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Sensitivity analysis for a new class of generalized parametric nonlinear ordered variational inequality problems in ordered Banach spaces

  • Kottakkaran Sooppy NisarEmail author
  • Mohd. Sarfaraz
  • Ahmed Morsy
  • Md. Kalimuddin Ahmad
Open Access
Research
  • 136 Downloads

Abstract

In this study, we introduce a class of new generalized parametric nonlinear ordered variational inequality problems and discuss its existence result. Also, we prove the sensitivity of the solution for the parametric inequality class with the help of B-restricted-accretive method in ordered Banach spaces. Some special cases of the main results are also discussed.

Keywords

Sensitivity analysis Variational inequality Parametric Ordered Banach spaces Restricted-accretive 

MSC

47J05 47J20 47J22 

1 Introduction

Ordered equations or inequalities have capacious significance to many fields, including neural networks, remote sensing, optimization of structures, optimization of electromagnetic systems, and many other applied sciences. Lately, much consideration has been given to the sensitivity analysis of variational inequalities. We comment that sensitivity analysis is imperative for a few reasons.

To begin with, since assessing issue information regularly presents estimation mistakes, sensitivity analysis soothes in distinguishing touchy factors that ought to be acquired with generally high exactness. Second, sensitivity analysis may anticipate in the doom changes of the steadiness because of alters in the dictating systems. Lastly, sensitivity gives valuable input for outlining or arranging different equilibrium in various frames. Moreover, from scientific and engineering perspectives, sensitivity analysis can give new bits of knowledge with respect to issues being considered and can fortify new thoughts for critical thinking.

During the most-recent decade, there has been expanding enthusiasm to concentrate on checking the sensitivity of different inequalities and inclusion systems. We contemplate the subjective conduct of the solution of the variational inequalities when the given operator and the feasible convex set change in a parameter. Such an investigation is known as sensitivity analysis, which is vital and significant. Sensitivity analysis gives us helpful data for outlining different equilibrium systems and for anticipating the coming alters of the equilibria because of the adjustments in the dictating systems.

Dafermos in [7] utilized the fixed-point technique to ponder the sensitivity analysis of Stampacchia’s variational inequalities. Time to time the model has been changed and stretched out by numerous writers for reviewing the sensitivity analysis of different classes of variational inequalities and inclusions problems; for more details, see [1, 2, 4, 5, 6, 12, 13, 14, 16, 17, 18].

Then again, in 1972, Amman [3] proposed the idea of finding the numbers of invariants under the nonlinear mapping in an ordered Banach space. Motivated by the idea of Amman, Li [9] instigated the work on the generalized nonlinear ordered variational inequalities and equation. By the B-restricted-accretive method for map A, Li initiated the study of existence and convergence result for approximation solution for the said problems in ordered Banach space. After that, in 2009, Li [10] started the study of a new class in ordered Banach space, and it was abbreviated as GNOVI. The sensitivity analysis was also investigated by Li [11], in which an existence result was proposed for another class of problems constricted as parametric GNOVI. Persuaded by the exploration in this tendency, we present a new class of generalized parametric nonlinear ordered variational inequalities involving ⊕ operator in ordered Banach space. We also prove the existence and continuity of the solution for the said problem.

2 Prelude

In this part of the paper, we review some basic facets which are supplementary for further processing.

Allow \((\mathcal{E}, \mbox{``}\leq\mbox{''})\) to be a real ordered Banach space with a norm \(\|\cdot \|\). Let \((\mathcal{C}, \mbox{``}\leq\mbox{''} )\) be a partial ordered cone having a normal constant N, θ is the zero member of \((\mathcal{E}, \mbox{``}\leq\mbox{''})\). Presume that \(\operatorname{glb}\{m,n\}\) and \(\operatorname{lub}\{m,n\}\) both exist.

Definition 2.1

\(\mathcal{C} \subset \mathcal{E}\), a non-void convex and closed subset, is termed cone if
  1. (i)

    as \(m\in \mathcal{C}\) and \(\kappa >0\), \(\kappa m \in \mathcal{C}\);

     
  2. (ii)

    if −m, \(m\in \mathcal{C}\), then \(m=\theta \).

     

Definition 2.2

([8])

A non-void subset \(\mathcal{C} \subset \mathcal{E}\) characterizes as normal cone iff \(\exists\ N > 0\) with \(0\leq m \leq n\) such that \(\|m\| \leq N \|n\|\).

Definition 2.3

([15])

Define a partial order \(\mbox{``}\leq\mbox{''}\) for any elements \(m, n \in \mathcal{E}\) as \(m \leq n\) iff \(m-n \in \mathcal{C}\). Then \(( \mathcal{E}, \mbox{``}\leq\mbox{''})\) is a real ordered Banach space.

Definition 2.4

([15])

Elements \(m, n \in \mathcal{E}\) related by the partial order defined above are called comparable elements.

Definition 2.5

([15])

Let ∨, ∧, and ⊕ be operations named OR, AND, and XOR and characterized respectively as follows:
  1. (i)

    \(m \vee n = \operatorname{lub}\{m, n\}\);

     
  2. (ii)

    \(m \wedge n = \operatorname{glb}\{m, n\}\);

     
  3. (iii)

    \(m \oplus n = (m-n) \vee (n-m)\).

     

Lemma 2.1

[15] For any members\(m, n, w \in \mathcal{E}\), the following relations hold:
  1. (i)

    if\(m\leq n\), then\(m \vee n = n\), \(m\wedge n=m\);

     
  2. (ii)

    if\(m \propto n\), then\(\theta \leq m \oplus n\);

     
  3. (iii)

    \((m+w) \vee (n+w)\)exists and\((m+w)\vee (n+w)=(m \vee n)+w\);

     
  4. (iv)

    \((m \wedge n)=(m+n)-(m \vee n)\);

     
  5. (v)

    for\(\lambda \geq 0\), one can have\(\lambda (m \vee n)= \lambda m \vee \lambda n\);

     
  6. (vi)

    for\(\lambda \leq 0\), one can have\(\lambda (m \wedge n)= \lambda m \vee \lambda n\);

     
  7. (vii)

    the converse part of (v) and (vi) holds if\(m \neq n\);

     
  8. (viii)

    either\(m \vee n\)or\(m \wedge n\)exists, then\(\mathcal{E}\)is a lattice;

     
  9. (ix)

    \((m+w) \wedge (n+w)\)exists and\((m+w)\wedge (n+w) = (m \wedge n)+ w\);

     
  10. (m)

    \((m \wedge n)=-(-m \vee -n)\);

     
  11. (xi)

    \((-m)\wedge (m) \leq \theta \leq (-m) \vee m\).

     

Proposition 1

([8])

For the comparable membersmandnin\(\mathcal{E}\), then\(m-n \propto n-m\), and\(\theta \leq (m-n)\vee (n-m)\).

Proposition 2

([8])

For any positive integerk, if\(m \propto n_{k}\)and\(n_{k} \rightarrow n^{\ast }\) (\(k \rightarrow \infty \)), then\(m \propto n^{\ast }\).

Lemma 2.2

([9])

For any comparable elements\(m,n,z,w \in \mathcal{E}\)and an operation ⊕, the following relations hold:
  1. (i)

    \(m \oplus n= n \oplus m\);

     
  2. (ii)

    \(m \oplus m =\theta \);

     
  3. (iii)

    \(\theta \leq m\oplus \theta \);

     
  4. (iv)

    let\(\lambda \in \mathbb{R}\), then\((\lambda m)\oplus ( \lambda n)=|\lambda |(m \oplus n)\);

     
  5. (v)

    \((m\oplus n) \leq (m \oplus w)+(w\oplus n)\);

     
  6. (vi)
    suppose\((m+n) \vee (p+q)\)exists, and if\(m \propto p\), qand\(n \propto p\), q, then
    $$ (m+n)\oplus (p+q) \leq (m \oplus p + n\oplus q)\wedge (m \oplus q + n \oplus p); $$
     
  7. (vii)
    supposem, n, z, wcan be comparative to each other, then
    $$ (m \wedge n)\oplus (z \wedge w)\leq \bigl((m\oplus z)\vee (n \oplus w)\bigr) \wedge \bigl((m\oplus w)\vee (n\oplus z)\bigr); $$
     
  8. (viii)

    if\(m \propto \theta \), then\(\alpha m \oplus \beta m=| \alpha -\beta |m +(\alpha \oplus \beta )m\).

     

Definition 2.6

([9])

Allow \(\mathcal{A}, \mathcal{B}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\) to be two parametric maps and \(\varSigma \subset \mathcal{E}\) be a non-void open subset in which ς lives.
  1. (i)

    \(\mathcal{A}\) is called comparison regarding the slot ς, if for any \(\varsigma \in \varSigma \) and each \(m(\varsigma ), n(\varsigma ) \in \mathcal{E}\) we have \(m(\varsigma ) \propto n( \varsigma )\), then \(\mathcal{A}(m(\varsigma ), \varsigma ) \propto \mathcal{A}(n(\varsigma ), \varsigma )\), \(m(\varsigma ) \propto \mathcal{A}(m(\varsigma ), \varsigma )\), and \(n(\varsigma ) \propto \mathcal{A}(n(\varsigma ),\varsigma )\).

     
  2. (ii)

    \(\mathcal{A}\) and \(\mathcal{B}\) are termed comparison regarding the spot ς to each other, if for each \(m(\varsigma ) \in \mathcal{E}\), \(\mathcal{A}(m(\varsigma ), \varsigma ) \propto \mathcal{B}(m(\varsigma ), \varsigma )\) (denoted by \(\mathcal{A} \propto \mathcal{B}\)).

     

Definition 2.7

([9])

Allow \(\mathcal{A}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\) to be a parametric map. \(\mathcal{A}\) is called β-ordered compression regarding the second slot ς, if \(\mathcal{A}\) is comparative with respect to ς, and \(\exists \beta \in (0, 1)\) such that, for any \(\varsigma \in \varSigma \),
$$ \mathcal{A}\bigl(m(\varsigma ), \varsigma \bigr) \oplus \mathcal{A}\bigl(n( \varsigma ), \varsigma \bigr) \leq \beta \bigl(m(\varsigma )\oplus n(\varsigma ) \bigr) $$
holds.

Definition 2.8

([9])

Let \((\mathcal{E},\mbox{``}\leq\mbox{''})\), \(\mathcal{C}\), N, and Σ have their predefined meanings. Allow \(\mathcal{A}, \mathcal{B}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\) to be two parametric maps, I stands for an identity map on \(\mathcal{E}\times \mathcal{E}\).
  1. (i)
    A map \(\mathcal{A}\) is referred as restricted-accretive regarding the slot ς if \(\mathcal{A}\) is comparative, and there exist two constants \(0 < \alpha _{1}, \alpha _{2} \leq 1\) such that, for any \(\varsigma \in \varSigma \) and arbitrary \(m(\varsigma ), n( \varsigma ) \in \mathcal{E}\),
    $$\begin{aligned}& \bigl(\mathcal{A}\bigl(m(\varsigma ),\varsigma \bigr) + I\bigl(m(\varsigma ), \varsigma \bigr)\bigr) \oplus \bigl(\mathcal{A}\bigl(n(\varsigma ),\varsigma \bigr) + I\bigl(n(\varsigma ), \varsigma \bigr)\bigr) \\& \quad \leq \alpha _{1} \bigl(\mathcal{A}\bigl(m(\varsigma ),\varsigma \bigr) \oplus \mathcal{A}\bigl(n( \varsigma ),\varsigma \bigr)\bigr) +\alpha _{2}\bigl(m(\varsigma ) \oplus n(\varsigma )\bigr) \end{aligned}$$
    holds.
     
  2. (ii)
    A map \(\mathcal{A}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\) is called a B-restricted-accretive map regarding the slot ς if \(\mathcal{A}\), \(\mathcal{B}\) and \(\mathcal{A} \wedge \mathcal{B}: \mathcal{E}\times \varSigma \rightarrow \mathcal{A}(m( \varsigma ), \varsigma )\wedge \mathcal{B}(m(\varsigma ),\varsigma ) \in \mathcal{E}\) all are comparative and they are comparison regarding the slot ς to each other, and ∃ constants \(0 < \alpha _{1}, \alpha _{2} \leq 1\) such that, for any \(\varsigma \in \varSigma \) and any \(m(\varsigma ), n(\varsigma ) \in \mathcal{E}\),
    $$\begin{aligned}& \bigl(\mathcal{A}\bigl(m(\varsigma ),\varsigma \bigr)\wedge \mathcal{B}\bigl(m( \varsigma ),\varsigma \bigr)+I\bigl(m(\varsigma ),\varsigma \bigr)\bigr) \\& \qquad {}\oplus \bigl(\mathcal{A}\bigl(n( \varsigma ),\varsigma \bigr)\wedge \mathcal{B}\bigl(n( \varsigma ),\varsigma \bigr)+I\bigl(n( \varsigma ),\varsigma \bigr)\bigr) \\& \quad \leq \alpha _{1} \bigl(\bigl(\mathcal{A}\bigl(m(\varsigma ),\varsigma \bigr)\wedge \mathcal{B}\bigl(m(\varsigma ),\varsigma \bigr)\bigr) \oplus \bigl( \mathcal{A}\bigl(n(\varsigma ),\varsigma \bigr)\wedge \mathcal{B}\bigl(n(\varsigma ),\varsigma \bigr)\bigr) \bigr) \\& \qquad {}+ \alpha _{2}\bigl(m(\varsigma )\oplus n(\varsigma )\bigr) \end{aligned}$$
    holds.
     

Proposition 3

([11])

Allow\(\mathcal{A}:\mathcal{E}\rightarrow \mathcal{E}\)to be a comparison map, then for each\(m, n \in \mathcal{E}\), the following relations hold:
  1. (i)

    \(\|\theta +\theta \|= \|\theta \|=\theta \);

     
  2. (ii)

    \(\|m \vee n\| \leq \|m\|\vee \|n\| \leq \|m\|+ \|n\|\);

     
  3. (iii)

    \(\|m \oplus n\|\leq \|m-n\|\leq N\|m \oplus n\|\);

     
  4. (iv)

    if\(m \propto n\), then\(\|m \oplus n\|= \|m-n\|\);

     
  5. (v)

    \(\lim_{m\rightarrow m_{0}}\|\mathcal{A}(m)-\mathcal{A}(m _{0})\|=0\)iff\(\lim_{m\rightarrow m_{0}} \mathcal{A}(m)\oplus \mathcal{A}(m_{0})=0\).

     

3 Formatting of the problem

Let \(\varSigma \subset \mathcal{E}\) be a non-void open subset in which the parameter ς lives. Let \(\mathcal{M}, \mathcal{A}, \mathcal{F}, g, h: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\) be single-valued compression maps and \(\operatorname{range} g(\cdot , \varsigma )\cap \operatorname{dom} \mathcal{A}(\cdot , \varsigma ) \neq \varPhi \), \(\operatorname{range} h(\cdot , \varsigma ) \cap \operatorname{dom} \mathcal{F}( \cdot , \varsigma ) \neq \varPhi \) for any \(\varsigma \in \varSigma \).

We contemplate the following problem:

Find \(m=m(\varsigma ): \varSigma \rightarrow \mathcal{E}\) such that
$$ \theta \leq \mathcal{M}(m, \varsigma )+ \mathcal{A}\bigl(g(m, \varsigma ), \varsigma \bigr)\oplus \mathcal{F}\bigl(h(m, \varsigma ),\varsigma \bigr). $$
(1)
Problem (1) is referred to as a new class of generalized parametric nonlinear ordered variational inequalities involving XOR operator (or in short GPNOVI).

3.1 Special cases

  1. Case 1:

    If \(\mathcal{F}(h(m, \varsigma ), \varsigma )\equiv \theta \) (the zero map), then problem (1) is reverted into problem (1.1) which was encountered by Li in [11].

     
  2. Case 2:

    If \(\mathcal{F}(h(m, \varsigma ), \varsigma )\equiv \theta \) and \(\mathcal{M}(m, \varsigma ) \equiv \theta \), then problem (1) is transformed into problem (2.1) which was encountered by Li in [9].

     

Lemma 3.1

Allow\(\mathcal{M}\), \(\mathcal{A}\), \(\mathcal{F}\), \((\mathcal{M}+ \mathcal{A}\oplus \mathcal{F})\), g, h, and\((\mathcal{M}+\mathcal{A} \oplus \mathcal{F})\wedge \mathcal{B}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\)be the comparison mappings. If\([ \mathcal{M}(m, \varsigma )+\mathcal{A}(g(m, \varsigma ), \varsigma ) \oplus \mathcal{F}(h(m, \varsigma ), \varsigma ) ]\wedge \mathcal{B}(m, \varsigma )=\theta \) (\(\theta \in \mathcal{E}\)) has an answer\(m^{\ast }\), then\(m^{\ast }\)will also be an answer to problem (1).

Proof

With the help of definitions and conditions on the mappings \(\mathcal{M}\), \(\mathcal{A}\), \(\mathcal{F}\), \((\mathcal{M}+\mathcal{A} \oplus \mathcal{F})\), g, h, and \((\mathcal{M}+\mathcal{A}\oplus \mathcal{F})\wedge \mathcal{B}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\), the proof follows. □

4 Existence result for generalized parametric nonlinear ordered variational inequality problem with ⊕ operator

In this part of the paper, we will prove the existence of solution of generalized parametric nonlinear ordered variational inequality problem (1).

Theorem 4.1

Let\((\mathcal{E},\textit{``}\leq\textit{''})\), \((\mathcal{C},\textit{``}\leq\textit{''})\), N, and\(\varSigma \subset \mathcal{E}\)have their predefined meaning, and let\(\mathcal{M}\), \(\mathcal{A}\), \(\mathcal{F}\), \(\mathcal{B}\), g, h, \(( \mathcal{M}+\mathcal{A}\oplus \mathcal{F})\), and\((\mathcal{M}+ \mathcal{A}\oplus \mathcal{F})\wedge \mathcal{B}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\)be comparison parametric mappings to each other. Suppose thatMis\(\lambda _{M}\)-ordered compression, Ais\(\lambda _{A}\)-ordered compression, Fis\(\lambda _{F}\)-ordered compression, Bis\(\lambda _{B}\)-ordered compression, gis\(\lambda _{g}\)-ordered compression, andhis\(\lambda _{h}\)-ordered compression mappings regarding the second slotς, respectively. Further, if\((\mathcal{M}+\mathcal{A}\oplus \mathcal{F}): \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\)is aB-restricted accretive map regarding the second slotςfor constants\(\alpha _{1}\)and\(\alpha _{2}\), and for any\(\tau > 0\), the given condition
$$ \tau \bigl[\lambda _{M}+(\lambda _{A} \lambda _{g}\oplus \lambda _{F}\lambda _{h}) \bigr]\vee \lambda _{B} < \frac{1-N\alpha _{2}}{N\alpha _{1}} $$
(2)
holds, then\(m^{\ast }\)is the answer to problem (1).

Proof

For any given \(\varsigma \in \varSigma \) and \(m_{1}=m_{1}(\varsigma )\) and \(m_{2}=m_{2}(\varsigma )\) in \(\mathcal{E}\), for \(\tau > 0\), let \(m_{1}(\varsigma )\propto m_{2}(\varsigma )\), then
$$\begin{aligned} \mathcal{F}\bigl(m_{i}(\varsigma ), \varsigma \bigr) &=\tau \bigl[\mathcal{M}\bigl(m _{i}(\varsigma ),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m_{i}(\varsigma ),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{i}(\varsigma ),\varsigma \bigr), \varsigma \bigr) \bigr] \\ &\quad {} \wedge \mathcal{B}\bigl(m_{i}(\varsigma ),\varsigma \bigr)+I \bigl(m_{i}( \varsigma ),\varsigma \bigr), \end{aligned}$$
(3)
where \(i=1,2\). It follows from the conditions, \(\mathcal{M}\), \(\mathcal{A}\), \(\mathcal{F}\), \(\mathcal{B}\), \((\mathcal{M}+\mathcal{A}\oplus \mathcal{F})\), g, h, and \((\mathcal{M}+\mathcal{A}\oplus \mathcal{F}) \wedge \mathcal{B}: \mathcal{E}\times \varSigma \rightarrow \mathcal{E}\) are comparison parametric mappings regarding the second slot ς to each other, and \(m_{1}(\varsigma )\propto m_{2}( \varsigma )\) that \(\mathcal{F}(m_{1}(\varsigma ),\varsigma ) \propto \mathcal{F}(m_{2}(\varsigma ),\varsigma )\). Using the conditions of restricted-accretive and the ordered-compression on suitable mappings regarding the second slot ς, respectively, and Proposition 2, we have
$$\begin{aligned} \theta &\leq \mathcal{F}(m_{1}, \varsigma )\oplus \mathcal{F}(m_{2}, \varsigma ) \\ &\leq \bigl[\tau \bigl(\mathcal{M}\bigl(m_{1}(\varsigma ),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m_{1}(\varsigma ),\varsigma \bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{1}(\varsigma ), \varsigma \bigr),\varsigma \bigr) \bigr) \\ &\quad {} \wedge \mathcal{B}\bigl(m_{1}(\varsigma ),\varsigma \bigr)+I \bigl(m_{1}(\varsigma ), \varsigma \bigr) \bigr] \\ &\quad {} \oplus \bigl[\tau \bigl(\mathcal{M}\bigl(m_{2}(\varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m_{2}(\varsigma ),\varsigma \bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{2}(\varsigma ), \varsigma \bigr),\varsigma \bigr) \bigr) \\ &\quad {} \wedge \mathcal{B}\bigl(m_{2}(\varsigma ),\varsigma \bigr)+I \bigl(m_{2}(\varsigma ), \varsigma \bigr) \bigr] \\ &\leq \alpha _{1}\tau \bigl[ \bigl( \bigl(\mathcal{M} \bigl(m_{1}(\varsigma ), \varsigma \bigr)+\mathcal{A}\bigl(g \bigl(m_{1}(\varsigma ),\varsigma \bigr),\varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m_{1}(\varsigma ),\varsigma \bigr),\varsigma \bigr) \bigr) \\ &\quad {} \wedge \mathcal{B}\bigl(m_{1}(\varsigma ),\varsigma \bigr) \bigr) \\ &\quad {} \oplus \bigl( \bigl(\mathcal{M}\bigl(m_{2}(\varsigma ),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m_{2}(\varsigma ),\varsigma \bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{2}(\varsigma ), \varsigma \bigr),\varsigma \bigr) \bigr) \\ &\quad {} \wedge \mathcal{B}\bigl(m_{2}(\varsigma ),\varsigma \bigr) \bigr) \bigr] +\alpha _{2}\bigl(m_{1}(\varsigma )\oplus m_{2}(\varsigma )\bigr) \\ &\leq \tau \alpha _{1} \bigl[ \bigl(\mathcal{M}\bigl(m_{1}( \varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m_{1}(\varsigma ), \varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{1}( \varsigma ),\varsigma \bigr),\varsigma \bigr) \bigr) \\ &\quad {} \oplus \bigl(\mathcal{M}\bigl(m_{2}(\varsigma ),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m_{2}(\varsigma ),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{2}(\varsigma ),\varsigma \bigr), \varsigma \bigr) \bigr) \\ &\quad {} \vee \bigl(\mathcal{B}\bigl(m_{1}(\varsigma ),\varsigma \bigr)\oplus \mathcal{B}\bigl(m_{2}(\varsigma ),\varsigma \bigr) \bigr) \bigr] + \alpha _{2}\bigl(m _{1}(\varsigma ) \oplus m_{2}( \varsigma )\bigr) \\ &\leq \tau \alpha _{1} \bigl[ \bigl(\bigl(\mathcal{M} \bigl(m_{1}(\varsigma ),\varsigma \bigr)\oplus \mathcal{M} \bigl(m_{2}(\varsigma ),\varsigma \bigr)\bigr) \\ &\quad {}+\bigl(\mathcal{A}\bigl(g \bigl(m _{1}(\varsigma ),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{1}( \varsigma ),\varsigma \bigr),\varsigma \bigr)\bigr) \\ & \quad {}\oplus \bigl(\mathcal{A}\bigl(g\bigl(m_{2}(\varsigma ),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m_{2}(\varsigma ), \varsigma \bigr),\varsigma \bigr)\bigr) \bigr) \\ &\quad {} \vee \lambda _{B}\bigl(m_{1}(\varsigma )\oplus m_{2}(\varsigma )\bigr) \bigr]+ \alpha _{2} \bigl(m_{1}(\varsigma )\oplus m_{2}(\varsigma )\bigr) \\ &\leq \tau \alpha _{1} \bigl[ \bigl(\lambda _{M} \bigl(m_{1}(\varsigma )\oplus m _{2}(\varsigma )\bigr)+( \lambda _{A}\lambda _{g}\oplus \lambda _{F} \lambda _{h}) \bigl(m_{1}(\varsigma )\oplus m_{2}( \varsigma )\bigr) \bigr) \\ &\quad {} \vee \lambda _{B}\bigl(m_{1}(\varsigma )\oplus m_{2}(\varsigma )\bigr) \bigr]+ \alpha _{2} \bigl(m_{1}(\varsigma )\oplus m_{2}(\varsigma )\bigr) \\ &\leq \tau \alpha _{1} \bigl[\bigl(\lambda _{M}+(\lambda _{A}\lambda _{g}\oplus \lambda _{F}\lambda _{h})\bigr)\vee \lambda _{B} \bigr]\bigl(m_{1}( \varsigma )\oplus m_{2}(\varsigma )\bigr) \\ & \quad {}+\alpha _{2}\bigl(m_{1}(\varsigma )\oplus m_{2}( \varsigma )\bigr) \\ &= \bigl[\tau \alpha _{1} \bigl(\bigl(\lambda _{M}+(\lambda _{A}\lambda _{g} \oplus \lambda _{F}\lambda _{h})\bigr)\vee \lambda _{B} \bigr)+\alpha _{2} \bigr]\bigl(m _{1}(\varsigma )\oplus m_{2}(\varsigma )\bigr). \end{aligned}$$
Using Definition 2.2 and Lemma 2.1, we obtain
$$ \bigl\Vert \mathcal{F}\bigl(m_{1}(\varsigma ), \varsigma \bigr)-\mathcal{F}\bigl(m_{2}(\varsigma ),\varsigma \bigr) \bigr\Vert \leq N \varPsi \bigl\Vert m_{1}(\varsigma )-m_{2}( \varsigma ) \bigr\Vert , $$
(4)
where \(\varPsi = [\tau \alpha _{1} ((\lambda _{M}+(\lambda _{A}\lambda _{g}\oplus \lambda _{F}\lambda _{h}))\vee \lambda _{B} )+\alpha _{2} ]\). It follows from condition (2) that \(0 < N \varPsi < 1\), we realize that \(\mathcal{F}(m(\varsigma ),\varsigma )\) is a contraction mapping, then \(\exists m^{\ast } \in \mathcal{E}\), which is invariant under \(\mathcal{F}(m(\varsigma ),\varsigma )\), and \(m^{\ast }\) is the answer of problem (1), i.e., \(m^{\ast }\) is the solution of the GPNOVI with ⊕ operator
$$ \bigl[\mathcal{M}(m, \varsigma )+\mathcal{A}\bigl(g(m,\varsigma ),\varsigma \bigr)\oplus \mathcal{F}\bigl(h(m,\varsigma ),\varsigma \bigr) \bigr]\wedge \mathcal{B}(m, \varsigma )=\theta $$
for any parametric \(\varsigma \in \varSigma \). By Lemma 3.1, then the generalized parametric nonlinear ordered variational inequality (1), there exists a solution \(m^{\ast }\). □

5 Sensitivity analysis for GPNOVI with ⊕ operator

In this part of the article, we will prove sensitivity analysis for generalized parametric nonlinear ordered variational inequality problem (1).

Theorem 5.1

Suppose\((\mathcal{E},\textit{``}\leq\textit{''})\), \((\mathcal{C}, \textit{``}\leq\textit{''})\), N, and\(\varSigma \subset \mathcal{E}\)have their predefined meanings. Let\(\mathcal{M}(\cdot , \varsigma )\), \(\mathcal{A}(\cdot , \varsigma )\), \(\mathcal{F}(\cdot , \varsigma )\), \(\mathcal{B}(\cdot , \varsigma )\), \(g( \cdot , \varsigma )\), and\(h(\cdot , \varsigma ): \mathcal{E}\times \varSigma : \rightarrow \mathcal{E}\)be the parametric mapping continuous regarding the slot\(\varsigma \in \varSigma \), \(\mathcal{M}\), \(\mathcal{A}\), \(\mathcal{F}\), \(\mathcal{B}\), \((\mathcal{M}+\mathcal{A}\oplus \mathcal{F})\), g, h, and\((\mathcal{M}+\mathcal{A}\oplus \mathcal{F})\wedge \mathcal{B}\)be comparison mappings to each other. Suppose\(\mathcal{M}\)is\(\lambda _{M}\)-ordered compression, \(\mathcal{A}\)is\(\lambda _{A}\)-ordered compression, \(\mathcal{F}\)is\(\lambda _{F}\)-ordered compression, \(\mathcal{B}\)is\(\lambda _{B}\)-ordered compression, gis\(\lambda _{g}\)-ordered compression, andhis\(\lambda _{h}\)-ordered compression mappings regarding second slotς, respectively. Further, if\((\mathcal{M}+\mathcal{A}\oplus \mathcal{F}):\mathcal{E}\times \varSigma \rightarrow \mathcal{E}\)is aB-restricted accretive mapping regarding the second slotς, with constants\(\alpha _{1}\)and\(\alpha _{2}\)and for any\(\tau > 0\),
$$ \tau \bigl[\lambda _{M}+(\lambda _{A} \lambda _{g}\oplus \lambda _{F}\lambda _{h}) \bigr]\vee \lambda _{B} < \frac{1-\alpha _{2}}{\alpha _{1}} $$
(5)
holds, then the answer\(m(\varsigma )\)of problem (1) is continuous inΣ.

Proof

For given \(\varsigma , \bar{\varsigma } \in \varSigma \), let \(m(\varsigma )\) and \(m(\bar{\varsigma })\) be two solutions of problem (1), then for any \(\tau > 0\), we have
$$\begin{aligned}& \begin{aligned} &\begin{aligned} m(\varsigma ) &= \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr) \\ &=\tau \bigl[\mathcal{M}\bigl(m(\varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g \bigl(m( \varsigma ),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h \bigl(m(\varsigma ), \varsigma \bigr),\varsigma \bigr) \bigr] \\ &\quad {} \wedge \mathcal{B}\bigl(m(\varsigma ),\varsigma \bigr)+ I\bigl(m(\varsigma ), \varsigma \bigr), \end{aligned} \\ &\begin{aligned} m(\bar{\varsigma }) &= \mathcal{F}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr) \\ &=\tau \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{\varsigma }\bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr),\bar{ \varsigma }\bigr) \bigr] \\ &\quad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+ I\bigl(m( \bar{ \varsigma }),\bar{\varsigma }\bigr). \end{aligned} \end{aligned} \end{aligned}$$
(6)
As per conditions that \(\mathcal{M}\), \(\mathcal{A}\), \(\mathcal{B}\), \(( \mathcal{M}+\mathcal{A}\oplus \mathcal{F})\), g, h, and \((\mathcal{M}+ \mathcal{A}\oplus \mathcal{F})\wedge \mathcal{B}\) are comparison mappings regarding the second slot ς with each other, respectively, and Lemma 2.2, we obtain
$$\begin{aligned} \theta &= m(\varsigma )\oplus m(\bar{\varsigma }) \\ &= \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \mathcal{F}\bigl(m( \bar{ \varsigma }),\bar{\varsigma }\bigr) \\ &\leq \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \theta \oplus \mathcal{F}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \\ &=\bigl[\mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \mathcal{F} \bigl(m(\bar{ \varsigma }),\varsigma \bigr)\bigr]\oplus \bigl[\mathcal{F}\bigl(m( \bar{\varsigma }), \varsigma \bigr)\oplus \mathcal{F}\bigl(m(\bar{\varsigma }), \bar{\varsigma }\bigr)\bigr]. \end{aligned}$$
(7)
Further, since \((\mathcal{M}+\mathcal{A}\oplus \mathcal{F})\) is a B-restricted-accretive mapping with constants \(\alpha _{1}\), \(\alpha _{2}\), \(\mathcal{M}\) is \(\lambda _{M}\)-ordered compression, \(\mathcal{A}\) is \(\lambda _{A}\)-ordered compression, \(\mathcal{F}\) is \(\lambda _{F}\)-ordered compression, \(\mathcal{B}\) is \(\lambda _{B}\)-ordered compression, g is \(\lambda _{g}\)-ordered compression, h is \(\lambda _{h}\)-ordered compression, regarding the slot ς, respectively, so by Theorem 4.1, we obtain
$$\begin{aligned}& \mathcal{F}\bigl(m(\varsigma ),\varsigma \bigr)\oplus \mathcal{F}\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \\& \quad \leq \bigl[\tau \bigl(\mathcal{M}\bigl(m(\varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\varsigma ),\varsigma \bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\varsigma ),\varsigma \bigr), \varsigma \bigr) \bigr) \\& \qquad {}\wedge \mathcal{B}\bigl(m(\varsigma ),\varsigma \bigr)+I\bigl(m(\varsigma ), \varsigma \bigr) \bigr]\oplus \bigl[\tau \bigl(\mathcal{M}\bigl(m(\bar{\varsigma }),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \\& \qquad {} \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \bigr) \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) + I\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \bigr] \\& \quad \leq \alpha _{1} \bigl[ \bigl(\tau \bigl[\mathcal{M} \bigl(m(\varsigma ),\varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\varsigma ), \varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\varsigma ), \varsigma \bigr),\varsigma \bigr) \bigr]\wedge \mathcal{B}\bigl(m(\varsigma ), \varsigma \bigr) \bigr) \\& \qquad {}\times \bigl\{ \tau \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\varsigma \bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr) \bigr]\wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \bigr\} \bigr] \\& \qquad {}+\alpha _{2}\bigl(m( \varsigma )\oplus m(\bar{\varsigma })\bigr) \\& \quad \leq \varPsi \bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr), \end{aligned}$$
(8)
where \(\varPsi =\alpha _{1} [\tau (\lambda _{M}+(\lambda _{A}\lambda _{g} \oplus \lambda _{F}\lambda _{h})) \vee \lambda _{B} ]+\alpha _{2} < 1\) for condition (5), and
$$\begin{aligned}& \mathcal{F}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{F}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr) \\& \quad \leq \bigl[\tau \bigl(\mathcal{M}\bigl(m(\bar{ \varsigma }),\varsigma \bigr)+\mathcal{A}\bigl(g \bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{F} \bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \bigr) \\& \qquad {}\wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)+I\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \bigr] \\& \qquad {}\oplus \bigl[\tau \bigl(\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr)+\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr) \bigr) \\& \qquad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+I\bigl(m( \bar{ \varsigma }),\bar{\varsigma }\bigr) \bigr] \\& \quad \leq \alpha _{1} \bigl[ \bigl(\tau \bigl[\mathcal{M}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }), \varsigma \bigr),\varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\bigr] \\& \qquad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \bigr) \\& \qquad {} \oplus \bigl(\tau \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{ \varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr] \\& \qquad {} \wedge \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr) \bigr]+ \alpha _{2}\bigl(m(\bar{\varsigma })\oplus m(\bar{\varsigma }) \bigr) \\& \quad \leq \alpha _{1} \bigl[\tau \bigl(\bigl[\mathcal{M}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)+\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }), \varsigma \bigr),\varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\bigr] \\& \qquad {} \oplus \bigl[\mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr)+ \mathcal{A}\bigl(g\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr),\bar{\varsigma }\bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\bar{ \varsigma }),\bar{\varsigma } \bigr),\bar{\varsigma }\bigr)\bigr] \bigr) \bigr] \\& \qquad {} \vee \bigl[\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr] \\& \quad \leq \alpha _{1} \bigl[\tau \bigl(\bigl[\mathcal{M}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{\varsigma }),\bar{ \varsigma }\bigr)\bigr]+\bigl[ \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }), \varsigma \bigr),\varsigma \bigr) \\& \qquad {} \oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{ \varsigma }\bigr)\bigr]\oplus \bigl[\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr] \bigr) \bigr] \\& \qquad {} \vee \bigl[\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr) \oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr]. \end{aligned}$$
(9)
Combining equations (7), (8), and (9), and by making use of Lemma 2.2, we obtain
$$\begin{aligned}& \begin{aligned} m(\varsigma )\oplus m(\bar{\varsigma }) &\leq \bigl[\varPsi \bigl(m(\varsigma ) \oplus m(\bar{\varsigma })\bigr)\bigr]\oplus \alpha _{1} \bigl[\tau \bigl(\bigl( \mathcal{M}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \\ &\quad {}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{\varsigma }\bigr)\bigr) \\ &\quad {}\oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \bigr) \\ &\quad {}\vee \bigl(\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \bigr], \end{aligned} \\& \begin{aligned} (1\oplus \varPsi ) \bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) &\leq \alpha _{1} \bigl[\tau \bigl(\bigl(\mathcal{M}\bigl(m(\bar{\varsigma }), \varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr) \\ &\quad {}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{\varsigma }\bigr)\bigr) \\ &\quad {}\oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \bigr) \\ &\quad {}\vee \bigl(\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \bigr], \end{aligned} \\& \bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq \biggl( \frac{\alpha _{1}}{1 \oplus \varPsi } \biggr) \bigl[\tau \bigl(\bigl(\mathcal{M}\bigl(m(\bar{\varsigma }), \varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{\varsigma }),\bar{\varsigma } \bigr)\bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{\varsigma }\bigr)\bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\varsigma \bigr), \varsigma \bigr) \oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\vee \bigl(\mathcal{B}\bigl(m(\bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{B}\bigl(m(\bar{ \varsigma }),\bar{\varsigma }\bigr)\bigr) \bigr] \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr)}\leq \biggl(\frac{\alpha _{1}}{1\oplus \varPsi } \biggr) \bigl[\tau (\bigl( \mathcal{M}\bigl(m( \bar{\varsigma }),\varsigma \bigr)\oplus \mathcal{M}\bigl(m(\bar{ \varsigma }), \bar{\varsigma }\bigr)\bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}+\bigl(\mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\varsigma \bigr),\varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \varsigma \bigr)\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\varsigma \bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\oplus \mathcal{A}\bigl(g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \bar{ \varsigma }\bigr)\bigr) \oplus \bigl(\mathcal{F}\bigl(h\bigl(m(\bar{ \varsigma }),\varsigma \bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m( \bar{\varsigma }),\bar{\varsigma }\bigr), \varsigma \bigr) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr), \varsigma \bigr)\oplus \mathcal{F}\bigl(h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr),\bar{ \varsigma }\bigr)\bigr) \big) \\& \hphantom{\bigl(m(\varsigma )\oplus m(\bar{\varsigma })\bigr) \leq{}}\vee \bigl(\mathcal{B}\bigl(m(\bar{ \varsigma }),\varsigma \bigr) \oplus \mathcal{B}\bigl(m(\bar{\varsigma }),\bar{ \varsigma }\bigr)\bigr) \bigr]. \end{aligned}$$
(10)
Using the continuity of the parametric mappings regarding the second slot \(\varsigma \in \varSigma \), we have
$$\begin{aligned}& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert g\bigl(m(\bar{\varsigma }), \varsigma \bigr)-g\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr\Vert = 0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert h\bigl(m(\bar{\varsigma }), \varsigma \bigr)-h\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{B}\bigl(m(\bar{ \varsigma }), \varsigma \bigr)-\mathcal{B}\bigl(m(\bar{\varsigma }),\bar{\varsigma }\bigr) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{A}\bigl(g\bigl(m(\bar{ \varsigma }),\varsigma \bigr),\varsigma \bigr)-\mathcal{A}\bigl(g\bigl(m(\bar{ \varsigma }), \varsigma \bigr),\bar{\varsigma }\bigr) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{F}(\cdot , \varsigma )- \mathcal{F}(\cdot , \bar{\varsigma }) \bigr\Vert =0, \\& \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert \mathcal{M}(\cdot , \varsigma )- \mathcal{M}(\cdot , \bar{\varsigma }) \bigr\Vert =0. \end{aligned}$$
From Proposition 3, we have
$$ \lim_{\varsigma \rightarrow \bar{\varsigma }}\bigl(m(\varsigma )\oplus m(\bar{ \varsigma }) \bigr)=\theta $$
and
$$ \lim_{\varsigma \rightarrow \bar{\varsigma }} \bigl\Vert m(\varsigma )-m(\bar{ \varsigma }) \bigr\Vert =0. $$
(11)
It ensures that the answer \(m(\varsigma )\) of problem (1) is continuous at \(\varsigma =\bar{\varsigma }\). □

Notes

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The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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The authors declare that they have no competing interests.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Kottakkaran Sooppy Nisar
    • 1
    Email author
  • Mohd. Sarfaraz
    • 2
  • Ahmed Morsy
    • 1
  • Md. Kalimuddin Ahmad
    • 2
  1. 1.Department of Mathematics, College of Arts and SciencesPrince Sattam Bin Abdulaziz UniversityWadi AldawasirKingdom of Saudi Arabia
  2. 2.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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