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Sensitivity analysis for generalized setvalued parametric ordered variational inclusion with $(\alpha ,\lambda )\text{}NODSM$ mappings in ordered Banach spaces
Fixed Point Theory and Applications volume 2014, Article number: 122 (2014)
Abstract
In this paper, a new class of general setvalued parametric ordered variational inclusions, $\theta \in M(x,g(x,\rho ),\rho )$, with $(\alpha ,\lambda )\text{}NODSM$ mappings is studied in ordered Banach spaces. Then, by using fixed point theory and the resolvent operator associated with $(\alpha ,\lambda )\text{}NODSM$ setvalued 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 [1–3]. 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 $x\in X$ such that $A(g(x))\ge \theta $ ($A(x)$ and $g(x)$ are singlevalued mappings), in ordered Banach spaces [4]. By using the Brestrictedaccretive method of the mapping A with constants ${\alpha}_{1}$, ${\alpha}_{2}$, 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 $RME$ setvalued mapping, the author introduced and studied a class of nonlinear inclusion problems for ordered MR setvalued mappings and the existence theorem of solutions and an approximation algorithm for this kind of nonlinear inclusion problems for ordered extended setvalued mappings in ordered Hilbert spaces [6]. In 2012, the author introduced and studied a class of nonlinear inclusion problems, to find $x\in X$ such that $0\in M(x)$ ($M(x)$ is a setvalued mapping) for ordered $(\alpha ,\lambda )\text{}NODM$ setvalued mappings, and he then, applying the resolvent operator associated with $(\alpha ,\lambda )\text{}NODM$ setvalued 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 $(\alpha ,\lambda )\text{}NODM$ 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 $x=x(\lambda ):\mathrm{\Omega}\to X$ such that $A(g(x,\lambda ),\lambda )+f(x,\lambda )\ge \theta $ ($A(x)$, $g(x)$ and $F(\cdot ,\cdot )$ are singlevalued 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 $({\alpha}_{A},\lambda )$weak$ANODD$ setvalued mappings, which was applied to finding an approximate solution for a new class of general nonlinear mixedorder quasivariational inclusions involving the ⊕ operator in ordered Banach spaces [9], and, applying the matrix analysis and the vectorvalued mapping fixed point analysis method, he introduced and studied a new class of generalized nonlinear mixedorder variational inequalities systems with ordered Brestrictedaccretive 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 quasivariational inclusions [11]. From 2000, Agarwal et al. have discussed a sensitivity analysis for strongly nonlinear quasivariations 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 $(\alpha ,\lambda )\text{}NODSM$ setvalued 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 $(\alpha ,\lambda )\text{}NODM$ setvalued mappings. For details, we refer the reader to [1–35] and the references therein.
Let X be a real ordered Banach space with a norm $\parallel \cdot \parallel $, 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 $\rho \in \mathrm{\Omega}$. Let $x=x(\rho )\in X$ ($\rho \in \mathrm{\Omega}$), $g(x,\rho ):X\times \mathrm{\Omega}\to X$ be a singlevalued mapping and $M(x,g(x,\rho ),\rho ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ be a setvalued mapping. We consider the following problem:
Find $x=x(\rho )\in X$ ($\rho \in \mathrm{\Omega}$) such that
and the solution $x(\rho )$ of the inclusion problem (1.1) is continuous from Ω and X.
Problem (1.1) is called a nonlinear generalized setvalued parametric ordered variational inclusions for ordered $(\alpha ,\lambda )\text{}NODSM$ setvalued mappings in ordered Banach spaces.
Remark 1.1 When mapping M is singlevalued 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 setvalued, 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 $(\alpha ,\lambda )\text{}NODSM$ mappings is studied in ordered Banach spaces. Then, by using the resolvent operator associated with $(\alpha ,\lambda )\text{}NODSM$ setvalued 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 $\parallel \cdot \parallel $, a zero θ, a normal cone P, a normal constant N and a partial ordering relation ≤ defined by the cone P. For arbitrary $x,y\in X$, $lub\{x,y\}$ 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.
Let X be a real Banach space with a norm $\parallel \cdot \parallel $, θ be a zero element in X.

(i)
A nonempty closed convex subset P of X is said to be a cone if (1) for any $x\in \mathbf{P}$ and any $\lambda >0$, $\lambda x\in \mathbf{P}$ holds, (2) if $x\in \mathbf{P}$ and $x\in \mathbf{P}$, then $x=\theta $;

(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 $\theta \le x\le y$, $\parallel x\parallel \le N\parallel y\parallel $ holds;

(iii)
for arbitrary $x,y\in X$, $x\le y$ if and only if $xy\in \mathbf{P}$;

(iv)
for $x,y\in X$, x and y are said to be comparative to each other, if and only if $x\le y$ (or $y\le x$) holds (denoted by $x\propto y$ for $x\le y$ and $y\le x$).
Lemma 2.2 If $x\propto y$, then $lub\{x,y\}$ and $glb\{x,y\}$ exist, $xy\propto yx$, and $\theta \le (xy)\vee (yx)$.
Proof If $x\propto y$, then $x\le y$ or $y\le x$. Let $x\le y$, then $lub\{x;y\}=y$ and $glb\{x;y\}=x$, and $xy\le \theta \le yx$. It follows that $lub\{x;y\}$ and $glb\{x;y\}$ exist, and $xy\propto yx$. $(xy)\vee (yx)=(yx)$, then $\theta \le (xy)\vee (yx)$. □
Lemma 2.3 If for any natural number n, $x\propto {y}_{n}$, and ${y}_{n}\to {y}^{\ast}$ ($n\to \mathrm{\infty}$), then $x\propto {y}^{\ast}$.
Proof If for any natural number n, $x\propto {y}_{n}$ and ${y}_{n}\to {y}^{\ast}$ ($n\to \mathrm{\infty}$), then $x{y}_{n}\in \mathbf{P}$ or ${y}_{n}x\in P$ for any natural number n. Since P is a nonempty closed convex subsets of X, we have $x{y}^{\ast}={lim}_{n\to \mathrm{\infty}}(x{y}_{n})\in \mathbf{P}$ or ${y}^{\ast}x={lim}_{n\to \mathrm{\infty}}({y}_{n}x)\in \mathbf{P}$. Therefore, $x\propto {y}^{\ast}$. □
Let X be an ordered Banach space, let P be a cone of X, let ≤ be a relation defined by the cone P in Definition 2.1(iii). For $x,y,v,u\in X$, the following relations hold:

(1)
the relation ≤ in X is a partial ordering relation in X;

(2)
$x\oplus y=y\oplus x$;

(3)
$x\oplus x=\theta $;

(4)
$\theta \le x\oplus \theta $;

(5)
let λ be a real, then $(\lambda x)\oplus (\lambda y)=\lambda (x\oplus y)$;

(6)
if x, y, and w can be comparative to each other, then $(x\oplus y)\le x\oplus w+w\oplus y$;

(7)
let $(x+y)\vee (u+v)$ exist, and if $x\propto u,v$ and $y\propto u,v$, then $(x+y)\oplus (u+v)\le (x\oplus u+y\oplus v)\wedge (x\oplus v+y\oplus u)$;

(8)
if x, y, z, w can be compared with each other, then $(x\wedge y)\oplus (z\wedge w)\le ((x\oplus z)\vee (y\oplus w))\wedge ((x\oplus w)\vee (y\oplus z))$;

(9)
if $x\le y$ and $u\le v$, then $x+u\le y+v$;

(10)
if $x\propto \theta $, then $x\oplus \theta \le x\le x\oplus \theta $;

(11)
if $x\propto y$, then $(x\oplus \theta )\oplus (y\oplus \theta )\le (x\oplus y)\oplus \theta =x\oplus y$;

(12)
$(x\oplus \theta )(y\oplus \theta )\le (xy)\oplus \theta $;

(13)
if $\theta \le x$ and $x\ne \theta $, and $\alpha >0$, then $\theta \le \alpha x$ and $\alpha x\ne \theta $.
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 $x=x(\rho )\in X$ ($\rho \in \mathrm{\Omega}$), $g(x,\rho ):X\times \mathrm{\Omega}\to X$ be a singlevalued mapping and $M(x,g(x,\rho ),\rho ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ be a setvalued mapping and $M(x,\cdot ,\rho )$ be a nonempty closed subset in X.

(1)
A setvalued mapping M is said to be a comparison mapping, if for any ${v}_{x}\in M(x,\cdot ,\cdot )$, $x\propto {v}_{x}$, and if $x\propto y$, then for any ${v}_{x}\in M(x,\cdot ,\cdot )$ and any ${v}_{y}\in M(y,\cdot ,\cdot )$, ${v}_{x}\propto {v}_{y}$ ($\mathrm{\forall}x,y\in X$).

(2)
A setvalued mapping M is said to be a comparison mapping with respect to g, if for any ${v}_{x}\in M(\cdot ,g(x),\cdot )$, $x\propto {v}_{x}$, and if $x\propto y$, then for any ${v}_{x}\in M(\cdot ,g(x),\cdot )$ and any ${v}_{y}\in M(\cdot ,g(y),\cdot )$, ${v}_{x}\propto {v}_{y}$ ($\mathrm{\forall}x,y\in X$).

(3)
A comparison mapping M is said to be an αnonordinary difference mapping, if there exists a constant $\alpha >0$, for each $x,y\in X$, ${v}_{x}\in M(x,\cdot ,\cdot )$, and ${v}_{y}\in M(y,\cdot ,\cdot )$ such that
$$({v}_{x}\oplus {v}_{y})\oplus \alpha (x\oplus y)=\theta .$$ 
(4)
A comparison mapping M is said to be λordered strongly monotonic increase mapping, if for $x\ge y$ there exists a constant $\lambda >0$ such that
$$\lambda ({v}_{x}{v}_{y})\ge xy\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in X,{v}_{x}\in M(x),{v}_{y}\in M(y,\cdot ,\cdot ).$$ 
(5)
A comparison mapping M is said to be a $(\alpha ,\lambda )\text{}NODSM$ mapping, if M is a αnonordinary difference and λordered strongly monotone increasing mapping, and $(I+\lambda M(x,\cdot ,\cdot ))(X)=X$ for $\alpha ,\lambda >0$.
Obviously, if M is a comparison mapping, then $M(x,\cdot ,\cdot )\propto I$ ($\mathrm{\forall}x\in X$).
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 $A:X\times X\to X$ is said to be βordered compression, if A is comparison, and there exists a constant $0<\beta <1$ such that
Definition 2.7 [4]
Let X be a real ordered Banach space. A mapping $A:X\times X\to X$ is said to be a restrictedaccretive mapping with constants $({\alpha}_{1},{\alpha}_{2})$, if A is a comparison, and there exist two constants $0<{\beta}_{1},{\beta}_{2}\le 1$ such that for arbitrary $x,y\in X$,
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 $x=x(\rho )\in X$ ($\rho \in \mathrm{\Omega}$). $x=x(\rho )$ is said to be a comparison element when, if ${\rho}_{1}\ne {\rho}_{2}$ then $x({\rho}_{1})\ne x({\rho}_{2})$ for any ${\rho}_{1},{\rho}_{2}\in \mathrm{\Omega}$.
Lemma 2.9 Let $M=M(x,\cdot ,\cdot ):X\times X\times X\to {2}^{X}$. If M is a αnonordinary difference mapping, then an inverse mapping ${J}_{M,\lambda}={(I+\lambda M)}^{1}:X\times X\times X\to {2}^{X}$ of $(I+\lambda M)$ is a singlevalued mapping ($\alpha ,\lambda >0$), where I is the identity mapping on X.
Proof Let $u\in X$, and x and y be two elements in ${(I+\lambda M)}^{1}(u)$. It follows that $ux\in \lambda M(x,\cdot ,\cdot )$ and $uy\in \lambda M(y,\cdot ,\cdot )$, and
Since M is a αnonordinary difference mapping, we have
and $x\oplus y=0$ from Lemma 2.4. Also, $x=y$ holds. Thus ${(I+\lambda M)}^{1}(u)$ is a singlevalued 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 $M=M(x,\cdot ,\cdot ):X\times X\times X\to {2}^{X}$ be a αnonordinary difference mapping. The resolvent operator ${J}_{M,\lambda}:X\times X\times X\to X$ of the $M(x,\cdot ,\cdot )$ is defined by
where $\lambda >0$ 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, let ≤ be an ordering relation defined by the cone P. If $M=M(x,\cdot ,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ is an αnonordinary difference mapping, then the inclusion problem (1.1) has a solution x if and only if $g(x,\cdot )={J}_{M(x,\cdot ,\cdot ),\lambda}g(x,\cdot )$ in X.
Proof This directly follows from the definition of the resolvent operator ${J}_{M,\lambda}$ of $M(x,\cdot ,\cdot )$. □
Theorem 3.2 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, let ≤ be an ordering relation defined by the cone P, the operator ⊕ be a $XOR$ operator. If $M=M(x,\cdot ,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ is an $(\alpha ,\lambda )\text{}NODSM$ mapping with respect to ${J}_{M,\lambda}$, then the resolvent operator ${J}_{M,\lambda}:X\to X$ is a comparison mapping.
Proof Since $M=M(x,\cdot ,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ is an αnonordinary difference mapping and a comparison mapping with respect to ${J}_{M,\lambda}$ so that $x\propto {J}_{M,\lambda}(x)$. For any $x,y\in X$, let $x\propto y$, and ${v}_{x}=\frac{1}{\lambda}(x{J}_{M,\lambda}(x))\in M({J}_{M(x,\cdot ),\lambda}(y))$ and ${v}_{y}=\frac{1}{\lambda}(y{J}_{M(x,\cdot ),\lambda}(y))\in M({J}_{M(x,\cdot ),\lambda}(y))$. Setting
by using the λorder strongly monotonicity of M, we have
and if $y\le x$ then $\lambda ({v}_{x}{v}_{y})(xy)\in \mathbf{P}$, and if $x\le y$ then $(xy)\lambda ({v}_{x}{v}_{y})\in \mathbf{P}$. Therefore ${J}_{M,\lambda}(y)\propto {J}_{M,\lambda}(x)$ 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, let ≤ be an ordering relation defined by the cone P. Let $M=M(\cdot ,x,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ be a $NODSM$ setvalued mapping with respect to ${J}_{M,\lambda}$. If $\alpha >\frac{1}{\lambda}>0$, then for the resolvent operator ${J}_{M,\lambda}:X\to X$, the following relation holds:
Proof Let $M=M(\cdot ,x,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ be a $NODSM$ setvalued mapping with respect to ${J}_{M,\lambda}$. For $y,z\in X$, let ${u}_{y}={J}_{M,\lambda}(y)\propto {u}_{z}={J}_{M,\lambda}(z)$, ${v}_{y}=\frac{1}{\lambda}(y{u}_{y})\in M(\cdot ,{u}_{y},\cdot )$ and ${v}_{z}=\frac{1}{\lambda}(z{u}_{z})\in M(\cdot ,{u}_{z},\cdot )$, then ${v}_{y}\propto {v}_{z}$ for $y\propto z$. Since $M(\cdot ,x,\cdot ):X\times X\times X\to {2}^{X}$ is an $(\alpha ,\lambda )\text{}NODSM$ mapping with respect to the ${J}_{M,\lambda}$, the following relation holds by Lemma 2.4 and the condition $({v}_{y}\oplus {v}_{z})\oplus \alpha ({u}_{y}\oplus {u}_{z})=\theta $:
It follows that $(\lambda \alpha 1)({u}_{y}\oplus {u}_{z})\le (y\oplus z)$ and ${J}_{M,\lambda}(y)\oplus {J}_{M,\lambda}(z)\le \frac{1}{(\alpha \lambda 1)}(y\oplus z)$ from the condition $\alpha >\frac{1}{\lambda}>0$. 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, let ≤ be an ordering relation defined by the cone P. Let $M=M(x,\cdot ,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ be an $(\alpha ,\lambda )\text{}NODSM$ setvalued mapping with respect to the first argument and $g:X\times \mathrm{\Omega}\to X$ be a γordered compression and an 1ordered strongly monotonic increase with respect to the first argument and $range(g)\cap domM(\cdot ,x,\cdot )\ne \mathrm{\varnothing}$, and ${J}_{M,\lambda}$ for M with respect to the first argument and $({J}_{M,\lambda}I)$ for M with respect to the second argument be two restrictedaccretive mappings with constants $({\xi}_{1},{\xi}_{2})$ and $({\beta}_{1},{\beta}_{2})$, respectively, and $g\propto {J}_{M,\lambda}$. Suppose that for any $x,y,z\in X$
and
hold. For any parametric $\rho \in \mathrm{\Omega}$, for the nonlinear parametric inclusion problem (1.1) there exists a solution ${x}^{\ast}$.
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 $M=M(x,\cdot ,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$, and for any given $\rho \in \mathrm{\Omega}$ and ${x}_{1}={x}_{1}(\rho ),{x}_{2}={x}_{2}(\rho )\in X$ for $\lambda >0$. If ${x}_{1}(\rho )\propto {x}_{2}(\rho )$, and setting
where $i=1,2$, by (3.1) and the λordered strongly monotonicity of M,
by $(I+\lambda M)(X)=X$, the comparability of ${J}_{M,\lambda}$, and the 1ordered monotonic increase of $g(x,\cdot )$, it follows from ${x}_{1}(\rho )\propto {x}_{2}(\rho )$ that $F({x}_{1}(\rho ),\rho )\propto F({x}_{2}(\rho ),\rho )$. Using (3.3), (3.5), Lemma 2.4, Theorem 3.3, and $\alpha >\frac{2}{\lambda}>0$, from the conditions that ${J}_{M,\lambda}$ for M with respect to the first argument and $({J}_{M,\lambda}I)$ for M with respect to the second argument are two restrictedaccretive mappings with constants $({\xi}_{1},{\xi}_{2})$ and $({\beta}_{1},{\beta}_{2})$, respectively, it follows that
and, by Definition 2.1(2), we obtain
where $h={\beta}_{2}+{\beta}_{1}(\gamma (\frac{{\xi}_{1}}{\alpha \lambda 1}\oplus {\xi}_{2}))\oplus \delta $. It follows from the condition (3.4) that $0<hN<1$, and $F(x(\rho ),\rho )$ has a fixed point ${x}^{\ast}\in X$ and the ${x}^{\ast}$ is a solution of the generalized nonlinear ordered parametric equation
Further, ${x}^{\ast}$ satisfies the generalized nonlinear ordered parametric equation
Then for the nonlinear parametric inclusion problems (1.1) there exists a solution ${x}^{\ast}\in X$ for any parametric $\rho \in \mathrm{\Omega}$. This completes the proof. □
Remark 3.5 Though the method of solving problem by the resolvent operator is the same as in [20, 25–28] and [34] for the nonlinear inclusion problem, the character of the ordered $(\alpha ,\lambda )\text{}ANODM$ setvalued mapping is different from the one of the $(A,\eta )$accretive mapping [25], the $(H,\eta )$monotone mapping [26], the $(G,\eta )$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, let ≤ be an ordering relation defined by the cone P. Let $M=M(x,\cdot ,\cdot ):X\times X\times \mathrm{\Omega}\to {2}^{X}$ be a $(\alpha ,\lambda )\text{}NODSM$ setvalued mapping and $g:X\times \mathrm{\Omega}\to X$ be a γordered compression, continuous and 1ordered monotonic increase of $g(x,\cdot )$ with respect to first argument $\rho \in \mathrm{\Omega}$, and $range(g)\cap domM(\cdot ,x,\rho )\ne \mathrm{\varnothing}$, and ${J}_{M,\lambda}$ for M with respect to first argument and $({J}_{M,\lambda}I)$ for M with respect to second argument be two restrictedaccretive mappings with constants $({\xi}_{1},{\xi}_{2})$ and $({\beta}_{1},{\beta}_{2})$, respectively, and $g\propto {J}_{M,\lambda}$. Suppose that for any $x,y,z\in X$
and
hold; if the solution $x(\rho )$ 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 $x(\rho )$, a comparison solution, is continuous on Ω.
Proof For any given $\rho ,\overline{\rho}\in \mathrm{\Omega}$, by Theorem 3.4, let $x(\rho )$ be a comparison solution, and $x(\rho )$ and $x(\overline{\rho})$ satisfy parametric problem (1.1), then for any $\lambda >0$, we have
By the condition that M, g, ${J}_{M,\lambda}$, and ${J}_{M,\lambda}I$ are comparisons for each other and by Lemma 2.4, we have
Further, ${J}_{M,\lambda}$ and $({J}_{M,\lambda}I)$ are two restrictedaccretive mappings with constants $({\xi}_{1},{\xi}_{2})$ and $({\beta}_{1},{\beta}_{2})$, respectively, so that from Lemma 2.4 and Theorem 3.3, $\alpha >\frac{2}{\lambda}>0$, and from (3.6), it follows that
where $h={\beta}_{2}+{\beta}_{1}({\gamma}_{1}(\frac{{\xi}_{1}}{\alpha \lambda 1}\oplus {\xi}_{2}))\oplus \delta <\frac{1}{N}$ for the condition (4.1), and
Combining (4.4), (4.5), and (4.6), and by using Lemma 2.4, we get
Therefore,
It follows that
By Lemma 2.4, ${\beta}_{2}\theta =\theta $, and continuity of g with respect to the first argument $\rho \in \mathrm{\Omega}$, we have
and
which implies that the solution $x(\rho )$ of problem (1.1) is continuous at $\rho =\overline{\rho}$. This completes the proof. □
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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 setvalued parametric ordered variational inclusion with $(\alpha ,\lambda )\text{}NODSM$ mappings in ordered Banach spaces. Fixed Point Theory Appl 2014, 122 (2014). https://doi.org/10.1186/168718122014122
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Keywords
 sensitivity analysis
 general setvalued parametric ordered variational inclusion
 $(\alpha ,\lambda )\text{}NODSM$ setvalued mapping
 ordered Banach spaces
 comparison solution