In this paper, we consider, under a hybrid iterative scheme with errors, a weak convergence theorem to a common element of the set of a finite family of asymptotically k -strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction, which is the approximation version of the corresponding results of Kumam et al. [1] . The equilibrium problems were introduced by Blum and Oettli [2] in 1994. Numerous problems in applied sciences, for example optimization problems, saddle point problems, variational inequality problems and Nash equilibria in noncoopera- tive games, are reduced to find a solution of the following equilibrium problem [2 , 3] ; finding x ∈ C such that where ϕ : C × C → ℝ is a bifunction. On the other hand, the problem of finding a common fixed point of a family of mappings is a classical problem in nonlinear analysis. Finding an optimal point in the set of common fixed points of a family of mappings is a task that occurs frequently in various areas of mathematical sciences and engineering. For example, the convex feasibility problem reduces to finding a point of the set of common fixed points of a family of nonexpansive mappings [4] . In 2009, Qin et al. considered the following weak convergence theorem to a common fixed point of a finite family of asymptotically k -strictly pseudo-contractive mappings under a hybrid iterative scheme. Theorem 1.1 ( [5] ). Assume the following conditions; Assume that F := ( F ( T_i )) ≠ . For any x₀ ∈ C , let { x_n } be a sequence generated by where { a_n } is a sequence in (0, 1) such that k + Ɛ ≤ a_n ≤ 1 − Ɛ   for some   Ɛ ∈ (0, 1) and n = ( h − 1) N + i ( n ≥ 1), where i = i ( n ) ∈ {1, 2, ⋯, N }, h = h ( n ) ≥ 1 is a positive integer and h ( n ) → ∞ as n → ∞. Then { x_n } converges weakly to an element of F . The existence of solutions of equilibrium problems and common fixed points of finite mappings are very important in nonlinear analysis with applications. Moreover, to find the intersection of solution sets of equilibrium problems and common fixed points of finite mappings and to apply the intersection are also important. Recently, there have been a few works for the intersection of the two sets to be the set of weakly convergent points of two given sequences in Hilbert spaces. In 2000, Kumam et al. considered the following weak convergence theorem to a given common element of the set of common xed points of a finite family of asymptotically k -strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction under a hybrid iterative scheme. Theorem 1.2 ( [1] ). Assume the conditions (1)-(5) in Theorem 1.1. Let ϕ : C × C → ℝ be a bifunction satisfying the followings; Assume that F := ( F ( T_i )) ∩ EP ( ϕ ) ≠ , where EP ( ϕ ) is a set of solutions of equilibrium problem (1.1). For any x ₀ ∈ C , let { x_n } and { υ_n } be sequences generated by where n = ( h − 1) N + i ( n ≥ 1), i = i ( n ) ∈ {1, 2,⋯, N }, h = h ( n ) ≥ 1 is a positive integer and h ( n ) → ∞ as n →∞. Let { a_n } and { r_n } satisfy the following conditions: Then { x_n } and { υ_n } converges weakly to an element of F . On the other hand, the fixed point iterative scheme with errors was introduced by Liu [6] . The idea of considering fixed point iterative scheme problems with errors which comes from practical numerical computation usually concerns the approximation fixed point and is related to the stability of fixed point iterative schemes. The idea of considering iterative scheme procedures with errors leads to finding the approximate solution to equilibrium problems. In 2005, Combettes and Hirstoga [3] introduced an iterative scheme for a problem of finding best approximate solutions to equilibrium problem and proved the strong convergence result. In this paper, we consider, under a hybrid iterative scheme with errors, a weak convergence theorem to a common element of the set of a finite family of asymptotically k -strictly pseudo-contractive mappings and a solution set of an equilibrium problem for a given bifunction, which is the approximation version of the corresponding results of Kumam et al. [1] . The following results will be needed in the main result. Lemma 1.1 ( [7 , 8] ). Let H be a real Hilbert space. There hold the following identities Lemma 1.2 ( [3] ). Assume that ϕ : C × C → ℝ satisfies (A1)-(A4). For r > 0 and x ∈ H , define a mapping S_r : H → C as follows; for all z ∈ H. Then the following hold; Definition 2.1. A mapping T : C → C is said to be asymptotically k-strictly pseudo-contractive if there exist a sequence { k_n } ⊂ [1, ∞) with k_n = 1 and k ∈ [0, 1) such that ∥ Tⁿx − Tⁿy ∥ ² ≤ ∥ x − y ∥ ² + k ∥( I − Tⁿ ) x − ( I − Tⁿ ) y ∥ ² , ∀ x , y ∈ C and n ∈ ℕ. The following proposition by Osilike and Igbokwe [8] was considered by using infinite terms of the given sequences based on Lemma 1 in [9] , but our proof is considered by using only finite terms of the given sequences based on the basic concepts of limit superior and limit inferior. Proposition 2.1.   Let { a_n }, { c_n } and { δ_n } be nonnegative real sequences satisfying the following condition: if and then a_n   exists . Proof . Consider Thus Since and for any ε > 0, take N ∈ ℕ such that and for n ≥ N . Thus, lim sup _m→∞   a_m ≤ e^εa_n + εe^ε . Letting ε → 0, we have the wanted result. Hence lim sup _m→∞   a_m ≤ lim inf _n→∞ a_n , which shows the existence of a_n                                  ⧠ Putting δ_n = 0( ∀ n ∈ ℕ), we have the following known lemma as a corollary; Lemma 2.1 ( [9] ). Let { a_n } and { b_n } be nonnegative real sequences satisfying the following condition: If then a_n exists . Now, we prove our main result. Theorem 2.1.   Assume the conditions (1)-(5) in Theorem 1.1. Let ϕ : C × C → ℝ be a bifunction satisfying (A1)-(A4). Assume that F := ( F ( T_i ))∩ EP ( ϕ )≠ . For any x ₀ ∈ C , let { x_n } and { v_n } be sequences generated by where { a_n }, { b_n } and { c_n } are sequences in [0; 1) such that a_n + b_n + c_n = 1, a_n ≥ k + ε , b_n ≥ ε for some ε ∈ (0, 1), { u_n } is a bounded sequence in C , { r_n } is a sequence in (0, ∞) such that inf r_n ≥ 0 and n = ( h −1) N + i ( n ≥ 1), where i = i ( n ) ∈ {1, 2,⋯, N }, h = h ( n ) ≥ 1 is a positive integer and h ( n ) → ∞ as n → ∞. Then { x_n } and { v_n } converges weakly to an element of F . Proof . Let p ∈ F . From (2.1) and Lemma 1.2, we have v _n−1 = S _rn−1 x _n−1 and From (2.1) and Lemma 1.1(ii), From Proposition 2.1, ∥ x_n − p ∥ exists. Observe (2.2) again Since a_n ≥ k + ε , b_n ≥ ε for all n ≥ 0 and some ε ∈ (0, 1), Taking the limits as n → ∞, we have Observe that It follows that Since S _rn−1 is firmly nonexpansive, we have and hence Using (2.2) and (2.7), we have hence Since ∥ x_n − p ∥ exists and k _h(n) = 1, From (2.6) and (2.8), we have It follows that Applying (2.8) and (2.9), we obtain which implies that ∥ x_n − x _n+j ∥ = 0, ∀ j ∈ {1, ⋯, N }. Since, for any positive integer n > N , it can be written as n = ( k ( n )−1) N + i ( n ), where i ( n ) ∈ {1, 2,⋯, N }, observe that Since, for each n > N , n ≡ n − N (mod N ) and n = ( k ( n ) − 1) N + i ( n ), we have n − N = ( k ( n )−1) N + i ( n ) = ( k ( n − N )−1) N + i ( n − N ), that is, i ( n − N ) = i ( n ), k ( n − N ) = k ( n ) − 1. Observe that and It follows from (2.11)-(2.13) that Applying (2.5) and (2.10) to (2.14), we obtain From (2.9) and (2.15), Also, we have for any j = 1, ⋯, N , which gives that Moreover, for each l ∈ {1, 2,⋯, N }, we have Put W ( x_n ) = { x ∈ H : x_ni ⇀ x for some subsequence { x_ni } of { x_n }}. Firstly, W ( x_n ) ≠ø. Indeed, since { x_n } is bounded and H is reflexive, W ( x_n ) ≠ø. Secondly, we claim that Let w ∈ W ( x_n ) be an arbitrary element. Then there exists a subsequence { x_ni } of { x_n } converging weakly to w . Applying (2.8), we can obtain that v_ni ⇀ w as i → ∞. It follows from ∥ v_n − T_lv_n ∥ = 0 that T_lv_ni → w , ∀ l ∈ {1, ⋯, N }. Let us show that w ∈ EP ( ϕ ). Since v_n = T_rn v_n , we have From (A2), we have and hence Since → 0 and as i → ∞, from (A4), we have For t ∈ (0; 1] and y ∈ C , let y_t = t_y +(1− t ) w . Since y ∈ C and w ∈ C , y_t ∈ C , and hence ϕ ( y_t , w ) ≤ 0. So, from (A1) and (A4), and hence 0 ≤ ϕ ( y_t , y ). From (A3), 0 ≤ ϕ ( w , y ), ∀ y ∈ C and hence w ∈ EP ( ϕ ) Next, we prove that w ∈ ( F ( T_i )). Suppose that w ∉ ( F ( T_i )). Then there exists l ∈ {1, ⋯, N } such that w ∉ F ( T_l ). From (2.16) and the Opial’s condition, which derives a contradiction. Hence w ∈ ( F ( T_i )). Finally, we show that { x_n } and { v_n } converge weakly to an element of F . Indeed, it is sufficient to show that W ( x_n ) is a single point set. We take w ₁ , w ₂ ∈ W ( x_n ) arbitrarily and let { x_ni } and { x_nj } be subsequences of { x_n } such that x_ni ⇀ w ₁ and x_nj ⇀ w ₂ . Since ∥ x_n − p ∥ exists for each p ∈ F and w ₁ , w ₂ ∈ F , by Lemma 1.1 (iii), we obtain Hence w ₁ = w ₂ , which shows that W ( x_n ) is single point set.                                 ⧠ Remark 2.1. (1) If c_n = 0(∀ n ∈ N ) in Theorem 2.1, then we obtain Theorem 1.2. (2) If ϕ ( x , y ) = 0(∀ x,y ∈ C ) and v_n = x_n , ∀ n ∈ ℕ in (2.1), then we obtain Theorem 1.1.