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Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with (α,λ)-NODSM mappings in ordered Banach spaces

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Abstract

In this paper, a new class of general set-valued parametric ordered variational inclusions, θM(x,g(x,ρ),ρ), with (α,λ)-NODSM mappings is studied in ordered Banach spaces. Then, by using fixed point theory and the resolvent operator associated with (α,λ)-NODSM 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.

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 xX 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 RME 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 xX such that 0M(x) (M(x) is a set-valued mapping) for ordered (α,λ)-NODM set-valued mappings, and he 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 this class of nonlinear inclusion problems, based on the existence theorem and the new (α,λ)-NODM 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):ΩX 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-ANODD 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)-NODSM 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)-NODM 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 ρΩ. Let x=x(ρ)X (ρΩ), g(x,ρ):X×ΩX 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 (ρΩ) such that

0M ( 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)-NODSM 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)-NODSM mappings is studied in ordered Banach spaces. Then, by using the resolvent operator associated with (α,λ)-NODSM 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,yX, 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 xP and any λ>0, λxP holds, (2) if xP and xP, 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 θxy, xNy holds;

  3. (iii)

    for arbitrary x,yX, xy if and only if xyP;

  4. (iv)

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

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

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

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 xP 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,uX, the following relations hold:

  1. (1)

    the relationin X is a partial ordering relation in X;

  2. (2)

    xy=yx;

  3. (3)

    xx=θ;

  4. (4)

    θxθ;

  5. (5)

    let λ be a real, then (λx)(λy)=|λ|(xy);

  6. (6)

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

  7. (7)

    let (x+y)(u+v) exist, and if xu,v and yu,v, then (x+y)(u+v)(xu+yv)(xv+yu);

  8. (8)

    if x, y, z, w can be compared with each other, then (xy)(zw)((xz)(yw))((xw)(yz));

  9. (9)

    if xy and uv, then x+uy+v;

  10. (10)

    if xθ, then xθxxθ;

  11. (11)

    if xy, then (xθ)(yθ)(xy)θ=xy;

  12. (12)

    (xθ)(yθ)(xy)θ;

  13. (13)

    if θx and xθ, and α>0, then θαx and αxθ.

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 (ρΩ), g(x,ρ):X×ΩX 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 xy, then for any v x M(x,,) and any v y M(y,,), v x v y (x,yX).

  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 xy, then for any v x M(,g(x),) and any v y M(,g(y),), v x v y (x,yX).

  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,yX, v x M(x,,), and v y M(y,,) such that

    ( v x v y )α(xy)=θ.
  4. (4)

    A comparison mapping M is said to be λ-ordered strongly monotonic increase mapping, if for xy there exists a constant λ>0 such that

    λ( v x v y )xyx,yX, v x M(x), v y M(y,,).
  5. (5)

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

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

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×XX 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(xy).

Definition 2.7 [4]

Let X be a real ordered Banach space. A mapping A:X×XX 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,yX,

( A ( x , ) + I ( x ) ) ( A ( y , ) + I ( y ) ) β 1 ( A ( x , ) A ( y , ) ) + β 2 (xy)

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 (ρΩ). 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+λM) is a single-valued mapping (α,λ>0), where I is the identity mapping on X.

Proof Let uX, and x and y be two elements in ( I + λ M ) 1 (u). It follows that uxλM(x,,) and uyλM(y,,), and

1 λ (ux) 1 λ (uy)= | 1 λ | (xy).

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 xy=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×XX of the M(x,,) is defined by

J M , λ (x)= ( I + λ M ) 1 (x)for all xX,

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 XOR operator. If M=M(x,,):X×X×Ω 2 X is an (α,λ)-NODSM mapping with respect to J M , λ , then the resolvent operator J M , λ :XX 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,yX, let xy, 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

θλ( v x v y )(xy)= J M , λ (y) J M , λ (x),
(3.1)

and if yx then λ( v x v y )(xy)P, and if xy then (xy)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 NODSM set-valued mapping with respect to J M , λ . If α> 1 λ >0, then for the resolvent operator J M , λ :XX, the following relation holds:

J M , λ (y) J M , λ (z) 1 ( α λ 1 ) (yz).
(3.2)

Proof Let M=M(,x,):X×X×Ω 2 X be a NODSM set-valued mapping with respect to J M , λ . For y,zX, 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 yz. Since M(,x,):X×X×X 2 X is an (α,λ)-NODSM 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 (λα1)( u y u z )(yz) and J M , λ (y) J M , λ (z) 1 ( α λ 1 ) (yz) 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)-NODSM set-valued mapping with respect to the first argument and g:X×ΩX be a γ-ordered compression and an 1-ordered strongly monotonic increase with respect to the first argument and range(g)domM(,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,zX

J M ( x , , ) , λ (z) J M ( y , , ) , λ (z)δ(xy)
(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 ρΩ, 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 ρΩ 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+λM)(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 ) hN 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<hN<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 ρΩ. 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)-ANODM 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)-NODSM set-valued mapping and g:X×ΩX be a γ-ordered compression, continuous and 1-ordered monotonic increase of g(x,) with respect to first argument ρΩ, and range(g)domM(,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,zX

J M ( x , , ) , λ (z) J M ( y , , ) , λ (z)δ(xy)
(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 ρΩ, 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. □

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Correspondence to Hong Gang Li.

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Authors’ contributions

The main idea of this paper was proposed by HGL, and HGL and LPL etc. prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

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Li, H.G., Li, L.P., Zheng, J.M. et al. Sensitivity analysis for generalized set-valued parametric ordered variational inclusion with (α,λ)-NODSM mappings in ordered Banach spaces. Fixed Point Theory Appl 2014, 122 (2014) doi:10.1186/1687-1812-2014-122

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Keywords

  • sensitivity analysis
  • general set-valued parametric ordered variational inclusion
  • (α,λ)-NODSM set-valued mapping
  • ordered Banach spaces
  • comparison solution