Vector Variational-Like Inequalities with Generalized SemimonotoneMappings

We introduce the concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity. We consider a class of generalized vector variationa-llike inequality problem involving generalized relaxed semimonotone mapping. By using KakutaniFan-Glicksberg�s �xed-point theorem, we prove the solvability for this class of vector variational-like inequality with relaxed monotonicity assumptions. e results presented in this paper generalize some known results for vector variational inequality in recent years.


Introduction
Vector variational inequalities were initially introduced and considered by Giannessi [1] in a �nite-dimensional Euclidean space in 1980.Due to its wide application the theory of the vector variational inequality is generalized in different directions and many existence results and algorithms for vector variational inequality problems have been established under various conditions; see for examples [2][3][4][5][6][7] and references therein.
e concept monotonicity and the compactness operators are very useful in nonlinear functional analysis and its applications.In 1968, Browder [8] �rst combined the compactness and accretion of operators and posed the concept of a semiaccretive operator.Motivated by this idea, Chen [9] studied the concept of a semimonotone operator, which combines the compactness and monotonicity of an operator and posed it to the study of variational inequalities.Recently in 2003, Fang and Huang [10] introduced relaxed --semimonotone mapping, a generalized concept of semimonotonicity, and they established several existence results for the variational-like inequality problem.
In this paper, we pose two new concepts of generalized relaxed monotonicity and generalized relaxed semimonotonicity as well as two classes of generalized vector variational-like inequalities with generalized relaxed monotone mappings and generalized relaxed semimonotone mappings.We investigate the solvability of vector variationallike inequalities with generalized relaxed semimonotone mappings by means of the Kakutani-Fan-Glicksberg �xedpoint theorem.e results presented in this paper generalize the results of Chen [9], Fang and Huang [10], Usman and Khan [11], and Zheng [7].

Preliminaries
roughout the paper unless otherwise speci�ed, let  and  be two real Banach spaces,    be a nonempty closed and convex subset of . is said to be a closed convex and pointed cone with its apex at the origin, if the following conditions hold: (i)    for all   , (ii)     , (iii)     .e partial order ≤  in , induced by the pointed cone , is de�ned by declaring  ≤   if and only if      for all   in .An ordered Banach space is a pair   with the partial order induced by .e weak order ≤ int  International Journal of Analysis in an ordered Banach space (  with int    is de�ned as  int   if and only if     int  for all   in , where int  denotes the interior of .Let (  denote the space of all continuous linear mappings from  into .A set-valued mapping       be such that for each    ( is a proper, closed, convex cone with int (   and let   = ⋂  (.
First, we give the concept of generalized relaxed monotone mapping.In order to do so, the de�nition of a vector monotone mapping is needed, which was posed by Chen [9].�e�nition � (see [9]).Let     (  be a mapping,    be a nonempty, closed and convex subset in .Let       be such that for each    ( is a proper, closed, convex cone with int (  . is said to be   -monotone on  if and only if it satis�es the following condition� where   = ⋂  (.
We now give the concept of generalized relaxed monotone mapping.�e�nition �.Let       be such that for each    ( is a proper, closed, convex cone with int (   and a mapping       .A mapping     (  is said to be generalized relaxed monotone, if where        is a mapping such that lim   0 + ((  + (   = 0.
then  is said to be relaxed --monotone, introduced and studied by Fang and Huang [10].
(ii) In the above inclusion (5), if we take (  =    for all    , then it reduces to       ≥      ∀    (6) then  is said to be -relaxed monotone.
We recall the following concepts and results which are needed in the sequel.

Main Results
roughout this section, let  be real re�exive Banach space,  a Banach space.Let    be a nonempty, bounded, closed, and convex subset of .Let       be such that for each     is a proper, closed, convex cone with int    and let  −  ⋂  .Some nonlinear mapping consisting with two variables, may be monotone with respect to the �rst variable and compact with respect to the second one.However, we cannot always expect for them to be monotone or compact with respect to the two variables simultaneously.Keeping this complexity in mind we are interested in the so called semimonotone mapping.We now give the concept of a generalized relaxed semimonotone mapping.
Hence, for every    (  is completely continuous.erefore, the mapping de�ned as above is a generalized relaxed semimonotone mapping.Now we will pose the main problem of our study.In this paper, we investigate the following generalized vector variational-like inequality problem (for short, GVVLIP) is to �nd a vector    satisfying  (    (  +  (  ≰ int (  ∀   (31) where       (  is a nonlinear mapping and       ,        are the two vector-valued bimappings. e GVVLIP (31) includes many variational inequality problems as special cases.
Some special cases of GVVLIP (31) are as follows.
(I) If (    −  and   , then the GVVLIP (31) reduces to the following variational inequality problem of �nding    such that which was introduced and studied by Zheng [7] (II) If    (   +, for all   .Also if (   −    and (  is the dual space  * of , then the GVVLIP (31) reduces to the following variational inequality problem of �nding    such that which was introduced and investigated by Chen [9].He obtained some existence results and discussed their applications in partial differential equations of divergence form.
We recall the following �xed-point theorem, by Zeidle [16], which will play an important role in establishing our existing results for GVVLIP (31).eorem 14 (see [16]).e set-valued mapping     2  has a �xed point, if the follo�ing conditions are satis�ed� (1)  is compact, convex, and nonempty set in locally convex space; (2) ( is convex, closed, and nonempty for every   ; (3)  is upper semicontinuous on .

Now
�e�nition �.A mapping      is said to be (i)   -convex, if ( + (      ( + (  ( for all       0 ; (ii)   -concave, if  is   -convex.Let  be real re�exive Banach space, and  be a Banach space.Let    be a nonempty, bounded, closed, and convex subset of .Let       be such that for each     is a proper, closed, convex cone with int   .Let      be  − -convex and upper semicontinuous in the �rst and second arguments, respectively, with the condition     for all   .From Lemma 6, we have     +    −    ≰ int   (16) that is,   , which follows   , for each    and so  is also a KKM mapping.Now we claim that for each       is closed in the weak topology of .Indeed suppose     , the weak closure of .Since  is re�exive, there is a se�uence {  } in  such that {  } converges weakly to   .en       +      −      ≰ int     (17) which implies that       +      −         − int      (18) Since   and   are lower and upper semicontinuous and   and  {− int   } are weakly lower semicontinuous, therefore [3]ma 6 (see[3]).Let (  be an ordered Banach space with a proper, closed, convex and solid cone ( then for all     , one has(i)  int(  and ≥ (    int ( , (ii)  ̸ ≥ int (  and  (    ̸ ≥ int ( .Lemma 7 (see [15]).Let  be a subset of a topological vector space  and let       be a KKM mapping.If for each   , ( is closed and for at least one   , ( is compact, then    (   (9) �e�nition �.A mapping     (  is said to be -hemicontinuous, if for any    , the mapping   ⟨ + (   (  is continuous at 0 + .Now, we have the following Minty's type Lemma.Lemma 9. Let    be a nonempty, closed, and convex subset of .Let       be such that for each    ( is a proper, closed, convex cone with int (  .Let        be   -convex in the �rst argument with the condition (  = 0 for all   .Suppose the following conditions hold: (i)        is an a�ne in the �rst argument with the condition (  + (  = 0 for all    ; (ii) the set-valued mapping       de�ned as ( =    int ( for all    is weakly upper semicontinuous; (iii)     (  is -hemicontinuous and generalized relaxed monotone mapping.en the following two problems are equivalent: (A    ⟨ (  + (  int ( 0 for all   , (B    ⟨ (  + (   (   int ( 0 for all   .Proof.Following the lines of proof given by Chen [9], one can easily prove.eorem10.Proof.�e�netwoset-valued mappings        as follows:            +   ≰ int    ∀              +   −    ≰ int    ∀   (11) en  and  are nonempty since   .We claim that  is a KKM mapping.If this is not true, then there exist a �nite set { 1  …    }   and   ≥    1 …   with Since  is generalized relaxed monotone, we have     +    ≥  −     +    +     (15)     +    −       {− int  }  (19) us we get     +    −    ≰ int  (20)and so   .is shows that  is weakly closed for each   .�ur claim is then veri�ed.Since  is re�exive and    is nonempty, bounded, closed, and convex,  is a weakly compact subset of  and so  is also weakly compact.According to Lemma 7,

�e�nition 11 .
Let    be a nonempty, closed, and convex subset of .Let        be a mapping and let        be a bifunction such that lim    + ((  + (1 −    and       .A mapping       (  is said to be generalized relaxed semimonotone mapping, if the following conditions hold:    (  is completely continuous, that is, when      (     (  (by the norm of operators), where   denotes the weak convergence.Remark 12.When    (   + , and (    −  for all     and   , then this is exactly the concept which was introduced by Chen [9].Example 13.Let      2     1   1.Let     2  be de�ned by  (   1   2      1 ≥   2 ≥      (25)  is de�ned as ‖‖  ++ for all   ( 1   2  and   ( 1   2   .Now, for �xed   , if    ,   , and     , it is easy to prove that      −  (  ⟶ Also     ,      are de�ne by (   − for all     and (   (− 2  for all    , respectively.Let us suppose       (  is de�ned by , we have the following existence results for GVVLIP (31) involving a generalized relaxed semimonotone mapping in re�exive �anach spaces.eorem 15.Let  be real re�exive Banach space, and let  be a Banach space.Let    be a nonempty, bounded, closed, and convex subset of .Let       be such that for each     is a proper, closed, convex cone with int   .Let         be a generalized relaxed semimonotone mapping.Suppose the following conditions hold: �or each �xed   ,          is continuous on each �nite dimensional subspace of .en there exists a  0   such that   0   0      0      0  ≰ int  0  0 ∀   (34) Proof.Let  be a �nite dimensional subspace of  and     ⋂   .For each     , we consider the following generalized vector variational-like inequality problem.Find  0    such that    0      0      0  ≰ int  0  0 ∀     (35) Since   ⊂  is bounded, closed, and convex,   is continuous on   and generalized relaxed monotone for each �xed   .From eorem 10, we know our problem has solution  0    .�e�ne a set-valued mapping         as follows:                   ≰ int  0 ∀      Since  is of �nite dimensional, hence   is compact.First, we claim that                    −      − int       (38) is convex.Indeed, let  1      and    0, such that     1         1    1 −  1      1                −           (39) Since   is affine and     are  − -convex, then from preceding two inclusions we have        1         1     −   1          1        (40) Since  is concave, we have  1      that is  is convex and our claim is then veri�ed.Now, we claim that  is closed.Let     such that    , then                −         − int                     −            (41) Since      is upper semicontinuous, also      and  is upper semicontinuous; therefore                −      ⟶            −        (42) is implies   ; hence  is closed.Next, we claim that  is upper semicontinuous.Let          , and    , then we have                  −            (43) From the complete continuity of   and the lower semicontinuity of  , we have            ⟶        (44) −    ≰ int  0 ∀      (37) Now we will use the �xed-point theorem to verify the existence of the solution of the problem in a �nite dimensional.which implies that            −        (45) that is,   ; thus our claim is then veri�ed.�ence  is upper semicontinuous.�y Fan-�licksberg �xed-point