Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem

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1 Available online at J. Nonlinear Sci. Appl. 9 (206), Research Article Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem Jong Soo Jung Department of Mathematics, Dong-A University, Busan 4935, Korea. Communicated by Y. J. Cho Abstract In this paper, we introduce two iterative algorithms based on the hybrid steepest descent method for solving the split feasibility problem. We establish results on the strong convergence of the sequences generated by the proposed algorithms to a solution of the split feasibility problem, which is a solution of a certain variational inequality. In particular, the minimum norm solution of the split feasibility problem is obtained. c 206 All rights reserved. Keywords: Split feasibility problem, nonexpansive mapping, variational inequality, minimum-norm, projection, bounded linear operator, ρ-lipschitzian, η-strongly monotone operator. 200 MSC: 47J20, 47J25, 47J05, 47H09, 47H0, 47H05.. Introduction We consider the split feasibility problem (SFP) which is formulated as finding a point x with property x C and Ax Q, (.) where C and Q are two nonempty closed convex subsets of real Hilbert spaces H and H 2, respectively, and A : H H 2 is a bounded linear operator. The SFP (.) in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [4] for modeling inverse problems which arise in phase retrievals and in medical image reconstruction []. In [3, 5, 6], it has been shown that the SPF (.) can also be used to model the intensity-modulated radiation therapy. address: jungjs@dau.ac.kr (Jong Soo Jung) Received

2 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), The SFP is said to be consistent if (.) has a solution. It is easy to see that SFP (.) is consistent if and only if the following fixed point problem has a solution (see Proposition 3.2 in [7]): find x C such that P C (I γa (I P Q )A)x = x, (.2) where P C and P Q are the projections onto C and Q, respectively, and A is the adjoint of A. It is well 2 known that if γ (0, ), then T = P A 2 C (I γa (I P Q )A) in the operator equation (.2) is nonexpansive ([6]). Various iterative algorithms have been studied to solve the SFP (.), see, e.g., [2, 4, 7, 9,, 4, 6, 7, 9, 23, 24] and references therein. In particular, in view of the fixed point formulation (.2) of the SFP (.), Xu [7] applied the following KM CQ algorithm to solve the SFP (.): where T is the averaged mapping given by x n+ = ( α n )x n + α n T x n, n 0, (.3) T = P C (I γa (I P Q )A) for γ (0, 2 A 2 ), and obtained weak convergence of the sequence {x n } generated by (.3) to a solution of SFP (.). Recently, some iterative algorithms for solving variational inclusions, mixed equilibrium problems, fixed point problems and for finding the minimum norm element in common solution set of the problems are considered by many authors. For instance, see [20 22] and references therein. On the other hand, Yamada [8] introduced the following hybrid steepest descent method for a nonexpansive mapping S for solving the variational inequality: x n+ = (I α n µf )Sx n, n 0, (.4) where F : H H is a κ-lipschitzian and η-strongly monotone operator with constants κ > 0 and η > 0; and 0 < µ < 2η. He proved that if {α κ 2 n } satisfies appropriate conditions, the sequence {x n } generated by (.4) converges strongly to the unique solution of the variational inequality related to F, of which the constraint set is the fixed point set F ix(s) of S. In this paper, as a continuation of study for solving the SFP (.) via fixed point methods, we present two iterative algorithms based on Yamada s hybrid steepest descent method [8] for solving the SFP (.). First, we introduce an implicit algorithm. Next, by discretizing the continuous implicit algorithm, we provide an explicit algorithm. Under some appropriate conditions, we show the strong convergence of proposed algorithms to some solution of the SFP (.) which solves a certain variational inequality. As special cases, we obtain two algorithms which converges strongly to the minimum norm solution of the SFP (.). 2. Preliminaries and lemmas Let H be a real Hilbert space with inner product, and norm, respectively, and let K be a nonempty closed convex of H. We recall that: () a mapping f : H H is k-contractive if fx fy k x y for some constant k [0, ) and x, y H; (2) a mapping V : H H is l-lipschitzian if V x V y l x y for some constant l [0, ) and x, y H; (3) a mapping T : H H is nonexpansive if T x T y x y, x, y H;

3 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), (4) a mapping T : H H is averaged if T = ( ν)i + νg, where ν (0, ) and G : H H is nonexpansive. In this case, we also say that T is ν-averaged; (5) a mapping A : H H is monotone if Ax Ay, x y 0, x, y H; (6) an operator F : H H is κ-lipschitzian and η-strongly monotone with constants κ > 0 and η > 0 if F x F y κ x y and F x F y, x y η x y 2, x, y H, respectively. Recall that the (nearest point or metric) projection from H onto K, denoted by P K, is defined in such a way that, for each x H, P K x is the unique point in K with the property and x P K (x) = min{ x y : y K}. It is well known that P K is nonexpansive, and for x H, Moreover, P K satisfies z = P K x x z, y z 0, y C. (2.) x y, P K x P K y P K x P K y 2, x, y H, x y 2 x P K x 2 + y P K x 2, x H and y K. It is also well known that P K is 2-averaged and composite of finite many averaged mappings is averaged. Throughout this paper, we will use the following notations: F ix(t ) stands for the set of fixed points of T ; x n x stands for the weak convergence of {x n } to x; x n x stands for the strong convergence of {x n } to x. We also need the following lemmas for the proof of our main results. Lemma 2. ([3]). In a real Hilbert space H, the following inequality holds: x + y 2 x y, x + y, x, y H. Lemma 2.2 ([8]). (Demiclosedness principle). Let C be a nonempty closed convex subset of a real Hilbert space H, and let S : C C be a nonexpansive mapping. Then, the mapping I S is demiclosed. That is, if {x n } is a sequence in C such that x n x and (I S)x n y, then (I S)x = y. Lemma 2.3 ([0]). Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that the mapping F : C H is monotone and weakly continuous along segments (that is, F (x + ty) F (x) as t 0). Then the variational inequality is equivalent to the dual variational inequality x C, F x, x x 0, x C, x C, F x, x x 0, x C. Lemma 2.4 ([2]). Let {x n } and {z n } be bounded sequences in a Banach space E and {γ n } be a sequence in [0, ] which satisfies the following condition: Suppose that x n+ = γ n x n + ( γ n )z n, n 0, and 0 < lim inf γ n lim sup γ n <. Then z n x n = 0. lim sup( z n+ z n x n+ x n ) 0.

4 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), Lemma 2.5 ([5]). Let {s n } be a sequence of non-negative real numbers satisfying where { n } and {δ n } satisfy the following conditions: (i) { n } [0, ] and n=0 n =, (ii) lim sup δ n 0 or n=0 n δ n <, Then lim s n = 0. s n+ ( n )s n + n δ n, n 0, The following lemma can be easily proven, and therefore, we omit the proof (see also [8]). Lemma 2.6. Let H be a real Hilbert space H. Let F : H H be a κ-lipschitzian and η-strongly monotone operator with constants κ > 0 and η > 0. Let 0 < µ < 2η and 0 < t < ξ. Then S := ξi tµf : H H κ 2 is a contractive mapping with constant ξ tτ, where τ = µ(2η µκ 2 ). 3. Iterative algorithms Throughout the rest of this paper, we always assume the following: H and H 2 are real Hilbert spaces; C and Q are nonempty closed convex subsets of H and H 2, respectively; A : H H 2 is a bounded linear operator and A is the adjoint of A; V : C H is l-lipschitzian with constant l [0, ); F : H H is a κ-lipschitzian and η-strongly monotone operator with constants κ > 0 and η > 0; constants µ, σ, l, τ, and γ satisfy 0 < µ < 2η κ 2, 0 < σl < τ = µ(2η µκ 2 ), and 0 < γ < 2 A 2. We use Γ to denote the solution set of the SFP (.), that is, Γ = {x C : Ax Q} = C A Q, and assume the consistency of (.) so that Γ is nonempty closed convex. First, we introduce the following iterative algorithm that generates a net {x t } t (0, ) in an implicit τ γl way: x t = P C [I γa (I P Q )A]P C [tσv x t + (I tµf )x t ]. (3.) We prove strong convergence of {x t } as t 0 to a x which is a solution of the the following variational inequality: x C A Q such that σv x µf x, x x 0, x C A Q. (3.2) Now, for t (0, τ σl ), consider a mapping W t : C C defined by W t x := P C [I γa (I P Q )A]P C [tσv x + (I tµf )x], x C. It is easy to see that W t is a contractive mapping with constant t(τ σl). Indeed, note that P C and I γa (I P Q )A are nonexpansive. Thus, by Lemma 2.6, we have for x, y C, W t x W t y = P C [I γa (I P Q )A]P C [tσv x + (I tµf )x] P C [I γa (I P Q )A]P C [tσv y + (I tµf )y] tσ V x V y + (I µtf )x (I µtf )y ] tσl x y + ( tτ) x y = [ t(τ σl)] x y.

5 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), Therefore W t is a contractive mapping when t (0, τ σl ). By the Banach Contraction Principle, W t has a unique fixed point in C, denoted by x t, that is, which is exactly (3.). We summarize the basic properties of {x t }. x t = P C [I γa (I P Q )A]P C [tσv x t + (I tµf )x t ], Proposition 3.. Assume that the SFP (.) is consistent. Let {x t } be defined via (3.). Then (i) {x t } is bounded for t (0, τ σl ); (ii) lim t 0 x t P C [I γa (I P Q )A]x t = 0; (iii) x t defines a continuous path from (0, τ σl ) into C. Proof. (i) Let x be any point in C A Q. Set Then, we can rewrite (3.) as It follows that Hence, U = I γa (I P Q )A. x t = P C [U]P C [tσv x t + (I tµf )x t ], x t x = P C [U]P C [tσv x t + (I tµf )x t ] x t ( ) 0,. τ σl tσ(v x t V x) + (I tµf )x t (I tµf ) x + tσv x tµf x tσ x t x + ( tτ) x t x + t σv x µf x = [ (τ σl)t] x t x + t σv x µf x. x t x σv x µf x. τ σl Then, {x t } is bounded and so are {V x t }, {Ux t } and {F x t }. (ii) From (3.), we have x t P C [I γa (I P Q )A]x t = x t P C [Ux t ] By boundedness of {V x t } and {F x t }, we obtain (iii) Let t, t 0 (0, τ σl ). We calculate = P C [U]P C [tσv x t + (I tµf )x t ] P C [Ux t ] t σv x t µf x t. lim x t P C [I γa (I P Q )A]x t = 0. t 0 x t x t0 = P C [I γa (I P Q )A]P C [tσv x t + (I tµf )x t ] P C [I γa (I P Q )A]P C [t 0 σv x t0 + (I tµf )x t0 ] tσv x t + (I tµf )x t (t 0 σv x t0 + (I tµf )x t0 ) tσv x t t 0 σv x t + (I tµf )x t (I tµf )x t0 + t 0 σv x t t 0 σv x t0 + (I tµf )x t0 (I t 0 µf )x t0 σ V x t t t 0 + ( tτ) x t x t0 + t 0 σl x t x t0 + µ F x t0 t t 0.

6 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), This implies that This completes the proof. x t x t0 σ V x t + µ F x t0 t t 0. tτ t 0 σl Theorem 3.2. Assume that the SFP (.) is consistent. Let the net {x t } be defined via (3.). Then x t converges strongly to a point x as t 0, which solves the variational inequality (3.2). Proof. First, we show easily the uniqueness of a solution of the variational inequality (3.2). In fact, noting that 0 σl < τ and µη τ κ η, it follows that (µf σv )x (µf σv )y, x y (µη σl) x y 2. That is, µf σv is strongly monotone for 0 σl < τ µη. So the variational inequality (3.2) has only one solution. Next, we show that {x t } is relatively norm-compact as t 0 +. To this end, set U = I γa (I P Q )A, and let {t n } (0, τ σl ) be such that t n 0 as n. Put x n := x tn. From Proposition (ii), we have lim x n P C [U]x n = 0. (3.3) Setting y t = P C [tσv x t + (I tµf )x t ] and z t = tσv x t + (I tµf )x t. We then have y t = P C [z t ], and for any x C A Q, y t x = y t z t + z t x = y t z t + tσ(v x t V x) + (I tµf )x t (I tµf ) x + t(σv x µf x). (3.4) By using the property (2.) of the metric projection, we have Combining (3.4) with (3.5) along with Lemma 2.6, we get y t x 2 = y t x, y t x = y t z t, y t x + tσ V x t V x, y t x y t z t, y t x 0. (3.5) + (I tµf )x t (I tµf ) x, y t x + t σv x µf x, y t x tσl x t x y t x + ( tτ) x t x y t x + t σv x µf x, y t x = [ (τ σl)t] x t x y t x + t σv x µf x, y t x (τ σl)t x t x y t x 2 + t σv x µf x, y t x. It follows that Thus, Hence, we obtain y t x 2 [ (τ σl)t] x t x 2 + 2t σv x µf x, y t x. x t x 2 = P C [U]y t x 2 y t x 2 [ (τ σl)t] x t x 2 + 2t σv x µf x, y t x. x t x 2 2 τ σl σv x µf x, y t x.

7 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), In particular, we have Note that x n x 2 2 τ σl σv x µf x, y n x, x C A Q. (3.6) x t y t = P C [x t ] P C [tσv x t + (I tµf )x t ] t σv x t µf x t 0 as t 0. So, x n y n 0 as n. Since {x n } is bounded, there exists a subsequence {x ni } of {x n } which converges weakly to a point x. Without loss of generality, we may assume that {x n } converges weakly to x (y n x ). Noticing (3.3), we can use Lemma 2.2 to get x C A Q. Therefore, we can substitute x for x in (3.6) to obtain x n x 2 2 τ σl σv x µf x, y n x. Consequently, y n x actually implies that x n x. This has proved the relative norm-compactness of the net {x t } as t 0 +. Letting n in (3.6), we have x x 2 2 τ σl σv x µf x, x x, x C A Q. This implies that x C A Q solves the variational inequality σv x µf x, x x 0, x C A Q. (3.7) By Lemma 2.3, equation (3.7) is equivalent to its dual variational inequality σv x µf x, x x 0, x C A Q. This is exactly (3.2). By uniqueness of the solution of the variational inequality (3.2), we deduce that each cluster point of {x t } as t 0 + equals to x. Therefore x t x as t 0 +. This completes the proof. Taking F = I and µ = in Theorem 3.2, we have the following corollary. Corollary 3.3. Assume that the SFP (.) is consistent. Let the net {x t } be defined by ( x t = P C [I γa (I P Q )A]P C [tσv x t + ( t)x t ], t 0, σl ). (3.8) Then, {x t } converges strongly as t 0 to a point x which is the unique solution of variational inequality x C A Q such that σv x x, x x 0, x C A Q. (3.9) Taking V = 0 in (3.8), we get the following corollary. Corollary 3.4 ([23]). Assume that the SFP (.) is consistent, and let the net {x t } be defined by x t = P C [I γa (I P Q )A]P C [( t)x t ], t (0, ). (3.0) Then, {x t } converges strongly as t 0 to a point x which is the minimum norm solution of the split feasibility problem (.).

8 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), Proof. If we take V = 0, then (3.8) reduces to (3.0). Thus, x t x C A Q which satisfies x, x x 0, x C A Q. Thus, x 2 x, x x x, x C A Q, which implies x x for all x C A Q. That is, x is the minimum norm solution of the split feasibility problem (.). This completes the proof. Next, we propose the following iterative algorithm which generates a sequence in an explicit way: x n+ = P C [I γa (I P Q )A]P C [α n σv x n + (I α n µf )x n ], n 0, (3.) where {α n } [0, ] and x 0 H is an arbitrary initial guess, and establish strong convergence of this sequence to a point x, which is also a solution of the variational inequality (3.2). Theorem 3.5. Assume that the SFP (.) is consistent. Let {x n } be the sequence generated by the explicit algorithm (3.), where {α n } satisfies the following conditions: (C) {α n } [0, ], lim α n = 0; (C2) n=0 α n =. Then, {x n } converges strongly to a point x C A Q as n, which solves the variational inequality (3.2). Proof. Let U = I γa (I P Q )A. It is clear that P C and U are averaged. Since the composite of finitely many averaged mappings is averaged, P C [U] is averaged mapping. Hence, there exists a positive constant (0, ) such that P C [U] = ( )I + G, where G is a nonexpansive mapping. Let x C A Q. We divide the proof into four steps as follows. Step. We show that {x n } is bounded. In fact, from (3.), we deduce x n+ x = P C [I γa (I P Q )A]P C [α n σv x n + (I α n µf )x n ] x α n σv x n + (I α n µf )x n x α n σ V x n V x + (I α n µf )x n (I α n µf ) x + α n σv x µf x α n l x n x + ( α n τ) x n x + α n σv x µf x σv x µf x = [ (τ σl)α n ] x n x + (τ σl)α n. τ σl It follows by induction that { x n+ x max x n x,.. { max x 0 x, } σv x µf x τ σl. σv x µf x τ σl This means that {x n } is bounded. It is easy to deduce that {V x n }, {Ux n }, and F x n } are also bounded. }. Step 2. We show that lim P C [U]x n x n = 0. To this end, set y n = α n σv x n + (I α n µf )x n for all n 0. Then, we can rewrite (3.) as x n+ = [( )I + G][α n σv x n + (I α n µf )x n ]

9 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), = ( )x n + α n ( )(σv x n µf x n ) + Gy n [ ] = ( )x n + α n(σv x n µf x n ) + Gy n = ( )x n + z n, (3.2) where z n = α n(σv x n µf x n ) + Gy n. It follows that Thus, z n+ z n = α n+(σv x n+ µf x n+ ) + Gy n+ α n(σv x n µf x n ) Gy n. z n+ z n Gy n+ Gy n + [α n+ σv x n+ µf x n+ + α n σv x n µf x n ] y n+ y n + [α n+ σv x n+ µf x n+ + α n σv x n µf x n ] = α n+ σv x n+ + (I α n+ µf )x n+ α n σv x n (I α n µf )x n + [α n+ σv x n+ µf x n+ + α n σv x n µf x n ] x n+ x n + α n+ σv x n+ µf x n+ + α n σv x n µf x n + [α n+ σv x n+ µf x n+ + α n σv x n µf x n ]. It follows that Therefore, z n+ z n x n+ x n α n+ σv x n+ µf x n+ + α n σv x n µf x n By Lemma 2.4, we get At the same time, we observe that + [α n+ σv x n+ µf x n+ + α n σv x n µf x n ]. lim sup( z n+ z n x n+ x n ) 0. lim z n x n = 0. (3.3) lim y n x n = lim α n σv x n µf x n = 0, (3.4) and lim z n Gy n = lim α n σv x n µf x n = 0. (3.5) From (3.3) (3.5), we deduce lim Gx n x n lim ( Gx n Gy n + Gy n z n + z n x n ) lim ( x n y n + Gy n z n + z n x n ) = 0. Since P C [U]x n x n = (Gx n x n ), we obtain lim P C[U]x n x n = Gx n x n = 0. Step 3. We show that lim sup σv x µf x, P C [y n ] x 0, where x is the unique solution of the variational inequality (3.2). Indeed, we can choose a subsequence {x ni } of {x n } such that lim sup σv x µf x, x n x = lim σv x µf x, x ni x. i

10 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), Since {x ni } is bounded, there exists a subsequence of {x ni } which converges weakly to a point x. Without loss of generality, we may assume that {x ni } converges weakly to x. Therefore, from Step 2 and Lemma 2.2, we have x ni x F ix(p C [U]). Therefore lim sup σv x µf x, x n x = lim σv x µf x, x ni x i = σv x µf x, x x 0. This together with (3.4) implies that lim sup σv x µf x, P C [y n ] x 0. Step 4. We show that lim x n = x. We observe that P C [y n ] x 2 = P C [y n ] y n, P C [y n ] x + y n x, P C [y n ] x. Since P C [y n ] y n, P C [y n ] x 0 by (2.), we get P C [y n ] x 2 y n x, P C [y n ] x It follows that = α n σ(v x n V x ) + (I α n µf )x n (I α n µf )x, P C [y n ] x + α n σv x µf x, P C [y n ] x (α n σl x n x + ( α n τ) x n x ) P C [y n ] x + α n σv x µf x, P C [y n ] x = ( α n (τ σl)) x n x P C [y n ] x + α n σv x µf x, P C [y n ] x α n(τ σl) x n x P C[y n ] x 2 + α n σv x µf x, P C [y n ] x. P C [y n ] x 2 [ α n (τ σl)] x n x 2 + 2α n σv x µf x, P C [y n ] x. (3.6) From (3.3) and (3.6), we have x n+ x 2 = P C [U]P C [y n ] x 2 P C [y n ] x 2 [ α n (τ σl)] x n x α n (τ σl) τ σl σv x µf x, P C [y n ] x. (3.7) Put n = α n (τ σl) and δ n = 2 τ σl σv x µf x, P C [y n ] x. It can be easily seen from Step 3 and conditions (C) and (C2) that n lim sup δ n 0. Since (3.7) reduces to 0, n=0 n = and x n+ x 2 ( n ) x n x 2 + n δ n, by Lemma 2.5, we conclude that lim x n x = 0. This completes the proof. Putting µ = and F = I in Theorem 3.5, we obtain the following corollary.

11 J. S. Jung, J. Nonlinear Sci. Appl. 9 (206), Corollary 3.6. Assume that the SFP (.) is consistent. Let {x n } be generated by the following algorithm: x n+ = P C [I γa (I P Q )A]P C [α n σv x n + ( α n )x n ], n 0. (3.8) Assume that the sequence {α n } [0, ] satisfies the conditions (C) and (C2) in Theorem 3.5 Then {x n } converges strongly to a point x C A Q which is the unique solution of the variational inequality (3.9). Putting V = 0 in (3.8), we get the following corollary. Corollary 3.7 ([23]). Assume that the SFP (.) is consistent. Let {x n } be generated by the following algorithm: x n+ = P C [I γa (I P Q )A]P C [( α n )x n ], n 0. Assume that the sequence {α n } satisfies the conditions (C) and (C2) in Theorem 3.5. Then, {x n } converges strongly to a point x which is the minimum norm solution of the split feasibility problem (.). Acknowledgment. This study was supported by research funds from Dong-A University. 2. The author would like to thank the anonymous referees for their careful reading and valuable suggests which improved the presentation of this manuscript, and the editor for his valuable comments along with providing some recent related papers. References [] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse problems, 8 (2002), [2] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), [3] Y. Censor, T. Bortfeld, B. Martin, A. Tronfimov, A unified approach for inversion problems in intensity modulated radiation therapy, Phy. Med. Biol., 5 (2006), [4] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (994), , [5] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 2 (2005), [6] Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the multiple-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), [7] Y. Dang, Y, Gao, The strong convergence of a KM-CQ-like algorithm for a split feasibility problem, Inverse Problems, 27 (20), 9 pages. [8] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, (990). 2.2 [9] A. Latif, D. R. Sahu, Q. H. Ansari, Variable KM-like algorithms for fixed point problems and split feasibility problems, Fixed Point Theory Appl., 204 (204), 20 pages. [0] G. J. Minty, On the generalization of a direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (967), [] B. Qu, N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, 2 (2005), [2] T. Suzuki, Strong convergence of Krasnoselskii and Mann s type sequences for one parameter nonexpansive semigroups without Bochner integral, J. Math. Anal. Appl., 305 (2005), [3] W. Takahashi, Introduction to Nonlinear and Convex Analysis, Yokohama Publishers, Yokohama, Japan, (2009). 2. [4] F. Wang, H.-K. Xu, Approximating curve and strong convergence of the CQ algorithms for the split feasibility problem, J. Inequal. Appl., 200 (200), 3 pages. [5] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), [6] H.-K. Xu, A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), [7] H.-K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Problems, 26 (200), 7 pages.,

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