In this section, we prove a weak convergence theorem by an extragradient methods for finding a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems in Hilbert spaces. The function f:H→ℝ is a continuous differentiable function with the minimization problem given by
(13)min x∈Cf(x):=12∥Ax-PQAx∥2.
In 2010, Xu [17] considered the following Tikhonov regularized problem:
(14)min x∈Cfα(x):=12∥Ax-PQAx∥2+12α∥x∥2,
where α>0 is the regularization parameter. The gradient given by
(15)∇fα(x)=∇f(x)+αI=A*(I-PQ)A+αI
is (α+∥A∥2)-Lipschitz continuous and α-strongly monotone (see [29] for the details).

Proof.
By Lemma 8, we have that PC(I-λ∇fα) is ξ-averaged for each λ∈(0,2/(α+∥A∥2)), where ξ=(2+λ(α+∥A∥2))/4. Hence, by Proposition 2.4, PC(I-λ∇fα) is nonexpansive. From {λn}⊂[a,b] and a,b∈(0,1/∥A∥2), we have a≤inf n≥0λn≤sup n≥0λn≤b<1/∥A∥2=lim n→∞1/(αn+∥A∥2). Without loss of generality, we assume that a≤inf n≥0λn≤sup n≥0λn≤b<1/(αn+∥A∥2), for all n≥0. Hence, for each n≥0,PC(I-λn∇fαn) is ξn-averaged with
(17)ξn=12+λn(αn+∥A∥2)2-12·λn(αn+∥A∥2)2=2+λn(αn+∥A∥2)4∈(0,1).
This implies that PC(I-λn∇fαn) is nonexpansive for all n≥0.

Next, we show that the sequence {xn} is bounded. Indeed, take a fixed p∈F(S)∩Γ∩Ω arbitrarily. Let T1,μ and T2,μ be defined as in Lemma 5 associated to F1 and F2, respectively. Thus, we get p=Sp=SPC(p), for all n≥0, p=PC(I-λ∇f)p, for all λ∈(0,2/∥A∥2) and p=T1,μ(T2,μp), for all μ>0. Put y*=T2,μp,zn=T1,μwn and wn=T2,μxn. From (29), we have
∥yn-p∥=∥PC(I-λn∇fαn)zn-PC(I-λn∇f)p∥≤∥PC(I-λn∇fαn)zn-PC(I-λn∇fαn)p∥ +∥PC(I-λn∇fαn)p-PC(I-λn∇f)p∥≤∥zn-p∥+∥(I-λn∇fαn)p-(I-λn∇f)p∥=∥zn-p∥+∥λnp(∇f-∇fαn)∥=∥zn-p∥+∥λnαnp∥=∥zn-p∥+λnαn∥p∥.
This implies that ∥zn-p∥=∥T1,μwn-T1,μy*∥≤∥wn-y*∥=∥T2,μxn-T2,μp∥≤∥xn-p∥. Thus, we obtain ∥yn-p∥≤∥xn-p∥+λnαn∥p∥. Put ln=PC(xn-λn∇fαn(yn)) for each n≥0. Then, by Proposition 1(ii), we have
(19)∥ln-p∥2≤∥xn-λn∇fαn(yn)-p∥2 -∥xn-λn∇fαn(yn)-ln∥2=∥xn-p∥2-∥xn-ln∥2+2λn〈∇fαn(yn),p-ln〉≤∥xn-p∥2-∥xn-ln∥2 +2λn[〈∇fαn(yn),yn-ln〉+〈∇fαnp,p-yn〉]≤∥xn-p∥2-∥xn-ln∥2 +2λn[αn〈p,p-yn〉+〈∇fαn(yn),yn-ln〉]≤∥xn-p∥2-∥xn-yn∥2 -2〈xn-yn,yn-ln〉-∥yn-ln∥2 +2λn[αn〈p,p-yn〉+〈∇fαn(yn),yn-ln〉]=∥xn-p∥2-∥xn-yn∥2-∥yn-ln∥2 +2〈xn-λn∇fαn(yn)-yn,ln-yn〉 +2λnαn〈p,p-yn〉.
Hence, by Proposition 1(i), we have
(20)〈xn-λn∇fαn(yn)-yn,ln-yn〉 =〈xn-λn∇fαn(xn)-yn,ln-yn〉 +〈λn∇fαn(xn)-λn∇fαn(yn),ln-yn〉 ≤〈λn∇fαn(xn)-λn∇fαn(yn),ln-yn〉 ≤λn∥∇fαn(xn)-λn∇fαn(yn)∥∥ln-yn∥ ≤λn(αn+∥A∥2)∥xn-yn∥∥ln-yn∥.
So, we have
(21)∥ln-p∥2≤∥xn-p∥2-∥xn-yn∥2-∥yn-ln∥2 +2λn(αn+∥A∥2)∥xn-yn∥∥ln-yn∥ +2λnαn∥p∥∥p-yn∥≤∥xn-p∥2-∥xn-yn∥2-∥yn-ln∥2 +λn2(αn+∥A∥2)2∥xn-yn∥2+∥ln-yn∥2 +2λnαn∥p∥∥p-yn∥=∥xn-p∥2+2λnαn∥p∥∥p-yn∥ +[λn2(αn+∥A∥2)2-1]∥xn-yn∥2≤∥xn-p∥2+2λnαn∥p∥∥p-yn∥≤∥xn-p∥2+2λnαn∥p∥[∥zn-p∥+λnαn∥p∥]≤∥xn-p∥2+2λnαn∥p∥[∥xn-p∥+λnαn∥p∥].
Then, from the last inequality we conclude that
∥xn+1-p∥2 =∥βnxn+(1-βn)S(ln)-p∥2 ≤βn∥xn-p∥2+(1-βn)∥S(ln)-p∥2 -βn(1-βn)∥xn-S(ln)∥2 ≤βn∥xn-p∥2+(1-βn)∥ln-p∥2 -βn(1-βn)∥xn-S(ln)∥2 ≤βn∥xn-p∥2 +(1-βn)[(αn+∥A∥2)2[λn2(αn+∥A∥2)2-1]∥xn-p∥2+2λnαn∥p∥∥p-yn∥ +(1-βn)m+[λn2(αn+∥A∥2)2-1]∥xn-yn∥2] -βn(1-βn)∥xn-S(ln)∥2 ≤∥xn-p∥2+2λnαn∥p∥∥p-yn∥ +(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2 -βn(1-βn)∥xn-S(ln)∥2 ≤∥xn-p∥2+αn(λn2∥p∥2+∥p-yn∥2) +(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2 -βn(1-βn)∥xn-S(ln)∥2 ≤∥xn-p∥2 +αn[λn2∥p∥2+(∥xn-p∥+λnαn∥p∥)2] +(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2 -βn(1-βn)∥xn-S(ln)∥2 ≤∥xn-p∥2 +αn[λn2∥p∥2+2∥xn-p∥2+2λn2αn2∥p∥2] +(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2 -βn(1-βn)∥xn-S(ln)∥2 =(1+2αn)∥xn-p∥2+αnλn2∥p∥2(1+2αn2) +(1-βn)(λn2(αn+∥A∥2)2-1)∥xn-yn∥2 -βn(1-βn)∥xn-S(ln)∥2 ≤(1+2αn)∥xn-p∥2+αnλn2∥p∥2(1+2αn2) =(1+δn)∥xn-p∥2+bn,
where δn=2αn and bn=αnλn2∥p∥2(1+2αn2). Since ∑n=0∞αn<∞ and {λn}⊂[a,b] for some a,b∈(0,1/∥A∥2), we conclude that ∑n=0∞δn<∞ and ∑n=0∞bn<∞. Therefore, by Lemma 7, we note that lim n→∞∥xn-p∥ exists for each p∈F(S)∩Γ∩Ω and hence the sequences {xn},{ln},{yn},{zn}, and {wn} are bounded. From (22), we also obtain
(23)(1-d)(1-b2(αn+∥A∥2)2)∥xn-yn∥2 +c(1-d)∥xn-S(ln)∥2 ≤(1-βn)(1-λn2(αn+∥A∥2)2)∥xn-yn∥2 +βn(1-βn)∥xn-S(ln)∥2 ≤(1+2αn)∥xn-p∥2-∥xn+1-p∥2 +αnλn2∥p∥2(1+2αn2),
where {λn}⊂[a,b] and {βn}⊂[c,d]. Since lim n→∞∥xn-p∥ exists and αn→0, it follows that
(24)lim n→∞∥xn-yn∥=lim n→∞∥xn-S(ln)∥≤lim n→∞[∥xn-p∥+∥p-S(ln)∥]≤lim n→∞[∥xn-p∥+∥p-ln∥]=0.
Similarly, from inequality (22), we have
(25)(1-d)(1-b2(αn+∥A∥2)2)∥xn-zn∥2 +c(1-d)∥xn-S(ln)∥2 ≤(1-βn)(1-λn2(αn+∥A∥2)2)∥xn-zn∥2 +βn(1-βn)∥xn-S(ln)∥2 ≤(1+αn)∥xn-p∥2-∥xn+1-p∥2 +αnλn2∥p∥2,
where {λn}⊂[a,b] and {βn}⊂[c,d]. Since lim n→∞∥xn-p∥ exists and αn→0, we obtain
(26)lim n→∞∥xn-zn∥=lim n→∞∥xn-S(ln)∥=0.
Moreover, we note that
(27)∥yn-ln∥=∥PC(zn-λn∇fαnzn)-PC(xn-λn∇fαn(yn))∥≤∥(zn-λn∇fαn(zn))-(xn-λn∇fαn(yn))∥=∥zn-xn-(λn∇fαn(zn)-λn∇fαn(yn))∥≤∥zn-xn∥+∥λn∇fαn(zn)-λn∇fαn(yn)∥≤∥zn-xn∥+λn[∥∇fαn(zn)-∇fαn(xn)∥ ∥zn-xn∥+λnm+∥∇fαn(xn)-∇fαn(yn)∥]≤∥zn-xn∥+λn[(αn+∥A∥2)∥xn-yn∥ ∥zn-xn∥+λnm+∥∇fαn(zn)-∇fαn(xn)∥].
From (26) and αn→0, it is implied that
(28)lim n→∞∥yn-ln∥=0.
Note that ∥ln-S(ln)∥≤∥ln-yn∥+∥yn-xn∥+∥xn-S(ln)∥. This together with (24) and (28) implies that lim n→∞∥ln-S(ln)∥=0. Also, from ∥xn-ln∥≤∥xn-yn∥+∥yn-ln∥, it follows that lim n→∞∥xn-ln∥=0. Since ∇f=A*(I-PC)A is a Lipschitz condition, where A* is the adjoint of A, we have lim n→∞∥∇f(yn)-∇f(ln)∥=0. Since {xn} is a bounded sequence, there exists a subsequence {xnj} of {xn} that converges weakly to some x^.

Next, we show that x^∈Γ. Since ∥xn-ln∥→0 and ∥yn-ln∥→0, it is known that lni⇀x^ and yni⇀x^. Let T:H→2H be a set value mappings defined by
(29)Tv={∇f(v)+NCv if v∈C,∅if v∉C,
where NCv={w∈H1:〈v-u,w〉≥0, for all u∈C}. Hence, by [34], T is maximal monotone and 0∈Tv if and only if v∈VI (C,∇f). Let (v,w)∈G(T). Then we have w∈Tv=∇f(v)+NCv and hence w-∇f(v)∈NCv. So, we obtain 〈v-u,w-∇f(v)〉≥0, for all u∈C. On the other hand, from ln=PC(xn-λn∇fαn(yn)) and v∈C, we have
(30)〈xn-λn∇fαn(yn)-ln,ln-v〉≥0,
and hence
(31)〈v-ln,ln-xnλn+∇fαn(yn)〉≥0.
Therefore, from w-∇f(v)∈NCv and lni∈C, we get
(32)〈v-lni,w〉 ≥〈v-lni,∇f(v)〉 ≥〈v-lni,∇f(v)〉 -〈v-lni,lni-xniλni+∇fαni(yni)〉 =〈v-lni,∇f(v)〉 -〈v-lni,lni-xniλni+∇f(yni)〉 -αni〈v-lni,yni〉 =〈v-lni,∇f(v)-∇f(lni)〉 +〈v-lni,∇f(lni)-∇f(yni)〉 -〈v-lni,lni-xniλni〉-αni〈v-lni,yni〉 ≥〈v-lni,∇f(lni)-∇f(yni)〉 -〈v-lni,lni-xniλni〉-αni〈v-lni,yni〉.
By taking i→∞, we obtain 〈v-x^,w〉≥0. Since T is maximal monotone, it follows that x^∈T-10 and hence x^∈VI (C,∇f). Therefore, by Lemma 8, x^∈Γ.

Next, we show that x^∈F(S). Since lni⇀x^ and ∥lni-S(lni)∥→0, it follows by the demiclosed principle that x^∈F(S). Hence, we have x^∈F(S)∩Γ.

Next, we show that x^∈Ω. Let G be a mapping which is defined as in Lemma 6, thus we have
(33)∥zn-G(zn)∥=∥T1,μT2,μxn-G(zn)∥=∥G(xn)-G(zn)∥≤∥xn-zn∥,
and hence
(34)∥xn-G(xn)∥≤∥xn-zn∥+∥zn-G(zn)∥ +∥G(zn)-G(xn)∥≤∥xn-zn∥+∥xn-zn∥+∥zn-xn∥=3∥xn-zn∥.
By taking n→∞, we have ∥xn-G(xn)∥→0. From lim n→∞∥xn-zn∥=0 and znj⇀x^, we obtain xnj⇀x^. According to demiclosedness and Lemma 6, we have x^∈Ω. Therefore, we have x^∈F(S)∩Γ∩Ω. Let {xnj} be another subsequence of {xn} such that xnj⇀x^. We show that x^=x~, suppose that x^=x~. Since lim n→∞∥xn-x^∥ exists for all x^∈F(S)∩Γ∩Ω, it follows by the Opial's condition that
(35)lim n→∞∥xn-x^∥=liminf i→∞ ∥xni-x^∥<liminf i→∞ ∥xni-x~∥=lim n→∞ ∥xn-x~∥=liminf j→∞ ∥xnj-x~∥<liminf j→∞ ∥xnj-x^∥=lim n→∞ ∥xnjxn-x^∥.
It is a contradiction. Thus, we have x^=x~ and so xn⇀x~∈F(S)∩Γ∩Ω. Further, from ∥xn-yn∥→0, it follows that yn⇀x~ and hence zn, wn. This completes the proof.