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**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.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

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.

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

**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.

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

**Definition 2.1 **The proximal normal cone of K at a point uH with uK is given by for some

Where α>0 is a constant and

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

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

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

such that

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

The Clarke normal cone denoted byis defined by

where co mean the closure of the convex hull.

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

**Definition 2.4** For any givena 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 andwith

**Lemma 2.5 **A closed setKH 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 cone(u) is closed as a set valued mapping. If we takeit is clear that rthen n=0:

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

Then the following statements are holds:

for all

for allis a Lipschitz continuous mapping with constanton

(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 thatand N : H X HH the mapping. For any constants n>0 and p>0, we consider the problem of finding such that and

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 findingsuch that

is called general Wiener-Hopf equations.

**Lemma 2.7 **is a solution of (2.1) if and only i fsatisfies the relationwhere 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 be an open subset of H in which parameter takes values. Let be the set valued mappings, N : H X Hand g; h :_ H→H the nonlinear single valued mappings such that** **and N : H X H→H the mapping. For any constants n > 0 and p > 0, we consider the problem of findingsuch that

and** **

The equation (2.1) is called general nonlinear nonconvexvariational inequalities.Now we consider the problem of solving general Wiener-Hopf equations. To be moreprecise, let 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 such that

is called general Wiener-Hopf equations.

**Lemma 2.7 **is a solution of (2.1) if and only if satisfies the relation

** (2.3)**

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

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

** (2.4)**

We also assume that for some problem has a unique solution 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 such that

** (2.5)**

where p > 0 is a constant and QKr (z) is define on the set withand 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

is a solution of (2.4), the mapping 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 if and only if parametric general Wiener-Hopf equation (2.5) has a solution

** (2.6)**

** (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 someproblem (2.5) has a solution and X is a closure of a ball in H centered at .We want to investigate those conditions under which for each is a neighbourhood of then (2.5) has a unique solution near** **and the function is (Lipschitz) continuous and differentiable

**Definition 2.10** For the mapping** **is said to be relaxed cocoercive with respect to first argument and with constants and Lipschitz continuous with respect to first and second argument if there exists a constants such that

And

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

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

where D is the Hausdorff metric.

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

and Lipschitz continuous if there exists a constant such that

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

** ** (3.1)

Where

(3.2)

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

**Lemma 3.1** Let P_{kr}be a Lipschitz continuous operator with constant Let be the Lipschitz continuous with first argument and second argument with Constants respectively. Let h; g :

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

respectively. We have

Where

(3.3)

(3.4)

For

(3.5)

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

(3.6)

Now

(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

(3.8)

And

(3.9)

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

(3.10)

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

(3.11)

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

(3.12)

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

(3.13)

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

(3.14)

Where From (3.13) and (3.14) we have

(3.15)

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

(3.16)

Where

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

**Remark 3.2 **From Lemma 3.1, we see that the map l define by (2.1) has a unique fixed point that is =Also by assumption the function for is a solution of parametric general Wiener-Hopf equations (2.5). Again by Lemma 3.1 we see that foris a fixed point ofand it is also a fixed point of Consequently, we conclude that

**Using Lemma 3.1 **we can prove the continuity of the solution of 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 are D-Lipschitz continuous and are Lipschitz continuous with respect to the parameter λ If the mapping N is Lipschitz continuous with first and second argument respectively, and the map are continuous (or Lipschitz continuous), the function satisfying the (2.3) is Lipschitz continuous at

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

** (3.17)**

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

** (3.18)**

Combining (3.17) and (3.18), we obtain

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 **be a solution of parametric general nonlinear nonconvexvariationalinequalities (2.4) and

be the solution of parametric general Wiener-Hopf equations (2.5 ) for** ** Let be _-relaxed cocoercive mapping and Lipschitz continuous mapping, and be the D-Lipschitz continuous mappings and N be )-relaxed coco-ercive mapping with respect to first argument and and be the Lipschitz continuous for all If the map are Lipschitz (continuous) mappings at then there exists a neighbourhood of such that for parametric general Wiener- Hopf equation (2.5) has a unique solution in the interior of and is (Lipschitz) continuous at

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

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