AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 309368 10.1155/2014/309368 309368 Research Article Strong Convergence Algorithms of the Split Common Fixed Point Problem for Total Quasi-Asymptotically Pseudocontractive Operators http://orcid.org/0000-0001-8438-8783 Wang Peiyuan Zhou Hy Janno Jaan Department of Mathematics, Ordnance Engineering College Shijiazhuang, Hebei 050003 China 2014 342014 2014 19 01 2014 02 03 2014 16 03 2014 3 4 2014 2014 Copyright © 2014 Peiyuan Wang and Hy Zhou. 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.

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.

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 H 2 be a bounded linear operator. To allow for constraints both in the domain and range of A , Censor and Elfving  originally formulated the split feasibility problem (SFP), which is to find a member of set (1) Ω = { x C : A x Q } . A recent generalization, due to Censor and Segal in , is called the split common fixed point problem (SCFPP), which is to find a point x * satisfying (2) x * C = i = 1 t Fix ( U i ) , A x * Q = j = 1 r Fix ( T j ) , where U i : H 1 H 1 ( i = 1,2 , t ) and T j : H 2 H 2 ( j = 1,2 , , r ) are some nonlinear operators and A : H 1 H 2 is also a bounded linear operator. Denote the solution set of SCFPP by (3) Γ = { x * C A x * Q } .

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  invented the following CQ-algorithm with directed operators to solve the two-set SCFPP: (4) x 0 H 1 , x n + 1 = U ( x n - γ A * ( I - T ) A x n ) , n 0 , 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  and , respectively: (5) u n = x n - γ A * ( I - T ) A x n , x n + 1 = ( 1 - α n ) u n + α n U ( u n ) , x 0 H 1 , n 0 . 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  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: (6) u n = x n - γ A * ( I - T n ) A x n , x n + 1 = ( 1 - α n ) u n + α n U n ( u n ) , x 1 H 1 , n 1 , 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: (7) u n = x n - γ A ( I - T n ) A x n , y n = ( 1 - β ) u n + β U n ( u n ) , x n + 1 = α n ψ ( y n ) + ( 1 - α n ) y n , n 1 , where ψ : H 1 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 · , · 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 [<xref ref-type="bibr" rid="B2">2</xref>, <xref ref-type="bibr" rid="B3">3</xref>, <xref ref-type="bibr" rid="B21">16</xref>, <xref ref-type="bibr" rid="B22">17</xref>]).

(i) Recalled that T : C C is said to be a directed or firmly quasi-nonexpansive operator; if p Fix ( T ) , then (8) T x - p 2 x - p 2 - x - T x 2 , x C .

(ii) Let D be a closed convex nonempty set of C ; T : C C is nonexpansive; we say that T is attracting with respect to D , if, for every x C D , p D , (9) T x - p < x - p .

(iii) A mapping T : C C is said to be paracontracting or quasi-nonexpansive; if p Fix ( T ) , then (10) T x - p x - p .

(iv) A mapping T : C C is said to be demicontractive or strictly quasi-pseudocontractive; for p Fix ( T ) , there exists a constant β [ 0,1 ) such that (11) T x - p 2 x - p 2 + β x - T x 2 , x C .

Definition 2 (see [<xref ref-type="bibr" rid="B11">11</xref>, <xref ref-type="bibr" rid="B16">18</xref>]).

(i) Let T : C C be a total quasi-asymptotically strictly pseudocontractive if F ( T ) , and there exist a constant β [ 0,1 ] , sequences { μ n } [ 0 , ) , and { ξ n } [ 0 , ) with μ n 0 and ξ n 0 as n such that (12) T n x - p 2 x - p 2 + β x - T n x 2 + μ n ϕ ( x - p ) + ξ n , h h h h h h h h h h h h h h h h h h h h h h h h h n 1 , x C , p Fix ( T ) , where ϕ : [ 0 , ) [ 0 , ) is a continuous and strictly increasing function with ϕ ( 0 ) = 0 .

(ii) A mapping T : C C is said to be total quasi-asymptotically pseudocontractive if F ( T ) , and there exist sequences { μ n } [ 0 , ) and { ξ n } [ 0 , ) with μ n 0 and ξ n 0 as n such that (13) T n x - p 2 x - p 2 + x - T n x 2 + μ n ϕ ( x - p ) + ξ n , h h h h h h h h h h h h h h h h h h h h h h h h h n 1 , x C , p Fix ( T ) , where ϕ : [ 0 , ) [ 0 , ) is a continuous and strictly increasing function with ϕ ( 0 ) = 0 .

(iii) A mapping T : C C is said to be quasi-pseudocontractive if Fix ( T ) , such that (14) T x - p 2 x - p 2 + x - T x 2 , x C , p Fix ( T ) .

(iv) A mapping T : C C is said to be uniformly k -Lipschitzian if there is a constant k > 0 , such that (15) T n x - T n y k x - y ,    n 1 , x , y C .

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 C be a total quasi-asymptotically pseudocontractive, if F ( T ) , for each x C and q Fix ( T ) ; from (13) we can easily obtain the following equivalent inequalities: (16) x - T n x , x - p - μ n 2 ϕ ( x - p ) - ξ n 2 ; (17) x - T n x , p - T n x x - T n x 2 + μ n 2 ϕ ( x - p ) + ξ n 2 ; (18) x - p , T n x - p x - p 2 + μ n 2 ϕ ( x - p ) + ξ n 2 .

Lemma 5 (see [<xref ref-type="bibr" rid="B17">19</xref>]).

Consider

(i) x ± y 2 = x 2 ± 2 x , y + 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 [<xref ref-type="bibr" rid="B16">18</xref>]).

Let C be a bounded and closed convex subset of a real Hilbert space H . Let T : C C be a uniformly L -Lipschitz and total quasi-asymptotically pseudocontractive mapping with Fix ( T ) . Suppose there exist positive constants M and M * , for the function ϕ in (13), ϕ ( ζ ) M * ζ 2 for all ζ M such that (19) ϕ ( ζ ) ϕ ( M ) + M * ζ 2 . Then Fix ( T ) is a closed convex subset of C .

Lemma 7 (see [<xref ref-type="bibr" rid="B18">20</xref>]).

A mapping I - T : C 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 0 as n ; then ( I - T ) x * = 0 .

Lemma 8 (see [<xref ref-type="bibr" rid="B20">21</xref>]).

Let { r n } , { s n } , and { t n } be sequences of nonnegative real numbers satisfying (20) r n + 1 ( 1 + t n ) r n + s n , n 1 . If n = 1 t n < and n = 1 s n < , then the limit lim n r n exists.

Lemma 9 (see [<xref ref-type="bibr" rid="B19">22</xref>]).

Let a sequence { t n } [ 0,1 ) satisfy lim n 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.

For all ɛ > 0 , there exists an integer N 1 such that, for all n N , (21) a n + 1 ( 1 - t n ) a n + t n ɛ ;

a n + 1 ( 1 - t n ) a n + o n ,   n 0 , where o n 0 satisfies lim n o n / t n = 0 ;

a n + 1 ( 1 - t n ) a n + t n c n , where lim - n c n 0 .

Then lim n 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 H 1 be a uniformly k 1 -Lipschitz and ( { μ n ( 1 ) } , { ξ n ( 1 ) } , ϕ 1 ) -total quasi-asymptotically pseudocontractive mapping, T : H 2 H 2 a uniformly k 2 -Lipschitz, and ( { μ n ( 2 ) } , { ξ n ( 2 ) } , ϕ 2 ) -total quasi-asymptotically pseudocontractive mappings satisfying the following conditions:

C = Fix ( U ) , Q = Fix ( T ) = ;

μ n = max { μ n ( 1 ) , μ n ( 2 ) } , ξ n = max { ξ n ( 1 ) , ξ n ( 2 ) } , n 1 , and n = 1 μ n < , n = 1 ξ n < ;

ϕ = max { ϕ 1 , ϕ 2 } and M , M * > 0 .

Let ψ : H 1 H 1 be a δ -contraction with δ ( 0,1 ) . Let A : H 1 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 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 , 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 (22) u n - p 2 = x n - p + γ A * ( T n - I ) A x n 2 = x n - p 2 + γ 2 A * ( I - T n ) A x n 2 + 2 γ A x n - A p , T n A x n - A x n (23) x n - p 2 + γ 2 L 2 ( I - T n ) A x n 2 + γ μ n ϕ ( A x n - A p ) + γ ξ n (24) ( 1 + γ μ n M * L ) x n - p 2 + γ 2 L 2 ( I - T n ) A x n 2 + γ μ n ϕ ( M ) + γ ξ n .

Since (25) ( I - T n ) A x n A x n - A p + T n A x n - A p ( A + k 2 A ) x n - p , substituting (25) into (24), we have (26) u n - p 2 [ 1 + γ μ n M * L + γ 2 L 2 ( A + k 2 A ) 2 ] × x n - p 2 + γ μ n ϕ ( M ) + γ ξ n = ( 1 + a n ) x n - p 2 + γ μ n ϕ ( M ) + γ ξ n , where a n = γ μ n M * L + γ 2 L 2 ( A + k 2 A ) 2 ; by condition ( C 2 ), we know (27) n = 1 a n < .

Next, from (7), (13), and Lemma 5, we can get (28) y n - p 2 = ( 1 - β ) ( u n - p ) + β ( U n ( u n ) - p ) 2 = ( 1 - β ) u n - p 2 + β U n ( u n ) - p 2 - β ( 1 - β ) u n - U n ( u n ) 2 ( 1 - β ) u n - p 2 + β [ u n - p 2 + u n - U n ( u n ) 2 + μ n ϕ ( u n - p ) + ξ n u n - p 2 ] - β ( 1 - β ) u n - U n ( u n ) 2 = u n - p 2 + β 2 u n - U n ( u n ) 2 + β μ n ϕ ( u n - p ) + β ξ n ; we also can see that (29) u n - U n ( u n ) u n - p + U n ( u n ) - p ( 1 + k 1 ) u n - p ; then substituting (29) into (28) and from (26), we have (30) y n - p 2 [ 1 + β 2 ( 1 + k 1 ) 2 ] u n - p 2 + β μ n ϕ ( u n - p ) + β ξ n [ 1 + β 2 ( 1 + k 1 ) 2 + β μ n M * ] u n - p 2 + β μ n ϕ ( M ) + β ξ n ( 1 + b n ) ( 1 + a n ) x n - p 2 + μ n ϕ ( M ) × [ ( 1 + b n ) γ + β ] + ξ n [ ( 1 + b n ) γ + β ] , where b n = β 2 ( 1 + k 1 ) 2 + β μ n M * , and we also know that (31) n = 1 b n < .

From (7) and Lemma 5, we also have (32) x n + 1 - p 2 = α n [ ψ ( y n ) - p ] + ( 1 - α n ) ( y n - p ) 2 α n ψ ( y n ) - p 2 + ( 1 - α n ) y n - p 2 ( 1 + 2 α n δ ) y n - p 2 + 2 α n ψ ( p ) - p 2 - α n y n - p 2 ( 1 + 2 α n δ ) y n - p 2 + 2 α n ψ ( p ) - p 2 .

Substituting (30) into (32) and simplifying it we have (33) x n + 1 - p 2 ( 1 + 2 α n δ ) ( 1 + b n ) ( 1 + a n ) × x n - p 2 + 2 α n ψ ( p ) - p 2 + ( 1 + 2 α n δ ) { μ n ϕ ( M ) [ ( 1 + b n ) γ + β ] + ξ n [ ( 1 + b n ) γ + β ] } = [ 1 + a n + ( 1 + a n ) ( b n + 2 a n δ + 2 a n b n δ ) ] × x n - p 2 + 2 α n ψ ( p ) - p 2 + ( 1 + 2 α n δ ) { μ n ϕ ( M ) [ ( 1 + b n ) γ + β ] + ξ n [ ( 1 + b n ) γ + β ] } ; set (34) t n = a n + ( 1 + a n ) ( b n + 2 a n δ + 2 a n b n δ ) , s n = 2 α n ψ ( p ) - p 2 + ( 1 + 2 α n δ ) × { μ n ϕ ( M ) [ ( 1 + b n ) γ + β ] + ξ n [ ( 1 + b n ) γ + β ] } ; (33) can be rewritten as (35) x n + 1 - p 2 ( 1 + t n ) x n - p 2 + s n ; 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: (36) lim n x n - p . 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 x n + 1 - x n = 0 , li m n u n + 1 - u n =0.

For each n 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 (37) u n - U n ( u n ) 2 = u n - U n ( u n ) , u n - U n ( u n ) = 1 β u n - y n , u n - U n ( u n ) = 1 β u n - U n ( u n ) - ( y n - U n ( y n ) ) , u n - y n + 1 β y n - U n ( y n ) , u n - y n 1 β ( u n - y n + U n ( u n ) - U n ( y n ) ) × u n - y n + 1 β u n - q , y n - U n ( y n ) + 1 β q - y n , y n - U n ( y n ) 1 + k 1 β u n - y n 2 + 1 β u n - q , y n - U n ( y n ) + 1 β [ μ n 2 ϕ ( y n - q ) + ξ n 2 ] ( 1 + k 1 ) β u n - U n ( u n ) 2 + 1 β u n - q , y n - U n ( y n ) + 1 β [ μ n 2 [ M * y n - q 2 + ϕ ( M ) ] + ξ n 2 ] ,

which implies that (38) β [ 1 - ( 1 + k 1 ) β ] u n - U n ( u n ) 2 u n - z , y n - U n ( y n ) + [ μ n 2 [ M * y n - z 2 + ϕ ( M ) ] + ξ n 2 ] . 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 (39) β [ 1 - ( 1 + k 1 ) β ] u n - U n ( u n ) 2 [ μ n 2 [ M * y n - z 2 + ϕ ( M ) ] + ξ n 2 ] . Letting n in (39), we have (40) lim n u n - U n ( u n ) = 0 .

From (7), we know that (41) x n + 1 - p 2 = y n - p + α n ( ψ ( y n ) - y n ) 2 = y n - p 2 + 2 α n y n - p , ψ ( y n ) - y n + α n 2 ψ ( y n ) - y n 2 . Letting n in (41) and by condition (i) in Theorem 10, we know (42) lim n x n - p = lim n y n - p . Similarly, (43) y n - p 2 = u n - p + β ( u n - U n ( u n ) ) 2 = u n - p 2 + 2 β u n - p , u n - U n ( u n ) + β 2 u n - U n ( u n ) 2 ; from (40) the limit of y n - p exists and (44) lim n x n - p = lim n y n - p = lim n u n - p .

Therefore, when we take limit on both sides of (22), we can deduce that (45) lim n A x n - T n A x n = 0 .

Then, (46) x n + 1 - x n y n - x n + α n ψ ( y n ) - y n u n - x n + β u n - U n ( u n ) + α n ψ ( y n ) - y n γ A A x n - T n A x n + β u n - U n ( u n ) + α n ψ ( y n ) - y n . In view of (40) and (45) we have that (47) lim n x n + 1 - x n = 0 . Similarly, it follows from (7), (45), and (47) that (48) u n + 1 - u n = x n + 1 - x n - γ A * ( I - T n + 1 ) A x n + 1 - γ A * ( I - T n ) A x n x n + 1 - x n + γ A * A x n + 1 T n + 1 - A x n + 1 + r A * A x n T n - A x n 0    ( n ) .

(3) Next we prove that x n - U ( u n ) 0 , as n .

From (40) and (48), we have (49) u n - U ( u n ) u n - U n ( u n ) + U n ( u n ) - U ( u n ) u n - U n ( u n ) + k 1 U n - 1 ( u n ) - u n u n - U n ( u n ) + k 1 [ U n - 1 ( u n ) - U n - 1 ( u n - 1 ) + U n - 1 ( u n - 1 ) - u n ] u n - U n ( u n ) + k 1 2 u n - u n - 1 + k 1 [ U n - 1 ( u n - 1 ) - u n - 1 + u n - 1 - u n ]    ( n ) .

By the same way, from (45) and (47) we can also prove that (50) A x n - T A x n 0 ,          n .

Therefore, from (44) and (49), we know (51) x n - U ( u n ) x n - u n + u n - U ( u n ) 0             ( n ) . 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 x * in norm. To do this, we calculate (52) x n + 1 - x * 2 = α n ψ ( y n ) + ( 1 - α n ) y n - x * , x n + 1 - x * = α n ψ ( y n ) + ( 1 - α n ) y n - x * , x n + 1 - x * = α n ψ ( y n ) - x * , x n + 1 - x * + ( 1 - α n ) y n - x * , x n + 1 - x * α n ψ ( y n ) - ψ ( x * ) , x n + 1 - x * + α n ψ ( x * ) - x * , x n + 1 - x * + 1 - α n 2 y n - x * 2 + 1 - α n 2 x n + 1 - x * 2 α n δ 2 y n - x * 2 + α n 2 x n + 1 - x * 2 + α n ψ ( x * ) - x * , x n + 1 - x * + 1 - α n 2 y n - x * 2 + 1 - α n 2 x n + 1 - x * 2 = 1 - ( 1 - δ ) α n 2 y n - x * 2 + 1 2 x n + 1 - x * 2 + α n ψ ( x * ) - x * , x n + 1 - x * . Therefore, we have (53) x n + 1 - x * 2 ( 1 - ( 1 - δ ) α n ) y n - x * 2 + 2 α n ψ ( x * ) - x * , x n + 1 - x * .

Substituting (23) into (28), we have (54) y n - x * 2 x n - p 2 + γ 2 L 2 ( I - T n ) A x n 2 + β 2 u n - U n ( u n ) 2 + μ n [ γ ϕ ( A x n - A p ) + β ϕ ( u n - p ) ] + ξ n ( γ + β ) . Since ( 1 - δ ) α n ( 0,1 ) and substituting (53) into (51), we get (55) x n + 1 - x * 2 ( 1 - ( 1 - δ ) α n ) x n - p 2 + ( 1 - ( 1 - δ ) α n ) × { γ 2 L 2 ( I - T n ) A x n 2 + β 2 u n - U n ( u n ) 2 + μ n [ γ ϕ ( A x n - A p ) + β ϕ ( u n - p ) ] + ξ n ( γ + β ) } . Let (56) o n = ( 1 - ( 1 - δ ) α n ) × { γ 2 L 2 ( I - T n ) A x n 2 + β 2 u n - U n ( u n ) 2 + μ n [ γ ϕ ( A x n - A p ) + β ϕ ( u n - p ) ] + ξ n ( γ + β ) } . Equation (55) can be rewritten as (57) x n + 1 - x * 2 ( 1 - ( 1 - δ ) α n ) x n - p 2 + o n . Evidently, from (40), (45), and Lemma 9 (ii), we can conclude that x n + 1 - x * 0    ( n ) .

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 H 1 be a uniformly k 1 -Lipschitz and quasi-pseudocontractive mapping with C = Fix ( U ) . Let T : H 2 H 2 be a uniformly k 2 -Lipschitz and quasi-pseudocontractive mapping with Q = Fix ( T ) = . Let ψ : H 1 H 1 be a δ -contraction with δ ( 0,1 ) . Let A : H 1 H 2 be a bounded linear operator. For x 1 H 1 , sequence { x n } can be generated by the iteration: (58) u n = x n - γ A ( I - T ) A x n , y n = ( 1 - β ) u n + β U ( u n ) , x n + 1 = α n ψ ( y n ) + ( 1 - α n ) y n , n 1 , where the sequence { α n } ( 0,1 ) satisfies (i) lim n 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 , 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 0 and ξ n 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 , Moudafi [3, 4], Mohammed [5, 6], Chang et al. [11, 13], Yang et al. , 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.

Conflict of Interests

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

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).

Censor Y. Elfving T. A multiprojection algorithm using Bregman projections in a product space Numerical Algorithms 1994 8 2 221 239 2-s2.0-0000424337 10.1007/BF02142692 MR1309222 ZBL0828.65065 Censor Y. Segal A. The split common fixed point problem for directed operators Journal of Convex Analysis 2009 16 2 587 600 2-s2.0-70349775725 MR2559961 ZBL1189.65111 Moudafi A. The split common fixed-point problem for demicontractive mappings Inverse Problems 2010 26 5 6 2-s2.0-77951628061 10.1088/0266-5611/26/5/055007 055007 MR2647149 ZBL1219.90185 Moudafi A. A note on the split common fixed-point problem for quasi-nonexpansive operators Nonlinear Analysis: Theory, Methods & Applications 2011 74 12 4083 4087 2-s2.0-79955767339 10.1016/j.na.2011.03.041 MR2802988 ZBL1232.49017 Mohammed L. B. A note on the split common fixed-point problem for strongly quasi-nonexpansive operator in Hilbert space International Journal of Innovative Research and Studies 2013 2 8 424 434 Mohammed L. B. Strong convergence of an algorithm about quasi-nonexpansive mappings for the split common fixed-pint problem in Hilbert space International Journal of Innovative Research and Studies 2013 2 8 298 306 Sheng D. Chen R. On the strong convergence of an algorithm about pseudo-demicontractive mappings for the split common fixed-point problem 2011, http://www.paper.edu.cn/ Yu Y. Sheng D. On the strong convergence of an algorithm about firmly pseudo-demicontractive mappings for the split common fixed-point problem Journal of Applied Mathematics 2012 2012 9 256930 10.1155/2012/256930 MR2915733 ZBL06063341 Gu G. Wang S. Cho Y. J. Strong convergence algorithms for hierarchical fixed points problems and variational inequalities Journal of Applied Mathematics 2011 2011 17 2-s2.0-81755179004 10.1155/2011/164978 164978 MR2830753 ZBL1227.49013 Zhang C. He S. Strong convergence theorems for the split common fixed point problem for countable family of nonexpansive operators Journal of Applied Mathematics 2012 2012 11 438121 MR2948099 ZBL06106214 Chang S. S. Wang L. Tang Y. K. Yang L. The split common fixed point problem for total asymptotically strictly pseudocontractive mappings Journal of Applied Mathematics 2012 2012 13 2-s2.0-84855588570 10.1155/2012/385638 385638 MR2861928 ZBL06021679 Yang L. Chang S. S. Cho Y. J. Multiple-set split feasibility problems for total asymptotically strict pseudocontraction mappings Fixed Point Theory and Applications 2011 2011, article 77 10.1186/1687-1812-2011-77 Chang S.-S. Cho Y. J. Kim J. K. Zhang W. B. Yang L. Multiple-set split feasibility problems for asymptotically strict pseudocontractions Abstract and Applied Analysis 2012 2012 12 2-s2.0-84858227267 10.1155/2012/491760 491760 MR2889090 ZBL1234.47047 Wang F. Xu H.-K. Cyclic algorithms for split feasibility problems in Hilbert spaces Nonlinear Analysis: Theory, Methods & Applications 2011 74 12 4105 4111 2-s2.0-79955759673 10.1016/j.na.2011.03.044 MR2802990 ZBL05907682 Tang Y.-C. Peng J.-G. Liu L.-W. A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces Mathematical Modelling and Analysis 2012 17 4 457 466 10.3846/13926292.2012.706236 MR2974828 ZBL1267.47104 Bauschke H. H. Borwein J. M. On projection algorithms for solving convex feasibility problems SIAM Review 1996 38 3 367 426 2-s2.0-0030246542 10.1137/S0036144593251710 MR1409591 ZBL0865.47039 Elsner L. Koltracht I. Neumann M. Convergence of sequential and asynchronous nonlinear paracontractions Numerische Mathematik 1992 62 1 305 319 2-s2.0-0000695406 10.1007/BF01396232 MR1169007 ZBL0763.65035 Wang Z.-M. Su Y. On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping Journal of Inequalities and Applications 2013 2013, article 375 9 10.1186/1029-242X-2013-375 MR3101922 Marino G. Xu H.-K. Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces Journal of Mathematical Analysis and Applications 2007 329 1 336 346 2-s2.0-33846303213 10.1016/j.jmaa.2006.06.055 MR2306805 ZBL1116.47053 Goebel K. Kirk W. A. Topics in Metric Fixed Point Theory 1990 28 Cambridge, UK Cambridge University Press Cambridge Studies in Advanced Mathematics 10.1017/CBO9780511526152 MR1074005 Aoyama K. Kimura Y. Takahashi W. Toyoda M. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space Nonlinear Analysis: Theory, Methods & Applications 2007 67 8 2350 2360 2-s2.0-34250705821 10.1016/j.na.2006.08.032 MR2338104 ZBL1130.47045 O'Hara J. G. Pillay P. Xu H.-K. Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces Nonlinear Analysis: Theory, Methods & Applications 2003 54 8 1417 1426 2-s2.0-0042128551 10.1016/S0362-546X(03)00193-7 MR1997227 ZBL1052.47049