First, we investigate relations between solution set of the WGVEP (SGVEP) and solution set of the DWGVEP (DSGVEP) when *K* is bounded.

**Theorem 3.1** *Let K* ⊆ *X be a nonempty and convex closed bounded set. If F* : *D × K × K →* 2^{Y}*satisfies the followings:*

*(i) F* (*u, x, x*) ⊆ *C*, ∀*x* ∈ *K, u* ∈ *T* (*x*)*;*

*(ii) the set* {(*u, x*), *u* ∈ *T* (*x*), *x* ∈ *K* : *F* (*u, x, y*) ⊈ -int *C*} *is closed for any y* ∈ *K;*

*(iii) F is weakly C-pseudomonotone;*

*(iv) the set* {*y* ∈ *K | F*(*u, x, y*) ⊈ int *C*} *is closed and F* (*u, x*, .) *is C-convex for any x* ∈ *K, u* ∈ *T*(*x*).

*Then the WGVEP has a nonempty solution set and x** ∈ *K is a solution of the WGVEP if and only if*

F\left(v,y,{x}^{*}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall y\in K,v\in T\left(y\right).

*Proof*. Set Γ: *D × K →* 2^{K}by

\Gamma \left(v,y\right)=\left\{x\in K|F\left(v,y,x\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C\right\},\forall y\in K,v\in T\left(y\right).

We claim that Γ is a KKM map. Suppose on the contrary, it does not hold, then there exists a finite set {*x*_{1}, . . ., *x*_{
n
}} ⊆ *K* and *z* ∈ *co*{*x*_{1}, . . ., *x*_{
n
}} such that z\notin {\bigcup}_{i=1}^{n}\Gamma \left(v,{x}_{i}\right). Thus, there exists *v*_{
i
} ∈ *T*(*x*_{
i
}) such that *F*(*v*_{
i
}*, x*_{
i
}*, z*) ⊆ int *C*, ∀*i* = 1, . . ., *n*. It follows from the weak *C*-pseudomonotonity of *F* that

F\left(u,z,{x}_{i}\right)\subseteq -\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall i=1,\dots ,n.

(3.1)

Taking into account that int *C* is convex, we obtain

{t}_{1}F\left(u,z,{x}_{1}\right)+\dots +{t}_{n}F\left(u,z,{x}_{n}\right)\subset -\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,

where z={\sum}_{1}^{n}{t}_{i}{x}_{i} and {\sum}_{1}^{n}{t}_{i}=1,{t}_{i}\ge 0,i=1,2,\dots ,n. For the above *t*_{
i
}, due to the convexity of *F*(*u, x*,.), one has

{t}_{1}F\left(u,z,{x}_{1}\right)+\dots \phantom{\rule{0.3em}{0ex}}{t}_{n}F\left(u,z,{x}_{n}\right)\subseteq F\left(u,z,z\right)+C\subseteq C+C\subseteq C,

which contradicts (3.1). By the condition (iv), we derive that the Γ is closed valued. Hence Γ is a KKM map. By the KKM Theorem, there exists *x**∈ *K* such that *x** ∈ ⋂_{v ∈ T(y), y ∈ K}Γ(*v, y*). That is, *F* (*v, y, x**) ⊈ int *C*,∀*y* ∈ *K, v* ∈ *T*(*y*).

Let us verify W{S}_{K}^{D}\subseteq W{S}_{K}. Take any *x**∈ *K*, obviously

F\left(v,y,{x}^{*}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall y\in K,v\in T\left(y\right).

(3.2)

For every *y* ∈ *K*, consider *x*_{
t
} = *x** + *t*(*y - x**), ∀*t* ∈ (0, 1). Clearly, *x*_{
t
} ∈ *K*. The *C*-convexity of *F* (*u, x*_{
t
},.) implies that

\left(1-t\right)F\left(u,{x}_{t},{x}^{*}\right)+tF\left(u,{x}_{t},y\right)\subseteq F\left(u,{x}_{t},{x}_{t}\right)+C\subseteq C+C\subseteq C.

Let us show *tF* (*u, x*_{
t
}*, y*) ⊈-int *C* by contradiction. Suppose on the contrary, then *tF* (*u, x*_{
t
}*, y*) ⊆ -int *C*. For any *p* ∈ *tF* (*u, x*_{
t
}*, y*), it holds

\left(1-t\right)F\left(u,{x}_{t},{x}^{*}\right)\subseteq C+p\subseteq C+\mathsf{\text{int}}C\subseteq \mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C.

So *F* (*u, x*_{
t
}*, x**) ⊆ int *C*, which contradicts (3.2). Noting that -int *C* is convex cone, we deduce

F\left(u,{x}_{t},y\right)\u2288-\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C.

(3.3)

Letting *t →* 0 in (3.3), we obtain by assumption (ii) and Lemma 2.4 that there exists *u** ∈ *T*(*x**) such that

F\left({u}^{*},{x}^{*},y\right)\u2288-\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall y\in K.

On the other hand, by the weak *C*-pseudomonotonity of *F*, we have W{S}_{K}\subseteq W{S}_{K}^{D}.

Hence, W{S}_{K}^{D}=W{S}_{K}.

**Theorem 3.2** *Let K* ⊆ *X be a nonempty and convex closed bounded set. If F* : *D × K × K →* 2^{Y}*satisfies the followings:*

*(i) F* (*u, x, x*) ⊆ *C*,∀*x* ∈ *K, u* ∈ *T*(*x*)*;*

*(ii) the set* {(*u, x*), *u* ∈ *T*(*x*), *x* ∈ *K | F* (*u, x, y*) ∩ -int *C* = ∅} *is closed for all y* ∈ *K;*

*(iii) F is strongly C-pseudomonotone;*

*(iv) the set* {*y* ∈ *K | F* (*u, x, y*) ∩ int *C* = ∅} *is closed and F* (*u, x*, .) *is C-convex for any x* ∈ *K, u* ∈ *T*(*x*).

*Then the SGVEP has a nonempty solution set and x** ∈ *K is a solution of the SGVEP if and only if*

F\left(v,y,{x}^{*}\right)\phantom{\rule{0.3em}{0ex}}\bigcap \phantom{\rule{0.3em}{0ex}}\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C=\varnothing ,\forall y\in K,v\in T\left(y\right).

*Proof*. Set Γ: *D × K →* 2^{K}by

\Gamma \left(v,y\right)=\left\{x\in K|F\left(v,y,x\right)\bigcap \mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C=\varnothing \right\},\forall y\in K,v\in T\left(y\right).

Following the similar arguments in the proof of Theorem 3.1, we can obtain the desired result.

In following sequel, we shall present some sufficient conditions for the nonemptiness and boundedness of the solution set of the WGVEP provided that it is strictly feasible in the strong sense.

**Theorem 3.3** *Let K* ⊆ *X be a nonempty, closed, convex and well-positioned set. If F* : *D × K × K →* 2^{Y}*satisfies the followings:*

*(i) F* (*u, x, x*) ⊆ *C*, ∀*x* ∈ *K, u* ∈ *T*(*x*)*;*

*(ii) the set* {(*u, x*), *u* ∈ *T* (*x*), *x* ∈ *K | F* (*u, x, y*) ⊈-int *C*} *is closed for any y* ∈ *K;*

*(iii) F is weakly C-pseudomonotone;*

*(iv) F* (*u, x*, .) *is C-convex and weakly lower semicontinuous for x* ∈ *K, u* ∈ *T*(*x*).

*Then the WGVEP has a nonempty bounded solution set whenever it is generalized strictly feasible in the strong sense*.

*Proof*. Suppose that the WGVEP is generalized strictly feasible in the strong sense. Then there exists *x*_{0} ∈*K* such that *x*_{0} ∈ *F*_{
s
}^{+}, i.e.,

F\left(u,{x}_{0},{x}_{0}+z\right)\subseteq \mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall u\in T\left({x}_{0}\right).

Set

D=\left\{x\in K|F\left(u,{x}_{0},x\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C\right\},\forall u\in T\left({x}_{0}\right).

By assumptions (i) and (iv), *x*_{0} ∈ *D* and *D* is weakly closed. We assert that *D* is bounded. Suppose on the contrary it does not holds, then there exists a sequence {*x*_{
n
}} ⊆ *M* with ||*x*_{
n
}|| *→* +*∞* as *n →* +*∞*. Since *X* is a reflexive Banach space, without loss of generality,

we may take a subsequence \left\{{{x}_{n}}_{{k}_{}}\right\} of {*x*_{
n
}} such that

\frac{1}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\in \left(0,1\right),\underset{k\to +\infty}{lim}\frac{{x}_{{n}_{k}}-{x}_{0}}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}=\underset{k\to +\infty}{lim}\frac{{x}_{{n}_{k}}}{\left|\right|{x}_{{n}_{k}}\left|\right|}\rightharpoonup z\in {K}_{\infty}.

By Lemma 2.3, *z* ≠ 0 since *K* is well-positioned. It follows from *x*_{0} ∈ *F*_{
s
}^{+} that

F\left(u,{x}_{0},{x}_{0}+z\right)\subseteq \mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C.

(3.4)

Noting that *F* (*u, x*, .) is C-convex, we have

\begin{array}{c}\left(1-\frac{1}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right)F\left(u,{x}_{0},{x}_{0}\right)+\frac{1}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}F\left(u,{x}_{0},{x}_{{n}_{k}}\right)\subseteq F\left(u,{x}_{0},\left(1-\frac{1}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right){x}_{0}+\frac{{x}_{{n}_{k}}}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right)+C\hfill \\ \phantom{\rule{2em}{0ex}}=F\left(u,{x}_{0},{x}_{0}+\frac{{x}_{{n}_{k}}-{x}_{0}}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right)+C.\hfill \end{array}

That is,

\frac{1}{\left|\right|{x}_{n}-{x}_{0}\left|\right|}F\left(u,{x}_{0},{x}_{{n}_{k}}\right)\subseteq F\left(u,{x}_{0},{x}_{0}+\frac{{x}_{{n}_{k}}-{x}_{0}}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right)+C.

We claim that F\left(u,{x}_{0},{x}_{0}+\frac{{x}_{{n}_{k}}-{x}_{0}}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C. Suppose on the contrary, F\left(u,{x}_{0},{x}_{0}+\frac{{x}_{{n}_{k}}-{x}_{0}}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right)\subseteq inC, we observe

\frac{1}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}F\left(u,{x}_{0},{x}_{{n}_{k}}\right)\subseteq F\left(u,{x}_{0},{x}_{0}+\frac{{x}_{{n}_{k}}-{x}_{0}}{\left|\right|{x}_{{n}_{k}}-{x}_{0}\left|\right|}\right)+C\subseteq inC+C\subseteq \mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,

which contradicts F\left(u,{x}_{0},{{x}_{n}}_{{k}_{}}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C. Taking into account the condition (iv), we obtain

F\left(u,{x}_{0},{x}_{0}+z\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C.

This is a contradiction to (3.4). Thus, *D* is bounded and it is weakly compact. For each *p* ∈ *K*, set

{D}_{p}=\left\{x\in D|F\left(v,p,x\right)\u2288\mathsf{\text{int}}C\right\},\forall p\in K,v\in T\left(p\right).

Then *D*_{
p
} ≠ ∅. Indeed, given *p* ∈ *K, v* ∈ *T* (*p*), set *K*_{0} = *conv* (*D* ⋃ *p*) ⊆ *K*, where *conv* means the convex hull of a set. Then *K*_{0} is nonempty, convex, and weakly compact. By Theorem 3.1, there exists \stackrel{\u0304}{x}\in {K}_{0} such that

F\left(v,y,\stackrel{\u0304}{x}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall y\in {K}_{0},v\in T\left(p\right).

Then F\left(u,{x}_{0},\stackrel{\u0304}{x}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}C implies \stackrel{\u0304}{x}\in D and F\left(v,p,\stackrel{\u0304}{x}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}C implies \stackrel{\u0304}{x}\in {D}_{p}. We obtain *D*_{
p
} ≠ ∅. Obviously, *D*_{
p
} is nonempty and weakly compact.

Next we prove that {*D*_{
p
} | *p* ∈ *K*} has the finite intersection property. For any finite set {*p*_{
i
} | *i* = 1, 2, . . ., *n*} ⊆ *K*, let *K*_{1} = *conv*{*D* ⋃ {*p*_{1}, *p*_{2}, . . ., *p*_{
n
}}}. Then *K*_{1} is weakly compact. By Theorem 3.1, there exists \widehat{x}\in {K}_{1} such that

F\left(v,y,\widehat{x}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall y\in {K}_{1},v\in T\left(p\right).

In particular, it holds

F\left(u,{x}_{0},\widehat{x}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,F\left(v,{p}_{i},\widehat{x}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,i=1,2,\dots ,n.

This means that \widehat{x}\in {\bigcap}_{i=1}^{n}{D}_{{p}_{i}}. Thus {*D*_{
p
} *| p* ∈ *K*} has the finite intersection property. Since *D* is weakly compact and *D*_{
p
} ⊆ *D* is weakly closed for all *p* ∈ *K, v* ∈ *T* (*p*), It follows that

\bigcap _{p\in K}{D}_{p}\ne \varnothing .

Let *x** ∈ ⋂_{p ∈ K}*D*_{
p
} It follows that

F\left(v,y,{x}^{*}\right)\u2288\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C,\forall y\in K,v\in T\left(y\right).

By Theorem 3.1, *x** is a solution of the WGVEP. As for the boundedness of the solution set of the WGVEP, it follows from Theorem 3.1 that the solution set of the WGVEP is a subset of *D*.

**Theorem 3.4** *Let K* ⊆ *X be a nonempty, closed, convex, and well-positioned set. If F* : *D × K × K →* 2^{Y}*satisfies the followings:*

*(i) F* (*u, x, x*) ⊆ *C*, ∀*x* ∈ *K, u* ∈ *T* (*x*)*;*

*(ii) the set* {(*u, x*), *u* ∈ *T*(*x*), *x* ∈ *K* | *F*(*u, x, y*) ⋂ - int *C* = ∅} *is closed for all y* ∈ *K;*

*(iii) F is strongly C-pseudomonotone;*

*(iv) F* (*u, x*, .) *is C-convex and weakly lower semicontinuous for x* ∈ *K, u* ∈ *T*(*x*);

*(v) F is positively homogeneous with degree α* > 0, *i.e., there exists α* > 0 *such that*

F\left(u,x,x+t\left(y-x\right)\right)={t}^{\alpha}F\left(u,x,y\right),\forall x,y\in K,u\in T\left(x\right),t\in \left(0,1\right).

*Then the SGVEP has a nonempty bounded solution set whenever it is generalized strictly feasible in the weak sense*.

*Proof*. Suppose that the SGVEP is generalized strictly feasible in the weak sense. Then there exists *x*_{0} ∈ *K* such that {x}_{0}\in {F}_{w}^{+}, i.e.,

F\left(u,{x}_{0},{x}_{0}+z\right)\phantom{\rule{0.3em}{0ex}}\bigcap \phantom{\rule{0.3em}{0ex}}\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C\ne \varnothing .

Set

D=\left\{x\in K|F\left(u,{x}_{0},x\right)\phantom{\rule{0.3em}{0ex}}\bigcap \phantom{\rule{0.3em}{0ex}}\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C=\varnothing \right\}.

By assumptions (i) and (iv), *x*_{0} ∈ *D* and *D* is weakly closed. We claim that *D* is bounded. Suppose on the contrary it does not holds, then there exists a sequence {*x*_{
n
}} ⊆ *M* with ||*x*_{
n
}|| *→* +*∞* as *n →* +*∞*. Since *X* is a reflexive Banach space, without loss of generality, we may take a subsequence \left\{{{x}_{n}}_{{k}_{}}\right\} of {*x*_{
n
}} such that

\frac{1}{\left|\right|{x}_{{n}_{k}}\left|\right|}\in \left(0,\phantom{\rule{0.3em}{0ex}}1\right),\underset{n\to +\infty}{lim}\frac{{x}_{{n}_{k}}}{\left|\right|{x}_{{n}_{k}}\left|\right|}\rightharpoonup z\in {K}_{\infty}.

By Lemma 2.3, *z* ≠ 0 since *K* is well-positioned. It follows from *x*_{0} ∈ *F*_{
w
}+ that

F\left(u,{x}_{0},{x}_{0}+z\right)\phantom{\rule{0.3em}{0ex}}\bigcap \phantom{\rule{0.3em}{0ex}}\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C\ne \varnothing .

(3.5)

Since {{x}_{n}}_{{k}_{}}\in D and *F* is positively homogenous with degree *α* > 0, it holds

F\left(u,{x}_{0},{x}_{0}+\frac{{x}_{{n}_{k}}-{x}_{0}}{\left|\right|{x}_{{n}_{k}}\left|\right|}\right)=\frac{1}{{\left|\right|{x}_{{n}_{k}}\left|\right|}^{\alpha}}F\left(u,{x}_{0},{x}_{{n}_{k}}\right)\phantom{\rule{0.3em}{0ex}}\bigcap \phantom{\rule{0.3em}{0ex}}\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C=\varnothing .

Taking into account the condition (iv), we obtain

F\left(u,{x}_{0},{x}_{0}+z\right)\phantom{\rule{0.3em}{0ex}}\bigcap \phantom{\rule{0.3em}{0ex}}\mathsf{\text{int}}\phantom{\rule{0.3em}{0ex}}C=\varnothing .

This is a contradiction to (3.5). Thus, *D* is bounded and it is weakly compact. Following the similar arguments in the proof of Theorem 3.3, we can prove the Theorem 3.4.

**Remark 3.1** *Assumption (v) of Theorem 3.4 is not new. Clearly, if F*(*x, y*) = 〈*u, y -x*〉, ∀*u* ∈ *T*(*x*), *then F is positively homogeneous with degree* = 1.

**Remark 3.2** *Since SS*_{
K
} ⊆ *WS*_{
K
}*, conditions for the solution set of the SGVEP to be nonempty and bounded are stronger than the WGVEP. Compared with Theorem 3.3, the condition that F is positively homogeneous in Theorem 3.4 is not dropped for the SGVEP*.

The following example shows that the converse of Theorem 3.3 or 3.4 is not true in general.

**Example 3.1** *Let X* = *R, K* = *R, D* = [0, 1], *Y* = *R*, C={R}_{+}^{2} *and*

T\left(x\right)=\left\{\begin{array}{cc}\left\{1\right\},\hfill & if\phantom{\rule{0.3em}{0ex}}x>0\hfill \\ \left\{0,1\right\},\hfill & if\phantom{\rule{0.3em}{0ex}}x=0.\hfill \end{array}\right.

*Let F* : *D × K × K →* 2^{Y}*be defined by*

F\left(u,x,y\right)=\left\{\begin{array}{cc}\u3008u,\left[\frac{\left({y}^{2}-{x}^{2}\right)}{2},\left({y}^{2}-{x}^{2}\right)\right]\u3009\hfill & \forall x,y\in K,u\in T\left(x\right);\hfill \\ \u3008u,\left(y-x\right)\u3009\hfill & \forall x,y\in K.\hfill \end{array}\right.

It is easily to see that *K* is well-positioned and *F* satisfies assumptions of Theorems 3.3 and 3.4. It can be verified that the WGVEP and the SGVEP have the same solution set {0}. On the other hand, it is easy to verify that {{F}_{w}}^{+}={{F}_{s}}^{+}=\varnothing .

For general generalized vector equilibrium problem, the following example shows *WS*_{
K
} ≠ ∅, but *SS*_{
K
} = ∅.

**Example 3.2** *Let X* = *R, K* = *R, D* = [-1, 1], *Y* = *R, C* = *R*_{+} *and*

\begin{array}{c}T\left(x\right)=\left\{-1,1\right\},\forall x\in K\\ F\left(u,x,y\right)=\left[-1,1\right],\forall x,y\in K,u\in T\left(x\right).\end{array}

It is obvious that the WGVEP has solution set *WS*_{
K
} = *R*, but solution set of the SGVEP *SS*_{
K
} = ∅.