Hybrid extragradient viscosity method for general system of variational inequalities

Transcription

1 Ceng et al. Journal of Inequalities and Applications (05) 05:50 DOI 0.86/s z R E S E A R C H Open Access Hybrid extragradient viscosity method for general system of variational inequalities Lu-Chuan Ceng,, Yeong-Cheng Liou 3,4*, Ching-Feng Wen 4 and Yuh-Jenn Wu 5 * Correspondence: simplex_liou@hotmail.com 3 Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan 4 Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 807, Taiwan Full list of author information is available at the end of the article Abstract In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of a general system of variational inequalities, the set of solutions of a split feasibility problem (SFP), and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. The iterative algorithm is based on Korpelevich s extragradient method, viscosity approximation method, Mann s iteration method, hybrid steepest-descent method and gradient-projection method (GPM) with regularization. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality. MSC: 49J30; 47H09; 47J0; 49M05 Keywords: Mann-type hybrid steepest-descent method; general mixed equilibrium; general system of variational inequalities; nonexpansive mapping; strict pseudocontraction; inverse-strongly monotone mapping Introduction Let H be a real Hilbert space with the inner product, and the norm, C be a nonempty closed convex subset of H and P C be the metric projection of H onto C. Let S : C C be a self-mapping on C. WedenotebyFix(S) thesetoffixedpointsofs and by R the set of all real numbers. A mapping A : C H is called L-Lipschitz continuous if there exists a constant L 0suchthat Ax Ay L x y, x, y C. In particular, if L =thenais called a nonexpansive mapping; if L [0, ) then A is called a contraction. A mapping T : C C is called ξ-strictly pseudocontractive if there exists a constant ξ [0, ) such that Tx Ty x y + ξ (I T)x (I T)y, x, y C. In particular, if ξ =0, then T is a nonexpansive mapping. 05 Ceng et al.; licensee Springer. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page of 43 Let A : C H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x C such that A x, y x 0, y C. (.) The solution set of VIP (.)isdenotedbyvi(c, A). VIP (.) was first discussed by Lions [] and now it is well known. Variational inequalities have extensively been investigated; see the monographs [ 6]. It is well known that if A is a strongly monotone and Lipschitz continuous mapping on C, thenvip(.) has a unique solution. In the literature, the recent research work shows that variational inequalities like VIP (.) cover several topics, for example, monotone inclusions, convex optimization and quadratic minimization over fixed point sets; see [7 ] formoredetails. In 976, Korpelevich [] proposed an iterative algorithm for solving VIP (.) inthe Euclidean space R n : { y n = P C (x n τax n ), x n+ = P C (x n τay n ), n 0, with τ > 0 a given number, which is known as the extragradient method. The literature on the VIP is vast, and Korpelevich s extragradient method has received great attention given by many authors who improved it in various ways; see, e.g.,[, 3 ] and the references therein, to name but a few. On the other hand, let C and Q be nonempty closed convex subsets of infinitedimensional real Hilbert spaces H and H, respectively. The split feasibility problem (SFP) is to find a point x with the property x C and Ax Q, (.) where A B(H, H) andb(h, H) denotes the family of all bounded linear operators from H to H.WedenotebyΓ the solution set of the SFP. In 994, the SFP was first introduced by Censor and Elfving [], in finite-dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as the SFP; see, e.g., [3] and the references therein. Recently, it has been found that the SFP can also be applied to study intensity-modulated radiation therapy (IMRT); see, e.g., [4, 5] and the references therein. In the recent past, a wide variety of iterative methods have been used in signal processing and image reconstruction and for solving the SFP; see, e.g.,[3, 5, 8, 9, 3 8] and the references therein. A seemingly more popular algorithm that solves the SFP is the CQ algorithm of Byrne [3, 7] which is found to be a gradient-projection method (GPM) in convex minimization. However, it remains a challenge how to implement the CQ algorithm in the case where the projections P C and/or P Q fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm. Very recently, Xu [6] 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 algo-

3 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 3 of 43 rithm which was proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained. Throughout this paper, assume that the SFP is consistent, that is, the solution set Γ of the SFP is nonempty. Let f : H R be a continuous differentiable function. The minimization problem min f (x):= x C Ax P QAx is ill-posed. Therefore, Xu [6]considered the following Tikhonov regularization problem: min x C f α(x):= Ax P QAx + α x, where α > 0 is the regularization parameter. Very recently, by combining the gradient-projection method with regularization and extragradient method due to Nadezhkina and Takahashi [4], Ceng et al. [9] proposed a Mann-type extragradient-like algorithm, and proved that the sequences generated by the proposed algorithm converge weakly to a common solution of SFP (.)andthefixed point problem of a nonexpansive mapping. Theorem CAY (see Theorem 3. in [9]) Let T : C C be a nonexpansive mapping such that Fix(T) Γ. Assume that 0<λ <, and let {x A n } and {y n } be the sequences in C generated by the following Mann-type extragradient-like algorithm: x 0 = x C chosenarbitrarily, y n =( β n )x n + β n P C (x n λ f αn (x n )), x n+ = γ n x n +( γ n )TP C (y n λ f αn (y n )), n 0, where the sequences of parameters {α n }, {β n } and {γ n } satisfy the following conditions: (i) n=0 α n < ; (ii) {β n } [0, ] and 0<lim inf n β n lim sup n β n <; (iii) {γ n } [0, ] and 0<lim inf n γ n lim sup n γ n <. Then both the sequences {x n } and {y n } converge weakly to an element z Fix(T) Γ. In this paper, we consider the following general mixed equilibrium problem (GMEP) (see also [9, 30]) of finding x C such that Θ(x, y)+h(x, y) 0, y C, (.3) where Θ, h : C C R are two bi-functions. We denote the set of solutions of GMEP (.3) by GMEP(Θ, h). GMEP (.3) is very general; for example, it includes the following equilibrium problems as special cases. As an example, in [6, 3, 3] the authors considered and studied the generalized equilibrium problem (GEP) which is to find x C such that Θ(x, y)+ax, y x 0, y C. The set of solutions of GEP is denoted by GEP(Θ, A).

4 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 4 of 43 In [9, 33, 34], the authors considered and studied the mixed equilibrium problem (MEP) which is to find x C such that Θ(x, y)+ϕ(y) ϕ(x) 0, y C. The set of solutions of MEP is denoted by MEP(Θ, ϕ). In [35 37], the authors considered and studied the equilibrium problem (EP) which is to find x C such that Θ(x, y) 0, y C. The set of solutions of EP is denoted by EP(Θ). It is worth to mention that the EP is a unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, etc. Throughout this paper, it is assumed as in [38] thatθ : C C R is a bi-function satisfying conditions (θ)-(θ3) and h : C C R is a bi-function with restrictions (h)- (h3), where (θ) Θ(x, x)=0for all x C; (θ) Θ is monotone (i.e., Θ(x, y)+θ(y, x) 0, x, y C) and upper hemicontinuous in the first variable, i.e.,foreachx, y, z C, lim sup Θ ( tz +( t)x, y ) Θ(x, y); t 0 + (θ3) Θ is lower semicontinuous and convex in the second variable; (h) h(x, x)=0for all x C; (h) h is monotone and weakly upper semicontinuous in the first variable; (h3) h is convex in the second variable. For r >0andx H,letT r : H C be a mapping defined by T r x = {z C : Θ(z, y)+h(z, y)+ r } y z, z x 0, y C called the resolvent of Θ and h. Assume that C is the fixed point set of a nonexpansive mapping T : H H, i.e., C = Fix(T).Let F : H H be η-stronglymonotoneand κ-lipschitzian with positive constants η, κ >0.Letu 0 H be given arbitrarily and {λ n } n= be a sequence in [0, ]. The hybrid steepest-descent method introduced by Yamada [39]isthealgorithm u n+ := T λ n+ u n =(I λ n+ μf)tu n, n 0, (.4) where I is the identity mapping on H. In 003, Xu and Kim [40] proved the following strong convergence result. Theorem XK (see Theorem 3. in [40]) Assume that 0<μ <η/κ. Assume also that the control conditions hold for {λ n } n= : lim n λ n =0, n= λ n = and lim n λ n /λ n+ =

5 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 5 of 43 (or equivalently, lim n (λ n λ n+ )/λ n+ =0).Then the sequence {u n } generated by algorithm (.4) converges strongly to the unique solution u in Fix(T) to the hierarchical VIP: Fu, v u 0, v Fix(T). (.5) Let F, F : C H be two mappings. Consider the following general system of variational inequalities (GSVI) of finding (x, y ) C C such that { ν F y + x y, x x 0, x C, (.6) ν F x + y x, x y 0, x C, where ν >0andν > 0 are two constants. The solution set of GSVI (.6) isdenotedby GSVI(C, F, F ). In particular, if F = F = A, then the GSVI (.6) reduces to the following problem of finding (x, y ) C C such that { ν Ay + x y, x x 0, x C, ν Ax + y x, x y 0, x C, which is defined by Verma [4] and it is called a new system of variational inequalities (NSVI). Further, if x = y additionally, then the NSVI reduces to the classical VIP (.). In 008, Ceng et al. [] transformed GSVI (.6) into the fixed point problem of the mapping G = P C (I ν F )P C (I ν F ), that is, Gx = x,wherey = P C (I ν F )x. Throughout this paper, the fixed point set of the mapping G is denoted by Ξ. On the other hand, if C is the fixed point set Fix(T) of a nonexpansive mapping T and S is another nonexpansive mapping (not necessarily with fixed points), then VIP (.)becomes the variational inequality problem of finding x Fix(T)suchthat (I S)x, x x 0, x Fix(T). (.7) This problem, introduced by Mainge and Moudafi [34, 36], is calledthehierarchical fixed pointproblem.itisclearthatifs has fixed points, then they are solutions of VIP (.7). If S is a ρ-contraction (i.e., Sx Sy ρ x y for some 0 ρ < ), the solution set of VIP (.7) is a singleton and it is well known as the viscosity problem. This was previously introduced by Moudafi [7] andalsodevelopedbyxu[8]. In this case, it is easy to see that solving VIP (.7) is equivalent to finding a fixed point of the nonexpansive mapping P Fix(T) S,whereP Fix(T) is the metric projection on the closed and convex set Fix(T). In 0, Marino et al. [4] introduced a multi-step iterative scheme Θ(u n, y)+h(u n, y)+ r n y u n, u n x n 0, y C, y n, = β n, S u n +( β n, )u n, y n,i = β n,i S i u n +( β n,i )y n,i, i =,...,N, x n+ = α n f (x n )+( α n )Ty n,n, (.8) with f : C C a ρ-contraction and {α n }, {β n,i } (0, ), {r n } (0, ), that generalizes the two-step iterative scheme in [0] for two nonexpansive mappings to a finite family of nonexpansive mappings T, S i : C C, i =,...,N, and proved that the proposed scheme (.8)

6 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 6 of 43 convergesstronglytoacommonfixedpointofthemappingsthatisalsoanequilibrium pointofgmep(.3). More recently, Marino, Muglia and Yao s multi-step iterative scheme (.8)wasextended to develop the following relaxed viscosity iterative algorithm. Algorithm CKW (see (3.) in [43]) Let f : C C be a ρ-contraction and T : C C be a ξ-strict pseudocontraction. Let S i : C C be a nonexpansive mapping for each i =,...,N. LetF j : C H be ζ j -inverse strongly monotone with 0 < ν j < ζ j for each j =,. Let Θ : C C R be a bi-function satisfying conditions (θ)-(θ3) and h : C C R be a bi-function with restrictions (h)-(h3). Let {x n } bethesequencegeneratedby Θ(u n, y)+h(u n, y)+ r n y u n, u n x n 0, y C, y n, = β n, S u n +( β n, )u n, y n,i = β n,i S i u n +( β n,i )y n,i, i =,...,N, y n = α n f (y n,n )+( α n )Gy n,n, x n+ = β n x n + γ n y n + δ n Ty n, n 0, where G = P C (I ν F )P C (I ν F ), {α n }, {β n } are sequences in (0, ) with 0 < lim inf n β n lim sup n β n <,{γ n }, {δ n } are sequences in [0, ] with lim inf n δ n >0 and β n +γ n +δ n =, n 0, {β n,i } is a sequence in (0, ) for each i =,...,N,(γ n +δ n )ξ γ n, n 0, and {r n } is a sequence in (0, )withlim inf n r n >0. The authors [43] proved that the proposed scheme (.9)converges strongly to a common fixed point of the mappings T, S i : C C, i =,...,N, that is also an equilibrium point of GMEP (.3)andasolutionofGSVI(.6). In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for finding a common element of the solution set GMEP(Θ, h) ofgmep(.3), the solution set GSVI(C, F, F )(i.e., Ξ) ofgsvi(.6), the solution set Γ of SFP (.), and the common fixed point set N i= Fix(S i) Fix(T) of finitely many nonexpansive mappings S i : C C, i =,...,N, and a strictly pseudocontractive mapping T : C C, in the setting of the infinite-dimensional Hilbert space. The iterative algorithm is based on Korpelevich s extragradient method, viscosity approximation method [7] (see also[8]), Mann s iteration method, hybrid steepest-descent method [40] and gradient-projection method (GPM) with regularization. Our aim is to prove that the iterative algorithm converges strongly to a common element of these sets, which also solves some hierarchical variational inequality. We observe that related results have been derived say in [0, 3, 8, 34, 36, 37, 4 54]. Preliminaries Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by, and,respectively.letc be a nonempty closed convex subset of H. Wewritex n x to indicate that the sequence {x n } converges weakly to x and x n x to indicate that the sequence {x n } converges strongly to x. Moreover,weuse ω w (x n ) to denote the weak ω-limit set of the sequence {x n } and ω s (x n )todenotethestrong ω-limit set of the sequence {x n }, i.e., ω w (x n ):= { x H : x ni x for some subsequence {x ni } of {x n } } (.9)

7 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 7 of 43 and ω s (x n ):= { x H : x ni x for some subsequence {x ni } of {x n } }. The metric (or nearest point) projection from H onto C is the mapping P C : H C which assigns to each point x H theuniquepointp C x C satisfying the property x P C x = inf x y =: d(x, C). y C The following properties of projections are useful and pertinent to our purpose. Proposition. Given any x Handz C, one has (i) z = P C x x z, y z 0, y C; (ii) z = P C x x z x y y z, y C; (iii) P C x P C y, x y P C x P C y, y H, which hence implies that P C is nonexpansive and monotone. Definition. A mapping T : H H is said to be (a) nonexpansive if Tx Ty x y, x, y H; (b) firmly nonexpansive if T I is nonexpansive, or equivalently, if T is -inverse strongly monotone (-ism), x y, Tx Ty Tx Ty, x, y H; alternatively, T is firmly nonexpansive if and only if T can be expressedas T = (I + S), where S : H H is nonexpansive; projections are firmly nonexpansive. Definition. A mapping A : C H is said to be (i) monotone if Ax Ay, x y 0, x, y C; (ii) η-strongly monotone if there exists a constant η >0such that Ax Ay, x y η x y, x, y C; (iii) α-inverse-strongly monotone if there exists a constant α >0such that Ax Ay, x y α Ax Ay, x, y C.

8 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 8 of 43 It can be easily seen that if T is nonexpansive, then I T is monotone. It is also easy to see that the projection P C is -ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields. On the other hand, it is obvious that if A : C H is α-inverse-strongly monotone, then A is monotone and -Lipschitz continuous. Moreover, we also have that, for all u, v C α and λ >0, (I λa)u (I λa)v u v + λ(λ α) Au Av. (.) So, if λ α,theni λa is a nonexpansive mapping from C to H. In 008, Ceng et al. [] transformedproblem(.6) into a fixed point problem in the following way. Proposition. (see []) For given x, ȳ C,( x, ȳ) is a solution of GSVI (.6) if and only if x is a fixed point of the mapping G : C Cdefinedby Gx = P C (I ν F )P C (I ν F )x, x C, where ȳ = P C (I ν F ) x. In particular, if the mapping F j : C H is ζ j -inverse-strongly monotone for j =,,then the mapping G is nonexpansive provided ν j (0, ζ j ]forj =,.WedenotebyΞ the fixed point set of the mapping G. The following result is easy to prove. Proposition.3 (see [8]) Given x H, the following statements are equivalent: (i) x solves the SFP; (ii) x solves the fixed point equation P C (I λ f )x = x, where λ >0, f = A (I P Q )A and A is the adjoint of A; (iii) x solves the variational inequality problem (VIP) of finding x C such that f ( x ), x x 0, x C. It is clear from Proposition. that Γ = Fix ( P C (I λ f ) ) = VI(C, f ), λ >0. Definition.3 A mapping T : H H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is, T ( α)i + αs, where α (0, ) and S : H H is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are -averaged mappings.

9 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 9 of 43 Proposition.4 (see [55]) Let T : H H be a given mapping. (i) T is nonexpansive if and only if the complement I T is -ism. (ii) If T is ν-ism, then for γ >0, γ T is ν γ -ism. (iii) T is averaged if and only if the complement I T is ν-ism for some ν > /. Indeed, for α (0, ), T is α-averaged if and only if I T is α -ism. Proposition.5 (see [55, 56]) Let S, T, V : H H be given operators. (i) If T =( α)s + αv for some α (0, ) and if S is averaged and V is nonexpansive, then T is averaged. (ii) T isfirmlynonexpansiveifandonlyifthecomplementi T is firmly nonexpansive. (iii) If T =( α)s + αv for some α (0, ) and if S is firmly nonexpansive and V is nonexpansive, then T isaveraged. (iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {T i } N i= is averaged, then so is the composite T T N. In particular, if T is α -averaged and T is α -averaged, where α, α (0, ), then the composite T T is α-averaged, where α = α + α α α. (v) If the mappings {T i } N i= are averaged and have a common fixed point, then N Fix(T i )=Fix(T T N ). i= The notation Fix(T) denotes the set of all fixed points of the mapping T, that is, Fix(T)={x H : Tx = x}. We need some facts and tools in a real Hilbert space H which are listed as lemmas below. Lemma. Let X be a real inner product space. Then there holds the following inequality: x + y x +y, x + y, x, y X. Lemma. Let H beareal Hilbertspace. Then the following hold: (a) x y = x y x y, y for all x, y H; (b) λx + μy = λ x + μ y λμ x y for all x, y H and λ, μ [0, ] with λ + μ =; (c) if {x n } is a sequence in H such that x n x, it follows that lim sup n x n y = lim sup x n x + x y, y H. n It is clear that, in a real Hilbert space H, T : C C is ξ-strictly pseudocontractive if and only if the following inequality holds: Tx Ty, x y x y ξ (I T)x (I T)y, x, y C. This immediately implies that if T is a ξ-strictly pseudocontractive mapping, then I T is ξ -inverse strongly monotone; for further details, we refer to [57] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the

10 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 0 of 43 class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions. Lemma.3 (see Proposition. in [57]) Let C be a nonempty closed convex subset of a real Hilbert space H and T : C C be a mapping. (i) If T is a ξ-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition Tx Ty +ξ x y, x, y C. ξ (ii) If T is a ξ-strictly pseudocontractive mapping, then the mapping I T is semiclosed at 0, that is, if {x n } is a sequence in C such that x n x and (I T)x n 0, then (I T) x =0. (iii) If T is ξ-(quasi-)strict pseudocontraction, then the fixed-point set Fix(T) of T is closed and convex so that the projection P Fix(T) is well defined. Lemma.4 (see [7]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C Cbeaξ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that (γ + δ)ξ γ. Then γ (x y)+δ(tx Ty) (γ + δ) x y, x, y C. Lemma.5 (see Demiclosedness principle in [58]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C with Fix(S). Then I Sisdemiclosed. That is, whenever {x n } is a sequence in C weakly converging to some x C and the sequence {(I S)x n } strongly converges to some y, it follows that (I S)x = y. Here I is the identity operator of H. Lemma.6 Let A : C H be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection (see Proposition.(i))implies u VI(C, A) u = P C (u λau), λ >0. Let C be a nonempty closed convex subset of a real Hilbert space H.Weintroducesome notations. Let λ be a number in (0, ] and let μ > 0. Associating with a nonexpansive mapping T : C C,wedefinethemappingT λ : C H by T λ x := Tx λμf(tx), x C, where F : C H is an operator such that, for some positive constants κ, η >0,F is κ-lipschitzian and η-strongly monotoneon C;thatis,F satisfies the conditions Fx Fy κ x y and Fx Fy, x y η x y for all x, y C.

11 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page of 43 Lemma.7 (see Lemma 3. in [40]) T λ is a contraction provided 0<μ < η ; that is, κ T λ x T λ y ( λτ) x y, x, y C, where τ = μ(η μκ ) (0, ]. Lemma.8 (see Lemma. in [59]) Let {a n } be a sequence of nonnegative real numbers satisfying a n+ ( β n )a n + β n γ n + δ n, n 0, where {β n }, {γ n } and {δ n } satisfy the following conditions: (i) {β n } [0, ] and n=0 β n = ; (ii) either lim sup n γ n 0 or n=0 β n γ n < ; (iii) δ n 0 for all n 0, and n= δ n <. Then lim n a n =0. In the sequel, we will indicate with GMEP(Θ, h) the solution set of GMEP (.3). Lemma.9 (see [38]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let Θ : C C R be a bi-function satisfying conditions (θ)-(θ3) and h : C C R be a bi-function with restrictions (h)-(h3).moreover, letussuppose that (H) for fixed r >0and x C, there exist bounded K C and ˆx K such that for all z C \ K, Θ(ˆx, z)+h(z, ˆx)+ ˆx z, z x <0. r For r >0and x H, the mapping T r : H C (i.e., the resolvent of Θ and h) has the following properties: (i) T r x ; (ii) T r x is a singleton; (iii) T r is firmly nonexpansive; (iv) GMEP(Θ, h)=fix(t r ) and it is closed and convex. Lemma.0 (see [38]) Let us suppose that (θ)-(θ3), (h)-(h3) and (H) hold. Let x, y H, r, r >0.Then T r y T r x y x + r r r T r y y. Lemma. (see [4]) Suppose that the hypotheses of Lemma.9 are satisfied. Let {r n } be asequencein(0, ) with lim inf n r n >0.Suppose that {x n } is a bounded sequence. Then the following statements are equivalent and true: (a) if x n T rn x n 0 as n, each weak cluster point of {x n } satisfies the problem Θ(x, y)+h(x, y) 0, y C, i.e., ω w (x n ) GMEP(Θ, h); (b) the demiclosedness principle holds in the sense that if x n x and x n T rn x n 0 as n, then (I T rk )x =0for all k.

12 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page of 43 Recall that a set-valued mapping T : D(T) H H is called monotone if for all x, y D(T), f Tx and g Ty imply f g, x y 0. A set-valued mapping T is called maximal monotone if T is monotone and (I + λt)d(t)= H for each λ >0,whereI is the identity mapping of H.WedenotebyG(T)thegraphofT. It is known that a monotone mapping T is maximal if and only if, for (x, f ) H H, f g, x y 0forevery(y, g) G(T) impliesf Tx. Next we provide an example to illustrate the concept of maximal monotone mapping. Let A : C H be a monotone, k-lipschitz-continuous mapping, and let N C v be the normal cone to C at v C, i.e., N C v = { u H : v p, u 0, p C }. Define Tv= { Av + NC v, ifv C,, ifv / C. Thenitisknownin[60]that T is maximal monotone and 0 Tv v VI(C, A). (.) 3 Main results We now propose the following hybrid extragradient viscosity iterative scheme: Θ(u n, y)+h(u n, y)+ r n y u n, u n x n 0, y C, y n, = β n, S u n +( β n, )u n, y n,i = β n,i S i u n +( β n,i )y n,i, i =,...,N, ỹ n,n = P C (y n,n λ n f αn (y n,n )), y n = P C [ɛ n γ Vy n,n +(I ɛ n μf)gp C (y n,n λ n f αn (ỹ n,n ))], x n+ = β n y n + γ n P C (y n,n λ n f αn (ỹ n,n )) + δ n TP C (y n,n λ n f αn (ỹ n,n )) (3.) for all n 0, where F : C H is a κ-lipschitzian and η-strongly monotone operator with positive constants κ, η >0and V : C C is an l-lipschitzian mapping with constant l 0; F j : C H is ζ j -inverse strongly monotone and G := P C (I ν F )P C (I ν F ) with ν j (0, ζ j ) for j =,; T : C C is a ξ-strict pseudocontraction and S i : C C is a nonexpansive mapping for each i =,...,N; Θ, h : C C R are two bi-functions satisfying the hypotheses of Lemma.9; {λ n } isasequencein(0, ) with 0<lim inf A n λ n lim sup n λ n < ; A 0<μ <η/κ and 0 γ l < τ with τ := μ(η μκ ); {α n } isasequencein(0, ) with n=0 α n < ; {ɛ n }, {β n } are sequences in (0, ) with 0<lim inf n β n lim sup n β n <;

13 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 3 of 43 {γ n }, {δ n } are sequences in [0, ] with β n + γ n + δ n =, n 0; {β n,i } N i= are sequences in (0, ) and (γ n + δ n )ξ γ n, n 0; {r n } is a sequence in (0, ) with lim inf n r n >0and lim inf n δ n >0. We start our main result from the following series of propositions. Proposition 3. Let us suppose that Ω = Fix(T) N i= Fix(S i) GMEP(Θ, h) Ξ Γ. Then the sequences {x n }, {y n }, {y n,i } for all i, {u n } are bounded. Proof Since 0 < lim inf n λ n lim sup n λ n < and 0 < lim inf A n β n lim sup n β n <, we may assume, without loss of generality, that {λ n } [a, b] (0, ) A and {β n } [c, d] (0, ). Now, let us show that P C (I λ f α )isσ-averaged for each λ (0, ), where α+ A σ = +λ(α + A ) 4 Indeed, it is easy to see that f = A (I P Q )A is (0, ). (3.) A -ism, that is, f (x) f (y), x y A f (x) f (y). (3.3) Observe that ( α + A ) f α (x) f α (y), x y = ( α + A )[ α x y + f (x) f (y), x y ] = α x y + α f (x) f (y), x y + α A x y + A f (x) f (y), x y α x y +α f (x) f (y), x y + f (x) f (y) = α(x y)+ f (x) f (y) = fα (x) f α (y). (3.4) Hence, it follows that f α = αi + A (I P Q )A is -ism. Thus, λ f α+ A α is -ism according to Proposition.4(ii). By Proposition.4(iii), the complement I λ f α is λ(α+ A ) λ(α+ A ) - averaged. Therefore, noting that P C is -averaged and utilizing Proposition.5(iv), we know that for each λ (0, ), P α+ A C (I λ f α )isσ-averaged with σ = + λ(α + A ) λ(α + A ) = +λ(α + A ) 4 (0, ). (3.5) This shows that P C (I λ f α ) is nonexpansive. Furthermore, for {λ n } [a, b] (0, ), A we have a inf λ n sup λ n b < n 0 n 0 A = lim n α n + A. (3.6)

14 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 4 of 43 Without loss of generality, we may assume that a inf λ n sup λ n b <, n 0. (3.7) n 0 α n + A n 0 Consequently, it follows that for each integer n 0, P C (I λ n f αn )isσ n -averaged with σ n = + λ n(α n + A ) λn(α n + A ) = +λ n(α n + A ) 4 This immediately implies that P C (I λ n f αn ) is nonexpansive for all n 0. For simplicity, we write t n = P C (y n,n λ n f αn (ỹ n,n )) and (0, ). (3.8) v n = ɛ n γ Vy n,n +(I ɛ n μf)gt n for all n 0. Then y n = P C v n and x n+ = β n y n + γ n t n + δ n Tt n. First of all, take a fixed p Ω arbitrarily. We observe that y n, p u n p x n p. For all from i =toi = N, by induction, one proves that y n,i p β n,i u n p +( β n,i ) y n,i p u n p x n p. Thus we obtain that for every i =,...,N, y n,i p u n p x n p. (3.9) For simplicity, we write p = P C (p ν F p), t n = P C (t n ν F t n )andz n = P C ( t n ν F t n ) for each n 0. Then z n = Gt n and p = P C (I ν F ) p = P C (I ν F )P C (I ν F )p = Gp. Since F j : C H is ζ j -inverse strongly monotone and 0 < ν j <ζ j for each j =,,weknow that for all n 0, z n p = Gt n p = P C (I ν F )P C (I ν F )t n P C (I ν F )P C (I ν F )p (I ν F )P C (I ν F )t n (I ν F )P C (I ν F )p = [ P C (I ν F )t n P C (I ν F )p ] [ ν F P C (I ν F )t n F P C (I ν F )p ] PC (I ν F )t n P C (I ν F )p + ν (ν ζ ) F P C (I ν F )t n F P C (I ν F )p (I ν F )t n (I ν F )p + ν (ν ζ ) F t n F p

15 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 5 of 43 = (tn p) ν (F t n F p) + ν (ν ζ ) F t n F p t n p + ν (ν ζ ) F t n F p + ν (ν ζ ) F t n F p t n p. (3.0) From (3.), (3.9) and the nonexpansivity of P C (I λ n f αn ), it follows that ỹ n,n p = PC (I λ n f αn )y n,n P C (I λ n f )p P C (I λ n f αn )y n,n P C (I λ n f αn )p + PC (I λ n f αn )p P C (I λ n f )p y n,n p + (I λ n f αn )p (I λ n f )p x n p + λ n α n p. (3.) Utilizing Lemma.,wealsohave ỹ n,n p = PC (I λ n f αn )y n,n P C (I λ n f )p = P C (I λ n f αn )y n,n P C (I λ n f αn )p + P C (I λ n f αn )p P C (I λ n f )p P C (I λ n f αn )y n,n P C (I λ n f αn )p + P C (I λ n f αn )p P C (I λ n f )p, ỹ n,n p y n,n p + P C (I λ n f αn )p P C (I λ n f )p ỹ n,n p x n p + (I λn f αn )p (I λ n f )p ỹn,n p x n p +λ n α n p ỹ n,n p. (3.) Furthermore, utilizing Proposition.(ii), we have t n p y n,n λ n f αn (ỹ n,n ) p y n,n λ n f αn (ỹ n,n ) t n = y n,n p y n,n t n +λ n fαn (ỹ n,n ), p t n = y n,n p y n,n t n ( +λ n fαn (ỹ n,n ) f αn (p), p ỹ n,n + ) f αn (p), p ỹ n,n + fαn (ỹ n,n ), ỹ n,n t n y n,n p y n,n t n ( ) +λ n fαn (p), p ỹ n,n + fαn (ỹ n,n ), ỹ n,n t n = y n,n p y n,n t n [ ] +λ n (αn I + f )p, p ỹ n,n + fαn (ỹ n,n ), ỹ n,n t n y n,n p y n,n t n [ +λ n αn p, p ỹ n,n + ] f αn (ỹ n,n ), ỹ n,n t n = y n,n p y n,n ỹ n,n y n,n ỹ n,n, ỹ n,n t n ỹ n,n t n [ +λ n αn p, p ỹ n,n + ] f αn (ỹ n,n ), ỹ n,n t n

16 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 6 of 43 = y n,n p y n,n ỹ n,n ỹ n,n t n + y n,n λ n f αn (ỹ n,n ) ỹ n,n, t n ỹ n,n +λn α n p, p ỹ n,n. (3.3) In the meantime, by Proposition.(i), we have yn,n λ n f αn (ỹ n,n ) ỹ n,n, t n ỹ n,n = y n,n λ n f αn (y n,n ) ỹ n,n, t n ỹ n,n + λ n f αn (y n,n ) λ n f αn (ỹ n,n ), t n ỹ n,n λ n f αn (y n,n ) λ n f αn (ỹ n,n ), t n ỹ n,n λ n f αn (y n,n ) f αn (ỹ n,n ) t n ỹ n,n ( λ n αn + A ) y n,n ỹ n,n t n ỹ n,n. (3.4) So, from (3.9)and(3.), we obtain t n p y n,n p y n,n ỹ n,n ỹ n,n t n + y n,n λ n f αn (ỹ n,n ) ỹ n,n, t n ỹ n,n +λn α n p, p ỹ n,n y n,n p y n,n ỹ n,n ỹ n,n t n ( +λ n αn + A ) y n,n ỹ n,n t n ỹ n,n +λ n α n p, p ỹ n,n y n,n p y n,n ỹ n,n ỹ n,n t n + λ n( αn + A ) yn,n ỹ n,n + ỹ n,n t n +λ n α n p, p ỹ n,n = y n,n p +λ n α n p p ỹ n,n + ( λ ( n αn + A ) ) yn,n ỹ n,n y n,n p +λ n α n p ỹ n,n p y n,n p +λ n α n p [ y n,n p + λ n α n p ] y n,n p + λ n α n p y n,n p +λ n α n p = ( y n,n p + λ n α n p ) ( x n p + λ n α n p ). (3.5) Hence, utilizing Lemma.7 we deduce from (3.9)and(3.5) that y n p = P C v n p ɛn γ (Vy n,n Vp)+(I ɛ n μf)gt n (I ɛ n μf)p + ɛ n (γ V μf)p ɛ n γ Vy n,n Vp + (I ɛ n μf)gt n (I ɛ n μf)p + ɛ n (γ V μf)p ɛ n γ l y n,n p +( ɛ n τ) t n p + ɛ n (γ V μf)p ɛ n γ l y n,n p +( ɛ n τ) [ y n,n p + λ n α n p ] + ɛ n (γ V μf)p ( ɛ n (τ γ l) ) y n,n p + ɛ n (γ V μf)p + λn α n p ( ɛ n (τ γ l) ) x n p + ɛ n (γ V μf)p + λn α n p

17 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 7 of 43 = ( ɛ n (τ γ l) ) (γ V μf)p x n p + ɛ n (τ γ l) τ γ l { max x n p, + λ n α n p } (γ V μf)p + λ n α n p. (3.6) τ γ l Since (γ n + δ n )ξ γ n for all n 0, utilizing Lemma.4, weobtainfrom(3.5) and(3.6) that x n+ p = βn (y n p)+γ n (t n p)+δ n (Tt n p) = β [ n(y n p)+(γ n + δ n ) γn (t n p)+δ n (Tt n p) ] γ n + δ n β n y n p +(γ n + δ n ) [ γn (t n p)+δ n (Tt n p) ] γ n + δ n β n y n p +(γ n + δ n ) t n p { } (γ V μf)p β n [max x n p, + ] λ n α n p τ γ l +( β n ) ( x n p + λ n α n p ) { (γ V μf)p = β n max x n p, τ γ l { } max x n p, + λ n α n p. By induction, we can prove { x n+ p max x 0 p, (γ V μf)p τ γ l } (γ V μf)p + τ γ l } +( β n ) x n p + λ n α n p n λk α k p, n 0. Since {λ n } [a, b] (0, )and A n=0 α n <,weknowthat{x n } is bounded, and so are the sequences {u n }, {v n }, {t n }, { t n }, {y n }, {ỹ n,n }, {y n,i } for each i =,...,N.Since Tt n p +ξ ξ t n p, {Tt n } is also bounded. Proposition 3. Let us suppose that Ω. Moreover, let us suppose that the following hold: (H0) lim n ɛ n =0and n=0 ɛ n = ; α (H) lim n α n λ n ɛ n =0and lim n λ n n ɛ n =0; β (H) lim n,i β n,i n ɛ n =0for each i =,...,N; (H3) n= ɛ ɛ n ɛ n < or lim n ɛ n n ɛ n =0; (H4) n= r r n r n < or lim n r n n ɛ n =0 (H5) n= β β n β n < or lim n β n n ɛ n =0; (H6) n= γ n β n γ n β n < or lim n ɛ n γ n β n γ n β n =0. If u n u n = o(ɛ n ), then lim n x n+ x n =0,i.e., {x n } is asymptotically regular. Proof First, it is known that {λ n } [a, b] (0, )and{β A n } [c, d] (0, ) as in the proof of Proposition 3.. Taking into account lim inf n r n >0,wemayassume,without loss of generality, that {r n } [ r, ) forsome r >0.First,wewritex n = β n y n +( k=0

18 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 8 of 43 β n )w n, n, where w n = x n β n y n β n. It follows that for all n, w n w n = x n+ β n y n β n x n β n y n β n = γ nt n + δ n Tt n β n γ n t n + δ n Tt n β n = γ n(t n t n )+δ n (Tt n Tt n ) + β n ( δn + δ n β n β n ( γn γ ) n t n β n β n Since (γ n + δ n )ξ γ n for all n 0, utilizing Lemma.4 we have ) Tt n. (3.7) γ n (t n t n )+δ n (Tt n Tt n ) (γ n + δ n ) t n t n. (3.8) Next, we estimate y n y n. Indeed, according to λ n (α n + A )<, t n t n ( y n,n λ n f αn (ỹ n,n ) ) ( y n,n λ n f αn (ỹ n,n ) ) y n,n y n,n + λ n f αn (ỹ n,n ) λ n f αn (ỹ n,n ) y n,n y n,n + λ n λ n fαn (ỹ n,n ) + λ fαn n (ỹ n,n ) f αn (ỹ n,n ) y n,n y n,n + λ n λ n f αn (ỹ n,n ) ( + λ fαn n (ỹ n,n ) f αn (ỹ n,n ) + fαn (ỹ n,n ) f αn (ỹ n,n ) ) y n,n y n,n + λ n λ n f αn (ỹ n,n ) [ + λ n αn α n ỹ n,n + ( α n + A ) ỹ n,n ỹ n,n ] = y n,n y n,n + λ n λ n fαn (ỹ n,n ) ( + λ n α n α n ỹ n,n + λ n αn + A ) ỹ n,n ỹ n,n y n,n y n,n + λ n λ n fαn (ỹ n,n ) + λ n α n α n ỹ n,n + ỹ n,n ỹ n,n (3.9) and ỹ n,n ỹ n,n = ( PC yn,n λ n f αn (y n,n ) ) ( P C yn,n λ n f αn (y n,n ) ) ( PC yn,n λ n f αn (y n,n ) ) ( P C yn,n λ n f αn (y n,n ) ) + ( P C yn,n λ n f αn (y n,n ) ) ( P C yn,n λ n f αn (y n,n ) ) y n,n y n,n + ( y n,n λ n f αn (y n,n ) ) ( y n,n λ n f αn (y n,n ) ) = y n,n y n,n + λn f αn (y n,n ) λ n f αn (y n,n ) y n,n y n,n + λ n α n λ n α n y n,n + λ n λ n f (yn,n )

19 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 9 of 43 y n,n y n,n + ( α n λ n λ n + λ n α n α n ) y n,n + λ n λ n f (y n,n ). (3.0) In the meantime, by the definition of y n,i one obtains that, for all i = N,...,, y n,i y n,i β n,i u n u n + S i u n y n,i β n,i β n,i +( β n,i ) y n,i y n,i. (3.) In the case i =,wehave y n, y n, β n, u n u n + S u n u n β n, β n, +( β n, ) u n u n = u n u n + S u n u n β n, β n,. (3.) Substituting (3.)inall(3.)-typeoneobtains, for i =,...,N, i y n,i y n,i u n u n + S k u n y n,k β n,k β n,k k= + S u n u n β n, β n,, (3.3) which together with (3.0)implies that ỹ n,n ỹ n,n ɛ n y ( ) n,n y n,n λ n λ n α n α n + α n + λ n y n,n ɛ n ɛ n ɛ n + λ n λ n f (y n,n ) ɛ n N u n u n ɛ n + k= + S u n u n β n, β n, S k u n y n,k β n,k β n,k ɛ n ( ) λ n λ n α n α n + α n + λ n y n,n ɛ n ɛ n ɛ n + λ n λ n ɛ n f (y n,n ). (3.4) Since u n u n = o(ɛ n ) and the sequences {u n }, {y n,i } N i= are bounded, we know that ỹ n,n ỹ n,n lim =0. n ɛ n On the other hand, we observe that { v n = ɛ n γ Vy n,n +(I ɛ n μf)z n, v n = ɛ n γ Vy n,n +(I ɛ n μf)z n, n.

20 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 0 of 43 Simple calculations show that v n v n =(I ɛ n μf)z n (I ɛ n μf)z n +(ɛ n ɛ n )(γ Vy n,n μfz n ) + ɛ n γ (Vy n,n Vy n,n ). Then, passing to the norm and using the nonexpansivity of G,we get y n y n v n v n (I ɛ n μf)z n (I ɛ n μf)z n + ɛ n ɛ n γ Vy n,n μfz n + ɛ n γ Vy n,n Vy n,n ( ɛ n τ) z n z n + M ɛ n ɛ n + ɛ n γ l y n,n y n,n ( ɛ n τ) t n t n + M ɛ n ɛ n + ɛ n γ l y n,n y n,n, (3.5) where sup n 0 γ Vy n,n μfz n M for some M >0.Also,itiseasytoseefrom(3.7)and (3.8)that w n w n γ n(t n t n )+δ n (Tt n Tt n ) β n + γ n γ n β n β t n n + δ n δ n β n β Tt n n (γ n + δ n ) t n t n + γ n β γ n n β n β t n n + γ n γ n β n β Tt n n = t n t n + γ n γ n β ( tn + Tt n ). (3.6) n Moreover, by Lemma.0, we know that u n u n x n x n + L r n, r n β n where L = sup n 0 u n x n. Further, we observe that { x n+ = β n y n +( β n )w n, x n = β n y n +( β n )w n, n. Simple calculations show that x n+ x n =( β n )(w n w n )+(β n β n )(y n w n )+β n (y n y n ).

21 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page of 43 Consequently, passing to the norm we get from (3.9), (3.3)and(3.5)-(3.6) x n+ x n ( β n ) w n w n + β n β n y n w n + β n y n y n [ ( β n ) t n t n + γ n γ n ( β tn + Tt n )] n β n [ + β n β n y n w n + β n ( ɛn τ) t n t n + M ɛ n ɛ n + ɛ n γ l y n,n y n,n ] ( β n ɛ n τ) t n t n + γ n γ n ( β tn + Tt n ) n β n + β n β n y n w n + M ɛ n ɛ n + β n ɛ n γ l y n,n y n,n ( β n ɛ n τ) [ y n,n y n,n + λ n λ n fαn (ỹ n,n ) + λ n α n α n ỹ n,n + ỹ n,n ỹ n,n ] + γ n γ n ( β tn + Tt n ) n β n + β n β n y n w n + M ɛ n ɛ n + β n ɛ n γ l y n,n y n,n ( β n ɛ n (τ γ l) ) y n,n y n,n + λ n λ n f αn (ỹ n,n ) + λ n α n α n ỹ n,n + ỹ n,n ỹ n,n + γ n γ n ( β tn + Tt n ) n β n + β n β n y n w n + M ɛ n ɛ n ( β n ɛ n (τ γ l) )[ u n u n + N S k u n y n,k β n,k β n,k ] + S u n u n β n, β n, + λ n λ n fαn (ỹ n,n ) k= + λ n α n α n ỹ n,n + ỹ n,n ỹ n,n + γ n γ n ( β tn + Tt n ) n β n + β n β n y n w n + M ɛ n ɛ n ( β n ɛ n (τ γ l) )[ x n x n + L r n N + S k u n y n,k β n,k β n,k k= + S u n u n β n, β n, + λ n λ n f αn (ỹ n,n ) + λ n α n α n ỹ n,n + ỹ n,n ỹ n,n + γ n γ n β n ] r n β n ( tn + Tt n )

22 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page of 43 + β n β n y n w n + M ɛ n ɛ n ( cɛ n (τ γ l) ) x n x n + L r n r n r + S u n u n β n, β n, + λ n λ n f αn (ỹ n,n ) + λ n α n α n ỹ n,n + ỹ n,n ỹ n,n + γ n γ n β n + β n β n y n w n + M ɛ n ɛ n ( ɛ n (τ γ l)c ) r n r n x n x n + M 0 r N + M 0 β n,k β n,k + M 0 β n, β n, k= + N S k u n y n,k β n,k β n,k k= β n ( tn + Tt n ) + M 0 λ n λ n + M 0 α n α n + M 0 ỹ n,n ỹ n,n + M 0 γ n β n γ n + M 0 β n β n + M 0 ɛ n ɛ n = ( ɛ n (τ γ l)c ) { r n r n N x n x n + M 0 + β n,k β n,k r k= + λ n λ n + α n α n + ỹ n,n ỹ n,n + γ n γ n β n β n } + β n β n + ɛ n ɛ n β n where sup n 0 = ( ɛ n (τ γ l)c ) x n x n { M 0 r n r n + ɛ n (τ γ l)c + (τ γ l)c ɛ n r + λ n λ n ɛ n + α n α n ɛ n + β n β n ɛ n N k= + γ n β n γ n β n + ɛ n ɛ n + ỹ n,n ỹ n,n ɛ n ɛ n ɛ n { N +L + M + S k u n y n,k + S u n u n k= β n,k β n,k ɛ n + f αn (ỹ n,n ) } + b ỹ n,n + t n + Tt n + y n w n M 0 }, (3.7) ỹ for some M 0 > 0. Noticing lim n,n ỹ n,n n ɛ n Lemma.8, we obtain the claim. = 0 and using hypotheses (H0)-(H6) and Proposition 3.3 Let us suppose that Ω. Let us suppose that {x n } is asymptotically regular. Then x n u n = x n T rn x n 0 and y n,n ỹ n,n 0 as n.

23 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 3 of 43 Proof Take fixed p Ω arbitrarily. We recall that, by the firm nonexpansivity of T rn,astandard calculation (see [44]) shows that for p GMEP(Θ, h), u n p x n p x n u n. (3.8) Utilizing Lemmas. and.7,weobtainfrom0 γ l < τ,(3.)and(3.0)that y n p = ɛ n γ (Vy n,n Vp)+(I ɛ n μf)z n (I ɛ n μf)p + ɛ n (γ V μf)p ɛ n γ (Vy n,n Vp)+(I ɛ n μf)z n (I ɛ n μf)p +ɛ n (γ V μf)p, yn p [ ɛ n γ Vy n,n Vp + (I ɛ n μf)z n (I ɛ n μf)p ] +ɛn (γ V μf)p, yn p [ ɛ n γ l y n,n p +( ɛ n τ) z n p ] +ɛn (γ V μf)p, yn p [ = ɛ n τ γ l ] τ y n,n p +( ɛ n τ) z n p +ɛ n (γ V μf)p, yn p (γ l) ɛ n τ τ y n,n p +( ɛ n τ) z n p +ɛ n (γ V μf)p, yn p ɛ n τ y n,n p + z n p +ɛ n (γ V μf)p yn p ɛ n τ y n,n p + t n p ν (ζ ν ) F t n F p ν (ζ ν ) F t n F p +ɛ n (γ V μf)p y n p. (3.9) Since (γ n + δ n )ξ γ n for all n 0, utilizing Lemma.4 we have from (3.), (3.9), (3.5), (3.8)and(3.9)that x n+ p = βn (y n p)+γ n (t n p)+δ n (Tt n p) = β [ n(y n p)+(γ n + δ n ) γn (t n p)+δ n (Tt n p) ] γ n + δ n β n y n p +(γ n + δ n ) [ γn (t n p)+δ n (Tt n p) ] γ n + δ n β n y n p +(γ n + δ n ) t n p = β n y n p +( β n ) t n p β n [ ɛn τ y n,n p + t n p ν (ζ ν ) F t n F p ν (ζ ν ) F t n F p +ɛ n (γ V μf)p yn p ] +( β n ) t n p t n p β n [ ν (ζ ν ) F t n F p + ν (ζ ν ) F t n F p ] + ɛ n τ y n,n p +ɛ n (γ V μf)p y n p y n,n p +λ n α n p p ỹ n,n + ( λ ( n αn + A ) ) yn,n ỹ n,n β n [ ν (ζ ν ) F t n F p + ν (ζ ν ) F t n F p ] + ɛ n τ y n,n p +ɛ n (γ V μf)p yn p

24 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 4 of 43 u n p +λ n α n p p ỹ n,n + ( λ ( n αn + A ) ) yn,n ỹ n,n β n [ ν (ζ ν ) F t n F p + ν (ζ ν ) F t n F p ] + ɛ n τ y n,n p +ɛ n (γ V μf)p yn p x n p x n u n +λ n α n p p ỹ n,n + ( λ ( n αn + A ) ) yn,n ỹ n,n β n [ ν (ζ ν ) F t n F p + ν (ζ ν ) F t n F p ] + ɛ n τ y n,n p +ɛ n (γ V μf)p yn p. (3.30) So, we deduce from {β n } [c, d] (0, ) and {λ n } [a, b] (0, A )that x n u n + ( b ( α n + A ) ) yn,n ỹ n,n + c [ ν (ζ ν ) F t n F p + ν (ζ ν ) F t n F p ] x n u n + ( λ ( n αn + A ) ) yn,n ỹ n,n [ + β n ν (ζ ν ) F t n F p + ν (ζ ν ) F t n F p ] x n p x n+ p +λ n α n p p ỹ n,n + ɛ n τ y n,n p +ɛ n (γ V μf)p y n p x n x n+ ( x n p + x n+ p ) +α n b p p ỹ n,n + ɛ n τ y n,n p +ɛ n (γ V μf)p y n p. By Propositions 3. and 3. we know that the sequences {x n }, {y n }, {y n,n } and {ỹ n,n } are bounded and that {x n } is asymptotically regular. Therefore, from α n 0andɛ n 0we obtain that lim x n u n = lim F t n F p = lim F t n F p = lim y n,n ỹ n,n =0. (3.3) n n n n Remark 3. By the last proposition we have ω w (x n )=ω w (u n )andω s (x n )=ω s (u n ), i.e.,the sets of strong/weak cluster points of {x n } and {u n } coincide. Of course, if β n,i β i 0asn, for all indices i, the assumptions of Proposition 3. are enough to assure that x n+ x n lim =0, i {,...,N}. n β n,i In the next proposition, we estimate the case in which at least one sequence {β n,k0 } is a null sequence. Proposition 3.4 Let us suppose that Ω. Let us suppose that (H0) holds. Moreover, for an index k 0 {,...,N}, lim n β n,k0 =0and the following hold:

25 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 5 of 43 (H7) for each index i {,...,N}, β n,i β n,i α n α n β n β n lim = lim = lim n ɛ n β n,k0 n ɛ n β n,k0 n ɛ n β n,k0 ɛ n ɛ n = lim = lim n ɛ n β n,k0 n λ n λ n = lim =0; n ɛ n β n,k0 (H8) there exists a constant δ >0such that If u n u n = o(ɛ n β n,k0 ), then x n+ x n lim =0. n β n,k0 Proof It is clear from (3.4)that ỹ n,n ỹ n,n ɛ n β n,k0 u n u n ɛ n β n,k0 + N k= ɛ n β n,k0 r n r n = lim n ɛ n β n,k0 γ n ɛ n β γ n n,k0 β n β n β n,k0 < δ for all n. S k u n y n,k β n,k β n,k ɛ n β n,k0 + S u n u n β n, β n, ɛ n β n,k0 ( ) λ n λ n α n α n + α n + λ n y n,n ɛ n β n,k0 ɛ n β n,k0 + λ n λ n ɛ n β n,k0 f (y n,n ). According to (H7) and u n u n = o(ɛ n β n,k0 ), we get ỹ n,n ỹ n,n lim =0. (3.3) n ɛ n β n,k0 Consider (3.7). Dividing both the terms by β n,k0,wehave x n+ x n β n,k0 So, by (H8) we have x n+ x n β n,k0 ( ɛ n (τ γ l)c ) x n x n + ɛ n (τ γ l)c β n,k0 M 0 (τ γ l)c { r n r n ɛ n β n,k0 r + N + λ n λ n ɛ n β n,k0 + α n α n ɛ n β n,k0 + β n β n ɛ n β n,k0 k= + γ n β n γ n β n + ɛ n ɛ n + ỹ n,n ỹ n,n ɛ n β n,k0 ɛ n β n,k0 ɛ n β n,k0 ( ɛ n (τ γ l)c ) x n x n β n,k0 + ( ɛ n (τ γ l)c ) x n x n β n,k0 β n,k0 β n,k β n,k ɛ n β n,k0 }.

26 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 6 of 43 + ɛ n (τ γ l)c M 0 { r n r n (τ γ l)c ɛ n β n,k0 r N β n,k β n,k + + λ n λ n + α n α n ɛ n β n,k0 ɛ n β n,k0 ɛ n β n,k0 k= + β n β n + γ n β n γ } n β n + ɛ n ɛ n + ỹ n,n ỹ n,n ɛ n β n,k0 ɛ n β n,k0 ɛ n β n,k0 ɛ n β n,k0 ( ɛ n (τ γ l)c ) x n x n + x n x n β n,k0 β n,k0 β n,k0 { M 0 r n r n + ɛ n (τ γ l)c (τ γ l)c ɛ n β n,k0 r + N k= β n,k β n,k ɛ n β n,k0 + λ n λ n ɛ n β n,k0 + α n α n ɛ n β n,k0 + β n β n + ɛ n β n,k0 γ n β n γ n β n + ɛ n ɛ n ɛ n β n,k0 ɛ n β n,k0 ( ɛ n (τ γ l)c ) x n x n + ɛ n δ x n x n + ɛ n (τ γ l)c + N k= β n,k0 M 0 { r n r n (τ γ l)c ɛ n β n,k0 r β n,k β n,k ɛ n β n,k0 + λ n λ n ɛ n β n,k0 + α n α n ɛ n β n,k0 + β n β n + ɛ n β n,k0 γ n β n γ n β n = ( ɛ n (τ γ l)c ) x n x n + ɛ n (τ γ l)c + ɛ n ɛ n ɛ n β n,k0 ɛ n β n,k0 β n,k0 { δ x n x n (τ γ l)c + ỹ n,n ỹ n,n ɛ n β n,k0 + ỹ n,n ỹ n,n ɛ n β n,k0 [ r n r n N + M 0 ɛ n β n,k0 r + β n,k β n,k + λ n λ n + α n α n ɛ n β n,k0 ɛ n β n,k0 ɛ n β n,k0 + β n β n + ɛ n β n,k0 k= γ n β n γ n β n + ɛ n ɛ n ɛ n β n,k0 ɛ n β n,k0 + ỹ n,n ỹ n,n ɛ n β n,k0 } } ]}. Therefore, utilizing Lemma.8, from(3.3), (H0), (H7) andtheasymptotical regularityof {x n } (due to Proposition 3.), we deduce that x n+ x n lim =0. n β n,k0 Proposition 3.5 Let us suppose that Ω. Let us suppose that (H0)-(H6) hold. If u n u n = o(ɛ n ), then z n t n 0 as n.

27 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 7 of 43 Proof Let p Ω. In terms of the firm nonexpansivity of P C and the ζ j -inverse strong monotonicity of F j for j =,,weobtainfromν j (0, ζ j ), j =,and(3.0)that t n p = PC (I ν F )t n P C (I ν F )p (I ν F )t n (I ν F )p, t n p = [ (I ν F )t n (I ν F )p + t n p (I ν F )t n (I ν F )p ( t n p) ] [ tn p + t n p (tn t n ) ν (F t n F p) (p p) ] = [ tn p + t n p (tn t n ) (p p) +ν (tn t n ) (p p), F t n F p ν F t n F p ] and z n p = PC (I ν F ) t n P C (I ν F ) p (I ν F ) t n (I ν F ) p, z n p = [ (I ν F ) t n (I ν F ) p + z n p (I ν F ) t n (I ν F ) p (z n p) ] [ t n p + z n p ( t n z n )+(p p) +ν F t n F p,( t n z n )+(p p) ν F t n F p ] [ tn p + z n p ( t n z n )+(p p) +ν F t n F p,( t n z n )+(p p) ]. Thus, we have t n p t n p (t n t n ) (p p) +ν (tn t n ) (p p), F t n F p ν F t n F p (3.33) and z n p t n p ( t n z n )+(p p) +ν F t n F p ( t n z n )+(p p). (3.34) Consequently, from (3.0), (3.5), (3.9), (3.30)and(3.33), it follows that x n+ p β n y n p +( β n ) t n p

28 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 8 of 43 which yields β n [ ɛn τ y n,n p + z n p +ɛ n (γ V μf)p yn p ] +( β n ) t n p β n [ ɛn τ y n,n p + t n p +ɛ n (γ V μf)p yn p ] +( β n ) t n p β n [ ɛn τ y n,n p + t n p (tn t n ) (p p) +ν (tn t n ) (p p) F t n F p +ɛ n (γ V μf)p yn p ] +( β n ) t n p t n p + ɛ n τ y n,n p β n (tn t n ) (p p) +ν (t n t n ) (p p) F t n F p +ɛ n (γ V μf)p y n p ( x n p + λ n α n p ) + ɛn τ y n,n p β n (tn t n ) (p p) +ν (t n t n ) (p p) F t n F p +ɛ n (γ V μf)p y n p, c (t n t n ) (p p) β n (t n t n ) (p p) ( x n p + λ n α n p ) xn+ p + ɛ n τ y n,n p +ν (t n t n ) (p p) F t n F p +ɛ n (γ V μf)p y n p ( x n x n+ + λ n α n p )( x n p + x n+ p + λ n α n p ) + ɛ n τ y n,n p +ν (tn t n ) (p p) F t n F p +ɛ n (γ V μf)p yn p. Since lim n α n =0,lim n ɛ n =0,lim n x n+ x n =0,and{x n }, {y n }, {y n,n }, {t n } and { t n } are bounded, we deduce from (3.3)that lim (tn t n ) (p p) =0. (3.35) n Furthermore, from (3.5), (3.9), (3.30) and(3.34), it followsthat x n+ p β n y n p +( β n ) t n p β n [ ɛn τ y n,n p + z n p +ɛ n (γ V μf)p yn p ] +( β n ) t n p β n [ ɛn τ y n,n p + t n p ( t n z n )+(p p) +ν F t n F p ( t n z n )+(p p) +ɛn (γ V μf)p yn p ] +( β n ) t n p t n p + ɛ n τ y n,n p β n ( t n z n )+(p p) +ν F t n F p ( t n z n )+(p p) +ɛn (γ V μf)p yn p ( x n p + λ n α n p ) + ɛn τ y n,n p β n ( t n z n )+(p p) +ν F t n F p ( t n z n )+(p p) +ɛn (γ V μf)p yn p,

29 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 9 of 43 which leads to c ( t n z n )+(p p) β n ( t n z n )+(p p) ( x n p + λ n α n p ) xn+ p + ɛ n τ y n,n p +ν F t n F p ( t n z n )+(p p) +ɛ n (γ V μf)p y n p ( x n x n+ + λ n α n p )( x n p + x n+ p + λ n α n p ) + ɛ n τ y n,n p +ν F t n F p ( t n z n )+(p p) +ɛ n (γ V μf)p y n p. Since lim n α n =0,lim n ɛ n =0,lim n x n+ x n =0,and{x n }, {y n }, {y n,n }, {z n } and { t n } are bounded, we deduce from (3.3)that lim ( t n z n )+(p p) =0. (3.36) n Note that t n z n (t n t n ) (p p) + ( t n z n )+(p p). Hence from (3.35)and(3.36)weget lim n t n z n = lim n t n Gt n = 0. (3.37) Proposition 3.6 Let us suppose that Ω. Let us suppose that 0<lim inf n β n,i lim sup n β n,i <for each i =,...,N. Moreover, suppose that u n u n = o(ɛ n ) and (H0)-(H6) are satisfied. Then lim n S i u n u n =0for each i =,...,Nprovided Ty n y n 0 as n. Proof Firstofall,itisclearthat t n ỹ n,n = PC ( yn,n λ n f αn (ỹ n,n ) ) P C ( yn,n λ n f αn (y n,n ) ) ( y n,n λ n f αn (ỹ n,n ) ) ( y n,n λ n f αn (y n,n ) ) = λ n fαn (ỹ n,n ) f αn (y n,n ) λ n ( αn + A ) ỹ n,n y n,n ỹ n,n y n,n. By Proposition 3.3,we get lim n t n ỹ n,n =0, which together with (3.3)implies that lim n t n y n,n =0. (3.38)

30 Ceng et al. Journal of Inequalities and Applications (05) 05:50 Page 30 of 43 Note that y n t n ɛn γ Vy n,n +(I ɛ n μf)z n t n ɛ n γ Vy n,n μfz n + z n t n. From Proposition 3.5 and ɛ n 0, we obtain lim y n t n =0. (3.39) n Also, observe that x n+ x n + x n y n = x n+ y n = γ n (t n y n )+δ n (Tt n y n ) = γ n (t n y n )+δ n (Tt n Ty n )+δ n (Ty n y n ). By Proposition 3. we know that {x n } is asymptotically regular. Utilizing Lemma.4 we have from (γ n + δ n )ξ γ n that y n x n = xn+ x n γ n (t n y n ) δ n (Tt n Ty n ) δ n (Ty n y n ) x n+ x n + γ n (t n y n ) δ n (Tt n Ty n ) + δ n Ty n y n x n+ x n +(γ n + δ n ) t n y n + δ n Ty n y n x n+ x n + t n y n + Ty n y n, which together with (3.39)and Ty n y n 0leadsto lim x n y n =0. (3.40) n Let us show that for each i {,...,N},onehas S i u n y n,i 0asn.Letp Ω. When i = N, by Lemma.(b)wehavefrom(3.9), (3.0), (3.5)and(3.9) y n p ɛ n τ y n,n p + z n p +ɛ n (γ V μf)p y n p ɛ n τ y n,n p + t n p +ɛ n (γ V μf)p y n p ɛ n τ y n,n p +ɛ n (γ V μf)p yn p +λ n α n p ỹ n,n p + y n,n p ɛ n τ y n,n p +ɛ n (γ V μf)p yn p +λ n α n p ỹ n,n p + β n,n S N u n p +( β n,n ) y n,n p β n,n ( β n,n ) S N u n y n,n ɛ n τ y n,n p +ɛ n (γ V μf)p yn p +λ n α n p ỹ n,n p + β n,n u n p +( β n,n ) u n p β n,n ( β n,n ) S N u n y n,n = ɛ n τ y n,n p +ɛ n (γ V μf)p y n p +λ n α n p ỹ n,n p + u n p β n,n ( β n,n ) S N u n y n,n