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 [7‐12] 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.
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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 \(x, y \in \mathcal{Y}\): where \(P^{0}\) denotes the interior of P.
(i)
\(x \leq _{P} y \Leftrightarrow y - x \in P\);
(ii)
\(x \nleq _{P} y \Leftrightarrow y - x \notin P\);
(iii)
\(x \leq _{P^{0}} y \Leftrightarrow y - x \in P^{0}\);
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
Denote by
the solution set of the problem (2.1).
$$ \bigl\langle Q(u,x) , \eta \bigl(y, g(x)\bigr)\bigr\rangle + \varphi \bigl(g(x),y\bigr) \nleq _{P^{0}}0, \quad \forall y \in \mathcal{D}. $$
(2.1)
$$ \varOmega = \bigl\{ x \in {\mathcal {{D}}}: \exists u \in T(x) \text{ such that } \bigl\langle Q(u,x), \eta \bigl(y,g(x)\bigr)\bigr\rangle + \varphi \bigl(g(x),y \bigr) \nleq _{P^{0}} 0, \forall y \in \mathcal{D}\bigr\} $$
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
$$ \phi \bigl(\mu x +(1-\mu )y\bigr) \geq _{P} \mu \phi (x)+(1-\mu ) \phi (y),\quad \forall x,y\in {\mathcal {{D}}},\mu \in [0,1]. $$
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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
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.
$$ \bigl\langle Q(u,\cdot ) - Q(v,\cdot ), \eta \bigl(g(x),y\bigr)\bigr\rangle - \alpha _{g}(x-y) \geq _{P} 0,\quad \forall x,y \in { \mathcal {{D}}}, u \in T(x), v \in T(y), $$
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
which is defined by
If
A, B
are compact sets in
\(\mathcal{X}\), then for each
\(u \in A\)
there exists
\(v \in B\)
such that
$$ d(u,v) = \Vert u - v \Vert , $$
$$ {\mathfrak{H}}(A,B ) = \max \Bigl\{ \sup_{u\in A}\inf _{v\in B} \Vert u-v \Vert , \sup_{v\in B} \inf_{u\in A} \Vert u-v \Vert \Bigr\} , \quad \forall A,B \in CB(\mathcal {{X}}). $$
$$ \Vert u - v \Vert \leq {\mathfrak{H}}(A,B). $$
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
where \({\mathfrak{H}}\) is the Hausdorff metric defined on \(CB(L( \mathcal {{X}},\mathcal {{Y}}))\).
$$ {\mathfrak{H}}\bigl(T\bigl(x + \tau (y-x)\bigr),T(x)\bigr) \to 0 \quad \text{as } \tau \to 0^{+}, \forall x,y \in {\mathcal {{D}}}, \tau \in (0,1), $$
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:
then the following two problems are equivalent:
(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;
(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
Corollary 3.3
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
and
$$ \varOmega _{\epsilon }\neq \emptyset , \quad \forall \epsilon > 0 $$
$$ \operatorname{diam} \varOmega _{\epsilon }\to 0 \quad \textit{as } \epsilon \to 0. $$
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
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
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
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
which contradicts to (3.1), for some \(\epsilon = r\).
$$ \operatorname{diam} \varOmega _{\epsilon }\nrightarrow 0 \quad \text{as } \epsilon \to 0, $$
$$ \bigl\Vert x_{n}-x^{\prime }_{n} \bigr\Vert > r, \quad \forall n \geq m. $$
(3.1)
$$\begin{aligned} &\bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}, \\ &\bigl\langle Q\bigl(u^{\prime }_{n},x^{\prime }_{n} \bigr), \eta \bigl(y, g\bigl(x^{\prime }_{n}\bigr)\bigr)\bigr\rangle + \varphi \bigl(g\bigl(x^{\prime }_{n}\bigr),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in { \mathcal {{D}}}. \end{aligned}$$
$$ \bigl\Vert x_{n}-x^{\prime }_{n} \bigr\Vert = \bigl\Vert x_{n}-x_{0} + x_{0}-x^{\prime }_{n} \bigr\Vert \leq \Vert x_{n}-x_{0} \Vert + \bigl\Vert x_{0}-x^{\prime }_{n} \bigr\Vert \leq \epsilon , $$
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
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
From the continuity of g, φ, η and \(\alpha _{g}\), we have
This together with (3.3) shows that
Taking the limit in (3.2), we have
Combining (3.4) and (3.5) and using Lemma 2.5(ii), we get
Thus, by Lemma 2.8, there exist \(x_{0} \in {\mathcal {{D}}}\) and \(u_{0} \in T(x_{0})\) such that
which implies that \(x_{0} \in \varOmega \). It remains to prove that \(x_{0}\) is a unique solution of the problem (2.1).
$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in \mathcal{D}, $$
(3.2)
$$\begin{aligned} &\bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) \\ &\quad \leq _{P} \bigl\langle Q(u,x_{n}), \eta \bigl(y,g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x _{n}),y\bigr) - \alpha _{g}(y-x_{n}). \end{aligned}$$
(3.3)
$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ & = \lim_{n\to \infty }\bigl\{ \bigl\langle Q(u,x_{n}), \eta \bigl(y,g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) -\alpha _{g}(y-x_{n}) \bigr\} . \end{aligned}$$
$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ &\quad \geq _{P} \lim_{n\to \infty }\bigl\{ \bigl\langle Q(u_{n},x_{n}) , \eta \bigl(y,g(x _{n})\bigr) \bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)\bigr\} . \end{aligned}$$
(3.4)
$$ \lim_{n\to \infty }\bigl\{ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)\bigr\} \nleq _{P^{0}} 0. $$
(3.5)
$$ \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) - \alpha _{g}(y-x_{0}) \nleq _{P^{0}} 0. $$
$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}, $$
Assume contrary that \(x_{1}\) and \(x_{2}\) are two distinct solutions of (2.1). Then
This is absurd and the proof is completed. □
$$ 0 < \Vert x_{1}-x_{2} \Vert \leq \operatorname{diam} \varOmega _{\epsilon }\to 0\quad \text{as } \epsilon \to 0. $$
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
and
$$ \varOmega \neq \emptyset $$
$$ \operatorname{diam} \varOmega _{\epsilon }\to 0\quad \textit{as } \epsilon \to 0. $$
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:
Then problem (2.1) is well-posed if and only if it has a unique solution.
(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
\(y\in {\mathcal {{D}}}\).
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
Let
and
Without loss of generality, we can assume that \(t_{n} \in (0,1)\) and
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
which implies that
Since \(x_{0}\) is a solution of (2.1), there exists \(u_{0}\in T(x_{0})\) such that
Combining (3.7) and (3.8) and, using Lemma 2.5(ii), we get
From the continuity of g, φ, η and \(\alpha _{g}\), we obtain
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
Using (3.9), (3.10) and Lemma 2.5(ii), we obtain
Therefore, by Lemma 2.8, there exist \(w \in {\mathcal {{D}}}\) and \(w_{0} \in T(w)\) such that
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
This implies that
Again from the continuity of g, φ, η and \(\alpha _{g}\), we have
This together with (3.11) shows that
By virtue of (3.6), we can obtain
From (3.12), (3.13) and Lemma 2.5(ii), we get
Thus, by Lemma 2.8, there exist \(\bar{x} \in {\mathcal {{D}}}\) and \(\bar{u} \in T(\bar{x})\) such that
which shows that x̄ is a solution to (2.1). Hence,
Since \(\{x_{n}\}\) is an approximating sequence, we have
The proof of Theorem 3.7 is completed. □
$$ \Vert x_{n} \Vert \to +\infty\quad \text{as } n \to +\infty . $$
$$ t_{n} = \frac{1}{ \Vert x_{n}-x_{0} \Vert } $$
$$ w_{n} = x_{0}+t_{n}(x_{n}-x_{0}). $$
$$ w_{n} \to w \neq x_{0}. $$
$$ \bigl\langle Q(u,x_{0}) - Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0})\bigr)\bigr\rangle - \alpha _{g}(y-x_{0}) \geq _{P}0, \quad \forall u_{0} \in T(x_{0}), u \in T(y), $$
$$\begin{aligned}& \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \\& \quad \leq _{P} \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x _{0}),y\bigr)-\alpha _{g}(y-x_{0}). \end{aligned}$$
(3.7)
$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall y \in \mathcal{D}. $$
(3.8)
$$ \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \nleq _{P^{0}} 0. $$
(3.9)
$$\begin{aligned} &\bigl\langle Q(u,w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y \bigr)-\alpha _{g}(y-w) \\ &\quad = \lim_{n\to \infty }\bigl\{ Q(u,w_{n}), \eta \bigl(y,g(w_{n})\bigr) + \varphi \bigl(g(w _{n}),y\bigr)- \alpha _{g}(y-w_{n})\bigr\} . \end{aligned}$$
$$\begin{aligned} &\bigl\langle Q(u,w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y \bigr)-\alpha _{g}(y-w) \\ &\quad \geq _{p} \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y \bigr)-\alpha _{g}(y-x_{0}). \end{aligned}$$
(3.10)
$$ \bigl\langle Q(u,w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y \bigr) - \alpha _{g}(y-w) \nleq _{P^{0}} 0. $$
$$ \bigl\langle Q(w_{0},w), \eta \bigl(y,g(w)\bigr)\bigr\rangle + \varphi \bigl(g(w),y\bigr)\leq _{P ^{0}} 0, \quad \forall y \in {\mathcal {{D}}}. $$
$$ \bigl\langle Q(u,y) - Q(u_{n_{\ell }},y) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle - \alpha _{g}(y-x_{n_{\ell }})) \geq _{P} 0,\quad \forall u_{n_{\ell }} \in T(x_{n_{\ell }}), u \in T(y). $$
$$\begin{aligned} &\bigl\langle Q(u_{n_{\ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }})\bigr) \bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y \bigr) \\ &\quad \leq _{P} \bigl\langle Q(u,x_{n_{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr) \bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y \bigr)-\alpha _{g}(y-x_{n_{\ell }}). \end{aligned}$$
(3.11)
$$\begin{aligned} &\bigl\langle Q(u,\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr)-\alpha _{g}(y -\bar{x}) \\ &\quad =\lim_{\ell \to \infty } \bigl\{ \bigl\langle Q(u,x_{n_{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y\bigr)-\alpha _{g}(y-x_{n_{\ell }}) \bigr\} . \end{aligned}$$
$$\begin{aligned} &\bigl\langle Q(u,\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) - \alpha _{g}(y-\bar{x}) \\ &\quad \geq _{P} \lim_{\ell \to \infty } \bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n _{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{ \ell }}),y\bigr)\bigr\} . \end{aligned}$$
(3.12)
$$ \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y\bigr)\bigr\} \nleq _{P^{0}} 0. $$
(3.13)
$$ \bigl\langle Q(u,\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g(\bar{x}),y\bigr)- \alpha _{g}(y-\bar{x}) \nleq _{P^{0}} 0. $$
$$ \bigl\langle Q(\bar{u},\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) \nleq _{P^{0}} 0, $$
$$ x_{n_{\ell }} \to \bar{x}, \quad \textit{i.e.},\quad x_{n_{\ell }}\to x_{0}. $$
$$ x_{n} \to x_{0}. $$
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:
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
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.
$$ \textstyle\begin{cases} T(x) = \{u :{\mathbb{R}} \to {\mathbb{R}} \mid u \text{ is a continuous linear mapping such that } u(x) = -x\}; \\ g(x)=x; \\ \varphi (g(x),y)= y - x; \\ \eta (y,g(x)) =\frac{1}{2}(y-x); \\ Q(v,y) = v; \\ \alpha _{g} = -x^{2}. \end{cases} $$
$$ \biggl\langle u, \frac{1}{2}(x -y)\biggr\rangle + y-x \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}. $$
(∗)
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
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
which implies
Since g, η, φ, \(\alpha _{g}\) are continuous,
Using the above inequality, we obtain
By virtue of (3.14), it can be written as
It follows from (3.15), (3.16) and Lemma 2.5(ii) that
Thus, by Lemma 2.8, there exist \(x_{0} \in {\mathcal {{D}}}\) and \(u_{0} \in T(x_{0})\) such that
This implies that \(x_{0} \in \varOmega \).
$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y, g(x_{n})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y \bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}. $$
(3.14)
$$\begin{aligned}& \bigl\langle Q(u,x_{n_{\ell }})- Q(u_{n_{\ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x _{n_{\ell }})\bigr)\bigr\rangle - \alpha _{g}(y-x_{n_{\ell }}) \\& \quad \geq _{P} 0,\quad \forall x_{n_{\ell }}\in {\mathcal {{D}}}, u_{n_{\ell }} \in T(x_{n _{\ell }}), u \in T(y), \end{aligned}$$
$$\begin{aligned} &\lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u,x_{n_{\ell }}), \eta \bigl(y,g(x_{n _{\ell }})\bigr)\bigr\rangle +\varphi \bigl(g(x_{n_{\ell }}),y \bigr)-\alpha _{g}(y-x_{n _{\ell }})\bigr\} \\ &\quad \geq _{P} \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{ \ell }},x_{n_{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle +\varphi \bigl(g(x _{n_{\ell }}),y\bigr)\bigr\} . \end{aligned}$$
$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle +\varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ &\quad = \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u,x_{n _{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle +\varphi \bigl(g(x_{n_{\ell }}),y\bigr)- \alpha _{g}(y-x_{n_{\ell }}) \bigr\} . \end{aligned}$$
$$\begin{aligned} &\bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \\ & \quad \geq _{P} \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n _{\ell }}) , \eta \bigl(y,g(x_{n_{\ell }}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{ \ell }}),y\bigr)\bigr\} . \end{aligned}$$
(3.15)
$$ \lim_{\ell \to \infty }\bigl\{ \bigl\langle Q(u_{n_{\ell }},x_{n_{\ell }}), \eta \bigl(y,g(x_{n_{\ell }})\bigr)\bigr\rangle + \varphi \bigl(g(x_{n_{\ell }}),y\bigr)\bigr\} \nleq _{P^{0}} 0. $$
(3.16)
$$ \bigl\langle Q(u,x_{0}), \eta \bigl(y,g(x_{0})\bigr) \bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr)- \alpha _{g}(y-x_{0}) \nleq _{P^{0}} 0. $$
$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0. $$
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:
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
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.
$$ \textstyle\begin{cases} T(x)=\{w,z:{\mathbb{R}}^{2}\to {\mathbb{R}}\mid w,z \text{ are continuous linear mappings} \\ \hphantom{T(x)={}}\text{such that } w(x_{1},x_{2})=x_{1}, z(x_{1},x_{2})=x_{2}\}; \\ g(x)=x; \\ \varphi (g(x),y)=y-x; \\ \eta (y,g(x)) = y-x; \\ Q(u,x)=-u; \\ \alpha _{g}=0. \end{cases} $$
$$ \langle -u, x-y\rangle + y-x \nleq _{P^{0}} 0, \quad \forall y\in {\mathcal {{D}}}. $$
(∗∗)
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
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
which implies that
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. □
$$ \varOmega _{\epsilon }\neq \emptyset $$
$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)+ \epsilon _{n} e \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}, $$
$$ x_{n} \in \varOmega _{\epsilon },\quad \forall n > m. $$
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:
where \(\varPsi :\mathcal {{D}} \to {\mathbf {{R}}}\) is a function, and Ω is the solution set of problem (2.1).
$$ \begin{aligned} &P \text{-minimize } \varPsi (x) \\ & \quad \text{subject to } x \in \varOmega , \end{aligned} $$
(4.1)
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
and
$$ \zeta (\delta ,\epsilon ) \neq \emptyset , \quad \forall \delta ,\epsilon > 0 $$
$$ \operatorname{diam} \zeta (\delta ,\epsilon ) \to 0 \quad \textit{as } (\delta , \epsilon ) \to (0,0). $$
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
which implies that
Since
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
Since Ψ is lower semicontinuous,
By using (4.1), the above inequality reduces to
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
This completes the proof. □
$$\begin{aligned}& \lim_{n\to \infty }\sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y), \end{aligned}$$
(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)
$$ x_{n} \in \zeta (\delta _{n},\epsilon _{n}), \quad \text{for } \delta _{n} \to 0. $$
$$ \operatorname{diam} \zeta (\delta ,\epsilon ) \to 0 \quad \text{as } (\delta , \epsilon ) \to (0,0), $$
$$ \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall u_{0} \in T(x_{0}), y \in \mathcal{D}. $$
(4.4)
$$ \varPsi (x_{0}) \leq _{P} \lim_{n\to \infty } \inf \varPsi (x_{n}) \leq _{P} \lim_{n\to \infty } \sup \varPsi (x_{n}). $$
$$ \varPsi (x_{0}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y). $$
(4.5)
$$ \operatorname{diam} \zeta (\delta ,\epsilon ) \to 0 \quad \text{as } (\delta , \epsilon ) \to (0,0). $$
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:
Consider the optimization problem with generalized \((\eta ,g,\varphi )\)-mixed vector variational-type inequality constraints:
where
We see that \(\varOmega =\{0\}\). Since
we have
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.
$$ \textstyle\begin{cases} \varPsi (x)= \vert x^{3} \vert ; \\ T(x)=\{u:{\mathbb{R}} \to {\mathbb{R}} \mid u \text{ is a continuous linear mapping such that } u(x)=-x \}; \\ g(x)=x; \\ \varphi (g(x),y)=y-x; \\ \eta (g(x),y)=\frac{1}{2}(y-x); \\ Q(v,x)=v; \\ \alpha _{g}=-x^{2}. \end{cases} $$
$$ \begin{aligned} &P\text{-minimize } \bigl\vert x^{3} \bigr\vert \\ &\quad \text{subject to } x \in \varOmega , \end{aligned} $$
(4.6)
$$ \varOmega = \biggl\{ x \in {\mathcal {{D}}}\Bigm| \exists u \in T(x) \text{ such that } \biggl\langle u, \frac{1}{2}(x-y)\biggr\rangle + y-x \nleq _{P^{0}} 0, \forall y \in {\mathcal {{D}}}\biggr\} . $$
$$ \zeta (\delta ,\epsilon )=\biggl\{ x \in {\mathcal {{D}}} \Bigm| \exists u \in T(x) \text{ such that } \bigl\vert x^{3} \bigr\vert \leq _{P} \delta \text{ and } (y-x) \biggl(1+\frac{x}{2} \biggr)+ \epsilon \nleq _{P^{0}} 0, \forall y \in {\mathcal {{D}}}\biggr\} , $$
$$ \operatorname{diam} \zeta (\delta ,\epsilon )\to 0 \quad \text{as } (\delta , \epsilon )\to (0,0). $$
Theorem 4.5
Proof
The necessary condition is obvious. Conversely, let (4.1) have a unique solution \(x_{0}\). Then
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
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
Since Ψ is lower semicontinuous, therefore,
Thus, from (4.7) and (4.8), we conclude that \(\bar{x} \in \zeta \), and the proof is completed. □
$$\begin{aligned}& \varPsi (x_{0})=\inf_{y\in \varOmega } \varPsi (y), \\& \bigl\langle Q(u_{0},x_{0}), \eta \bigl(y,g(x_{0}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{0}),y\bigr) \nleq _{P^{0}} 0, \quad \forall u_{0}\in T(x_{0}), y \in {\mathcal {{D}}}. \end{aligned}$$
$$\begin{aligned}& \lim_{n\to \infty }\sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y), \\& \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr)+ \epsilon _{n} e \nleq _{P^{0}} 0,\quad \forall y \in {\mathcal {{D}}}. \end{aligned}$$
$$ \bigl\langle Q(\bar{u},\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) \nleq _{P^{0}} 0, \quad \forall \bar{u} \in T(\bar{x}), y \in {\mathcal {{D}}}. $$
(4.7)
$$ \varPsi (\bar{x}) \leq _{P} \lim_{\ell \to \infty } \inf \varPsi (x_{n_{ \ell }}) \leq _{P} \lim_{\ell \to \infty } \sup \varPsi (x_{n_{\ell }}) \leq _{P} \inf _{y\in \varOmega } \varPsi (y). $$
(4.8)
Theorem 4.6
Proof
Let \(\epsilon > 0\) such that
and suppose \(\{x_{n}\}\) is an approximating sequence of problem (2.1). Then 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. □
$$ \zeta (\epsilon ,\epsilon ) \neq \emptyset $$
(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
which implies that for some positive integer m
$$ \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}, n \in {\mathbb{N}}, $$
$$ x_{n} \in \zeta (\epsilon ,\epsilon ),\quad \forall n > m. $$
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
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
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.,
Since Ψ is lower semicontinuous, we have
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,
Hence, (4.1) is well-posed. The proof is completed. □
$$\begin{aligned}& \lim_{n\to \infty }\sup \varPsi (x_{n}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y), \\& \bigl\langle Q(u_{n},x_{n}), \eta \bigl(y,g(x_{n}) \bigr)\bigr\rangle + \varphi \bigl(g(x_{n}),y\bigr) + \epsilon _{n} e \nleq _{P^{0}} 0, \quad \forall y \in {\mathcal {{D}}}. \end{aligned}$$
$$ \bigl\langle Q(\bar{u},\bar{x}), \eta \bigl(y,g(\bar{x})\bigr)\bigr\rangle + \varphi \bigl(g( \bar{x}),y\bigr) \nleq _{P^{0}} 0,\quad \forall \bar{u} \in T(\bar{x}), y \in {\mathcal {{D}}}. $$
(5.1)
$$ \varPsi (\bar{x}) \leq _{P} \lim_{\ell \to \infty }\inf \varPsi (x_{n_{ \ell }}) \leq _{P} \lim_{\ell \to \infty } \sup \varPsi (x_{n_{\ell }}) \leq _{P} \inf_{y\in \varOmega } \varPsi (y). $$
(5.2)
$$ \bar{x} = x_{0} \quad \text{and}\quad x_{n} \to x_{0}. $$
Acknowledgements
The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article.
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The data sets used during the current study are available from the corresponding author on request.
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The authors declare that they have no competing interests.
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