Existence of Solutions to a Generalized System *

In this paper, we introduce a generalized system (for short, GS) in real Banach spaces. Using Brouwer’s fixed point theorem, we establish some existence theorems for the generalized system without monotonicity. Further, we extend the concept of C-strong pseudomonotonicity and extend Minty’s lemma for the generalized system. And using the Minty lemma and KKM-Fan lemma, we establish an existence theorem for the generalized system with monotonicity in real reflexive Banach spaces. As the continuation of existing studies, our paper present a series of extended results based on existing corresponding results.


Introduction
Variational inequality theory has played a fundamental and important role in the study of a wide range of problems arising in physics, mechanics, differential equations, contact problems in elasticity, optimization, economics and engineering sciences, etc.As a useful and important branch of variational inequality theory, vector variational inequalities were initially introduced and considered by Giannessi [1] in a finite-dimensional space in 1980.Ever since then, vector variational inequalities have been extensively studied and generalized in infinite-dimensional spaces.
Very recently, Fang and Huang [2] studied some existence results for strong vector variational inequalities in Banach spaces.Long, Huang and Teo [3] established an existence theorem of solutions for a generalized strong vector quasi-equlilbrium problem by using Kakutani-Fan-Glicksberg fixed point theorem.Li and He [4] studied the existence of solutions for VVI with a single-valued function and a continuous selection theorem and obtained the existence theorem for the GVVI under the assumption of -pseudomonotonicity.
Motivated and inspired by research works mentioned above, in this paper, we consider a generalized system, which includes as special cases the strong vector variational inequalities [1,2] and problems so called equilibrium problems [5,6].In order to derive the existence of solutions for the generalized system, Using Brouwer's fixed point theorem, we obtain some existence results for the generalized system without monotonicity.Furthermore, by the concepts of -C-continuous, C-strong pseudomonotonicity and hemicontinuity, we extend Minty's lemma, moreover, with the help of Minty's lemma and KKM-Fan lemma, we establish an existence theorem for the generalized system with monotonicity in real reflexive Banach spaces.The results presented in this paper extend the corresponding results of [2,6], and Theorems 3.1-3.3can be considered as a generalization of Theorems 2.1-2.3 in [6].

Preliminaries
Throughout the paper, otherwise special statement, let X and Y be two real Banach spaces, let K X  Y be a nonempty, closed and convex set, let be a solid, pointed, closed and convex cone with apex at the origin.Let  for all x K  and s Tx  .Then we consider the following generalized systems (for short, GS):

And find
We know that a solution    is called a weak solution and a strong solution to GS(1) and GS(2), respectively.It is easy to see that a solution to GS(2) is a solution to GS (1), but in general the converse is not true.
 is a single-valued mapping,  is a nonlinear mapping, then GS(1)and GS(2) both reduces to the following strong vector variational-like inequality problem (for short, SVVLIP): Find Now, we recall the following definitions.Definition 2.1 [7].Let K be a subset of a topological vector space X.A set-valued mapping is called a KKM mapping if, for each nonempty finite subset , , where CoA denotes the convex hull of the set A.
Definition 2.2.Let X and Y be topological vector spaces, let K be a nonempty, convex subset of X and let If C contains (or is equal to,or is contained in) the non-negative orthant, then the C-convex function is called C-convex (or convex, or strictly Cconvex); 2) If C contains (or is equal to, or is contained in) the nonpositive orthant, then the C-convex function is called C-concave (or concave, or strictly C-concave).
Theorem 2.1.(Brouwer's fixed point theorem [8]) Let K be a nonempty, compact and convex subset of a finite-dimensional space X and let be a continuous mapping.Then, there exists Theorem 2.2.(KKM-Fan Lemma [7]) Let K be a subset of a topological vector space X and let be a KKM mapping.If, for each is closed and for at least one

Main Results
In this section, we shall obtain the following existence theorems for GS (1).
Theorem 3.1.Let K be a nonempty, compact and convex subset of X and let the mapping be a C-convex function in the second argument.Assume that, for every y K  , the set Proof.We proceed by contradiction.Assume that GS(1) admits no solution, then for each 0 x K  , there exists some y K  and for all For every y K  , define the set y N as follows: By given assumption, the set y N is open in K and hence from (4), it follows that is an open cover of K. Since K is compact, there exists a finite set , ,   . So there exists a continuous partition of unity   Define a mapping by : Since j  is continuous for each j, it follows from (6) that h is also continuous.Let .Then S is a simplex of a finite dimensional space and h maps S into S.By Theorem 1.1, there exists some For any given x K  , let 4) and ( 7) that for all , , \ 0 , a contradiction.Hence, GS(1) has a solution.This completes the proof.
Then it is easy to see that f is a C-convex function in the second argument, and the is open in K. Now we could say that all conditions of Theorem 2.1 hold.And also, the solutions set for GS (1) in Theorem 2.1 is Theorem 3.2.Let K be a nonempty, closed and convex subset of X and let the mapping 1) For every y K  , the set 2) K is locally compact and there is an and 0 0 r  x K  , 0 x r  , such that, for all y K  , y r  , there exists an s Tx  such that  0 , ,  f s x y  C .Then, GS(1) has a solution.Proof.Let  : : K is compact; hence, it follows from Theorem 2.1 that there exists an We claim that x is the desired solution of GS (1).Indeed: 1) If x r  , by the assumption 2), there exists For any y K  , choose Then, from (7), it follows that Implying that This completes the proof.
The following example shows that the assumption that the set  , is not trivial.For two similar examples for a vector-valued function, see [2,6].
f is continuous and monotone in the sense that and hence is open in K.
Next, we define following concepts which will be used in the sequel.Definition 3.1.A mapping v Ty ,   .Remark 3.1.This definition generalizes corresponding definitions of [9,10].
Hence, f is C-strong pseudomonotone.For a similar example for a vector-valued function, see [6].
Now we prove the existence result for GS(2) with monotonicity.First, we prove the following Minty's type lemma for GS (2).Lemma 3.2.Let K be a nonempty closed and convex subset of X, Y be a real Banach space ordered by a nonempty closed convex pointed cone C with apex at the origin and int be a nonempty compact set-valued mapping.Suppose the following conditions hold: if and only if there exists Proof.(9)  (10): It follows from the C-strong pseudomonotonicity of f respect to T.
we will obtain a contradiction between ( 16) and ( 12), as C is convex pointed cone.That will complete the proof.
In fact, suppose on the contrary that Since C is closed, we know that there exists a origin neighbourhood V in Y, such that From 4), there exists neighbourhood , , be a C-convex function in the second argument and C-strong pseudomonotone.
be a nonempty compact setvalued mapping.Then, GS(2) has a solution.
Proof.Define two set-valued mappings, , : for some , .We claim that A is KKM-mapping.If assertion were false, then there would exists   1 2 , , , n x x x   K and 0 for all y K  , and hence B is also a KKM-mapping.By Lemma 2.1, we see that Next we claim that for each fixed y K , , and , , Since f is C-convex function in second argument, we have that, for  B y for all y K .
convex for each fixed y K  .The continuity of f in the second argument and closedness of -C give the closedness of .We now equip X with that weak topology.Since is closed, bounded and convex subset of the reflexive Banach space X, then it turns out to be weakly compact for all -valued mapping, : K K X    is a nonlinear mapping, then GS(1) and GS(2) both reduces to the SVVLIP(3).As applications, we have the following existence result for SVVLIP(3).

Corollary 3 . 1 .ForCorollary 3 . 2 .
Let K, X, T,  be as in Theorem 3.1 and Remark 3.2, let  be affine in the first argument.open in K. Then the SVVLIP(3) has a solution.Let K, X, T,  be as in Corollary 3.1.SVVLIP(3) has a solution.
on K.
 if, for each neighbourhood V of Tx, there exists a neighbourhood U of x, such that   T U V  .If T is upper semicontinuous at each point of W, then T is upper semicontinuous on W. Lemma 3.1 [11].Let W, E be two topological spaces, a