The Strong Convergence Theorems for the Split Equality Fixed Point Problems in Hilbert Spaces
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1 Applied Mathematical Sciences, Vol. 2, 208, no. 6, HIKARI Ltd, The Strong Convergence Theorems for the Split Equality Fixed Point Problems in Hilbert Spaces Jun Niu, Zheng Zhou, Jian-Qiang Zhang, and Li-Juan Qin 2 College of Statistics and Mathematics Yunnan University of Finance and Economics Kunming, Yunnan, 65022, P.R. China Corresponding author 2 Department of Mathematics, Kunming University Kunming, 65024, P.R. China Copyright c 208 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to investigate the split equality fixed point problem for quasi-asymptotically pseudo-contractive mappings in Hilbert spaces. And without assumption of semi-compactness, the strong convergence of the sequence generated by the proposed iterative scheme is obtained. The results presented in this paper improve and extend some recent corresponding results announced. Keywords: Split equality fixed point problems; Hilbert spaces; quasiasymptotically pseudo-contractive mappings; Strong convergence Introduction The split feasibility problem (SF P ) was first introduced in 994 by Censor and Elfving [] in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been shown the it can be used in many areas of applications such as imagine restoration, computer tomograph and radiation therapy treatment planning [3-5].
2 256 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H 2, respectively, and A : H H 2 be a bounded linear operator. The split feasibility problem (SF P ) is formulated as: finding x C, such that Ax Q. (.) It is easy to see that x C is a solution of (SF P ) if and only if it solves the following fixed point equation x = P C (I γa (I P Q )A)x, x C, where P C (resp. P Q ) is the (orthogonal) projection from H (resp. H 2 ) onto C (resp. Q), γ > 0, and A is the adjoint of A. In 203, Moudafi [7,8] proposed a new split feasibility problem which is also called split equality problem (SEP ). Let H, H 2 and H 3 be real Hilbert spaces. A : H H 3 and B : H 2 H 3 be two bounded linear operators, C H and Q H 2 be two nonempty closed convex sets. The split equality problem (SEP ) is formulated as: finding x C, y Q such that Ax = By. (.3) Obviously, if B = I (identity mapping on H 2 ) and H 3 = H 2, then the split equality problem (SEP ) reduces to the split feasibility problem (SF P ). In (.3), when C and Q are the sets of fixed points of two nonlinear operator T and S, and C and Q are nonempty closed convex,respectively, the split equality problem is called split equality fixed point problem (SEF P P ). This is finding x C = F (T ), y Q = F (S) such that Ax = By. (.4) Since each closed and convex subset of a Hilbert space may be considered as a fixed point set of a projection on the subset, hence the split equality fixed point problem is a generalization of the split equality problem. The split equality problem (SEP ) and split equality fixed point problem (SEF P P ) have been studied by many authors [9-7]. To solve the (SEF P P ), Modaufi [9] presented the following simultaneous iterative method and obtained weak convergence theorem: { xn+ = T (x n γa (Ax n By n )), y n+ = S(y n + βb (Ax n By n )), (.5) where T and S are two firmly quasi-nonexpansive operators.
3 Strong convergence theorems 257 In 205, Che and Li [6] proposed the following iterative algorithm for finding a solution of the (SEF P P ) of strictly pseudo-nonexpansive mappings: u n = x n γ n A (Ax n By n ), x n+ = α n x n + ( α n )T u n, v n = y n + γ n B (Ax n By n ), y n+ = α n y n + ( α n )Sv n. They also obtained the weak convergence of the iterative scheme (.6). (.6) In 205, Chang et al. [7] proposed an iterative algorithm to establish the strong convergence and weak convergence results of the (SEF P P ) of L- Lipschitzian and quasi-pseudo-contractive mappings: u n = x n γ n A (Ax n By n ), x n+ = α n x n + ( α n )(( ξ n )I + ξ n T (( η n )I + η n T ))u n, v n = y n + γ n B (Ax n By n ), y n+ = α n y n + ( α n )(( ξ n )I + ξ n S(( η n )I + η n S))v n, (.7) In the above research work, to prove the strong convergence, the semicompactness on mappings is needed. In 205, Zhang et al. [8] introduced an new iterative algorithm to solve split common fixed point problem of asymptotically nonexpansive mappings and proved its strong convergence without assumption of semi-compactness on mappings in Hilbert spaces: z n = x n + λa (T n 2 I)Ax n, y n = α n z n + ( α n )T n z n, C n+ = {v C n : y n v k n z n v, z n v k n x n v }, x n+ = P Cn+ x, n. (.8) In 206, Tang et al. [9] used the following hybrid projection algorithm to solve split equality fixed point problem (SEF P P ) for L-Lipschitzian and quasi-pseudo-contractive mappings in Hilbert spaces and proved its strong convergence theorem without assumption of semi-compactness on mappings: u n = x n γ n A (Ax n By n ), w n = α n x n + ( α n )(( ξ)i + ξt (( η)i + ηt ))u n, v n = y n + γ n B (Ax n By n ), z n = α n y n + ( α n )(( ξ)i + ξs(( η)i + ηs))v n, C n+ Q n+ = {(p, q) C n Q n : w n p 2 + z n q 2 x n p 2 + y n q 2 }, x n+ = P Cn+ x, y n+ = P Qn+ y, (.9)
4 258 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Motivated and inspired by the researches going on this direction, the purpose of this paper is to introduce an new iterative algorithm to solve split equality fixed point problem for quasi-asymptotically pseudo-contractive mappings in Hilbert spaces. And without assumption of semi-compactness on mappings, the strong convergence of the sequence generated by the proposed iterative scheme is obtained. The results presented in this paper improve and extend some recent corresponding results announced. 2 Preliminaries Throughout this paper, we use the notations to denote the strong convergence, and to denote the weak convergence. Let C be a closed convex subset of H. For every point x H, there exists a unique nearest point in C, denoted by P C x satisfying x P C x x y, y C. (2.) The operator P C is called the metric projection mapping of H onto C. The metric projection P C is characterized by the following inequalities: and y P C x, x P C x 0, x H, y C, (2.2) y P C x 2 + x P C x 2 x y 2, y C, x H. (2.3) In a real Hilbert space H, it is also well known that for any x, y H λx+( λ)y 2 = λ x 2 +( λ) y 2 λ( λ) x y 2, λ [0, ] (2.4) and 2 x, y = x 2 + y 2 x y 2. Definition 2.. An operator T : C C is said to be (I) pseudo-contractive if T x T y, x y x y 2, for all x, y C, or equivalently, T x T y 2 x y 2 + (I T )x (I T )y 2, x, y C. (II) quasi-pseudo-contractive if F (T ) and T x p 2 x p 2 + T x x 2, x, y C, p F (T ). (III) asymptotically pseudo-contractive if there exists a sequence {l n } [, + ) with lim l n = such that T n x T n y 2 l n x y 2 + (I T n )x (I T n )y 2, x, y C, for each n,
5 Strong convergence theorems 259 or equivalently, T n x T n y, x y l n + x y 2, x, y C, for each n. 2 (IV) quasi-asymptotically pseudo-contractive if F (T ) and there exists a sequence {l n } [, + ) with lim l n = such that T n x p 2 l n x p 2 + T n x x 2, x, y C, p F (T ), for each n, Definition 2.2. An operator T : C C is said to be uniformly L- Lipschitzian, if there exists some L > 0 such that T n x T n y L x y, x, y C, for each n. Definition 2.3. An operator T : C C is said to be demi-closed at zero, if for any sequence {x n } with x n x and lim x n T (x n ) = 0, then x = T x. Lemma 2.4. Let H be a real Hilbert space, T : H H be a uniformly L- Lipschitzian and {l n }-quasi-asymptotically pseudo-contractive mapping with L and {l n } [, ) and lim l n =. Let {K n : H H} be a sequence of mappings defined by: If 0 < a < ξ < η < b < conclusions hold: K n := ( ξ)i + ξt n (( η)i + ηt n ). (2.5) M+ + 2 and M = sup l n, then the following +L 4 2 n (M+) 2 () F (T ) = F (T n (( η)i + ηt n )) = F (K n ) for each n ; (2) If T is demi-closed at zero, then K is also demi-closed at zero; (3) For each n and x H, u F (T ) = F (K n ), K n x u k n x u, where k n = + ξ(l n )( + ηl n ), {k n } [, + ) and lim k n =. Proof. () If u F (T ), i.e., u = T u, we have T n (( η)i + ηt n )u = T n (( η)u + ηt n u) = T n u = u. This shows that u F (T n (( η)i + ηt n )). Conversely, if u F (T n (( η)i + ηt n )) for all n, i.e., u = T n (( η)i + ηt n )u. Put (( η)i + ηt n ))u = v, then T n v = u. Now we prove that u = v. In fact, we have u v = u (( η)i +ηt n ))u = η u T n u = η T n v T n u Lη u v.
6 260 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Since 0 < η < M+ + 2 (M+) 2 4 +L 2 < L, we have 0 < Lη <. Then we easily obtain u = v, i.e., u F (T ). This shows that F (T ) = F (T n (( η)i + ηt n )) for all n. It is obvious that u F (K n ) if and only if u F (T n (( η)i + ηt n )). Then the conclusion () is proved. (2) For any sequence {u n } satisfying u n u and u n K u n 0. We show that u F (K ). From conclusion (), it suffices to prove u F (T ). In fact, since T is L-Lipchitz, we have u n T u n u n T (( η)i + ηt )u n + T (( η)i + ηt )u n T u n ξ u n ( ξ)u n ξt (( η)i + ηt )u n + Lη u n T u n = ξ u n K u n + Lη u n T u n. Simplifying it, we have u n T u n ξ( Lη) u n K u n 0. Since T is demi-closed at 0, we have u F (T ) = F (K). The conclusion (2) is proved. (3) For all u F (T ), it follows from Definition 2.(IV) that T n (( η)i + ηt n )x u 2 l n ( η)x + ηt n x u 2 + T n (( η)i + ηt n )x (( η)i + ηt n )x 2 = l n ( η)(x u) + η(t n x u) 2 + T n (( η)i + ηt n )x (( η)i + ηt n )x 2. (2.6) Since T is L-Lipschitz, we have T n (( η)i+ηt n )x T n x L ( η)x+ηt n x x = Lη T n x x. (2.7) From (2.4) and (2.7) we have ( η)(x u) + η(t n x u) 2 = ( η) x u 2 + η T n x u 2 η( η) T n x x 2 ( η) x u 2 + η(l n x u 2 + T n x x 2 ) η( η) T n x x 2 = ( + η(l n )) x u 2 + η 2 T n x x 2. (2.8)
7 Strong convergence theorems 26 Using (2.4), we have T n (( η)i + ηt n )x (( η)i + ηt n )x 2 = ( η)(t n (( η)x + ηt n x) x) + η(t n (( η)x + ηt n x) T n x) 2 = ( η) T n (( η)x + ηt n x) x 2 + η T n (( η)x + ηt n x) T n x 2 η( η) T n x x 2 ( η) T n (( η)x + ηt n x) x 2 η( η η 2 L 2 ) T n x x 2. (2.9) Substituting (2.8) and (2.9) into (2.6), we obtain T n (( η)i + ηt n )x u 2 l n ( + η(l n )) x u 2 + l n η 2 T n x x 2 +( η) T n (( η)x + ηt n x) x 2 η( η η 2 L 2 ) T n x x 2 = l n ( + η(l n )) x u 2 + ( η) T n (( η)x + ηt n x) x 2 η( η η 2 L 2 l n η 2 ) T n x x 2. (2.0) Since η < get M+ + 2 (M+) 2 4 +L 2 we deduce η η 2 L 2 l n η 2 > 0. From (2.0) we T n (( η)i + ηt n )x u 2 l n ( + η(l n )) x u 2 Combine (2.4) and (2.) we have +( η) T n (( η)x + ηt n x) x 2. (2.) K n x u 2 = ( ξ)x + ξt n (( η)x + ηt n x) u 2 = ( ξ) x u 2 + ξ T n (( η)x + ηt n x) u 2 ξ( ξ) T n (( η)x + ηt n x) x 2 ( ξ) x u 2 + ξl n ( + η(l n )) x u 2 +(ξ( η) ξ( ξ)) T n (( η)x + ηt n x) x 2 = ( + ξ( + ηl n )(l n )) x u 2 ξ(η ξ) T n (( η)x + ηt n x) x 2. This together with ξ < η implies that K n x u 2 k n x u 2, x H, u F (K n ), n, where k n = + ξ(l n )( + ηl n ). In view of that {l n } [, + ) and lim l n =, we have {k n } [, + ) and lim k n =. The conclusion (3) is proved.
8 262 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin 3 Main Results Theorem 3.. Let H,H 2 and H 3 be three real Hilbert spaces, A : H H 3 and B : H 2 H 3 be two bounded linear operators with adjoints A and B, respectively. Let T : H H and S : H 2 H 2 be two uniformly L- Lipschitzian and {l n }-quasi-asymptotically pseudo-contractive mappings with F (T ) and F (S). For any given initial points x C = H, y Q = H 2, the sequence {(x n, y n )} is defined as follows: u n = x n γ n A (Ax n By n ), v n = y n + γ n B (Ax n By n ), z n = α n x n + ( α n )K n u n, s n = α n y n + ( α n )G n v n, C n+ Q n+ = {(x, y) C n Q n : z n x 2 + s n y 2 ( + ( α n )(kn 2 ))( x n x 2 + y n y 2 )}, x n+ = P Cn+ x, y n+ = P Qn+ y, (3.) where K n = ( ξ)i +ξt n (( η)i +ηt n ), G n = ( ξ)i +ξs n (( η)i +ηs n ), k n = + ξ(l n )( + ηl n ) and the following conditions are satisfied: () L, and n= (l n ) < ; (2) {α n } (0, ), lim inf α n > 0; (3) γ n (0, max( A 2, B 2 )) with lim inf γ n > 0, n ; (4) 0 < a < ξ < η < b < M+ + 2, M = sup l n, n. +L 4 2 n (M+) 2 If Ω = {(p, q) F (T ) F (S) such that Ap = Bq}, T and S are demiclosed at zero, then the sequences {(x n, y n )} converges strongly to a point (x, y ) Ω. Proof. Since k n = + ξ(l n )( + ηl n ), ξ <, {l n } [, ) and n= (l n ) <, we have n= (k n ) = n= ξ(l n )( + ηl n ) < n= (l n )(l n +) <. Further, due to {α n } (0, ), we also have α n )(k 2 n ) n= (k n )(k n + ) <. We shall divide the proof into five steps. Step. We show that C n Q n is closed and convex for each n. n= ( Putting ρ n = + ( α n )(k 2 n ). Since C = H and Q = H 2, so C Q is closed and convex. Suppose that C n Q n is closed and convex. For any (x, y) C n+ Q n+, we have z n x 2 + s n y 2 ( + ( α n )(k 2 n ))( x n x 2 + y n y 2 ),
9 Strong convergence theorems 263 which is equivalent to 2ρ n x n ρ n x 2z n + x, x ρ n x n 2 + ρ n y n y 2 z n 2. So, we know that C n+ is closed. Similarly,we can prove that Q n+ is closed. Therefore C n+ Q n+ is closed. Besides, it is easy to prove that C n+ Q n+ also is a convex. Therefore C n+ Q n+ is a closed and convex for any n. Step 2. We prove that Ω C n Q n for any n. For any given (p, q) Ω, then p F (T ), Q F (S) and Ap = Bq. It follows from (3.), we have u n p 2 = x n γ n A (Ax n By n ) p 2 = x n p 2 + γ n A (Ax n By n ) 2 2γ n x n p, A (Ax n By n ) x n p 2 + γ 2 n A 2 Ax n By n 2 2γ n x n p, A (Ax n By n ) = x n p 2 + γ 2 n A 2 Ax n By n 2 2γ n Ax n Ap, Ax n By n.(3.4) Similarly, from (3.), we have v n q 2 y n q 2 +γ 2 n B 2 Ax n By n 2 +2γ n By n Bq, Ax n By n. Adding up (3.4) and (3.5) and noting Ap = Bq, we have that (3.5) u n p 2 + v n q 2 x n p 2 + y n q 2 + γ 2 n( A 2 + B 2 ) Ax n By n 2 2γ n Ax n Ap By n + Bq, Ax n By n = x n p 2 + y n q 2 +γ n (γ n ( A 2 + B 2 ) 2) Ax n By n 2. (3.6) Since γ n (0, max( A 2, B 2 )), γ n A 2 < and γ n B 2 <. This implies that γ n ( A 2 + B 2 ) 2 < 0. Therefore (3.6) can be written as u n p 2 + v n q 2 x n p 2 + y n q 2. (3.7) According to condition (4) and Lemma 2.4, we know F (T ) = F (K n ), F (S) = F (G n ), n ; K n u n p k n u n p, G n v n q k n v n q, n. Then, it follows from (3.), condition (2) and Lemma 2.4, we obtain that z n p 2 = α n x n + ( α n )K n u n p 2 = α n x n p 2 + ( α n ) K n u n p 2 α n ( α n ) K n u n x n 2 α n x n p 2 + ( α n )k 2 n u n p 2 α n ( α n ) K n u n x n 2 α n x n p 2 + ( α n )k 2 n u n p 2. (3.8)
10 264 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Similarly, it follows from (3.), condition (2) and Lemma 2.4 that s n q 2 α n y n q 2 + ( α n )k 2 n v n q 2 α n ( α n ) G n v n y n 2 α n y n q 2 + ( α n )k 2 n v n q 2. (3.9) Adding up (3.8) and (3.9), we have z n p 2 + s n q 2 α n ( x n p 2 + y n q 2 ) Substituting (3.7) into (3.0), we can obtain that +( α n )k 2 n( u n p 2 + v n q 2 ). (3.0) z n p 2 + s n q 2 α n ( x n p 2 + y n q 2 ) +( α n )k 2 n( x n p 2 + y n q 2 ) = (α n + ( α n )k 2 n)( x n p 2 + y n q 2 ) = ( + ( α n )(k 2 n ))( x n p 2 + y n q 2 ). Therefore, we know that (p, q) C n Q n and Ω C n Q n for any n. Step 3. sequences. We show that the sequences {x n } and {y n } are two Cauchy Since Ω C n+ Q n+ C n Q n and (x n+, y n+ ) = (P Cn+ x, P Qn+ y ) C n+ Q n+ C n Q n, then for any n and (p, q) Ω, we have x n+ x p x, n, (3.) y n+ y q y, n. (3.2) Hence, {x n } and {y n } are bounded. For any n, by using (2.3), we have x n+ x n 2 + x x n 2 = x n+ P Cn x 2 + x P Cn x 2 x n+ x 2, and y n+ y n 2 + y y n 2 = y n+ P Qn y 2 + y P Qn y 2 y n+ y 2, which imply that { x n x } and { y n y } are nondecreasing. By virtue of the boundedness of {x n } and {y n }, lim x n x and lim y n y exist. For positive integers m, n with m n, from (x n, y n ) = (P Cn x, P Qn y ) C m Q m and (2.3), we have x m x n 2 + x x n 2 = x m P Cn x 2 + x P Cn x 2 x m x 2, (3.3) y m y n 2 + y y n 2 = y m P Qn y 2 + y P Qn y 2 y m y 2. (3.4)
11 Strong convergence theorems 265 Since lim x n x and lim y n y exist, it follows from (3.3) and (3.4) that x m x n 2 x m x 2 x x n 2 0, y m y n 2 y m y 2 y y n 2 0. Therefore {x n } and {y n } are two Cauchy sequences. Step 4. We show that lim Ax n By n = 0, lim K n u n x n = 0 and lim G nv n y n = 0. Since (x n+, y n+ ) = (P Cn+ x, P Qn+ y ) C n+ Q n+ C n Q n, we have z n x n 2 + s n y n 2 ( z n x n+ + x n+ x n ) 2 +( s n y n+ + y n+ y n ) 2 2 z n x n x n+ x n 2 +2 s n y n y n+ y n 2 (2 + 2ρ n )( x n+ x n 2 + y n+ y n 2 ) 0. So we know that lim z n x n = 0 and lim s n y n = 0. Again since {x n } and {y n } are bounded, we know that {z n } and {s n } are bounded. From (3.6), (3.8) and (3.9), we have z n p 2 + s n q 2 α n x n p 2 + ( α n )kn u 2 n p 2 α n ( α n ) K n u n x n 2 + α n y n q 2 +( α n )kn v 2 n q 2 α n ( α n ) G n v n y n 2 = α n ( x n p 2 + y n q 2 ) +( α n )kn( u 2 n p 2 + v n q 2 ) α n ( α n )( K n u n x n 2 + G n v n y n 2 ) (α n + ( α n )kn)( x 2 n p 2 + y n q 2 ) +( α n )knγ 2 n (γ n ( A 2 + B 2 ) 2) Ax n By n 2 α n ( α n )( K n u n x n 2 + G n v n y n 2 ) = ( + ( α n )(kn 2 ))( x n p 2 + y n q 2 ) +( α n )knγ 2 n (γ n ( A 2 + B 2 ) 2) Ax n By n 2 α n ( α n )( K n u n x n 2 + G n v n y n 2 ). (3.5)
12 266 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin It follows from (3.5) that ( α n )k 2 nγ n (2 γ n ( A 2 + B 2 )) Ax n By n 2 +α n ( α n )( K n u n x n 2 + G n v n y n 2 ) ( + ( α n )(k 2 n ))( x n p 2 + y n q 2 ) z n p 2 s n q 2 = ( α n )(k 2 n )( x n p 2 + y n q 2 ) + x n p 2 + y n q 2 z n p 2 s n q 2 = ( α n )(k 2 n )( x n p 2 + y n q 2 ) +( x n p + z n p )( x n p z n p ) +( y n q + s n q )( y n q s n q ) ( α n )(k 2 n )( x n p 2 + y n q 2 ) + ( x n p + z n p )( x n z n ) +( y n q + s n q )( y n s n ). Since lim k n = and {α n } (0, ), we have lim ( α n )(kn 2 ) = 0. By virtue of the boundedness of {z n }, {s n }, {x n } and {y n }, lim z n x n = 0 and lim s n y n = 0, we get ( α n )(kn 2 )( x n p 2 + y n q 2 )+( x n p + z n p )( x n z n )+( y n q + s n q )( y n s n ) 0. In addition, it follows from Conditions (2) and (3) that ( α n )knγ 2 n (2 γ n ( A 2 + B 2 )) > 0 and α n ( α n ) > 0. Thus we may get lim Ax n By n = 0; (3.6) lim K nu n x n = 0, for each n =, 2, 3, ; (3.7) lim G nv n y n = 0, for each n =, 2, 3,. (3.8) Step 5. We show that {(x n, y n )} converges strongly to an element of Ω. By (3.6), we have u n x n + v n y n = γ n A (Ax n By n ) + γ n B (Ax n By n ) So, we know γ n ( A + B ) Ax n By n 0. lim u n x n = 0, lim v n y n = 0. (3.9) From (3.7), (3.8) and (3.9), we have K n u n u n K n u n x n + u n x n 0; (3.20) G n v n v n G n v n y n + v n y n 0. (3.2) Since {x n } and {y n } are two Cauchy sequences, there exists x H and y H 2 such that x n x and y n y. From (3.9) we also have u n x
13 Strong convergence theorems 267 and v n y. So it follows from (3.20), (2.2) and Lemma 2.4 that x F (K ) = F (T ) and y F (G ) = F (S). On the other hand, since A and B are two bounded linear operators, we have that Ax n By n Ax By. By using the weakly lower semi-continuity of squared norm, we have Ax By 2 lim inf Ax n By n 2 = lim Ax n By n 2 = 0, thus Ax = By. Therefore, {(x n, y n )} converges strongly to (x, y ) Ω. completed. The proof is Remark 3.2. Since a quasi-pseudo-contractive mapping is a quasi-aymptotically pseudo-contractive mapping, the Theorem 3. extends the main results in [7] and [9] from quasi-pseudo-contractive mappings to quasi-aymptotically pseudo-contractive mappings. The following corollary may be directly concluded from Theorem 3.. Corollary 3.3. Let H,H 2 and H 3 be three real Hilbert spaces, A : H H 3 and B : H 2 H 3 be two bounded linear operators with their adjoints A and B, respectively. Let T : H H and S : H 2 H 2 be two uniformly L- Lipschitzian and quasi-pseudo-contractive mappings, F (T ), and F (S). For given initial value x C = H, y Q = H 2, and let {(x n, y n )} be defined as follows: u n = x n γ n A (Ax n By n ), v n = y n + γ n B (Ax n By n ), z n = α n x n + ( α n )Ku n, s n = α n y n + ( α n )Gv n, (3.23) C n+ Q n+ = {(x, y) C n Q n : z n x 2 + s n y 2 x n x 2 + y n y 2 }, x n+ = P Cn+ x, y n+ = P Qn+ y, where K = ( ξ)i + ξt (( η)i + ηt ), G = ( ξ)i + ξs(( η)i + ηs), and the following conditions are satisfied: () α n (0, ), lim inf α n > 0, L ; (2) γ n (0, max( A 2, B 2 )), lim inf γ n > 0; (3) 0 < a < ξ < η < b < + +L 2. If Ω = {(p, q) F (T ) F (S) such that Ap = Bq}, T and S are demiclosed at zero, then the sequences {(x n, y n )} converges strongly to a point (x, y ) Ω.
14 268 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Acknowledgements. This work was supported by the Scientific Research Foundation of Postgraduate of Yunnan University of Finance and Economics (207YUFEYC035) and the Scientific Research Foundation of Yunnan Province Education Department (204C00Y). References [] Y. Censor, T. Elfving, A mulitprojection algorithm using Bregman projections in a product space, Number. Algorithms, 8 (994), [2] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Problems, 8 (2002), [3] Y. Censor, T. Bortfeld, N. Martin, A. Trofimov, A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol., 5 (2006), [4] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The mulitple-sets split feasibility problem and its applocations for inverse problems, Inverse Problems, 2 (2005), [5] Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the mulit-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), [6] Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 6 (2009), [7] A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Analysis: Theory, Methods and Applications, 79 (203), [8] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 5 (204), no. 4, [9] A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problems and applications, Trans. Math. Program. Appl., (203), -.
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