Open Access

Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with ( α , λ ) - N O D S M mappings in ordered Banach spaces

Fixed Point Theory and Applications20142014:122

https://doi.org/10.1186/1687-1812-2014-122

Received: 24 October 2013

Accepted: 21 April 2014

Published: 17 May 2014

Abstract

In this paper, a new class of general set-valued parametric ordered variational inclusions, θ M ( x , g ( x , ρ ) , ρ ) , with ( α , λ ) - N O D S M mappings is studied in ordered Banach spaces. Then, by using fixed point theory and the resolvent operator associated with ( α , λ ) - N O D S M set-valued mappings, an existence theorem and a sensitivity analysis of the solution set for this kind of parametric variational inclusion is proved and discussed in ordered Banach spaces. The obtained results seem to be general in nature.

MSC:49J40, 47H06.

Keywords

sensitivity analysis general set-valued parametric ordered variational inclusion ( α , λ ) - N O D S M set-valued mapping ordered Banach spaces comparison solution

1 Introduction

Generalized nonlinear ordered variational inequalities and inclusions (ordered equation) have wide applications in many fields including, for example, mathematics, physics, optimization and control, nonlinear programming, economics, and engineering sciences etc. In recent years, nonlinear mapping fixed point theory and applications have been extensively studied in ordered Banach spaces [13]. In 2008 the author introduced and studied the approximation algorithm and the approximation solution for a class of generalized nonlinear ordered variational inequalities and ordered equations, to find x X such that A ( g ( x ) ) θ ( A ( x ) and g ( x ) are single-valued mappings), in ordered Banach spaces [4]. 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 in ordered Banach spaces [5]. By using the resolvent operator associated with an R M E set-valued mapping, the author introduced and studied a class of nonlinear inclusion problems for ordered MR set-valued mappings 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 [6]. In 2012, the author introduced and studied a class of nonlinear inclusion problems, to find x X such that 0 M ( x ) ( M ( x ) is a set-valued mapping) for ordered ( α , λ ) - N O D M set-valued mappings, and he then, applying the resolvent operator associated with ( α , λ ) - N O D M set-valued mappings, established the existence theorem on the solvability and a general algorithm applied to the approximation solvability of this class of nonlinear inclusion problems, based on the existence theorem and the new ( α , λ ) - N O D M model in ordered Hilbert space [7]. For Banach spaces, the author made a 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 2012 [8]. In this field, the obtained results seem to be general in nature. In 2013, the author introduced and studied characterizations of ordered ( α A , λ ) -weak- A N O D D set-valued mappings, which was applied to finding an approximate solution for a new class of general nonlinear mixed-order quasi-variational inclusions involving the operator in ordered Banach spaces [9], and, applying the matrix analysis and the vector-valued mapping fixed point analysis method, he introduced and studied a new class of generalized nonlinear mixed-order variational inequalities systems with ordered B-restricted-accretive mappings for ordered Lipschitz continuous mappings in ordered Banach spaces [10].

On the other hand, as everyone knows, the sensitivity analysis for a class of general nonlinear variational inequalities (inclusions) has wide applications to many fields. In 1999, Noor and Noor have studied a sensitivity analysis for strongly nonlinear quasi-variational inclusions [11]. From 2000, Agarwal et al. have discussed a sensitivity analysis for strongly nonlinear quasi-variations in Hilbert spaces by using the resolvent operator technique [12]; furthermore, Bi et al. [13], Lan et al. [14, 15], Dong et al. [16], Jin [17], Verma [18], Li et al. [9], and Li [19] have shown the existence of solutions and made a sensitivity analysis for a class of nonlinear variational inclusions involving generalized nonlinear mappings in Banach spaces, respectively. Recently, it has become of the highest interest that we are studying a new class of nonlinear ordered inclusion problems for ordered ( α , λ ) - N O D S M set-valued mappings and a sensitivity analysis of the solution set for this kind of parametric variational inclusions in ordered Banach spaces by using the resolvent operator technique [20] associated with ordered ( α , λ ) - N O D M set-valued mappings. For details, we refer the reader to [135] and the references therein.

Let X be a real ordered Banach space with a norm , zero θ, and a partial ordering relation ≤ defined by the normal cone P, and a normal constant N of P [4]. Let Ω be a nonempty open subset of X and we have the parametric ρ Ω . Let x = x ( ρ ) X ( ρ Ω ), g ( x , ρ ) : X × Ω X be a single-valued mapping and M ( x , g ( x , ρ ) , ρ ) : X × X × Ω 2 X be a set-valued mapping. We consider the following problem:

Find x = x ( ρ ) X ( ρ Ω ) such that
0 M ( x , g ( x , ρ ) , ρ ) ,
(1.1)

and the solution x ( ρ ) of the inclusion problem (1.1) is continuous from Ω and X.

Problem (1.1) is called a nonlinear generalized set-valued parametric ordered variational inclusions for ordered ( α , λ ) - N O D S M set-valued mappings in ordered Banach spaces.

Remark 1.1 When mapping M is single-valued and M ( x , y ) = A ( g ( x ) ) , then the problem (1.1) reduces to problem (2.1) in [4].

When the mapping M ( x , y ) = M ( x ) is set-valued, then the problem (1.1) reduces to problem (1.1) in [7].

Inspired and motivated by recent research work in this field, in this paper, a new class of nonlinear generalized parametric ordered variational inclusions with ( α , λ ) - N O D S M mappings is studied in ordered Banach spaces. Then, by using the resolvent operator associated with ( α , λ ) - N O D S M set-valued mappings, an existence theorem of this class of nonlinear inclusions is established, and a sensitivity analysis of the solution set for this kind of parametric variational inclusions is proved and discussed in ordered Banach spaces. The obtained results seem to be general in nature.

2 Preliminaries

Let X be a real ordered Banach space with a norm , a zero θ, a normal cone P, a normal constant N and a partial ordering relation ≤ defined by the cone P. For arbitrary x , y X , lub { x , y } and glb { x , y } express the least upper bound of the set { x , y } and the greatest lower bound of the set { x , y } on the partial ordering relation ≤, respectively. Suppose that lub { x , y } and glb { x , y } exist. Let us recall some concepts and results.

Definition 2.1 [4, 24]

Let X be a real Banach space with a norm , θ be a zero element in X.
  1. (i)

    A nonempty closed convex subset P of X is said to be a cone if (1) for any x P and any λ > 0 , λ x P holds, (2) if x P and x P , then x = θ ;

     
  2. (ii)

    P is said to be a normal cone if and only if there exists a constant N > 0 , a normal constant of P such that for θ x y , x N y holds;

     
  3. (iii)

    for arbitrary x , y X , x y if and only if x y P ;

     
  4. (iv)

    for x , y X , x and y are said to be comparative to each other, if and only if x y (or y x ) holds (denoted by x y for x y and y x ).

     

Lemma 2.2 If x y , then lub { x , y } and glb { x , y } exist, x y y x , and θ ( x y ) ( y x ) .

Proof If x y , then x y or y x . Let x y , then lub { x ; y } = y and glb { x ; y } = x , and x y θ y x . It follows that lub { x ; y } and glb { x ; y } exist, and x y y x . ( x y ) ( y x ) = ( y x ) , then θ ( x y ) ( y x ) . □

Lemma 2.3 If for any natural number n, x y n , and y n y ( n ), then x y .

Proof If for any natural number n, x y n and y n y ( n ), then x y n P or y n x P for any natural number n. Since P is a nonempty closed convex subsets of X, we have x y = lim n ( x y n ) P or y x = lim n ( y n x ) P . Therefore, x y . □

Lemma 2.4 [46]

Let X be an ordered Banach space, let P be a cone of X, letbe a relation defined by the cone P in Definition  2.1(iii). For x , y , v , u X , the following relations hold:
  1. (1)

    the relationin X is a partial ordering relation in X;

     
  2. (2)

    x y = y x ;

     
  3. (3)

    x x = θ ;

     
  4. (4)

    θ x θ ;

     
  5. (5)

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

     
  6. (6)

    if x, y, and w can be comparative to each other, then ( x y ) x w + w y ;

     
  7. (7)

    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 ) ;

     
  8. (8)

    if x, y, z, w can be compared with each other, then ( x y ) ( z w ) ( ( x z ) ( y w ) ) ( ( x w ) ( y z ) ) ;

     
  9. (9)

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

     
  10. (10)

    if x θ , then x θ x x θ ;

     
  11. (11)

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

     
  12. (12)

    ( x θ ) ( y θ ) ( x y ) θ ;

     
  13. (13)

    if θ x and x θ , and α > 0 , then θ α x and α x θ .

     

Proof (1)-(8) come from Lemma 2.5 in [4] and Lemma 2.3 in [5], and (8)-(13) directly follow from (1)-(8). □

Definition 2.5 Let X be a real ordered Banach space, let Ω be a nonempty open subset of X in which the parametric ρ takes values, let x = x ( ρ ) X ( ρ Ω ), g ( x , ρ ) : X × Ω X be a single-valued mapping and M ( x , g ( x , ρ ) , ρ ) : X × X × Ω 2 X be a set-valued mapping and M ( x , , ρ ) be a nonempty closed subset in X.
  1. (1)

    A set-valued mapping M is said to be a comparison mapping, if for any v x M ( x , , ) , x v x , and if x y , then for any v x M ( x , , ) and any v y M ( y , , ) , v x v y ( x , y X ).

     
  2. (2)

    A set-valued mapping M is said to be a comparison mapping with respect to g, if for any v x M ( , g ( x ) , ) , x v x , and if x y , then for any v x M ( , g ( x ) , ) and any v y M ( , g ( y ) , ) , v x v y ( x , y X ).

     
  3. (3)
    A comparison mapping M is said to be an α-non-ordinary difference mapping, if there exists a constant α > 0 , for each x , y X , v x M ( x , , ) , and v y M ( y , , ) such that
    ( v x v y ) α ( x y ) = θ .
     
  4. (4)
    A comparison mapping M is said to be λ-ordered strongly monotonic increase mapping, if for x y there exists a constant λ > 0 such that
    λ ( v x v y ) x y x , y X , v x M ( x ) , v y M ( y , , ) .
     
  5. (5)

    A comparison mapping M is said to be a ( α , λ ) - N O D S M mapping, if M is a α-non-ordinary difference and λ-ordered strongly monotone increasing mapping, and ( I + λ M ( x , , ) ) ( X ) = X for α , λ > 0 .

     

Obviously, if M is a comparison mapping, then M ( x , , ) I ( x X ).

Definition 2.6 [4]

Let X be a real ordered Banach space, P be a normal cone with a normal constant N in X; a mapping A : X × X X is said to be β-ordered compression, if A is comparison, and there exists a constant 0 < β < 1 such that
( A ( x , ) A ( y , ) ) β ( x y ) .

Definition 2.7 [4]

Let X be a real ordered Banach space. A mapping A : X × X X is said to be a restricted-accretive mapping with constants ( α 1 , α 2 ) , if A is a comparison, and there exist two constants 0 < β 1 , β 2 1 such that for arbitrary x , y X ,
( A ( x , ) + I ( x ) ) ( A ( y , ) + I ( y ) ) β 1 ( A ( x , ) A ( y , ) ) + β 2 ( x y )

holds, where I is the identity mapping on X.

Definition 2.8 Let X be a real ordered Banach space, let Ω be a nonempty open subset of X in which the parametric ρ takes values, let x = x ( ρ ) X ( ρ Ω ). x = x ( ρ ) is said to be a comparison element when, if ρ 1 ρ 2 then x ( ρ 1 ) x ( ρ 2 ) for any ρ 1 , ρ 2 Ω .

Lemma 2.9 Let M = M ( x , , ) : X × X × X 2 X . If M is a α-non-ordinary difference mapping, then an inverse mapping J M , λ = ( I + λ M ) 1 : X × X × X 2 X of ( I + λ M ) is a single-valued mapping ( α , λ > 0 ), where I is the identity mapping on X.

Proof Let u X , and x and y be two elements in ( I + λ M ) 1 ( u ) . It follows that u x λ M ( x , , ) and u y λ M ( y , , ) , and
1 λ ( u x ) 1 λ ( u y ) = | 1 λ | ( x y ) .
Since M is a α-non-ordinary difference mapping, we have
0 = ( 1 λ ( u x ) 1 λ ( u y ) ) α ( x y ) = | 1 λ | ( x y ) α ( x y ) = | | 1 λ | + α | ( x y )

and x y = 0 from Lemma 2.4. Also, x = y holds. Thus ( I + λ M ) 1 ( u ) is a single-valued mapping. The proof is completed. □

Definition 2.10 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in the X, let M = M ( x , , ) : X × X × X 2 X be a α-non-ordinary difference mapping. The resolvent operator J M , λ : X × X × X X of the M ( x , , ) is defined by
J M , λ ( x ) = ( I + λ M ) 1 ( x ) for all  x X ,

where λ > 0 is a constant.

3 Existence theorem of the solution

In this section, we will show an existence theorem on the solvability of this class of nonlinear inclusion problems (1.1).

Theorem 3.1 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in X, letbe an ordering relation defined by the cone P. If M = M ( x , , ) : X × X × Ω 2 X is an α-non-ordinary difference mapping, then the inclusion problem (1.1) has a solution x if and only if g ( x , ) = J M ( x , , ) , λ g ( x , ) in X.

Proof This directly follows from the definition of the resolvent operator J M , λ of M ( x , , ) . □

Theorem 3.2 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, letbe an ordering relation defined by the cone P, the operator be a X O R operator. If M = M ( x , , ) : X × X × Ω 2 X is an ( α , λ ) - N O D S M mapping with respect to J M , λ , then the resolvent operator J M , λ : X X is a comparison mapping.

Proof Since M = M ( x , , ) : X × X × Ω 2 X is an α-non-ordinary difference mapping and a comparison mapping with respect to J M , λ so that x J M , λ ( x ) . For any x , y X , let x y , and v x = 1 λ ( x J M , λ ( x ) ) M ( J M ( x , ) , λ ( y ) ) and v y = 1 λ ( y J M ( x , ) , λ ( y ) ) M ( J M ( x , ) , λ ( y ) ) . Setting
v x v y = 1 λ ( x y + J M ( x , ) , λ ( y ) J M ( x , ) , λ ( x ) ) ,
by using the λ-order strongly monotonicity of M, we have
θ λ ( v x v y ) ( x y ) = J M , λ ( y ) J M , λ ( x ) ,
(3.1)

and if y x then λ ( v x v y ) ( x y ) P , and if x y then ( x y ) λ ( v x v y ) P . Therefore J M , λ ( y ) J M , λ ( x ) for Lemma 2.4. □

Theorem 3.3 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in X, letbe an ordering relation defined by the cone P. Let M = M ( , x , ) : X × X × Ω 2 X be a N O D S M set-valued mapping with respect to J M , λ . If α > 1 λ > 0 , then for the resolvent operator J M , λ : X X , the following relation holds:
J M , λ ( y ) J M , λ ( z ) 1 ( α λ 1 ) ( y z ) .
(3.2)
Proof Let M = M ( , x , ) : X × X × Ω 2 X be a N O D S M set-valued mapping with respect to J M , λ . For y , z X , let u y = J M , λ ( y ) u z = J M , λ ( z ) , v y = 1 λ ( y u y ) M ( , u y , ) and v z = 1 λ ( z u z ) M ( , u z , ) , then v y v z for y z . Since M ( , x , ) : X × X × X 2 X is an ( α , λ ) - N O D S M mapping with respect to the J M , λ , the following relation holds by Lemma 2.4 and the condition ( v y v z ) α ( u y u z ) = θ :
1 λ ( ( y z ) + ( u y u z ) ) v y v z = α ( u y u z ) .

It follows that ( λ α 1 ) ( u y u z ) ( y z ) and J M , λ ( y ) J M , λ ( z ) 1 ( α λ 1 ) ( y z ) from the condition α > 1 λ > 0 . The proof is completed. □

Theorem 3.4 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, letbe an ordering relation defined by the cone P. Let M = M ( x , , ) : X × X × Ω 2 X be an ( α , λ ) - N O D S M set-valued mapping with respect to the first argument and g : X × Ω X be a γ-ordered compression and an 1-ordered strongly monotonic increase with respect to the first argument and range ( g ) dom M ( , x , ) , and J M , λ for M with respect to the first argument and ( J M , λ I ) for M with respect to the second argument be two restricted-accretive mappings with constants ( ξ 1 , ξ 2 ) and ( β 1 , β 2 ) , respectively, and g J M , λ . Suppose that for any x , y , z X
J M ( x , , ) , λ ( z ) J M ( y , , ) , λ ( z ) δ ( x y )
(3.3)
and
γ ( ξ 1 α λ 1 ξ 2 ) δ < 1 N β 2 N β 1
(3.4)

hold. For any parametric ρ Ω , for the nonlinear parametric inclusion problem (1.1) there exists a solution x .

Proof Let X be a real ordered Banach space, let P be a normal cone with normal constant N in the X, let ≤ be an ordering relation defined by the cone P, let Ω be a nonempty open subset of X in which the parametric ρ takes values, let M = M ( x , , ) : X × X × Ω 2 X , and for any given ρ Ω and x 1 = x 1 ( ρ ) , x 2 = x 2 ( ρ ) X for λ > 0 . If x 1 ( ρ ) x 2 ( ρ ) , and setting
F ( x i ( ρ ) , ρ ) = x i ( ρ ) g ( x i , ρ ) + J M , λ ( g ( x i , ρ ) ) ,
(3.5)
where i = 1 , 2 , by (3.1) and the λ-ordered strongly monotonicity of M,
F ( x 1 ( ρ ) , ρ ) F ( x 2 ( ρ ) , ρ ) = ( x 1 ( ρ ) x 2 ( ρ ) ) + ( g ( x 2 , ρ ) g ( x 1 , ρ ) ) + ( J M , λ ( g ( x 1 , ρ ) ) J M , λ ( g ( x 2 , ρ ) ) ) = ( x 1 ( ρ ) x 2 ( ρ ) ) + λ ( v g ( x 2 ) v g ( x 1 ) ) ( x 1 ( ρ ) x 2 ( ρ ) ) ( g ( x 1 , ρ ) g ( x 2 , ρ ) ) θ ;
by ( I + λ M ) ( X ) = X , the comparability of J M , λ , and the 1-ordered monotonic increase of g ( x , ) , it follows from x 1 ( ρ ) x 2 ( ρ ) that F ( x 1 ( ρ ) , ρ ) F ( x 2 ( ρ ) , ρ ) . Using (3.3), (3.5), Lemma 2.4, Theorem 3.3, and α > 2 λ > 0 , from the conditions that J M , λ for M with respect to the first argument and ( J M , λ I ) for M with respect to the second argument are two restricted-accretive mappings with constants ( ξ 1 , ξ 2 ) and ( β 1 , β 2 ) , respectively, it follows that
θ F ( x 1 ( ρ ) , ρ ) F ( x 2 ( ρ ) , ρ ) ( x 1 ( ρ ) g ( x 1 , ρ ) + J M , λ ( g ( x 1 , ρ ) ) ) ( x 2 ( ρ ) g ( x 2 , ρ ) + J M , λ ( g ( x 2 , ρ ) ) ) β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 [ ( J M ( x 1 ( ρ ) , , ρ ) , λ ( g ( x 1 ( ρ ) , ρ ) ) g ( x 1 ( ρ ) , ρ ) ) ( J M ( x 2 ( ρ ) , , ρ ) , λ ( g ( x 2 ( ρ ) , ρ ) ) g ( x 2 ( ρ ) , ρ ) ) ] β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 { [ ( J M ( x 1 ( ρ ) , , ρ ) , λ ( g ( x 1 ( ρ ) , ρ ) ) g ( x 1 ( ρ ) , ρ ) ) ( J M ( x 2 ( ρ ) , , ρ ) , λ ( g ( x 1 ( ρ ) , ρ ) ) g ( x 1 ( ρ ) , ρ ) ) ] [ ( J M ( x 2 ( ρ ) , , ρ ) , λ ( g ( x 1 ( ρ ) , ρ ) ) g ( x 1 ( ρ ) , ρ ) ) ( J M ( x 2 ( ρ ) , , ρ ) , λ ( g ( x 2 ( ρ ) , ρ ) ) g ( x 2 ( ρ ) , ρ ) ) ] } β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 { δ ( x 1 ( ρ ) x 2 ( ρ ) ) [ ξ 2 ( g ( x 1 ( ρ ) , ρ ) g ( x 2 ( ρ ) , ρ ) ) + ξ 1 ( J M ( x 2 ( ρ ) , , ρ ) , λ ( g ( x 1 ( ρ ) , ρ ) ) J M ( x 2 ( ρ ) , , ρ ) , λ ( g ( x 2 ( ρ ) , ρ ) ) ) ] } β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 { δ ( x 1 ( ρ ) x 2 ( ρ ) ) [ ξ 2 γ ( x 1 ( ρ ) x 2 ( ρ ) ) + ξ 1 α λ 1 ( g ( x 1 ( ρ ) , ρ ) g ( x 2 ( ρ ) , ρ ) ) ] } β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 { δ ( x 1 ( ρ ) x 2 ( ρ ) ) [ ξ 2 γ ( x 1 ( ρ ) x 2 ( ρ ) ) + ξ 1 α λ 1 ( γ ( x n x n 1 ) ) ] } β 2 ( x n x n 1 ) + β 1 { δ ( x 1 ( ρ ) x 2 ( ρ ) ) [ ( ξ 2 ξ 1 α λ 1 ) γ ( x 1 ( ρ ) x 2 ( ρ ) ) ] } β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 ( | δ ( ξ 2 ξ 1 α λ 1 ) γ | ( x 1 ( ρ ) x 2 ( ρ ) ) ) β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 [ ( ξ 2 ξ 1 1 α λ 1 ) γ δ ] ( x 1 ( ρ ) x 2 ( ρ ) ) β 2 ( x 1 ( ρ ) x 2 ( ρ ) ) + β 1 [ ( ξ 1 α λ 1 ξ 2 ) γ δ ] ( x 1 ( ρ ) x 2 ( ρ ) ) [ β 2 + β 1 ( γ ( ξ 1 α λ 1 ξ 2 ) δ ) ] ( x 1 ( ρ ) x 2 ( ρ ) ) ,
(3.6)
and, by Definition 2.1(2), we obtain
F ( x 1 ( ρ ) , ρ ) F ( x 2 ( ρ ) , ρ ) h N x 1 ( ρ ) x 2 ( ρ ) ,
(3.7)
where h = β 2 + β 1 ( γ ( ξ 1 α λ 1 ξ 2 ) ) δ . It follows from the condition (3.4) that 0 < h N < 1 , and F ( x ( ρ ) , ρ ) has a fixed point x X and the x is a solution of the generalized nonlinear ordered parametric equation
x ( ρ ) = x ( ρ ) g ( x ( ρ ) , ρ ) + J M , λ ( g ( x ( ρ ) , ρ ) ) .
Further, x satisfies the generalized nonlinear ordered parametric equation
g ( x ( ρ ) , ρ ) = J M , λ ( g ( x ( ρ ) , ρ ) ) .

Then for the nonlinear parametric inclusion problems (1.1) there exists a solution x X for any parametric ρ Ω . This completes the proof. □

Remark 3.5 Though the method of solving problem by the resolvent operator is the same as in [20, 2528] and [34] for the nonlinear inclusion problem, the character of the ordered ( α , λ ) - A N O D M set-valued mapping is different from the one of the ( A , η ) -accretive mapping [25], the ( H , η ) -monotone mapping [26], the ( G , η ) -monotone mapping [27] and the monotone mapping [34].

4 Sensitivity analysis of the solution

Theorem 4.1 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, letbe an ordering relation defined by the cone P. Let M = M ( x , , ) : X × X × Ω 2 X be a ( α , λ ) - N O D S M set-valued mapping and g : X × Ω X be a γ-ordered compression, continuous and 1-ordered monotonic increase of g ( x , ) with respect to first argument ρ Ω , and range ( g ) dom M ( , x , ρ ) , and J M , λ for M with respect to first argument and ( J M , λ I ) for M with respect to second argument be two restricted-accretive mappings with constants ( ξ 1 , ξ 2 ) and ( β 1 , β 2 ) , respectively, and g J M , λ . Suppose that for any x , y , z X
J M ( x , , ) , λ ( z ) J M ( y , , ) , λ ( z ) δ ( x y )
(4.1)
and
γ ( ξ 1 α λ 1 ξ 2 ) δ < 1 N β 2 N β 1
(4.2)

hold; if the solution x ( ρ ) of the nonlinear parametric inclusion problem (1.1) is a comparison element, which is said to be a comparison solution of the nonlinear parametric inclusion problem (1.1), then x ( ρ ) , a comparison solution, is continuous on Ω.

Proof For any given ρ , ρ ¯ Ω , by Theorem 3.4, let x ( ρ ) be a comparison solution, and x ( ρ ) and x ( ρ ¯ ) satisfy parametric problem (1.1), then for any λ > 0 , we have
x ( ρ ) = F ( x ( ρ ) , ρ ) = x ( ρ ) g ( x ( ρ ) , ρ ) + J M , λ ( g ( x ( ρ ) , ρ ) ) , x ( ρ ¯ ) = F ( x ( ρ ¯ ) , ρ ¯ ) = x ( ρ ¯ ) g ( x ( ρ ¯ ) , ρ ¯ ) + J M , λ ( g ( x ( ρ ¯ ) , ρ ¯ ) ) .
(4.3)
By the condition that M, g, J M , λ , and J M , λ I are comparisons for each other and by Lemma 2.4, we have
θ x ( ρ ) x ( ρ ¯ ) F ( x ( ρ ) , ρ ) F ( x ( ρ ¯ ) , ρ ¯ ) F ( x ( ρ ) , ρ ) θ F ( x ( ρ ¯ ) , ρ ¯ ) [ F ( x ( ρ ) , ρ ) F ( x ( ρ ¯ ) , ρ ) ] [ F ( x ( ρ ¯ ) , ρ ) F ( x ( ρ ¯ ) , ρ ¯ ) ] .
(4.4)
Further, J M , λ and ( J M , λ I ) are two restricted-accretive mappings with constants ( ξ 1 , ξ 2 ) and ( β 1 , β 2 ) , respectively, so that from Lemma 2.4 and Theorem 3.3, α > 2 λ > 0 , and from (3.6), it follows that
F ( x ( ρ ) , ρ ) F ( x ( ρ ¯ ) , ρ ) ( x ( ρ ) g ( x ( ρ ) , ρ ) + J M , λ ( g ( x ( ρ ) , ρ ) ) ) ( x ( ρ ¯ ) g ( x ( ρ ¯ ) , ρ ) + J M , λ ( g ( x ( ρ ¯ ) , ρ ) ) ) h ( x ( ρ ) x ( ρ ¯ ) ) ,
(4.5)
where h = β 2 + β 1 ( γ 1 ( ξ 1 α λ 1 ξ 2 ) ) δ < 1 N for the condition (4.1), and
F ( x ( ρ ¯ ) , ρ ) F ( x ( ρ ¯ ) , ρ ¯ ) ( x ( ρ ¯ ) g ( x ( ρ ¯ ) , ρ ) + J M , λ ( g ( x ( ρ ¯ ) , ρ ) ) ) ( x ( ρ ¯ ) g ( x ( ρ ¯ ) , ρ ¯ ) + J M , λ ( g ( x ( ρ ¯ ) , ρ ¯ ) ) ) β 2 θ + β 1 [ ( J M ( x ( ρ ¯ ) , , ρ ) , λ ( g ( x ( ρ ¯ ) , ρ ) ) g ( x ( ρ ¯ ) , ρ ) ) ( J M ( x ( ρ ¯ ) , , ρ ¯ ) , λ ( g ( x ( ρ ¯ ) , ρ ¯ ) ) g ( x ( ρ ¯ ) , ρ ¯ ) ) ] β 2 θ + β 1 [ ξ 2 ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) + ξ 1 ( J M ( x ( ρ ¯ ) , , ρ ) , λ ( g ( x ( ρ ¯ ) , ρ ) ) J M ( x ( ρ ¯ ) , , ρ ¯ ) , λ ( g ( x ( ρ ¯ ) , ρ ¯ ) ) ) ] β 2 θ + β 1 [ ξ 2 ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) + ξ 1 ( ( ρ ρ ¯ ) ( J M ( x ( ρ ¯ ) , , ρ ¯ ) , λ ( g ( x ( ρ ¯ ) , ρ ) ) J M ( x ( ρ ¯ ) , , ρ ¯ ) , λ ( g ( x ( ρ ¯ ) , ρ ¯ ) ) ) ) ] β 2 θ + β 1 [ ξ 2 ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) + ξ 1 ( ( ρ ρ ¯ ) ( δ ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) ) ) ] β 2 θ + β 1 ξ 2 ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) + β 1 ξ 1 ( ( ρ ρ ¯ ) ( δ ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) ) ) .
(4.6)
Combining (4.4), (4.5), and (4.6), and by using Lemma 2.4, we get
( x ( ρ ) x ( ρ ¯ ) ) h ( x ( ρ ) x ( ρ ¯ ) ) [ β 2 θ + β 1 ξ 2 ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) + β 1 ξ 1 ( ( ρ ρ ¯ ) ( δ ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) ) ) ] .
Therefore,
( x ( ρ ) x ( ρ ¯ ) ) h ( x ( ρ ) x ( ρ ¯ ) ) β 2 θ + β 1 ξ 2 ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) + β 1 ξ 1 ( ( ρ ρ ¯ ) ( δ ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) ) ) .
It follows that
( x ( ρ ) x ( ρ ¯ ) ) 1 1 h [ β 2 θ + β 1 ξ 2 ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) + β 1 ξ 1 ( ( ρ ρ ¯ ) ( δ ( g ( x ( ρ ¯ ) , ρ ) g ( x ( ρ ¯ ) , ρ ¯ ) ) ) ) ] .
(4.7)
By Lemma 2.4, β 2 θ = θ , and continuity of g with respect to the first argument ρ Ω , we have
lim ρ ρ ¯ x ( ρ ) x ( ρ ¯ ) = θ
and
lim ρ ρ ¯ x ( ρ ) x ( ρ ¯ ) = 0 ,
(4.8)

which implies that the solution x ( ρ ) of problem (1.1) is continuous at ρ = ρ ¯ . This completes the proof. □

Declarations

Authors’ Affiliations

(1)
Institute of Applied Mathematics Research, Chongqing University of Posts and Telecommunications
(2)
Institute of Nonlinear Analysis Research, Changjiang Normal University

References

  1. Du YH: Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal. 1990, 38: 1–20. 10.1080/00036819008839957View ArticleMathSciNetGoogle Scholar
  2. 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 ArticleGoogle Scholar
  3. Ge DJ: Fixed points of mixed monotone operators with applications. Appl. Anal. 1988, 31: 215–224. 10.1080/00036818808839825View ArticleMathSciNetGoogle Scholar
  4. 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.MathSciNetGoogle Scholar
  5. Li H-g: 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.MathSciNetGoogle Scholar
  6. Li H-g: Nonlinear inclusion problem for ordered RME set-valued mappings in ordered Hilbert space. Nonlinear Funct. Anal. Appl. 2011, 16(1):1–8.Google Scholar
  7. Li H-g:Nonlinear inclusion problem involving ( α , λ ) - N O D M set-valued mappings in ordered Hilbert space. Appl. Math. Lett. 2012, 25: 1384–1388. 10.1016/j.aml.2011.12.007View ArticleMathSciNetGoogle Scholar
  8. Li H-g: Sensitivity analysis for general nonlinear ordered parametric variational inequality with restricted-accretive mapping in ordered Banach space. Nonlinear Funct. Anal. Appl. 2011, 1(17):109–118.Google Scholar
  9. Li H-g, Qiu D, Zou Y:Characterizations of weak- A N O D D set-valued mappings with applications to approximate solution of GNMOQV inclusions involving operator in order Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 241Google Scholar
  10. 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., 2013: Article ID 281Google Scholar
  11. Noor MA, Noor KI: Sensitivity analysis for strongly nonlinear quasi-variational inclusions. J. Math. Anal. Appl. 1999, 236: 290–299. 10.1006/jmaa.1999.6424View ArticleMathSciNetGoogle Scholar
  12. Agarwal RP, Cho YJ, Huang NJ: Sensitivity analysis for strongly nonlinear quasi-variations. Appl. Math. Lett. 2000, 13: 19–24.View ArticleMathSciNetGoogle Scholar
  13. Bi ZS, Han Z, Fang YP: Sensitivity analysis for nonlinear variational inclusions involving generalized m -accretive mapping. J. Sichuan Univ. 2003, 40(2):240–243.MathSciNetGoogle Scholar
  14. Lan HY, Cho YJ, Verma RU:On nonlinear relaxed cocoercive inclusions involving ( A , η ) -accretive mappings in Banach spaces. Math. Appl. Comput. 2006, 51: 1529–1538. 10.1016/j.camwa.2005.11.036View ArticleMathSciNetGoogle Scholar
  15. 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.MathSciNetGoogle Scholar
  16. Dong H, Lee BS, Hang NJ: Sensitivity analysis for generalized quasi-variational-like inclusions. Nonlinear Anal. Forum 2001, 6(2):313–320.MathSciNetGoogle Scholar
  17. Jin MM: Sensitivity analysis for quasi-variational inequalities. Nonlinear Anal. Forum 2003, 8(1):93–99.MathSciNetGoogle Scholar
  18. Verma RU: Sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions. J. Appl. Math. Stoch. Anal. 2006., 2006: Article ID 52041Google Scholar
  19. Li HG:Sensitivity analysis for generalized set-valued parametric mixed quasi-variational inclusion with ( A , η ) -accretive mappings. Nonlinear Anal. Forum 2008, 13(1):27–37.View ArticleMathSciNetGoogle Scholar
  20. Fang YP, Huang NJ: H -Accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Appl. Math. Lett. 2004, 17(6):647–653. 10.1016/S0893-9659(04)90099-7View ArticleMathSciNetGoogle Scholar
  21. Li H-g, 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
  22. 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 ArticleMathSciNetGoogle Scholar
  23. Li HG, Pan XB: Sensitivity analysis for general nonlinear parametric set-valued variational inclusions in Banach spaces. J. Southwest Univ. (Natur. Sci. Ed.) 2009, 32(2):29–33.Google Scholar
  24. Schaefer HH: Banach Lattices and Positive Operators. Springer, Berlin; 1974.View ArticleGoogle Scholar
  25. Lan HY, Cho YJ, Verma RU:On nonlinear relaxed cocoercive inclusions involving ( A , η ) -accretive mappings in Banach spaces. Comput. Math. Appl. 2006, 51: 1529–1538. 10.1016/j.camwa.2005.11.036View ArticleMathSciNetGoogle Scholar
  26. 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
  27. 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
  28. 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
  29. 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 ArticleMathSciNetGoogle Scholar
  30. 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 ArticleMathSciNetGoogle Scholar
  31. 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 ArticleMathSciNetGoogle Scholar
  32. Deng Z, Huang Y: Existence and multiplicity of symmetric solutions for a class of singular elliptic problems. Nonlinear Anal., Real World Appl. 2012, 13: 2293–2303. 10.1016/j.nonrwa.2012.01.024View ArticleMathSciNetGoogle Scholar
  33. Qiu D, Shu L: Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings. Inf. Sci. 2008, 178: 3595–3604. 10.1016/j.ins.2008.05.018View ArticleMathSciNetGoogle Scholar
  34. 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 ArticleMathSciNetGoogle Scholar
  35. 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

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/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.