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

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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.

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.

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 \(x, y \in \mathcal{Y}\):

  1. (i)

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

  2. (ii)

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

  3. (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

  1. (i)

    P-convex, if

    $$ \phi \bigl(\mu x+(1-\mu )y\bigr) \leq _{P} \mu \phi (x) + (1-\mu ) \phi (y),\quad \forall x,y \in {\mathcal {{D}}}, \mu \in [0,1]; $$
  2. (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

  1. (i)

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

  2. (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:

  1. (i)

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

  2. (ii)

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

  3. (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:

  1. (a)

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

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

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

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

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

  1. (i)

    there exists a unique solution \(x_{0}\) of problem (2.1);

  2. (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

  1. (i)

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

  2. (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:

  1. (i)

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

  2. (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 \(\varphi (g(x),x)=0\), \(\forall x \in {\mathcal {{D}}}\);

  3. (iii)

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

  4. (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;

  5. (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 (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 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. □

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

  1. (i)

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

  2. (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 , 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

  1. (i)

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

  2. (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. □

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 (say) as \(\ell \to \infty \). Since problem (2.1) is well-posed, 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 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. □

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Acknowledgements

The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article.

Availability of data and materials

The data sets used during the current study are available from the corresponding author on request.

Funding

This work was supported by the Natural Science Foundation of China (No. 11361170) and supported by the Center for General Education, China Medical University, Taichung, Taiwan; and supported by the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (No. 2018JY0334).

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Correspondence to Shih-sen Chang.

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Chang, S., Salahuddin, Wang, L. et al. Well-posedness for generalized \((\eta ,g,\varphi )\)-mixed vector variational-type inequality and optimization problems. J Inequal Appl 2019, 238 (2019) doi:10.1186/s13660-019-2194-4

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MSC

  • 49K40
  • 54C60
  • 90C33
  • 47H09
  • 47J20
  • 54H25

Keywords

  • Generalized \((\eta ,g,\varphi )\)-mixed vector variational-type inequality problems
  • Optimization problems
  • Well-posedness
  • Relaxed η-\(\alpha _{g}\)-P-monotonicity