Vector and Ordered Variational Inequalities and Applications to Order-Optimization Problems on Banach Lattices

The variational inequality problem VI(C, f) has been extensively studied by many authors. This theory has been widely applied to optimization theory, game theory, economic equilibrium,mechanics, and so forth. It has been recognized as an important branch in nonlinear analysis (see, e.g., [1–5]). The theory of variational inequality was first extended by Giannessi [6] in 1980 to vector variational inequality problems in finite dimensional vector spaces. Since then, it has been deeply studied by many authors such as Giannessi [6–10], Agarwal et al. [11], Mastroeni and Pellegrini [12], Li and Huang [13], Luc [14], and Facchinei and Pang [15] and also has been applied to many fields such as vector optimization problems and vector equilibrium problems. In 2005, Huang and Fang [10] generalized the vector variational inequality problems from finite dimensional vector spaces to the Banach spaces. We recall the extension as follows. For any positive integer n, let Rn denote the ndimensional Euclidean space. Let C be a convex and pointed cone ofR and let ≽ c be the partial order onR induced byC. Let f : Rm → Rn×m be a matrix-valued function. LetK be a closed convex subset ofR. The vector variational inequality problem associated withf,C, andK is to find an x∗ ∈ K such that


Introduction
Let  be a real Banach space with its norm dual   .Let  be a nonempty convex subset of  and  :  →   a single valued mapping.The variational inequality problem associated with  and , simply denoted as VI(, ), is to find an  * ∈  such that ⟨ ( * ) , ( −  * )⟩ ≥ 0, ∀ ∈ . ( The variational inequality problem VI(, ) has been extensively studied by many authors.This theory has been widely applied to optimization theory, game theory, economic equilibrium, mechanics, and so forth.It has been recognized as an important branch in nonlinear analysis (see, e.g., [1][2][3][4][5]).The theory of variational inequality was first extended by Giannessi [6] in 1980 to vector variational inequality problems in finite dimensional vector spaces.Since then, it has been deeply studied by many authors such as Giannessi [6][7][8][9][10], Agarwal et al. [11], Mastroeni and Pellegrini [12], Li and Huang [13], Luc [14], and Facchinei and Pang [15] and also has been applied to many fields such as vector optimization problems and vector equilibrium problems.In 2005, Huang and Fang [10] generalized the vector variational inequality problems from finite dimensional vector spaces to the Banach spaces.We recall the extension as follows.
For any positive integer , let R  denote the dimensional Euclidean space.Let  be a convex and pointed cone of R  and let ≽  be the partial order on R  induced by .Let  : R  → R × be a matrix-valued function.Let  be a closed convex subset of R  .The vector variational inequality problem associated with , , and  is to find an  * ∈  such that  ( * ) ( −  * ) ⊀  0, ∀ ∈ . ( As a special case, one may consider finding an  * ∈  such that In some economics circumstances, the preferences of a certain type of outcomes may be described by a special partial order, in particular a lattice order, on the space of outcomes.In this case, any preference inequalities and optimal problems must be defined under the given partial order that describes the preferences.Based on this motivation, Li and Wen [16] recently extended the variational inequality problem (1) to the following case: let  be a Banach space and let (; ≽  ) be a Banach lattice, where  is considered as the income domain and (; ≽  ) is considered as the production outcome space.
Let  be a nonempty closed convex subset of  and let  be a mapping from  to (, ).Then the ordered variational From the Choquet-Kendall theorem, for a given convex and pointed cone  in the -dimensional Euclidean space R  , the partial order ≽  on R  induced by  may not be a lattice order on R  .Furthermore, we can provide counterexamples to show that even in the case that ≽  on R  induced by  is a lattice order on R  , (R  , ≽  ) may not be a Banach lattice.Hence problems ( 5) and ( 4) are generalizations of ( 2) and (3), respectively, only if (R  , ≽  ) is considered as a Banach lattice.In this paper, we investigate the connections between problems (2), (3) and problems (4), (5).
In [16], Li and Wen proved some solvability theorems about the ordered variational inequality problems ( 4) and ( 5) by applying the KKM mappings and the Fan-KKM theorem.In this paper, we use some fixed point theorem to prove the solvability of the ordered variational inequality problems (4) and ( 5) and the existence of solutions to some orderoptimization problems in the Banach lattices.

Preliminaries
In this section, we recall some concepts and properties of the Banach lattices.These properties will be frequently used throughout this paper.For more details, the readers are referred to [17].We also recall the concept of vector variational inequality problems in the -dimensional Euclidean space R  and the variational order inequality problems in the Banach lattices.
Theorem 1 (see [16,17]).Both of the positive and negative cones of any Banach lattice are norm closed, so they are weakly closed.
Lemma 2 (see [16]).Let (; ≽  ) be an arbitrary Banach lattice.Then both of the positive and negative cones X + and X − are order-closed.That is, for any net {  } in  + (or in  − ), which order-converges to x,  ∈  + (or in  − ).
For any pair of elements  and  in a Banach lattice (; ≽  ), we say that ≻   whenever ≽   and  ¡ ≽  .Define the strictly positive and the strictly negative cones of the Banach lattice (; ≽  ), respectively, by These two cones possess the following properties: In general, both of  ++ and  −− are neither closed nor open in .This can be demonstrated by the following example.
Example 3. Let (R 2 , ≽ 2 ) be the Hilbert lattice with the coordinate lattice order ≽ 2 on R 2 ; that is, for any pair of points Any two elements  and  in a Banach lattice (; ≽  ) are said to be noncomparable under the order ≽  , if neither  ≽   nor  ≽   holds, which is denoted by  ⋈  .The nonzero-comparable set of a Banach lattice (; ≽  ), denoted by  ⋈ , is the subset { ∈  :  ⋈  0}.
From Theorem 3.46 [17] and Lemma 2.3 [16] recalled in this section, the union of the positive and negative cones of a Banach lattice is norm closed.It immediately implies the following lemma.

Lemma 4. The nonzero-comparable set 𝑋
Then the lemma follows from that both  + and  − are closed.Definition 5. Let  be a Banach space and let (; ≽  ) be a Banach lattice.Let  be a nonempty convex subset of  and  :  → (, ) a mapping, where (, ) denotes the Banach space of continuous linear operators from  to .The ordered variational inequality problem associated with , , and , denoted by VOI(, , ), is to find an  * ∈  such that where, as usual, 0 denotes the origin of .If  is linear, then the problem VOI(, , ) is called a linear ordered variational inequality problem; otherwise, it is called a nonlinear ordered variational inequality problem.
Many authors study the solvability of variational inequalities by applying the Fan-KKM theorem and fixed point theorems (see [18]).In [16], Li and Wen used the Fan-KKM theorem to prove the existence of solutions for some ordered variational inequalities.In this paper, we use fixed point theorems to show the solvability of some ordered variational inequalities.We first recall the concept of upper semicontinuous mappings from some topological spaces to topological spaces.Definition 6 (see [11]).Let ,  be Hausdorff topological spaces.Let  :  → 2  \ {} be a set valued mapping.For a point  0 ∈ , if, for any neighborhood   (( 0 )) of the set ( 0 ) in , there exists a neighborhood   ( 0 ) of the point  0 in , such that then  is said to be upper semicontinuous at point  0 .If  is upper semicontinuous at every point in , then  is said to be upper semicontinuous on .
The following theorem provides a criterion to check if a mapping is upper semi-continuous, which is useful in the following contents (see Agarwal et al. [11], for details).
Theorem 7 (see [11]).Let ,  be Hausdorff topological spaces with  compact.Let  :  → 2  \{} be a set valued mapping.If  has a closed graph, then  is upper semi-continuous on .

Bohnenblust-Karlin Fixed Point Theorem (1950).
Let  be a Banach space and  a nonempty compact convex subset of .Let  :  →  be an upper semi-continuous mapping with closed convex values.Then  has a fixed point.

Connections between Vector Variational Inequalities and Ordered Variational Inequalities
In finite dimensional real vector spaces, the vector variational inequalities defined by ( 2) and ( 3) and the ordered variational inequalities defined in ( 9) are all generalizations of variational inequalities defined in (1).In this section, we investigate the connections between these generalizations.We first recall a useful type of partial order on a vector space that is induced by a cone in this space.
Let  be a closed convex cone in a topological linear space .The partial order ≽  on  induced by  is defined as follows: Hence, the topological linear space  equipped with the partial order ≽  is a partially ordered topological linear space, which is denoted by (, ≽  ) and is said to be induced by the cone .
It is important to notice that the partial order ≽  on the partially ordered topological linear space (, ≽  ) is far away from being a lattice order; that is, the partially ordered topological linear space (, ≽  ) may not be a Riesz space (vector lattice).This can be demonstrated by the following simple example.
Example 8.In R 3 , take  to be the closed convex cone as follows: We claim that the partial order ≽  on R 3 is not a lattice order.
It immediately implies that  ∨  does not exist.Hence ≽  is not a lattice order on R 3 and (R 3 , ≽  ) is not a vector lattice.
Here we consider some special cones in real vector spaces such that their induced partial order is a lattice order.For more details, the reader is referred to Kendall [19].
Let  be a nonempty convex set in a real vector space .The subset { + (1 − ) :  real; ,  ∈ } of  is called the minimal affine extension of .Suppose that the minimal affine extension of  does not contain the zero vector.Let  = { :  ⩾ 0,  ∈ }.Then  is a pointed convex cone in .
Kinderlehrer and Stampacchia [4] calls a nonempty convex set  in a real vector space  a simplex when the intersection of two positively homothetic images of ,  + ,  +  (,  ∈ ; ,  ⩾ 0) , is empty or is a set  +  ( ∈ , V ⩾ 0) of the same nature.
Choquet-Kendall Theorem (see [19]).Let  be a nonempty convex set spanning a real vector space  and suppose that its minimal affine extension does not contain the zero vector.Let  be the pointed positive convex cone = { :  ⩾ 0,  ∈ } having  as a base (of ).Then , partly ordered by , will be a vector lattice if and only if both Now we just prove that previous statement." ⇒" Suppose that the inequality ⟨, ⟩ ⩾ 0 holds for all ,  ∈ .Then, for every ,  ∈ R  with 0 ≼  ≼  , we have It implies ‖‖ ≤ ‖‖, which is condition (b3) for the Hilbert lattices.
The following example shows that condition (3 ∘ ) in Theorem 9 is necessary for (R  , ≽  ) to be a Hilbert lattice.
Example 10.Let  be the closed segment in R 2 with ending points at (1, 0) and (−1, 1).One can show that  is a simplex in R 2 under Choquet's definition (9).Then the closed pointed convex cone  generated by  is the closed convex cone in R 2 bounded by the two rays with origin as the ending point and passing through points (1, 0) and (−1, 1), respectively.It can be seen that  is a convex subset spanning R 2 and satisfies the two conditions in the Choquet-Kendall theorem.Hence R 2 , partly ordered by , is a vector lattice (Riesz space).Take two vectors ,  in  with origin as the initial point and with ending points at (1, 0) and (−1, 1), respectively.It is clear that ⟨, ⟩ = −1 < 0. So  does not satisfy condition (3 ∘ ) in Theorem 9, and hence (R 2 , ≽  ) is not a Hilbert lattice.To more precisely show that (R 2 , ≽  ) is not a Hilbert lattice, we can show that ≽  does not satisfy condition (b3) of the Banach lattices recalled in Section 2. To this end, take  to be the vector in R 2 with origin as the initial point and with ending points at (0, 1) (in fact,  ∈ ).We have − =  ∈ .It yields that ≽  ≽  0. On the other hand, we see that ‖‖ = 1 and ‖‖ = √ 2. Hence, the vectors  and  do not satisfy condition (b3) for the Hilbert lattices.
To summarize, we state the connections between vector variational inequalities and ordered variational inequalities as a proposition.
Proposition 11.If (R  , ≽  ) is a Hilbert lattice, that is, the convex cone  in R  is generated by a simplex  which spans R  and satisfies the three conditions in Theorem 9, then the ordered variational inequalities (5) and (4) in the Hilbert lattice (R  , ≽  ) coincide with vector variational inequalities (2) and (3), respectively.

Existence of Solutions of Ordered Variational Inequalities on the Banach Lattices
The convexity and concavity of functions play important roles in nonlinear analysis.In this section, we extend these concepts to order-convexity and order-concavity of mappings in Banach lattices, which will be applied in the proofs of the existence of solutions of ordered variational inequalities in the Banach lattices.
Definition 12. Let  be a Banach space and let (; ≽  ) be a Banach lattice.Let  be a nonempty convex subset of .A mapping  :  →  is said to be (1) order-convex on  if, for every pair of points ,  ∈ , the following order inequality holds: (2) order-concave on  if, for every pair of points ,  ∈ , the following order inequality holds: The following result is the main theorem of this paper.
Theorem 13.Let  be a Banach space and let (; ≽  ) be a Banach lattice.Let  be a nonempty compact convex subset of .Let  :  → (, ) be a continuous (with respect to the norms) mapping.If () :  →  has a lower order closed range, that is, ⋀ ∈ ()() exists and satisfies then the problem VOI(, , ) has a solution.
We obtain that (, V) ∈ graph Γ, and hence graph Γ is closed.
To summarize, we obtain that Γ is an upper semicontinuous set valued mapping with nonempty compact convex values.From the Bohnenblust-Karlin fixed point theorem, the set of fixed points of Γ is nonempty.Taking a fixed point  * of , we have  * ∈ Γ( * ).That is, This implies  ( * ) ( * ) ≼   ( * ) () , ∀ ∈ .
Noticing that ( * ) is linear, from the properties of the lattice order ≼  , it is equivalent to Hence  * is a solution to the ordered variational inequality problem VOI(, , ).This theorem is proved.

Applications to Order-Optimization Problems
As some applications of Theorem 13, we solve some orderoptimization problems in the Banach lattices in this section.
Definition 14.Let  be a Banach space and let (; ≽  ) be a Banach lattice.Let  be a nonempty subset of .Let  :  → (, ) be a mapping.F is said to take an associated orderminimum value at a point  * ∈  if it satisfies is said to take an associated order-maximum value at a point From the proof of Theorem 13 and from the linearity of (), for all  ∈ , it is clear that (32) is equivalent to (9).Hence, a point  * ∈  is a solution to the problem VOI(, , ) if and only if, at the point  * ,  takes an associated order-minimum value.So the following corollary is an immediate consequence of Theorem 13.

Corollary 15.
Let  be a Banach space and let (; ≽  ) be a Banach lattice.Let  be a nonempty compact convex subset of .Let  :  → (, ) be a continuous (with respect to the norms) mapping.If () :  →  has a lower order closed range, that is, ⋀ ∈ ()() exists and satisfies ⋀ ∈  () () ∈  () () , for every  ∈ , (34 then there is a point  * ∈ , at which F takes an associated order-minimum value.
The existence of associated order-maximum value problem is also considered as a consequence of Theorem 13.We provide a solution of this problem as follows as a corollary of Theorem 13.
Hence,  takes an associated order-maximum value at the point  * .
Now by applying the Choquet-Kendall theorem, we prove the following theorem in R  .Proof.Note that the partial order ≽  on R  induced by the cone  always satisfies the first two conditions of Hilbert lattices (b1) and (b2) recalled in Section 2. From the Choquet-Kendall theorem, we only need to show that ∘ ) each B-segment { + (1 − ) : 0 ≤  ≤ 1; ,  ∈ } is contained in a maximal B-segment.⟨, ⟩ ⩾ 0, ∀,  ∈ , if and only if 0 ≼  ≼   implies ‖‖ ≤          , for every ,  ∈ R  .