive operators.

Remark 4.

Let T : C [arrow right] C be a total quasi-asymptotically pseudocontractive, if F ( T ) ...0; ∅ , for each x ∈ C and q ∈ Fix ... ( T ) ; from (13) we can easily obtain the following equivalent inequalities: [figure omitted; refer to PDF]

Lemma 5 (see [19]).

Consider

(i) ^{|| x ± y || 2} = ^{|| x || 2} ± 2 Y9; x , y YA; + ^{|| y || 2} , for all x , y ∈ H ;

(ii) ^{|| ( 1 - t ) x + t y || 2} = ( 1 - t ) ^{|| x || 2} + t ^{|| y || 2} - t ( 1 - t ) ^{|| x - y || 2} , for all x , y ∈ H and t ∈ ... .

Lemma 6 (see [18]).

Let C be a bounded and closed convex subset of a real Hilbert space H . Let T : C [arrow right] C be a uniformly L -Lipschitz and total quasi-asymptotically pseudocontractive mapping with Fix ... ( T ) ...0; ∅ . Suppose there exist positive constants M and ^{M *} , for the function [varphi] in (13), [varphi] ( ζ ) ...4; ^{M * ζ 2} for all ζ ...5; M such that [figure omitted; refer to PDF] Then Fix ... ( T ) is a closed convex subset of C .

Lemma 7 (see [20]).

A mapping I - T : C [arrow right] C is said to be demiclosed at zero, if for any sequence { _{x n} } ∈ C , such that _{x n} ... ^{x *} ∈ C and ( I - T ) _{x n} [arrow right] 0 as n [arrow right] ∞ ; then ( I - T ) ^{x *} = 0 .

Lemma 8 (see [21]).

Let { _{r n} } , { _{s n} } , and { _{t n} } be sequences of nonnegative real numbers satisfying [figure omitted; refer to PDF] If ^{∑ n = 1 ∞} ... _{t n} < ∞ and ^{∑ n = 1 ∞} ... _{s n} < ∞ , then the limit _{lim ... n [arrow right] ∞ r n} exists.

Lemma 9 (see [22]).

Let a sequence { _{t n} } ∈ [ 0,1 ) satisfy _{lim ... n [arrow right] ∞ t n} = 0 and ^{∑ n = 1 ∞} ... _{t n} = ∞ . Let { _{a n} } be a sequence of nonnegative real numbers that satisfies any of the following conditions.

(i) For all [varepsilon] > 0 , there exists an integer N ...5; 1 such that, for all n ...5; N , [figure omitted; refer to PDF]

(ii) _{a n + 1} ...4; ( 1 - _{t n} ) _{a n} + _{o n} , n ...5; 0 , where _{o n} ...5; 0 satisfies _{lim ... n [arrow right] ∞ o n} / _{t n} = 0 ;

(iii): _{a n + 1} ...4; ( 1 - _{t n} ) _{a n} + _{t n c n} , where _{lim ... - n [arrow right] ∞ c n} ...4; 0 .

Then _{lim ... n [arrow right] ∞ a n} = 0 .

3. Main Results

In this section, we will prove the strong convergence of (7) to solve the two-set SCFPP.

Theorem 10.

Let C and Q be nonempty closed convex subsets of real Hilbert spaces _{H 1} and _{H 2} , respectively. Let U : _{H 1} [arrow right] _{H 1} be a uniformly _{k 1} -Lipschitz and ( { ^{μ n ( 1 )} } , { ^{ξ n ( 1 )} } , _{[varphi] 1} ) -total quasi-asymptotically pseudocontractive mapping, T : _{H 2} [arrow right] _{H 2} a uniformly _{k 2} -Lipschitz, and ( { ^{μ n ( 2 )} } , { ^{ξ n ( 2 )} } , _{[varphi] 2} ) -total quasi-asymptotically pseudocontractive mappings satisfying the following conditions:

( _{C 1} ) : C ... = Fix ... ( U ) ...0; ∅ , Q ... = Fix ... ( T ) = ∅ ;

( _{C 2} ) : _{μ n} = max ... { ^{μ n ( 1 )} , ^{μ n ( 2 )} } , _{ξ n} = max ... { ^{ξ n ( 1 )} , ^{ξ n ( 2 )} } , n ...5; 1 , and ^{∑ n = 1 ∞} ... _{μ n} < ∞ , ^{∑ n = 1 ∞} ... _{ξ n} < ∞ ;

( _{C 3} ) : [varphi] = max ... { _{[varphi] 1} , _{[varphi] 2} } and ∃ M , ^{M *} > 0 .

Let ψ : _{H 1} [arrow right] _{H 1} be a δ -contraction with δ ∈ ( 0,1 ) . Let A : _{H 1} [arrow right] _{H 2} be a bounded linear operator. For ∀ _{x 1} ∈ _{H 1} , sequence { _{x n} } can be generated by the iteration (7), where the sequence { _{α n} } ⊂ ( 0,1 ) satisfies (i) _{lim ... n [arrow right] ∞ α n} = 0 and (ii) ^{∑ n = 0 ∞} ... _{α n} = ∞ , { β } ⊂ [ a , b ] with a , b ∈ ( 0 , 1 / ( 1 + _{k 1} ) ) , and { γ } ⊂ ( 0,2 / L ) with L being the largest eigenvalue of the matrix ^{A T} A . Assume that I - U and I - T are demiclosed at zero. If Γ ...0; ∅ , then { _{x n} } generated by (7) converges strongly to a solution of the two-set SCFPP.

Proof.

(1) First of all, we show that, for ∀ p ∈ Γ , { _{x n} } generated by (7) is bounded.

From (7), (16), and Lemma 6, we have [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] substituting (25) into (24), we have [figure omitted; refer to PDF] where _{a n} = γ _{μ n} ^{M *} L + ^{γ 2 L 2 ( || A || + _{k 2} || A || ) 2} ; by condition ( _{C 2} ), we know [figure omitted; refer to PDF]

Next, from (7), (13), and Lemma 5, we can get [figure omitted; refer to PDF] we also can see that [figure omitted; refer to PDF] then substituting (29) into (28) and from (26), we have [figure omitted; refer to PDF] where _{b n} = ^{β 2} ( 1 + _{k 1} ^{) 2} + β _{μ n} ^{M *} , and we also know that [figure omitted; refer to PDF]

From (7) and Lemma 5, we also have [figure omitted; refer to PDF]

Substituting (30) into (32) and simplifying it we have [figure omitted; refer to PDF] set [figure omitted; refer to PDF] (33) can be rewritten as [figure omitted; refer to PDF] by condition ( _{C 2} ), (27), and (31), we know that ^{∑ n = 1 ∞} ... _{t n} < ∞ and ^{∑ n = 1 ∞} ... _{s n} < ∞ . Thus it follows from Lemma 8 that the following limit exists: [figure omitted; refer to PDF] Therefore, we obtain that { _{x n} } is bounded, so is { _{u n} } . Set _{z n} = ^{U n} ( _{u n} ) . Then { _{z n} } is also bounded.

(2) Next we prove li _{m n [arrow right] ∞} || _{x n + 1} - _{x n} || = 0 , li _{m n [arrow right] ∞} || _{u n + 1} - _{u n} || =0.

For each n ...5; 1 , ∀ _{u n} ∈ _{H 1} , assume there exists ^{v i ( n )} ∈ C ( i = 1,2 ) such that _{u n} = w ^{v 1 ( n )} + ( 1 - w ) ^{v 2 ( n )} for w ∈ ( 0,1 ) . Then for all q ∈ C , and by virtue of (16), we have [figure omitted; refer to PDF]

which implies that [figure omitted; refer to PDF] Now we take q = ^{v i ( n )} ( i = 1,2 ) in (38); multiplying w and ( 1 - w ) on the two side of (38), respectively, and then adding up, we can obtain [figure omitted; refer to PDF] Letting n [arrow right] ∞ in (39), we have [figure omitted; refer to PDF]

From (7), we know that [figure omitted; refer to PDF] Letting n [arrow right] ∞ in (41) and by condition (i) in Theorem 10, we know [figure omitted; refer to PDF] Similarly, [figure omitted; refer to PDF] from (40) the limit of || _{y n} - p || exists and [figure omitted; refer to PDF]

Therefore, when we take limit on both sides of (22), we can deduce that [figure omitted; refer to PDF]

Then, [figure omitted; refer to PDF] In view of (40) and (45) we have that [figure omitted; refer to PDF] Similarly, it follows from (7), (45), and (47) that [figure omitted; refer to PDF]

(3) Next we prove that || _{x n} - U ( _{u n} ) || [arrow right] 0 , as n [arrow right] ∞ .

From (40) and (48), we have [figure omitted; refer to PDF]

By the same way, from (45) and (47) we can also prove that [figure omitted; refer to PDF]

Therefore, from (44) and (49), we know [figure omitted; refer to PDF] Since { _{x n} } is bounded, there exists a subsequence { _{x n i} } of { _{x n} } which converges weakly to a point ^{x *} . Without loss of generality, we may assume that { _{x n} } converges weakly to ^{x *} . Therefore, from (49)-(51) and Lemma 7, we have ^{x *} ∈ Fix ... ( U ) .

(4) Finally, we prove that _{x n} [arrow right] ^{x *} in norm. To do this, we calculate [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF]

Substituting (23) into (28), we have [figure omitted; refer to PDF] Since ( 1 - δ ) _{α n} ∈ ( 0,1 ) and substituting (53) into (51), we get [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Equation (55) can be rewritten as [figure omitted; refer to PDF] Evidently, from (40), (45), and Lemma 9 (ii), we can conclude that _{x n + 1} - ^{x *} [arrow right] 0 ( n [arrow right] ∞ ) .

This completes the proof.

The following theorem can be concluded from Theorem 10 immediately.

Theorem 11.

Let C and Q be nonempty closed convex subsets of real Hilbert spaces _{H 1} and _{H 2} , respectively. Let U : _{H 1} [arrow right] _{H 1} be a uniformly _{k 1} -Lipschitz and quasi-pseudocontractive mapping with C ... = Fix ... ( U ) ...0; ∅ . Let T : _{H 2} [arrow right] _{H 2} be a uniformly _{k 2} -Lipschitz and quasi-pseudocontractive mapping with Q ... = Fix ... ( T ) = ∅ . Let ψ : _{H 1} [arrow right] _{H 1} be a δ -contraction with δ ∈ ( 0,1 ) . Let A : _{H 1} [arrow right] _{H 2} be a bounded linear operator. For ∀ _{x 1} ∈ _{H 1} , sequence { _{x n} } can be generated by the iteration: [figure omitted; refer to PDF] where the sequence { _{α n} } ⊂ ( 0,1 ) satisfies (i) _{lim ... n [arrow right] ∞ α n} = 0 and (ii) ^{∑ n = 0 ∞} ... _{α n} = ∞ , { β } ⊂ [ a , b ] with a , b ∈ ( 0 , 1 / ( 1 + _{k 1} ) ) , and { γ } ⊂ ( 0,2 / L ) with L being the largest eigenvalue of the matrix ^{A T} A . Assume that I - U and I - T are demiclosed at zero. If Γ ...0; ∅ , then { _{x n} } generated by (58) converges strongly to a solution of the two-set SCFPP.

Proof.

For each p ∈ Γ , if we take T = ^{T n} , U = ^{U n} , _{μ n} [arrow right] 0 and _{ξ n} [arrow right] 0 , and follow the proof of Theorem 10, we can also prove that { _{x n} } converges strongly to ^{x *} ∈ Γ by the same way.

Remark 12.

Algorithm (7) and Theorems 10 and 11 improve and extend the corresponding results of Censor and Segal [2], Moudafi [3, 4], Mohammed [5, 6], Chang et al. [11, 13], Yang et al. [12], and others.

4. Concluding Remarks

In this work, we develop the split common fixed point problem with more general classes of total quasi-asymptotically pseudocontractive and quasi-pseudocontractive operators; corresponding algorithms are improved based on the viscosity iteration; thus we can obtain strong convergence without more constraints on operators.

Acknowledgments

The authors would like to thank the associate editor and the referees for their comments and suggestions. This research was supported by the National Natural Science Foundation of China (11071053).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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