Generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map

Abstract

In this paper, we introduce and study a new class of generalized nonlinear vector mixed quasi-variational-like inequalities governed by a multi-valued map in Hausdorff topological vector spaces which includes generalized vector mixed general quasi-variational-like inequalities, generalized nonlinear mixed variational-like inequalities, and so on. By using the fixed point theorem, we prove some existence theorems for the proposed inequality.

1 Introduction

Variational inequality theory has appeared as an effective and powerful tool to study and investigate a wide class of problems arising in pure and applied sciences including elasticity, optimization, economics, transportation, and structural analysis; see, for instance, [1–4] and the references therein. A vector variational inequality in a finite-dimensional Euclidean space was first introduced by Giannessi [5]. This is a generalization of scalar variational inequality to the vector case by virtue of multi-criterion consideration. In 1966, Browder [6] first introduced and proved the basic existence theorems of solutions to a class of nonlinear variational inequalities. The Browder’s results was extended to more generalized nonlinear variational inequalities by Liu et al. [7], Ahmad and Irfan [8], Husain and Gupta [9] and Xiao et al. [10], Zhao et al. [11].

In this paper, we consider a generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map and establish some existence results in locally convex topological vector spaces by using the fixed point theorem.

Let Y be a locally convex Hausdorff topological vector space (l.c.s., in short) and let K be a nonempty convex subset of a Hausdorff topological vector space (t.v.s., in short) E. We denote by $L(E,Y)$ the space of all continuous linear operators from E into Y, where $L(E,Y)$ is equipped with a σ-topology, and by $〈l,x〉$ the evaluation of $l\in L(E,Y)$ at $x\in E$. Let $X\subseteq L(E,Y)$. From the corollary of Schaefer [12], $L(E,Y)$ becomes a l.c.s. By Ding and Tarafdar [13], we have the bilinear map $〈\cdot ,\cdot 〉:L(K,Y)\times K\to Y$ is continuous. Let intA and $co(A)$ represent the interior and convex hull of a set A, respectively. Let $C:K\to 2^Y$ be a set-valued mapping such that $intC(x)\ne \mathrm{\varnothing }$ for each $x\in K$, let $\eta :K\times K\to E$ be a vector-valued mapping.

Let $N:L(E,Y)\times L(E,Y)\times L(E,Y)\to 2^{L(E,Y)}$ be a set-valued mapping, $H:K\times K\to 2^Y$, $D:K\to 2^K$ and $T,A,M:K\to 2^X$ be set-valued mappings. For each ${\omega }^{\ast }\in L(E,Y)$ and $g:K\to K$ a single-valued mapping, we consider the following class of generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map :

$$\text{(}\mathcal{P}\text{)}\{\begin{array}{c}\text{find }u\in K\text{ such that }u\in D(u)\text{ and for each }v\in D(u),\hfill \\ \text{there exist }x\in T(u),y\in A(u)\text{ and }z\in M(u)\text{ satisfying}\hfill \\ 〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)⊈-intC(u).\hfill \end{array}$$

(1.1)

The problem ($\mathcal{P}$) encompasses many models of variational inequality problems. The following problems are the special cases of ($\mathcal{P}$).

(a) If $N:L(E,Y)\times L(E,Y)\times L(E,Y)\to L(E,Y)$ and $H:K\times K\to Y$ are two single-valued mappings, $N(x,y,z)=A(x)$, where $A:L(E,Y)\to L(E,Y)$ and ${\omega }^{\ast }=0$, then the problem ($\mathcal{P}$) reduces to the following generalized vector mixed general quasi-variational-like inequality problem for finding $u\in K$ such that $u\in D(u)$ and for each $v\in D(u)$, there exists $x\in T(u)$ satisfying

$$〈A(x),\eta (v,g(u))〉+H(g(u),v)\notin -intC(u).$$

(1.2)

The problem (1.2) was studied by Ding and Salahuddin [14]. Some existence results of solutions are established under suitable assumptions without monotonicity and compactness.

(b) If g is an identity mapping and ${\omega }^{\ast }=0$, then the problem ($\mathcal{P}$) reduces to the following generalized nonlinear vector quasi-variational-like inequality problem for finding $(u,x,y,z)\in K\times U\times V\times W$ such that $u\in D(u)$ and for each $v\in D(u)$, there exist $x\in T(u)$, $y\in A(u)$ and $z\in M(u)$ satisfying

$$〈N(x,y,z),\eta (v,u)〉+H(u,v)⊈-intC(u).$$

(1.3)

The problem (1.3) was studied by Husain and Gupta [15].

(c) If $D(u)=K$, then the problem (1.3) reduces to the problem of finding $u\in K$ such that there exist $x\in T(u)$, $y\in A(u)$ and $z\in M(u)$ satisfying

$$〈N(x,y,z),\eta (v,u)〉+H(u,v)⊈-intC(u),\mathrm{\forall }v\in K,$$

(1.4)

which is introduced and studied by Xiao et al. [5]. When $N:L(E,Y)\times L(E,Y)\times L(E,Y)\to L(E,Y)$ and $H:K\times K\to Y$ are two single-valued mappings, the problem (1.4) includes some generalized variational inequality problems investigated in [8, 11, 16–19] as special cases.

(d) If $T(u)=A(u)=\mathrm{\varnothing }$ for all $u\in K$, and N is an identity mapping, the problem (1.3) reduces to the problem of finding $u\in K$ such that $u\in D(u)$ and for all $v\in D(u)$,

$$〈T(u),\eta (v,u)〉+H(u,v)⊈-intC(u),$$

which is introduced and studied by Peng and Yang [20].

For suitable and appropriate conditions imposed on the mappings C, N, H, D, T, A, M, η and g and by means of the fixed point theorem, we establish some existence results of solutions for the problem ($\mathcal{P}$). It is clear that the problem ($\mathcal{P}$) is the most general and unifying one, which is also one of the main motivations of this paper.

Definition 1.1 [21]

Let A and B be two topological vector spaces and let $T:A\to 2^B$ be a multi-valued mapping, then

(i) T is said to be upper semicontinuous if for any $x_0\in A$ and for each open set U in B containing $T(x_0)$, there is a neighborhood V of $x_0$ in A such that $T(x)\subset U$ for all $x\in V$.

(ii) T is said to have open lower sections if the set $T^{-1}(y)=\{x\in A:y\in T(x)\}$ is open in X for each $y\in B$.

(iii) T is said to be closed if any net $\{x_{\alpha }\}$ in A such that $x_{\alpha }\to x$ and any $\{y_{\alpha }\}$ in B such that $y_{\alpha }\to y$ and $y_{\alpha }\in T(x_{\alpha })$ for any α, we have $y\in T(x)$.

(iv) T is said to be lower semicontinuous if for any $x_0\in A$ and for each open set U in B containing $T(x_0)$, there is a neighborhood V of $x_0$ in A such that $T(x)\cap U\ne \mathrm{\varnothing }$ for all $x\in V$.

(v) T is said to be continuous if it is both lower and upper semicontinuous.

Lemma 1.2 [22]

Let A and B be two topological spaces. Suppose $T:A\to 2^B$ and $H:A\to 2^B$ are multi-valued mappings having open lower sections, then

(i) $G:A\to 2^B$ defined by, for each $x\in A$, $G(x)=co(T(x))$ has open lower sections;

(ii) $\rho :A\to 2^B$ defined by, for each $x\in A$, $\rho (x)=T(x)\cap H(x)$ has open lower sections.

Lemma 1.3 [23]

Let A and B be two topological spaces. If $T:A\to 2^B$ is an upper semicontinuous mapping with closed values, then T is closed.

Lemma 1.4 [24]

Let A and B be two topological spaces and let $T:A\to 2^B$ be an upper semicontinuous mapping with compact values. Suppose $\{x_{\alpha }\}$ is a net in A such that $x_{\alpha }\to x_0$. If $y_{\alpha }\in T(x_{\alpha })$ for each α, then there is a $y_0\in T(x_0)$ and a subset $\{y_{\beta }\}$ of $\{y_{\alpha }\}$ such that $y_{\beta }\to y_0$.

Let I be an index set, $E_i$ be a Hausdorff topological vector space for each $i\in I$. Let $K_i$ be a family of nonempty compact convex subsets in $E_i$. Let $K={\prod }_{i\in I}{K}_i$ and $E={\prod }_{i\in I}{E}_i$.

Lemma 1.5 [8]

For each $i\in I$, let $T_i:K\to 2^{K_i}$ be a set-valued mapping. Assume that the following conditions hold.

(i) For each $i\in I$, $T_i$ is a convex set-valued mapping;

(ii) $K=\cup \{int{T}_i^{-1}(x_i):x_i\in K_i\}$.

Then there exists $\overline{x}\in K$ such that $\overline{x}\in T(\overline{x})={\prod }_{i\in I}{T}_i({\overline{x}}_i)$, that is, ${\overline{x}}_i\in T_i({\overline{x}}_i)$ for each $i\in I$, where ${\overline{x}}_i$ is the projection of $\overline{x}$ onto $K_i$.

2 Main results

In this section, we shall derive the solvability for the problem ($\mathcal{P}$) under certain conditions.

First, we give the concept of 0-diagonally convex which is useful for establishing the existence theorem for the problem ($\mathcal{P}$).

Definition 2.1 Let K be a convex subset of a t.v.s. E and Y be a t.v.s. Let $C:K\to 2^Y$ be a set-valued mapping and $g:K\to K$ be a single-valued mapping. Then the multi-valued mapping $H:K\times K\to 2^Y$ is said to be 0-diagonally convex with respect to g in the second variable if for any finite subset $\{x_1,\dots ,x_n\}$ of K and any $x={\sum }_{i=1}^n{\alpha }_i{x}_i$ with ${\alpha }_i\ge 0$ for $i=1,\dots ,n$, and ${\sum }_{i=1}^n{\alpha }_i=1$,

$$\sum _{i=1}^n{\alpha }_{i}H(g(x),x_i)⊈-intC(x).$$

Remark 2.2

(i) If g is an identity mapping, then the concept in Definition 2.1 reduces to the corresponding concept of 0-diagonal convexity in [25].

(ii) If $H:K\times K\to Y$ is a single-valued mapping, then the concept in Definition 2.1 reduces to the corresponding concept of 0-diagonally convex with respect to g in the second variable in [14].

Theorem 2.3 Let Y be a l.c.s., K be a nonempty convex subset of a Hausdorff t.v.s. E, X be a nonempty compact convex subset of $L(E,Y)$, which is equipped with a σ-topology. Let $g:K\to K$, ${\omega }^{\ast }\in L(E,Y)$ and $T,A,M:K\to 2^X$ be upper semicontinuous set-valued mappings with nonempty compact values. Assume that the following conditions are satisfied:

(i) $D:K\to 2^K$ is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each $v\in K$, the mapping

$$〈N(\cdot ,\cdot ,\cdot )-{\omega }^{\ast },\eta (v,\cdot )〉+H(\cdot ,v):L(E,Y)\times L(E,Y)\times L(E,Y)\times K\times K\to 2^Y$$

is an upper semicontinuous set-valued mapping with compact values;

(iii) $C:K\to 2^Y$ is a convex set-valued mapping with $intC(u)\ne \mathrm{\varnothing }$ for all $u\in K$;

(iv) $\eta :K\times K\to E$ is affine in the first argument and for all $u\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to 2^Y$ is generalized vector 0-diagonally convex with respect to g;

(vi) $g:K\to K$ is continuous;

(vii) for each $u\in K$, the set $\{u\in K:co\mathrm{\Lambda }(u)\cap D(u)\ne \mathrm{\varnothing }\}$ is closed in K, where $\mathrm{\Lambda }(u)$ is defined as

$$\begin{array}{rcl}\mathrm{\Lambda }(u)& =& \{v\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\}.\end{array}$$

Then the problem ($\mathcal{P}$) admits at least one solution.

Proof Let ${\omega }^{\ast }\in L(E,Y)$. Define a set-valued mapping $Q:K\to 2^K$ by

$$\begin{array}{rcl}Q(u)& =& \{v\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\}\end{array}$$

for all $u\in K$. We first prove that $u\notin coQ(u)$ for all $u\in K$. To see this, suppose, by the method of contradiction, that there exists some point $\overline{u}\in K$ such that $\overline{u}\in coQ(\overline{u})$. Then there exists a finite subset $\{v_1,v_2,\dots ,v_n\}\subset Q(\overline{u})$, for $\overline{u}\in co\{v_1,v_2,\dots ,v_n\}$, such that

$$〈N(\overline{x},\overline{y},\overline{z})-{\omega }^{\ast },\eta (v_i,g(\overline{u}))〉+H(g(\overline{u}),v_i)\subseteq -intC(\overline{u}),i=1,2,\dots ,n.$$

Since $intC(\overline{u})$ is a convex set and η is affine in the first argument, for $i=1,2,\dots ,n$, ${\alpha }_i\ge 0$ with ${\sum }_{i=1}^n{\alpha }_i=1$, $\overline{u}={\sum }_{i=1}^n{\alpha }_i{v}_i$, we have

$$〈N(\overline{x},\overline{y},\overline{z})-{\omega }^{\ast },\eta (\sum _{i=1}^n{\alpha }_i{v}_i,g(\overline{u}))〉+\sum _{i=1}^n{\alpha }_{i}H(g(\overline{u}),v_i)\subseteq -intC(\overline{u}).$$

Since $\eta (u,g(u))=0$, for all $u\in K$, we have

$$\sum _{i=1}^n{\alpha }_{i}H(g(\overline{u}),v_i)\subseteq -intC(\overline{u}),$$

which contradicts the condition (v), so that $u\notin coQ(u)$ for all $u\in K$.

We now prove that

$$\begin{array}{rcl}{Q}^{-}(v)& =& \{u\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\}\end{array}$$

is open for all $v\in K$, that is, Q has open lower sections.

Consider a set-valued mapping $J:K\to 2^K$ is defined by

$$\begin{array}{rcl}J(v)& =& \{u\in K:\mathrm{\exists }x\in T(u),y\in A(u),z\in M(u)\text{ such that}\\ 〈N(x,y,z,)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)⊈-intC(u)\}.\end{array}$$

We only need to prove that $J(v)$ is closed for all $v\in K$. Let $\{u_{\alpha }\}$ be a net in $J(v)$ such that

$$u_{\alpha }\to u^{\ast }.$$

Since g is continuous, we have

$$g(u_{\alpha })\to g\left(u^{\ast }\right).$$

Then there exist $x_{\alpha }\in T(u_{\alpha })$, $y_{\alpha }\in A(u_{\alpha })$ and $z_{\alpha }\in M(u_{\alpha })$ such that

$$〈N(x_{\alpha },y_{\alpha },z_{\alpha },)-{\omega }^{\ast },\eta (v_{\alpha },g(u_{\alpha }))〉+H(g(u_{\alpha }),v_{\alpha })⊈-intC(u_{\alpha }).$$

Since T, A, M are upper semicontinuous set-valued mappings with compact values, by Lemma 1.4, $\{x_{\alpha }\}$, $\{y_{\alpha }\}$, $\{z_{\alpha }\}$ have convergent subnets with limits, say $x^{\ast }$, $y^{\ast }$, $z^{\ast }$ and $x^{\ast }\in T(u^{\ast })$, $y^{\ast }\in A(u^{\ast })$ and $z^{\ast }\in M(u^{\ast })$. Without loss of generality, we may assume that $x_{\alpha }\to x^{\ast }$, $y_{\alpha }\to y^{\ast }$ and $z_{\alpha }\to z^{\ast }$. Suppose that

$$m_{\alpha }\in \{〈N(x_{\alpha },y_{\alpha },z_{\alpha },)-{\omega }^{\ast },\eta (v_{\alpha },g(u_{\alpha }))〉+H(g(u_{\alpha }),v_{\alpha })⊈-intC(u_{\alpha })\}.$$

Since $〈N(\cdot ,\cdot ,\cdot )-{\omega }^{\ast },\eta (v,\cdot )〉+H(\cdot ,v)$ is upper semicontinuous with compact values, by Lemma 1.4, there exist $m^{\ast }\in 〈N(x^{\ast },y^{\ast },z^{\ast })-{\omega }^{\ast },\eta (v^{\ast },g(u^{\ast }))〉+H(g(u^{\ast }),v^{\ast })$ and a subnet $\{m_{\beta }\}$ of $\{m_{\alpha }\}$ such that $m_{\beta }\to m^{\ast }$. Hence $J(v)$ is closed in K. So that $Q^{-}(v)$ is open for each $v\in K$. Therefore Q has open lower sections.

Consider a set-valued mapping $G:K\times U\times V\times W\to 2^K$ defined by

$$G(u)=coQ(u)\cap D(u),\mathrm{\forall }u\in K.$$

Since D has open lower sections by hypothesis (i), we may apply Lemma 1.2 to assert that the set-valued mapping G has also open lower sections. Let

$$Z=\{u\in K:G(u)\ne \mathrm{\varnothing }\}.$$

There are two cases to consider. In the case $Z=\mathrm{\varnothing }$, we have

$$coQ(u)\cap D(u)=\mathrm{\varnothing }\text{for each }u\in K.$$

This implies that for each $u\in K$,

$$Q(u)\cap D(u)=\mathrm{\varnothing }.$$

On the other hand, by the condition (i), and the fact that K is a compact convex subset of Y, we can apply Lemma 1.5, in this case that $I=\{1\}$, to assert the existence of a fixed point $u^{\ast }\in D(u^{\ast })$, we have

$$Q\left(u^{\ast }\right)\cap D\left(u^{\ast }\right)=\mathrm{\varnothing }.$$

This implies $\mathrm{\forall }v\in D(u^{\ast })$, $v\notin Q(u^{\ast })$. Hence, in this particular case, the assertion of the theorem holds.

We now consider the case $Z\ne \mathrm{\varnothing }$. Define a set-valued mapping $S:K\to 2^K$ by

$$S(u)=\{\begin{array}{cc}G(u),\hfill & u\in Z;\hfill \\ D(u),\hfill & u\in K\setminus Z.\hfill \end{array}$$

Then, for each $u\in K$, $S(u)$ is a convex set and for each $t\in K$,

$$S^{-}(t)=G^{-}(t)\cup ((K\setminus Z)\cap (D^{-}(t))).$$

Since $D^{-}(t)$, $co{Q}^{-}(t)$ are open in K and $K\setminus Z$ is open in K by the condition (vii), we have $S^{-}(t)$ is open in K. This implies that S has open lower sections. Therefore, there exists $u^{\ast }\in K$ such that $u^{\ast }\in S(u^{\ast })$. Suppose that $u^{\ast }\in Z$, then

$$u^{\ast }\in coQ\left(u^{\ast }\right)\cap D\left(u^{\ast }\right),$$

so that $u^{\ast }\in coQ(u^{\ast })$. This is a contradiction. Hence, $u^{\ast }\notin Z$. Therefore,

$$u^{\ast }\in D\left(u^{\ast }\right)\text{and}G\left(u^{\ast }\right)=\mathrm{\varnothing }.$$

Thus

$$u^{\ast }\in D\left(u^{\ast }\right)\text{and}coQ\left(u^{\ast }\right)\cap D\left(u^{\ast }\right)=\mathrm{\varnothing }.$$

This implies

$$Q\left(u^{\ast }\right)\cap D\left(u^{\ast }\right)=\mathrm{\varnothing }.$$

Consequently, the assertion of the theorem holds in this case. The problem ($\mathcal{P}$) admits at least one solution. □

Corollary 2.4 Let Y be a l.c.s., K be a nonempty convex subset of a Hausdorff t.v.s. E, X be a nonempty compact convex subset of $L(E,Y)$, which is equipped with a σ-topology. Assume that N and H are single-valued mappings and $T,A,M:K\to 2^X$ are upper semicontinuous set-valued mappings with nonempty compact values. Let ${\omega }^{\ast }\in L(E,Y)$ and $g:K\to K$. Assume that the following conditions are satisfied:

(i) $D:K\to 2^K$ is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each $v\in K$, the mapping

$$〈N(\cdot ,\cdot ,\cdot )-{\omega }^{\ast },\eta (v,\cdot )〉+H(\cdot ,v):L(E,Y)\times L(E,Y)\times L(E,Y)\times K\times K\to 2^Y$$

is continuous;

(iii) $C:K\to 2^Y$ is a convex set-valued mapping with $intC(u)\ne \mathrm{\varnothing }$ for all $u\in K$;

(iv) $\eta :K\times K\to E$ is affine in the first argument and for all $u\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to 2^Y$ is vector 0-diagonally convex with respect to g;

(vi) $g:K\to K$ is continuous;

(vii) for each $u\in K$, the set $\{u\in K:co\mathrm{\Lambda }(u)\cap D(u)\ne \mathrm{\varnothing }\}$ is closed in K, where $\mathrm{\Lambda }(u)$ is defined as

$$\begin{array}{rcl}\mathrm{\Lambda }(u)& =& \{v\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\};\end{array}$$

(viii) $Y\setminus \{-intC(u)\}$ is an upper semicontinuous set-valued mapping.

Then there exists a point $\overline{u}\in K$ such that $\overline{u}\in D(\overline{u})$ and for each $v\in D(\overline{u})$, there exist $\overline{x}\in T(\overline{u})$, $\overline{y}\in A(\overline{u})$ and $\overline{z}\in M(\overline{u})$ such that

$$〈N(\overline{x},\overline{y},\overline{z})-{\omega }^{\ast },\eta (v,g(\overline{u}))〉+H(g(\overline{u}),v)\notin -intC(\overline{u}).$$

Proof Define a set-valued mapping $Q:K\to 2^K$ by

$$\begin{array}{rcl}Q(u)& =& \{v\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\in -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\}\end{array}$$

for all $u\in K$. We now prove that $Q^{-}(v)$ is open for each $v\in K$, that is,

$$\begin{array}{rl}{(Q^{-1}(v))}^c=& \{u\in K:\mathrm{\exists }x\in T(u),y\in A(u),z\in M(u)\text{ such that}\\ 〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\cap Y\setminus \{-intC(u)\}\ne \mathrm{\varnothing }\}\end{array}$$

is closed in K. Let $\{u_t\}$ be a net in ${(Q^{-1}(v))}^c$ such that

$$g(u_t)\to g\left(u^{\ast }\right)\in K.$$

Then there exist $x_t\in T(u_t)$, $y_t\in A(u_t)$ and $z_t\in M(u_t)$ such that

$$〈N(x_t,y_t,z_t)-{\omega }^{\ast },\eta (v,g(u_t))〉+H(g(u_t),v)\in Y\setminus \{-intC(u_t)\}.$$

The upper semicontinuity, compact values of T, A, M and Lemma 1.4 imply that there exist convergent subnets $\{x_{t_j}\}$, $\{y_{t_j}\}$ and $\{z_{t_j}\}$ such that

$$x_{t_j}\to x^{\ast },y_{t_j}\to y^{\ast }\text{and}{z}_{t_j}\to z^{\ast }$$

for some $x^{\ast }\in T(u),y^{\ast }\in A(u)$ and $z^{\ast }\in M(u)$. Since $〈N(\cdot ,\cdot ,\cdot )-{\omega }^{\ast },\eta (v,\cdot )〉+H(\cdot ,v)$ is continuous, we have

$$\begin{array}{r}〈N(x_{t_j},y_{t_j},z_{t_j})-{\omega }^{\ast },\eta (v,g(u_{t_j}))〉+H(g(u_{t_j}),v)\\ \to 〈N(x^{\ast },y^{\ast },z^{\ast })-{\omega }^{\ast },\eta (v,g\left(u^{\ast }\right))〉+H(g\left(u^{\ast }\right),v).\end{array}$$

From Lemma 1.3 and upper semicontinuity of $Y\setminus (-intC(u))$, we have

$$〈N(x^{\ast },y^{\ast },z^{\ast })-{\omega }^{\ast },\eta (v,g\left(u^{\ast }\right))〉+H(g\left(u^{\ast }\right),v)\in Y\setminus (-intC\left(u^{\ast }\right)),$$

and hence $u^{\ast }\in {(Q^{-1}(v))}^c$, which gives that ${(Q^{-1}(v))}^c$ is closed. Therefore Q has open lower sections. For the remainder of the proof, we can just follow that of Theorem 2.3. This completes the proof. □

Theorem 2.5 Let Y be a l.c.s., K be a nonempty convex subset of a Hausdorff t.v.s. E, X be a nonempty compact convex subset of $L(E,Y)$, which is equipped with a σ-topology. Let ${\omega }^{\ast }\in L(E,Y)$, $g:K\to K$ and $T,A,M:K\to 2^X$ be upper semicontinuous set-valued mappings. Assume that the following conditions are satisfied.

(i) $D:K\to 2^K$ is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each $y\in K$, the mapping

$$〈N(\cdot ,\cdot ,\cdot )-{\omega }^{\ast },\eta (v,\cdot )〉+H(\cdot ,v):L(E,Y)\times L(E,Y)\times L(E,Y)\times K\times K\to 2^Y$$

is upper semicontinuous;

(iii) $C:K\to 2^Y$ is a convex set-valued mapping with $intC(u)\ne \mathrm{\varnothing }$ for all $u\in K$;

(iv) $\eta :K\times K\to E$ is affine in the first argument and for all $x\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to 2^Y$ is generalized vector 0-diagonally convex with respect to g;

(vi) $g:K\to K$ is continuous;

(vii) For each $u\in K$, the set $\{u\in K:co\mathrm{\Lambda }(u)\cap D(u)\ne \mathrm{\varnothing }\}$ is closed in K, where $\mathrm{\Lambda }(u)$ is defined as

$$\begin{array}{rcl}\mathrm{\Lambda }(u)& =& \{v\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\};\end{array}$$

(viii) for a given $u\in K$, and a neighborhood O of u, for all $t\in O$, $intC(u)=intC(t)$.

Then the problem ($\mathcal{P}$) admits at least one solution.

Proof Define a set-valued mapping $Q:K\to 2^K$ by

$$\begin{array}{rcl}Q(u)& =& \{v\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\}\end{array}$$

for all $u\in K$. We now prove that for each $v\in K$,

$$\begin{array}{rcl}{Q}^{-1}(v)& =& \{u\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\}\end{array}$$

is open. That is, Q has open lower sections in K. Indeed, let $\overline{u}\in Q^{-}(v)$, that is,

$$〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(\overline{u}))〉+H(g(\overline{u}),v)\subseteq -intC(\overline{u}).$$

Since $〈N(\cdot ,\cdot ,\cdot )-{\omega }^{\ast },\eta (y,g(\cdot ))〉+H(g(\cdot ),y)$ is upper semicontinuous, there exists a neighborhood O of $\overline{u}$ such that

$$〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\mathrm{\forall }u\in O.$$

By (vii),

$$〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(\overline{u}),\mathrm{\forall }u\in O.$$

Hence, $O\subset Q^{-}(v)$. This implies $Q^{-}(v)$ is open for each $v\in K$, and so Q has open lower sections. For the remainder of the proof, we can just follow that of Theorem 2.3. This completes the proof. □

Corollary 2.6 Let Y be a l.c.s., K be a nonempty convex subset of a Hausdorff t.v.s. E, X be a nonempty compact convex subset of $L(E,Y)$, which is equipped with a σ-topology. Let ${\omega }^{\ast }\in L(E,Y)$, $g:K\to K$ and $T,A,M:K\to 2^X$ be upper semicontinuous set-valued mappings. Assume that the following conditions are satisfied.

(i) $D:K\to 2^K$ is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each $y\in K$, the mapping

$$〈N(\cdot ,\cdot ,\cdot )-{\omega }^{\ast },\eta (v,g(\cdot ))〉+H(g(\cdot ),v):L(E,Y)\times L(E,Y)\times L(E,Y)\times K\times K\to 2^Y$$

is upper semicontinuous;

(iii) $C:K\to 2^Y$ is a convex set-valued mapping such that for each $u\in K$, $C(u)=C$ is a convex cone with $intC(u)\ne \mathrm{\varnothing }$ for all $u\in K$;

(iv) $\eta :K\times K\to E$ is affine in the first argument and for all $u\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to 2^Y$ is generalized vector 0-diagonally convex with respect to g;

(vi) $g:K\to K$ is continuous;

(vii) for each $u\in K$, the set $\{u\in K:co\mathrm{\Lambda }(u)\cap D(u)\ne \mathrm{\varnothing }\}$ is closed in K, where $\mathrm{\Lambda }(u)$ is defined as

$$\begin{array}{rcl}\mathrm{\Lambda }(u)& =& \{v\in K:〈N(x,y,z)-{\omega }^{\ast },\eta (v,g(u))〉+H(g(u),v)\subseteq -intC(u),\\ \mathrm{\forall }x\in T(u),y\in A(u),z\in M(u)\}.\end{array}$$

Then the problem ($\mathcal{P}$) admits at least one solution.

Proof By hypothesis (iii), the condition (vii) in Theorem 2.5 is satisfied. Hence, all the conditions in Theorem 2.5 are satisfied. □