Motivated by Takahashi and Zembayashi [22], and Ceng and Yao [23], we next prove the following crucial lemma concerning the GEP in a strictly convex, reflexive, and smooth Banach space.
Theorem 3.1. Let C be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C Ã— C to satisfying (A1)(A4), where
(A1) f(x, x) = 0 for all x âˆˆ C;
(A2) f is monotone, i.e. f(x, y) + f(y, x) â‰¤ 0 for all x, y âˆˆ C;
(A3) for all y âˆˆ C, f(., y) is weakly upper semicontinuous;
(A4) for all x âˆˆ C, f(x,.) is convex.
Let A be Î±inverse strongly monotone of C into E*. For all r > 0 and Ã— âˆˆ E, define the mapping S_{
r
} : E â†’ 2^{C} as follows:
Then, the following statements hold:

(1)
for each x âˆˆ E, S_{
r
}(x) â‰ âˆ…;

(2)
S
_{
r
}
is singlevalued;

(3)
ã€ˆS_{
r
}(x)  S_{
r
}(y), J(S_{
r
}x  x)ã€‰ â‰¤ ã€ˆS_{
r
}(x)  S_{
r
}(y), J(S_{
r
}y  y)ã€‰ for all x, y âˆˆ E;

(4)
F (S_{
r
}) = GEP (f);

(5)
GEP(f) is nonempty, closed, and convex.
Proof. (1) Let x_{0} be any given point in E. For each y âˆˆ C, we define the mapping G : C â†’ 2^{E} by
It is easily seen that y âˆˆ G(y), and hence G(y). â‰ âˆ…

(a)
First, we will show that G is a KKM mapping. Suppose that there exists a finite subset {y_{1}, y_{2},..., y_{
m
}} of C and Î±_{
i
} > 0 with such that for all i = 1, 2,..., m. It follows that
By (A1) and (A4), we have
which is a contradiction. Thus, G is a KKM mapping on C.

(b)
Next, we show that G(y) is closed for all y âˆˆ C. Let {z_{
n
}} be a sequence in G(y) such that z_{
n
} â†’ z as n â†’ âˆž. It then follows from z_{
n
} âˆˆ G(y) that,
By (A3), the continuity of J, and the lower semicontinuity of  Â· ^{2}, we obtain from (3.2) that
This shows that z âˆˆ G(y), and hence G(y) is closed for all y âˆˆ C.

(c)
We prove that G(y) is weakly compact. We now equip E with the weak topology. Then, C, as closed, bounded convex subset in a reflexive space, is weakly compact. Hence, G(y) is also weakly compact.
Using (a), (b), and (c) and Lemma 2.5, we have â‹‚_{xâˆˆC}G(y) â‰ âˆ…. It is easily seen that
Hence, s_{
r
}(x_{0}) â‰ âˆ…. Since x_{0} is arbitrary, we can conclude that s_{
r
}(x) â‰ âˆ… for all x âˆˆ E.
(2) We prove that S_{
r
} is singlevalued. In fact, for x âˆˆ C and r > 0, let z_{1}, z_{2} âˆˆ S_{
r
}(x). Then,
and
Adding the two inequalities and from the condition (A2) and monotonicity of A, we have
and hence,
Hence,
Since J is monotone and E is strictly convex, we obtain that z_{1}  x = z_{2}  x and hence z_{1} = z_{2}.
Therefore S_{
r
} is singlevalued.
(3) For x, y âˆˆ C, we have
and
Again, adding the two inequalities, we also have
It follows from monotonicity of A that
(4) It is easy to see that
Hence, F (S_{
r
}) = GEP (f).
(5) Finally, we claim that GEP (f) is nonempty, closed, and convex. For each y âˆˆ C, we define the mapping Î˜ : C â†’ 2^{E} by
Since y âˆˆ Î˜ (y), we have Î˜(y) â‰ âˆ… We prove that Î˜ is a KKM mapping on C. Suppose that there exists a finite subset {z_{1}, z_{2},..., z_{
m
}} of C and Î±_{
i
} > 0 with such that for all i = 1, 2,..., m. Then,
From (A1) and (A4), we have
which is a contradiction. Thus, Î˜ is a KKM mapping on C.
Next, we prove that Î˜ (y) is closed for each y âˆˆ C. For any y âˆˆ C, let {x_{
n
}} be any sequence in Î˜ (y) such that x_{
n
} â†’ x_{0}. We claim that x_{0} âˆˆ Î˜ (y). Then, for each y âˆˆ C, we have
By (A3), we see that
This shows that x_{0} âˆˆ Î˜ (y) and Î˜(y) is closed for each y âˆˆ C. Thus, is also closed.
We observe that Î˜ (y) is weakly compact. In fact, since C is bounded, closed, and convex, we also have Î˜(y) is weakly compact in the weak topology. By Lemma 2.5, we can conclude that .
Finally, we prove that GEP (f) is convex. In fact, let u, v âˆˆ F (S_{
r
}) and z_{
t
} = tu+(1  t)v for t âˆˆ (0, 1). From (3), we know that
This yields that
Similarly, we also have
It follows from (3.4) and (3.5) that
Hence, z_{
t
} âˆˆ F (S_{
r
}) = GEP (f) and hence GEP (f) is convex. This completes the proof.