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On the existence and essential components of solution sets for systems of generalized quasi-variational relation problems

Abstract

In this paper, we study the existence of a solution for a system of quasi-variational relation problems (in short, (SQVR)). Moreover, we discuss the existence of essentially connected components of the solution set for (SQVR). Then the obtained results are applied to systems of quasi-variational inclusions and to systems of weak vector quasi-equilibrium problems. The results presented in the paper improve and extend many results from the literature. Some examples are given to illustrate our results.

MSC:47J20, 49J40.

1 Introduction and preliminaries

Variational relation problems were first introduced and studied by Luc in [1]. These problems include as special cases variational inclusion problems, vector equilibrium problems, vector variational inequality problems and vector optimization problems, etc. Later, the results of many authors had been extended and studied as regards the existence and stability of solutions in different models; see for example [211] and the references therein.

In 1950, Fort [12] first introduced the notion of essential fixed points of a continuous mapping from a compact metric space into itself and proved that any mapping can be approximately closed by a mapping whose fixed points are all essential. Later, Kinoshita [13] introduced the notion of essential components of the set of fixed points of a single-valued map and proved that there exists at least one essential component of the set of its fixed points. Recently, the essential components of the solution set have been studied for vector equilibrium problems [14, 15], vector variational inequality problems [16], etc. Very recently, Yang and Pu [17] introduced and studied the system of strong vector quasi-equilibrium problems (in short, (SSVQEP)) and also obtained the existence of essential components for these problems.

Motivated by the research works mentioned above, in this paper, we introduce the system of generalized quasi-variational relation problems. Then we establish some existence theorems of solution sets for this problem. Moreover, we also obtain an existence theorem for essentially connected components of the set of solutions for a system of generalized quasi-variational relation problems. These results are then applied to systems of quasi-variational inclusion problems and systems of weak vector quasi-equilibrium problems.

Now, we pass to our problem setting. Let I={1,,n} be an index set. For each iI, let X i and Y i be two real locally convex Hausdorff topological vector spaces and K i a nonempty convex compact subset of X i . Denote

K i ˆ = j I , j i K j ,K= i I K i = K i × K i ˆ ,X= i I X i .

For each xK, we can write x=( x i , x i ˆ ). For each iI, let S i , T i :K 2 K i be set-valued mappings, and let R i ( x i , x i ˆ , y i ) be a relation linking x i K i , x i ˆ K i ˆ and y i K i . We consider the following system of generalized quasi-variational relation problems (in short, (SQVR)).

(SQVR): Find ( x ¯ i , x ¯ i ˆ )K such that, for each iI, x ¯ i S i ( x ¯ ) and

R i ( x ¯ i , x ¯ i ˆ , y i ) holds, y i T i ( x ¯ ),

where x ¯ is a solution of (SQVR). We denote by Σ(R) the solution set of (SQVR).

Next, we recall some basic definitions and some of their properties.

Let X, Y be two topological vector spaces; let A be a nonempty subset of X and F:A 2 Y be a multifunction.

  1. (i)

    F is said to be lower semicontinuous (lsc) at x 0 A if F( x 0 )U for some open set UY implies the existence of a neighborhood N of x 0 such that F(x)U, xN.

  2. (ii)

    F is said to be upper semicontinuous (usc) at x 0 A if, for each open set UG( x 0 ), there is a neighborhood N of x 0 such that UF(x), xN.

  3. (iii)

    F is said to be continuous at x 0 A if it is both lsc and usc at x 0 A.

  4. (iv)

    F is said to be closed if Graph(F)={(x,y):xA,yF(x)} is a closed subset in A×Y.

Let A, B, C be convex sets in topological vector spaces and R(x,y,z) be a relation between elements of the three sets. The relation R is said to be closed if the set {(x,y,z)A×B×C:R(x,y,z) holds} is closed. The relation R is said to be convex in the first variable if whenever R( x 1 ,y,z) holds and R( x 2 ,y,z) holds, then R(λ x 1 +(1λ) x 2 ,y,z) holds, for all x 1 , x 2 A, λ[0,1].

Definition 1.1 ([18])

Let X, Y be two topological vector spaces, A is a nonempty subset of X, and F:A 2 Y be a multifunction; and CY is a nonempty, closed, and convex cone. F is called upper C-continuous at x 0 A, if, for any neighborhood U of the origin in Y, there is a neighborhood V of x 0 such that

F(x)F( x 0 )+U+C,xV.

Definition 1.2 ([18])

Let X and Y be two topological vector spaces and A be a nonempty convex subset of X. A set-valued mapping F:A 2 Y is said to be properly C-quasiconvex if, for any x,yA and λ[0,1], we have

either F ( x ) F ( t x + ( 1 t ) y ) + C or F ( y ) F ( t x + ( 1 t ) y ) + C .

Lemma 1.1 ([18])

Let X, Y be two Hausdorff topological vector spaces and F:X 2 Y be a multivalued map.

  1. (i)

    If F is upper semicontinuous with closed values, then F is closed.

  2. (ii)

    If F is closed and Y is compact, then F is upper semicontinuous.

Lemma 1.2 ([19])

Let X, Y be two Hausdorff topological vector spaces and F:X 2 Y be a set-valued mapping with compact values. Then F is upper semicontinuous x 0 X if and only if, for each net { x α }X which converges to x 0 X and for each net { y α }F( x α ), there exist y 0 F( x 0 ) and a subnet { y β } of { y α } such that y β y 0 .

Lemma 1.3 ([20])

Let A be a nonempty compact convex subset of a Hausdorff topological vector space X. Suppose that M:A 2 A {} be a set-valued map with the following conditions:

  1. (i)

    for each at xA, M(x) is convex;

  2. (ii)

    for each at xA, xM(x);

  3. (iii)

    for each at yA, M 1 (y)={xA:yM(x)} is open in A.

Then there exists x 0 A such that M( x 0 )=.

Lemma 1.4 (Kakutani-Fan-Glicksberg [21])

Let A be a nonempty compact convex subset of a locally convex Hausdorff vector topological space X. If F:A 2 A is upper semicontinuous and for any xA, F(x) is nonempty, convex, and closed, then there exists an x A such that x F( x ).

2 Existence of solutions

In this section, we establish an existence theorem of solutions for system of generalized quasi-variational relation problems.

Theorem 2.1 For each iI, assume that

  1. (i)

    S i is upper semicontinuous in K with nonempty compact convex values;

  2. (ii)

    T i is lower semicontinuous in K with nonempty convex values;

  3. (iii)

    for all ( x i , x i ˆ ) K i × K i ˆ , R i ( x i , x i ˆ , x i ) holds;

  4. (iv)

    the set {( x i ˆ , y i ) K i ˆ × K i : R i (, x i ˆ , y i ) does not hold} is convex in K i ;

  5. (v)

    the relation R i is convex in the first variable and closed.

Then the (SQVR) has a solution, i.e., there exists ( x ¯ i , x ¯ i ˆ )K such that, for each iI, x ¯ i S i ( x ¯ ) and

R i ( x ¯ i , x ¯ i ˆ , y i ) holds, y i T i ( x ¯ ).

Moreover, the solution set of the (SQVR) is closed.

Proof We define a set-valued map: Φ:K 2 K by Φ(x)= i I Φ i (x), where

Φ i (x)= { a i S i ( x ) : R i ( a i , x i ˆ , y i )  holds , y i T i ( x ) } ,xK.

(The map Φ is called the best-reply map; see [14, 17].)

For each iI:

  1. (I)

    For any xK, we show that Φ i (x) is nonempty.

Indeed, for all xK, we define a set-valued map M i : S i (x) 2 S i ( x ) {} by

M i (x)= { y i T i ( x ) : R i ( x i , x i ˆ , y i )  does not hold } ,for each  x i S i (x).
  1. (a)

    For each x i S i (x), by condition (iv), M i (x) is a convex set.

  2. (b)

    For each x i S i (x), by condition (iii), x i M i ( x i ).

  3. (c)

    For each y i T i (x), by condition (v), the set M i 1 ( y i )={ x i S i (x): y i M i (x)}={ x i S i (x): R i ( x i , x i ˆ , y i ) does not hold} is open in S i (x).

By Lemma 1.3, there exists x ¯ i S i (x) such that M i ( x ¯ i )=, i.e., Φ i (x).

  1. (II)

    We show that Φ i (x) is convex.

Let a i 1 , a i 2 Φ i (x) and λ[0,1] and put a i =λ a i 1 +(1λ) a i 2 . Since a i 1 , a i 2 S i (x) and S i (x) is convex, we have a i S i (x). Thus, for a i 1 , a i 2 Φ i (x), it follows that

R i ( a i , x i ˆ , y i ) holds, y i T i (x).

By (v), R i is convex in the first variable, we have

R i ( λ a i 1 + ( 1 λ ) a i 2 , x i ˆ , y i )  holds,λ[0,1], y i T i (x),

i.e., a i Φ i (x). Therefore, Φ i (x) is convex.

  1. (III)

    We will prove that Φ i is upper semicontinuous in K with nonempty compact values.

Since K is a compact set, it suffices to show that Φ i is a closed mapping. Indeed, let { ( a i α , x α ) } α Λ be any a net in Graph( Φ i ) such that ( a i α , x α )( a i 0 , x 0 ). Now, we need only prove that a i 0 Φ i ( x 0 ). Since a i α S i ( x α ) and S i is upper semicontinuous at x 0 K with nonempty compact values, we have S i is closed at x 0 K, thus, a i 0 S i ( x 0 ). Suppose to the contrary a i 0 Φ i ( x 0 ). Then y i 0 T i ( x 0 ) such that

R i ( a i 0 , x i ˆ , y i 0 )  does not hold.
(2.1)

By the lower semicontinuity of T i , there is a net { y i α } with y i α T i ( x α ) such that y i α y i 0 . Since a i α Φ i ( x α ),

R i ( a i α , x i ˆ α , y i α )  holds.
(2.2)

By condition (v) and (2.2),

R ( a i 0 , x i ˆ , y i 0 )  holds.
(2.3)

This is a contradiction between (2.1) and (2.3). Thus, a i 0 Φ i ( x 0 ). Hence, Φ i is upper semicontinuous in K with nonempty compact values.

By the definition of the mapping Φ is upper semicontinuous with nonempty compact values. By Lemma 1.4, there exists ( x ¯ i , x ¯ i ˆ )K such that, for each iI, x ¯ i S i ( x ¯ ) and

R i ( x ¯ i , x ¯ i ˆ , y i ) holds, y i T i ( x ¯ ).
  1. (IV)

    Now we prove that Σ(R) is closed.

Let a net { x α ,αI}Σ(R): x α x 0 . We need to prove that x 0 Σ(R). Indeed, by the lower semicontinuity of T i , for any y i 0 T i ( x 0 ), there exists y i α T i ( x α ) such that y i α y i 0 . As x α Σ(R),

R i ( x i α , x i ˆ α , y i α )  holds.

Since S i is upper semicontinuous with nonempty and closed values, by Lemma 1.1(i), we find that S i is closed. Thus, x 0 S i ( x 0 ). By condition (v),

R i ( x i 0 , x i ˆ 0 , y i 0 )  holds.

This means that x 0 Σ(R). Thus Σ(R) is a closed set. □

If we let S i (x)= T i (x)= K i for each iI and xX, then (SQVR) becomes the following system of variational relation problems (in short, (SVR)). So, we obtain following result.

Corollary 2.1 For each iI, assume that

  1. (i)

    for all ( x i , x i ˆ ) K i × K i ˆ , R i ( x i , x i ˆ , x i ) holds;

  2. (ii)

    the set {( x i ˆ , y i ) K i ˆ × K i : R i (, x i ˆ , y i ) does not hold} is convex in K i ;

  3. (iii)

    the relation R i is convex in the first variable and closed.

Then the (SVR) has a solution, i.e., there exists ( x ¯ i , x ¯ i ˆ )K such that, for each iI,

R i ( x ¯ i , x ¯ i ˆ , y i ) holds, y i K i .

Moreover, the solution set of the (SVR) is closed.

If I is a singleton, S i (x)=S(x), T i (x)=T(x), then (SQVR) reduces to the following quasi-variational relation problem (in short, (QVR)). So, we also obtain the following result.

Corollary 2.2 Assume that

  1. (i)

    S is upper semicontinuous in K with nonempty compact convex values;

  2. (ii)

    T is lower semicontinuous in K with nonempty convex values;

  3. (iii)

    for all xK, R(x,x) holds;

  4. (iv)

    the set {yK:R(,y) does not hold} is convex in K;

  5. (v)

    the relation R is convex in the first variable and closed.

Then the (QVR) has a solution, i.e., there exists x ¯ K such that x ¯ S( x ¯ ) and

R( x ¯ ,y) holds,yT( x ¯ ).

Moreover, the solution set of the (QVR) is closed.

Remark 2.1

  1. (i)

    For each iI, let X i , Y i and K i , K i ˆ , K, T i = S i be as in (SQVR). Let F i : K i × K i ˆ × K i 2 Y i be a set-valued mapping, C i Y i be a nonempty, closed, and convex cone, and the relation R i be defined as follows:

    R i ( x i , x i ˆ , y i ) holdsiff F i ( x i , x i ˆ , y i ) C i .

    Then (SQVR) becomes the system of strong vector quasi-equilibrium problem (in short, (SSVQEP)) studied in [17].

  2. (ii)

    Let S i (x)= T i (x)= K i for each iI and xX and let F i , C i as in Remark 2.1(i). Then (SVR) becomes the system of strong vector equilibrium problems (in short, (SSVEP)) studied in [17].

  3. (iii)

    If I is a singleton, S i (x)=S(x), T i (x)=T(x), and let F:K×K 2 Y be a set-valued mapping, CY be a nonempty, closed, and convex cone, and the relation R be defined as follows:

    R(x,y) holdsiffF(x,y)C.

    Then (QVR) becomes the strong vector quasi-equilibrium problem (in short, (SQVEP)) studied in [17].

  4. (iv)

    Yang and Pu [17] obtained some existence results for the system of strong vector quasi-equilibrium problems. However, the assumptions of theorems in [17] are different from the assumptions of Theorem 2.1, Corollary 2.1, and Corollary 2.2.

  5. (v)

    Corollary 2.2 is a particular case of Theorems 3.1 and 3.3 in [2]. However, the assumptions and proof methods in Theorems 3.1 and 3.3 in [2] are different from the assumptions and proof methods of Corollary 2.2.

In the next example all assumptions of Corollary 2.2 are satisfied, but Theorem 3.3 in [17] is not applicable. The reason is that F is neither upper C-continuous nor properly C-quasiconvex.

Example 2.1 Let I={1}, X i = Y i =R, K i =[0,1], C= R + , and let S i , T i : K i × K i 2 K i and F: K i × K i 2 Y i be defined by

S i ( x , y ) = T i ( x , y ) = [ 0 , 1 ] , F i ( x i , x i ˆ , y i ) = F ( x , y ) = { [ 1 , 2 ] if  x 0 = 1 2 , [ 1 2 , 1 ] otherwise .

We let the relation R i be defined by R i ( x i , x i ˆ , y i ) holding iff F i ( x i , x i ˆ , y i )=F(x,y) R + . We show that all assumptions of Corollary 2.2 are satisfied. However, F is neither upper C-continuous nor properly C-quasiconvex at x 0 = 1 2 .

Firstly, we prove that F is not upper C-continuous at x 0 = 1 2 . Indeed, we let a neighborhood U=[ 1 6 , 1 6 ] of the origin in Z, then for any neighborhood V=[ 1 2 ε, 1 2 +ε] of x 0 = 1 2 , where ε>0, we choose 1 2 x V and y= 1 2 . Then

F ( x , y ) = F ( x , 1 2 ) = [ 1 2 , 1 ] F ( x 0 , y ) + U + C = F ( 1 2 , 1 2 ) + [ 1 6 , 1 6 ] + R + = [ 1 , 2 ] + [ 1 6 , 1 6 ] + R + = [ 5 6 , 13 6 ] + R + .

Next, we show that F is not properly C-quasiconvex at x 0 = 1 2 . Indeed, we let y= 1 2 , λ= 1 2 , and x 1 =0, x 2 =1. Then

F ( x 1 , y ) = F ( 0 , 1 2 ) = [ 1 2 , 1 ] F ( x 1 λ + ( 1 λ ) x 2 , y ) + C F ( x 1 , y ) = F ( 1 2 , 1 2 ) + R + = [ 1 , 2 ] + R + , F ( x 2 , y ) = F ( 1 , 1 2 ) = [ 1 2 , 1 ] F ( x 1 λ + ( 1 λ ) x 2 , y ) + C F ( x 2 , y ) = F ( 1 2 , 1 2 ) + R + = [ 1 , 2 ] + R + .

Thus, it gives also the case where Corollary 2.2 can be applied, but Theorem 3.3 in [17] does not work.

3 Essential components

In this section, we discuss the existence of essential components for (SQVR).

First, we recall some notions; see [17, 22, 23]. Let A be a nonempty and compact subset of a linear normed X. Denote by M the set of all upper semicontinuous maps R:A 2 A with nonempty convex compact values. For any R,PM, we define ξ(R,P)= sup x A H(R(x),P(x)), where H is the Hausdorff metric defined in A. It is easy to verify that (M,ξ) is a metric space. For each RM, we denote by Ξ(R) the set of all fixed points of R. By Kakutani-Fan-Glicksberg’s fixed point theorem, Ξ(R) is a nonempty compact set.

For each RM, the connected component of a point xΞ(R) is the union of all the connected subsets of Ξ(R) containing x. Note that the connected components are connected closed subsets of Ξ(R), and, since A is compact, thus, all the connected components are connected compact. It is easy to see that the connected components of two distinct points of Ξ(R) either coincide or are disjoint, so that all connected components constitute a decomposition of Ξ(R) into connected pairwise disjoint compact subsets, i.e.,

Ξ(R)= α Λ Ξ α (R),

where Λ is an index set. For each αΛ, Ξ α (R) is a nonempty connected compact subset of Ξ(R), and for any α,βΛ (αβ), Ξ α (R) Ξ β (R)=.

Definition 3.1 ([22])

Let RM and E be a nonempty and closed subset of Ξ(R). E is said to be an essential set of Ξ(R) if, for each open set OE, there exists an open neighborhood U of R in M such that for any R U with Ξ( R )O. If a connected component Ξ α (R) of Ξ(R) is an essential set with respect to M, then Ξ α (R) is said be an essential component of Ξ(R) with respect to M.

Lemma 3.1 ([23])

For any RM, there is at least one essential component of Ξ(R) with respect to M.

Next, we discuss the existence of essential components for (SQVR).

Let Ω be the collection of all (SQVR) satisfying the conditions of Theorem 2.1. For each ωΩ, denote by Ψ(ω) the solution set of ω. It is easy to see that Ψ(ω)=Ξ(Φ), where Φ is the best-reply map of ω. By the proof of Theorem 2.1, we know ΦM.

Definition 3.2 Let ωΩ and Ξ α a connected component of Ψ(ω). Ξ α is said to be essential if it, as a connected component of Ξ(Φ), is an essential component of Ξ(Φ) with respect to M, where Φ is the best-reply map of ω.

By Lemma 3.1, we obtain the following result.

Theorem 3.1 For any ωΩ, there is at least one essential component of Ψ(ω).

Remark 3.1

  1. (i)

    In the special case of Remark 2.1, Theorem 3.1 improves and extends Theorems 4.1 and 4.2 in [17].

  2. (ii)

    In 2011, Khanh and Quan [24] studied the existence of essential components for generalized KKM points. Moreover, Khanh and Quan also applied these results to optimization-related problems. However, the assumptions and proof methods are very different from the assumptions and proof methods in Theorem 3.1.

  3. (iii)

    If I is a singleton, then (SQVR) becomes the variational relation problem (in short, (VR)) studied by Khanh and Luc in [3]. However, Khanh and Luc discussed some kinds of semicontinuous sets as outer-continuous, inner-continuous, inner-open, and outer-open solution sets for (VR), while our Theorem 3.1 discusses the existence of essential components for (SQVR) by using Kakutani-Fan-Glicksberg’s fixed point theorem.

4 Applications (I): system of quasi-variational inclusion problems

For each iI, let X i , Y i and K i , K i ˆ , K, T i = S i be as in (SQVR). Let F i : K i × K i ˆ × K i 2 Y i be a set-valued mapping. We consider the following system of quasi-variational inclusion problems (in short, (SQIP)): Find ( x ¯ i , x ¯ i ˆ )K such that, for each iI, x ¯ i S i ( x ¯ ) and

0 F i ( x ¯ i , x ¯ i ˆ , y i ), y i S i ( x ¯ ),

where x ¯ is a solution of (SQIP).

Definition 4.1 Let X, Y, Z be topological vector spaces. Suppose F:X×Y×X 2 Z is a multifunction. F is said to be generalized {0}-quasiconvex (in the first variable) in a convex set AX, if, whenever 0F( x 1 ,y,z) and 0F( x 2 ,y,z), then 0F(λ x 1 +(1λ) x 2 ,y,z), x 1 , x 2 A, λ[0,1].

Theorem 4.1 For each iI, assume that

  1. (i)

    S i is continuous in K with nonempty compact convex values;

  2. (ii)

    for all ( x i , x i ˆ ) K i × K i ˆ , 0 F i ( x i , x i ˆ , x i );

  3. (iii)

    the set {( x i ˆ , y i ) K i ˆ × K i :0 F i (, x i ˆ , y i )} is convex in K i ;

  4. (iv)

    for all ( x i ˆ , y i ) K i ˆ × K i , F i (, x i ˆ , y i ) is generalized {0}-quasiconvex in K i ;

  5. (v)

    the set {( x i , x i ˆ , y i ) K i × K i ˆ × K i :0 F i ( x i , x i ˆ , y i )} is closed.

Then the (SQIP) has a solution, i.e., there exists ( x ¯ i , x ¯ i ˆ )K such that, for each iI, x ¯ i S i ( x ¯ ) and

0 F i ( x ¯ i , x ¯ i ˆ , y i ), y i S i ( x ¯ ).

Moreover, the solution set of the (SQIP) is closed.

Proof Let the relation R i be defined as follows:

R i ( x i , x i ˆ , y i ) holdsiff0 F i ( x i , x i ˆ , y i ).

Then the problem (SQIP) becomes a particular case of (SQVR) and Theorem 4.1 is a direct consequence of Theorem 2.1. □

Next, we discuss the existence of essential components for (SQIP).

Let Θ be the collection of all (SQIP) satisfying the conditions of Theorem 4.1. For each θΘ, denote by Δ(θ) the solution set of θ. It is easy to see that Δ(θ)=Ξ(Φ), where Φ is the best-reply map of θ. By the proof of Theorem 4.1, we know ΦM.

Definition 4.2 Let θΘ and Ξ α be a connected component of Δ(θ). Ξ α is said to be essential if it, as a connected component of Ξ(Φ), is an essential component of Ξ(Φ) with respect to M, where Φ is the best-reply map of θ.

By Lemma 3.1, we also obtain the following result.

Theorem 4.2 For any θΘ, there is at least one essential component of Δ(θ).

5 Applications (II): system of weak vector quasi-equilibrium problems

For each iI, let X i , Y i and K i , K i ˆ , K, T i = S i be as in (SQVR). Let f i : K i × K i ˆ × K i Y i be a vector function and C i Y i be a nonempty, closed, and convex cone. We consider the following weak system of quasi-equilibrium problems (in short, (WSQVEP)).

(WSQVEP): Find ( x ¯ i , x ¯ i ˆ )K such that, for each iI, x ¯ i S i ( x ¯ ) and

f i ( x ¯ i , x ¯ i ˆ , y i )int C i , y i S i ( x ¯ ).

Definition 5.1 Let X, Y, Z be topological vector spaces and CZ be a nonempty, closed, and convex cone. Suppose f:X×Y×XZ is a vector function. f is said to be weakly C-quasiconvex (in the first variable) in a convex set AX, if whenever f( x 1 ,y,z)intC and f( x 2 ,y,z)intC, then f(λ x 1 +(1λ) x 2 ,y,z)intC, x 1 , x 2 A, λ[0,1].

Theorem 5.1 For each iI, assume that

  1. (i)

    S i is continuous in K with nonempty compact convex values;

  2. (ii)

    for all ( x i , x i ˆ ) K i × K i ˆ , f i ( x i , x i ˆ , x i )int C i ;

  3. (iii)

    the set {( x i ˆ , y i ) K i ˆ × K i : f i (, x i ˆ , y i )int C i } is convex in K i ;

  4. (iv)

    for all ( x i ˆ , y i ) K i ˆ × K i , f i (, x i ˆ , y i ) is weakly C i -quasiconvex in K i ;

  5. (v)

    the set {( x i , x i ˆ , y i ) K i × K i ˆ × K i : f i ( x i , x i ˆ , y i )int C i } is closed.

Then the (WSQVEP) has a solution, i.e., there exists ( x ¯ i , x ¯ i ˆ )K such that, for each iI, x ¯ i S i ( x ¯ ) and

f i ( x ¯ i , x ¯ i ˆ , y i )int C i , y i S i ( x ¯ ).

Moreover, the solution set of the (WSQVEP) is closed.

Proof Let the relation R i be defined as follows:

R i ( x i , x i ˆ , y i ) holdsiff f i ( x i , x i ˆ , y i )int C i .

Then the problem (WSQVEP) becomes a particular case of (SQVR) and Theorem 5.1 is a direct consequence of Theorem 2.1. □

Definition 5.2 ([14])

Let X and Z be two topological vector spaces and AX be nonempty convex set, and CZ be a nonempty, closed, and convex cone. Suppose f:AZ is a vector function. f is called C-continuous at x 0 A if, for any open neighborhood V of the zero element θ in Z, there exists an open neighborhood U of x 0 in A such that

f(x)f( x 0 )+V+C,xU;

and C-continuous in A if it is C-continuous at every point of A.

Remark 5.1 Lin et al. [14] obtained some existence results of (WSVQEP). However, the assumptions of Theorems 3.1, 3.3, and 3.4 in [14] are different from the assumptions in Theorem 5.1. Example 5.1 shows that all the assumptions of Theorem 5.1 are satisfied. But Theorem 3.1 in [14] does not work. The reason is that f i is not- C i -continuous.

Example 5.1 Let I={1}, X i = Y i = Z i =R, K i =[0,1], C i = R + , and let S i : K i 2 K i and f: K i R be defined by

S i ( x ) = [ 0 , 1 ] , f ( x , y ) = f ( x ) = { [ 0 , 1 2 ] if  x 0 = 1 6 , [ 1 , 2 ] otherwise .

We show that all assumptions of Theorem 5.1 are satisfied. However, f is not-C-continuous at x 0 = 1 6 . Thus, it gives the case where Theorem 5.1 can be applied but Theorems 3.1, 3.3, and 3.4 in [14] do not work.

Remark 5.2 If we let S i (x)= K i for each iI. Then (WSQVEP) becomes the system of vector equilibrium problem (in short, (SVEP)). If we let I={1}, then (WSQVEP) becomes the strong vector equilibrium problem (in short, (VEP)). The problems (SVEP) and (VEP) are studied in [14].

Definition 5.3 ([14])

Let X and Z be two topological vector spaces and AX be a nonempty convex set, and CZ be a nonempty, closed, and convex cone. Suppose f:AZ is a vector function. f is called C-quasiconvex-like if, for any x i , x 2 A and each λ[0,1], we have

either f ( λ x 1 + ( 1 λ ) x 2 ) f ( x 1 ) C or f ( λ x 1 + ( 1 λ ) x 2 ) f ( x 2 ) C .

Example 5.2 shows that in the special cases of Remarks 5.1 and 5.2, all the assumptions of Theorem 5.1 are satisfied. But Theorems 3.1, 3.3, and 3.4 in [14] do not work. The reason is that f i is not- C i -quasiconvex-like.

Example 5.2 Let X i , Y i , Z i , K i , C i be as in Example 5.1, and let S i :[0,1] 2 K i and f:[0,1]R be defined by

S i ( x ) = [ 0 , 1 ] , f ( x ) = { [ 1 3 , 1 2 ] if  x = 1 6 , [ 2 , 5 ] otherwise .

We show that all assumptions of Theorem 5.1 are satisfied. However, f is not-C-quasiconvex-like at x 0 = 1 6 . Indeed, let λ= 1 2 and x 1 =0, x 2 = 1 3 . Then

f ( λ x 1 + ( 1 λ ) x 2 ) = f ( 1 6 ) = [ 1 3 , 1 2 ] f ( x 1 ) + C = f ( 0 ) + C f ( λ x 1 + ( 1 λ ) x 2 ) = [ 2 , 5 ] + [ 0 , + ) , f ( λ x 1 + ( 1 λ ) x 2 ) = f ( 1 6 ) = [ 1 3 , 1 2 ] f ( x 2 ) + C = f ( 1 3 ) + C f ( λ x 1 + ( 1 λ ) x 2 ) = [ 2 , 5 ] + [ 0 , + ) .

Thus, Theorem 5.1 can be applied but Theorems 3.1, 3.3, and 3.4 in [14] do not work.

Theorem 5.2 Assume for the problem (WSQVEP) assumptions (i), (ii), (iii), and (iv) are as in Theorem  5.1 and replace (v) by (v′)

(v′) f i is continuous in K i × K i ˆ × K i .

Then the (WSQVEP) has a solution. Moreover, the solution set of the (WSQVEP) is closed.

Proof We omit the proof since the technique is similar to that for Theorem 5.1 with suitable modifications. □

Next, we discuss the existence of essential components for (WSQVEP).

Let ϒ be the collection of all (WSQVEP) satisfying the conditions of Theorem 5.1. For each γϒ, denote by Π(γ) the solution set of γ. It is easy to see that Π(γ)=Ξ(Φ), where Φ is the best-reply map of γ. By the proof of Theorem 5.1, we know that ΦM.

Definition 5.4 Let γϒ and Ξ α a connected component of Π(γ). Ξ α is said to be essential if it, as a connected component of Ξ(Φ), is an essential component of Ξ(Φ) with respect to M, where Φ is the best-reply map of γ.

By Lemma 3.1, we also obtain the following result.

Theorem 5.3 For any γϒ, there is at least one essential component of Π(γ).

Remark 5.3 In the special case of Remarks 5.1 and 5.2, Theorem 5.3 improves and extends Theorems 4.1 and 4.4 in [14].

References

  1. Luc DT: An abstract problem in variational analysis. J. Optim. Theory Appl. 2008, 138: 65–76. 10.1007/s10957-008-9371-9

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal RP, Balaj M, O’Regan D: A unifying approach to variational relation problems. J. Optim. Theory Appl. 2012, 155: 417–429. 10.1007/s10957-012-0090-x

    Article  MathSciNet  MATH  Google Scholar 

  3. Khanh PQ, Luc DT: Stability of solutions in parametric variational relation problems. Set-Valued Anal. 2008, 16: 1015–1035. 10.1007/s11228-008-0101-0

    Article  MathSciNet  MATH  Google Scholar 

  4. Balaj M, Lin LJ: Existence criteria for the solutions of two types of variational relation problems. J. Optim. Theory Appl. 2013, 156: 232–246. 10.1007/s10957-012-0136-0

    Article  MathSciNet  MATH  Google Scholar 

  5. Latif A, Luc DT: Variational relation problems: existence of solutions and fixed points of contraction mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 315

    Google Scholar 

  6. Balaj M, Luc DT: On mixed variational relation problems. Comput. Math. Appl. 2010, 60: 2712–2722. 10.1016/j.camwa.2010.09.026

    Article  MathSciNet  MATH  Google Scholar 

  7. Yang Z: On existence and essential stability of solutions of symmetric variational relation problems. J. Inequal. Appl. 2014., 2014: Article ID 5

    Google Scholar 

  8. Hung NV: Continuity of solutions for parametric generalized quasi-variational relation problems. Fixed Point Theory Appl. 2012., 2012: Article ID 102

    Google Scholar 

  9. Hung NV: Sensitivity analysis for generalized quasi-variational relation problems in locally G -convex spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 158

    Google Scholar 

  10. Lin LJ, Ansari QH: System of quasi-variational relations with applications. Nonlinear Anal. 2010, 72: 1210–1220. 10.1016/j.na.2009.08.005

    Article  MathSciNet  MATH  Google Scholar 

  11. Pu YJ, Yang Z: Stability of solutions for variational relation problem with applications. Nonlinear Anal. 2012, 75: 1758–1767. 10.1016/j.na.2011.09.007

    Article  MathSciNet  MATH  Google Scholar 

  12. Fort MK: Essential and nonessential fixed points. Am. J. Math. 1950, 72: 315–322. 10.2307/2372035

    Article  MathSciNet  MATH  Google Scholar 

  13. Kinoshita S: On essential components of the set of fixed points. Osaka J. Math. 1952, 4: 19–22.

    MathSciNet  MATH  Google Scholar 

  14. Lin Z, Yang H, Yu J: On existence and essential components of the solution set for the system of vector quasi-equilibrium problems. Nonlinear Anal. 2005, 63: 2445–2452. 10.1016/j.na.2005.03.049

    Article  Google Scholar 

  15. Lin Z, Yu J: The existence of solutions for the system of generalized vector quasi-equilibrium problems. Appl. Math. Lett. 2005, 18: 415–422. 10.1016/j.aml.2004.07.023

    Article  MathSciNet  MATH  Google Scholar 

  16. Hou SH, Gong XH, Yang XM: Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions. J. Optim. Theory Appl. 2010, 146: 387–398. 10.1007/s10957-010-9656-7

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang Z, Pu YJ: On the existence and essential components for solution set for system of strong vector quasi-equilibrium problems. J. Glob. Optim. 2013, 55: 253–259. 10.1007/s10898-011-9830-y

    Article  MathSciNet  MATH  Google Scholar 

  18. Luc DT Lecture Notes in Economics and Mathematical Systems. In Theory of Vector Optimization. Springer, Berlin; 1989.

    Chapter  Google Scholar 

  19. Ferro F: Optimization and stability results through cone lower semicontinuity. Set-Valued Anal. 1997, 5: 365–375. 10.1023/A:1008653120360

    Article  MathSciNet  MATH  Google Scholar 

  20. Yannelis N, Prabhakar ND: Existence of maximal elements and equilibria in linear topological spaces. J. Math. Econ. 1983, 12: 233–245. 10.1016/0304-4068(83)90041-1

    Article  MathSciNet  MATH  Google Scholar 

  21. Holmes RB: Geometric Functional Analysis and Its Application. Springer, New York; 1975.

    Book  MATH  Google Scholar 

  22. Hillas J: On the definition of the strategic stability of equilibria. Econometrica 1990, 58: 1365–1390. 10.2307/2938320

    Article  MathSciNet  MATH  Google Scholar 

  23. Jiang JH: Essential component of the set of fixed points of the multivalued mappings and its application to the theory of games. Sci. Sin. 1963, 12: 951–964.

    MathSciNet  MATH  Google Scholar 

  24. Khanh PQ, Quan NH: Generic stability and essential components of generalized KKM points and applications. J. Optim. Theory Appl. 2011, 148: 488–504. 10.1007/s10957-010-9764-4

    Article  MathSciNet  MATH  Google Scholar 

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The authors thank the editor and the two anonymous referees for their valuable remarks and suggestions, which helped them to considerably improve the article.

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Hung, N.V., Kieu, P.T. On the existence and essential components of solution sets for systems of generalized quasi-variational relation problems. J Inequal Appl 2014, 250 (2014). https://doi.org/10.1186/1029-242X-2014-250

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