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Secondorder composed contingent epiderivatives and setvalued vector equilibrium problems
Journal of Inequalities and Applications volume 2014, Article number: 406 (2014)
Abstract
In this paper, we introduce the concept of a secondorder composed contingent epiderivative for setvalued maps and discuss some of its properties. Then, by virtue of the secondorder composed contingent epiderivative, we establish secondorder sufficient optimality conditions and necessary optimality conditions for the weakly efficient solution of setvalued vector equilibrium problems with unconstraints and setvalued vector equilibrium problems with constraints, respectively.
MSC:90C46, 91B50.
1 Introduction
The vector equilibrium problem, which contains vector optimization problems, vector variational inequality problems and vector complementarity problems as special case, has been studied (see [1–14]). But so far, most papers focused mainly on the existence of solutions and the properties of the solutions, there are a few papers which deal with the optimality conditions. Giannessi et al. [15] turned the vector variational inequalities with constraints into another vector variational inequalities without constraints. They gave the sufficient conditions for the efficient solution and the weakly efficient solution of the vector variational inequalities in finite dimensional spaces. Morgan and Romaniello [16] gave the scalarization and KuhnTuckerlike conditions for weak vector generalized quasivariational inequalities in Hilbert space by using the subdifferential of the function. Gong [17] presented the necessary and sufficient conditions for the weakly efficient solution, the Henig efficient solution and the superefficient solution for vector equilibrium problems with constraints under the condition of coneconvexity. Qiu [18] presented the necessary and sufficient conditions for globally efficient solutions of vector equilibrium problems under generalized conesubconvexlikeness. Gong and Xiong [19] weakened the convexity assumption in [17] and obtained the necessary and sufficient conditions for weakly efficient solutions of vector equilibrium problems. Under the nearly conesubconvexlikeness, Long et al. [20] obtained the necessary and sufficient conditions for the Henig efficient solution and the superefficient solution to the vector equilibrium problems with constraints. By using the concept of Fréchet differentiability of mapping, Wei and Gong [21] obtained the KuhnTucker optimality conditions for weakly efficient solutions, Henig efficient solutions, superefficient solutions and globally efficient solutions to the vector equilibrium problems with constraints. Ma and Gong [22] obtained the firstorder necessary and sufficient conditions for the weakly efficient solution, the Henig efficient solution, and the globally proper efficient solution to the vector equilibrium problems with constraints.
It is well known that the secondorder tangent sets and higherorder tangent sets introduced in [23], in general, are not cones and convex sets, there are some difficulties in studying secondorder and higherorder optimality conditions for general setvalued optimization problems. Until now, there are some papers to deal with higherorder optimality conditions by virtue of the higherorder derivatives or epiderivatives introduced by the higherorder tangent sets (see [24–32]). However, as far as we know the secondorder optimality conditions of the solutions remain unstudied in setvalued vector equilibrium problems.
Motivated by the work reported in [22, 24, 26–31, 33], we introduce a new secondorder derivative called secondorder composed contingent epiderivative for setvalued maps and obtain some of its properties. By virtue of the secondorder composed contingent epiderivative, we obtain secondorder sufficient optimality conditions and necessary optimality conditions for the weakly efficient solution of setvalued vector equilibrium problems.
The rest of the paper is organized as follows. In Section 2, we recall some notions. In Section 3, we introduce secondorder composed contingent epiderivatives for setvalued maps and discuss some of its properties. In Section 4, we establish secondorder necessary and sufficient optimality conditions for weakly efficient solutions to setvalued vector equilibrium problems.
2 Preliminaries and notations
Throughout this paper, let X, Y, and Z be three real normed spaces, {Y}^{\ast} and {Z}^{\ast} be the topological dual spaces of Y and Z, respectively. {0}_{X}, {0}_{Y} and {0}_{Z} denote the origins of X, Y, and Z, respectively. Let C\subset Y and D\subset Z be closed convex pointed cones in Y and Z, respectively. Let M be a nonempty subset in Y. The cone hull of M is defined by cone(M)=\{tyt\ge 0,y\in M\}. Let {C}^{\ast} be the dual cone of coneC, defined by
Let E be a nonempty subset of X, G:E\to {2}^{Z} be a setvalued map. The domain, the graph and the epigraph of G are defined, respectively, by
Denote
A setvalued map W:X\to {2}^{Y} is said to be

(i)
strictly positive homogeneous if
W(\alpha x)=\alpha W(x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}\alpha >0,\mathrm{\forall}x\in X; 
(ii)
subadditive if
W({x}_{1})+W({x}_{2})\subseteq W({x}_{1}+{x}_{2})+C.
Let E\subset X be a nonempty convex set and G:E\to {2}^{Z} be a setvalued map with G(x)\ne \mathrm{\varnothing}, for all x\in E. G is said to be Dconvex on E, if for any {x}_{1},{x}_{2}\in E and \lambda \in (0,1),
Let X be a normed space supplied with a distance d and K be a subset of X. We denote by d(x,K)={inf}_{y\in K}d(x,y) the distance from x to K, where we set d(x,\mathrm{\varnothing})=+\mathrm{\infty}.
Let K be a nonempty subset of X and x\in K, u\in X. The contingent cone of K at x is
Proposition 2.1 (see [36])
Let K\subseteq X and x\in K. The following statements are equivalent:

(i)
u\in T(K,x);

(ii)
there exist sequences \{{\lambda}_{n}\} with {\lambda}_{n}\to +\mathrm{\infty} and \{{x}_{n}\} with {x}_{n}\in K and {x}_{n}\to x such that {\lambda}_{n}({x}_{n}x)\to v.
Proposition 2.2 (see [23])
Let K\subseteq X and x\in K. Then T(K,x) is a closed cone.
Proposition 2.3 (see [37])
Let K\subseteq X be a convex set, x\in K, and u\in T(K,x). Then
Let E be a nonempty subset of X, F:E\times E\to {2}^{Y} be a setvalued bifunction, F({x}_{1},{x}_{2})\ne \mathrm{\varnothing}, for all {x}_{1},{x}_{2}\in E. We suppose that {0}_{Y}\in F(x,x), for any x\in E.
Let {x}_{0}\in E be given. {F}_{{x}_{0}}:E\to {2}^{Y} is the setvalued map defined by
The set
is called the epigraph of {F}_{{x}_{0}}. Denote
Let G:E\to {2}^{Z} be a setvalued map with G(x)\ne \mathrm{\varnothing}, for all x\in E.
In this paper, we consider the setvalued vector equilibrium problem with unconstraints (USVVEP): find {x}_{0}\in E such that
where {A}_{0}=A\setminus \{{0}_{Y}\}, A is a convex cone in Y.
We also consider the setvalued vector equilibrium problem with constraints (CSVVEP): find {x}_{0}\in K such that
where {A}_{0}=A\setminus \{{0}_{Y}\}, A is a convex cone in Y, and K:=\{x\in E:G(x)\cap (D)\ne \mathrm{\varnothing}\}.
Definition 2.4 Let intC\ne \mathrm{\varnothing}.

(i)
A vector {x}_{0}\in E is called a weakly efficient solution of (USTVEP) if
F({x}_{0},E)\cap (intC)=\mathrm{\varnothing}. 
(ii)
A vector {x}_{0}\in K is called a weakly efficient solution of (CSTVEP) if
F({x}_{0},K)\cap (intC)=\mathrm{\varnothing}.
3 Secondorder composed contingent epiderivatives
Let E\subset X. Let F:E\to {2}^{Y} be a setvalued map, {y}_{0}\in F({x}_{0}), and (u,v)\in X\times Y. We first recall the definition of the generalized secondorder composed contingent epiderivative introduced by Li et al. [24].
Definition 3.1 (see [24])
The generalized secondorder composed contingent epiderivative {D}_{g}^{\prime \prime}F({x}_{0},{y}_{0},u,v) of F at ({x}_{0},{y}_{0}) in the directive (u,v) is the setvalued map from X to Y defined by
Now we introduce the following secondorder composed contingent epiderivatives of setvalued maps, and then we investigate some of its properties.
Definition 3.2 Let ({x}_{0},{y}_{0})\in graph(F), (u,v)\in X\times Y. The secondorder composed contingent epiderivative {D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v) of F at ({x}_{0},{y}_{0}) in the directive (u,v) is the setvalued map from X to Y defined by
Proposition 3.1 Let (\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})\in graph(F), (\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})\in X\times Y, and M:=dom{D}^{\u2033}{F}_{+}(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y},\stackrel{\u02c6}{u},\stackrel{\u02c6}{v}). Then
Proof Let x\in M, y\in {D}^{\u2033}{F}_{+}(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y},\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})(x)+C. Then there exist (x,\overline{y})\in T(T(epiF,(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})),(\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})), and c\in C, such that (x,y)=(x,\overline{y}+c). Since (x,\overline{y})\in T(T(epiF,(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})),(\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})), there exist sequences ({x}_{n},{y}_{n})\to (x,\overline{y}) and {t}_{n}\downarrow {0}^{+}, such that
Moreover, \mathrm{\forall}n\in N, there exist sequences ({x}_{n}^{k},{y}_{n}^{k})\to (\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})+{t}_{n}({x}_{n},{y}_{n}) and {t}_{n}^{k}\downarrow {0}^{+}, such that (\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})+{t}_{n}^{k}({x}_{n}^{k},{y}_{n}^{k})\in epiF, \mathrm{\forall}k\in N. Then we have
Since c\in C, combine with (2), we have \stackrel{\u02c6}{y}+{t}_{n}^{k}({y}_{n}^{k}+{t}_{n}c)=\stackrel{\u02c6}{y}+{t}_{n}^{k}{y}_{n}^{k}+{t}_{n}^{k}{t}_{n}c\in F(\stackrel{\u02c6}{x}+{t}_{n}^{k}{x}_{n}^{k})+C, \mathrm{\forall}n,k\in N. That is (\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})+{t}_{n}^{k}({x}_{n}^{k},{y}_{n}^{k}+{t}_{n}c)\in epi{F}_{+}, \mathrm{\forall}n,k\in N. Since ({x}_{n}^{k},{y}_{n}^{k})\to (\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})+{t}_{n}({x}_{n},{y}_{n}), we have ({x}_{n}^{k},{y}_{n}^{k}+{t}_{n}c)\to (\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})+{t}_{n}({x}_{n},{y}_{n}+c) as k\to +\mathrm{\infty}. Thus,
Simultaneously, ({x}_{n},{y}_{n}+c)\to (x,\overline{y}+c), since ({x}_{n},{y}_{n})\to (x,\overline{y}) as n\to +\mathrm{\infty}. Together with (x,y)=(\overline{x},\overline{y}+c), we have (x,y)=T(T(epiF,(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y})),(\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})), which implies
So
Naturally, {D}^{\u2033}{F}_{+}(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y},\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})(x)\subseteq {D}^{\u2033}{F}_{+}(\stackrel{\u02c6}{x},\stackrel{\u02c6}{y},\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})(x)+C. Thus (1) holds, and this completes the proof. □
By definitions and Proposition 3.1, we can conclude that the following result holds.
Proposition 3.2 Let F:E\to {2}^{Y}, ({x}_{0},{y}_{0})\in graph(F), (u,v)\in T(epi(F),({x}_{0},{y}_{0})), and x\in X. Then {D}_{g}^{\prime \prime}F({x}_{0},{y}_{0},u,v)(x)+C\subseteq {D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v)(x).
Remark 3.1 The inclusion relation
may not hold.
Now we give the following example to explain Remark 3.1.
Example 3.1 Let C={R}_{+}^{2} and F(x)=\{({y}_{1},{y}_{2})\in {R}^{2}{y}_{1}\ge {x}^{2},{y}_{2}\in R\}, \mathrm{\forall}x\in {R}_{+}. Let ({x}_{0},{y}_{0})=(0,(0,0)), and (u,v)=(1,(0,1)). Then
Therefore, for any x\in R, we have
and
And then, for any x\in R, we have
Now we discuss some crucial properties of the secondorder composed contingent epiderivative.
Proposition 3.3 Let ({x}_{0},{y}_{0})\in graphF, (u,v)\in T(epiF,({x}_{0},{y}_{0})) with v\in C and E\subset X be convex. If F is Cconvex on E, then for all x\in E,
Proof Since F is Cconvex on E, epiF is a convex set. So it follows from Proposition 2.3 that
Since for every c\in C, x\in S and y\in F(x), one has
Then it follows from (3) that
Thus, by the definition of the secondorder composed contingent epiderivative, we have
and then
The proof is complete. □
Proposition 3.4 Let ({x}_{0},{y}_{0})\in graph(F), (u,v)\in T(epiF,({x}_{0},{y}_{0})). Then

(i)
{D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v) is strictly positive homogeneous.
Moreover, if F is Cconvex on a nonempty convex set E, then

(ii)
{D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v) is subadditive.
Proof (i) Let \alpha >0 and x\in X.
If y\in {D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v)(x), then there exist sequences \{{h}_{n}\} with {h}_{n}\to {0}^{+} and \{({x}_{n},{y}_{n})\} with ({x}_{n},{y}_{n})\in T(epiF,({x}_{0},{y}_{0})) such that
and then
So (\alpha x,\alpha y)\in T(T(epiF,({x}_{0},{y}_{0})),u,v), and then we can obtain
Thus
The proof of
follows along the lines of (4). So {D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v) is strictly positive homogeneous.

(ii)
Let {x}_{1},{x}_{2}\in X, {y}_{1}\in {D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v)({x}_{1}), {y}_{2}\in {D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v)({x}_{2}). Then one has
({x}_{1},{y}_{1})\in T(T(epiF,({x}_{0},{y}_{0})),(u,v)),\phantom{\rule{2em}{0ex}}({x}_{2},{y}_{2})\in T(T(epiF,({x}_{0},{y}_{0})),(u,v)).
Since F is Cconvex on S, epiF is convex, and then T(T(epiF,({x}_{0},{y}_{0})),(u,v)) is a close and convex cone. Thus we have
and then
Thus
and the proof is complete. □
By the proof of Proposition 3.4, we can conclude that the following result holds.
Proposition 3.5 Let ({x}_{0},{y}_{0})\in graph(F), (u,v)\in T(epiF,({x}_{0},{y}_{0})), M:=dom{D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v). If F is Cconvex on a nonempty convex set E, then {D}^{\u2033}{F}_{+}({x}_{0},{y}_{0},u,v)(M) is a convex cone.
4 Secondorder optimality conditions of weakly efficient solutions
Throughout this section, let {x}_{0}\in K, {y}_{0}={0}_{Y}\in {F}_{{x}_{0}}({x}_{0}), intC\ne \mathrm{\varnothing}, and intD\ne \mathrm{\varnothing}. Firstly, we recall a definition and a result in [25].
Let K\subset X and {x}_{0}\in K. The interior tangent cone of K at {x}_{0} defined as
where {B}_{X}(u,\delta ) stands for the closed ball centered at u\in X and of radius δ.
Lemma 4.1 (see [25])
If K\subset X is convex, {x}_{0}\in K, and intK\ne \mathrm{\varnothing}, then
Theorem 4.1 Let {x}_{0} be a weakly efficient solution of the problem (USVVEP). Then, for every (u,v)\in T(epi{F}_{{x}_{0}},({x}_{0},{y}_{0})) with v\in \partial C, we have
for every x\in dom{D}^{\u2033}{({F}_{{x}_{0}})}_{+}({x}_{0},{y}_{0},u,v).
Proof Suppose to the contrary that there exists an x\in dom{D}^{\u2033}{({F}_{{x}_{0}})}_{+}({x}_{0},{y}_{0},u,v) such that (5) does not hold. Then there exist \lambda \in R and
such that
Let us consider two possible cases for λ.
Case 1: If \lambda >0, then it follows from Proposition 3.1 and (6) that {y}^{\prime}\in {D}^{\u2033}{({F}_{{x}_{0}})}_{+}({x}_{0},{y}_{0},u,v)(x). So
By definition, there exist sequences {\lambda}_{n}\to +\mathrm{\infty} and ({u}_{n},{v}_{n})\in T(epiF,({x}_{0},{y}_{0})) such that ({u}_{n},{v}_{n})\to (u,v) and
It follows from (9), (7), and v\in C that there exists {N}_{1}\in \mathcal{N} such that
Since ({u}_{n},{v}_{n})\in T(epi{F}_{{x}_{0}},({x}_{0},{y}_{0})), for every n\in \mathcal{N}, there exist a sequence {\lambda}_{n}^{k}\to +\mathrm{\infty} as k\to +\mathrm{\infty} and a sequence ({x}_{n}^{k},{y}_{n}^{k})\in epiF, such that ({x}_{n}^{k},{y}_{n}^{k})\to ({x}_{0},{y}_{0}) and
It follows from (10) that there exists {N}_{1}(n)\in \mathcal{N} such that
which implies
Since ({x}_{n}^{k},{y}_{n}^{k})\in epi{F}_{{x}_{0}}, there exists {\overline{y}}_{n}^{k}\in {F}_{{x}_{0}}({x}_{n}^{k}) such that {y}_{n}^{k}\in \{{\overline{y}}_{n}^{k}\}+C. Then, by (12), we have
Therefore
which contradicts that ({x}_{0},{y}_{0}) is a weakly efficient solution of the problem (USVVEP).
Case 2: If \lambda \le 0, then it follows from Proposition v\in C and (7) that y\in intC and
By a similar proof method to case 1, there exist consequences {\overline{x}}_{n}^{k} and {N}_{2},{N}_{2}(n)\in \mathcal{N} such that
which contradicts that {x}_{0} is a weakly efficient solution of the problem (USVVEP). Thus (5) holds, and the proof is complete. □
Remark 4.1 In Theorem 4.1, we cannot use v\in intC instead of v\in \partial C. Since ({x}_{0},{y}_{0}) is a weakly efficient solution of the problem (USVVEP),
It follows from (u,v)\in T(epi({F}_{{x}_{0}}),({x}_{0},{y}_{0})), and {y}_{0}=0 that there exist sequences \{{\lambda}_{n}\} with {\lambda}_{n}\to +\mathrm{\infty} and ({x}_{n},{y}_{n})\in epi({F}_{{x}_{0}}) with ({x}_{n},{y}_{n})\to ({x}_{0},0) such that
So, it follows from v\in intC that there exists N\in \mathcal{N} such that
which implies
Since ({x}_{n},{y}_{n})\in epi({F}_{{x}_{0}}), there exists {\overline{y}}_{n}\in {F}_{{x}_{0}}({x}_{n}) such that {y}_{n}\in \{{\overline{y}}_{n}\}+C. Then, combined with (14), we have
Therefore
which contradicts (13).
Next, we give an example to illustrate Theorem 4.1.
Example 4.1 Let F(x)=\{({y}_{1},{y}_{2})\in {R}^{2}{y}_{1}\in R,{y}_{2}\ge {x}^{2}\}, \mathrm{\forall}x\in {R}_{+}, ({x}_{0},{y}_{0})=(0,(0,0)), and C={R}_{+}^{2}. Then T(epi(F),({x}_{0},{y}_{0}))=\{(u,({v}_{1},{v}_{2}))\in R\times {R}^{2}x\in {R}_{+},{v}_{1}\in R,{v}_{2}\ge 0\}. Take (u,v)=(1,(1,0)). Then
Therefore, for any x\in R, we have
And then, for any x\in R, we have
which shows that Theorem 4.1 holds.
Theorem 4.2 Let (u,v)\in T(epi{F}_{{x}_{0}}({x}_{0},{y}_{0})) with v\in C and E\subset X be convex. If {F}_{{x}_{0}} is Cconvex on E, and for all x\in E,
then {x}_{0} is a weakly efficient solution of the problem (USVVEP).
Proof It follows from Proposition 3.3 that
Then, from (15), we have
So {x}_{0} is a weakly efficient solution of (USVVEP), and the proof is complete. □
Theorem 4.3 Let (u,v,w)\in T(epi({F}_{{x}_{0}},G),{x}_{0},{y}_{0},{z}_{0}) with v\in \partial C and w\in D. If {x}_{0} is a weakly efficient solution of (CSVVEP), then for any {z}_{0}\in G({x}_{0})\cap (D),
for all x\in \mathrm{\Omega}:=dom[{D}^{\u2033}{({F}_{{x}_{0}},G)}_{+}({x}_{0},{y}_{0},{z}_{0},u,v,w+{z}_{0})].
Proof To prove the result by contradiction, suppose that there exists an x\in \mathrm{\Omega} such that (16) does not hold, that is, there exists a (y,z)\in Y\times Z such that
and
Then, by the definition of secondorder composed contingent epiderivatives, there exist sequences {\lambda}_{n}\to +\mathrm{\infty} and ({u}_{n},{v}_{n},{w}_{n})\in T(epi({F}_{{x}_{0}},G),({x}_{0},{y}_{0},{z}_{0})) such that ({u}_{n},{v}_{n},{w}_{n})\to (u,v,w+{z}_{0}) and
It follows from (17) that there exist \mu >0, \nu \in R, c\in intC, and d\in intD such that
Let us consider two possible cases for ν.
Case 1: If \nu \le 0, then, from (19), v\in C, and w,{z}_{0}\in D, we have y\in intC and z\in intD. Thus, by (18), there exists {N}_{1}\in \mathcal{N} such that
Thus, it follows from v\in C and w\in D that
Since ({u}_{n},{v}_{n},{w}_{n})\in T(epi({F}_{{x}_{0}},G),({x}_{0},{y}_{0},{z}_{0})), for every n\in \mathcal{N}, there exist a sequence \{{\lambda}_{n}^{k}\} with {\lambda}_{n}^{k}\to +\mathrm{\infty} as k\to +\mathrm{\infty} and a sequence ({x}_{n}^{k},{y}_{n}^{k},{z}_{n}^{k})\in epi({F}_{{x}_{0}},G), such that ({x}_{n}^{k},{y}_{n}^{k},{z}_{n}^{k})\to ({x}_{0},{y}_{0},{z}_{0}) and
It follows from (20) and (21) that there exists {N}_{1}(n)\in \mathcal{N} such that
which implies
Since ({x}_{n}^{k},{y}_{n}^{k},{z}_{n}^{k})\in epi({F}_{{x}_{0}},G), there exist {\overline{y}}_{n}^{k}\in {F}_{{x}_{0}}({x}_{n}^{k}), {\overline{z}}_{n}^{k}\in G({x}_{n}^{k}), c\in C and d\in D such that {y}_{n}^{k}={\overline{y}}_{n}^{k}+c and {z}_{n}^{k}={\overline{z}}_{n}^{k}+d. Then, by (22) and {y}_{0}=0, we have
So
which contradicts that {x}_{0} is a weakly efficient solution of (CSVVEP).
Case 2: If \nu >0, then, from (19), we get y=\mu \nu (\frac{1}{\nu}c+v) and z=\mu \nu (\frac{1}{\nu}d+(w+{z}_{0})). So it follows from c\in intC and d\in intD that
Then, by Lemma 4.1 and (23), we get y\in IT(intC,v) and z=IT(intD,w+{z}_{0}). Therefore, there exists \delta >0 such that
For this δ, it follows from (18) that there exists {N}_{2}\in \mathcal{N} such that
Then, by (24) and (25), we have
Thus, from v\in C, w,{z}_{0}\in D, and \delta {\lambda}_{n}>1, \mathrm{\forall}n>{N}_{2}, we have
By a similar proof method to case 1, there exists {N}_{2}(n)\in \mathcal{N} such that
which contradicts that {x}_{0} is a weakly efficient solution of (CSVVEP). Thus (16) holds and the proof is complete. □
Theorem 4.4 Let (u,v,w)\in T(epi({F}_{{x}_{0}},G),{x}_{0},{y}_{0},{z}_{0}) with v\in \partial C and w\in D. If {x}_{0} is a weakly efficient solution of (CSVVEP), then for any {z}_{0}\in G({x}_{0})\cap (D),
for all x\in \mathrm{\Omega}:=dom[{D}^{\u2033}{({F}_{{x}_{0}},G)}_{+}({x}_{0},{y}_{0},{z}_{0},u,v,w+{z}_{0})].
Proof To prove the result by contradiction, suppose that there exists an x\in \mathrm{\Omega} such that (26) does not hold, that is, there exists (y,z)\in Y\times Z such that
Then, by the definition of secondorder composed contingent epiderivatives, there exist sequences {\lambda}_{n}\to +\mathrm{\infty} and ({u}_{n},{v}_{n},{w}_{n})\in T(epi({F}_{{x}_{0}},G),({x}_{0},{y}_{0},{z}_{0})) such that ({u}_{n},{v}_{n},{w}_{n})\to (u,v,w+{z}_{0}) and
It follows from (27) that there exist c\in intC and d\in intD such that
Thus, by (28), there exists {N}_{1}\in \mathcal{N} such that
Thus, it follows from v\in C and w,{z}_{0}\in D that
Since ({u}_{n},{v}_{n},{w}_{n})\in T(epi({F}_{{x}_{0}},G),({x}_{0},{y}_{0},{z}_{0})), for every n\in \mathcal{N}, there exist a sequence \{{\lambda}_{n}^{k}\} with {\lambda}_{n}^{k}\to +\mathrm{\infty} as k\to +\mathrm{\infty} and a sequence ({x}_{n}^{k},{y}_{n}^{k},{z}_{n}^{k})\in epi({F}_{{x}_{0}},G), such that ({x}_{n}^{k},{y}_{n}^{k},{z}_{n}^{k})\to ({x}_{0},{y}_{0},{z}_{0}) and
It follows from (30) and (31) that there exists {N}_{1}(n)\in \mathcal{N} such that
which implies
Since ({x}_{n}^{k},{y}_{n}^{k},{z}_{n}^{k})\in epi({F}_{{x}_{0}},G), there exist {\overline{y}}_{n}^{k}\in {F}_{{x}_{0}}({x}_{n}^{k}), {\overline{z}}_{n}^{k}\in G({x}_{n}^{k}), c\in C, and d\in D such that {y}_{n}^{k}={\overline{y}}_{n}^{k}+c and {z}_{n}^{k}={\overline{z}}_{n}^{k}+d. Then, by (32) and {y}_{0}=0, we have
So
which contradicts that {x}_{0} is a weakly efficient solution of (CSVVEP). Thus (26) holds and the proof is complete. □
Theorem 4.5 Let E\subset X be a nonempty convex set, {z}_{0}\in G({x}_{0})\cap (D) and (u,v,w)\in X\times (C)\times (D). Suppose that the following conditions are satisfied:

(i)
({F}_{{x}_{0}},G) is C\times Dconvex on E.

(ii)
{x}_{0} is a weakly efficient solution of (CSVVEP).
Then there exist \varphi \in {C}^{\ast} and \psi \in {D}^{\ast}, not both zero functionals, such that
where A:={\bigcup}_{x\in \mathrm{\Omega}}{D}^{\u2033}{({F}_{{x}_{0}},G)}_{+}({x}_{0},{y}_{0},{z}_{0},u,v,w+{z}_{0})(x) and \mathrm{\Omega}:=dom[{D}^{\u2033}{({F}_{{x}_{0}},G)}_{+}({x}_{0},{y}_{0},{z}_{0},u,v,w+{z}_{0})].
Proof Define M=A+({0}_{Y},{z}_{0}). By Proposition 3.5, we see that M is a convex set. By Theorem 4.4, we get
By the separation theorem of convex sets, there exist \varphi \in {Y}^{\ast} and \psi \in {Z}^{\ast}, not both zero functionals, such that
Since intC\cup \{{0}_{Y}\} and intD\cup \{{0}_{Z}\} are cones, by (33), we have
and
From (34), we find that ψ is bounded below on intD. Then \psi (z)\ge 0, for all z\in intD. Naturally \psi \in {D}^{\ast}.
By a similar line of proof to \psi \in {D}^{\ast}, we can obtain \varphi \in {C}^{\ast}.
It follows from Proposition 3.5 that ({0}_{Y},{0}_{Z})\in A, and then, from {z}_{0}\in D, \psi \in {D}^{\ast}, and (35), we obtain
The proof is complete. □
Theorem 4.6 Let E\subset X be a nonempty convex set, (u,v,w)\in T(epi({F}_{{x}_{0}},G),{x}_{0},{y}_{0},{z}_{0}) with v\in C and w\in D and {z}_{0}\in G({x}_{0})\cap (D). Suppose that the following conditions are satisfied:

(i)
({F}_{{x}_{0}},G) is C\times Dconvex on E;

(ii)
there exist \varphi \in {C}^{\ast}\setminus \{0\} and \psi \in {D}^{\ast} such that
inf\{\bigcup _{(y,z)\in V}\varphi (y)+\psi (z)\}=0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\psi ({z}_{0})=0,
where U:={\bigcup}_{x\in \mathrm{\Omega}}{D}^{\u2033}{({F}_{{x}_{0}},G)}_{+}({x}_{0},{y}_{0},{z}_{0},u,v,w)(x) and V:=dom[{D}^{\u2033}{({F}_{{x}_{0}},G)}_{+}({x}_{0},{y}_{0},{z}_{0},u,v,w)].
Then {x}_{0} is a weakly efficient solution of (CSVVEP).
Proof To prove the result by contradiction, suppose that {x}_{0} is not a weakly efficient solution of (CSVVEP). Then there exist {x}^{\prime}\in K and {y}^{\prime}\in F({x}_{0},{x}^{\prime}) such that {y}^{\prime}\in intC. Since {x}^{\prime}\in K, there exists {z}^{\prime}\in G({x}^{\prime})\cap (D). It follows from assumption (i) and Proposition 3.3 that we have
and then, from assumption (ii), we obtain
Since {y}^{\prime}{y}_{0}={y}^{\prime}\in intC, \varphi \in {C}^{\ast}\setminus \{0\}, \varphi ({y}^{\prime}{y}_{0})<0. It follows from {z}^{\prime}\in G({x}^{\prime})\cap (D), \psi \in {D}^{\ast}, and \psi ({z}_{0})=0 that \psi ({z}^{\prime}{z}_{0})\le 0, thus
which contradicts (36). So {x}_{0} is a weakly efficient solution of (CSVVEP), and this completes the proof. □
5 Conclusions
In this paper, we propose a new concept of a secondorder derivative for setvalued maps, which is called the secondorder composed contingent epiderivative, and we investigate some of its properties. Simultaneously, by virtue of the derivative, we obtain secondorder sufficient optimality conditions and necessary optimality conditions for setvalued equilibrium problems.
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Acknowledgements
The authors would like to thank anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by Science and Technology Research Project of Chongqing Municipal Education Commission (KJ130414), Chongqing Natural Science Foundation Project of CQ CSTC(cstc2012jjA00038) and the National Natural Science Foundation of China (Nos: 11171362, 11271389, 11201509, and 11301571).
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Wang, Q., Lin, Z., Zeng, J. et al. Secondorder composed contingent epiderivatives and setvalued vector equilibrium problems. J Inequal Appl 2014, 406 (2014). https://doi.org/10.1186/1029242X2014406
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DOI: https://doi.org/10.1186/1029242X2014406
Keywords
 setvalued vector equilibrium problems
 secondorder composed contingent epiderivatives
 weakly efficient solutions
 secondorder optimality conditions