Open Access

Solving GNOVI frameworks involving ( γ G , λ ) -weak-GRD set-valued mappings in positive Hilbert spaces

  • Hong Gang Li1,
  • Xian Bing Pan2Email author,
  • Zhi Ying Deng1 and
  • Chang You Wang1
Fixed Point Theory and Applications20142014:146

https://doi.org/10.1186/1687-1812-2014-146

Received: 4 March 2014

Accepted: 25 June 2014

Published: 22 July 2014

Abstract

First, a new concept, positive Hilbert spaces, is introduced and some fundamental inequalities which are applied to studying the properties of the resolvent operator associated for ( γ G , λ ) -weak-GRD set-valued mappings are introduced and discussed in positive Hilbert spaces. Next, by using the resolvent operator and fixed point theory, an existence theorem and an approximation algorithm to solve a new class of general nonlinear ordered inclusions are established and suggested. In this field, the results obtained seem to be general in nature.

MSC:49J40, 47H06.

Keywords

general nonlinear ordered inclusion frameworks positive Hilbert spaces inequalities ( γ G , λ ) -weak-GRD set-valued mapping approximation algorithm

1 Introduction

Generalized nonlinear variational inclusion was introduced and studied by Hassouni and Moudafi [1]; it is useful and important in, for example, optimization and control, nonlinear programming, economics, mathematics, physics and engineering sciences. From 1989, Chang and Zhu [2], Chang and Huang [3], Ding and Jong [4], Ding and Luo [5], Jin [6], Li [7], Ahmad and Bazán [8], Chang [9], Cho et al. [10] and in recent years, Huang and Fang [11, 12], Chang and Huang [13], Fang et al. [14], Lan et al. [15] and others studied the properties of many kinds of resolvent operators (generalized m-accretive mappings, generalized monotone mappings, maximal η-monotone mappings, H-monotone operators, ( H , η ) -monotone operators, ( A , η ) -accretive mappings) and variational inequalities (inequalities, equalities, quasi-variational inclusions, quasi-complementarity) for fuzzy mappings, generalized random multivalued mappings etc.

On the other hand, in 1972, a number of solutions of nonlinear equations were introduced and studied by Amann [16]; and in recent years, the nonlinear mapping fixed point theory and application have been intensively studied in ordered Banach spaces [1719]. Therefore, it is very important and natural for generalized nonlinear ordered variational inequalities (ordered equation) to be studied and discussed.

In 2008, the author introduced the generalized nonlinear ordered variational inequalities (the ordered equations) and studied an approximation algorithm and an approximation solution for a class of generalized nonlinear ordered variational inequalities and ordered equations in ordered Banach spaces [20]. In 2009, by using the B-restricted-accretive method of the mapping A with constants α 1 , α 2 , the author introduced and studied a new class of general nonlinear ordered variational inequalities and equations and established an existence theorem and an approximation algorithm of solutions for this kind of generalized nonlinear ordered variational inequalities (equations) in ordered Banach spaces [21]. In 2011, by using the resolvent operator associated with an RME set-valued mapping, the author introduced and studied a class of nonlinear inclusion problems for ordered RME set-valued mappings to find x X such that 0 M ( x ) ( M ( x ) is a set-valued mapping), and the existence theorem of solutions and an approximation algorithm for this kind of nonlinear inclusion problems for ordered extended set-valued mappings in ordered Hilbert spaces [22]. In 2012, the author introduced and studied a class of nonlinear inclusion problems for ordered ( α , λ ) -NODM set-valued mappings and then, applying the resolvent operator associated with ( α , λ ) -NODM set-valued mappings, established the existence theorem on the solvability and a general algorithm applied to the approximation solvability of the nonlinear inclusion problem of this class of nonlinear inclusion problems, based on the existence theorem and the new ( α , λ ) -NODM model in an ordered Hilbert space [23]. In Banach spaces, the author proved sensitivity analysis of the solution for a new class of general nonlinear ordered parametric variational inequalities to find x = x ( λ ) : Ω X such that A ( g ( x , λ ) , λ ) + f ( x , λ ) θ ( A ( x ) , g ( x ) and F ( , ) are single-valued mappings) in 2011 [24]. In this field, the obtained results seem to be general in nature.

Very recently, in 2013, the author introduced and studied characterizations of ordered ( α A , λ ) -weak-ANODD set-valued mappings, which was applied to solving an approximate solution for a new class of general nonlinear mixed order quasi-variational inclusions involving operator in ordered Banach spaces [25] and GNM ordered variational inequality system with ordered Lipschitz continuous mappings in ordered Banach spaces [26]. In 2014, a class of nonlinear mixed ordered inclusion problems for ordered ( α A , λ ) -ANODM set-valued mappings with strong comparison mapping A [27] and sensitivity analysis for GSV parametric OVI with ( α , λ ) -NODSM mappings in ordered Banach spaces [28] were introduced and studied. Now, it is excellent that we are introducing positive Hilbert spaces and studying the properties of ( γ G , λ ) -weak ordered GRD set-valued mappings, which is applied to finding a solution for a new class of general nonlinear ordered inclusion frameworks involving a strong comparison mapping in positive Hilbert spaces. For details, we refer the reader to [150] and the references therein.

2 Fundamental inequalities in positive Hilbert spaces

In this paper, unless specified otherwise, X expresses a real ordered Hilbert space with an inner product , , a norm , a zero element θ, a normal cone P with normal constant N > 0 and a partially ordered relation ≤ defined by a normal cone P. For x , y X , x and y are said to be comparable to each other if and only if x y (or y x ) holds (denoted by x y for x y and y x ) [23]. C B ( X ) expresses the family of all nonempty closed bounded subsets of X.

Lemma 2.1 ([25])

Let X be an ordered Hilbert space andbe a partially ordered relation.
  1. (i)

    If x y , then lub { x , y } and glb { x , y } exist, x y y x , and θ ( x y ) ( y x ) ;

     
  2. (ii)

    If x y = lub { x , y } , x y = glb { x , y } , x y = ( x y ) ( y x ) , x y = ( x y ) ( y x ) , then the following relations hold:

     
  3. (1)

    x y = y x , x x = θ , x y = y x = ( x y ) ;

     
  4. (2)

    let λ be real, then ( λ x ) ( λ y ) = | λ | ( x y ) ;

     
  5. (3)

    let ( x + y ) ( u + v ) exist, and if x u , v and y u , v , then ( x + y ) ( u + v ) ( x u + y v ) ( x v + y u ) ;

     
  6. (4)

    if x y and u v , then x + u y + v ;

     
  7. (5)

    if x y , then x y = θ if and only if x = y ;

     
  8. (6)

    x y = x + y ( x y ) ;

     
  9. (7)

    α x β x = | α β | x if x θ .

     

Definition 2.2 An ordered Hilbert space X is said to be a positive Hilbert space with a partially ordered relation ≤ (denoted by X P ) if for any x , y X , x θ and y θ , then x , y 0 .

Example 2.3 Let X = R n be a real n-dimensional ordered inner product space with orthogonal basis { α i } i = 1 n . Setting P = { β = i = 1 n k i α i | k i 0 , k i R n ( 1 i n ) } , it is a normal cone, then R P n is a positive Hilbert space.

Theorem 2.4 (Inequalities I)

If X is an ordered Hilbert space, for x , y , z , w X , then
  1. (1)

    x x y , y x y , x y x , x y y , x y x y ;

     
  2. (2)

    x y if and only if y x ;

     
  3. (3)

    x ( y z ) ( x y ) ( x z ) , x + ( y z ) ( x + y ) ( x + z ) ;

     
  4. (4)

    x ( y z ) ( x y ) ( x z ) , x + ( y z ) ( x + y ) ( x + z ) ;

     
  5. (5)

    if θ x , θ y , then x y x y , x y x y ;

     
  6. (6)

    ( x + y ) ( z + w ) ( ( x z ) ( y w ) ) ( ( x w ) ( y z ) ) .

     

Proof Obviously, (1)-(5) hold for Lemma 2.1 and Definition 2.2.

For x , y , z , w X , we have ( x + y ) ( z + w ) + ( y w ) ( x + y z w ) ( z + w x y ) + ( y w ) ( w y ) ( x + y z w ) + ( w y ) x z ; in the same way, ( x + y ) ( z + w ) + ( y w ) z x . Therefore,
( x + y ) ( z + w ) ( x z ) ( y w ) ,

and hence (6) holds for x + y = y + x and x y = y x . □

Theorem 2.5 (Inequalities II)

If X p is a positive Hilbert space, for x , y , z , w X , then
  1. (1)

    if x y , θ z , then y , z x , z ;

     
  2. (2)

    if θ z , then x y , z x , z y , z , x , z y , z x y , z ;

     
  3. (3)

    if θ z , then x + y , z x , z y , z + x y , z ;

     
  4. (4)

    if θ z , then x y , z x , z + y , z x , z y , z ;

     
  5. (5)

    if θ z , then x y , z x , z y , z .

     
Proof From Lemma 2.1, Definition 2.2 and Theorem 2.4 it follows that (1)-(4) hold. Let θ z , by (6) in Lemma 2.1 and (1)-(3) in Theorem 2.4, hold
x y , z = ( x y ) ( y x ) , z = ( x y ) , z + ( y x ) , z ( x y ) ( y x ) , z = ( x y ) ( y x ) , z [ ( x y ) , z ( y x ) , z ] = [ ( x , z y , z ) ( y , z x , z ) ] = ( x , z y , z ) ( y , z x , z ) = x , z y , z .

It follows that (5) holds. □

3 Properties of ( γ G , λ ) -weak-GRD set-valued mappings in positive Hilbert spaces

Definition 3.1 Let X be a real ordered Hilbert space, let G : X X be a strong comparison and β-ordered compressed mapping [23], and let M : X C B ( X ) be a set-valued mapping.
  1. (1)

    [22]M is said to be an ordered rectangular mapping if for each x , y X , any v x M ( x ) and any v y M ( y ) , v x v y , ( x y ) = 0 holds;

     
  2. (2)
    M is said to be a γ G -ordered rectangular mapping with respect to G if there exists a constant γ G 0 ; for any x , y X , there exist v x M ( G ( x ) ) and v y M ( G ( y ) ) such that
    v x v y , ( G ( x ) ( y ) ) γ G G ( x ) G ( y ) 2
     
holds, where v x and v y are said to be γ G -elements, respectively;
  1. (3)

    M is said to be a weak comparison mapping with respect to G if for any x , y X , x y , then there exist v x M ( G ( x ) ) and v y M ( G ( y ) ) such that x v x , y v y and v x v y , where v x and v y are said to be weak comparison elements, respectively;

     
  2. (4)

    M with respect to G is said to be a λ-weak ordered different comparison mapping with respect to G if there exists a constant λ > 0 such that for any x , y X , there exist v x M ( G ( x ) ) , v y M ( G ( y ) ) , λ ( v x v y ) x y holds, where v x and v y are said to be λ-elements, respectively;

     
  3. (5)

    A weak comparison mapping M with respect to B is said to be a ( γ G , λ ) -weak-GRD mapping with respect to B if M is a γ G -ordered rectangular and λ-weak ordered different comparison mapping with respect to B and ( G + λ M ) ( X ) = X for λ > 0 , and there exist v x M ( G ( x ) ) and v y M ( G ( y ) ) such that v x and v y are ( γ G , λ ) -elements, respectively.

     
Remark 3.2 Let X be a real ordered Hilbert space, let G : X X be a single-valued mapping, and let M : X C B ( X ) be a set-valued mapping, then the following obviously hold:
  1. (i)

    A λ-ordered monotone mapping must be λ-weak ordered different comparison [22];

     
  2. (ii)

    If G = I (identical mapping), then a γ I -ordered rectangular mapping must be ordered rectangular in [22];

     
  3. (iii)

    An ordered RME mapping must be λ-weak-GRD in [22].

     
Theorem 3.3 Let X P be a real positive Hilbert space with normal constant N, let G be a strong comparison and β-ordered compressed mapping, and let M : X C B ( X ) be an α I -weak ordered rectangular set-valued mapping that I is an identical mapping. Let a mapping J M , λ G = ( G + λ M ) 1 : X 2 X be an inverse mapping of ( G + λ M ) .
  1. (1)

    If α I λ > β > 0 , then J M , λ G is a single-valued mapping;

     
  2. (2)
    If λ ( α I γ G ) > β > 0 , and M : X C B ( X ) is a ( γ G , λ ) -weak-GRD set-valued mapping with respect to J M , λ G , and v x M ( J M , λ G ( x ) ) and v y M ( J M , λ G ( y ) ) are α I , γ G and λ-elements, respectively, then the resolvent operator J M , λ G of M is a comparison, and
    J M , λ G ( x ) J M , λ G ( y ) 1 γ G λ β x y .
    (3.1)
     
Proof Certificate (1): Let u X and x , y J M , λ G ( u ) = ( G + λ M ) 1 ( u ) . Since M is an α I -weak ordered rectangular mapping so that there exist v x = 1 λ ( u G ( x ) ) M ( x ) and v y = 1 λ ( u G ( y ) ) M ( y ) such that
v x v y , ( x y ) α x y 2 ,

where v x and v y are α I -elements, respectively.

Since G is a β-ordered compressed mapping so that
v x v y , ( x y ) = 1 λ ( u G ( x ) ) 1 λ ( u G ( y ) ) , ( x y ) = 1 λ ( G ( x ) G ( y ) ) , ( x y ) 1 λ β ( x y ) , ( x y ) = β λ x y 2 ,

and α I x y 2 β λ x y 2 for Theorems 2.4 and 2.5. It follows that x = y = J M , λ G ( u ) and J M , λ G ( u ) is a single-valued mapping from α I λ > β > 0 .

Certificate (2): Since M : X C B ( X ) still is an λ-weak ordered different comparison mapping so that λ ( v x v y ) ( x y ) and x J M , λ G ( x ) , where v x and v y are α I and λ-elements ( x , y X ), such that v x = 1 λ ( x G ( J M , λ G ( x ) ) ) M ( J M , λ G ( x ) ) and v y = 1 λ ( y G ( J M , λ G ( y ) ) ) M ( J M , λ G ( y ) ) , respectively, then
λ ( v x v y ) ( x y ) = G ( J M , λ G ( x ) ) G ( J M , λ G ( y ) ) .

Hence, G ( J M , λ G ( y ) ) G ( J M , λ G ( x ) ) , and J M , λ G ( y ) J M , λ G ( x ) by strong comparability of G.

Let M be a ( γ G , λ ) -weak-GRD mapping with respect to J M , λ G ( x ) , then for any x , y X and λ > 0 , v x = 1 λ ( x G ( J M , λ G ( x ) ) ) M ( J M , λ G ( x ) ) and v y = 1 λ ( y G ( J M , λ G ( y ) ) ) M ( J M , λ G ( y ) ) are α I , λ and γ G -elements, respectively. Hence, by Definition 3.1(4), Theorems 2.4 and 2.5 and the comparability of J M , λ G , we have
γ G J M , λ G ( x ) J M , λ G ( y ) 2 1 λ ( x G ( J M , λ G ( x ) ) ) 1 λ ( y G ( J M , λ G ( y ) ) ) , ( J M , λ G ( x ) J M , λ G ( y ) ) = 1 λ [ ( x G ( J M , λ G ( x ) ) ) ( y G ( J M , λ G ( y ) ) ) ] , ( J M , λ G ( x ) J M , λ G ( y ) ) 1 λ [ ( G ( J M , λ G ( x ) ) G ( J M , λ G ( y ) ) ) + ( x y ) ] , ( J M , λ G ( x ) J M , λ G ( y ) ) = 1 λ ( G ( J M , λ G ( x ) ) G ( J M , λ G ( y ) ) ) , ( J M , λ G ( x ) J M , λ G ( y ) ) + 1 λ ( x y ) , ( J M , λ G ( x ) J M , λ G ( y ) ) β λ ( J M , λ G ( x ) J M , λ G ( y ) ) , J M , λ G ( x ) J M , λ G ( y ) + 1 λ ( x y ) , ( J M , λ G ( x ) J M , λ G ( y ) ) .
It follows that
λ γ G J M , λ G ( x ) J M , λ G ( y ) 2 β J M , λ G ( x ) J M , λ G ( y ) 2 + ( x y ) , ( J M , λ G ( x ) J M , λ G ( y ) )
and
( γ G λ β ) J M , λ G ( x ) J M , λ G ( y ) 2 x y J M , λ G ( x ) J M , λ G ( y ) ,
by the condition λ ( α I γ G ) > β > 0 , then there is
J M , λ G ( x ) J M , λ G ( y ) 1 γ G λ β x y .

 □

4 Approximation solution for GNOVI frameworks

In this section, by using Theorems 2.4 and 2.5 and Theorem 3.3, we study a new class of GNOVI frameworks in positive Hilbert spaces.

Let X P be a real positive Hilbert space with a normal constant N, a norm , an inner product , and zero θ. Let M : X C B ( X ) and ρ M ( x ) = { ρ v | v M ( x ) } be two set-valued mappings. We consider the problem: For w X and ρ > 0 , find x X such that
w ρ M ( x ) ,
(4.1)

which is called a new class of general nonlinear ordered variational inclusion frameworks (GNOVI) in positive Hilbert spaces.

Remark 4.1 If M ( x ) = A ( g ( x ) ) is single-valued, w = θ and ρ = 1 , then (4.1) reduces to (2.1) in [20]; when M ( x ) = A ( x ) F ( x , g ( x ) ) , w = θ and ρ = 1 , then (4.1) reduces to (1.1) in [21]; if w = θ , then (1.1) in [22] or [23] can be obtained as special cases of (4.1) as ρ = 1 .

Lemma 4.2 Let X P be a real positive Hilbert space with normal constant N, let G be a strong comparison and β-ordered compressed mapping, and let M : X C B ( X ) be a ( γ G , λ ) -weak ordered GRD set-valued mapping with respect to J M , λ G . Then the inclusion problem (2) has a solution x if and only if x = J M , λ G ( G ( x ) + λ ρ w ) in X.

Proof For ρ > 0 , take notice of the fact that w ρ M ( x ) if and only if w ρ M ( x ) , this directly follows from the definition of J M , λ G and problem (4.1). □

Theorem 4.3 Let X P be a real positive Hilbert space with normal constant N, let G be a strong comparison and β-ordered compressed mapping, and let M : X C B ( X ) be an α I -ordered rectangular and ( γ G , λ ) -weak-GRD set-valued mapping with respect to J M , λ G ( x ) . Let v x M ( J M , λ G ( x ) ) and v y M ( J M , λ G ( y ) ) be α I , λ and γ G -elements, respectively. If β satisfies
0 < β < λ ( γ G 2 α I ) 1 ,
(4.2)
β + a N 1 N ( 1 a ) β 2 < γ G λ ( 0 < a < 1 ) ,
(4.3)

then there exists a solution x of GNOVI (4.1), which is a fixed point of J M , λ G , that is converged strongly by a sequence { x n } n = 0 generated by the following algorithm:

For any given x 0 X and any 0 < a < 1 , set
x n + 1 = ( 1 a ) x n + a J M , λ ( G ( x n ) + λ ρ w ) ( n = 1 , 2 , ) .
(4.4)

Proof Let X P be a positive Hilbert space with normal constant N, let G be a strong comparison and β-ordered compression mapping, and let M ( x ) = { v | v M ( x ) } : X C B ( X ) ( ρ > 0 ) be a ( γ G , λ ) -weak-GRD set-valued mapping with respect to J M , λ G .

Since α I , β , γ G , λ > 0 and by condition (4.2) we have
λ ( α I γ G ) λ ( γ G 2 α ) = λ γ G 2 λ α I > β > 0 and 1 > β γ G λ β > 0 .
By Theorem 3.3(1), if x 1 x 2 , then J M , λ G ( G + λ ρ w ) ( x 1 ) J M , λ G ( G + λ ρ w ) ( x 2 ) for x 1 , x 2 X , and
J ρ M , λ G ( G + λ ρ w ) ( x 1 ) J ρ M , λ G ( G + λ ρ w ) ( x 2 ) 1 γ G λ β ( G + λ ρ w ) ( x 1 ) ( G + λ ρ w ) ( x 2 ) 1 γ G λ β G ( x 1 ) G ( x 2 ) β γ G λ β x 1 x 2 .
(4.5)

It follows that J M , λ G ( G + λ ρ w ) has a fixed point x , which is a solution x for GNOVI (4.1), from Lemma 4.2 and (4.5).

For any x 0 X and 0 < a < 1 , by using (4.4), (4.5) and Theorem 3.3, the following hold:
θ x n + 1 x n = ( ( 1 a ) x n + a J M , λ ( G ( x n ) + λ ρ w ) ) ( ( 1 a ) x n 1 + a J M , λ ( G ( x n 1 ) + λ ρ w ) )
and
x n + 1 x n = ( 1 a ) ( x n x n 1 ) + a ( J M , λ ( G ( x n ) + λ ρ w ) J M , λ ( G ( x n 1 ) + λ ρ w ) ) N [ ( 1 a ) x n x n 1 + a β γ G λ β G ( x n ) G ( x n 1 ) ] δ N x n x n 1 ,
(4.6)
where δ = 1 a + a β 2 γ G λ β . It follows that x m x n i = n m 1 x i + 1 x i N x 1 x 0 i = n m 1 δ i N i for any m > n > 0 , 1 > δ N > 0 and (4.6), and hence { x n } n = 0 is a Cauchy sequence in a complete space X by condition (4.3) and x n x n 1 δ n N n x 1 x 0 . Let x n x as n ( x X ), by (4.2) we get
x = lim n x n + 1 = lim n J M , λ ( G ( x n ) + λ ρ w ) = J M , λ ( G ( x ) + λ ρ w ) ,

then the sequence { x n } n = 0 converges strongly to a solution x of problem (4.1), which is generated by (4.4). This completes the proof. □

Remark 4.4 (i) For a suitable choice of the mappings G, M and constant ρ, we can obtain several known results of [20] and [22] as special cases of Theorem 4.3.

(ii) There exists β > 0 satisfying (4.3). In fact, if we change (4.3) to β + N β 2 < γ G λ as 0 < a 1 , then 1 + 4 N γ G λ 1 2 > β > 0 holds.

Declarations

Acknowledgements

The authors acknowledge the support of the Educational Science Foundation of Chongqing (KJ1400426) and the support of the Scientific and Technological Research Program of Chongqing Municipal Education Commission of China (KJ120520).

Authors’ Affiliations

(1)
Nonlinear Analysis Institute, Institute of Applied Mathematics Research, College of Mathematics and Physics, Chongqing University of Posts and Telecommunications
(2)
College of Mobile Telecommunications, Chongqing University of Posts and Telecommunications

References

  1. Hassouni A, Moudafi A: A perturbed algorithms for variational inequalities. J. Math. Anal. Appl. 2001, 185: 706–712.View ArticleMathSciNetMATHGoogle Scholar
  2. Chang SS, Zhou HY: On variational inequalities for fuzzy mappings. Fuzzy Sets Syst. 1989, 32: 359–367. 10.1016/0165-0114(89)90268-6View ArticleMATHMathSciNetGoogle Scholar
  3. Chang SS, Huang NJ: Generalized complementarity problem for fuzzy mappings. Fuzzy Sets Syst. 1993, 55: 227–234. 10.1016/0165-0114(93)90135-5View ArticleMathSciNetMATHGoogle Scholar
  4. Ding XP, Jong YP: A new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings. J. Comput. Appl. Math. 2002, 138: 243–257. 10.1016/S0377-0427(01)00379-XView ArticleMathSciNetMATHGoogle Scholar
  5. Ding XP, Luo CL: Perturbed proximal point algorithms for generalized quasi-variational-like inclusions. J. Comput. Appl. Math. 2000, 210: 153–165.View ArticleMathSciNetMATHGoogle Scholar
  6. Jin MM: Generalized nonlinear implicit quasi-variational inclusions with relaxed monotone mappings. Adv. Nonlinear Var. Inequal. 2004, 7(2):173–181.MathSciNetMATHGoogle Scholar
  7. Li HG:Iterative algorithm for a new class of generalized nonlinear fuzzy set-valued variational inclusions involving ( H , η ) -monotone mappings. Adv. Nonlinear Var. Inequal. 2007, 10(1):89–100.MathSciNetGoogle Scholar
  8. Ahmad R, Bazán FF: An iterative algorithm for random generalized nonlinear mixed variational inclusions for random fuzzy mappings. Appl. Math. Comput. 2005, 167: 1400–1411. 10.1016/j.amc.2004.08.025View ArticleMathSciNetMATHGoogle Scholar
  9. Chang SS: Variational Inequality and Complementarity Problem Theory with Applications. Shanghai Scientific and Tech. Literature Publishing House, Shanghai; 1991.Google Scholar
  10. Cho YJ, Huang NJ, Kang SM: Random generalized set-valued strongly nonlinear implicit quasi-variational inequalities. J. Inequal. Appl. 2000, 5: 515–531.MathSciNetMATHGoogle Scholar
  11. Huang NJ, Fang YP: Generalized m -accretive mappings in Banach spaces. J. Sichuan Univ. 2001, 38(4):591–592.MATHGoogle Scholar
  12. Huang NJ, Fang YP: A new class of general variational inclusions involving maximal η -monotone mappings. Publ. Math. (Debr.) 2003, 62(1–2):83–98.MathSciNetMATHGoogle Scholar
  13. Chang SS, Huang NJ: Generalized random multivalued quasi-complementarity problem. Indian J. Math. 1993, 33: 305–320.MathSciNetMATHGoogle Scholar
  14. Fang YP, Huang NJ, Thompson HB:A new system of variational inclusions with ( H , η ) -monotone operators in Hilbert spaces. Comput. Math. Appl. 2005, 49: 365–374. 10.1016/j.camwa.2004.04.037View ArticleMathSciNetMATHGoogle Scholar
  15. Lan, HY, Cho, YJ, Verma, RU: On nonlinear relaxed cocoercive inclusions involving ( A , η ) -accretive mappings in Banach spaces. Comput. Math. Appl. 51 (2006)Google Scholar
  16. Amann H: On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal. 1972, 11: 346–384. 10.1016/0022-1236(72)90074-2View ArticleMathSciNetMATHGoogle Scholar
  17. Du YH: Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal. 1990, 38: 1–20. 10.1080/00036819008839957View ArticleMathSciNetMATHGoogle Scholar
  18. Ge DJ, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. TMA 1987, 11: 623–632. 10.1016/0362-546X(87)90077-0View ArticleMATHMathSciNetGoogle Scholar
  19. Ge DJ: Fixed points of mixed monotone operators with applications. Appl. Anal. 1988, 31: 215–224. 10.1080/00036818808839825View ArticleMathSciNetGoogle Scholar
  20. Li HG: Approximation solution for general nonlinear ordered variational inequalities and ordered equations in ordered Banach space. Nonlinear Anal. Forum 2008, 13(2):205–214.MathSciNetMATHGoogle Scholar
  21. Li HG: Approximation solution for a new class of general nonlinear ordered variational inequalities and ordered equations in ordered Banach space. Nonlinear Anal. Forum 2009, 14: 89–97.MathSciNetMATHGoogle Scholar
  22. Li HG: Nonlinear inclusion problem for ordered RME set-valued mappings in ordered Hilbert space. Nonlinear Funct. Anal. Appl. 2011, 16(1):1–8.MATHGoogle Scholar
  23. Li HG:Nonlinear inclusion problem involving ( α , λ ) -NODM set-valued mappings in ordered Hilbert space. Appl. Math. Lett. 2012, 25: 1384–1388. 10.1016/j.aml.2011.12.007View ArticleMathSciNetMATHGoogle Scholar
  24. Li HG: Sensitivity analysis for general nonlinear ordered parametric variational inequality with restricted-accretive mapping in ordered Banach space. Nonlinear Funct. Anal. Appl. 2011, 17(1):109–118.MATHGoogle Scholar
  25. Li HG, Qiu D, Jin M: GNM order variational inequality system with ordered Lipschitz continuous mappings in ordered Banach space. J. Inequal. Appl. 2013. 10.1186/1029-242X-2013-514Google Scholar
  26. Li HG, Qiu D, Zou Y: Characterizations of weak-ANODD set-valued mappings with applications to an approximate solution of GNMOQV inclusions involving operator in ordered Banach spaces. Fixed Point Theory Appl. 2012. 10.1186/1687-1812-2012-195Google Scholar
  27. Li HG, Li PL, Jin MM: A class of nonlinear mixed ordered inclusion problems for ordered ( α A , λ ) -ANODM set-valued mappings with strongly comparison mapping A . Fixed Point Theory Appl. 2014. 10.1186/1687-1812-2014-79Google Scholar
  28. Li HG, Li PL, Jin MM, Zheng JM:Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with ( α , λ ) -NODSM mappings in ordered Banach spaces. Fixed Point Theory Appl. 2014. 10.1186/1687-1812-2014-122Google Scholar
  29. Fang YP, Huang NJ: H -Monotone operator and resolvent operator technique for variational inclusions. Appl. Math. Comput. 2003, 145: 795–803. 10.1016/S0096-3003(03)00275-3View ArticleMathSciNetMATHGoogle Scholar
  30. Schaefer HH: Banach Lattices and Positive Operators. Springer, Berlin; 1974.View ArticleMATHGoogle Scholar
  31. Lan HY, Cho YJ, Verma RU:Nonlinear relaxed cocoercive variational inclusions involving ( A , η ) -accretive mappings in Banach spaces. Comput. Math. Appl. 2006, 51: 1529–1538. 10.1016/j.camwa.2005.11.036View ArticleMathSciNetMATHGoogle Scholar
  32. Li HG, Xu AJ, Jin MM:A hybrid proximal point three-step algorithm for nonlinear set-valued quasi-variational inclusions system involving ( A , η ) -accretive mappings. Fixed Point Theory Appl. 2010., 2010: Article ID 635382 10.1155/2010/635382Google Scholar
  33. Li HG, Xu AJ, Jin MM:A Ishikawa-hybrid proximal point algorithm for nonlinear set-valued inclusions problem based on ( A , η ) -accretive framework. Fixed Point Theory Appl. 2010., 2010: Article ID 501293 10.1155/2010/501293Google Scholar
  34. Lan HY, Cho YJ, Verma RU: On solution sensitivity of generalized relaxed cocoercive implicit quasivariational inclusions with A -monotone mappings. J. Comput. Anal. Appl. 2006, 8: 75–87.MathSciNetMATHGoogle Scholar
  35. Lan HY, Cho YJ, Huang NJ: Stability of iterative procedures for a class of generalized nonlinear quasi-variational-like inclusions involving maximal η -monotone mappings. Fixed Point Theory Appl. 2006, 6: 107–116.MathSciNetGoogle Scholar
  36. Lan HY, Kim JH, Cho YJ: On a new system of nonlinear A -monotone multivalued variational inclusions. J. Math. Anal. Appl. 2007, 327: 481–493. 10.1016/j.jmaa.2005.11.067View ArticleMathSciNetMATHGoogle Scholar
  37. Cho YJ, Lan HY: A new class of generalized nonlinear multi-valued quasi-variational-like-inclusions with H -monotone mappings. Math. Inequal. Appl. 2007, 10: 389–401.MathSciNetMATHGoogle Scholar
  38. Lan HY, Kang JI, Cho YJ:Nonlinear ( A , η ) -monotone operator inclusion systems involving non-monotone set-valued mappings. Taiwan. J. Math. 2007, 11: 683–701.MathSciNetMATHGoogle Scholar
  39. Cho YJ, Qin XL, Shang MJ, Su YF:Generalized nonlinear variational inclusions involving ( A , η ) -monotone mappings in Hilbert spaces. Fixed Point Theory Appl. 2007., 2007: Article ID 29653Google Scholar
  40. Cho YJ, Lan HY:Generalized nonlinear random ( A , η ) -accretive equations with random relaxed cocoercive mappings in Banach spaces. Comput. Math. Appl. 2008, 55: 2173–2182. 10.1016/j.camwa.2007.09.002View ArticleMathSciNetMATHGoogle Scholar
  41. Cho YJ, Qin X: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Anal. 2008, 69: 4443–4451. 10.1016/j.na.2007.11.001View ArticleMathSciNetMATHGoogle Scholar
  42. Alimohammady M, Balooee J, Cho YJ, Roohi M:A new system of nonlinear fuzzy variational inclusions involving ( A , η ) -accretive mappings in uniformly smooth Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 806727 10.1155/2009/806727Google Scholar
  43. Alimohammady M, Balooee J, Cho YJ, Roohi M: Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities. Nonlinear Anal. 2010, 73: 3907–3923. 10.1016/j.na.2010.08.022View ArticleMathSciNetMATHGoogle Scholar
  44. Alimohammady M, Balooee J, Cho YJ, Roohi M: New perturbed finite step iterative algorithms for a system of extended generalized nonlinear mixed quasi-variational inclusions. Comput. Math. Appl. 2010, 60: 2953–2970. 10.1016/j.camwa.2010.09.055View ArticleMathSciNetMATHGoogle Scholar
  45. Yao Y, Cho YJ, Liou Y: Iterative algorithms for variational inclusions, mixed equilibrium problems and fixed point problems approach to optimization problems. Cent. Eur. J. Math. 2011, 9: 640–656. 10.2478/s11533-011-0021-3View ArticleMathSciNetMATHGoogle Scholar
  46. Yao Y, Cho YJ, Liou Y: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 2011, 212: 242–250. 10.1016/j.ejor.2011.01.042View ArticleMathSciNetMATHGoogle Scholar
  47. Pan XB, Li HG, Xu AJ: The over-relaxed A -proximal point algorithm for general nonlinear mixed set-valued inclusion framework. Fixed Point Theory Appl. 2011., 2011: Article ID 840978 10.1155/2011/840978Google Scholar
  48. Verma RU:A hybrid proximal point algorithm based on the ( A , η ) -maximal monotonicity framework. Appl. Math. Lett. 2008, 21: 142–147. 10.1016/j.aml.2007.02.017View ArticleMathSciNetMATHGoogle Scholar
  49. Li HG, Qiu M: Ishikawa-hybrid proximal point algorithm for NSVI system. Fixed Point Theory Appl. 2012., 2012: Article ID 195 10.1186/1687-1812-2012-195Google Scholar
  50. Li HG, Qiu D, Zheng JM, Jin MM: Perturbed Ishikawa-hybrid quasi-proximal point algorithm with accretive mappings for a fuzzy system. Fixed Point Theory Appl. 2013. 10.1186/1687-1812-2013-281Google Scholar

Copyright

© Li et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.