Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 RESEARCH Open Access Mann’s type extragradient for solving split feasibility and fixed point problems of Lipschitz asymptotically quasi-nonexpansive mappings Jitsupa Deepho and Poom Kumam* * Correspondence: poom.kum@kmutt.ac.th Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thrung Khru, Bangkok, 10140, Thailand Abstract The purpose of this paper is to introduce and analyze Mann’s type extragradient for finding a common solution set of the split feasibility problem and the set Fix(T) of fixed points of Lipschitz asymptotically quasi-nonexpansive mappings T in the setting of infinite-dimensional Hilbert spaces Consequently, we prove that the sequence generated by the proposed algorithm converges weakly to an element of Fix(T) ∩ under mild assumption The result presented in the paper also improves and extends some result of Xu (Inverse Probl 26:105018, 2010; Inverse Probl 22:2021-2034, 2006) and some others MSC: 49J40; 47H05 Keywords: split feasibility problems; fixed point problems; extragradient methods; asymptotically quasi-nonexpansive mappings; maximal monotone mappings Introduction The split feasibility problem (SFP) in finite dimensional spaces was first introduced by Censor and Elfving [] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [] Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning [–] The split feasibility problem in an infinitedimensional Hilbert space can be found in [, , –] and references therein Throughout this paper, we always assume that H , H are real Hilbert spaces, ‘→’, ‘ ’ denote strong and weak convergence, respectively, and F(T) is the fixed point set of a mapping T Let C and Q be nonempty closed convex subsets of infinite-dimensional real Hilbert spaces H and H , respectively, and let A ∈ B(H , H ), where B(H , H ) denotes the class of all bounded linear operators from H to H The split feasibility problem (SFP) is finding a point xˆ with the property xˆ ∈ C, Aˆx ∈ Q (.) ©2013Deepho and Kumam; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 In the sequel, we use Page of 19 to denote the set of solutions of SFP (.), i.e., = {ˆx ∈ C : Aˆx ∈ Q} Assuming that the SFP is consistent (i.e., (.) has a solution), it is not hard to see that x ∈ C solves (.) if and only if it solves the fixed-point equation x = PC I – γ A∗ (I – PQ )A x, x ∈ C, (.) where PC and PQ are the (orthogonal) projections onto C and Q, respectively, γ > is any positive constant, and A∗ denotes the adjoint of A To solve (.), Byrne [] proposed his CQ algorithm, which generates a sequence (xk ) by xk+ = PC I – γ A∗ (I – PQ )A xk , k ∈ N, (.) where γ ∈ (, /λ), and again λ is the spectral radius of the operator A∗ A The CQ algorithm (.) is a special case of the Krasnonsel’skii-Mann (K-M) algorithm The K-M algorithm generates a sequence {xn } according to the recursive formula xn+ = ( – αn )xn + αn Txn , where {αn } is a sequence in the interval (, ) and the initial guess x ∈ C is chosen arbitrarily Due to the fixed point for formulation (.) of the SFP, we can apply the K-M algorithm to the operator PC (I – γ A∗ (I – PQ )A) to obtain a sequence given by xk+ = ( – αk )xk + αk PC I – γ A∗ (I – PQ )A xk , k ∈ N, (.) where γ ∈ (, /λ), and again λ is the spectral radius of the operator A∗ A Then, as long as (xk ) satisfies the condition ∞ k= αk ( – αk ) = +∞, we have weak convergence of the sequence generated by (.) Very recently, Xu [] gave a continuation of the study on the CQ algorithm and its convergence He applied Mann’s algorithm to the SFP and proposed an averaged CQ algorithm which was proved to be weakly convergent to a solution of the SFP He derived a weak convergence result, which shows that for suitable choices of iterative parameters (including the regularization), the sequence of iterative solutions can converge weakly to an exact solution of the SFP He also established the strong convergence result, which shows that the minimum-norm solution can be obtained On the other hand, Korpelevich [] introduced an iterative method, the so-called extragradient method, for finding the solution of a saddle point problem He proved that the sequences generated by the proposed iterative algorithm converge to a solution of a saddle point problem Motivated by the idea of an extragradient method in [], Ceng [] introduced and analyzed an extragradient method with regularization for finding a common element of the solution set of the split feasibility problem and the set Fix(T) of a nonexpansive mapping T in the setting of infinite-dimensional Hilbert spaces Chang [] introduced an algorithm for solving the split feasibility problems for total quasi-asymptotically nonexpansive mappings in infinite-dimensional Hilbert spaces Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page of 19 The purpose of this paper is to study and analyze a Mann’s type extragradient method for finding a common element of the solution set of the SFP and the set Fix(T) of asymptotically quasi-nonexpansive mappings and Lipshitz continuous mappings in a real Hilbert space We prove that the sequence generated by the proposed method converges weakly to an element xˆ in Fix(T) ∩ Preliminaries We first recall some definitions, notations and conclusions which will be needed in proving our main results Let H be a real Hilbert space with the inner product ·, · and · , and let C be a nonempty closed and convex subset of H Let E be a Banach space A mapping T : E → E is said to be demi-closed at origin if for x∗ and (I – T)xn → , then x∗ = Tx∗ any sequence {xn } ⊂ E with xn x∗ , A Banach space E is said to have the Opial property if for any sequence {xn } with xn then lim inf xn – x∗ < lim inf xn – y , n→∞ n→∞ ∀y ∈ E with y = x∗ Remark . It is well known that each Hilbert space possesses the Opial property Proposition . For given x ∈ H and z ∈ C: (i) z = PC x if and only if x – z, y – z ≤ for all y ∈ C (ii) z = PC x if and only if x – z ≤ x – y – y – z for all y ∈ C (iii) For all x, y ∈ H, PC x – PC y, x – y ≥ PC x – PC y Definition . Let C be a nonempty, closed and convex subset of a real Hilbert space H We denote by F(T) the set of fixed points of T, that is, F(T) = {x ∈ C : x = Tx} Then T is said to be () nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C; () asymptotically nonexpansive if there exists a sequence kn ≥ , limn→∞ kn = and T n x – T n y ≤ kn x – y (.) for all x, y ∈ C and n ≥ ; () asymptotically quasi-nonexpansive if there exists a sequence kn ≥ , limn→∞ kn = and T n x – p ≤ kn x – p (.) for all x ∈ C, p ∈ F(T) and n ≥ ; () uniformly L-Lipschitzian if there exists a constant L > such that T nx – T ny ≤ L x – y for all x, y ∈ C and n ≥ Remark . By the above definitions, it is clear that: (.) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page of 19 (i) a nonexpansive mapping is an asymptotically quasi-nonexpansive mapping; (ii) a quasi-nonexpansive mapping is an asymptotically-quasi nonexpansive mapping; (iii) an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive mapping Proposition . (see []) We have the following assertions () T is nonexpansive if and only if the complement I – T is -ism () If T is ν-ism and γ > , then γ T is γν -ism () T is averaged if and only if the complement I – T is ν-ism for some ν > -ism Indeed, for α ∈ (, ), T is α-averaged if and only if I – T is α Proposition . (see [, ]) Let S, T, V : H → H be given operators We have the following assertions () If T = ( – α)S + αV for some α ∈ (, ), S is averaged and V is nonexpansive, then T is averaged () T is firmly nonexpansive if and only if the complement I – T is firmly nonexpansive () If T = ( – α)S + αV for some α ∈ (, ), S is firmly nonexpansive and V is nonexpansive, then T is averaged () The composite of finite many averaged mappings is averaged That is, if each of the mappings {Ti }ni= is averaged, then so is the composite T ◦ T ◦ · · · ◦ TN In particular, if T is α -averaged and T is α -averaged, where α , α ∈ (, ), then the composite T ◦ T is α-averaged, where α = α + α – α α () If the mappings {Ti }ni= are averaged and have a common fixed point, then n Fix(Ti ) = Fix(T · · · TN ) i= The notation Fix(T) denotes the set of all fixed points of the mapping T , that is, Fix(T) = {x ∈ H : Tx = x} Lemma . (see [], demiclosedness principle) Let C be a nonempty closed and convex subset of a real Hilbert space H, and let T : C → C be a nonexpansive mapping with Fix(S) = ∅ If the sequence {xn } ⊆ C converges weakly to x and the sequence {(I – S)xn } converges strongly to y, then (I – S)x = y; in particular, if y = , then x ∈ Fix(S) ∞ Lemma . (see []) Let {an }∞ n= and {bn }n= be two sequences of nonnegative numbers satisfying the inequality an+ ≤ an + bn , if ∞ n= bn ∀n ≥ , converges, then limn→∞ an exists The following lemma gives some characterizations and useful properties of the metric projection PC in a Hilbert space For every point x ∈ H, there exists a unique nearest point in C, denoted by PC x, such that x – PC x ≤ x – y , ∀y ∈ C, (.) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page of 19 where PC is called the metric projection of H onto C We know that PC is a nonexpansive mapping of H onto C Lemma . (see []) Let C be a nonempty closed and convex subset of a real Hilbert space H, and let PC be the metric projection from H onto C Given x ∈ H and z ∈ C, then z = PC x if and only if the following holds: x – z, y – z ≤ , ∀y ∈ C (.) Lemma . (see []) Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let PC : H → C be the metric projection from H onto C Then the following inequality holds: y – PC x + x – PC x ≤ x – y , ∀x ∈ H, ∀y ∈ C (.) Lemma . (see []) Let H be a real Hilbert space Then the following equations hold: (i) x – y = x – y – x – y, y for all x, y ∈ H; (ii) tx + ( – t)y = t x + ( – t) y – t( – t) x – y for all t ∈ [, ] and x, y ∈ H Throughout this paper, we assume that the SFP is consistent, that is, the solution set of the SFP is nonempty Let f : H → R be a continuous differentiable function The minimization problem f (x) := x∈C Ax – PQ Ax (.) is ill-posed Therefore (see []) consider the following Tikhonov regularized problem: fα (x) := x∈C Ax – PQ Ax + α x , (.) where α > is the regularization parameter We observe that the gradient ∇fα = ∇f + αI = A∗ (I – PQ )A + αI (.) is (α + A )-Lipschitz continuous and α-strongly monotone Let C be a nonempty closed convex subset of a real Hilbert space H, and let F : C → H be a monotone mapping The variational inequality problem (VIP) is to find x ∈ C such that Fx, y – x ≥ , ∀y ∈ C The solution set of the VIP is denoted by VIP(C, F) It is well known that x ∈ VI(C, F) ⇔ x = PC (x – λFx), ∀λ > Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page of 19 A set-valued mapping T : H → H is called monotone if for all x, y ∈ H, f ∈ Tx and g ∈ Ty imply x – y, f – g ≥ A monotone mapping T : H → H is called maximal if its graph G(T) is not properly contained in the graph of any other monotone mapping It is known that a monotone mapping T is maximal if and only if, for (x, f ) ∈ H × H, x – y, f – g ≥ for every (y, g) ∈ G(T) implies f ∈ Tx Let F : C → H be a monotone and k-Lipschitz continuous mapping, and let NC v be the normal cone to C at v ∈ C, that is, NC v = w ∈ H : v – u, w ≥ , ∀u ∈ C Define Tv = Fv + NC v if v ∈ C, ∅ if v ∈/ C Then T is maximal monotone and ∈ Tv if and only if v ∈ VI(C, F); see [] for more details We can use fixed point algorithms to solve the SFP on the basis of the following observation Let λ > and assume that x∗ ∈ Then Ax∗ ∈ Q, which implies that (I – PQ )Ax∗ = , and thus λA∗ (I – PQ )Ax∗ = Hence, we have the fixed point equation (I – λA∗ (I – PQ )A)x∗ = x∗ Requiring that x∗ ∈ C, we consider the fixed point equation PC (I – λ∇f )x∗ = PC I – λA∗ (I – PQ )A x∗ = x∗ (.) It is proved in [, Proposition .] that the solutions of fixed point equation (.) are exactly the solutions of the SFP; namely, for given x∗ ∈ H , x∗ solves the SFP if and only if x∗ solves fixed point equation (.) Proposition . (see []) Given x∗ ∈ H , the following statements are equivalent (i) x∗ solves the SFP; (ii) x∗ solves fixed point equation (.); (iii) x∗ solves the variational inequality problem (VIP) of finding x∗ ∈ C such that ∇f x∗ , x – x∗ ≥ , ∀x ∈ C, (.) where ∇f = A∗ (I – PQ )A and A∗ is the adjoint of A Proof (i) ⇔ (ii) See the proof in ([], Proposition .) (ii) ⇔ (iii) Observe that PC I – λA∗ (I – PQ )A x∗ = x∗ ⇔ I – λA∗ (I – PQ )A x∗ – x∗ , x – x∗ ≤ , ∀x ∈ C Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 ⇔ –λ A∗ (I – PQ )Ax∗ , x – x∗ ≤ , ⇔ ∇f x∗ , x – x∗ ≥ , Page of 19 ∀x ∈ C ∀x ∈ C, where ∇f = A∗ (I – PQ )A Remark . It is clear from Proposition . that = Fix PC (I – λ∇f ) = VI(C, ∇f ), for any λ > , where Fix(PC (I – λ∇f )) and VI(C, ∇f ) denote the set of fixed points of PC (I – λ∇f ) and the solution set of VIP Proposition . (see []) There hold the following statements: (i) the gradient ∇fα = ∇f + αI = A∗ (I – PQ )A + αI is (α + A )-Lipschitz continuous and α-strongly monotone; (ii) the mapping PC (I – λ∇fα ) is a contraction with coefficient – λ α – λ A +α ≤ √ – αλ ≤ – αλ , where < λ ≤ ( A α +α) ; (iii) if the SFP is consistent, then the strong limα→ xα exists and is the minimum norm solution of the SFP Main result Theorem . Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let T : C → C be an uniformly L-Lipschitzian and asymptotically quasi-nonexpansive mapping with Fix(T) ∩ = ∅ and {kn } ⊂ [, ∞) for all n ∈ N such that limn→∞ kn = Let {xn }, {yn } and {un } be the sequences in C generated by the following algorithm: ⎧ x = x ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ ⎨y = P (x – λ ∇f x ), n C n n αn n ⎪un = PC (xn – λn ∇fαn yn ), ⎪ ⎪ ⎩ xn+ = βn un + ( – βn )T n un , (.) where ∇fαn = ∇f + αn I = A∗ (I – PQ )A + αn I, and the sequences {αn }, {λn } and {βn } satisfy the following conditions: (i) < lim infn→∞ βn ≤ lim supn→∞ βn < , (ii) {λn } ∈ (, A ) and ∞ n= λn < ∞, ∞ (iii) n= αn < ∞ Then the sequence {xn } converges weakly to an element xˆ ∈ Fix(T) ∩ Proof We first show that PC (I – λ∇fα ) is ζ -averaged for each λn ∈ (, α+ A ), where ζ= + λ(α + A ) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page of 19 Indeed, it is easy to see that ∇f = A* (I – PQ )A is ∇f (x) – ∇f (y), x – y ≥ A -ism, A that is, ∇f (x) – ∇f (y) Observe that α+ A ∇fα (x) – ∇fα (y), x – y = α+ A = α x – y +α A α x–y ≥ α x – y + ∇f (x) – ∇f (y), x – y + α ∇f (x) – ∇f (y), x – y x–y + A ∇f (x) – ∇f (y), x – y + α ∇f (x) – ∇f (y), x – y + ∇f (x) – ∇f (y) = α(x – y) + ∇f (x) – ∇f (y) = ∇f (x) – ∇f (y) Hence, it follows that ∇fα = αI + A* (I – PQ )A is -ism Thus, λ∇fα is λ(α+ A ) -ism By α+ A Proposition .(iii) the composite (I – λ∇fα ) is λ(α+A ) -averaged Therefore, noting that PC is -averaged and utilizing Proposition .(iv), we know that for each λ ∈ (, α+ A ), PC (I – λ∇fα ) is ζ -averaged with ζ= λ(α + A ) λ(α + A ) + λ(α + A ) + – · = ∈ (, ) This shows that PC (I – λ∇fα ) is nonexpansive Furthermore, for {λn } ∈ [a, b] with a, b ∈ (, A ), utilizing the fact that limn→∞ α + A = A , we may assume that n < a ≤ λn ≤ b < A = lim n→∞ αn + A , ∀n ≥ Without loss of generality, we may assume that < a ≤ λn ≤ b < αn + A , ∀n ≥ Consequently, it follows that for each integer n ≥ , PC (I – λn ∇fαn ) is ζn -averaged with ζn = λn (αn + A ) λn (αn + A ) + λn (αn + A ) + – · = ∈ (, ) This immediately implies that PC (I – λn ∇fαn ) is nonexpansive for all n ≥ We divide the remainder of the proof into several steps Step We will prove that {xn } is bounded Indeed, we take fixed p ∈ Fix(T) ∩ arbitrarily Then we get PC (I – λn ∇f )p = p for λn ∈ (, A ) Since PC and (I – λn ∇fαn ) are Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page of 19 nonexpansive mappings, then we have yn – p = PC (I – λn ∇fαn )xn – PC (I – λn ∇f )p ≤ PC (I – λn ∇fαn )xn – PC (I – λn ∇fαn )p + PC (I – λn ∇fαn )p – PC (I – λn ∇f )p ≤ xn – p + (I – λn ∇fαn )p – (I – λn ∇f )p = xn – p + p – λn ∇fαn p – (p – λn ∇fp) = xn – p + λn ∇fp – λn ∇fαn p = xn – p + λn ∇fp – ∇fαn p = xn – p + λn ∇fp – ∇fp – αn p = x n – p + λ n αn p (.) and un – p = PC (xn – λn ∇fαn yn ) – p = PC (xn – λn ∇fαn yn ) – PC (I – λn ∇f )p ≤ (xn – λn ∇fαn yn ) – (p – λn ∇f )p = (xn – p) + (λn ∇fp – λn ∇fαn yn ) = (xn – p) + λn (∇fp – ∇fαn yn ) = (xn – p) + λn (∇fp – ∇fαn p + ∇fαn p – ∇fαn yn ) = (xn – p) + λn ∇fp – (∇fp + αn p) + λn (∇fαn p – ∇fαn yn ) ≤ xn – p + λn αn p + λn ∇fαn (p) – ∇fαn (yn ) ≤ x n – p + λ n αn p + λ n αn + A p – yn (.) Substituting (.) into (.) and simplifying, we have un – p ≤ xn – p + λn αn p + λn αn + A = x n – p + λ n αn p + λ n αn + A = xn – p + λ n α n p + λ n α n + A p – yn x n – p + λ n αn p xn – p + λn αn αn + A = xn – p + λ n αn p + λ n αn xn – p + λ n A + λn αn A xn – p + λn αn p p = + λn αn + λ n A x n – p + λ n α n p + λn α n + λ n A Since un = PC (xn – λn ∇fαn yn ) for each n ≥ , then by Proposition .(ii) we have un – p ≤ xn – λn ∇fαn (yn ) – p – xn – λn ∇fαn (yn ) – un + λn ∇fαn (yn ), p – un = xn – p p – xn – un (.) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 = xn – p + λn ∇fαn (yn ) – ∇fαn (p), p – yn – xn – un Page 10 of 19 + ∇fαn (p), p – yn + ∇fαn (yn ), yn – un ≤ xn – p – xn – un + λn ∇fαn (p), p – yn + ∇fαn (yn ), yn – un = xn – p – xn – un + λn (αn I + ∇f )p, p – yn + ∇fαn (yn ), yn – un ≤ xn – p – xn – un + λn αn p, p – un + ∇fαn (yn ), yn – un = xn – p – xn – yn + yn – un = xn – p – xn – yn + λn αn p, p – un + ∇fαn (yn ), yn – un – xn – yn , yn – un – yn – un + λn αn p, p – un + ∇fαn (yn ), yn – un = xn – p – xn – yn – yn – un + xn – λn ∇fαn (yn ) – yn , un – yn + λn αn + A + λn αn p, p – un Furthermore, by Proposition .(i) we have xn – λn ∇fαn (yn ) – yn , un – yn = xn – λn ∇fαn (xn ) – yn , un – yn + λn ∇fαn (xn ) – λn ∇fαn (yn ), un – yn ≤ λn ∇fαn (xn ) – λn ∇fαn (yn ), un – yn ≤ λn ∇fαn (xn ) – ∇fαn (yn ) un – yn ≤ λ n αn + A un – yn xn – yn So, we obtain un – p ≤ xn – p – xn – yn – yn – un + λn αn p p – un xn – yn un – yn (.) Consider λn αn + A = λn αn + A xn – yn – un – yn – λn αn + A xn – yn un – yn + un – yn , xn – yn it follows that λn αn + A = λn αn + A xn – yn – λn αn + A ≤ λn αn + A un – yn xn – yn + un – yn xn – yn – un – yn xn – yn + un – yn (.) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page 11 of 19 Substituting (.) into (.) and simplifying, we have un – p ≤ xn – p – xn – yn + un – yn = xn – p + λn αn + A – yn – un xn – yn + λn αn p p – un – xn – yn + λn αn + A xn – yn + λn αn p p – un = xn – p + λn αn + A – xn – yn + λn αn p p – un (.) Substituting (.) into (.) and simplifying, we have un – p ≤ xn – p + λn αn + A + λn αn p – xn – yn + λn αn + λ n A + λ n α n p + λn α n + λ n A = xn – p + λn αn + A xn – p – xn – yn + λn αn p + λn αn + λn A + λn αn p + λn α n + λ n A xn – p (.) Consequently, utilizing Lemma .(ii) and the last relations, we conclude that xn+ – p = βn un + ( – βn )T n un – βn + ( – βn ) p = βn un – βn p + ( – βn )T n un – ( – βn )p = βn (un – p) + ( – βn ) T n un – p = βn un – p + ( – βn ) T n un – p ≤ βn un – p + ( – βn )kn un – p = βn + ( – βn )kn un – p = βn + ( – βn )kn + λn αn + λ n A = βn + ( – βn )kn xn – p – βn ( – βn ) un – T n un – βn ( – βn ) un – T n un + λn αn + A + λn αn p + λn αn + λn A + λn αn p – βn ( – βn ) un – T n un xn – p – xn – yn – βn ( – βn ) un – T n un + kn – βn kn – xn – p – xn – yn + λn αn βn + ( – βn )kn + λn αn + αn A = kn – βn kn – + βn + ( – βn )kn λn αn p p xn – p + λn α n + λ n A λn αn + A xn – p + βn + ( – βn )kn λn αn + A – βn ( – βn ) un – T n un – xn – yn + λn αn kn – βn (kn – ) + λn αn + αn A p xn – p Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 + kn – βn kn – λn αn p Page 12 of 19 + λn αn + λ n A – βn ( – βn ) un – T n un (.) Since limn→∞ kn = , (i)-(iii) and by Corollary ., we deduce that lim xn – p exists for each p ∈ Fix(T) ∩ , (.) n→∞ and the sequences {xn }, {un } and {yn } are bounded It follows that T n x n – p ≤ k n xn – p Hence {T n xn – p} is bounded Step We will prove that lim un – Tun = n→∞ From (.) we have xn+ – p ≤ kn – βn kn – xn – p + kn – βn kn – λn αn + A + λn αn kn – βn (kn – ) + λn αn + αn A + kn – βn kn – λn αn p – βn ( – βn ) un – T n un = kn – βn kn – xn – p + kn – βn kn – – xn – yn p xn – p + λn αn + λ n A λn αn + A – xn – yn + αn kn – βn kn – M + αn kn – βn kn – M – βn ( – βn ) un – T n un = kn – βn kn – xn – p – kn – βn kn – – λn αn + A xn – yn + αn kn – βn kn – (M + M ) – βn ( – βn ) un – T n un , where M = supn≥ {λn ( + λn αn + αn A ) p xn – p } < ∞ and M = sup λn αn p n≥ + λn αn + λ n A < ∞ So, kn – βn kn – – λn αn + A ≤ kn – βn kn – xn – p xn – yn – xn+ – p + βn ( – βn ) un – T n un + αn kn – βn kn – (M + M ) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page 13 of 19 Since limn→∞ kn = , αn → , (i) and from (.), we have lim xn – yn = lim un – T n un = n→ n→ (.) Furthermore, we obtain yn – un = PC xn – λn ∇fαn (xn ) – PC xn – λn ∇fαn (yn ) ≤ xn – λn ∇fαn (xn ) – xn – λn ∇fαn (yn ) = λn ∇fαn (xn ) – ∇fαn (yn ) ≤ λ n αn + A xn – yn This together with (.) implies that lim yn – un = (.) n→ Also, xn – un ≤ xn – yn + yn – un together with (.) and (.) implies that lim xn – un = (.) n→ We can rewrite (.) from (.) by lim xn – T n un = n→ (.) Consider xn+ – xn = βn un + ( – βn )T n un – xn ≤ βn un – xn + ( – βn ) T n un – xn From (.) and (.), we obtain xn+ – xn → (as n → ∞) (.) Next, we will show that (.) implies that lim un – Tun = n→ We compute that yn+ – yn = PC (xn+ – λn+ ∇fαn+ xn+ ) – PC (xn – λn ∇fαn xn ) = PC (I – λn+ ∇fαn+ )xn+ – PC (I – λn ∇fαn )xn (.) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page 14 of 19 ≤ PC (I – λn+ ∇fαn+ )xn+ – PC (I – λn+ ∇fαn+ )xn + PC (I – λn+ ∇fαn+ )xn – PC (I – λn ∇fαn )xn ≤ xn+ – xn + (I – λn+ ∇fαn+ )xn – (I – λn ∇fαn )xn = xn+ – xn + xn – λn+ ∇fαn+ xn – (xn – λn ∇fαn xn ) = xn+ – xn + λn ∇fαn xn – λn+ ∇fαn+ xn = xn+ – xn + λn (∇f + αn )xn – λn+ (∇f + αn+ )xn = xn+ – xn + λn ∇fxn + λn αn xn – (λn+ ∇fxn + λn+ αn+ xn ) = xn+ – xn + (λn – λn+ )∇fxn + λn αn xn – λn+ αn+ xn = xn+ – xn + (λn – λn+ )∇fxn + λn αn xn – λn αn+ xn + λn αn+ xn – λn+ αn+ xn = xn+ – xn + (λn – λn+ )∇fxn + λn (αn – αn+ )xn + (λn – λn+ )αn+ xn ≤ xn+ – xn + |λn – λn+ | ∇fxn + λn |αn – αn+ | xn + αn+ |λn – λn+ | xn From conditions (ii), (iii) and (.), we obtain that yn+ – yn → (as n → ∞) (.) and un+ – un = PC (xn+ – λn+ ∇fαn+ yn+ ) – PC (xn – λn ∇fαn yn ) ≤ (xn+ – λn+ ∇fαn+ yn+ ) – (xn – λn ∇fαn yn ) = (xn+ – xn ) + (λn ∇fαn yn – λn+ ∇fαn + yn+ ) ≤ xn+ – xn + λn ∇fαn yn – λn+ ∇fαn+ yn+ = xn+ – xn + λn (∇f + αn )yn – λn+ (∇f + αn+ )yn+ = xn+ – xn + λn ∇fyn + λn αn yn – (λn+ ∇fyn+ + λn+ αn+ yn+ ) = xn+ – xn + (λn ∇fyn – λn+ ∇fyn+ ) + λn αn yn – λn+ αn+ yn+ ≤ xn+ – xn + λn ∇fyn – λn+ ∇fyn+ + λn αn yn – λn+ αn+ yn+ = xn+ – xn + (λn ∇fyn – λn ∇fyn+ ) + (λn ∇fyn+ – λn+ ∇fyn+ ) + (λn αn yn – λn αn yn+ ) + (λn αn yn+ – λn+ αn+ yn+ ) ≤ xn+ – xn + λn ∇fyn – ∇fyn+ + |λn – λn+ | ∇fyn+ + λn αn yn – yn+ + |λn αn – λn+ αn+ | yn+ From conditions (ii), (iii), (.) and (.), we obtain that un+ – un → (as n → ∞) (.) Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page 15 of 19 Since T is uniformly L-Lipschitzian continuous, then un – Tun ≤ un – un+ + un+ – T n+ un+ + T n+ un+ – T n+ un + T n+ un – Tun ≤ un – un+ + un+ – T n+ un+ + L un – un+ + L T n un – un Since limn→∞ un+ – un = and limn→∞ un – T n un = , it follows that lim un – Tun = (.) n→∞ Step We will show that xˆ ∈ Fix(T) ∩ We have from (.) xn – yn → (as n → ∞) (.) Since ∇f = A∗ (I – PQ )A is Lipschitz continuous and from (.), we have lim ∇f (xn ) – ∇f (yn ) = n→∞ Since {xn } is bounded, there is a subsequence {xni } of {xn } that converges weakly to some xˆ xˆ First, we show that xˆ ∈ Since xn – yn → , it is known that yni Put Aw = ∇fw + NC w if w ∈ C, ∅ if w ∈/ C, where NC w = {z ∈ H : w – v, z ≥ , ∀v ∈ C} Then A is maximal monotone and ∈ Aw if and only if w ∈ VI(C, ∇f ); see [] for more details Let (w, z) ∈ G(A), we have z ∈ Aw = ∇fw + NC w, and hence z – ∇fw ∈ NC w So, we have w – v, z – ∇fw ≥ , ∀v ∈ C On the other hand, from un = PC (xn – λn ∇fαn yn ) and w ∈ C, we have xn – λn ∇fαn yn – un , un – w ≥ , Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page 16 of 19 and hence w – un , un – xn + ∇fαn yn ≥ λn Therefore from z – ∇fw ∈ NC w and {uni } ∈ C it follows that w – uni , z ≥ w – uni , ∇fw ≥ w – uni , ∇fw – w – uni , uni – xni + ∇fαni yni λni = w – uni , ∇fw – w – uni , uni – xni + ∇fyni – αni w – uni , yni λni = w – uni , ∇fw – ∇funi + w – uni , ∇funi – ∇fyni – w – uni , uni – xni – αni w – uni , yni λni ≤ w – uni , ∇funi – ∇fyni – w – uni , uni – xni λni – αni w – uni , yni Hence, we obtain w – xˆ , z ≥ as i → ∞ ˆ ∈ VI(C, ∇f ) Thus it is clear Since A is maximal monotone, we have xˆ ∈ A– , and hence x that xˆ ∈ Next, we show that xˆ ∈ Fix(T) Indeed, since yni xˆ and uni – Tuni → , by (.) and Lemma ., we get xˆ ∈ Fix(T) Therefore, we have xˆ ∈ Fix(T) ∩ Now we prove that xn xˆ and yn xˆ Suppose the contrary and let {xnk } be another subsequences of {xn } such that {xnk } x∗ Then x∗ ∈ Fix(T) ∩ Let us show that xˆ = x∗ Assume that xˆ = x∗ From the Opial condition [], we have lim xn – xˆ = lim inf xnk – xˆ n→∞ k→∞ < lim inf xnk – x∗ k→∞ = lim xn – x∗ n→∞ = lim inf xnk – x∗ k→∞ < lim inf xnk – xˆ k→∞ = lim xn – xˆ n→∞ This is a contradiction Thus, we have xˆ = x∗ This implies xn xˆ ∈ Fix(T) ∩ Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 Page 17 of 19 Further, from xn – yn → it follows that yn xˆ This shows that both sequences {yn } and {un } converge weakly to xˆ ∈ Fix(T) ∩ This completes the proof Utilising Theorem ., we have the following new results in the setting of real Hilbert spaces Take T n ≡ T in Theorem . Therefore the conclusion follows Corollary . Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let T : C → C be an uniformly L-Lipschitzian and quasi-nonexpansive mapping with Fix(T) ∩ = ∅ Let {xn }, {yn } and {un } be the sequences in C generated by the following algorithm: ⎧ x = x ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ ⎨y = P (x – λ ∇f x ), n C n n αn n ⎪ u = P (x – λ ∇f n C n n αn yn ), ⎪ ⎪ ⎩ xn+ = βn un + ( – βn )T n un , (.) where ∇fαn = ∇f + αn I = A∗ (I – PQ )A + αn I, and the sequences {αn }, {λn } and {βn } satisfy the following conditions: (i) < lim infn→∞ βn ≤ lim supn→∞ βn < , (ii) {λn } ∈ (, A ) and ∞ n= λn < ∞, ∞ (iii) n= αn < ∞ Then the sequence {xn } converges weakly to an element xˆ ∈ Fix(T) ∩ Take T n ≡ I (identity mappings) in Theorem . Therefore the conclusion follows Corollary . Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let T : C → C be an uniformly L-Lipschitzian with Fix(T) ∩ = ∅ Let {xn }, {yn } and {un } be the sequences in C generated by the following algorithm: ⎧ x = x ∈ C chosen arbitrarily, ⎪ ⎪ ⎪ ⎨y = P (x – λ ∇f x ), n C n n αn n ⎪un = PC (xn – λn ∇fαn yn ), ⎪ ⎪ ⎩ xn+ = βn un + ( – βn )T n un , (.) where ∇fαn = ∇f + αn I = A∗ (I – PQ )A + αn I, and the sequences {αn }, {λn } and {βn } satisfy the following conditions: (i) < lim infn→∞ βn ≤ lim supn→∞ βn < , (ii) {λn } ∈ (, A ), ∞ (iii) n= αn < ∞ Then the sequence {xn } converges weakly to an element xˆ ∈ Fix(T) ∩ Remark . Theorem . improves and extends [, Theorem .] in the following respects: (a) The iterative algorithm [, Theorem .] is extended for developing our Mann’s type extragradient algorithm in Theorem . Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 (b) The technique of proving weak convergence in Theorem . is different from that in [, Theorem .] because our technique uses asymptotically quasi-nonexpansive mappings and the property of maximal monotone mappings (c) The problem of finding a common element of Fix(T) ∩ for asymptotically quasi-nonexpansive mappings is more general than that for nonexpansive mappings and the problem of finding a solution of the (SFP) in [, Theorem .] Competing interests The authors declare that they have no competing interests Authors’ contributions All authors contributed equally and significantly in writing this article All authors read and approved the final manuscript Acknowledgements The authors thank the referees for comments and suggestions on this manuscript The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D Program (Grant No PHD/0033/2554) and the King Mongkut’s University of Technology Thonburi Received: 16 September 2013 Accepted: 14 November 2013 Published: 20 Dec 2013 References Censor, Y, Elving, T: A multiprojection algorithm using Bregman projections in product space Numer Algorithms 8, 221-239 (1994) Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem Inverse Probl 18, 441-453 (2002) Censor, Y, Bortfeld, T, Martin, B, Trofimov, A: A unified approach for inversion problem in intensity-modulated radiation therapy Phys Med Biol 51, 2353-2365 (2006) Censor, Y, Elfving, T, Kopf, N, Bortfeld, T: The multiple-sets split feasibility problem and its applications Inverse Probl 21, 2071-2084 (2005) Censor, Y, Motova, A, Segal, A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem J Math Anal Appl 327, 1244-1256 (2007) Deepho, J, Kumam, P: A modified Halpern’s iterative scheme for solving split feasibility problems Abstr Appl Anal 2012, Article ID 876069 (2012) Sunthrayuth, P, Cho, YJ, Kumam, P: General iterative algorithms approach to variational inequalities and minimum-norm fixed point for minimization and split feasibility problems Opsearch (2013, in press) doi:10.1007/s12597-013-0150-5 Xu, HK: A variable Krasnosel’skill-Mann algorithm and the multiple-set split feasibility problem Inverse Probl 22, 2021-2034 (2006) Yang, Q: The relaxed CQ algorithm for solving the split feasibility problem Inverse Probl 20, 1261-1266 (2004) 10 Zhao, J, Yang, Q: Several solution methods for the split feasibility problem Inverse Probl 21, 1791-1799 (2005) 11 Korpelevich, GM: An extragradient method for finding saddle points and for other problems Èkon Mat Metody 12, 747-756 (1976) 12 Phiangsungnoen, S, Kumam, P: A hybrid extragradient method for solving Ky Fan inequalities, variational inequalities and fixed point problems In: Proceedings of the International MultiConference of Engineers and Computer Scientists 2013, vol II, pp 1042-1047 IMECS 2013, March 13-15, Hong Kong (2013) 13 Ceng, LC, Ansari, QH, Yao, JC: An extragradient method for solving split feasibility and fixed point problems Comput Math Appl 64(4), 633-642 (2012) 14 Chang, S: Split feasibility problems total quasi-asymptotically nonexpansive mappings Fixed Point Theory Appl 2012, Article ID 151 (2012) doi:10.1186/1687-1812-2012-151 15 Byrne, C: A unified treatment of some iterative algorithms in signal processing and image reconstruction Inverse Probl 26, 103-120 (2004) 16 Combettes, PL: Solving monotone inclusions via compositions of nonexpansive averaged operator Optimization 53(5-6), 475-504 (2004) 17 Geobel, K, Kirk, WA: Topics in Metric Fixed Point Theory Cambridge Studies in Advanced Mathematics, vol 28 Cambridge University Press, Cambridge (1990) 18 Tan, KK, Xu, HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process J Math Anal Appl 178, 301-308 (1993) 19 Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert space J Math Anal Appl 329, 336-346 (2007) 20 Nakajo, K, Takahashi, W: Strong convergence theorem for nonexpansive mappings and nonexpansive semigroups J Math Anal Appl 279, 372-379 (2003) 21 Rockafellar, RT: On the maximality of sums of nonlinear monotone operators Trans Am Math Soc 149, 75-88 (1970) 22 Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings Bull Am Math Soc 73, 591-597 (1967) Page 18 of 19 Deepho and Kumam Fixed Point Theory and Applications 2013, 2013:349 http://www.fixedpointtheoryandapplications.com/content/2013/1/349 10.1186/1687-1812-2013-349 Cite this article as: Deepho and Kumam: Mann’s type extragradient for solving split feasibility and fixed point problems of Lipschitz asymptotically quasi-nonexpansive mappings Fixed Point Theory and Applications 2013, 2013:349 Page 19 of 19 ... and Kumam: Mann? ? ?s type extragradient for solving split feasibility and fixed point problems of Lipschitz asymptotically quasi- nonexpansive mappings Fixed Point Theory and Applications 2013, 2013:349... algorithm for solving the split feasibility problems for total quasi -asymptotically nonexpansive mappings in infinite-dimensional Hilbert spaces Deepho and Kumam Fixed Point Theory and Applications 2013,... quasi- nonexpansive mapping; (ii) a quasi- nonexpansive mapping is an asymptotically- quasi nonexpansive mapping; (iii) an asymptotically nonexpansive mapping is an asymptotically quasi- nonexpansive