Proof.
We define a set-valued mapping Ai:X→2Xi by
(7)Ai(x)={⋃xi∈Si(x)ui∈Si(x)∣maxξei(Fi(ui,xi))vvvvvvvvv=min⋃xi∈Si(x)maxξei(Fi(xi,xi))}.

It follows from [12, pages 110–119, Propositions 6 and 21] that maxξei(Fi(·,xi)) is upper semicontinuous for each fixed xi∈Xi. By [12, page 112, Proposition 11], the set
(8)⋃θ∈S(x)maxξei(F(θ,y))
is compact. Therefore, Ai(x) is nonempty for every x∈X.

Let
(9){xn}∈X, xn⟶x0,ui,n∈Ai(xn), ui,n⟶ui,0.
We must show that ui,0∈Ai(x0). First, note that ui,n∈Ai(xn) and then ui,n∈Si(xn). As Si(·) is upper semicontinuous and the set Si(x0) is compact, it follows that ui,0∈Si(x0). Suppose that ui,0∉Ai(x0). Then, there exists a vector wi,0∈Si(x0) satisfying
(10)maxξei(Fi(wi,0,x0i))<maxξei(Fi(ui,0,x0i)).
As Si(·) is lower semicontinuous, there exists wi,n∈Si(xn), such that wi,n→wi,0. It follows from compactness of Fi(wi,n,xni) that there exists zi,n∈Fi(wi,n,xni) such that
(11)ξei(zi,n)=maxξei(Fi(wi,n,xni)).

It follows from the upper semicontinuity of Fi(·,·) and the compactness of Xi×Xi that Fi(xi,xi) is compact. Hence, for the net {zi,n}, there exists a subnet of {zi,n} converging to zi,0. Without loss of generality, assume zi,n→zi,0. Now we prove that
(12)ξei(zi,0)=maxξei(Fi(wi,0,x0i)).
Since the mapping Fi(·,·) is upper semicontinuous and the set Fi(wi,0,x0i) is compact, we have ξei(zi,0)∈ξei(Fi(wi,0,x0i)).

Now, suppose that ξei(zi,0)≠maxξei(Fi(wi,0,x0i)). Namely, there exists vi,0∈Fi(wi,0,x0i) such that ξei(vi,0)>ξei(zi,0). As Fi(·,·) is lower semicontinuous, there exists vi,n∈Fi(wi,n,xni) such that vi,n→vi,0. Since ξei(·) is continuous, for n large enough,
(13)ξei(vi,n)>ξei(zi,n),
which is a contradiction to (11).

From the compactness of Fi(ui,n,xni), we take z~i,n∈F(ui,n,xni) such that
(14)ξei(z~i,n)=maxξei(Fi(ui,n,xni)).
By the compactness of Fi(xi,xi), we can choose a converging subnet of {z~i,n}, which is denoted without loss of generality by the original net {z~i,n}. Assume z~i,n→z~i,0. Similar to the preceding proof, we have
(15)ξei(z~i,0)=maxξei(F(ui,0,x0i)).
Then, by (10), ξei(zi,0)<ξei(z~i,0).

It follows from the continuity of ξei(·) that ξei(zi,n)→ξei(zi,0) and ξei(z~i,n)→ξei(z~i,0). Therefore, ξei(zi,n)<ξei(z~i,n), when n is large enough. It is said that
(16)maxξei(Fi(wi,n,xni))<maxξei(Fi(ui,n,xni)).
By the definition of Ai(·) and ui,n∈Ai(xn), we have
(17)maxξei(Fi(ui,n,xni))=min⋃xi,n∈Si(xn)maxξei(Fi(xi,n,xni)).
This, however, contradicts the fact ui,n∈Ai(xn). Therefore, the mapping Ai(·) is closed.

Let ui,1,ui,2∈Ai(x), λ∈(0,1), and
(18)α0=min⋃θi∈Si(x)maxξei(Fi(θi,xi)).
From the definition of Ai(·), we have ui,1,ui,2∈Si(x) and
(19)maxξei(F(ui,1,xi))=maxξei(F(ui,2,xi))=α0.
As Si(x) is convex-valued, λui,1+(1-λ)ui,2∈Si(x).

According to the generalized Luc's quasi-Ci-convexity of Fi(·,xi), we get that, for all zi′∈Fi(λui,1+(1-λ)ui,2,xi), there exist zi,1∈Fi(ui,1,xi) and zi,2∈Fi(ui,2,xi) such that
(20)zi′∈zi-Ci, ∀zi∈C(zi,1,zi,2).
Without loss of generality, suppose l1=ξei(zi,1) and l2=ξei(zi,2), l1≥l2; we have zi,1∈l1ei-Ci and zi,2∈l2ei-Ci⊂l1ei-Ci. From (20), zi′∈l1ei-Ci. By the monotonicity of ξei(·),
(21)ξei(zi′)≤ξei(l1ei)=l1.
As
(22)l1≤max(maxξei(F(ui,1,xi)),maxξei(F(ui,2,xi)))=α0.
therefore, ξei(zi′)≤α0.

Since
(23)zi′∈Fi(λui,1+(1-λ)ui,2,xi)
is arbitrary, we have
(24)maxξei(Fi(λui,1+(1-λ)ui,2,xi))≤α0.
By the fact that Fi(λui,1+(1-λ)ui,2,xi) is compact and ξei(·) is continuous, there exists
(25)z~i∈Fi(λui,1+(1-λ)ui,1,xi)
such that
(26)ξei(z~i)=maxξei(Fi(λui,1+(1-λ)ui,2,xi)).
Thus, ξei(z~i)≤α0. It follows from the definition of α0 that
(27)maxξei(Fi(λui,1+(1-λ)ui,2,xi))=α0.
Thus, λui,1+(1-λ)ui,2∈Ai(x); namely, Ai(x) is a convex set.

Define A:X→2X by A(x)=Πi∈IAi(x),∀x∈X. Therefore, A(x) is a nonempty, convex, and closed subset of X for each x∈X. Since Ai(·) is closed, so is A(·), and since A(x)⊆X, X is compact, by [12, page 111, Corollary 9], A(·) is upper semicontinuous. By Lemma 5, there exists a point x¯∈X such that x¯∈A(x¯).

By the definition of A(·), we have(28)x¯i∈Si(x¯),maxξei(Fi(xi,x¯i))≥maxξei(F(x¯i,x¯i))maxξei(Fi(xi,x¯i))vv∀xi∈Si(x¯), i∈I.

From (28), ∀z¯i∈Fi(x¯i,x¯i),
(29)maxξei(Fi(xi,x¯i))≥ξei(z¯i).
By the compactness of Fi(xi,x¯i) and the continuity of ξei(·), there exists zi∈Fi(xi,x¯i), such that ξei(zi)=maxξei(Fi(xi,x¯i)). Thus, for all z¯i∈F(x¯i,x¯i), there exists zi∈Fi(xi,x¯i) such that ξei(z¯i)≤ξei(zi). Then, it follows from the subadditivity of ξei(·) that
(30)ξei(zi-z¯i)≥0.
By Lemma 4, we get
(31)zi-z¯i∉-intP.
So x¯ is a solution of (CWNEP) and this completes the proof.