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Sensitivity Analysis for General Nonlinear Nonconvex Variational Inequalities | OMICS International
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Sensitivity Analysis for General Nonlinear Nonconvex Variational Inequalities

Salahuddin*

Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia

*Corresponding Author:
Salahuddin
Department of Mathematics
Jazan University
Jazan, Kingdom of Saudi Arabia
Tel: 0922-5291501-502
E-mail:[email protected]

Received Date: December 09, 2014; Accepted Date: February 19, 2015; Published Date: February 28, 2015

Citation: Salahuddin (2015) Sensitivity Analysis for General Nonlinear Nonconvex Variational Inequalities. J Appl Computat Math 4:206. doi: 10.4172/2168-9679.1000206

Copyright: © 2015 Salahuddin. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In this communication, we proved that the parametric general nonlinear nonconvex variational inequalities are equivalent to the parametric general Wiener-Hopf equations. We use this alternative equivalence formulation to studied the sensitivity analysis for general nonlinear nonconvex variational inequalities without assuming the differentiability of the given data.

Keywords

Sensitivity analysis; Parametric general nonlinear nonconvex variational inequalities; Fixed point; Parametric general Wiener-Hopf equations; (Equation )-relaxed cocoercive mapping; Lipschitz continuous mappings; uniformly r-prox regular sets; Hilbert spaces

AMS Mathematics Subject Classification: 49J40, 47H06

Historical background

The variational inequality theory was introduced by Stampacchia [1] has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, finance, transportation, networks, structural analysis and optimizations [2-5]. It should be pointed that almost all the results regarding the existence and iterative scheme for solving variational inequalities and related optimization problems are being considered in the convex setting. Consequently all the techniques are based on the properties of the projection operators are convex sets which may not hold in general when the sets are nonconvex. It is known that the uniformly r-prox regular sets are nonconvex and included the convex sets as a special cases [6-9].

Over the last decade there has been increasing interest in studying the sensitivity analysis of variational inequalities and variational inclusions. Sensitivity analysis for variational inclusions and inequalities have been studied extensively [2,3,10-13].

The techniques suggested so far vary with the problems being studied. Dafermos used the fixed point formulation to considered the sensitivity analysis of the classical variational inequalities. These techniques have been modified and extended by many authors for studying the sensitivity analysis of the other classes of variational inequalities and variational inclusions. It is known that the variational inequalities are equivalent to Wiener-Hopf equations [14]. This alternative equivalence formulation has been used by Noor [15-17] to developed the sensitivity analysis frame work for various classes of (quasi) variational inequalities.

In this paper we develop the general frame work of sensitivity analysis for general non-linear nonconvex variational inequalities. First we establish the equivalence between the parametric general nonlinear nonconvex variational inequalities and the parametric general Wiener- Hopf equations by using the projection techniques. By using the fixed point formulation, we obtain an approximate rearrangement of the Wiener-Hopf equations. We use this equivalence to developed the sensitivity analysis for general nonlinear nonconvex variational inequalities without assuming the differentiability of the given data.

Preliminaries

Let H be a real Hilbert space whose inner product and norm are denoted byN Equationand Equation respectively. Let K be a nonempty closed subset of H.

Definition 2.1 The proximal normal cone of K at a point uEquationH with uEquationK is given by Equationfor some Equation

Where α>0 is a constant andEquation

Where dK(.) or d(.;K) is the usual distance function to the subset of K, that is

Equation

The proximal normal coneEquation(u) has the following characterizations:

Lemma 2.2 Let K be a nonempty closed subset in H. ThenEquationif and only if there exists a constant

Equation such that

Equation

Lemma 2.3 Let K be a nonempty closed and convex subset in H. ThenEquation

Equation

The Clarke normal cone denoted byEquationis defined by

Equation

where co mean the closure of the convex hull.

ClearlyEquationbut the converse is not true in general. Note thatEquationis always closed and convex cone where asEquationis convex but may not be closed, see [5, 12].

Definition 2.4 For any givenEquationa subset Kr of H is said to be normalized uniformly r-prox regular (or uniformly r-prox regular) if and only if every nonzero proximal normal to Kr can be realized by an r-ball that is for all EquationandEquationwithEquation

Equation

Lemma 2.5 A closed setKEquationH is convex if and only if it is proximally smooth of radius r for every r>0:

If r=1 then uniformly r-prox regularity of Kr is equivalent to the convexity of K: If Kr is uniformly r-prox regular set, then the proximal normal coneEquation(u) is closed as a set valued mapping. If we takeEquationit is clear that rEquationthen n=0:

Proposition 2.6 [12] For r>0, let Kr be a nonempty closed and uniformly r-prox regular subset of H. Set

Equation

Then the following statements are holds:

for all Equation

for allEquationis a Lipschitz continuous mapping with constantEquationon

Equation

(i) the proximal normal cone is closed as a set valued mapping.

Assume that F; T : H ! 2H are set valued mappings, g; h : H ! H the nonlinear single valued mappings such thatEquationand N : H X HEquationH the mapping. For any constants n>0 and p>0, we consider the problem of finding Equation such thatEquation and

Equation

The equation (2.1) is called general nonlinear nonconvex variational inequalities. Now we consider the problem of solving general Wiener- Hopf equations. To be more precise, let QKr = I – h-1 PKr where PKr is the projection operator, h-1 is the inverse of nonlinear mapping h and I is an identity mapping. For given nonlinear mappings T; F; h; g; consider the problem of findingEquationsuch thatEquation

is called general Wiener-Hopf equations.

Lemma 2.7 Equationis a solution of (2.1) if and only i fEquationsatisfies the relationEquationwhere PKr is a projection of H onto the uniformly r-prox regular set Kr:

Lemma 2.7 implies that the general nonlinear nonconvex variational inequality (2.1) is equivalent to the fixed point problem (2.3).

Now we consider the parametric version of equations (2.1), (2.2) and (2.3). To for- mulate the problem, let Equationbe an open subset of H in which parameter Equationtakes values. Let Equationbe the set valued mappings, N : H X HEquationand g; h :Equation_ H→H the nonlinear single valued mappings such that Equationand N : H X H→H the mapping. For any constants n > 0 and p > 0, we consider the problem of findingEquationsuch that Equation

and Equation

The equation (2.1) is called general nonlinear nonconvexvariational inequalities.Now we consider the problem of solving general Wiener-Hopf equations. To be moreprecise, letEquation is the projection operator, h-1 is the inverseof nonlinear mapping h and I is an identity mapping. For given nonlinear mappingsT; F; h; g; consider the problem of finding Equationsuch that

Equation

is called general Wiener-Hopf equations.

Lemma 2.7 Equationis a solution of (2.1) if and only if Equationsatisfies the relation

Equation         (2.3)

where PKr is a projection of H onto the uniformly r-prox regular set Kr: the single valued mappings. We defineEquation

unless otherwise specified. The parametric general non-linear nonconvex variational inequality is to find Equationsuch that

Equation         (2.4)

We also assume that for some Equation problem has a unique solution Equation Related to the parametric general nonlinear nonconvex variational inequality (2.4), we consider the parametric general Wiener-Hopf equation. We consider the problem of finding Equationsuch that

Equation         (2.5)

where p > 0 is a constant and QKr (z) is define on the set EquationwithEquationand takes values in H. The equation (2.5) is called parametric general Wiener-Hopf equation.

Lemma 2.8 If H is a real Hilbert space. Than the following two statements are equivalent: an element Equation

is a solution of (2.4), the mappingEquation has a fixed point.

One can established the equivalence between (2.4) and (2.5), by using the projection techniques, see Noor [10,11].

Lemma 2.9 Parametric general nonlinear nonconvex variational inequality (2.4) has a Solution Equationif and only if parametric general Wiener-Hopf equation (2.5) has aEquation solution

Equation         (2.6)

Equation         (2.7)

From Lemma 2.9, we see that Parametric general nonlinear nonconvex variational inequalities (2.4) and parametric general Wiener-Hopf equations (2.5) are equivalent. We use these equivalence to study the sensitivity analysis of general nonlinear non-convex variational inequalities. We assume that for someEquationproblem (2.5) has a solution Equationand X is a closure of a ball in H centered at Equation.We want to investigate those conditions under which for each Equation is a neighbourhood of Equationthen (2.5) has a unique solution Equationnear Equationand the function Equationis (Lipschitz) continuous and differentiable

Definition 2.10 For Equation the mapping Equationis said to be Equation relaxed cocoercive with respect to first argument and Equationwith constants Equation and Lipschitz continuous with respect to first and second argument if there exists a constants Equationsuch that

Equation

Equation

And

Equation

Equation

Definition 2.11 A single valued mapping Equationis said to be Lipschitz continuous if there exists a constant Equation such that

Equation

Definition 2.12 The set valued mapping Equationis said to be D-Lipschitz continuous if there exists a constant v > 0 such that

Equation

where D is the Hausdorff metric.

Definition 2.13 Let Equationbe a single valued mapping. Then h Equation is said to be Equation-relaxed cocoercive if there exists a constant Equationsuch that

Equation

and Lipschitz continuous if there exists a constant Equation such that

Equation

Main Results

In this section, we consider the case when the solution of the parametric general Wiener-Hopf equations (2.5) lies in the interior of X. We consider the map

Equation         (3.1)

Where

Equation         (3.2)

We have to show that the map Equation has a fixed point which is a solution of parametric general Wiener-Hopf equations (2.5). First of all we prove the map Equationdefined by (3.1) is a contraction map with respect to z uniformly in Equation by using the techniques of Noor [10].

Lemma 3.1 Let Pkrbe a Lipschitz continuous operator with constant EquationLet Equationbe the Lipschitz continuous with first argument and second argument with Constants Equation respectively. Let h; g :

Equationbe the Lipschitz continuous with constants Equation respectively and Equation be the Equationrelaxed cocoercive with respect to the constant Equation<0: Let T; F : EquationH→2H be the D-Lipschitz continuous with constants v; x > 0; respectively. Let N be the Equation relaxed cocoercive with respect to first argument and Equationwith constants Equation

respectively. We have

Equation

Where

Equation         (3.3)

Equation         (3.4)

For

Equation

Equation

Equation

Equation         (3.5)

Proof. For all z1; z2 Equation from (3.1) we have

Equation         (3.6)

Now

Equation         (3.7)

Since N is Lipschitz continuous with respect to _rst and second argument and T; F are D-Lipschitz continuous with constants v; x > 0 respectively, we have

Equation         (3.8)

And

Equation         (3.9)

From the Equation)-relaxed cocoercive mapping of N with respect to first argument and Equationwe have

Equation         (3.10)

Hence from (3.7)-(3.10), we have

Equation         (3.11)

Therefore from (3.6) and (3.11), we have

Equation         (3.12)

Also from (3.2) and Lipschitz continuity of projection mapping PKr with constant δ;we Have

Equation         (3.13)

Since h is Lipschitz continuous with constant Equation and Equation relaxed Equation we have cocoercive with constant

Equation         (3.14)

Where Equation From (3.13) and (3.14) we have

Equation         (3.15)

Combining (3.12),(3.15) and using (3.3) we have

Equation         (3.16)

Where

Equation

From (3.5) it follows that Equationand consequently the map Equation de_ne by (3.12) is a contraction map and has a fixed point Equation which is the solution of parametric general Wiener-Hopf equations (2.5).

Remark 3.2 From Lemma 3.1, we see that the map Equationl define by (2.1) has a unique fixed point Equationthat is Equation=EquationAlso by assumption the function Equationfor Equationis a solution of parametric general Wiener-Hopf equations (2.5). Again by Lemma 3.1 we see that EquationforEquationis a fixed point ofEquationand it is also a fixed point of EquationConsequently, we conclude that Equation

Using Lemma 3.1 we can prove the continuity of the solution Equationof parametric general Wiener-Hopf equations (2.5). However for the sake of completeness and to convey the idea of the techniques involved, we give the proof.

Lemma 3.3 Assume that the mappings Equationare D-Lipschitz continuous and Equationare Lipschitz continuous with respect to the parameter λ If the mapping N is Lipschitz continuous with first and second argument respectively, and the map Equationare continuous (or Lipschitz continuous), the function Equationsatisfying the (2.3) is Lipschitz continuous at Equation

Proof. For all Equationinvoking Lemma 3.1 and the triangle inequality, we have

Equation         (3.17)

From (3.1) and the fact that the mappingEquationand Equationare Lipschitz continuous with respect to the parameter λ we have

Equation         (3.18)

Combining (3.17) and (3.18), we obtain

Equation

from which the required results follows.

We now state and prove the main result of this paper which is motivation of the next result.

Theorem 3.4 Let Equationbe a solution of parametric general nonlinear nonconvexvariationalinequalities (2.4) and Equation

be the solution of parametric general Wiener-Hopf equations (2.5 ) for Equation Let Equation be Equation_-relaxed cocoercive mapping and Lipschitz continuous mapping, and Equationbe the D-Lipschitz continuous mappings and N be Equation)-relaxed coco-ercive mapping with respect to first argument and Equation and Equation be the Lipschitz continuous for all Equation If the map Equationare Lipschitz (continuous) mappings at Equationthen there exists a neighbourhood Equationof Equationsuch that for Equationparametric general Wiener- Hopf equation (2.5) has a unique solution Equationin the interior of Equationand Equationis (Lipschitz) continuous at Equation

Proof. Its proof follows from Lemma 3.1, 3.3 and Remark 3.2.

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