Parametric Extended General Mixed Variational Inequalities

Abstract

It is well known that the resolvent equations are equivalent to the extended general mixed variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the extended general mixed variational inequalities without assuming the differentiability of the given data. Since the extended general mixed variational inequalities include extended general variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results.

1. Introduction

In recent years, much attention have been given to investigate the behaviour of the changes of the data of the given problems. The study of these changes is known as the sensitivity analysis. We remark that sensitivity analysis is important for several reasons. First, since estimating problem data often introduces measurement errors, sensitivity analysis helps in identifying sensitive parameters that should be obtained with relatively high accuracy. Second, sensitivity analysis may help to predict the future changes of the equilibrium as a result of changes in the governing systems. Third, from mathematical and engineering points of view, sensitivity analysis can provide new insight regarding problems being studied and can stimulate new ideas for problem solving. 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 has been studied by many authors including Noor et al. [1], Kyparisis [2, 3], Dafermos [4], Qiu and Magnanti [5], Tobin [6], Noor [7–10], Moudafi and Noor [11], M. A. Noor and K. I. Noor [12], and Liu [13] using quite different techniques. The techniques suggested so far vary with the problem being studied. Dafermos [4] used the fixed-point formulation to consider the sensitivity analysis of the classical variational inequalities. This technique has been modified and extended by many authors for studying the sensitivity analysis of other classes of variational inequalities and variational inclusions, see [1, 7, 8, 11, 12, 14–16] and the references therein.

In this paper, we develop a sensitivity framework for the extended general mixed variational inequalities, which were introduced and studied by Noor [17] and Noor et al. [18] in conjunction with the optimality conditions of the differentiable nonconvex functions. This class is quite general and includes the extended general variational inequalities and related optimization problems as special cases. We first establish the equivalence between the extended general mixed variational inequalities and the resolvent equations by using the resolvent operator method. This fixed-point formulation is obtained by a suitable and appropriate rearrangement of the resolvent equations. We would like to point out that the resolvent equations technique is quite general, unified, and flexible and provides us with a new approach to study the sensitivity analysis of variational inclusions and related optimization problems. We use this alternative equivalent formulation to develop sensitivity analysis for the extended general mixed variational inequalities without assuming the differentiability of the given data. Our results can be considered as significant extensions of the results of Dafermos [4], Moudafi and Noor [11], Noor [9], and others in this area.

2. Preliminaries

Let 𝐾 be a nonempty closed and convex set in a real Hilbert space 𝐻, whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝑇∶𝐻→𝐻 be a nonlinear operator and 𝑆 be a nonexpansive operator. Let 𝑃_𝐾 be the projection of 𝐻 onto the convex set 𝐾. Let 𝜑∶𝐻→𝑅∪{∞} be a continuous function.

For given nonlinear operators 𝑇,𝑔,ℎ∶𝐻→𝐻, consider the problem of finding 𝑢∈𝐻 such that 𝑇𝑢,ℎ(𝑣)−𝑔(𝑢)+𝜑(ℎ(𝑣))−𝜑𝑔(𝑢)≥0,∀𝑣∈𝐻. (2.1) Inequality of type (2.1) is called the extended general mixed variational inequality involving four operators and is quite different than all other classes of variational inequalities. Extended general mixed variational inequalities were introduced by Noor [17]. A wide class of problems arising in pure and applied sciences can be studied via the extended general mixed variational inequalities (2.1), see [18].

Example 2.1 (see [17]). As an application of problem (2.1), we show that the optimality condition for the minimum of sum of differentiable and nondifferentiable nonconvex functions on a nonconvex set 𝐾 in 𝐻 can be characterized by the general mixed variational inequality of type (2.1). This result is due to Noor [17]. We include some details to convey an idea of the technique.

For this purpose, we recall the following well-known concepts, see [7].

Definition 2.2. Let 𝐾 be any set in 𝐻. The set 𝐾 is said to be 𝑔ℎ-convex if there exist functions 𝑔,ℎ∶𝐻→𝐻 such that ℎ(𝑢)+𝑡𝑔(𝑣)−ℎ(𝑢)∈𝐾,∀𝑢,𝑣∈𝐻∶ℎ(𝑢),𝑔(𝑣)∈𝐾,𝑡∈[0,1]. (2.2) Note that every convex set is 𝑔ℎ-convex but the converse is not true, see [7]. If 𝑔=ℎ=𝐼, then the 𝑔ℎ-convex set 𝐾 is called the convex set.

Definition 2.3 (see [7]). The function 𝐹∶𝐾→𝐻 is said to be 𝑔ℎ-convex if there exist functions 𝑔,ℎ such that 𝐹𝑔(𝑢)+𝑡ℎ(𝑣)−𝑔(𝑢)≤(1−𝑡)𝐹𝑔(𝑢)+𝑡𝐹(ℎ(𝑣)),∀𝑢,𝑣∈𝐻∶ℎ(𝑢),𝑔(𝑣)∈𝐾,𝑡∈[0,1]. (2.3) Clearly every convex function is 𝑔ℎ-convex, but the converse is not true. For the properties and various classes of the 𝑔ℎ-convex functions, see Noor [7, 8, 19, 20]. We note that if the 𝑔ℎ-convex function is differentiable, then 𝐹𝑔(𝑣)−𝐹(ℎ(𝑢))≥𝐹^′(ℎ(𝑢)),𝑔(𝑣)−ℎ(𝑢),𝑢,𝑣∈𝐻∶ℎ(𝑢),𝑔(𝑣)∈𝐾, (2.4) and conversely, where 𝐹^′(ℎ(𝑢)) is the differential of the 𝑔ℎ-convex function at the point ℎ(𝑢).
For a given differentiable 𝑔ℎ-convex function 𝐹 and a nondifferentiable 𝑔ℎ-convex function 𝜑, we consider the functional of the type𝐼[𝑣]=𝐹(𝑣)+𝜑(𝑣),∀𝑣∈𝐾. (2.5) One can prove that the minimum of the functional 𝐼[𝑣] on the 𝑔ℎ-convex set 𝐾 can be characterized by a class of variational inequalities (2.1). This result is due to Noor [17].

Lemma 2.4 (see [17, 18]). Let 𝐹 be a differentiable 𝑔ℎ-convex function and 𝜑 be a nondifferentiable 𝑔ℎ-convex function on the 𝑔ℎ-convex set 𝐾. Then 𝑢∈𝐾 is the minimum of 𝐼[𝑣], defined by (2.5), on 𝐾⊂𝑔(𝐻) if and only if 𝑢∈𝐻∶𝑔(𝑢)∈𝐾 satisfies the inequality 𝐹^′𝑔(𝑢),ℎ(𝑣)−𝑔(𝑢)+𝜑(ℎ(𝑣))−𝜑𝑔(𝑢)≥0,∀𝑣∈𝐻∶𝑔(𝑣)∈𝐾, (2.6) where 𝐹^′(𝑔(𝑢)) is the differential of 𝐹 at 𝑔(𝑢)∈𝐾.

From Lemma 2.4, we see that the extended general mixed variational inequalities arise as a minimization of the sum of the differentiable and nondifferentiable 𝑔ℎ-convex functions on the 𝑔ℎ-convex set. This shows that the nonconvexity plays an important part in the study of the extended general mixed variational inequalities.

We would like to point out that the extended general mixed variational inequality (2.1) can be written in the equivalent form as: Find 𝑢∈𝐻 such that 𝜌𝑇𝑢+ℎ(𝑢)−𝑔(𝑢),ℎ(𝑣)−𝑔(𝑢)+𝜌𝜑(ℎ(𝑣))−𝜌𝜑𝑔(𝑢)≥0,∀𝑣∈𝐻, (2.7) where 𝜌>0 is a constant.

This equivalent formulation plays important part in developing iterative methods for solving the general mixed variational inequalities.

If 𝑔=ℎ=𝐼, the identity operator, then the extended general mixed variational inequalities (2.1) and (2.7) are equivalent to finding that 𝑢∈𝐻 such that ⟨𝑇𝑢,𝑣−𝑢⟩+𝜑(𝑣)−𝜑(𝑢)≥0,∀𝑣∈𝐻, (2.8) which is known as the mixed variational inequality introduced or variational inequality of the second type. We note that if the function 𝜑 in the extended general mixed variational inequality (2.7) is a proper, convex, and lower semicontinuous, then problem (2.7) is equivalent to finding 𝑢∈𝐻 such that 0∈𝜌𝑇𝑢+ℎ(𝑢)−𝑔(𝑢)+𝜌𝜕𝜑𝑔(𝑢), (2.9) which is known as the problem of finding a zero of sum of two (or more) monotone operators. It is well known that a large class of problems arising in industry, ecology, finance, economics, transportation, network analysis and optimization can be formulated and studied in the framework of (2.1) and (2.9), see the references therein.

If 𝜑 is an indicator function of a closed convex set 𝐾 in 𝐻, that is, 𝜑(𝑢)=𝐼_𝐾(𝑣)=0,if𝑣∈𝐾+∞,otherwise, (2.10) then the extended general mixed variational inequalities (2.1) are equivalent to finding 𝑢∈𝐻∶𝑔(𝑢)∈𝐾 such that Tu,ℎ(𝑣)−𝑔(𝑢)≥0,∀𝑣∈𝐻∶ℎ(𝑣)∈𝐾, (2.11) which is called the extended general variational inequality, introduced and studied by Noor [1, 7–10, 12, 15–21]. From Lemma 2.2, we see that the minimum of a class of differentiable nonconvex function on the nonconvex set can be characterized by the extended general variational inequalities of the type (2.11). For applications, numerical methods and other aspects of the extended general variational inequalities (2.11), see [1, 7–10, 12, 15–21].

We note that for ℎ=𝑔, (2.1) is equivalent to finding 𝑢∈𝐻∶𝑔(𝑢)∈𝐾 such that 𝑇(𝑢),𝑔(𝑣)−𝑔(𝑢)≥0,∀𝑣∈𝐻∶𝑔(𝑣)∈𝐾, (2.12) which is known as the general variational inequality and has been studied extensively in recent years. For the formulation, numerical methods, sensitivity analysis, and other aspects of the general variational inequalities, see [1, 7–10, 12, 15–21].

If 𝑔=ℎ=𝐼, then problems (2.11) and (2.12) reduce to finding 𝑢∈𝐾 such that ⟨𝑇𝑢,𝑣−𝑢⟩≥0,∀𝑣∈𝐾, (2.13) which is known as the classical variational inequality, introduced and studied by Stampacchia [22] in 1964. For the numerical methods, formulations and applications of the mixed variational inequalities, readers may consult the the recent state-of-the-art papers [1–29] and the references therein.

We now recall some well-known concepts and results.

Definition 2.5 (see [23]). For any maximal operator 𝑇, the resolvent operator associated with 𝑇, for any 𝜌>0, is defined as 𝐽_𝑇(𝑢)=𝐼+𝜌𝑇⁻¹(𝑢),∀𝑢∈𝐻. (2.14)
It is well known that an operator 𝑇 is maximal monotone if and only if its resolvent operator 𝐽_𝑇 is defined everywhere. It is single valued and nonexpansive. If 𝜑(.) is a proper, convex and lower-semicontinuous function, then its subdifferential 𝜕𝜑(.) is a maximal monotone operator. In this case, we can define the resolvent operator𝐽_𝜑(𝑢)=𝐼+𝜌𝜕𝜑⁻¹(𝑢),∀𝑢∈𝐻 (2.15) associated with the subdifferential 𝜕𝜑(.). The resolvent operator 𝐽_𝜑 has the following useful characterization.

Lemma 2.6. For a given 𝑧∈𝐻,𝑢∈𝐻 satisfies the inequality ⟨𝑢−𝑧,𝑣−𝑢⟩+𝜌𝜑(𝑣)−𝜌𝜑(𝑢)≥0,∀𝑣∈𝐻 (2.16) if and only if 𝑢=𝐽_𝜑(𝑧), (2.17) where 𝐽_𝜑=(𝐼+𝜌𝜕𝜑)⁻¹ is the resolvent operator.
It is well known the resolvent operator 𝐽_𝜑 is nonexpansive, that is, 𝐽_𝜑𝑢−𝐽_𝜑𝑣≤‖𝑢−𝑣‖,∀𝑢,𝑣∈𝐻. (2.18)

We now consider to the problem of solving the resolvent equations. To be more precise, let 𝑅_𝜑=𝐼−𝑔ℎ⁻¹𝐽_𝜑, where 𝐼 is the identity operator, and 𝑔,ℎ are given nonlinear operator. For given nonlinear operators 𝑇,𝑔,ℎ, we consider the problem of finding 𝑧∈𝐻 such that 𝑇ℎ⁻¹𝐽_𝜑𝑧+𝜌⁻¹𝑅_𝜑𝑧=0, (2.19)

which is called the extended general resolvent equation. We note that if 𝑔=ℎ=𝐼, then one can obtain the original resolvent equations. It has been shown that the resolvent equations have played an important and significant role in developing several numerical techniques for solving extended general mixed variational inequalities and related optimization problems.

If the proper, convex, and lower-semicontinuous function 𝜑 is an indicator function of a closed convex set 𝐾, then 𝐽_𝜑≡𝑃_𝐾, the projection of 𝐻 onto the closed convex set 𝐾. Consequently, the extended general resolvent equations (2.19) are equivalent to finding, 𝑧∈𝐻 such that 𝑇ℎ⁻¹𝑃_𝐾𝑧+𝜌⁻¹𝑄_𝐾𝑧=0, (2.20)

which are called the extended general Wiener-Hopf equations, see Noor [8], where 𝑄_𝐾=𝐼−𝑔ℎ⁻¹𝑃_𝐾. For 𝑔=ℎ=𝐼, one can obtain the original Wiener-Hopf equations of Shi [27].

We now consider the parametric versions of the problem (2.7) and (2.19). To formulate the problem, let 𝑀 be an open subset of 𝐻 in which the parameter 𝜆 takes values. Let 𝑇(𝑢,𝜆) be given operator defined on 𝐻×𝑀 and take value in 𝐻.

From now onward, we denote 𝑇_𝜆(.)≡𝑇(.,𝜆), unless otherwise specified.

The parametric general variational inequality problem is to find (𝑢,𝜆)∈𝐻×𝑀 such that 𝜌𝑇_𝜆𝑢+ℎ(𝑢)−𝑔(𝑣),𝑔(𝑣)−ℎ(𝑢)+𝜑(ℎ(𝑣))−𝜑𝑔(𝑢)≥0,∀𝑣∈𝐻∶𝑔(𝑣)∈𝐾. (2.21) We also assume that, for some 𝜆∈𝑀, problem (2.19) has a unique solution 𝑢.

Related to the parametric extended general mixed variational inequality (2.21), we consider the parametric resolvent equations. We consider the problem of finding (𝑧,𝜆),(𝑢,𝜆)∈𝐻×𝑀, such that 𝑇_𝜆ℎ⁻¹𝐽_𝜑𝑧+𝜌⁻¹𝑅_𝜑𝑧=0, (2.22) where 𝜌>0 is a constant and 𝐽_𝜑 is defined on the set of (𝑧,𝜆) with 𝜆∈𝑀 and takes values in 𝐻. The equations of the type (2.22) are called the parametric resolvent equations.

One can establish the equivalence between the problems (2.21) and (2.22) by using the resolvent operator technique, see Noor [9, 10].

Lemma 2.7. The parametric extended general mixed variational inequality (2.21) has a solution (𝑢,𝜆)∈𝐻×𝑀 if and only if the parametric resolvent equations (2.22) have a solution (𝑧,𝜆),(𝑢,𝜆)∈𝐻×𝑀, where ℎ(𝑢)=𝐽_𝜑,𝑧=𝑔(𝑢)−𝜌𝑇_𝜆(𝑢). (2.23)

From Lemma 2.7, we see that the parametric extended general mixed variational inequalities (2.21) and the parametric resolvent equations (2.22) are equivalent. We use this equivalence to study the sensitivity analysis of the extended general mixed variational inequalities. We assume that for some 𝜆∈𝑀, problem (2.22) has a solution 𝑧, and 𝑋 is a closure of a ball in 𝐻 centered at 𝑧. We want to investigate those conditions under which, for each 𝜆 in a neighborhood of 𝜆, problem (2.22) has a unique solution 𝑧(𝜆) near 𝑧, and the function 𝑧(𝜆) is (Lipschitz) continuous and differentiable.

Definition 2.8. Let 𝑇_𝜆(.) be an operator on 𝑋×𝑀. Then, the operator 𝑇_𝜆(.) is said to the following:(a)locally strongly monotone if there exists a constant 𝛼>0 such that𝑇_𝜆(𝑢)−𝑇_𝜆(𝑣),𝑢−𝑣≥𝛼‖𝑢−𝑣‖²,∀𝜆∈𝑀,𝑢,𝑣∈𝑋, (2.24)(b)locally Lipschitz continuous if there exists a constant 𝛽>0 such that𝑇_𝜆(𝑢)−𝑇_𝜆(𝑣)≤𝛽‖𝑢−𝑣‖,∀𝜆∈𝑀,𝑢,𝑣∈𝑋. (2.25)

3. Main Results

We consider the case, when the solutions of the parametric resolvent equations (2.22) lie in the interior of 𝑋. Following the ideas of Dafermos [4] and Noor [8, 10], we consider the map 𝐹_𝜆(𝑧)=𝐽_𝜑𝑧−𝜌𝑇_𝜆(𝑢)()(𝑢),∀𝑧,𝜆∈𝑋×𝑀=𝑔−𝜌𝑇_𝜆(𝑢), (3.1) where ℎ(𝑢)=𝑃_𝐾𝑧. (3.2) We have to show that the map 𝐹_𝜆(𝑧) has a fixed point, which is a solution of the resolvent equations (2.13). First of all, we prove that the map 𝐹_𝜆(𝑧), defined by (3.1), is a contraction map with respect to 𝑧 uniformly in 𝜆∈𝑀.

Lemma 3.1. Let 𝑇_𝜆(.) be a locally strongly monotone with constant 𝛼>0 and locally Lipschitz continuous with constant 𝛽>0. If that the operators 𝑔,ℎ are strongly monotone with constants 𝜎>0,𝜇>0 and Lipschitz continuous with constants 𝛿>0,𝜂>0, respectively, then, for all 𝑧₁,𝑧₂∈𝑋 and 𝜆∈𝑀, we have 𝐹_𝜆𝑧₁−𝐹_𝜆𝑧₂≤𝜃𝑧₁−𝑧₂, (3.3) where 𝜃=1−2𝜎+𝛿²+1−2𝛼𝜌+𝛽²𝜌²1−1−2𝜇+𝜂² (3.4) for 𝛼𝜌−𝛽²<𝛼²−𝛽²𝑘(2−𝑘)𝛽²,𝛼>𝛽𝑘(2−𝑘),𝑘<1, (3.5) where 𝑘=1−2𝜎+𝛿²+1−2𝜇+𝜂², (3.6)

Proof. For all 𝑧₁,𝑧₂∈𝑋, 𝜆∈𝑀, we have, from (3.1), 𝐹_𝜆𝑧₁−𝐹_𝜆𝑧₂=𝑔𝑢₁−𝑔𝑢₂−𝜌𝑇_𝜆𝑢₁−𝑇_𝜆𝑢₂≤𝑢₁−𝑢₂−𝑔𝑢₁−𝑔𝑢₂+𝑢₁−𝑢₂−𝜌𝑇_𝜆𝑢₁−𝑇_𝜆𝑢₂. (3.7) Using the strongly monotonicity and Lipschitz continuity of the operator 𝑔, we have 𝑢₁−𝑢₂−(𝑔(𝑢₁)−𝑔(𝑢₂))²≤𝑢₁−𝑢₂²−2𝑢₁−𝑢₂,𝑔𝑢₁−𝑔𝑢₂+𝑔𝑢₁−𝑔𝑢₂²≤1−2𝜎+𝛿²𝑢₁−𝑢₂². (3.8) In a similar way, we have 𝑢₁−𝑢₂−𝜌𝑇_𝜆𝑢₁−𝑇_𝜆𝑢₂²≤1−2𝜌𝛼+𝛽²𝜌²𝑢₁−𝑢₂², (3.9) where 𝛼>0 is the strongly monotonicity constant and 𝛽>0 is the Lipschitz continuity constant of the operator 𝑇_𝜆 respectively.
From (3.7), (3.8), and (3.9), we have𝐹_𝜆𝑧₁−𝐹_𝜆𝑧₂≤1−2𝜎+𝛿²+1−2𝛼𝜌+𝛽²𝜌²𝑢₁−𝑢₂. (3.10) From (3.2) and using the nonexpansivity of the operator 𝐽_𝜑, we have ‖𝑢₁−𝑢₂‖≤𝑢₁−𝑢₂−ℎ𝑢₁−ℎ𝑢₂+𝐽_𝜑𝑧₁−𝐽_𝜑𝑧₂≤1−2𝜇+𝜂²𝑢₁−𝑢₂+𝑧₁−𝑧₂, (3.11) from which we obtain that 𝑢₁−𝑢₂≤11−1−2𝜇+𝜂²𝑧₁−𝑧₂, (3.12) where 𝜇>0 is the strongly monotonicity constant, and 𝜂>0 is the Lipschitz continuity constant of the operator ℎ, respectively.
Combining (3.10) and (3.12), we have𝐹_𝜆𝑧₁−𝐹_𝜆𝑧₂≤1−2𝜎+𝛿²+1−2𝜌𝛼+𝜌²𝛽²1−1−2𝜇+𝜂²𝑧₁−𝑧₂=𝜃𝑧₁−𝑧₂, (3.13) where 𝜃=(1−2𝜎+𝛿²+1−2𝛼𝜌+𝛽²𝜌²)/1−1−2𝜇+𝜂².
Now consider 𝜃<1. Using (3.5), we have𝑘+1−2𝜌𝛼+𝜌²𝛽²<1, (3.14) which shows that (3.5) holds. Consequently, from (3.5), it follows that 𝜃<1 and consequently the map 𝐹_𝜆(𝑧) defined by (3.1) is a contraction map and has a fixed point 𝑧(𝜆), which is the solution of the resolvent equations (2.22).

Remark 3.2. From Lemma 3.1, we see that the map 𝐹_𝜆(𝑧) defined by (3.1) has a unique fixed point 𝑧(𝜆), that is, 𝑧(𝜆)=𝐹_𝜆(𝑧). Also, by assumption, the function 𝑧, for 𝜆=𝜆 is a solution of the parametric resolvent equations (2.22). Again using Lemma 3.1, we see that 𝑧, for 𝜆=𝜆, is a fixed point of 𝐹_𝜆(𝑧) and it is also a fixed point of 𝐹_𝜆(𝑧). Consequently, we conclude that 𝑧𝜆=𝑧=𝐹_𝜆𝑧𝜆. (3.15) Using Lemma 3.1, we can prove the continuity of the solution 𝑧(𝜆) of the parametric resolvent equations (2.22) using the technique of Noor [9, 10]. However, for the sake of completeness and to convey an idea of the techniques involved, we give its proof.

Lemma 3.3. Assume that the operator 𝑇_𝜆(.) is locally Lipschitz continuous with respect to the parameter 𝜆. If the operator 𝑇_𝜆(.) is locally Lipschitz continuous and the map 𝜆→𝐽_𝜑𝜆𝑧 is continuous (or Lipschitz continuous), then the function 𝑧(𝜆) is (Lipschitz) continuous at 𝜆=𝜆.

Proof. For all 𝜆∈𝑀, invoking Lemma 3.1 and the triangle inequality, we have 𝑧(𝜆)−𝑧𝜆≤𝐹_𝜆(𝑧(𝜆))−𝐹_𝜆𝑧𝜆+‖𝐹_𝜆𝑧𝜆−𝐹_𝜆𝑧𝜆‖≤𝜃‖𝑧(𝜆)−𝑧𝜆‖+‖𝐹_𝜆𝑧𝜆−𝐹_𝜆𝑧𝜆‖. (3.16) From (3.16) and the fact that the operator 𝑇_𝜆 is a Lipschitz continuous with respect to the parameter 𝜆, we have 𝐹_𝜆𝑧𝜆−𝐹_𝜆𝑧𝜆=𝑢𝜆−𝑢𝜆+𝜌𝑇_𝜆𝑢𝜆,𝑢𝜆−𝑇_𝜆𝑢𝜆,𝑢𝜆≤𝜌𝜇𝜆−𝜆. (3.17) Combining (3.16) and (3.17), we obtain 𝑧(𝜆)−𝑧𝜆≤𝜌𝜇1−𝜃𝜆−𝜆,∀𝜆,𝜆∈𝑀, (3.18) from which the required result follows.

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

Theorem 3.4. Let 𝑢 be the solution of the parametric extended generalized variational inequality (2.21) and 𝑧 be the solution of the parametric resolvent equations (2.22) for 𝜆=𝜆. Let 𝑇_𝜆(𝑢) be the locally strongly monotone Lipschitz continuous operator for all 𝑢,𝑣∈𝑋. If the the map 𝜆→𝐽_𝜑 is (Lipschitz) continuous at 𝜆 = 𝜆, then there exists a neighborhood 𝑁⊂𝑀 of 𝜆 such that, for 𝜆∈𝑁, the parametric resolvent equations (2.22) have a unique solution 𝑧(𝜆) in the interior of 𝑋,𝑧(𝜆)=𝑧 and 𝑧(𝜆) is (Lipschitz) continuous at 𝜆=𝜆.

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

4. Conclusion

In this paper, we have developed a general framework of the sensitivity analysis for the extended general mixed variational inequalities. Several special cases are also discussed. Results proved in this paper may be extended for the multivalued general variational inequalities and related optimization problems. This is an interesting and fascinating problem for future research.

Acknowledgment

The author would like to thank Dr. S. M. Junaid Zaidi, Rector, and CIIT for providing excellent research facilities.