Well-posedness for generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality and optimization problems

Abstract

The purpose of this paper is to focus on the well-posedness for a generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality and optimization problems with a constraint. We establish a metric characterization of well-posedness in terms of an approximate solution set. Also we prove that well-posedness of optimization problem is closely related to that of generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality problems.

1 Introduction

The theory of variational inequality for multi-valued mappings has been studied by several authors (see [1, 4, 9, 14, 16, 25]). Since variational inequality theory is closely related to mathematical programming problems under mild conditions, consequently the concept of Tykhonov well-posedness has also been generalized to variational inequalities [7,8,9,10,11,12] and equilibrium problems, fixed point problems, optimization problems, mixed quasi-variational-like inequality with constraints etc. [15, 17, 18, 24, 26].

In 2000, Lignola and Morgan [20] defined the parametric well-posedness for optimization problems with variational inequality constraints by using the approximating sequences. Lignola [19] discussed the well-posedness, L-well-posedness and metric characterizations of well-posedness for quasi-variational-inequality problems. Ceng and Yao [3] extended these concepts to derive the conditions under which the generalized mixed variational inequality problems are well-posed. Thereafter, Lin and Chuang [21] established well-posedness for variational inclusion, and optimization problems with variational inclusion and scalar equilibrium constraints in a generalized sense. In 2010, Fang et al. [11] extended the notion of well-posedness by perturbations to a mixed variational inequality problem in a Banach space. Recently, Ceng et al. [2] suggested the conditions of well-posedness for hemivariational inequality problems involving Clarkes generalized directional derivative under different types of monotonicity assumptions.

Inspired and motivated by recent work [6, 7, 13,14,15,16, 23, 25], we consider and study well-posedness for generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality problems and optimization problems with constrained involving a relaxed η-$\alpha _{g}$-P-monotone operator.

2 Preliminaries

Assume that $\mathcal{X}$ and $\mathcal{Y}$ are two real Banach spaces. Let $\mathcal{D} \subset \mathcal{X}$ be a nonempty closed convex subset of $\mathcal{X}$ and $P\subset \mathcal{Y}$ a closed convex and proper cone with nonempty interior. Throughout this paper, we shall use the following inequalities. For all $x, y \in \mathcal{Y}$:

• (i) $x \leq _{P} y \Leftrightarrow y - x \in P$;

• (ii) $x \nleq _{P} y \Leftrightarrow y - x \notin P$;

• (iii) $x \leq _{P^{0}} y \Leftrightarrow y - x \in P^{0}$;

where $P^{0}$ denotes the interior of P.

If $\leq _{P}$ is a partial order, then $(\mathcal{Y},\leq _{P})$ is called an ordered Banach space ordered by P. Let $T:\mathcal{X} \to 2^{L( \mathcal{X},\mathcal{Y})}$ be a set-valued mapping where $L(\mathcal{X},\mathcal{Y})$ denotes the space of all continuous linear mappings from $\mathcal{X}$ into $\mathcal{Y}$. Assume that $Q:L(\mathcal{X},\mathcal{Y})\times \mathcal{D} \to L(\mathcal{X}, \mathcal{Y})$, $\varphi :\mathcal{D} \times \mathcal{D} \to \mathcal{Y}$, $\eta : \mathcal{X} \times \mathcal{X} \to \mathcal{X}$ are bi-mappings and $g:\mathcal{D}\to \mathcal{D}$ is single-valued mapping. We consider the following generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality problem for finding $x \in \mathcal{D}$ and $u \in T(x)$ such that

$$ \bigl\langle Q(u,x) , \eta \bigl(y, g(x)\bigr)\bigr\rangle + \varphi \bigl(g(x),y\bigr) \nleq _{P^{0}}0, \quad \forall y \in \mathcal{D}. $$

(2.1)

Denote by

$$ \varOmega = \bigl\{ x \in {\mathcal {{D}}}: \exists u \in T(x) \text{ such that } \bigl\langle Q(u,x), \eta \bigl(y,g(x)\bigr)\bigr\rangle + \varphi \bigl(g(x),y \bigr) \nleq _{P^{0}} 0, \forall y \in \mathcal{D}\bigr\} $$

the solution set of the problem (2.1).

Definition 2.1

A mapping $\phi : \mathcal{D} \to \mathcal{Y}$ is said to be

• (i) P-convex, if

  $$ \phi \bigl(\mu x+(1-\mu )y\bigr) \leq _{P} \mu \phi (x) + (1-\mu ) \phi (y),\quad \forall x,y \in {\mathcal {{D}}}, \mu \in [0,1]; $$

• (ii) P-concave, if

  $$ \phi \bigl(\mu x +(1-\mu )y\bigr) \geq _{P} \mu \phi (x)+(1-\mu ) \phi (y),\quad \forall x,y\in {\mathcal {{D}}},\mu \in [0,1]. $$

Definition 2.2

([25])

A set-valued mapping $T:\mathcal{D} \to 2^{L(\mathcal{X},\mathcal{Y})}$ is said to be monotone with respect to the first variable of Q, if

$$ \bigl\langle Q(u,\cdot ) - Q(v,\cdot ), x - y\bigr\rangle \geq _{P} 0,\quad \forall x,y \in {\mathcal {{D}}}, u \in T(x), v\in T(y). $$

Definition 2.3

Let $g:\mathcal{D}\to \mathcal{D}$ be a single-valued mapping. A set-valued mapping $T: \mathcal {{D}}\to 2^{L( \mathcal{X},\mathcal{Y})}$ is said to be relaxed η-$\alpha _{g}$-P-monotone with respect to the first variable of Q and g, if

$$ \bigl\langle Q(u,\cdot ) - Q(v,\cdot ), \eta \bigl(g(x),y\bigr)\bigr\rangle - \alpha _{g}(x-y) \geq _{P} 0,\quad \forall x,y \in { \mathcal {{D}}}, u \in T(x), v \in T(y), $$

where $\alpha _{g}:\mathcal{X} \to \mathcal{Y}$ is a mapping such that $\alpha _{g}(tz)=t^{p}\alpha _{g}(z)$, $\forall t>0$, $z\in {\mathcal {{X}}}$, and $p>1$ is a constant.

Definition 2.4

A mapping $\gamma : \mathcal{X}\times \mathcal{X} \to \mathcal{X}$ is said to be affine with respect to the first variable if, for any $x_{i} \in \mathcal{D}$ and $\lambda _{i} \geq 0$ ($1 \leq i \leq n$) with $\sum_{i=1}^{n} \lambda _{i} = 1$ and for any $y \in \mathcal{D}$,

$$ \gamma \Biggl(\sum^{n}_{i=1}\lambda _{i} x_{i} , y\Biggr)=\sum ^{n}_{i=1} \lambda _{i} \gamma (x_{i} , y). $$

Lemma 2.5

([5])

Let $(\mathcal{Y},P)$ be an ordered Banach space with closed convex pointed cone P and $P^{0}\neq \emptyset$. Then, for all $x,y,z \in \mathcal{Y}$, we have

• (i) $z \nleq _{P^{0}} x$, $x \geq _{P} y \Rightarrow z \nleq _{P^{0}} y$;

• (ii) $z \ngeq _{P^{0}} x$, $x \leq _{P} y \Rightarrow z \ngeq _{P^{0}} y$.

Lemma 2.6

([22])

Let $(\mathcal {{X}},\|\cdot \|)$ be a normed linear space and ${\mathfrak{H}}$ be a Hausdorff metric on the collection $CB(\mathcal {{X}})$ of all nonempty, closed and bounded subsets of $\mathcal {{X}}$ induced by metric

$$ d(u,v) = \Vert u - v \Vert , $$

which is defined by

$$ {\mathfrak{H}}(A,B ) = \max \Bigl\{ \sup_{u\in A}\inf _{v\in B} \Vert u-v \Vert , \sup_{v\in B} \inf_{u\in A} \Vert u-v \Vert \Bigr\} , \quad \forall A,B \in CB(\mathcal {{X}}). $$

If A, B are compact sets in $\mathcal{X}$, then for each $u \in A$ there exists $v \in B$ such that

$$ \Vert u - v \Vert \leq {\mathfrak{H}}(A,B). $$

Definition 2.7

A set-valued mapping $T:\mathcal {{D}} \to 2^{L(\mathcal {{X}},\mathcal {{Y})}}$ is said to be ${\mathfrak{H}}$-hemicontinuous, if

$$ {\mathfrak{H}}\bigl(T\bigl(x + \tau (y-x)\bigr),T(x)\bigr) \to 0 \quad \text{as } \tau \to 0^{+}, \forall x,y \in {\mathcal {{D}}}, \tau \in (0,1), $$

where ${\mathfrak{H}}$ is the Hausdorff metric defined on $CB(L( \mathcal {{X}},\mathcal {{Y}}))$.

Lemma 2.8

Let $\mathcal{D}$ be a closed convex subset of a real Banach space $\mathcal {{X}}$, $\mathcal {{Y}}$ be a real Banach space ordered by a nonempty closed convex pointed cone P with apex at the origin and $P^{0} \neq \emptyset$. Assume that $Q:L(\mathcal {{X}}, \mathcal {{Y})}\to L(\mathcal {{X}},\mathcal {{Y})}$ is a continuous mapping and $T:\mathcal {{D}} \to 2^{L(\mathcal {{X}}, \mathcal {{Y}})}$ is a nonempty compact set-valued mapping. If the following conditions are satisfied:

• (i) $\varphi :\mathcal {{D}}\times \mathcal {{D}} \to {\mathcal {{Y}}}$ is a P-convex in the second variable with $\varphi (x,x)=0$, $\forall x \in {\mathcal {{D}}}$;

• (ii) $\eta :\mathcal{X} \times \mathcal{X} \to \mathcal{X}$ is an affine mapping in the first variable with $\eta (x,x)=0$, $\forall x \in \mathcal{D}$;

• (iii) $T:\mathcal{D} \to 2^{L(\mathcal{X},\mathcal{Y})}$ is $\mathfrak{H}$-hemicontinuous and relaxed η-α-P-monotone with respect to Q;

then the following two problems are equivalent:

• (a) there exist $x_{0} \in \mathcal{D}$ and $u_{0} \in T(x_{0})$ such that

  $$ \bigl\langle Q(u_{0}), \eta (y,x_{0})\bigr\rangle + \varphi (x_{0},y) \nleq _{P^{0}} 0,\quad \forall y \in \mathcal{D}, $$

• (b) there exists $x_{0} \in \mathcal{D}$ such that

  $$ \bigl\langle Q(v), \eta (y,x_{0})\bigr\rangle + \varphi (x_{0},y)-\alpha (y-x _{0}) \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}, v \in T(y). $$

3 Well-posedness for problem (2.1)

In this section, we established the well-posedness for problem (2.1) with relaxed η-$\alpha _{g}$-P-monotone operator.

Definition 3.1

A sequence $\{x_{n}\} \in \mathcal{D}$ is said to be an approximating sequence for problem (2.1) if, there exist $u_{n}\in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n} \to 0$ such that

$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}, e\in \operatorname{int} P. $$

Definition 3.2

The generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality problem is said to be well-posed if

• (i) there exists a unique solution $x_{0}$ of problem (2.1);

• (ii) every approximating sequence of problem (2.1) converges to $x_{0}$.

Corollary 3.3

From Definition 3.2, it follows that if the generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality problem is well-posed, then

• (i) the solution set Ω of problem (2.1) is nonempty;

• (ii) every approximating sequence has a subsequence that converges to some point of Ω.

To investigate well-posedness of problem (2.1), we denote the approximate solution set of problem (2.1) by

$$\begin{aligned} \varOmega _{\epsilon } =& \bigl\{ x \in {\mathcal {{D}}}: \exists u \in T(x) \text{ such that} \\ &\bigl\langle Q(u,x), \eta \bigl(y,g(x)\bigr)\bigr\rangle +\varphi \bigl(g(x),y \bigr)+\epsilon e \nleq _{P^{0}} 0, \forall y \in {\mathcal {{D}}}, \epsilon \geq 0\bigr\} . \end{aligned}$$

Remark 3.4

We note that, if $\epsilon =0$ then $\varOmega = \varOmega _{\epsilon }$, and if $\epsilon >0$ then $\varOmega \subseteq \varOmega _{\epsilon }$.

Denote by $\operatorname{diam} \mathcal{B}$ the diameter of a set $\mathcal{B}$ which is defined as

$$ \operatorname{diam} \mathcal {{B}} = \sup_{a,b\in {\mathcal {{B}}}} \Vert a - b \Vert . $$

Theorem 3.5

Let $g: \mathcal {{D}}\to {\mathcal {{D}}}$ and $Q:L(\mathcal {{X}}, \mathcal {{Y}})\times \mathcal {{D}} \to L( \mathcal {{X}},\mathcal {{Y}})$ be two continuous mappings. Let $\varphi (\cdot ,y)$, $\eta (y,\cdot )$ and $\alpha _{g}$ be continuous functions for all $y \in {\mathcal {{D}}}$. If the conditions in Lemma 2.8 are satisfied, then problem (2.1) is well-posed if and only if

$$ \varOmega _{\epsilon }\neq \emptyset , \quad \forall \epsilon > 0 $$

and

$$ \operatorname{diam} \varOmega _{\epsilon }\to 0 \quad \textit{as } \epsilon \to 0. $$

Proof

Assume that problem (2.1) is well-posed, then it has a unique solution $x_{0} \in \varOmega$. Since $\varOmega \subseteq \varOmega _{\epsilon }$, $\forall \epsilon > 0$, this implies that $\varOmega _{\epsilon }\neq \emptyset$, $\forall \epsilon > 0$. On the contrary, if

$$ \operatorname{diam} \varOmega _{\epsilon }\nrightarrow 0 \quad \text{as } \epsilon \to 0, $$

then there exist $r > 0$, m (a positive integer), and a sequence $\{\epsilon _{n} > 0\}$ with $\epsilon _{n} \to 0$ and $x_{n}, x^{ \prime }_{n} \in \varOmega _{\epsilon _{n}}$ such that

$$ \bigl\Vert x_{n}-x^{\prime }_{n} \bigr\Vert > r, \quad \forall n \geq m. $$

(3.1)

Since $x_{n}, x^{\prime }_{n} \in \varOmega _{\epsilon _{n}}$, there exist $u_{n} \in T(x_{n})$ and $u^{\prime }_{n} \in T(x^{\prime }_{n})$ such that

$$\begin{aligned} &\bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}, \\ &\bigl\langle Q\bigl(u^{\prime }_{n},x^{\prime }_{n} \bigr), \eta \bigl(y, g\bigl(x^{\prime }_{n}\bigr)\bigr)\bigr\rangle + \varphi \bigl(g\bigl(x^{\prime }_{n}\bigr),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in { \mathcal {{D}}}. \end{aligned}$$

Since the problem is well-posed, the approximating sequences $\{x_{n}\}$ and $\{x^{\prime }_{n}\}$ of problem (2.1) converge to $x_{0}$. Therefore we have

$$ \bigl\Vert x_{n}-x^{\prime }_{n} \bigr\Vert = \bigl\Vert x_{n}-x_{0} + x_{0}-x^{\prime }_{n} \bigr\Vert \leq \Vert x_{n}-x_{0} \Vert + \bigl\Vert x_{0}-x^{\prime }_{n} \bigr\Vert \leq \epsilon , $$

which contradicts to (3.1), for some $\epsilon = r$.

Conversely, assume that $\{x_{n}\}$ is an approximating sequence of problem (2.1). Then there exist $u_{n} \in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n} \to 0$ such that

$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in \mathcal{D}, $$

(3.2)

which implies that $x_{n} \in \varOmega _{\epsilon _{n}}$. Since $\operatorname{diam} \varOmega _{\epsilon _{n}} \to 0$ as $\epsilon _{n} \to 0$, $\{x_{n}\}$ is a Cauchy sequence, which converges to some $x_{0} \in \mathcal{D}$ (because $\mathcal {{D}}$ is closed). Again since T is relaxed η-$\alpha _{g}$-P-monotone with respect to the first variable of Q and g on $\mathcal{D}$, it follows from Definition 2.3, for any $y \in {\mathcal {{D}}}$ and $u \in T(y)$, we have

$$\begin{aligned} &\bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) \\ &\quad \leq _{P} \bigl\langle Q(u,x_{n}), \eta \bigl(y,g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x _{n}),y\bigr) - \alpha _{g}(y-x_{n}). \end{aligned}$$

(3.3)

From the continuity of g, φ, η and $\alpha _{g}$, we have

$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ & = \lim_{n\to \infty }\bigl\{ \bigl\langle Q(u,x_{n}), \eta \bigl(y,g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) -\alpha _{g}(y-x_{n}) \bigr\} . \end{aligned}$$

This together with (3.3) shows that

$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ &\quad \geq _{P} \lim_{n\to \infty }\bigl\{ \bigl\langle Q(u_{n},x_{n}) , \eta \bigl(y,g(x _{n})\bigr) \bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)\bigr\} . \end{aligned}$$

(3.4)

Taking the limit in (3.2), we have

$$ \lim_{n\to \infty }\bigl\{ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)\bigr\} \nleq _{P^{0}} 0. $$

(3.5)

Combining (3.4) and (3.5) and using Lemma 2.5(ii), we get

$$ \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) - \alpha _{g}(y-x_{0}) \nleq _{P^{0}} 0. $$

Thus, by Lemma 2.8, there exist $x_{0} \in {\mathcal {{D}}}$ and $u_{0} \in T(x_{0})$ such that

$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}, $$

which implies that $x_{0} \in \varOmega$. It remains to prove that $x_{0}$ is a unique solution of the problem (2.1).

Assume contrary that $x_{1}$ and $x_{2}$ are two distinct solutions of (2.1). Then

$$ 0 < \Vert x_{1}-x_{2} \Vert \leq \operatorname{diam} \varOmega _{\epsilon }\to 0\quad \text{as } \epsilon \to 0. $$

This is absurd and the proof is completed. □

Corollary 3.6

Assume that all assumptions of Lemma 2.8 hold and $g, \varphi (\cdot ,y)$, $\eta (y,\cdot )$ and $\alpha _{g}$ are continuous functions for all $y\in {\mathcal {{D}}}$. Then the problem (2.1) is well-posed if and only if

$$ \varOmega \neq \emptyset $$

and

$$ \operatorname{diam} \varOmega _{\epsilon }\to 0\quad \textit{as } \epsilon \to 0. $$

Theorem 3.7

Let $\mathcal {{D}}$ be a closed convex subset of a real Banach space $\mathcal {{X}}$. Let $\mathcal {{Y}}$ be a real Banach space ordered by a nonempty closed convex pointed cone P with the apex at the origin and $P^{0} \neq \emptyset$. Assume that $Q:L(\mathcal {{X}}, \mathcal {{Y}})\times \mathcal {{D}} \to L( \mathcal {{X}}, \mathcal {{Y}})$ is a continuous mapping and $T:\mathcal {{D}} \to 2^{L(\mathcal {{X}},\mathcal {{Y})}}$ is a nonempty compact set-valued mapping. If the following conditions are satisfied:

• (i) $g: \mathcal {{D}}\to {\mathcal {{D}}}$ is continuous and P-convex;

• (ii) $\varphi :\mathcal {{D}} \times \mathcal {{D}} \to {\mathcal {{Y}}}$ is P-convex in the second variable and P-concave in the first argument with $\varphi (g(x),x)=0$, $\forall x \in {\mathcal {{D}}}$;

• (iii) $\eta : \mathcal {{X}} \times \mathcal {{X}} \to {\mathcal {{X}}}$ is an affine mapping in the first and second variables with $\eta (g(x),x)=0$, $\forall x \in {\mathcal {{D}}}$;

• (iv) $T:\mathcal {{D}} \to 2^{L(\mathcal {{X}},\mathcal {{Y}})}$ is $\mathfrak{H}$-hemicontinuous and relaxed η-$\alpha _{g}$-P-monotone with respect to first the variable of Q and g;

• (v) $\varphi (\cdot ,y)$, $\eta (y,\cdot )$ and $\alpha _{g}$ are continuous functions for all $y\in {\mathcal {{D}}}$.

Then problem (2.1) is well-posed if and only if it has a unique solution.

Proof

Assume that problem (2.1) is well-posed, then it has a unique solution.

Conversely, let (2.1) have a unique solution $x_{0}$. If the problem (2.1) is not well-posed, then there exists an approximating sequence $\{x_{n}\}$ of (2.1) which does not converge to $x_{0}$. Since $\{x_{n}\}$ is an approximating sequence, there exist $u_{n} \in T(x _{n})$ and a sequence of positive real numbers $\epsilon _{n} \to 0$ such that

$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)+ \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}. $$

(3.6)

Now, we prove that $\{x_{n}\}$ is bounded. Suppose that $\{x_{n}\}$ is not bounded. Then, without loss of generality, we can suppose that

$$ \Vert x_{n} \Vert \to +\infty\quad \text{as } n \to +\infty . $$

Let

$$ t_{n} = \frac{1}{ \Vert x_{n}-x_{0} \Vert } $$

and

$$ w_{n} = x_{0}+t_{n}(x_{n}-x_{0}). $$

Without loss of generality, we can assume that $t_{n} \in (0,1)$ and

$$ w_{n} \to w \neq x_{0}. $$

By the hypothesis, T is relaxed $\eta -\alpha _{g}-P$-monotone with respect to the first variable of Q and g; therefore, for any $x,y \in {\mathcal {{D}}}$, we have

$$ \bigl\langle Q(u,x_{0}) - Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0})\bigr)\bigr\rangle - \alpha _{g}(y-x_{0}) \geq _{P}0, \quad \forall u_{0} \in T(x_{0}), u \in T(y), $$

which implies that

$$\begin{aligned}& \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \\& \quad \leq _{P} \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x _{0}),y\bigr)-\alpha _{g}(y-x_{0}). \end{aligned}$$

(3.7)

Since $x_{0}$ is a solution of (2.1), there exists $u_{0}\in T(x_{0})$ such that

$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall y \in \mathcal{D}. $$

(3.8)

Combining (3.7) and (3.8) and, using Lemma 2.5(ii), we get

$$ \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \nleq _{P^{0}} 0. $$

(3.9)

From the continuity of g, φ, η and $\alpha _{g}$, we obtain

$$\begin{aligned} &\bigl\langle Q(u,w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y \bigr)-\alpha _{g}(y-w) \\ &\quad = \lim_{n\to \infty }\bigl\{ Q(u,w_{n}), \eta \bigl(y,g(w_{n})\bigr) + \varphi \bigl(g(w _{n}),y\bigr)- \alpha _{g}(y-w_{n})\bigr\} . \end{aligned}$$

Since η is affine in the second variable, φ is P-concave in the first variable and using $w_{n} = x_{0} + t_{n}(x _{n}-x_{0})$, the above equation can be rewritten as

$$\begin{aligned} &\bigl\langle Q(u,w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y \bigr)-\alpha _{g}(y-w) \\ &\quad \geq _{p} \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y \bigr)-\alpha _{g}(y-x_{0}). \end{aligned}$$

(3.10)

Using (3.9), (3.10) and Lemma 2.5(ii), we obtain

$$ \bigl\langle Q(u,w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y \bigr) - \alpha _{g}(y-w) \nleq _{P^{0}} 0. $$

Therefore, by Lemma 2.8, there exist $w \in {\mathcal {{D}}}$ and $w_{0} \in T(w)$ such that

$$ \bigl\langle Q(w_{0},w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y\bigr)\leq _{P ^{0}} 0, \quad \forall y \in {\mathcal {{D}}}. $$

The above inequality implies that w is also a solution of (2.1), which contradicts the uniqueness of $x_{0}$. Hence, $\{x_{n}\}$ is a bounded sequence having a convergent subsequence $\{x_{n_{\ell }}\}$ which converges to x̄ (say) as $\ell \to \infty$. Therefore from the definition of relaxed η-$\alpha _{g}$-P-monotonicity, for any $x_{n_{\ell }} , y \in {\mathcal {{D}}}$, we have

$$ \bigl\langle Q(u,y) - Q(u_{n_{\ell }},y) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle - \alpha _{g}(y-x_{n_{\ell }})) \geq _{P} 0,\quad \forall u_{n_{\ell }} \in T(x_{n_{\ell }}), u \in T(y). $$

This implies that

$$\begin{aligned} &\bigl\langle Q(u_{n_{\ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }})\bigr) \bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y \bigr) \\ &\quad \leq _{P} \bigl\langle Q(u,x_{n_{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr) \bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y \bigr)-\alpha _{g}(y-x_{n_{\ell }}). \end{aligned}$$

(3.11)

Again from the continuity of g, φ, η and $\alpha _{g}$, we have

$$\begin{aligned} &\bigl\langle Q(u,\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr)-\alpha _{g}(y -\bar{x}) \\ &\quad =\lim_{\ell \to \infty } \bigl\{ \bigl\langle Q(u,x_{n_{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y\bigr)-\alpha _{g}(y-x_{n_{\ell }}) \bigr\} . \end{aligned}$$

This together with (3.11) shows that

$$\begin{aligned} &\bigl\langle Q(u,\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) - \alpha _{g}(y-\bar{x}) \\ &\quad \geq _{P} \lim_{\ell \to \infty } \bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n _{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{ \ell }}),y\bigr)\bigr\} . \end{aligned}$$

(3.12)

By virtue of (3.6), we can obtain

$$ \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y\bigr)\bigr\} \nleq _{P^{0}} 0. $$

(3.13)

From (3.12), (3.13) and Lemma 2.5(ii), we get

$$ \bigl\langle Q(u,\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g(\bar{x}),y\bigr)- \alpha _{g}(y-\bar{x}) \nleq _{P^{0}} 0. $$

Thus, by Lemma 2.8, there exist $\bar{x} \in {\mathcal {{D}}}$ and $\bar{u} \in T(\bar{x})$ such that

$$ \bigl\langle Q(\bar{u},\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) \nleq _{P^{0}} 0, $$

which shows that x̄ is a solution to (2.1). Hence,

$$ x_{n_{\ell }} \to \bar{x}, \quad \textit{i.e.},\quad x_{n_{\ell }}\to x_{0}. $$

Since $\{x_{n}\}$ is an approximating sequence, we have

$$ x_{n} \to x_{0}. $$

The proof of Theorem 3.7 is completed. □

Example 3.8

Let $\mathcal {{X}} = \mathcal {{Y}} = {\mathbb{R}}$, $\mathcal {{D}} = [0, 1]$ and $P = [0,\infty )$. Let us define the mappings $T:\mathcal {{D}} \to 2^{L(\mathcal {{X}},\mathcal {{Y}})}$, $\varphi :\mathcal {{D}}\times \mathcal {{D}} \to {\mathcal {{Y}}}$, $\eta : \mathcal {{X}} \times \mathcal {{X}} \to {\mathcal {{X}}}$, and $Q:L(\mathcal {{X}},\mathcal {{Y}})\times \mathcal {{D}} \to L(\mathcal {{X}},\mathcal {{Y}})$ as follows:

$$ \textstyle\begin{cases} T(x) = \{u :{\mathbb{R}} \to {\mathbb{R}} \mid u \text{ is a continuous linear mapping such that } u(x) = -x\}; \\ g(x)=x; \\ \varphi (g(x),y)= y - x; \\ \eta (y,g(x)) =\frac{1}{2}(y-x); \\ Q(v,y) = v; \\ \alpha _{g} = -x^{2}. \end{cases} $$

In this case, the generalized $(\eta , g, \varphi )$-mixed vector variational-type inequality problem (2.1) is to find $x \in {\mathcal {{D}}}$ and $u\in T(x)$ such that

$$ \biggl\langle u, \frac{1}{2}(x -y)\biggr\rangle + y-x \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}. $$

(∗)

It easy to see that $\varOmega = \{0\}$ and T is relaxed η-$\alpha _{g}$-P-monotone with respect to the first variable of Q and g, and all conditions in Theorem 3.7 are satisfied. Therefore the problem (∗) is well-posed.

Theorem 3.9

Suppose that all the conditions in Lemma 2.8 are satisfies. Further, assume that $\mathcal {{D}}$ is a compact set and $g, \varphi (\cdot ,y)$, $\eta (y,\cdot )$, $\alpha _{g}$ are continuous functions for all $y \in {\mathcal {{D}}}$. Then problem (2.1) is well-posed if and only if the solution set Ω is nonempty.

Proof

Suppose that problem (2.1) is well-posed. Then its solution set Ω is nonempty. Conversely, let $\{x_{n}\}$ be an approximating sequence of problem (2.1). Then there exist $u_{n} \in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n} \to 0$ such that

$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y, g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y \bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}. $$

(3.14)

By the hypothesis, Ω is compact; hence, $\{x_{n}\}$ has a subsequence $\{x_{n_{\ell }}\}$ converging to some point $x_{0} \in {\mathcal {{D}}}$. Since T is relaxed η-$\alpha _{g}$-P-monotone with respect to the first variable of Q and g, by Definition 2.3, for any $y \in {\mathcal {{D}}}$, we have

$$\begin{aligned}& \bigl\langle Q(u,x_{n_{\ell }})- Q(u_{n_{\ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x _{n_{\ell }})\bigr)\bigr\rangle - \alpha _{g}(y-x_{n_{\ell }}) \\& \quad \geq _{P} 0,\quad \forall x_{n_{\ell }}\in {\mathcal {{D}}}, u_{n_{\ell }} \in T(x_{n _{\ell }}), u \in T(y), \end{aligned}$$

which implies

$$\begin{aligned} &\lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u,x_{n_{\ell }}), \eta \bigl(y,g(x_{n _{\ell }})\bigr)\bigr\rangle +\varphi \bigl(g(x_{n_{\ell }}),y \bigr)-\alpha _{g}(y-x_{n _{\ell }})\bigr\} \\ &\quad \geq _{P} \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{ \ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle +\varphi \bigl(g(x _{n_{\ell }}),y\bigr)\bigr\} . \end{aligned}$$

Since g, η, φ, $\alpha _{g}$ are continuous,

$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle +\varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ &\quad = \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u,x_{n _{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle +\varphi \bigl(g(x_{n_{\ell }}),y\bigr)- \alpha _{g}(y-x_{n_{\ell }}) \bigr\} . \end{aligned}$$

Using the above inequality, we obtain

$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ & \quad \geq _{P} \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n _{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{ \ell }}),y\bigr)\bigr\} . \end{aligned}$$

(3.15)

By virtue of (3.14), it can be written as

$$ \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n_{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y\bigr)\bigr\} \nleq _{P^{0}} 0. $$

(3.16)

It follows from (3.15), (3.16) and Lemma 2.5(ii) that

$$ \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \nleq _{P^{0}} 0. $$

Thus, by Lemma 2.8, there exist $x_{0} \in {\mathcal {{D}}}$ and $u_{0} \in T(x_{0})$ such that

$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0. $$

This implies that $x_{0} \in \varOmega$.

The proof is completed. □

Example 3.10

Let $\mathcal {{X}}=\mathcal {{Y}}= {\mathbb{R}} ^{2}$, $\mathcal {{D}}=[0,1]\times [0,1]$ and $P=[0,\infty )\times [0, \infty )$. Let us define the mappings $T: \mathcal {{D}} \to 2^{L( \mathcal {{X}},\mathcal {{Y}})}$, $\varphi :\mathcal {{D}}\times \mathcal {{D}} \to {\mathcal {{Y}}}$, $\eta :\mathcal {{X}}\times \mathcal {{X}} \to {\mathcal {{X}}}$, and $Q:L(\mathcal {{X}},\mathcal {{Y}})\times \mathcal {{D}} \to L(\mathcal {{X}},\mathcal {{Y}})$ as follows:

$$ \textstyle\begin{cases} T(x)=\{w,z:{\mathbb{R}}^{2}\to {\mathbb{R}}\mid w,z \text{ are continuous linear mappings} \\ \hphantom{T(x)={}}\text{such that } w(x_{1},x_{2})=x_{1}, z(x_{1},x_{2})=x_{2}\}; \\ g(x)=x; \\ \varphi (g(x),y)=y-x; \\ \eta (y,g(x)) = y-x; \\ Q(u,x)=-u; \\ \alpha _{g}=0. \end{cases} $$

In this case, the generalized $(\eta , g, \varphi )$-mixed vector variational-type inequality problem (2.1) is to find $x\in {\mathcal {{D}}}$ and $u\in T(x)$ such that

$$ \langle -u, x-y\rangle + y-x \nleq _{P^{0}} 0, \quad \forall y\in {\mathcal {{D}}}. $$

(∗∗)

Clearly, $\varOmega =[0,1]\times [0,1]$. It can be easily verified that T is relaxed η-$\alpha _{g}$-P-monotone with respect to the first variable of Q and g, and all conditions in Theorem 3.9 are satisfies. Hence, problem (∗∗) is well-posed.

Theorem 3.11

Assume that all conditions in Lemma 2.8 are satisfied and assume that $g, \varphi (\cdot ,y)$, $\eta (y,\cdot )$, $\alpha _{g}$ are continuous functions for all $y\in {\mathcal {{D}}}$. If there exists some $\epsilon > 0$ such that $\varOmega _{\epsilon } \neq \emptyset$ and is bounded. Then problem (2.1) is well-posed.

Proof

Let $\epsilon > 0$ such that

$$ \varOmega _{\epsilon }\neq \emptyset $$

and suppose $\{x_{n}\}$ is an approximating sequence of problem (2.1). Then there exist $u_{n}\in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n}\to 0$ such that

$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)+ \epsilon _{n} e \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}, $$

which implies that

$$ x_{n} \in \varOmega _{\epsilon },\quad \forall n > m. $$

Therefore, $\{x_{n}\}$ is a bounded sequence which has a convergent subsequence $\{x_{n_{\ell }}\}$ converging to $x_{0}$ as $\ell \to \infty$. Following lines similar to the proof of Theorem 3.9, we get $x_{0} \in \varOmega$. The proof is completed. □

4 Well-posedness of optimization problems with constraints

This section is devoted to a study of the well-posedness of optimization problems with generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality constraints:

$$ \begin{aligned} &P \text{-minimize } \varPsi (x) \\ & \quad \text{subject to } x \in \varOmega , \end{aligned} $$

(4.1)

where $\varPsi :\mathcal {{D}} \to {\mathbf {{R}}}$ is a function, and Ω is the solution set of problem (2.1).

Denote by ζ the solution set of (4.1), i.e.,

$$\begin{aligned} \zeta =& \Bigl\{ x \in {\mathcal {{D}}} \bigm| \exists u\in T(x) \text{ such that } \varPsi (x) \leq _{P} \inf_{y\in \varOmega }\varPsi (y) \text{ and} \\ &\bigl\langle Q(u,x), \eta \bigl(y,g(x)\bigr)\bigr\rangle + \varphi \bigl(g(x),y\bigr) \nleq _{P ^{0}} 0, \forall y \in {\mathcal {{D}}}\Bigr\} . \end{aligned}$$

Definition 4.1

A sequence $\{x_{n}\} \in {\mathcal {{D}}}$ is said to be an approximating sequence for problem (4.1), if

• (i) $\lim_{n\to \infty }\sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega }\varPsi (y)$,

• (ii) there exist $u_{n}\in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n} \to 0$ such that

  $$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)+ \epsilon _{n} e \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}. $$

For $\delta , \epsilon \geq 0$, we denote the approximating solution set of (4.1) by $\zeta (\delta ,\epsilon )$, i.e.,

$$\begin{aligned} \zeta (\delta ,\epsilon ) =&\Bigl\{ x \in {\mathcal {{D}}} \bigm| \exists u \in T(x) \text{ such that } \varPsi (x) \leq _{P} \inf _{y\in \varOmega }\varPsi (y) + \delta \text{ and} \\ &\bigl\langle Q(u,x), \eta \bigl(y,g(x)\bigr)\bigr\rangle + \varphi \bigl(g(x),y\bigr) + \epsilon e \nleq _{P^{0}} 0, \forall y \in {\mathcal {{D}}}\Bigr\} . \end{aligned}$$

Remark 4.2

It is obvious that $\zeta = \zeta (\delta , \epsilon )$ when $(\delta ,\epsilon )=(0,0)$ and

$$ \zeta \subseteq \zeta (\delta ,\epsilon ),\quad \forall \delta , \epsilon > 0. $$

Theorem 4.3

Assume that all assumptions of Theorem 3.5 are satisfies and Ψ is lower semicontinuous. Then (4.1) is well-posed if and only if

$$ \zeta (\delta ,\epsilon ) \neq \emptyset , \quad \forall \delta ,\epsilon > 0 $$

and

$$ \operatorname{diam} \zeta (\delta ,\epsilon ) \to 0 \quad \textit{as } (\delta , \epsilon ) \to (0,0). $$

Proof

The necessary part directly follows from the proof of Theorem 3.5, so it is omitted. Conversely, suppose that $\{x_{n}\}$ is an approximating sequence of (4.1). Then there exist $u_{n}\in T(x _{n})$ and a sequence of positive real number $\epsilon _{n} \to 0$ such that

$$\begin{aligned}& \lim_{n\to \infty }\sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y), \end{aligned}$$

(4.2)

$$\begin{aligned}& \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in \mathcal{D}, \end{aligned}$$

(4.3)

which implies that

$$ x_{n} \in \zeta (\delta _{n},\epsilon _{n}), \quad \text{for } \delta _{n} \to 0. $$

Since

$$ \operatorname{diam} \zeta (\delta ,\epsilon ) \to 0 \quad \text{as } (\delta , \epsilon ) \to (0,0), $$

and $\{x_{n}\}$ is a Cauchy sequence converging to $x_{0} \in {\mathcal {{D}}}$ (because $\mathcal {{D}}$ is closed). By the same argument as in Theorem 3.5, we get

$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall u_{0} \in T(x_{0}), y \in \mathcal{D}. $$

(4.4)

Since Ψ is lower semicontinuous,

$$ \varPsi (x_{0}) \leq _{P} \lim_{n\to \infty } \inf \varPsi (x_{n}) \leq _{P} \lim_{n\to \infty } \sup \varPsi (x_{n}). $$

By using (4.1), the above inequality reduces to

$$ \varPsi (x_{0}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y). $$

(4.5)

Thus, from (4.3) and (4.4), we conclude that $x_{0}$ solve (4.1). The uniqueness of $x_{0}$ directly follows from the assumption

$$ \operatorname{diam} \zeta (\delta ,\epsilon ) \to 0 \quad \text{as } (\delta , \epsilon ) \to (0,0). $$

This completes the proof. □

Example 4.4

Let $\mathcal {{X}}= \mathcal {{Y}}={\mathbb{R}}$, $\mathcal {{D}}=[0,1]$ and $P=[0,\infty )$. Let us define the mappings $\varPsi :\mathcal {{D}}\to {\mathbf {{R}}}$, $T: \mathcal {{D}} \to 2^{L(\mathcal {{X}},\mathcal {{Y}})}$, $\varphi : \mathcal {{D}} \times \mathcal {{D}} \to {\mathcal {{Y}}}$, $\eta : \mathcal {{X}} \times \mathcal {{X}} \to {\mathcal {{X}}}$, and $Q:L(\mathcal {{X}},\mathcal {{Y}})\times \mathcal {{D}} \to L(\mathcal {{X}},\mathcal {{Y}})$ as follows:

$$ \textstyle\begin{cases} \varPsi (x)= \vert x^{3} \vert ; \\ T(x)=\{u:{\mathbb{R}} \to {\mathbb{R}} \mid u \text{ is a continuous linear mapping such that } u(x)=-x \}; \\ g(x)=x; \\ \varphi (g(x),y)=y-x; \\ \eta (g(x),y)=\frac{1}{2}(y-x); \\ Q(v,x)=v; \\ \alpha _{g}=-x^{2}. \end{cases} $$

Consider the optimization problem with generalized $(\eta ,g,\varphi )$-mixed vector variational-type inequality constraints:

$$ \begin{aligned} &P\text{-minimize } \bigl\vert x^{3} \bigr\vert \\ &\quad \text{subject to } x \in \varOmega , \end{aligned} $$

(4.6)

where

$$ \varOmega = \biggl\{ x \in {\mathcal {{D}}}\Bigm| \exists u \in T(x) \text{ such that } \biggl\langle u, \frac{1}{2}(x-y)\biggr\rangle + y-x \nleq _{P^{0}} 0, \forall y \in {\mathcal {{D}}}\biggr\} . $$

We see that $\varOmega =\{0\}$. Since

$$ \zeta (\delta ,\epsilon )=\biggl\{ x \in {\mathcal {{D}}} \Bigm| \exists u \in T(x) \text{ such that } \bigl\vert x^{3} \bigr\vert \leq _{P} \delta \text{ and } (y-x) \biggl(1+\frac{x}{2} \biggr)+ \epsilon \nleq _{P^{0}} 0, \forall y \in {\mathcal {{D}}}\biggr\} , $$

we have

$$ \operatorname{diam} \zeta (\delta ,\epsilon )\to 0 \quad \text{as } (\delta , \epsilon )\to (0,0). $$

It is easily verified that T is relaxed η-$\alpha _{g}$-P-monotone with respect to the first variable of Q and g, and all assumptions of Theorem 4.3 are satisfied. Hence (4.6) is well-posed.

Theorem 4.5

Let all conditions in Theorem 3.7 hold and let Ψ be lower semicontinuous. Then the problem (4.1) is well-posed if and only if it has a unique solution.

Proof

The necessary condition is obvious. Conversely, let (4.1) have a unique solution $x_{0}$. Then

$$\begin{aligned}& \varPsi (x_{0})=\inf_{y\in \varOmega } \varPsi (y), \\& \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall u_{0}\in T(x_{0}), y \in {\mathcal {{D}}}. \end{aligned}$$

Let $\{x_{n}\}$ be an approximating sequence. Then there exist $u_{n}\in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n}\to 0$ such that

$$\begin{aligned}& \lim_{n\to \infty }\sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y), \\& \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)+ \epsilon _{n} e \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}. \end{aligned}$$

Now, following lines similar to the proof of Theorem 3.7, we find that the sequence $\{x_{n}\}$ has a subsequence $\{x_{n_{\ell }}\}$ converging to x̄, for any $\bar{x} \in {\mathcal {{D}}}$ and

$$ \bigl\langle Q(\bar{u},\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) \nleq _{P^{0}} 0, \quad \forall \bar{u} \in T(\bar{x}), y \in {\mathcal {{D}}}. $$

(4.7)

Since Ψ is lower semicontinuous, therefore,

$$ \varPsi (\bar{x}) \leq _{P} \lim_{\ell \to \infty } \inf \varPsi (x_{n_{ \ell }}) \leq _{P} \lim_{\ell \to \infty } \sup \varPsi (x_{n_{\ell }}) \leq _{P} \inf _{y\in \varOmega } \varPsi (y). $$

(4.8)

Thus, from (4.7) and (4.8), we conclude that $\bar{x} \in \zeta$, and the proof is completed. □

Theorem 4.6

Assume that all assumptions of Theorem 4.5 are satisfies and Ψ is lower semicontinuous, and there exists some $\epsilon > 0$ such that $\zeta (\epsilon ,\epsilon ) \neq \emptyset$, and it is bounded. Then (4.1) is well-posed.

Proof

Let $\epsilon > 0$ such that

$$ \zeta (\epsilon ,\epsilon ) \neq \emptyset $$

and suppose $\{x_{n}\}$ is an approximating sequence of problem (2.1). Then

• (i) $\lim_{n\to \infty } \sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y)$,

• (ii) there exist $u_{n}\in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n} \to 0$ such that

  $$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}, n \in {\mathbb{N}}, $$

  which implies that for some positive integer m

  $$ x_{n} \in \zeta (\epsilon ,\epsilon ),\quad \forall n > m. $$

Therefore, $\{x_{n}\}$ is a bounded sequence and there exists a subsequence $\{x_{n_{\ell }}\}$ such that $\{x_{n_{\ell }}\}$ converges to $x_{0}$ as $\ell \to \infty$. Following the lines similar to the proof of Theorem 4.5, we conclude that $x_{0} \in \zeta$. Hence, (4.1) is well-posed and the proof is completed. □

5 Well-posedness of optimization problems by using well-posedness of constraints

In this section, we derive the well-posedness of problem (4.1) by using the well-posedness of problem (2.1).

Theorem 5.1

Let $\mathcal {{D}}$ be a nonempty compact set and Ψ be lower semicontinuous. Suppose problem (4.1) has a unique solution. If problem (2.1) is well-posed, then problem (4.1) is also well-posed.

Proof

If problem (4.1) has a unique solution $x_{0}$, and $\{x_{n}\}$ is an approximating sequence for problem (4.1), then there exist $u_{n} \in T(x_{n})$ and a sequence of positive real numbers $\epsilon _{n} \to 0$ such that

$$\begin{aligned}& \lim_{n\to \infty }\sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y), \\& \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}. \end{aligned}$$

Since $\mathcal {{D}}$ is compact, there exists a subsequence $\{x_{n_{\ell }}\}$ of $\{x_{n}\}$ such that $\{x_{n_{\ell }}\}$ converges to a x̄ (say) as $\ell \to \infty$. Since problem (2.1) is well-posed, x̄ solves (2.1), i.e.,

$$ \bigl\langle Q(\bar{u},\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) \nleq _{P^{0}} 0,\quad \forall \bar{u} \in T(\bar{x}), y \in {\mathcal {{D}}}. $$

(5.1)

Since Ψ is lower semicontinuous, we have

$$ \varPsi (\bar{x}) \leq _{P} \lim_{\ell \to \infty }\inf \varPsi (x_{n_{ \ell }}) \leq _{P} \lim_{\ell \to \infty } \sup \varPsi (x_{n_{\ell }}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y). $$

(5.2)

Thus, from (5.1) and (5.2) we conclude that x̄ solves problem (4.1). But (4.1) has a unique solution $x_{0}$; therefore,

$$ \bar{x} = x_{0} \quad \text{and}\quad x_{n} \to x_{0}. $$

Hence, (4.1) is well-posed. The proof is completed. □