Abstract
The stability for a class of generalized Minty variational-hemivariational inequalities has been considered in reflexive Banach spaces. We demonstrate the equivalent characterizations of the generalized Minty variational-hemivariational inequality. A stability result is presented for the generalized Minty variational-hemivariational inequality with \((f,J)\)-pseudomonotone mapping.
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1 Introduction
Let X be a real Banach space with its dual \(X^{*}\). Let \(K\subset X\) be a nonempty, closed, and convex set. Let \(F:K\to2^{X^{*}}\) be a set-valued mapping. Let \(A:K\to X^{*}\) be a single-valued mapping. Let \(f:K\subset X\to\mathbf{R}\cup\{+\infty\}\) be a proper, convex, and lower semicontinuous functional. Let \(J:X\to\mathbf{R}\) be a locally Lipschitz functional. We use \(J^{\circ}(\cdot,\cdot)\) to denote Clarke’s generalized directional derivative of J. Recall that the variational-hemivariational inequality [1] can mathematically be formulated as the problem of finding a point \(u\in K\) such that
In particular, if \(J=0\), then the \(\operatorname{VHVI}(A,J,K)\) reduces to the following mixed variational inequality of finding \(u\in K\) such that
MVI has been studied extensively in the literature, see, for instance, [2–6].
Under some suitable conditions, (1.2) is equivalent to the following Minty mixed variational inequality [7–15] which is to find \(u\in K\) such that
In the present paper, we consider the following generalized Minty variational-hemivariational inequality of finding \(u\in K\) such that
Special cases: (i) If \(J=0\), then (1.4) reduces to the following generalized Minty mixed variational inequality of finding \(u\in K\) such that
(ii) If \(F=A\) and \(f=0\), then (1.5) reduces to the following classical Minty variational inequality of finding \(u\in K\) such that
Let \((Z_{1},d_{1})\) and \((Z_{2},d_{2})\) be two metric spaces. \(L:Z_{1}\to2^{X}\) be a set-valued mapping with nonempty, closed, and convex values. Let \(F:X\times Z_{2}\to2^{X^{*}}\) be a set-valued mapping. Let \(f:X\to\mathbf {R}\cup\{+\infty\}\) be a proper, convex, and lower semicontinuous functional. Next, we consider the following parameter generalized Minty variational-hemivariational inequality which is to find \(x\in L(u)\) such that
In particular, if \(J=0\), then (1.7) reduces to the following parameter generalized Minty mixed variational inequality: find \(x\in K\) such that
It is well known that the variational inequality theory has wide applications in finance, economics, transportation, optimization, operations research, and engineering sciences, see [16–25]. In 2010, Zhong and Huang [19] studied the stability of solution sets for the generalized Minty mixed variational inequality in reflexive Banach spaces.
Inspired and motivated by the above work of Zhong and Huang [19], we investigate the stability of solution sets for the generalized Minty variational-hemivariational inequality in reflexive Banach spaces. We first present several equivalent characterizations for the generalized Minty variational-hemivariational inequality. Consequently, we show the stability of a solution set for the generalized Minty variational-hemivariational inequality with \((f,J)\)-pseudomonotone mapping in reflexive Banach spaces. As an application, we give the stability result for a generalized variational-hemivariational inequality. The results presented in this paper extend the corresponding results of Zhong and Huang [19] from the generalized mixed variational inequalities to the generalized variational-hemivariational inequalities.
2 Preliminaries
Let X be a real reflexive Banach space. Let \(J:X\to\mathbf{R}\) be a locally Lipschitz function on X. Clarke’s generalized directional derivative of J at x in the direction y, denoted by \(J^{\circ}(x,y)\), is defined by
Let \(f:X\to\mathbf{R}\cup\{+\infty\}\) be a proper, convex, and lower semicontinuous function. Denote by \(\partial f:X\to2^{X^{*}}\) and \(\overline{\partial}J:X\to2^{X^{*}}\) the subgradient of f and Clarke’s generalized gradient of J (see [26]), respectively. That is,
and
It is known that \(\overline{\partial}J(x)=\partial(J^{\circ}(x,\cdot ))(0)\), see [27].
Proposition 2.1
([1])
Let X be a Banach space and J be a locally Lipschitz functional on X. Then we have:
-
(i)
The function \(y\mapsto J^{\circ}(x,y)\) is finite, convex, positively homogeneous, and subadditive;
-
(ii)
\(J^{\circ}(x,y)\) is upper semicontinuous and is Lipschitz continuous on the second variable;
-
(iii)
\(J^{\circ}(x,-y)=(-J)^{\circ}(x,y)\);
-
(iv)
\(\overline{\partial}J(x)\) is a nonempty, convex, bounded, and weak ∗-compact subset of \(X^{*}\);
-
(v)
For every \(y\in X\), \(J^{\circ}(x,y)=\max\{\langle\xi,y\rangle:\xi\in\overline{\partial}J(x)\}\);
-
(vi)
The graph of \(\overline{\partial}J(x)\) is closed in \(X\times (w^{*}-X^{*})\) topology, where \((w^{*}-X^{*})\) denotes the space \(X^{*}\) equipped with weak ∗ topology, i.e., if \(\{x_{n}\}\subset X\) and \(\{x^{*}_{n}\}\subset X^{*}\) are sequences such that \(x^{*}_{n}\in\overline{\partial}J(x_{n}), x_{n}\to x\) in X and \(x^{*}_{n}\to x^{*}\) weakly ∗ in \(X^{*}\), then \(x^{*}\in\overline{\partial}J(x)\).
Let K be a nonempty, closed, and convex subset of X. Let Y be a topological space. We use \(\operatorname{barr}(K)\) to denote the barrier cone of K which is defined by \(\operatorname{barr}(K):=\{x^{*}\in X^{*}:\sup_{x\in K}\langle x^{*},x\rangle<\infty\} \). The recession cone of K, denoted by \(K_{\infty}\), is defined by \(K_{\infty}:=\{d\in X:x_{0}+\mu d\in K, \forall\mu>0, \forall x_{0}\in K\}\). The negative polar cone \(K^{-}\) of K is defined by \(K^{-}:=\{x^{*}\in X^{*}:\langle x^{*},x\rangle\leq0, \forall x\in K\}\). The positive polar cone of K is defined as \(K^{+}:=\{x^{*}\in X^{*}:\langle x^{*},x\rangle\geq0, \forall x\in K\}\).
Let \(f:K\to\mathbf{R}\cup\{+\infty\}\) be a proper, convex, and lower semicontinuous function. The recession function \(f_{\infty}\) of f is defined by
where \(x_{0}\in\operatorname{Dom}f\).
It is known that
and \(f_{\infty}(\cdot)\) satisfies \(f_{\infty}(\lambda x)=\lambda f_{\infty}(x)\) for all \(x\in X, \lambda \geq0\). According to Proposition 2.5 in [28], we deduce
where \(\{x_{n}\}\) is any sequence in X converging weakly to x and \(t_{n}\to+\infty\).
Definition 2.2
A set-valued mapping \(F:K\subset X\to2^{X^{*}}\) is said to be
-
(i)
upper semicontinuous at \(x_{0}\in K\) iff, for any neighborhood \(\mathrm{N}(F(x_{0}))\) of \(F(x_{0})\), there exists a neighborhood \(\mathrm{N}(x_{0})\) of \(x_{0}\) such that
$$ F(x)\subset\mathrm{N}\bigl(F(x_{0})\bigr), \quad\forall x\in\mathrm{N}(x_{0})\cap K; $$ -
(ii)
lower semicontinuous at \(x_{0}\in K\) iff, for any \(y_{0}\in F(x_{0})\) and any neighborhood \(\mathrm{N}(y_{0})\) of \(y_{0}\), there exists a neighborhood \(\mathrm{N}(x_{0})\) of \(x_{0}\) such that
$$ F(x)\cap\mathrm{N}(y_{0})\neq\emptyset, \quad\forall x\in\mathrm{N}(x_{0})\cap K. $$
F is said to be continuous at \(x_{0}\) iff it is both upper and lower semicontinuous at \(x_{0}\); and F is continuous on K iff it is both upper and lower semicontinuous at every point of K.
Definition 2.3
The mapping F is said to be
-
(i)
monotone on K iff, for all \((x,x^{*}),(y,y^{*})\) in the \(\operatorname{graph}(F)\),
$$ \bigl\langle y^{*}-x^{*},y-x\bigr\rangle \geq0; $$ -
(ii)
pseudomonotone on K iff, for all \((x,x^{*}),(y,y^{*})\) in the \(\operatorname{graph}(F)\),
$$ \bigl\langle x^{*},y-x\bigr\rangle \geq0 \quad\text{implies that}\quad \bigl\langle y^{*},y-x \bigr\rangle \geq0; $$ -
(iii)
stably pseudomonotone on K with respect to a set \(U\subset X^{*}\) iff F and \(F(\cdot)-u\) are pseudomonotone on K for every \(u\in U\);
-
(iv)
f-pseudomonotone on K iff, for all \((x,x^{*}),(y,y^{*})\) in the \(\operatorname{graph}(F)\),
$$ \bigl\langle x^{*},y-x\bigr\rangle +f(y)-f(x)\geq0 \quad\Rightarrow\quad\bigl\langle y^{*},x-y \bigr\rangle +f(x)-f(y)\leq0; $$ -
(v)
\((f,J)\)-pseudomonotone on K iff, for all \((x,x^{*}),(y,y^{*})\) in the \(\operatorname{graph}(F)\),
$$ \bigl\langle x^{*},y-x\bigr\rangle +J^{\circ}(x,y-x)+f(y)-f(x)\geq0 \quad\Rightarrow\quad \bigl\langle y^{*},x-y\bigr\rangle +J^{\circ}(y,x-y)+f(x)-f(y) \leq0. $$
Definition 2.4
Let \(\{A_{n}\}\subset X\) be a sequence. Define
Definition 2.5
Let \(\psi:X\times X\to\mathbf{R}\) be a function. ψ is said to be bi-sequentially weakly lower semicontinuous iff, for any sequences \(\{ x_{n}\}\) and \(\{y_{n}\}\) with \(x_{n}\rightharpoonup x_{0}\) and \(y_{n}\rightharpoonup y_{0}\), one has
Lemma 2.6
([29])
Let \(K\subset X\) be a nonempty, closed, and convex set with \(\operatorname{int}(\operatorname{barr}(K))\neq\emptyset\). Then there exists no sequence \(\{x_{n}\}\subset K\) satisfying \(\|x_{n}\|\to\infty\) and \(\frac{x_{n}}{\|x_{n}\| }\rightharpoonup0\). If K is a cone, then there exists no sequence \(\{d_{n}\}\subset K\) with \(\|d_{n}\|=1\) satisfying \(d_{n}\rightharpoonup0\).
Lemma 2.7
([30])
Let \(K\subset X\) be a nonempty, closed, and convex set with \(\operatorname{int}(\operatorname{barr}(K))\neq\emptyset\). Then there exists no sequence \(\{d_{n}\}\subset K_{\infty}\) with \(\|d_{n}\|=1\) satisfying \(d_{n}\rightharpoonup0\).
Lemma 2.8
([30])
Let \((Z,d)\) be a metric space and \(u_{0}\in Z\) be a given point. Let \(L:Z\to2^{X}\) be a set-valued mapping with nonempty values, and let L be upper semicontinuous at \(u_{0}\). Then there exists a neighborhood U of \(u_{0}\) such that \((L(u))_{\infty}\subset(L(u_{0}))_{\infty}\) for all \(u\in U\).
Lemma 2.9
([31])
Let E be a Hausdorff topological vector space and \(K\subset E\) be a nonempty and convex set. Let \(G:K\to2^{E}\) be a set-valued mapping satisfying the following conditions:
-
(i)
G is a KKM mapping, i.e., for every finite subset A of K, \(\operatorname{conv}(A)\subset\bigcup_{x\in A}G(x)\);
-
(ii)
\(G(x)\) is closed in E for every \(x\in K\);
-
(iii)
\(G(x_{0})\) is compact in E for some \(x_{0}\in K\).
Then \(\bigcap_{x\in K}G(x)\neq\emptyset\).
3 Boundedness of solution sets
In this section, we introduce several characterizations for the solution set D of \(\operatorname{GMVHVI}(F,J,K)\).
Let \(K\subset X\) be a nonempty, closed, and convex set. Let \(F:K\to 2^{X^{*}}\) be a set-valued mapping with nonempty values, \(J:X\to\mathbf {R}\) be a locally Lipschitz functional, and \(f:K\subset X\to\mathbf{R}\) be a convex and lower semicontinuous function.
Theorem 3.1
Suppose \(D\ne\emptyset\). Then
Proof
Define a function \(\varPhi:X\to\mathbf{R}\cup\{+\infty\}\) by
where \(\varphi(y,y^{*}):=\max\{\|y^{*}\|,1\}\max\{\|y\|,1\}\max\{|f(y)|,1\} \). Clearly, Φ is a proper, convex, and lower semicontinuous function and so \(\varPhi_{\infty}\) is well defined on X.
Let \(D=\{x\in K:\varPhi(x)\leq0\}\). It is clear that D is nonempty. According to formula (2.29) in [32], \(\{x\in X:\varPhi(x)\leq r\} _{\infty}=\{d\in X:\varPhi_{\infty}(d)\leq0\}\). Hence
It remains to prove that
Let \(d\in\{d\in X:\langle y^{*},d\rangle+J^{\circ}(y,d)+f_{\infty}(d)\leq0, \forall y^{*}\in F(y),y\in K\}\) and \(x_{0}\in X\) with \(\varPhi(x_{0})<\infty\). By virtue of the subadditivity and positive homogeneousness of the function \(y\mapsto J^{\circ}(x,y)\), we have
This implies that
and so
Therefore,
Conversely, if \(d\notin\{d\in X:\langle y^{*},d\rangle+J^{\circ}(y,d)+f_{\infty}(d)\leq0, \forall y^{*}\in F(y),y\in K\}\), then there exist \(y\in K\) and \(y^{*}\in F(y)\) such that \(\langle y^{*},d\rangle+J^{\circ}(y,d)+f_{\infty}(d)>0\). Hence,
This yields that
and hence the converse inclusion is true. This completes the proof. □
Corollary 3.2
Suppose \(D\ne\emptyset\). Then
Proof
If \(J=0\), then \(J^{\circ}=0\). In this case, \(\operatorname{GMVHVI}(F,J,K)\) reduces to \(\operatorname{GMMVI}(F,K)\). Utilizing Theorem 3.1, we immediately deduce Corollary 3.2. □
Remark 3.3
It is known that if \(J=0\) then Theorem 3.1 reduces to Zhong and Huang’s one [19, Theorem 3.1]. Thus, Theorem 3.1 generalizes and extends Theorem 3.1 in Zhong and Huang [19] from \(\operatorname{GMMVI}(F,K)\) to \(\operatorname{GMVHVI}(F,J,K)\). If \(f=0\) additionally, then \(f_{\infty}=0\) and so
Hence, Zhong and Huang’s Theorem 3.1 in [19] is a generalization of Lemma 3.1 in [29].
Theorem 3.4
Suppose the following statements hold:
-
(i)
D is nonempty and bounded;
-
(ii)
\(K_{\infty}\cap\{d\in X:\langle y^{*},d\rangle+J^{\circ}(y,d)+f_{\infty}(d)\leq0, \forall y^{*}\in F(y),y\in K\}=\{0\}\);
-
(iii)
There exists a bounded set \(C\subset K\) such that, for every \(x\in K\setminus C\), there exists some \(y\in C\) satisfying
$$ \sup_{y^{*}\in F(y)}\bigl\langle y^{*},x-y\bigr\rangle +J^{\circ}(y,x-y)+f(x)-f(y)>0. $$
Then (i)⇒(ii). (ii)⇒(iii) if \(\operatorname{barr}(K)\) has nonempty interior. (iii)⇒(i) if F is \((f,J)\)-pseudomonotone on K.
Proof
The relationship (i)⇒(ii) can be deduced from Theorem 3.1.
Next, we first prove that (ii)⇒(iii). If (iii) does not hold, then there exists a sequence \(\{x_{n}\}\subset K\) such that, for each \(n, \|x_{n}\|\geq n\) and \(\sup_{y^{*}\in F(y)}\langle y^{*},x_{n}-y\rangle +J^{\circ}(y,x_{n}-y)+f(x_{n})-f(y)\leq0\) for every \(y\in K\) with \(\|y\|\leq n\). Without loss of generality, we may assume that \(d_{n}=x_{n}/\|x_{n}\|\) weakly converges to d. Then \(d\in K_{\infty}\). By Lemma 2.7, we get \(d\neq0\). Let \(y\in K\) and \(y^{*}\in F(y)\). Then, for all \(n>\|y\|\), we have
This together with (2.2) implies that
and so
This implies that
a contradiction to (ii).
It remains to prove that (iii) implies (i) under the assumption that F is \((f,J)\)-pseudomonotone on K. Indeed, let \(G:K\to2^{K}\) be a set-valued mapping defined by
Firstly, we show that \(G(y)\) is a closed subset of K. In fact, for any \(x_{n}\in G(y)\) with \(x_{n}\to x_{0}\), we have
From the lower semicontinuity of f and the Lipschitz continuity of \(J^{\circ}(\cdot,\cdot)\) in the second variable, it follows that
This shows that \(x_{0}\in G(y)\) and so \(G(y)\) is closed.
Next we prove that \(G:K\to K\) is a KKM mapping. If it is not so, then there exist \(t_{1},t_{2},\ldots,t_{n}\in[0,1], y_{1},y_{2},\ldots,y_{n}\in K\), and \(\bar{y}=t_{1}y_{1}+t_{2}y_{2}+\cdots+t_{n}y_{n}\in\operatorname{conv}\{y_{1},y_{2},\ldots ,y_{n}\}\) such that \(\bar{y}\notin\bigcup_{i\in\{1,2,\ldots,n\}}G(y_{i})\). Hence,
By the \((f,J)\)-pseudomonotonicity of F, we get
Since \(y\mapsto J^{\circ}(x,y)\) is convex, we deduce
which yields
It follows that
and hence
which is a contradiction. Therefore, G is a KKM mapping.
Assume that C is a bounded, closed, and convex (otherwise, we can use the closed convex hull of C instead of C). Let \(\{y_{1},\ldots,y_{m}\}\) be a finite number of points in K, and let \(M:=\operatorname{conv}(C\cup\{y_{1},\ldots,y_{m}\})\). It is obvious that M is weakly compact and convex. Let \(G'(y):=G(y)\cap M\) for all \(y\in M\). Then \(G'(y)\) is a weakly compact and convex subset of M and \(G'\) is a KKM mapping. We claim that
Indeed, by Lemma 2.9, the intersection in (3.1) is nonempty. Moreover, if there exists some \(x_{0}\in\bigcap_{y\in M}G'(y)\) but \(x_{0}\notin C\), then by (iii) we have
for some \(y\in C\). Thus, \(x_{0}\notin G(y)\) and so \(x_{0}\notin G'(y)\), which is a contradiction to the choice of \(x_{0}\).
Let \(z\in\bigcap_{y\in M}G'(y)\). Then \(z\in C\) by (11) and so \(z\in \bigcap^{m}_{i=1}(G(y_{i})\cap C)\). This shows that the collection \(\{G(y)\cap C:y\in K\}\) has the finite intersection property. For each \(y\in K\), it follows from the weak compactness of \(G(y)\cap C\) that \(\bigcap_{y\in K}(G(y)\cap C)\) is nonempty, which coincides with the solution set of \(\operatorname{GMVHVI}(F,J,K)\). This completes the proof. □
Corollary 3.5
Suppose the following statements hold:
-
(i)
D is nonempty and bounded;
-
(ii)
\(K_{\infty}\cap\{d\in X:\langle y^{*},d\rangle+f_{\infty}(d)\leq 0, \forall y^{*}\in F(y),y\in K\}=\{0\}\);
-
(iii)
There exists a bounded set \(C\subset K\) such that, for every \(x\in K\setminus C\), there exists some \(y\in C\) satisfying
$$ \sup_{y^{*}\in F(y)}\bigl\langle y^{*},x-y\bigr\rangle +f(x)-f(y)>0. $$
Then (i)⇒(ii). (ii)⇒(iii) if \(\operatorname{barr}(K)\) has nonempty interior. (iii)⇒(i) if F is \((f,J)\)-pseudomonotone on K.
Remark 3.6
It is known that if \(J=0\) then Theorem 3.4 reduces to Theorem 3.2 in Zhong and Huang [19]. Thus, Theorem 3.4 generalizes and extends Theorem 3.2 in Zhong and Huang [19] from \(\operatorname{GMMVI}(F,K)\) to \(\operatorname{GMVHVI}(F,J,K)\). If \(f=0\) additionally, then \(f_{\infty}=0\). Consequently, statements (i), (ii), and (iii) in [19, Theorem 3.2] reduce to (i), (ii), and (iii) in [29, Theorem 3.1], respectively. Thus, Zhong and Huang’s Theorem 3.2 in [19] is a generalization of Theorem 3.1 in [29].
4 Stability of solution sets
In this section, we will establish the stability of solution sets for the generalized Minty variational-hemivariational inequality \(\operatorname{GMVHVI}(F,J,K)\) and the generalized variational-hemivariational inequality \(\operatorname{GVHVI}(F,J,K)\) with \((f,J)\)-pseudomonotone mappings.
Let \((Z_{1},d_{1})\) and \((Z_{2},d_{2})\) be two metric spaces, \(u_{0}\in Z_{1}\) and \(v_{0}\in Z_{2}\) be given points. Let \(L:Z_{1}\to2^{X}\) be a continuous set-valued mapping with nonempty, closed, and convex values and \(\operatorname{int}(\operatorname{barr}L(u_{0}))\neq\emptyset\). Suppose that there exists a neighborhood \(U\times V\) of \((u_{0},v_{0})\) such that \(M=\bigcup_{u\in U}L(u)\), \(F:M\times V\to 2^{X^{*}}\) is a lower semicontinuous set-valued mapping with nonempty values, and let \(f:M\subset X\to\mathbf{R}\) be a convex and lower semicontinuous function. Let \(J:X\to\mathbf{R}\) be a locally Lipschitz functional such that \(J^{\circ}:M\times M\subset X\times X\to\mathbf{R}\) is bi-sequentially weakly lower semicontinuous.
Theorem 4.1
If
then there exists a neighborhood \(U'\times V'\) of \((u_{0},v_{0})\) with \(U'\times V'\subset U\times V\) such that
for all \((u,v)\in U'\times V'\).
Proof
Assume that the conclusion does not hold. Then there exists a sequence \(\{(u_{n},v_{n})\}\) in \(Z_{1}\times Z_{2}\) with \((u_{n},v_{n})\to(u_{0},v_{0})\) such that
Since \(f_{\infty}(\lambda x)=\lambda f_{\infty}(x)\) for all \(x\in X \) and \(\lambda\geq0\), we deduce that
is a cone. Thus, we can select a sequence \(\{d_{n}\}\) such that
satisfying \(\|d_{n}\|=1\) for every \(n=1,2,\ldots\) . Without loss of generality, we can assume that \(d_{n}\rightharpoonup d_{0}\ne0\) by Lemma 2.7. By the upper semicontinuity of L and Lemma 2.8, we have \((L(u_{n}))_{\infty}\subset(L(u_{0}))_{\infty}\) for large enough n and so \(d_{n}\in (L(u_{0}))_{\infty}\) for large enough n. Since \((L(u_{0}))_{\infty}\) is weakly closed, we have \(d_{0}\in(L(u_{0}))_{\infty}\). Take any fixed \(y\in L(u_{0})\) and \(y^{*}\in F(y,v_{0})\). From the lower semicontinuity of L, there exists \(y_{n}\in L(u_{n})\) such that \(y_{n}\to y\). Hence, \((y_{n},v_{n})\to (y,v_{0})\). By the lower semicontinuity of F, there exists \(y^{*}_{n}\in F(y_{n},v_{n})\) such that \(y^{*}_{n}\to y^{*}\). Since
we have
Combining with \(y_{n}\to y, y^{*}_{n}\to y^{*}, d_{n}\rightharpoonup d_{0}\), the bi-sequential weak lower semicontinuity of \(J^{\circ}\) and the weak lower semicontinuity of \(f_{\infty}\), it follows that \(\langle y^{*},d_{0}\rangle+J^{\circ}(y,d_{0})+f_{\infty}(d_{0})\leq0\). Since \(y\in L(u_{0})\) and \(y^{*}\in F(y,v_{0})\) are arbitrary, from the above discussion, we have
and so
with \(d_{0}\neq0\), which contradicts the assumption. This completes the proof. □
Corollary 4.2
If
then there exists a neighborhood \(U'\times V'\) of \((u_{0},v_{0})\) with \(U'\times V'\subset U\times V\) such that
for all \((u,v)\in U'\times V'\).
Proof
Whenever \(J=0\), we know that \(J^{\circ}=0\) and hence \(J^{\circ}\) is bi-sequentially weakly lower semicontinuous. In this case, (4.1) and (4.2) in Theorem 4.1 reduce to (4.3) and (4.4), respectively. Utilizing Theorem 4.1, we immediately deduce Corollary 4.2. □
Remark 4.3
It is known that if \(J=0\) then Theorem 4.1 reduces to Theorem 4.1 in Zhong and Huang [19]. Thus, Theorem 4.1 generalizes and extends Zhong and Huang’s Theorem 4.1 [19] to the case of Clarke’s generalized directional derivative of a locally Lipschitz functional. If \(f=0\) additionally, then \(f_{\infty}=0\). Thus, (4.3) and (4.4) in Corollary 4.2 reduce to (3.1) and (3.2) in [30, Theorem 3.1], respectively. Therefore, Zhong and Huang’s Theorem 4.1 in [19] is a generalization of Theorem 3.1 in [30].
Theorem 4.4
Assume that all the conditions of Theorem 4.1 are satisfied. Suppose that
-
(i)
for each \(v\in V\), the mapping \(x\mapsto F(x,v)\) is \((f,J)\)-pseudomonotone on M;
-
(ii)
the solution set of \(\operatorname{GMVHVI}(F(\cdot,v_{0}),J,L(u_{0}))\) is nonempty and bounded.
Then there exists a neighborhood \(U'\times V'\) of \((u_{0},v_{0})\) with \(U'\times V'\subset U\times V\) such that, for every \((u,v)\in U'\times V'\), the solution set of \(\operatorname{GMVHVI}(F(\cdot,v),J,L(u))\) is nonempty and bounded. Moreover, if f is continuous on \(M=\bigcup_{u\in U}L(u)\) and \(J^{\circ}:M\times(M-M)\to \mathbf{R}\) is continuous, then ω-\(\limsup_{(u,v)\to (u_{0},v_{0})}S_{\mathrm{GM}}(u,v) \subset S_{\mathrm{GM}}(u_{0},v_{0})\), where \(S_{\mathrm{GM}}(u,v)\) and \(S_{\mathrm {GM}}(u_{0},v_{0})\) are the solution sets of \(\operatorname{GMVHVI}(F(\cdot,v),J,L(u))\) and \(\operatorname{GMVHVI}(F(\cdot,v_{0}),J,L(u_{0}))\), respectively.
Proof
By Theorem 3.1, we get
It follows from Theorem 4.1 that there exists a neighborhood \(U'\times V'\) of \((u_{0},v_{0})\) with \(U'\times V'\subset U\times V\) such that
for all \((u,v)\in U'\times V'\). Since F is \((f,J)\)-pseudomonotone, Theorem 3.4 implies that the solution set of \(\operatorname{GMVHVI}(F(\cdot,v),J,L(u))\) is nonempty and bounded for every \((u,v)\in U'\times V'\).
Next, we prove that ω-\(\limsup_{(u,v)\to(u_{0},v_{0})}S_{\mathrm {GM}}(u,v)\subset S_{\mathrm{GM}}(u_{0},v_{0})\). For \(\{(u_{n},v_{n})\}\subset U'\times V'\) with \((u_{n},v_{n})\to(u_{0},v_{0})\), we need to prove that ω-\(\limsup_{n\to\infty}S_{\mathrm{GM}}(u_{n},v_{n})\subset S_{\mathrm{GM}}(u_{0},v_{0})\). For any \(n=0,1,2,\ldots\) , define a function \(\varPhi_{n}:X\to\mathbf{R}\) by
where
Let \(A_{n}:=\{x\in L(u_{n}):\varPhi_{n}(x)\leq0\}\) for every non-negative integer n. By the definition of \(\varPhi_{n}\), it is easy to see that \(A_{n}=\{x\in L(u_{n}):\varPhi_{n}(x)\leq0\}\) coincides with the solution set \(S_{\mathrm{GM}}(u_{n},v_{n})\) of \(\operatorname{GMVHVI}(F(\cdot,v),J,L(u))\) for all \(n=0,1,2,\ldots\) . Thus, \(A_{n}\) is nonempty and bounded by condition (ii) for every non-negative integer n. From the above discussion, we need only to prove that ω-\(\limsup_{n\to\infty}A_{n}\subset A_{0}\). Let \(x\in\omega \)-\(\limsup_{n\to\infty}A_{n}\). Then there exists a sequence \(\{x_{n_{j}}\}\) with each \(x_{n_{j}}\in A_{n_{j}}\) such that \(x_{n_{j}}\) weakly converges to x. We claim that there exists \(z_{n_{j}}\in L(u_{0})\) such that \(\lim_{j\to\infty}\|x_{n_{j}}-z_{n_{j}}\|=0\). Indeed, if the claim does not hold, then there exist a subsequence \(\{x_{n_{j_{k}}}\}\) of \(\{ x_{n_{j}}\}\) and some \(\varepsilon_{0}>0\) such that
This implies that \(x_{n_{j_{k}}}\notin L(u_{0})+\varepsilon_{0}B(0,1)\) and so \(L(u_{n_{j_{k}}})\not\subset L(u_{0})+\varepsilon_{0}B(0,1)\), which contradicts the upper semicontinuity of \(L(\cdot)\). Moreover, we obtain \(x\in L(u_{0})\) as \(L(u_{0})\) is a closed and convex subset of X and hence weakly closed. Next we prove that \(\varPhi_{0}(x)\leq0\) and hence \(x\in A_{0}\). In fact, for any fixed \(y\in L(u_{0})\) and \(y^{*}\in F(y,v_{0})\), since L is lower semicontinuous and \(u_{n}\to u_{0}\), we know that there exists \(y_{n}\in L(u_{n})\) for every \(n=1,2,\ldots\) such that \(\lim_{n\to\infty}y_{n}=y\). Since F is lower semicontinuous, it follows that there exists a sequence of elements \(y^{*}_{n}\in F(y_{n},v_{n})\) such that \(y^{*}_{n}\to y^{*}\). Now \(x_{n_{j}}\in A_{n_{j}}\) implies that \(\varPhi_{n_{j}}(x_{n_{j}})\leq0\) and so
Since f is continuous on \(M=\bigcup_{u\in U}L(u)\) and \(J^{\circ}:M\times (M-M)\to\mathbf{R}\) is also continuous, letting \(j\to\infty\), we have
Since \(y\in L(u_{0})\) and \(y^{*}\in F(y,v_{0})\) are arbitrary, we know that \(\varPhi_{0}(x)\leq0\) and hence \(x\in A_{0}\). This completes the proof. □
Corollary 4.5
Assume that all the conditions of Corollary 4.2 are satisfied. Suppose that
-
(i)
for each \(v\in V\), the mapping \(x\mapsto F(x,v)\) is f-pseudomonotone on M;
-
(ii)
the solution set of \(\operatorname{GMMVI}(F(\cdot,v_{0}),L(u_{0}))\) is nonempty and bounded.
Then there exists a neighborhood \(U'\times V'\) of \((u_{0},v_{0})\) with \(U'\times V'\subset U\times V\) such that, for every \((u,v)\in U'\times V'\), the solution set of \(\operatorname{GMMVI}(F(\cdot,v),L(u))\) is nonempty and bounded. Moreover, if f is continuous on \(M=\bigcup_{u\in U}L(u)\), then ω-\(\limsup_{(u,v)\to (u_{0},v_{0})}S_{M}(u,v)\subset S_{M}(u_{0},v_{0})\), where \(S_{M}(u,v)\) and \(S_{M}(u_{0},v_{0})\) are the solution sets of \(\operatorname{GMMVI}(F(\cdot,v),L(u))\) and \(\operatorname{GMMVI}(F(\cdot ,v_{0}),L(u_{0}))\), respectively.
Proof
Whenever \(J=0\), we know that \(J^{\circ}=0\), \(\operatorname{GMVHVI}(F(\cdot ,v),J,L(u))\) (resp., \(\operatorname{GMVHVI}(F(\cdot, v_{0}),J,L(u_{0}))\)) reduces to \(\operatorname{GMMVI}(F(\cdot,v),L(u))\) (resp., \(\operatorname{GMMVI}(F(\cdot,v_{0}),L(u_{0}))\)), \(S_{\mathrm{GM}}(u,v)\) (resp., \(S_{\mathrm{GM}}(u_{0},v_{0})\)) reduces to \(S_{M}(u,v)\) (resp., \(S_{M}(u_{0},v_{0})\)), and the \((f,J)\)-pseudomonotonicity of F in the first variable reduces to the f-pseudomonotonicity of F in the first variable. Utilizing Theorem 4.9, we immediately deduce Corollary 4.5. □
Remark 4.6
It is known that if \(J=0\) then Theorem 4.4 reduces to Theorem 4.2 in Zhong and Huang [19]. Thus, Theorem 4.4 generalizes and extends Theorem 4.2 in Zhong and Huang [19] from the generalized Minty mixed variational inequality to the generalized Minty variational-hemivariational inequality. If \(f=0\) additionally, then \(f_{\infty}=0\), and so the generalized Minty mixed variational inequality \(\operatorname{GMMVI}(F,K)\) reduces to the generalized Minty variational inequality. Hence, Zhong and Huang’s Theorem 4.2 [19] generalizes [30, Theorem 3.2] from the generalized Minty variational inequality to the generalized Minty mixed variational inequality. In addition, for the case of \(J=f=0\), He [29] obtained the corresponding result of Zhong and Huang’s Theorem 4.2 [19] when either the mapping or the constraint set is perturbed (see Theorems 4.1 and 4.4 of [29]). Therefore, Zhong and Huang’s Theorem 4.2 [19] is a generalization of Theorems 4.1 and 4.4 in [29].
In the following, as an application of Theorem 4.4, we will consider the stability behavior for the following generalized variational-hemivariational inequality, denoted by \(\operatorname{GVHVI}(F,J,K)\), which is to find \(x\in K\) and \(x^{*}\in F(x)\) such that
If \(J=0\), then \(\operatorname{GVHVI}(F,J,K)\) reduces to the generalized mixed variational inequality, which is to find \(x\in K\) and \(x^{*}\in F(x)\) such that
If F is single-valued, then (4.5) reduces to (1.1). Furthermore, if \(f=0\), then (4.6) reduces to the following generalized variational inequality of finding \(x\in K\) and \(x^{*}\in F(x)\) such that
Next we consider the parametric generalized variational-hemivariational inequality, denoted by \(\operatorname{GVHVI}(F(\cdot,v),J,L(u))\), which is to find \(x\in L(u)\) and \(x^{*}\in F(x,v)\) such that
In particular, if \(J=0\), then (4.8) reduces to the following parametric generalized mixed variational inequality, which is to find \(x\in L(u)\) and \(x^{*}\in F(x,v)\) such that
The following lemma shows that \(\operatorname{GVHVI}(F,J,K)\) is closely related to its generalized Minty variational-hemivariational inequality.
Lemma 4.7
(i) If F is \((f,J)\)-pseudomonotone on K, then every solution of \(\operatorname{GVHVI}(F,J,K)\) solves \(\operatorname{GMVHVI}(F,J,K)\). (ii) If F is upper hemicontinuous on K with nonempty values, then every solution of \(\operatorname{GMVHVI}(F,J,K)\) solves \(\operatorname{GVHVI}(F,J,K)\).
Proof
(i) The conclusion is obvious. Now we prove (ii). Suppose that x is a solution of \(\operatorname{GMVHVI}(F,J,K)\), but it is not a solution of \(\operatorname{GVHVI}(F,J,K)\). Then there exists some \(y\in K\) such that
Since the set \(\{x^{*}\in X^{*}:\langle x^{*},y-x\rangle+J^{\circ}(x,y-x)+f(y)-f(x)<0\}\) is a weakly∗ open neighborhood of \(F(x)\) and F is upper hemicontinuous, setting \(x_{t}=ty+(1-t)x\) for \(t>0\) small enough, we deduce from the positive homogeneousness of \(J^{\circ}\) in the second variable
It follows that, for any \(t>0\),
By the convexity of f, we have
and so \(f(x_{t})-f(x)\leq t(f(y)-f(x))\). Utilizing (4.10) and the subadditivity of \(J^{\circ}\) in the second variable, we obtain that
which immediately leads to
This contradicts the fact that x is a solution of \(\operatorname{GMVHVI}(F,J,K)\). Hence, the conclusion of (ii) holds. This completes the proof. □
Corollary 4.8
(i) If F is f-pseudomonotone on K, then every solution of \(\operatorname{GMVI}(F,K)\) solves \(\operatorname{GMMVI}(F,K)\). (ii) If F is upper hemicontinuous on K with nonempty values, then every solution of \(\operatorname{GMMVI}(F,K)\) solves \(\operatorname{GMVI}(F,K)\).
Proof
Whenever \(J=0\), we know that \(J^{\circ}=0\), \(\operatorname{GMVHVI}(F,J,K)\) (resp., \(\operatorname{GVHVI}(F,J,K)\)) reduces to \(\operatorname{GMMVI}(F,K)\) (resp., \(\operatorname{GMVI}(F,K)\)), and the \((f,J)\)-pseudomonotonicity of F reduces to the f-pseudomonotonicity of F. Utilizing Lemma 4.7, we immediately deduce Corollary 4.8. □
Lemma 4.9
Let K be a nonempty, closed, and convex subset in a reflexive Banach space X, \(f:K\subset X\to\mathbf{R}\) be a convex and lower semicontinuous function, and \(J:X\to\mathbf{R}\) be a locally Lipschitz functional. Suppose that F is upper hemicontinuous and \((f,J)\)-pseudomonotone on K with nonempty values. Consider the following statements:
-
(i)
the solution set of \(\operatorname{GVHVI}(F,J,K)\) is nonempty and bounded;
-
(ii)
the solution set of \(\operatorname{GMVHVI}(F,J,K)\) is nonempty and bounded;
-
(iii)
\(K_{\infty}\cap\{d\in X:\langle y^{*},d\rangle+J^{\circ}(y,d)+f_{\infty}(d)\leq0, \forall y^{*}\in F(y), y\in K\}=\{0\}\).
Then (i)⇔(ii) and (ii)⇒(iii); moreover, if \(\operatorname{int}(\operatorname{barr}(K))\neq\emptyset\), then (iii)⇒(ii) and hence they all are equivalent.
Proof
Under the assumptions of F, the equivalence of (i) and (ii) is stated in Lemma 4.7. Then the conclusion follows from Theorem 3.4. □
Corollary 4.10
Let K be a nonempty, closed, and convex subset in a reflexive Banach space X and \(f:K\subset X\to\mathbf{R}\) be a convex and lower semicontinuous function. Suppose that F is upper hemicontinuous and f-pseudomonotone on K with nonempty values. Consider the following statements:
-
(i)
the solution set of \(\operatorname{GMVI}(F,K)\) is nonempty and bounded;
-
(ii)
the solution set of \(\operatorname{GMMVI}(F,J,K)\) is nonempty and bounded;
-
(iii)
\(K_{\infty}\cap\{d\in X:\langle y^{*},d\rangle+f_{\infty}(d)\leq 0, \forall y^{*}\in F(y), y\in K\}=\{0\}\).
Then (i)⇔(ii) and (ii)⇒(iii); moreover, if \(\operatorname{int}(\operatorname{barr}(K))\neq\emptyset\), then (iii)⇒(ii) and hence they all are equivalent.
Proof
Whenever \(J=0\), we know that \(J^{\circ}=0\), the \((f,J)\)-pseudomonotonicity of F reduces to the f-pseudomonotonicity of F, and statements (i), (ii), and (iii) in Lemma 4.9 reduce to (i), (ii), and (iii) in Corollary 4.10. Utilizing Lemma 4.9, we deduce the desired result. □
Remark 4.11
It is known that if \(J=0\) then Lemmas 4.7 and 4.9 reduce to Lemmas 4.1 and 4.2 in [19], respectively. Thus, Lemmas 4.7 and 4.9 generalize and extend Lemmas 4.1 and 4.2 in [19] from the generalized mixed variational inequality to the generalized variational-hemivariational inequality. If \(f=0\) additionally, then Lemma 4.2 in [19] reduces to Theorem 3.2 of [29]. Therefore, Lemma 4.2 in [19] generalizes Theorem 3.2 of [29] from the generalized variational inequality to the generalized mixed variational inequality.
From Theorem 4.4 and Lemma 4.9, we can easily establish the following stability result for the generalized variational-hemivariational inequality.
Theorem 4.12
Assume that all the conditions of Theorem 4.1 are satisfied. Suppose that
-
(i)
for each \(v\in V\), the mapping \(x\mapsto F(x,v)\) is upper hemicontinuous and \((f,J)\)-pseudomonotone on M;
-
(ii)
the solution set of \(\operatorname{GVHVI}(F(\cdot,v_{0}),J,L(u_{0}))\) is nonempty and bounded.
Then there exists a neighborhood \(U'\times V'\) of \((u_{0},v_{0})\) with \(U'\times V'\subset U\times V\) such that, for every \((u,v)\in U'\times V'\), the solution set of \(\operatorname{GVHVI}(F(\cdot,v),J,L(u))\) is nonempty and bounded. Moreover, if f is continuous on \(M=\bigcup_{u\in U}L(u)\) and \(J^{\circ}:M\times(M-M)\to\mathbf{R}\) is continuous, then ω-\(\limsup_{(u,v)\to(u_{0},v_{0})}S_{G}(u,v) \subset S_{G}(u_{0},v_{0})\), where \(S_{G}(u,v)\) and \(S_{G}(u_{0},v_{0})\) are the solution sets of \(\operatorname{GVHVI}(F(\cdot,v),J,L(u))\) and \(\operatorname{GVHVI}(F(\cdot,v_{0}),J,L(u_{0}))\), respectively.
Proof
Since F is upper hemicontinuous with nonempty values and \((f,J)\)-pseudomonotone on M, it follows from Lemma 4.9 that the solution set of \(\operatorname{GMVHVI}(F(\cdot,v),J,L(u))\) coincides with that of \(\operatorname{GVHVI}(F(\cdot,v),J,L(u))\), and so the result follows directly from Theorem 4.4. This completes the proof. □
Change history
05 April 2019
The authors have retracted this article [1] because it significantly overlaps with the previously published article by Zhong and Huang [2]. All authors agree to this retraction.
References
Xiao, Y.B., Huang, N.J., Wong, M.M.: Well-posedness of hemivariational inequalities and inclusion problems. Taiwan. J. Math. 15(3), 1261–1276 (2011)
Brezis, H.: Operateurs Maximaux Monotone et Semigroups de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)
Salmon, G., Strodiot, J.J., Nguyen, V.H.: A bundle method for solving variational inequalities. SIAM J. Optim. 14, 869–893 (2004)
Crouzeix, J.P.: Pseudomonotone variational variational problems: existence od solutions. Math. Program. 78, 305–314 (1997)
Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. 86(2), 433–438 (1999)
Yao, J.C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 83, 391–403 (1994)
Yuan, G.X.Z.: KKM Theory and Application to Nonlinear Analysis. Dekker, New York (1999)
Minty, G.J.: On the generalization of a direct method of the calculus of variations. Bull. Am. Math. Soc. 73, 315–321 (1967)
Giannessi, F.: On Minty variational principle. In: New Trends in Mathematical Programming, pp. 93–99. Kluwer Academic, Boston (1998)
Fang, Y.P., Huang, N.J., Yao, J.C.: Well-posedness by perturbations of mixed variational inequalities in Banach spaces. Eur. J. Oper. Res. 201, 682–692 (2010)
Grespi, G.P., Ginchev, I., Rocca, M.: Minty variational inequalities, increase along rays property and optimization. J. Optim. Theory Appl. 123, 479–496 (2004)
John, R.: A note on Minty variational inequality and generalized monotonicity. In: Lecture Notes in Economics and Mathematical Systems, vol. 502, pp. 240–246. Springer, Berlin (2001)
Yang, X.M., Yang, X.Q., Teo, K.L.: Some remarks on the Minty vector variational inequality. J. Optim. Theory Appl. 121, 193–201 (2004)
Yao, Y., Noor, M.A., Liou, Y.-C., Kang, S.M.: Iterative algorithms for general multi-valued variational inequalities. Abstr. Appl. Anal. 2012, Article ID 768272 (2012). https://doi.org/10.1155/2012/768272
Ceng, L.C., Liou, Y.-C., Yao, J.-C., Yao, Y.: Well-posedness for systems of time-dependent hemivariational inequalities in Banach spaces. J. Nonlinear Sci. Appl. 10, 4318–4336 (2017)
Gowda, M.S., Pang, J.S.: Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory. Math. Oper. Res. 19, 831–879 (1994)
Cho, S.-Y., Qin, X., Yao, J.-C., Yao, Y.-H.: Viscosity approximation splitting methods for monotone and nonexpansive operators in Hilbert spaces. J. Nonlinear Convex Anal. 19, 251–264 (2018)
Yao, Y.H., Liou, Y.C., Yao, J.C.: Iterative algorithms for the split variational inequality and fixed point problems under nonlinear transformations. J. Nonlinear Sci. Appl. 10, 843–854 (2017)
Zhong, R.Y., Huang, N.J.: Stability analysis for Minty mixed variational inequality in reflexive Banach spaces. J. Optim. Theory Appl. 147, 454–472 (2010)
Yao, Y., Noor, M.A., Liou, Y.C.: Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities. Abstr. Appl. Anal. 2012, Article ID 817436 (2012). https://doi.org/10.1155/2012/817436
Yao, Y., Shahzad, N.: Strong convergence of a proximal point algorithm with general errors. Optim. Lett. 6, 621–628 (2012)
Zegeye, H., Shahzad, N., Yao, Y.H.: Minimum-norm solution of variational inequality and fixed point problem in Banach spaces. Optimization 64, 453–471 (2015)
Yao, Y.H., Chen, R.D., Xu, H.K.: Schemes for finding minimum-norm solutions of variational inequalities. Nonlinear Anal. 72, 3447–3456 (2010)
Yao, Y.H., Liou, Y.C., Kang, S.M.: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Math. Appl. 59, 3472–3480 (2010)
Yao, Y.H., Qin, X., Yao, J.C.: Projection methods for firmly type nonexpansive operators. J. Nonlinear Convex Anal. 19, 407–415 (2018)
Clarke, F.H.: Optimization and Nonsmooth Analysis. SIAM, Philadelphia (1990)
Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications. Springer, Berlin (2005)
Baiocchi, C., Buttazzo, G., Gastaldi, F., Tomarelli, F.: General existence theorems for unilateral problems in continuum mechanics. Arch. Ration. Mech. Anal. 100, 149–189 (1988)
He, Y.R.: Stable pseudomonotone variational inequality in reflexive Banach spaces. J. Math. Anal. Appl. 330, 352–363 (2007)
Fan, J.H., Zhong, R.Y.: Stability analysis for variational inequality in reflexive Banach spaces. Nonlinear Anal. 69, 2566–2574 (2008)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)
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Ceng, LC., Agarwal, R.P., Yao, JC. et al. RETRACTED ARTICLE: On stability analysis for generalized Minty variational-hemivariational inequality in reflexive Banach spaces. J Inequal Appl 2018, 298 (2018). https://doi.org/10.1186/s13660-018-1890-9
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DOI: https://doi.org/10.1186/s13660-018-1890-9