Now, we are in a position to state and prove the main strong convergence theorem for the given iterative scheme.

Proof.
The proof is divided into several steps as follows.

Step 1. For every x*∈C and every k, we show that
(25)∥xk+1-x*∥2≤∥xk-x*∥2+2αkf(xk,x*)+2αkϵk+2βk2,
and there exists the limit
(26)c≔limk→∞∥xk-x*∥.
For each k≥1, let
(27)S¯k≔∑i=1pηk,iSi.
By Proposition 2, we see that S¯k is a L¯-strict pseudocontraction on C and the sequence {xk} generated by Algorithm 9 can be rewritten as
(28)xk+1≔δkwk+(1-δk)S¯k(wk), ∀k≥1.
Then, for all k≥1, we have
(29)∥xk+1-x*∥2 =∥δkwk+(1-δk)S¯k(wk)-x*∥2 =∥δk(wk-x*)+(1-δk)(S¯k(wk)-x*)∥2 =δk∥wk-x*∥2+(1-δk)∥S¯k(wk)-S¯k(x*)∥2 -δk(1-δk)∥wk-S¯k(wk)∥2 ≤δk∥wk-x*∥2+(1-δk) ×(∥wk-x*∥2+L¯∥(I-S¯k)(wk)-(I-S¯k)(x*)∥2) -δk(1-δk)∥wk-S¯k(wk)∥2 =∥wk-x*∥2+(1-δk)(L¯∥S¯k(wk)-wk∥2) -δk(1-δk)∥wk-S¯k(wk)∥2 =∥wk-x*∥2+(1-αk)(L¯-δk)∥S¯k(wk)-wk∥2 ≤∥xk-x*∥2-∥xk-wk∥2+2〈xk-wk,x*-wk〉 + (1-δk)(L¯-δk)∥S¯k(wk)-wk∥2(30) ≤∥xk-x*∥2-∥xk-wk∥2+2〈xk-wk,x*-wk〉 ≤∥xk-x*∥2+2〈xk-wk,x*-wk〉.
Since wk=PC(xk-αkyk) and x*∈C, we have
(31)〈xk-wk,x*-wk〉≤αk〈yk,x*-wk〉.
Combining this inequality with (30) yields
(32)∥xk+1-x*∥2≤∥xk-x*∥2+2〈xk-wk,x*-wk〉≤∥xk-x*∥2+2αk〈yk,x*-wk〉=∥xk-x*∥2+2αk〈yk,x*-xk〉 +2αk〈yk,xk-wk〉≤∥xk-x*∥2+2αk〈yk,x*-xk〉 +2αk∥yk∥∥xk-wk∥=∥xk-x*∥2+2αk〈yk,x*-xk〉 +2βkmax{λk,∥yk∥}∥yk∥∥xk-wk∥≤∥xk-x*∥2+2αk〈yk,x*-xk〉 +2βk∥xk-wk∥.
Using again wk=PC(xk-αkyk) and xk∈C, we have
(33)∥xk-wk∥2≤αk〈yk,x*-wk〉≤αk∥yk∥∥xk-wk∥=βkmax{λk,∥yk∥}∥yk∥∥xk-wk∥≤βk∥xk-wk∥,

which gives that ∥xk-wk∥≤βk. Consequently,
(34)limk→∞∥xk-wk∥=0.

This together with (32) implies that
(35)∥xk+1-x*∥2≤∥xk-x*∥2+2αk〈yk,x*-xk〉+2βk2.
Since yk∈∂ϵkf(xk,·)(xk), x*∈C and f(x,x)=0 for all x∈C, we have
(36)〈yk,x*-xk〉≤f(xk,x*)-f(xk,xk)+ϵk≤f(xk,x*)+ϵk.
Combining (35) and (36), we obtain that
(37)∥xk+1-x*∥2≤∥xk-x*∥2+2αkf(xk,x*) +2αkϵk+2βk2.
On the other hand, since x*∈Sol(C,f), that is, f(x*,x)≥0 for all x∈C, by pseudomonotonicity of f with respect to x*, we have f(x,x*)≤0 for all x∈C. Replacing x by xk∈C, we get f(xk,x*)≤0. Then, from (37), it follows that
(38)∥xk+1-x*∥≤∥xk-x*∥2+2αkϵk+2βk2.
By Assumptions 8 (ii) and (iii), we found that ∑k=1∞(2αkϵk+2βk2)<∞. Using Lemma 3 and (38), we arrive at the existence of
(39)c≔limk→∞∥xk-x*∥.

Step 2. For every x*∈C, we show that
(40)limsupk→∞f(xk,x*)=0.
Since f is pseudomonotone on C and f(x*,xk)≥0, we have -f(xk,x*)≥0. By Step 1,
(41)∥xk+1-x*∥2≤∥xk-x*∥2+2αkf(xk,x*)+2αkϵk+2βk2.
We have
(42)2αk[-f(xk,x*)]≤∥xk-x*∥2 -∥xk+1-x*∥2+2αkϵk+2βk2.
Summing up the above inequalities for every k, we obtain that
(43)0≤2∑k=0∞αk[-f(xk,x*)]≤∥x0-x*∥2 +2∑k=0∞αkϵk+2∑k=0∞βk2<+∞.
It follows from the boundedness of the sequences {yk} and {λk} that we can assume that
(44)max{λk,∥yk∥}≤M,
for a constant M≥0. Thus,
(45)αk=βkγk=βkmax{λk,∥yk∥}≥βkM,
which together with (43) implies
(46)0≤2M∑k=0∞βk[-f(xk,x*)]≤2∑k=0∞αk[-f(xk,x*)]<+∞.
Thus,
(47)∑k=0∞βk[-f(xk,x*)]<+∞.
Then, by ∑k=0∞βk=∞ and -f(xk,x*)≥0, we can deduce that limsupk→∞f(xk,x*)=0 as desired.

Step 3. For any x*∈Ω, suppose that {xkj} is the subsequence of {xk} such that
(48)limsupk→∞f(xk,x*)=limj→∞f(xkj,x*),
and, without loss of generality, we may assume that xkj⇀x¯ as j→∞ for some x¯∈C. We show that x¯ solves EP(C,f). To this end, since f(·,x*) is weakly upper semicontinuous, we have
(49)f(x¯,x*)≥limsupj→∞f(xkj,x*)=limj→∞f(xkj,x*)=limsupj→∞f(xk,x*)=0.
On the other hand, since f is pseudomonotone with respect to x* and f(x*,x¯)≥0, we have
(50)f(x¯,x*)≤0.
From (49) and (50), we can conclude that f(x¯,x*)=0. By Assumption 6, we can deduce that x¯ is a solution of EP(f,C) as well.

Step 4. We prove that any weakly cluster point of the sequence {xk} is a common fixed point of Li-strict pseudocontraction, for each i=1,2,…,p. In particular, x¯∈⋂i=1pFix(Si,C). Let y¯ be any weakly cluster point of {xk} and let {xkm} be a subsequence of {xk}⊂C weakly converging to y¯. By convexity and the closedness of C, C is weakly closed. Hence, y¯∈C. We first show that
(51)limm→∞∥xkm-S(xkm)∥=0.
It follows from (29) that
(52)(1-δk)(δk-L¯)∥S¯k(wk)-wk∥2 ≤∥xk-x*∥2-∥xk+1-x*∥2+2αk〈yk,x*-xk〉 +2βk∥xk-wk∥ ≤∥xk-x*∥2-∥xk+1-x*∥2+2αk∥yk∥∥x*-xk∥ +2βk∥xk-wk∥,
which gives that
(53)limk→∞∥S¯k(wk)-wk∥=0.
Consequently,
(54)∥S¯k(wk)-xk∥≤∥S¯k(wk)-wk∥ +∥xk-wk∥⟶0 as k⟶∞.
By Proposition 2 (i), we arrive at the following:
(55)∥S¯k(xk)-xk∥≤∥S¯k(xk)-S¯k(wk)∥+∥S¯k(wk)-xk∥≤1+L¯1-L¯∥xk-wk∥+∥S¯k(wk)-xk∥≤1+L¯1-L¯βk+∥S¯k(wk)-xk∥.
Thus, we obtain
(56)limk→∞∥S¯k(xk)-xk∥=0.
For each i=1,…,p, we suppose that {ηkm,i} converges to ηi as m→∞ such that ∑i=1pηi=1. Then, for each i=1,2,…,p and x∈C, we have
(57)S¯km(x)≔∑m=1pηkm,iSi(x)⟶∑i=1pηiSi(x)≔S(x) as m⟶∞.
It follows from (56) that
(58)∥xkm-S(xkm)∥≤∥xkm-S¯km(xkm)∥ +∥S¯km(xkm)-S(xkm)∥=∥xkm-S¯km(xkm)∥ +∥∑i=1pηkm,iSi(xkm)-∑i=1pηiSi(xkm)∥=∥xkm-S¯km(xkm)∥ +∥∑i=1p(ηkm,i-ηi)Si(xkm)∥≤∥xkm-S¯km(xkm)∥ +∑i=1p|ηkm,i-ηi|∥Si(xkm)∥.
We obtain that
(59)limj→∞∥xkm-S(xkm)∥=0.
By Proposition 2 (ii), we have
(60)y¯∈Fix(S)=Fix(∑i=1pηiSi)x.
It then follows from Proposition 2 (v) that we have
(61)y¯∈⋂i=1pFix(Si,C).
In particular, we conclude that x¯∈⋂i=1pFix(Si,C).

Step 5. Finally, we prove that
(62)limk→∞xk=limk→∞wk=limk→∞PΩ(xk)=x¯.
It follows from (38) that, for all x*∈Ω,
(63)∥xk+1-x*∥2≤∥xk-x*∥2+μk,
where μk≔2αkϵk+2βk2>0 for all k≥0 and ∑k=0∞μk<+∞. Now, using property (14) of the metric projection, we have
(64)∥xk+1-PΩ(xk+1)∥2=∥δk(wk-PΩ(xk))+(1-δk)(S¯k(wk)-PΩ(xk))∥2≤δk∥wk-PΩ(xk)∥2+(1-δk)∥S¯k(wk)-PΩ(xk)∥2≤δk[∥wk-xk∥+∥xk-PΩ(xk)∥]2 +(1-δk)∥S¯k(wk)-xk∥2-(1-δk)∥xk-PΩ(xk)∥2=δk∥wk-xk∥2+2δk∥wk-xk∥∥xk-PΩ(xk)∥ +δk∥xk-PΩ(xk)∥2 +(1-δk)∥S¯k(wk)-xk∥2-(1-δk)∥xk-PΩ(xk)∥2=δk∥wk-xk∥2+2δk∥wk-xk∥∥xk-PΩ(xk)∥ +(2δk-1)∥xk-PΩ(xk)∥2+(1-δk)∥S¯k(wk)-xk∥2.
Since δk→1/2, ∥wk-xk∥→0, and ∥S¯k(wk)-xk∥→0 as k→∞, we obtain that
(65)limk→∞∥xk+1-PΩ(xk+1)∥=0.
For the simplicity of notation, let zk≔PΩ(xk) for each k≥1. Then, for all m>k, since Ω is convex, we have (1/2)(zm+zk)∈Ω, and therefore
(66)∥zm-zk∥2=2∥xm-zm∥2+2∥xm-zk∥2 -4∥xm-12(zm+zk)∥2≤2∥xm-zm∥2+2∥xm-zk∥2 -4∥xm-zm∥2=2∥xm-zk∥2-2∥xm-zm∥2.
Replacing x* with zk in (63), we can obtain the following:
(67)∥xm-zk∥2≤∥xm-1-zk∥2+μm-1≤∥xm-2-zk∥2+μm-1+μm-2≤⋯≤∥xk-zk∥2+∑j=km-1μj,
Combining this inequality with (66), we have
(68)∥zm-zk∥2≤2∥xk-zk∥2+2∑j=km-1μj-2∥xm-zm∥2,
which gives that
(69)limm→∞,k→∞∥zm-zk∥=0,
which implies also that {zk} is a Cauchy sequence. Hence, {zk} strongly converges to some point z¯∈Ω. However, since zki≔PΩ(xki), letting i→∞, we obtain in the limit that
(70)z¯=limi→∞PΩ(xki)=PΩ(x¯)=x¯∈Ω.
Therefore, zk≔PΩ(xk)→z¯=x¯∈Ω. Then, from (65), we can conclude that xk→x¯. Finally, since limk→∞∥xk-wk∥=0, we have limk→∞wk=x¯.