Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 DOI 10.1186/s13660-015-0567-x RESEARCH Open Access The common solution for a generalized equilibrium problem, a variational inequality problem and a hierarchical fixed point problem Ibrahim Karahan1 , Aydin Secer2* , Murat Ozdemir3 and Mustafa Bayram2 * Correspondence:
[email protected]2 Department of Mathematical Engineering, Faculty of Chemistry-Metallurgical, Yildiz Technical University, Istanbul, Turkey Full list of author information is available at the end of the article Abstract The present paper aims to deal with a new iterative method to find a common solution of a generalized equilibrium problem, a variational inequality problem and a hierarchical fixed point problem for a sequence of nearly nonexpansive mappings. It is proved that the proposed method converges strongly to a common solution of above problems under some assumptions. The results here improve and extend some recent corresponding results by many other authors. MSC: 90C33; 49J40; 47H10; 47H05 Keywords: generalized equilibrium problem; variational inequality; hierarchical fixed point; projection method; nearly nonexpansive mappings 1 Introduction Let H be a real Hilbert space whose inner product and norm are denoted by ·, · and · , respectively, C be a nonempty, closed, and convex subset of H. It is well known that for any x ∈ H, there exists a unique point y ∈ C such that x – y = inf x – y : y ∈ C . Here, y is denoted by PC x, where PC is called the metric projection of H onto C. Let us recall some kinds of nonlinear mappings as follows, which are needed in the next sections. A mapping T : C → H is called L-Lipschitzian if there exists a constant L > such that Tx – Ty ≤ Lx – y, ∀x, y ∈ C. In particular, if L ∈ [, ), then T is said to be a contraction; if L = , then T is called a nonexpansive mapping. Let us fix a sequence {an } in [, ∞) with an → . If the inequality T n x – T n y ≤ x – y + an holds for all x, y ∈ C and n ≥ , then T is said to be nearly nonexpansive [, ] with respect to {an }. Let {Tn } be a sequence of mappings from C into H. Then the sequence {Tn } is called a sequence of nearly nonexpansive mappings [, ] with respect to a sequence {an } if Tn x – Tn y ≤ x – y + an , ∀x, y ∈ C, ∀n ≥ . (.) © 2015 Karahan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 2 of 25 It is obvious that the sequence of nearly nonexpansive mappings is a wider class of sequence of nonexpansive mappings. A mapping A : C → H is called α-inverse strongly monotone if there exists a positive real number α > such that Ax – Ay, x – y ≥ αAx – Ay , ∀x, y ∈ C, and a mapping F : C → H is called η-strongly monotone if there exists a constant η ≥ such that Fx – Fy, x – y ≥ ηx – y , ∀x, y ∈ C. In particular, if η = , then F is said to be monotone. Let G : C × C → R be a bifunction and B be a nonlinear mapping. The generalized equilibrium problem, denoted by GEP, is to find a point x ∈ C such that G(x, y) + Bx, y – x ≥ (.) for all y ∈ C, and the solution of the problem (.) is denoted by GEP(G), i.e., GEP(G) = x ∈ C : G(x, y) + Bx, y – x ≥ , ∀y ∈ C . If B = , then the GEP is reduced to equilibrium problem, denoted by EP, which is to find a point x ∈ C such that G(x, y) ≥ for all y ∈ C. The set of solutions of EP is denoted by EP(G). In the case of G = , then GEP is equivalent to find a x ∈ C such that Bx, y – x ≥ (.) for all y ∈ C. The problem (.) is called variational inequality problem, denoted by VI(C, B), and the solution of VI(C, B) is denoted by , i.e., = x ∈ C : Bx, y – x ≥ , ∀y ∈ C . The generalized equilibrium problem includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity, the Nash equilibrium problem in noncooperative games, the vector optimization problem, etc. Hence, the existence of solutions of generalized equilibrium problems has been extensively studied by many authors in the literature (see, e.g., [–]). Let S : C → H be a nonexpansive mapping. The following problem is called a hierarchical fixed point problem: Finding x∗ ∈ Fix(T) such that ∗ x – Sx∗ , x – x∗ ≥ , x ∈ Fix(T), (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 3 of 25 where Fix(T) is the set of fixed points of T, i.e., Fix(T) = {x ∈ C : Tx = x}. The problem (.) is equivalent to the following fixed point problem: Finding an x∗ ∈ C that satisfies x∗ = PFix(T) Sx∗ . Since Fix(T) is closed and convex, the metric projection PFix(T) is well defined. It is well known that the hierarchical fixed point problem (.) links with some monotone variational inequalities and convex programming problems; see [–]. Therefore, there exist various methods to solve the hierarchical fixed point problem; see Yao and Liou in [], Xu in [], Marino and Xu in [] and Bnouhachem and Noor in []. Now, we give some iteration schemes which are related with the problems (.), (.), and (.). In , Ceng et al. [] investigated the following iterative method: xn+ = PC αn ρVxn + ( – αn μF)Txn , ∀n ≥ , (.) where F is a L-Lipschitzian and η-strongly monotone operator with constants L, η > andV is a γ -Lipschitzian (possibly non-self-)mapping with constant γ ≥ such that < μ < Lη and ≤ ργ < – – μ(η – μL ). They proved that under some approximate assumptions on the operators and parameters, the sequence {xn } generated by (.) converges strongly to the unique solution of the variational inequality (ρV – μF)x∗ , x – x∗ ≤ , (.) ∀x ∈ Fix(T). Recently, in , Sahu et al. [] introduced the following iterative process for the sequence of nearly nonexpansive mappings {Tn } defined by (.): ⎧ ⎨y = ( – β )x + β S x , n n n n n n n i= (αi– – αi )Ti yn ], ⎩xn+ = PC [αn fxn + (.) ∀n ≥ , where f is a contraction and {Sn } is a sequence of nonexpansive mappings from C into itself. They proved that the sequence {xn } generated by (.) converges strongly to the unique solution of the following variational inequality: ∗ ∗ ∗ (I – f )x + (I – S)x , x – x ≥ , τ ∀x ∈ ∞ Fix(Tn ). i= In the same year, Bnouhachem and Noor [] introduced a new iterative scheme to find a common solution of a variational inequality, a generalized equilibrium problem and a hierarchical fixed point problem. Their scheme is as follows: ⎧ ⎪ ⎪ ⎪G(un , y) + Bx, y – un + rn y – un , un – xn ≥ , ⎪ ⎪ ⎨z = P (u – λ Au ), n C n n n ⎪ ⎪ yn = PC (βn Sxn + ( – βn )zn ), ⎪ ⎪ ⎪ ⎩ xn+ = PC (αn fxn + ni= (αi– – αi )Vi yn ), ∀y ∈ C, (.) ∀n ≥ , where Vi = ki I + ( – ki )Ti , ≤ ki < , {Ti }∞ i= : C → C is a countable family of ki strict pseudo-contraction mappings, A and B are inverse strongly monotone mappings. Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 4 of 25 They proved that the sequence {xn } generated by (.) converges strongly to a point z ∈ P∩GEP(G)∩Fix(T) f (z) which is the unique solution of the following variational inequality: (I – f )z, x – z ≥ , ∀x ∈ ∩ GEP(G) ∩ Fix(T), where Fix(T) = ∞ i= Fix(Ti ). In , Bnouhachem and Chen [] introduced the following iterative method: ⎧ ⎪ F (un , y) + Dxn , y – un + ϕ(y) – ϕ(un ) + rn y – un , un – xn ≥ , ⎪ ⎪ ⎪ ⎪ ⎨z = P (u – λ Au ); n C n n n ⎪ ⎪ yn = βn Sxn + ( – βn )zn ; ⎪ ⎪ ⎪ ⎩ xn+ = PC [αn ρUxn + γn xn + (( – γn )I – αn μF)(T(yn ))], ∀y ∈ C; (.) ∀n ≥ , where D, A : C → H are inverse strongly monotone mappings, F : C × C → R is a bifunction, ϕ : C → R is a proper lower semicontinuous and convex function, S, T : C → C are nonexpansive mappings, F : C → C is Lipschitzian and a strongly monotone mapping and U : C → C is a Lipschitzian mapping. The authors proved the strong convergence of the sequence generated by (.) to a common solution of a variational inequality, a generalized mixed equilibrium problem, and a hierarchical fixed point problem. In addition to all these papers, similar problems are considered in several papers; see, e.g., [–]. In this paper, motivated by the above works and by the recent work going in this direction, we introduce an iterative projection method and prove a strong convergence theorem based on this method for computing an approximate element of the common set of solution of a generalized equilibrium problem, a variational inequality problem and a fixed point problem for a sequence of nearly nonexpansive mappings defined by (.). The proposed method improves and extends many known results; see, e.g., [, , , , ] and the references therein. 2 Preliminaries Let {xn } be a sequence in a Hilbert space H and x ∈ H. Throughout this paper, xn → x denotes the strong convergence of {xn } to x and xn x denotes the weak convergence. Let C be a nonempty subset of a real Hilbert space H. For solving an equilibrium problem for a bifunction G : C × C → R, let us assume that G satisfies the following conditions: (A) G(x, x) = , ∀x ∈ C, (A) G is monotone, i.e. G(x, y) + G(y, x) ≤ , ∀x, y ∈ C, (A) ∀x, y, z ∈ C, lim+ G tz + ( – t)x, y ≤ G(x, y), t→ (A) ∀x ∈ C, y −→ G(x, y) is convex and lower semicontinuous. Lemma [] Let C be a nonempty, closed, and convex subset of H, and let G be a bifunction from C × C into R satisfying (A)-(A). Let r > and x ∈ H. Then there exists z ∈ C Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 5 of 25 such that G(z, y) + y – z, z – x ≥ r (.) for all x ∈ C. Lemma [] Suppose that G : C × C → R satisfies (A)-(A). For r > and x ∈ H, define a mapping Tr : H → C as follows: Tr (x) = z ∈ C : G(z, y) + y – z, z – x ≥ , ∀y ∈ C r for all z ∈ H. Then the following hold: () Tr is single valued, () Tr is firmly nonexpansive i.e. Tr x – Tr y ≤ Tr x – Tr y, x – y, ∀x, y ∈ H, () Fix(Tr ) = EP(G), () EP(G) is closed and convex. Let T , T : C → H be two mappings. We denote B (C), the collection of all bounded subsets of C. The deviation between T and T on B ∈ B (C), denoted by DB (T , T ), is defined by DB (T , T ) = sup T x – T x : x ∈ B . The following lemmas will be used in the next section. Lemma [] Let C be a nonempty, closed, and bounded subset of a Banach space X and {Tn } be a sequence of nearly nonexpansive self-mappings on C with a sequence {an } such that DC (Tn , Tn+ ) < ∞. Then, for each x ∈ C, {Tn x} converges strongly to some point of C. Moreover, if T is a mapping from C into itself defined by Tz = limn→∞ Tn z for all z ∈ C, then T is nonexpansive and limn→∞ DC (Tn , T) = . Lemma [] Let V : C → H be a γ -Lipschitzian mapping with a constant γ ≥ and let F : C → H be a L-Lipschitzian and η-strongly monotone operator with constants L, η > . Then for ≤ ργ < μη, (μF – ρV )x – (μF – ρV )y, x – y ≥ (μη – ργ )x – y , ∀x, y ∈ C. That is, μF – ρV is strongly monotone with coefficient μη – ργ . Lemma [] Let C be a nonempty subset of a real Hilbert space H. Suppose that λ ∈ (, ) and μ > . Let F : C → H be a L-Lipschitzian and η-strongly monotone operator on C. Define the mapping G : C → H by Gx = x – λμFx, ∀x ∈ C. Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 6 of 25 Then G is a contraction that provided μ < Lη . More precisely, for μ ∈ (, Lη ), Gx – Gy ≤ ( – λν)x – y, where ν = – ∀x, y ∈ C, – μ(η – μL ). Lemma [] Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and T be a nonexpansive self-mapping on C. If Fix(T) = ∅, then I – T is demiclosed; that is whenever {xn } is a sequence in C weakly converging to some x ∈ C and the sequence {(I – T)xn } strongly converges to some y, it follows that (I – T)x = y. Here I is the identity operator of H. Lemma [] Assume that {xn } is a sequence of nonnegative real numbers satisfying the conditions xn+ ≤ ( – αn )xn + αn βn , ∀n ≥ , where {αn } and {βn } are sequences of real numbers such that (i) {αn } ⊂ [, ] and ∞ αn = ∞, n= (ii) lim sup βn ≤ . n→∞ Then limn→∞ xn = . 3 Main results Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A, B : C → H be α, θ -inverse strongly monotone mappings, respectively. Let G : C × C → R be a bifunction satisfying assumptions (A)-(A), S : C → H be a nonexpansive mapping and {Tn } be a sequence of nearly nonexpansive mappings with the sequence {an } such that F := Fix(T) ∩ ∩ GEP(G) = ∅ where Tx = limn→∞ Tn x for all x ∈ C and Fix(T) = ∞ n= Fix(Tn ). It is clear that the mapping T is nonexpansive. Let V : C → H be a γ -Lipschitzian mapping, F : C → H be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy < μ < Lη , ≤ ργ < ν, where ν = – – μ(η – μL ). For an arbitrarily initial value x , define the sequence {xn } in C generated by ⎧ ⎪ G(un , y) + Bxn , y – un + rn y – un , un – xn ≥ , ⎪ ⎪ ⎪ ⎪ ⎨z = P (u – λ Au ), n C n n n ⎪ ⎪ yn = PC [βn Sxn + ( – βn )zn ], ⎪ ⎪ ⎪ ⎩ xn+ = PC [αn ρVxn + (I – αn μF)Tn yn ], ∀y ∈ C, (.) n ≥ , where {λn } ⊂ (, α), {rn } ⊂ (, θ ), {αn } and {βn } are sequences in [, ]. As can be seen, the convergence of the sequence {xn } generated by (.) depends on the choice of the control sequences and mappings. We list the following hypotheses Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 7 of 25 on them: (C) (C) lim αn = and n→∞ lim lim (C) αn = ∞; n= an n→∞ αn n→∞ ∞ = , lim βn n→∞ αn |βn – βn– | = , αn = , and lim DB (Tn , Tn+ ) = and n→∞ lim n→∞ |αn – αn– | = , αn lim n→∞ |λn – λn– | = ; αn lim |rn – rn– | = ; αn lim DB (Tn , Tn+ ) = for each B ∈ B (C). αn n→∞ n→∞ Now, we need the following lemmas to prove our main theorem. Lemma Assume that the conditions (C), (C) hold and p ∈ F . Then the sequences {xn }, {yn }, {zn }, and {un } generated by (.) are bounded. Proof It is easy to see that the mapping I – rn B is nonexpansive, so the mapping I – λn A is also nonexpansive. From Lemma , we have un = Trn (xn – rn Bxn ). Let p ∈ F . So, we get p = Trn (p – rn Bp). Then we obtain un – p = Trn (xn – rn Bxn ) – Trn (p – rn Bp) ≤ (xn – rn Bxn ) – (p – rn Bp) = xn – p – rn xn – p, Bxn – Bp + rn Bxn – Bp ≤ xn – p – rn (θ – rn )Bxn – Bp ≤ xn – p . (.) From (.), we get zn – p = PC (un – λn Aun ) – PC (p – λn Ap) ≤ un – p – λn (Aun – Ap) ≤ un – p – λn (α – λn )Aun – Ap ≤ un – p ≤ xn – p . (.) It follows from (.) that yn – p = PC βn Sxn + ( – βn )xn – PC p ≤ βn Sxn + ( – βn )zn – p ≤ ( – βn )zn – p + βn Sxn – p ≤ ( – βn )xn – p + βn Sxn – Sp + βn Sp – p ≤ xn – p + βn Sp – p. (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 8 of 25 Since limn→∞ αβnn = , without loss of generality, we can assume that βn ≤ αn , for all n ≥ . This gives us limn→∞ βn = . Let tn = αn ρVxn + (I – αn μF)Tn yn . Then we get xn+ – p = PC tn – PC p ≤ tn – p = αn ρVxn + (I – αn μF)Tn yn – p ≤ αn ρVxn – μFp + (I – αn μF)Tn yn – (I – αn μF)Tn p ≤ αn ργ xn – p + αn ρVp – μFp + ( – αn ν) yn – p + an . (.) From (.) and (.), we get xn+ – p ≤ αn ργ xn – p + αn ρVp – μFp + ( – αn ν) xn – p + βn Sp – p + an ≤ – αn (ν – ργ ) xn – p an + αn ρVp – μFp + Sp – p + αn ≤ – αn (ν – ργ ) xn – p an . ρVp – μFp + Sp – p + + αn (ν – ργ ) (ν – ργ ) αn (.) From condition (C), there exists a constant M > such that ρVp – μFp + Sp – p + an ≤ M , αn ∀n ≥ . Thus, from (.) we have xn+ – p ≤ – αn (ν – ργ ) xn – p + αn (ν – ργ ) M . (ν – ργ ) By induction, we get xn+ – p ≤ max x – p, M . (ν – ργ ) Hence, we find that {xn } is bounded. So, the sequences {yn }, {zn }, and {un } are bounded. Lemma Assume that (C)-(C) hold. Let p ∈ F and {xn } be the sequence generated by (.). Then the follow hold: (i) limn→∞ xn+ – xn = . (ii) ww (xn ) ⊂ Fix(T) where ww (xn ) is the weak w-limit set of {xn }, i.e., ww (xn ) = {x : xni x}. Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 9 of 25 Proof (i) Since the mappings PC and (I – λn A) are nonexpansive, we get zn – zn– = PC (un – λn Aun ) – PC (un– – λn– Aun– ) ≤ (un – λn Aun ) – (un– – λn– Aun– ) = un – un– – λn (Aun – Aun– ) – (λn – λn– )Aun– ≤ un – un– – λn (Aun – Aun– ) + |λn – λn– |Aun– ≤ un – un– + |λn – λn– |Aun– , (.) and so yn – yn– = PC βn Sxn + ( – βn )zn – PC βn– Sxn– – ( – βn– )zn– ≤ βn Sxn + ( – βn )zn – βn– Sxn– + ( – βn– )zn– ≤ βn (Sxn – Sxn– ) + (βn – βn– )Sxn– + ( – βn )(zn – zn– ) + (βn– – βn )zn– ≤ βn xn – xn– + ( – βn )zn – zn– + |βn – βn– | Sxn– + zn– ≤ βn xn – xn– + ( – βn ) un – un– + |λn – λn– |Aun– + |βn – βn– | Sxn– + zn– . (.) On the other hand, since un = Trn (xn – rn Bxn ) and un– = Trn– (xn– – rn– Bxn– ), we have G(un , y) + Bxn , y – un + y – un , un – xn ≥ , rn ∀y ∈ C, (.) and G(un– , y) + Bxn– , y – un– + rn– y – un– , un– – xn– ≥ , ∀y ∈ C. (.) If we take y = un– and y = un in (.) and (.), respectively, then we get G(un , un– ) + Bxn , un– – un + un– – un , un – xn ≥ rn (.) and G(un– , un ) + Bxn– , un – un– + un – un– , un– – xn– ≥ . rn– (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 10 of 25 It follows from (.), (.), and monotonicity of the function G that Bxn– – Bxn , un – un– + un – un– , un– – xn– un – xn ≥ . – rn– rn The last inequality implies that rn ≤ un – un– , rn (Bxn– – Bxn ) + (un– – xn– ) – (un – xn ) rn– rn un– = un– – un , un – un– + – rn– rn + (xn– – rn Bxn– ) – (xn – rn Bxn ) – xn– + xn– rn– rn un– + (xn– – rn Bxn– ) = un– – un , – rn– rn xn– – un – un– – (xn – rn Bxn ) – xn– + rn– rn = un– – un , – (un– – xn– ) rn– + (xn– – rn Bxn– ) – (xn – rn Bxn ) – un – un– rn un– – xn– ≤ un– – un – rn– + (xn– – rn Bxn– ) – (xn – rn Bxn ) – un – un– rn ≤ un– – un – un– – xn– rn– + xn– – xn – un – un– . (.) From (.), we have rn un– – un ≤ – un– – xn– + xn– – xn . rn– Without loss of generality, we can assume that there exists a real number μ such that rn > μ > for all positive integers n. Then we obtain un– – un ≤ xn– – xn + |rn– – rn |un– – xn– . μ From (.) and (.), we get yn – yn– ≤ βn xn – xn– + ( – βn ) xn– – xn + |rn– – rn |un– – xn– μ (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 + |λn – λn– |Aun– + |βn – βn– | Sxn– + zn– = xn – xn– + ( – βn ) |rn– – rn |un– – xn– μ + |λn – λn– |Aun– + |βn – βn– | Sxn– + zn– . Then we have xn+ – xn = PC tn – PC tn– ≤ tn – tn– = αn ρVxn + (I – αn μF)Tn yn – αn– ρVxn– + (I – αn– μF)Tn– yn– ≤ αn ρV (xn – xn– ) + (αn – αn– )ρVxn– + (I – αn μF)Tn yn – (I – αn μF)Tn yn– + Tn yn– – Tn– yn– + αn– μFTn– yn– – αn μFTn yn– ≤ αn ργ xn – xn– + γ |αn – αn– |Vxn– + ( – αn ν)Tn yn – Tn yn– + Tn yn– – Tn– yn– + μαn– FTn– yn– – αn FTn yn– ≤ αn ργ xn – xn– + γ |αn – αn– |Vxn– + ( – αn ν) yn – yn– + an + Tn yn– – Tn– yn– + μαn– (FTn– yn– – FTn yn– ) – (αn – αn– )FTn yn– ≤ αn ργ xn – xn– + γ |αn – αn– |Vxn– + ( – αn ν) xn – xn– + ( – βn ) |rn– – rn |un– – xn– + |λn – λn– |Aun– μ + |βn – βn– | Sxn– + zn– + ( – αn ν)an + DB (Tn , Tn– ) + μαn– LDB (Tn , Tn– ) + |αn – αn– |FTn yn– ≤ – αn (ν – ργ ) xn – xn– + |αn – αn– | γ Vxn– + FTn yn– + ( + μαn– L)DB (Tn , Tn– ) + an |rn– – rn |un– – xn– + |λn – λn– |Aun– μ + |βn – βn– | Sxn– + zn– + Page 11 of 25 Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 12 of 25 ≤ – αn (ν – ργ ) xn – xn– + ( + μαn– L)DB (Tn , Tn– ) + M |αn – αn– | + |rn– – rn | μ + |λn – λn– | + |βn – βn– | + an , (.) where M = max sup γ Vxn– + FTn yn– , sup un– – xn– , n≥ n≥ sup Aun– , sup Sxn– + zn– . n≥ n≥ Hence, we write xn+ – xn ≤ – αn (ν – ργ ) xn – xn– + αn (ν – ργ )δn , (.) where DB (Tn , Tn– ) ( + μαn– L) δn = (ν – ργ ) αn |αn – αn– | |rn– – rn | |λn – λn– | |βn – βn– | an . + + M + + + αn αn μ αn αn αn From conditions (C) and (C), we get lim sup δn ≤ . (.) n→∞ So, it follows from (.), (.), and Lemma that lim xn+ – xn = . n→∞ (.) (ii) First, we show that limn→∞ un – xn = . Since p ∈ F , from (.) and (.), we obtain xn+ – p ≤ tn – p = αn ρVxn + (I – αn μF)Tn yn – p = αn ρVxn – αn μFp + (I – αn μF)Tn yn – (I – αn μF)Tn p ≤ αn ρVxn – μFp + ( – αn ν) yn – p + an = αn ρVxn – μFp + ( – αn ν) yn – p + an yn – p + an = αn ρVxn – μFp + ( – αn ν)yn – p + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + ( – αn ν) βn Sxn – p + ( – βn )zn – p + ( – αn ν)an yn – p + ( – αn ν)an Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 13 of 25 = αn ρVxn – μFp + ( – αn ν)βn Sxn – p + ( – αn ν)( – βn )zn – p + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + ( – αn ν)βn Sxn – p + ( – αn ν)( – βn ) xn – p – rn (θ – rn )Bxn – Bp – λn (α – λn )Aun – Ap + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + βn Sxn – p + xn – p – ( – αn ν)( – βn ) rn (θ – rn )Bxn – Bp + λn (α – λn )Aun – Ap + ( – αn ν)an yn – p + ( – αn ν)an . (.) Then, from (.), we get ( – αn ν)( – βn ) rn (θ – rn )Bxn – Bp + λn (α – λn )Aun – Ap ≤ αn ρVxn – μFp + βn Sxn – p + xn – p – xn+ – p + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + βn Sxn – p + xn – p + xn+ – p xn+ – p + ( – αn ν)an yn – p + ( – αn ν)an . It follows from (.) and from conditions (C) and (C) that limn→∞ Bxn – Bp = and limn→∞ Aun – Ap = . Since Trn is firmly nonexpansive mapping, we have un – p = Trn (xn – rn Bxn ) – Trn (p – rn Bp) ≤ un – p, (xn – rn Bxn ) – (p – rn Bp) = un – p + (xn – rn Bxn ) – (p – rn Bp) – un – p – (xn – rn Bxn ) – (p – rn Bp) . Therefore, we get un – p ≤ (xn – rn Bxn ) – (p – rn Bp) – un – xn – rn (Bxn – Bp) ≤ xn – p – un – xn – rn (Bxn – Bp) ≤ xn – p – un – xn + rn un – xn Bxn – Bp. (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 14 of 25 Then, from (.), (.), and (.), we obtain xn+ – p ≤ αn ρVxn – μFp + ( – αn ν) βn Sxn – p + ( – βn )zn – p + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + ( – αn ν) βn Sxn – p + ( – βn )un – p + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + ( – αn ν) βn Sxn – p + ( – βn ) xn – p – un – xn + rn un – xn Bxn – Bp + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + βn Sxn – p + xn – p – ( – αn ν)( – βn )un – xn + rn un – xn Bxn – Bp + ( – αn ν)an yn – p + ( – αn ν)an . The last inequality implies that ( – αn ν)( – βn )un – xn ≤ αn ρVxn – μFp + βn Sxn – p + xn – p – xn+ – p + rn un – xn Bxn – Bp + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + βn Sxn – p + xn – p + xn+ – p xn+ – xn + rn un – xn Bxn – Bp + ( – αn ν)an yn – p + ( – αn ν)an . Since limn→∞ Bxn – Bp = and {yn – p} is a bounded sequence, by using (.) and conditions (C), (C), we obtain lim un – xn = . n→∞ On the other hand, since a metric projection PC satisfies u – v, PC u – PC v ≥ PC u – PC v , we write zn – p = PC (un – λn Aun ) – PC (p – λn Ap) ≤ zn – p, (un – λn Aun ) – (p – λn Ap) (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 = ≤ ≤ Page 15 of 25 zn – p + un – p(Aun – Ap) – un – p – λn (Aun – Ap) – (zn – p) zn – p + un – p – un – zn – λn (Aun – Ap) zn – p + un – p – un – zn + λn un – zn , Aun – Ap ≤ zn – p + un – p – un – zn + λn un – zn Aun – Ap . So, we get zn – p ≤ un – p – un – zn + λn un – zn Aun – Ap ≤ xn – p – un – zn + λn un – zn Aun – Ap. By using (.) and (.), we have xn+ – p ≤ αn ρVxn – μFp + ( – αn ν) βn Sxn – p + ( – βn )zn – p + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + ( – αn ν) βn Sxn – p + ( – βn ) xn – p – un – zn + λn un – zn Aun – Ap + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + βn Sxn – p + xn – p – ( – αn ν)βn un – zn + λn un – zn Aun – Ap + ( – αn ν)an yn – p + ( – αn ν)an . Therefore, we get ( – αn ν)βn un – zn ≤ αn ρVxn – μFp + βn Sxn – p + xn – p – xn+ – p + λn un – zn Aun – Ap + ( – αn ν)an yn – p + ( – αn ν)an ≤ αn ρVxn – μFp + βn Sxn – p (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 16 of 25 + xn – p + xn+ – p xn+ – xn + λn un – zn Aun – Ap + ( – αn ν)an yn – p + ( – αn ν)an . Since limn→∞ Aun – Ap = and {yn – p} is a bounded sequence, by using (.) and conditions (C), (C), we obtain lim un – zn = . (.) n→∞ Also, from (.) and (.), we have lim xn – zn = . (.) n→∞ On the other hand, we get xn – yn ≤ xn – un + un – zn + zn – yn = xn – un + un – zn + βn (Sxn – zn ). Since limn→∞ βn = , again from (.) and (.), we obtain lim xn – yn = . (.) n→∞ Now, we show that limn→∞ xn – Txn = . Before that we need to show that limn→∞ xn – Tn xn = : xn – Tn xn ≤ xn – xn+ + xn+ – Tn xn ≤ xn – xn+ + PC tn – PC Tn xn ≤ xn – xn+ + αn ρVxn + (I – αn μF)Tn yn – Tn xn ≤ xn – xn+ + αn (ρVxn – μFTn yn ) + Tn yn – Tn xn ≤ xn – xn+ + αn ρVxn – μFTn yn + yn – xn + an . Since an → , by using (.), (.), and condition (C), we obtain lim xn – Tn xn = . (.) n→∞ Hence, from (.) and condition (C), we have xn – Txn ≤ xn – Tn xn + Tn xn – Txn ≤ xn – Tn xn + DB (Tn , T) → as n → ∞. Since {xn } is bounded, there exists a weak convergent subsequence {xnk } of {xn }. Let xnk w. So, it follows from Lemma that w as k → ∞. From the Opial condition, we get xn w ∈ Fix(T). Therefore, ww (xn ) ⊂ Fix(T). Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 17 of 25 Theorem Assume that (C)-(C) hold. Then the sequence {xn } generated by (.) converges strongly to x∗ ∈ F , which is the unique solution of the variational inequality (ρV – μF)x∗ , x – x∗ ≤ , ∀x ∈ F . (.) Proof Since the mapping T is defined by Tx = limn→∞ Tn x for all x ∈ C, by Lemma , T is a nonexpansive mapping, and Fix(T) = ∅. Moreover, since the operator μF – ρV is (μη – ργ )-strongly monotone by Lemma , we get the uniqueness of the solution of the variational inequality (.). Let us denote this solution by x∗ ∈ Fix(T) = F . Now, we divide our proof into three steps. w. Step . From Lemma , since {xn } is bounded, there exists an element w such that xn First, we show that w ∈ F = Fix(T) ∩ ∩ GEP(G). It follows from Lemma that w ∈ Fix(T) = ∞ n= Fix(Tn ). Next we show that w ∈ . Let NC v be the normal cone to C at v ∈ C, i.e., NC v = w ∈ H : v – u, w ≥ , ∀u ∈ C . Let ⎧ ⎨Av + N v, v ∈ C, C Hv = ⎩∅, v ∈/ C. Then H is maximal monotone mapping. Let (v, u) ∈ G(H). Since u – Av ∈ NC v and zn ∈ C, we get v – zn , u – Av ≥ . (.) On the other hand, from the definition of zn , we have v – zn , zn – un – λn Aun ≥ and hence, v – zn , zn – un + Aun ≥ . λn Therefore, using (.), we get v – zni , u ≥ v – zni , Av zn – uni + Auni ≥ v – zni , Av – v – zni , i λni zni – uni = v – zni , Av – Auni – λni = v – zni , Av – Azni + v – zni , Azni – Auni – v – zni , ≥ v – zni , Azni – Auni – v – zni , zni – uni . λni zni – uni λni (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 By using (.), (.), and (.), we get uni (.) we have Page 18 of 25 w and zni w for i → ∞. Hence, from v – w, u ≥ . Since H is maximal monotone, we have w ∈ H – and hence w ∈ . Finally, we show that w ∈ GEP(G). By using un = Trn (xn – rn Bxn ), we get G(un , y) + Bxn , y – un + y – un , un – xn ≥ , rn ∀y ∈ C. Also, from the monotonicity of G, we have Bxn , y – un + y – un , un – xn ≥ G(y, un ), rn ∀y ∈ C, and unk – xnk ≥ G(y, unk ), Bxnk , y – unk + y – unk , rnk ∀y ∈ C. (.) Let y ∈ C and yt = ty + ( – t)w, for t ∈ (, ]. Then yt ∈ C. From (.), we get Byt , yt – unk ≥ Byt , yt – unk – Bxnk , yt – unk un – xnk + G(yt , unk ) – yt – unk , k rnk = Byt – Bxnk , yt – unk + Bunk – Bxnk , yt – unk un – xnk – yt – unk , k + G(yt , unk ). rnk (.) Since B is Lipschitz continuous, using (.) we obtain limk→∞ Bunk –Bxnk = . It follows w and the monotonicity of B that from (.), unk Byt , yt – w ≥ G(yt , w). (.) Therefore, from assumptions (A)-(A) and (.), we have = G(yt , yt ) ≤ tG(yt , y) + ( – t)G(yt , w) ≤ tG(yt , y) + ( – t)Byt , yt – w ≤ tG(yt , y) + ( – t)tByt , y – w. The last inequality implies that G(yt , y) + ( – t)Byt , y – w ≥ . If we take the limit t → + , we get G(w, y) + Bw, y – w ≥ , ∀y ∈ C. Hence, we have w ∈ GEP(G). Thus, we obtain w ∈ F = Fix(T) ∩ ∩ GEP(G). Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 19 of 25 Step . We show that lim supn→∞ (ρV – μF)x∗ , xn – x∗ ≤ , where x∗ is the unique solution of variational inequality (.). Since the sequence {xn } is bounded, it has a weak convergent subsequence {xnk } such that lim sup (ρV – μF)x∗ , xn – x∗ = lim sup (ρV – μF)x∗ , xnk – x∗ . n→∞ Let xnk k→∞ w, as k → ∞. It follows from Step that w ∈ F . Hence lim sup (ρV – μF)x∗ , xn – x∗ = (ρV – μF)x∗ , w – x∗ ≤ . n→∞ Step . Finally, we show that the sequence {xn } generated by (.) converges strongly to the point x∗ . By using the iteration (.), we have xn+ – x∗ = PC tn – x∗ , xn+ – x∗ = PC tn – tn , xn+ – x∗ + tn – x∗ , xn+ – x∗ . Since the metric projection PC satisfies the inequality x – PC x, y – PC x ≤ , ∀x ∈ H, y ∈ C, from (.), we get xn+ – x∗ ≤ tn – x∗ , xn+ – x∗ = αn ρVxn + (I – αn μF)Tn yn – x∗ , xn+ – x∗ = αn ρVxn – μFx∗ + (I – αn μF)Tn yn – (I – αn μF)Tn x∗ , xn+ – x∗ = αn ρ Vxn – Vx∗ , xn+ – x∗ + αn ρVx∗ – μFx∗ , xn+ – x∗ + (I – αn μF)Tn yn – (I – αn μF)Tn x∗ , xn+ – x∗ . Hence, from Lemma , we obtain xn+ – x∗ ≤ αn ργ xn – x∗ xn+ – x∗ + αn ρVx∗ – μFx∗ , xn+ – x∗ + ( – αn ν) yn – x∗ + an xn+ – x∗ ≤ αn ργ xn – x∗ xn+ – x∗ + αn ρVx∗ – μFx∗ , xn+ – x∗ + ( – αn ν) βn xn – x∗ + βn Sx∗ – x∗ + ( – βn )zn – x∗ + an xn+ – x∗ ≤ αn ργ xn – x∗ xn+ – x∗ + αn ρVx∗ – μFx∗ , xn+ – x∗ + ( – αn ν) βn xn – x∗ + βn Sx∗ – x∗ + ( – βn )xn – x∗ + an xn+ – x∗ ≤ – αn (ν – ργ ) xn – x∗ xn+ – x∗ (.) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 ≤ Page 20 of 25 + αn ρVx∗ – μFx∗ , xn+ – x∗ + ( – αn ν)βn Sx∗ – x∗ xn+ – x∗ + ( – αn ν)an xn+ – x∗ ( – αn (ν – ργ )) xn – x∗ + xn+ – x∗ + αn ρVx∗ – μFx∗ , xn+ – x∗ + ( – αn ν)βn Sx∗ – x∗ xn+ – x∗ + ( – αn ν)an xn+ – x∗ . The last inequality implies that xn+ – x∗ ≤ ( – αn (ν – ργ )) xn – x∗ ( + αn (ν – ργ )) αn + ρVx∗ – μFx∗ , xn+ – x∗ ( + αn (ν – ργ )) ∗ βn Sx – x∗ xn+ – x∗ + ( + αn (–ργ )) an xn+ – x∗ + ( + αn (ν – ργ )) ≤ – αn (ν – ργ ) xn – x∗ + αn (ν – ργ )θn , βn an ρVx∗ – μFx∗ , xn+ – x∗ + M + xn+ – x∗ , θn = ( + αn (ν – ργ ))(ν – ργ ) αn αn and sup Sx∗ – x∗ xn+ – x∗ ≤ M . n≥ Since αβnn → and αann → , we get lim sup θn ≤ . n→∞ So, it follows from Lemma that the sequence {xn } generated by (.) converges strongly to x∗ ∈ F which is the unique solution of variational inequality (.). Putting A = in Theorem , we have the following corollary. Corollary Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let B : C → H be θ -inverse strongly monotone mapping, G : C × C → R be a bifunction satisfying assumptions (A)-(A), S : C → H be a nonexpansive mapping and {Tn } be a sequence of nearly nonexpansive mappings with the sequence {an } such that F := Fix(T) ∩ ∩ GEP(G) = ∅ where Tx = limn→∞ Tn x for all x ∈ C and Fix(T) = ∞ n= Fix(Tn ). Let V : C → H be a γ -Lipschitzian mapping, F : C → H be a L-Lipschitzian and ηstrongly monotone operator such that these coefficients satisfy < μ < Lη , ≤ ργ < ν, where Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 21 of 25 ν = – – μ(η – μL ). For an arbitrarily initial value x ∈ C, consider the sequence {xn } in C generated by ⎧ ⎪ ⎪G(un , y) + Bxn , y – un + rn y – un , un – xn ≥ , ⎨ yn = PC [βn Sxn + ( – βn )un ], ⎪ ⎪ ⎩ xn+ = PC [αn ρVxn + (I – αn μF)Tn yn ], ∀y ∈ C, (.) n ≥ , where {rn } ⊂ (, θ ), {αn } and {βn } are sequences in [, ] satisfying the conditions (C)n– | (C) except the condition limn→∞ |λn –λ = . Then the sequence {xn } generated by (.) αn ∗ ∗ converges strongly to x ∈ F , where x is the unique solution of variational inequality (.). In Theorem , if we take A = and βn = for all n ≥ , then we have the following corollary. Corollary Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let B : C → H be θ -inverse strongly monotone mapping, G : C × C → R be a bifunction satisfying assumptions (A)-(A), {Tn } be a sequence of nearly nonexpansive mappings with the sequence {an } such that F := Fix(T) ∩ ∩ GEP(G) = ∅ where Tx = limn→∞ Tn x for all x ∈ C and Fix(T) = ∞ n= Fix(Tn ). Let V : C → H be a γ -Lipschitzian mapping, F : C → H be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy < μ < Lη , ≤ ργ < ν, where ν = – – μ(η – μL ). For an arbitrarily initial value x ∈ C, consider the sequence {xn } in C generated by ⎧ ⎨G(u , y) + Bx , y – u + y – u , u – x ≥ , n n n rn n n ⎩xn+ = PC [αn ρVxn + (I – αn μF)Tn un ], n n ≥ , ∀y ∈ C, (.) where {rn } ⊂ (, θ ), {αn } is a sequence in [, ] satisfying the conditions (C)-(C) except n– | n– | = and limn→∞ |βn –β = . Then the sethe conditions limn→∞ αβnn = , limn→∞ |λn –λ αn αn ∞ ∗ quence {xn } generated by (.) converges strongly to x ∈ n= Fix(Tn ) ∩ ∩ GEP(G), where x∗ is the unique solution of variational inequality (.). Putting A = and B = , we have the following corollary, which gives us an iterative scheme to find a common solution of an equilibrium problem and a hierarchical fixed point problem. Corollary Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let G : C × C → R be a bifunction satisfying assumptions (A)-(A), S : C → H be a nonexpansive mapping and {Tn } be a sequence of nearly nonexpansive mappings with the sequence {an } such that F := Fix(T) ∩ ∩ GEP(G) = ∅ where Tx = limn→∞ Tn x for all x ∈ C and Fix(T) = ∞ n= Fix(Tn ). Let V : C → H be a γ -Lipschitzian mapping, F : C → H be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy < μ < Lη , ≤ ργ < ν, where ν = – – μ(η – μL ). For an arbitrarily initial value x , define the sequence {xn } in C generated by ⎧ ⎪ ⎪ ⎨G(un , y) + rn y – un , un – xn ≥ , ∀y ∈ C, yn = PC [βn Sxn + ( – βn )un ], ⎪ ⎪ ⎩ xn+ = PC [αn ρVxn + (I – αn μF)Tn yn ], (.) n ≥ , Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 22 of 25 where {rn } ⊂ (, ∞), {αn } and {βn } are sequences in [, ] satisfying the conditions (C)-(C) n– | = . Then the sequence {xn } generated by (.) conexcept the condition limn→∞ |λn –λ αn ∞ ∗ verges strongly to x ∈ n= Fix(Tn ) ∩ EP(G), where x∗ is the unique solution of variational inequality (.). Corollary Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A, B : C → H be α, θ -inverse strongly monotone mappings, respectively. G : C × C → R be a bifunction satisfying assumptions (A)-(A), S : C → H be a nonexpansive mapping and {Tn } be a sequence of nonexpansive mappings such that F := Fix(T) ∩ ∩ GEP(G) = ∅ where Tx = limn→∞ Tn x for all x ∈ C and Fix(T) = ∞ n= Fix(Tn ). Let V : C → H be a γ Lipschitzian mapping, F : C → H be a L-Lipschitzian and η-strongly monotone operator such that these coefficients satisfy < μ < Lη , ≤ ργ < ν, where ν = – – μ(η – μL ). For an arbitrarily initial value x ∈ C, consider the sequence {xn } in C generated by (.) where {λn } ⊂ (, α), {rn } ⊂ (, θ ), {αn } and {βn } are sequences in [, ] satisfying the conditions (C)-(C) of Theorem except the condition limn→∞ αann = . Then the sequence {xn } converges strongly to x∗ ∈ F , where x∗ is the unique solution of variational inequality (.). Remark Our results can be reduced to some corresponding results in the following ways: () In our iterative process (.), if we take G(x, y) = for all x, y ∈ C, B = , and rn = for all n ≥ , then we derive the iterative process xn+ = PC αn ρVxn + (I – αn μF)Tn xn , n ≥ , which is studied by Sahu et al. []. Therefore, Theorem generalizes the main result of Sahu et al. [, Theorem .]. So, our results extend the corresponding results of Ceng et al. [] and of many other authors. () If we take S as a nonexpansive self-mapping on C and Tn = T for all n ≥ such that T is a nonexpansive mapping in (.), then it is clear that our iterative process generalizes the iterative process of Wang and Xu []. Hence, Theorem generalizes the main result of Wang and Xu [, Theorem .]. So, our results extend and improve the corresponding results of [, ]. () The problem of finding the solution of variational inequality (.) is equivalent to finding the solutions of hierarchical fixed point problem (I – S)x∗ , x∗ – x ≤ , ∀x ∈ F , where S = I – (ρV – μF). Example Let H = R and C = [, ]. Let G : C × C → R, G(x, y) = y + xy – x , S = I, A : C → H, Ax = x, B : C → H, Bx = x – , Vx = x + , Fx = x, and ⎧ ⎨ – x, if x ∈ [, ), Tn x = ⎩an , if x = , Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 23 of 25 for all x ∈ C. It is clear that G(x, y) is a bifunction satisfying the assumptions (A)-(A), S is nonexpansive mapping, A is -inverse strongly monotone mapping, B is -inverse strongly monotone mapping, V is γ -Lipschitzian mapping with γ = , F is L-Lipschitzian and η-strongly monotone operator with L = η = and {Tn } is a sequence of nearly nonexpansive mappings with respect to the sequence an = n – . Define sequences {αn } and {βn } in [, ] by αn = n and βn = n+ for all n ≥ and take μ = ρ = , ν = , rn = n+ , and λn = n+ . It is easy to see that all conditions of Theorem are satisfied. First, we find the sequence {un } which satisfies the following generalized equilibrium problem for all y ∈ C: G(un , y) + Bxn , y – un + y – un , un – xn ≥ . rn For all n ≥ , we get G(un , y) + Bxn , y – un + y – un , un – xn ≥ rn (y – un )(un – xn ) ≥ rn ⇒ y + un y – un + (xn – )(y – un ) + ⇒ y rn + y(un r + xn rn + un – rn – xn ) – un rn – xn un rn + un rn – un + un xn ≥ . Put K(y) = y rn + y(un r + xn rn + un – rn – xn ) – un rn – xn un rn + un rn – un + un xn . Then K is a quadratic function of y with coefficients a = rn , b = un rn + xn rn + un – rn – xn , and c = –un rn – xn un rn + un rn – un + un xn . Next, we compute the discriminant of K as follows: = b – ac = (un r + xn rn + un – rn – xn ) – rn –un rn – xn un rn + un rn – un + un xn = (un – rn – xn + rn un + rn xn ) . We know that K(y) ≥ for all y ∈ C = [, ]. If it has most one solution in [, ], so ≤ n (–rn ) n and hence un = rn +x+r = +nx . By using this equation, the sequence {xn } generated by n+ n the iterative scheme (.) becomes ⎧ ⎪ y + un y – un + (xn – )(y – un ) + (n + )(y – un )(un – xn ) ≥ , ⎪ ⎪ ⎪ ⎪ ⎨z = u – u , n n n+ n ⎪yn = ⎪ x + ( – n+ )zn , ⎪ n + n ⎪ ⎪ ⎩ xn+ = n (xn + ) + ( – n )( – yn ), ∀n ≥ , ∀y ∈ C, (.) for all n ≥ , and it converges strongly to x∗ = . which is the unique common fixed point of the sequence {Tn } and the unique solution of the variational inequality (.) over ∞ n= Fix(Tn ). Some of the values of the iterative scheme (.) for the different initial values x = ., x = ., and x = . are as in Table . Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 24 of 25 Table 1 Some of the values of the iterative scheme (3.37) x2 x3 x4 x5 x6 x7 x8 x9 x10 .. . x100 . .. x1000 x1 = 1.000000E–01 x1 = 4.000000E–01 x1 = 7.000000E–01 4.800000E–01 6.520000E–01 5.392000E–01 5.534400E–01 5.257984E–01 5.319411E–01 5.191295E–01 5.226747E–01 5.151936E–01 .. . 5.015339E–01 . .. 5.001506E–01 7.200000E–01 6.280000E–01 5.488000E–01 5.481600E–01 5.291776E–01 5.295757E–01 5.208866E–01 5.213129E–01 5.162830E–01 .. . 5.015208E–01 . .. 5.001503E–01 9.600000E–01 6.040000E–01 5.584000E–01 5.428800E–01 5.325568E–01 5.272102E–01 5.226438E–01 5.199510E–01 5.173725E–01 .. . 5.015075E–01 . .. 5.001503E–01 Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, Turkey. 2 Department of Mathematical Engineering, Faculty of Chemistry-Metallurgical, Yildiz Technical University, Istanbul, Turkey. 3 Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, Turkey. Received: 19 November 2014 Accepted: 16 January 2015 References 1. Agarwal, RP, O’Regan, D, Sahu, DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 8(1), 61-79 (2007) 2. Agarwal, RP, O’Regan, D, Sahu, DR: Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications. Springer, New York (2009) 3. Wong, NC, Sahu, DR, Yao, JC: A generalized hybrid steepest-descent method for variational inequalities in Banach spaces. Fixed Point Theory Appl. 2011, Article ID 754702 (2011) 4. Sahu, DR, Kang, SM, Sagar, V: Approximation of common fixed points of a sequence of nearly nonexpansive mappings and solutions of variational inequality problems. J. Appl. Math. 2012, Article ID 902437 (2012) 5. Sanhan, S, Inchan, I, Sanhan, W: Weak and strong convergence theorem of iterative scheme for generalized equilibrium problem and fixed point problems of asymptotically strict pseudo-contraction mappings. Appl. Math. Sci. 5, 1977-1992 (2011) 6. Kangtunyakarn, A: Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions. Fixed Point Theory Appl. 2011, 23 (2011) doi:10.1186/1687.1812.2011.23 7. Min, L, Shisheng, Z: A new iterative method for common states of generalized equilibrium problem, fixed point problem of infinite κ -strict pseudo-contractive mappings, and quasi-variational inclusion problem. Acta Math. Sci. 32B(2), 499-519 (2012) 8. Wang, Y, Xu, HK, Yin, X: Strong convergence theorems for generalized equilibrium, variational inequalities and nonlinear operators. Arab. J. Math. 1, 549-568 (2012) 9. Razani, A, Yazdı, M: A new iterative method for generalized equilibrium and fixed point problem of nonexpansive mappings. Bull. Malays. Math. Soc. 35(4), 1049-1061 (2012) 10. Cianciaruso, F, Marino, G, Muglia, L, Yao, Y: On a two-steps algorithm for hierarchical fixed point problems and variational inequalities. J. Inequal. Appl. 2009, 13 (2009) 11. Tian, M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal., Theory Methods Appl. 73(3), 689-694 (2010) 12. Yao, Y, Cho, YJ, Liou, YC: Iterative algorithms for hierarchical fixed points problems and variational inequalities. Math. Comput. Model. 52(9-10), 1697-1705 (2010) 13. Gu, G, Wang, S, Cho, YJ: Strong convergence algorithms for hierarchical fixed points problems and variational inequalities. J. Appl. Math. 2011, 1-17 (2011) 14. Yao, Y, Chen, R: Regularized algorithms for hierarchical fixed-point problems. Nonlinear Anal. 74, 6826-6834 (2011) 15. Tian, M, Huang, LH: Iterative methods for constrained convex minimization problem in Hilbert spaces. Fixed Point Theory Appl. 2013, 105 (2013) 16. Yao, Y, Liou, YC: Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Problems 24, 015015 (2008) 17. Xu, HK: Viscosity method for hierarchical fixed point approach to variational inequalities. Taiwan. J. Math. 14(2), 463-478 (2010) Karahan et al. Journal of Inequalities and Applications ( 2015) 2015:53 Page 25 of 25 18. Marino, G, Xu, HK: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149(1), 61-78 (2011) 19. Bnouhachem, A, Noor, MA: An iterative method for approximating the common solutions of a variational inequality, a mixed equilibrium problem and a hierarchical fixed point problem. J. Inequal. Appl. 2013, 490 (2013) 20. Bnouhachem, A, Chen, Y: An iterative method for a common solution of a generalized mixed equilibrium problems, variational inequalities, and a hierarchical fixed point problems. Fixed Point Theory Appl. 2014, 155 (2014) 21. Ceng, LC, Ansari, QH, Yao, JC: Hybrid pseudoviscosity approximation schemes for equilibrium problems, and fixed point problems of infinitely many nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 4, 743-754 (2010) 22. Ceng, LC, Ansari, QH, Schaible, S, Yao, JC: Iterative methods for generalized equilibrium problems, systems of general generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert space. Fixed Point Theory 12(2), 293-308 (2011) 23. Ceng, LC, Ansari, QH: Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems. J. Glob. Optim. 53, 69-96 (2012) 24. Latif, A, Ceng, LC, Ansari, QH: Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of an equilibrium problem and fixed point problems. Fixed Point Theory Appl. 2012, 186 (2012) 25. Ceng, LC, Ansari, QH, Yao, JC: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal. 74, 5286-5302 (2011) 26. Sahu, DR, Kang, SM, Sagar, V: Iterative methods for hierarchical common fixed point problems and variational inequalities. Fixed Point Theory Appl. 2013, 299 (2013) 27. Marino, G, Xu, HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43-52 (2006) 28. Wang, Y, Xu, W: Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities. Fixed Point Theory Appl. 2013, 121 (2013) 29. Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123-145 (1994) 30. Combettes, PL, Hirstoaga, A: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117-136 (2005) 31. Yamada, I: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D, Censor, Y, Reich, S (eds.) Inherently Parallel Algorithms and Optimization and Their Applications, pp. 473-504. North-Holland, Amsterdam (2001) 32. Goebel, K, Kirk, WA: Topics on Metric Fixed-Point Theory. Cambridge University Press, Cambridge (1990) 33. Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119(1), 185-201 (2003)