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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 863450, 14 pages
http://dx.doi.org/10.1155/2012/863450
Research Article

Regularized Mixed Variational-Like Inequalities

1Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
2Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received 13 December 2011; Accepted 22 December 2011

Academic Editor: Yonghong Yao

Copyright © 2012 Muhammad Aslam Noor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use auxiliary principle technique coupled with iterative regularization method to suggest and analyze some new iterative methods for solving mixed variational-like inequalities. The convergence analysis of these new iterative schemes is considered under some suitable conditions. Some special cases are also discussed. Our method of proofs is very simple as compared with other methods. Our results represent a significant refinement of the previously known results.

1. Introduction

Variational inequalities are being used to study a wide class of diverse unrelated problems arising in various branches of pure and applied sciences in a unified framework. Various generalizations and extensions of variational inequalities have been considered in different directions using a novel and innovative technique. A useful and important generalization of the variational inequalities is called the variational-like inequality, which has been studied and investigated extensively. It has been shown [13] that the minimum of the differentiable preinvex (invex) functions on the preinvex sets can be characterized by the variational-like inequalities. Note that the preinvex functions may not be convex functions and the invex sets may not be convex sets. This implies that the concept ofinvxesityplays same roles in the variational-like inequalities as the convexity plays the role in the variational inequalities.We would like to point out that the variational-like inequalities are quite different then variational inequalities in several aspects. For example, one can prove that the variational inequalities are equivalent to the fixed point problems, whereas variational-like inequalities are not equivalent to the fixed point problems. However, if the invex set is equivalent to the convex set, then variational-like inequalities collapse to the variational inequalities. This shows that variational-like inequalities include variational inequalities as a special case. Authors are advised to see the delicate difference between these two different problems. For other kind of variational inequalities involving two and three operators, see Noor [47] and Noor et al. [813].

There is a substantial number of numerical methods including the projection technique and its variant forms including the Wiener-Hopf equations, auxiliary principle, and resolvent equations methods for solving variational inequalities and related optimization problems. However, it is known that the projection method, Wiener-Hopf equations, and resolvent equations techniques cannot be extended to suggest and analyze similar iterative methods for solving variational-like inequalities due to the presence of the bifunction 𝜂(,). This fact motivated us to use the auxiliary principle technique of Glowinski et al. [14]. In this technique, one consider an auxiliary problem associated with the original problem. This way, one defines a mapping and shows that this mapping has a fixed point, which is a solution of the original problem. This fact enables us to suggest and analyze some iterative methods for solving the original problem. This technique has been used to suggest and analyze several iterative methods for solving various classes of variational inequalities and their generalizations, see [1, 2, 434] and the references therein.

The principle of iterative regularization is also used for solving variational inequalities. It was introduced by Bakušinskiĭ [16] in connection with variational inequalities in 1979. An important extension of this approach is presented by Alber and Ryazantseva[15].In this approach, the regularized parameter is changed at each iteration which is in contrast with the common practice for parameter identification of using a fixed regularization parameter throughout the minimization process. One can combine these two different techniques for solving the variational inequalities and related optimization problems. This approach was used by Khan and Rouhani [22] and Noor et al. [9, 10] for solving the mixed variational inequalities.

Motivated and inspired by the these activities, we suggest and analyze some iterative algorithms based on auxiliary principle and principle of iterative regularization for solving a class of mixed variational-like inequalities. For the convergence analysis of the explicit version of this iterative algorithm, we use partially relaxed strongly monotone operator which is a weaker condition than strongly monotonicity used by Khan and Rouhani [22]. We also suggest a new implicit iterative algorithm, the convergence of which requires only the monotonicity, which is weaker condition than strongly monotonicity. Results proved in this paper represent a significant improvement of the previously known results. The comparison of these methods with other methods is an interesting problem for future research.

2. Preliminaries

Let 𝐻 be a real Hilbert space, whose inner product and norm are denoted by , and , respectively. Let 𝐾 be a nonempty closed set in 𝐻. Let 𝑓𝐾𝑅 and 𝜂(,)𝐾×𝐾𝐻 be mappings. First of all, we recall the following well-known results and concepts; see [13, 21, 33].

Definition 2.1. Let 𝑢𝐾. Then the set 𝐾 is said to be invex at 𝑢 with respect to 𝜂(,), if []𝑢+𝑡𝜂(𝑣,𝑢)𝐾,𝑢,𝑣𝐾,𝑡0,1.(2.1)
𝐾 is said to be an invex set with respect to 𝜂(,), if 𝐾 is invex at each 𝑢𝐾. The invex set 𝐾 is also called 𝜂-connected set. Clearly, every convex set is an invex set with 𝜂(𝑣,𝑢)=𝑣𝑢,  for all 𝑢,𝑣𝐾, but the converse is not true; see [3, 33].

From now onwards,𝐾is a nonempty closed and invex set in 𝐻 with respect to 𝜂(,), unless otherwise specified.

Definition 2.2. A function 𝑓𝐾𝑅 is said to be preinvex with respect to 𝜂(,), if []𝑓(𝑢+𝑡𝜂(𝑣,𝑢))(1𝑡)𝑓(𝑢)+𝑡𝑓(𝑣),𝑢,𝑣𝐾,𝑡0,1.(2.2)
Note that every convex function is a preinvex function, but the converse is not true; see [3, 33].

Definition 2.3. A function 𝑓 is said to be a strongly preinvex function on 𝐾 with respect to the function 𝜂(,) with modulus 𝜇, if 𝑓(𝑢+𝑡𝜂(𝑣,𝑢))(1𝑡)𝑓(𝑢)+𝑡𝑓(𝑣)𝑡(1𝑡)𝜇𝜂(𝑣,𝑢)2[],𝑣,𝑢𝐾,𝑡0,1.(2.3)
Clearly, a differentiable strongly preinvex function 𝑓 is a strongly invex function with constant 𝜇>0, that is, 𝑓𝑓(𝑣)𝑓(𝑢)(𝑢),𝜂(𝑣,𝑢)+𝜇𝜂(𝑣,𝑢)2,𝑣,𝑢𝐾,(2.4) and the converse is also true under certain conditions.

We remark that if 𝑡=1, then Definitions 2.2 and 2.3 reduce to𝑓(𝑢+𝜂(𝑣,𝑢))𝑓(𝑣),𝑢,𝑣𝐾.(2.5) One can easily show that the minimum of the differentiable preinvex function on the invex set 𝐾 is equivalent to finding 𝑢𝐾 such that 𝑓(𝑢),𝜂(𝑣,𝑢)0,𝑣𝐾,(2.6) which is known as the variational-like inequality. This shows that the preinvex functions play the same role in the study of variational-like inequalities as the convex functions play in the theory of variational inequalities. For other properties of preinvex functions, see [3, 30, 33] and the references therein.

Let 𝐾 be a nonempty closed and invex set in 𝐻. For given nonlinear operator 𝑇𝐾𝐻 and a continuous function 𝜙(), we consider the problem of finding 𝑢𝐾 such that𝑇𝑢,𝜂(𝑣,𝑢)+𝜙(𝑣)𝜙(𝑢)0,𝑣𝐾,(2.7) which is called the mixed variational-like inequality introduced and studied by [1]. It has been shown in [13] that a minimum of differentiable preinvex functions 𝑓(𝑢) on the invex sets in the normed spaces can be characterized by a class of variational-like inequalities (2.7) with 𝑇𝑢=𝑓(𝑢) where 𝑓(𝑢) is the differential of a preinvex function 𝑓(𝑢). This shows that the concept of variational-like inequalities is closely related to the concept of invexity. For the applications, numerical methods, and other aspects of the mixed variational-like inequalities, see [1, 2, 29] and the references therein.

We note that if 𝜂(𝑣,𝑢)=𝑣𝑢, then the invex set 𝐾 becomes the convex set 𝐾 and problem (2.7) is equivalent to finding 𝑢𝐾 such that𝑇𝑢,𝑣𝑢+𝜙(𝑣)𝜙(𝑢)0,𝑣𝐾,(2.8) which is known as a mixed variational inequality. It has been shown [114, 1735] that a wide class of problems arising in elasticity, fluid flow through porous media and optimization can be studied in the general framework of problems (2.7) and (2.8).

If 𝜙()=0, then problem (2.7) is equivalent to finding 𝑢𝐾 such that 𝑇𝑢,𝜂(𝑣,𝑢)0,𝑣𝐾,(2.9) which is known as the variational-like inequality and has been studied extensively in recent years. For 𝜂(𝑣,𝑢)=𝑣𝑢, the variational-like inequality (2.9) reduces to the original variational inequality, which was introduced and studied by Stampacchi [32] in 1964. For the applications, numerical methods, dynamical system, and other aspects of variational inequalities and related optimization problems, see [135] and the references therein.

Definition 2.4. An operator 𝑇𝐾𝐾 is said to be(i)𝜂-Monotone, if and only if, 𝑇𝑢,𝜂(𝑣,𝑢)+𝑇𝑣,𝜂(𝑢,𝑣)0, for all  𝑢,𝑣𝐾.(ii)Partially relaxed strongly 𝜂-monotone, if there exists a constant 𝛼>0 such that𝑇𝑢,𝜂(𝑣,𝑢)+𝑇𝑧,𝜂(𝑢,𝑣)𝛼𝜂(𝑧,𝑢)2,𝑢,𝑣,𝑧𝐾.(2.10)
Note that for 𝑧=𝑣, partially relaxed strong 𝜂-monotonicity reduces to 𝜂-monotonicity of the operator 𝑇. For 𝜂(𝑣,𝑢)=𝑣𝑢, the invex set 𝐾 becomes the convex set and consequently Definition 2.4 collapses to the well concept of monotonicity and partial relaxed strongly monotonicity of the operator.

Assumption 2.5. Assume that the bifunction 𝜂𝐾×𝐾𝐻 satisfies the condition 𝜂(𝑢,𝑣)=𝜂(𝑢,𝑧)+𝜂(𝑧,𝑣),𝑢,𝑣,𝑧𝐾.(2.11) In particular, it follows that 𝜂(𝑢,𝑢)=0 and 𝜂(𝑢,𝑣)+𝜂(𝑢,𝑣)=0,𝑢,𝑣𝐻.(2.12) Assumption 2.5 has been used to suggest and analyze some iterative methods for various classes of variational-like inequalities.

3. Auxiliary Principle Technique/Principle of Iterative Regularization

In this section, we will discuss the solution of mixed variational-like inequality (2.7) using its regularized version. We will use auxiliary principle technique [14] coupled with principle of iterative regularization for solving the mixed variational-like inequalities.

For a given 𝑢𝐾 satisfying (2.7), we consider the problem of finding 𝑧𝐾 such that𝑇𝑤+𝐸(𝑤)𝐸(𝑢),𝜂(𝑣,𝑤)+𝜙(𝑣)𝜙(𝑤)0,𝑣𝐾.(3.1) Note that, if 𝑤=𝑢, then (3.1) reduces to (2.7). Using (3.1), we suggest an iterative scheme for solving (2.7). For a given 𝑢𝐾, consider the problem of finding a solution 𝑧𝐾 satisfying the auxiliary variational-like inequality𝜌𝑛𝑇𝑤+𝜀𝑛𝑤+𝐸(𝑤)𝐸(𝑢),𝜂(𝑣,𝑤)+𝜌𝑛𝜙(𝑣)𝜌𝑛𝜙(𝑤)0,𝑣𝐾,(3.2) where {𝜌𝑛}𝑛=1 be a sequence of positive real, and {𝜀𝑛}𝑛=1 be a decreasing sequence of positive real such that 𝜀𝑛0 as 𝑛. Clearly, if 𝑤=𝑢 and 𝜀𝑛0 as 𝑛, then 𝑤 is a solution of (2.7).

Now, we consider the regularized version of (2.7). For a fixed but arbitrary 𝑛𝑁 and for 𝜀𝑛>0, find 𝑢𝜀𝑛𝐾 such that𝑇𝑢𝜀𝑛+𝜀𝑛𝑢𝜀𝑛,𝜂𝑣,𝑢𝜀𝑛𝑢+𝜙(𝑣)𝜙𝜀𝑛0,𝑣𝐾.(3.3)

Algorithm 3.1. For a given 𝑢0𝐾, compute 𝑢𝑛+1𝐾 from the iterative scheme 𝜌𝑛𝑇𝑛𝑢𝑛+1+𝜀𝑛𝑢𝑛+1+𝐸𝑢𝑛+1𝐸𝑢𝑛,𝜂𝑣,𝑢𝑛+1+𝜌𝑛𝜙(𝑣)𝜌𝑛𝜙𝑢𝑛+10,𝑣𝐾,(3.4) where {𝜌𝑛}𝑛=1 be a sequence of positive real and {𝜀𝑛}𝑛=1 be a decreasing sequence of positive reals such that 𝜀𝑛0 as 𝑛.

We now study the convergence analysis of Algorithm 3.1.

Theorem 3.2. Let T be a monotone operator.For the approximation 𝑇𝑛 of 𝑇, assume that there exists {𝛿𝑛} such that 𝛿𝑛>0 such that 𝑇𝑛(𝑢)𝑇(𝑣)𝑐𝛿𝑛(1+𝜂(𝑢,𝑣)),𝑢𝐾,where𝑐isaconstant.(3.5) Also for the sequences {𝜀𝑛}, {𝛿𝑛}, and {𝜌𝑛}, one has 𝑛=0𝛿2𝑛<,𝑛=0𝜌2𝑛+𝛿2𝑛<,𝑛=0𝜀𝑛𝜌𝑛<,𝑛=0𝛼𝑛𝜌𝑛<.(3.6) Then the approximate solution 𝑢𝑛+1 obtained fromAlgorithm 3.1converges to an exact solution 𝑢𝐾 satisfying (2.7).

Proof. Let 𝑢𝜀𝑛𝐾 satisfying the regularized mixed variational-like inequality (3.3). Then replacing 𝑣 by 𝑢𝑛+1 in (3.3), we have 𝜌𝑛𝑇𝑢𝜀𝑛+𝜀𝑛𝑢𝜀𝑛𝑢,𝜂𝑛+1,𝑢𝜀𝑛+𝜌𝑛𝜙𝑢𝑛+1𝜌𝑛𝜙𝑢𝜀𝑛0.(3.7) Let 𝑢𝑛+1𝐾 be the approximate solution obtained from (3.4). Replacing 𝑣 by 𝑢𝜀𝑛, we have 𝜌𝑛𝑇𝑛𝑢𝑛+1+𝜀𝑛𝑢𝑛+1+𝐸𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1+𝜌𝑛𝜙𝑢𝜀𝑛𝜌𝑛𝜙𝑢𝑛+10.(3.8) For the sake of simplicity, we have 𝑇+𝜀𝑛=𝐹𝑛 and 𝐹𝑛+𝜀𝑛=𝐹𝑛 in (3.7) and (3.8), respectively, and then adding the resultant inequalities, we have 𝐸𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝜌𝑛𝐹𝑛𝑢𝜀𝑛𝜌𝑛𝐹𝑛𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1.(3.9) We consider the Bregman function: 𝐵𝐸(𝑢,𝑤)=𝐸(𝑢)𝐸(𝑤)(𝑢),𝜂(𝑤,𝑢)𝜇𝜂(𝑤,𝑢)2.(3.10) Now 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝑢=𝐸𝜀𝑛1𝑢𝐸𝑛𝐸𝑢𝑛𝑢,𝜂𝜀𝑛1,𝑢𝑛𝑢𝐸𝜀𝑛𝑢+𝐸𝑛+1+𝐸𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝑢=𝐸𝜀𝑛1𝑢𝐸𝜀𝑛+𝐸𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝑢+𝐸𝑛+1𝑢𝐸𝑛𝐸𝑢𝑛𝑢,𝜂𝜀𝑛1,𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝑛+1,𝑢𝑛𝑢𝐸𝜀𝑛1𝑢𝐸𝜀𝑛+𝐸𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛1,𝑢𝑛+1𝜂𝑢+𝜇𝑛+1,𝑢𝑛2𝑢𝐸𝜀𝑛1𝑢𝐸𝜀𝑛+𝐸𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1+𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝜀𝑛1𝜂𝑢+𝜇𝑛+1,𝑢𝑛2𝑢𝐸𝜀𝑛1𝑢𝐸𝜀𝑛+𝜌𝑛𝐹𝑛𝑢𝜀𝑛𝐹𝑛𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1+𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝜀𝑛1𝜂𝑢+𝜇𝑛+1,𝑢𝑛2𝜂𝑢𝜇𝑛+1,𝑢𝑛2𝜂𝑢+𝜇𝜀𝑛1,𝑢𝜀𝑛2+𝐸𝑢𝑛𝐸𝑢𝜀𝑛𝑢,𝜂𝜀𝑛,𝑢𝜀𝑛1+𝜌𝑛𝐹𝑛𝑢𝜀𝑛𝐹𝑛𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1.(3.11)
Since 𝑇 is a monotone operator, 𝐹𝑛=𝑇+𝜀𝑛 is strongly monotone with constant (𝛼+𝜀𝑛)=𝛼𝑛  (say), we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝜂𝑢𝜇𝑛+1,𝑢𝑛2𝜂𝑢+𝜇𝜀𝑛1,𝑢𝜀𝑛2+𝐸𝑢𝑛𝐸𝑢𝜀𝑛𝑢,𝜂𝜀𝑛,𝑢𝜀𝑛1+𝜌𝑛𝛼𝑛𝜂𝑢𝑛,𝑢𝜀𝑛2𝜌𝑛𝐹𝑛𝑢𝑛𝐹𝑛𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1,(3.12) from which, we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝜇𝜂(𝑢𝜀𝑛1,𝑢𝜀𝑛)2+𝜇𝜂(𝑢𝑛+1,𝑢𝑛)2+𝜌𝑛𝛼𝑛𝜂(𝑢𝜀𝑛,𝑢𝑛+1)2+𝜏1+𝜏2,(3.13) where 𝜏1=𝐸𝑢𝑛𝐸𝑢𝜀𝑛𝑢,𝜂𝜀𝑛,𝑢𝜀𝑛1.(3.14) Using Lemma  2.1and Lipschitz continuity of operator 𝐸, we have 𝛽2𝜀22𝜂(𝑢𝑛,𝑢𝜀𝑛)212𝜀2𝜂(𝑢𝜀𝑛1,𝑢𝜀𝑛)2.(3.15) Thus 𝜏1𝜀𝑛𝜌𝑛2𝜂𝑢𝑛,𝑢𝜀𝑛2𝛽22𝜀𝑛𝜌𝑛𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2.(3.16) Solving for 𝜏2, where 𝜏2𝜌𝑛𝐹𝑛𝑢𝑛+1𝐹𝑛𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝜀2𝜌𝑛2𝐹𝑛𝑢𝑛+1𝐹𝑛𝑢𝑛2𝜀2𝜌𝑛2𝐹𝑛𝑢𝑛𝐹𝑛𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12.(3.17) Using (3.5), we obtain 𝜏2𝑐2𝛿2𝑛𝜀2𝜌𝑛2𝜂𝑢1+𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12,(3.18) where we have used the Lipschitz continuity of 𝐹𝑛(=𝑇+𝜀𝑛) with constant 𝛾𝑛(=𝛾+𝜀𝑛).
Now using Assumption 2.5, we have 𝜏2𝑐2𝛿2𝑛𝜀2𝜌𝑛2𝜂𝑢1+𝑛,𝑢𝜀𝑛𝑢+𝜂𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12𝑐2𝛿2𝑛𝜀2𝜌𝑛2𝜂𝑢𝑡+𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12.=𝑐2𝛿2𝑛𝜀2𝜌𝑛𝑡2𝑐2𝛿2𝑛𝜀2𝜌𝑛𝜂𝑢𝜀𝑛,𝑢𝑛+12𝜂𝑢,𝑡1+𝑛,𝑢𝜀𝑛𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12.(3.19) From (3.13), (3.16) and (3.19), we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝜂𝑢𝜇𝑛+1,𝑢𝑛2𝜂𝑢+𝜇𝜀𝑛1,𝑢𝜀𝑛2+𝜌𝑛𝛼𝑛𝜂𝑢𝜀𝑛,𝑢𝑛+12𝜀𝑛𝜌𝑛2𝜂𝑢𝑛+1,𝑢𝜀𝑛2𝛽22𝜀𝑛𝜌𝑛𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2𝑐2𝛿2𝑛𝜀2𝜌𝑛𝑡2𝑐2𝛿2𝑛𝜀2𝜌𝑛𝜂𝑢𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12𝛾𝜇𝑛2𝜂𝑢𝑛+1,𝑢𝑛2+𝐶1𝜀𝑛𝜌𝑛𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2+𝜌𝑛𝛼𝑛𝜂𝑢𝑛,𝑢𝜀𝑛2𝜀𝑛𝜌𝑛2𝜂𝑢𝑛,𝑢𝜀𝑛2𝑐2𝛿2𝑛𝑡2𝛾𝑛𝐶2𝛿2𝑛+𝜌𝑛2𝜂𝑢𝜀𝑛,𝑢𝑛+12.(3.20) Using conditions (3.6), we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝛾𝜇𝑛2𝜂𝑢𝑛+1,𝑢𝑛2.(3.21) If 𝑢𝑛+1=𝑢𝑛, it is easily shown that 𝑢𝑛 is a solution of the variational-like inequality (2.7).
Otherwise, the assumption 𝛾𝑛>2𝜇 implies that 𝐵(𝑢𝜀𝑛1,𝑢𝑛)𝐵(𝑢𝜀𝑛,𝑢𝑛+1) is nonnegative and we must have lim𝑛𝜂𝑢𝑛+1,𝑢𝑛=0.(3.22) From (3.22), it follows that the sequence {𝑢𝑛} is bounded. Let ̂𝑢𝐾 be a cluster point of the sequence {𝑢𝑛}and let the subsequence {𝑢𝑛𝑖} of this sequence converges to ̂𝑢𝐾. Now essentially using the technique of Zhu and Marcotte [35], it can be shown that the entire sequence {𝑢𝑛} converges to the cluster point ̂𝑢𝐾 satisfying the variational-like inequality (2.7).

To implement the proximal method, one has to calculate the solution implicitly, which is itself a difficult problem. We again use the auxiliary principle technique to suggest another iterative method, the convergence of which requires only the partially relaxed strongly monotonicity of the operator. For this, we rewrite (3.1) as follows.

For a given 𝑢𝐾, consider the problem of finding 𝑧𝐾such that𝑇𝑢+𝐸(𝑧)𝐸(𝑢),𝜂(𝑣,𝑧)+𝜙(𝑣)𝜙(𝑧)0,𝑣𝐾.(3.23) Note that if 𝑧=𝑢, then (3.23) reduces to (2.7). Using (3.23), we develop an iterative scheme for solving (2.7).

For a given 𝑢𝐾, consider the problem of finding a solution 𝑧𝐾 satisfying the auxiliary variational-like inequality𝜌𝑛𝑇𝑢+𝜀𝑛𝑢+𝐸(𝑧)𝐸(𝑢),𝜂(𝑣,𝑧)+𝜌𝑛𝜙(𝑣)𝜌𝑛𝜙(𝑧)0,𝑣𝐾,(3.24) where {𝜌𝑛}𝑛=1 be a sequence of positive reals, and {𝜀𝑛}𝑛=1 be a decreasing sequence of positive reals such that 𝜀𝑛0 as 𝑛.

Note that if 𝑧=𝑢 and 𝜀𝑛0 as 𝑛, then 𝑧 is a solution of (2.7).

Algorithm 3.3. For a given 𝑢0𝐾, compute 𝑢𝑛+1𝐾 from the iterative scheme 𝜌𝑛𝑇𝑛𝑢𝑛+𝜀𝑛𝑢𝑛+𝐸𝑢𝑛+1𝐸𝑢𝑛,𝜂𝑣,𝑢𝑛+1+𝜌𝑛𝜙(𝑣)𝜌𝑛𝜙𝑢𝑛+10,𝑣𝐾,𝑛=0,1,2,,(3.25) where {𝜌𝑛}𝑛=1 be a sequence of positive and {𝜀𝑛}𝑛=1 be a decreasing sequence of positive such that 𝜀𝑛0 as 𝑛.

Using the technique of Theorem 3.2, one can prove the convergence of Algorithm 3.3. We include its proof for the sake of completeness.

Theorem 3.4. Let 𝑇 be a partially relaxed strongly monotone operator with constant 𝛼>0. For the approximation 𝑇𝑛 of 𝑇, let (3.5) holds. Also for the sequences {𝜀𝑛}, {𝛿𝑛} and {𝜌𝑛}, (3.6) is satisfied. Then the approximate solution 𝑢𝑛+1 obtained from Algorithm 3.3 converges to an exact solution 𝑢𝐾 satisfying (2.7).

Proof. Let 𝑢𝜀𝑛𝐾 satisfying the regularized mixed variational-like inequality (3.3), then replacing 𝑣 by 𝑢𝑛+1, we have 𝜌𝑛𝑇𝑢𝜀𝑛+𝜀𝑛𝑢𝜀𝑛𝑢,𝜂𝑛+1,𝑢𝜀𝑛+𝜌𝑛𝜙𝑢𝑛+1𝜌𝑛𝜙𝑢𝜀𝑛0.(3.26) Let 𝑢𝑛+1𝐾 be the approximate solution obtained from (3.25). Replacing 𝑣 by 𝑢𝜀𝑛, we have 𝜌𝑛𝑇𝑛𝑢𝑛+𝜀𝑛𝑢𝑛+𝐸𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1+𝜌𝑛𝜙𝑢𝜀𝑛𝜌𝑛𝜙𝑢𝑛+10.(3.27) For the sake of simplicity, we have 𝑇+𝜀𝑛=𝐹𝑛 and 𝑇𝑛+𝜀𝑛=𝐹𝑛 in (3.26) and (3.27), respectively, and then adding the resultant inequalities, we have 𝜌𝑛𝑇𝑛𝑢𝑛𝜌𝑛𝑇𝑛𝑢𝑛+𝐸𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+10,(3.28) from which, we have 𝐸𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝜌𝑛𝑇𝑛𝑢𝜀𝑛𝜌𝑛𝑇𝑛𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1.(3.29) We consider the Bregman function: 𝐵𝐸(𝑢,𝑤)=𝐸(𝑢)𝐸(𝑤)(𝑢),𝑤𝑢.(3.30) Now, we investigate the difference. Using the strongly preinvexity of 𝐸, we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝑢=𝐸𝜀𝑛1𝑢𝐸𝑛𝐸𝑢𝑛𝑢,𝜂𝜀𝑛1,𝑢𝑛𝑢𝐸𝜀𝑛𝑢+𝐸𝑛+1+𝐸𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝑢𝐸𝜀𝑛1𝑢𝐸𝜀𝑛+𝐸𝑢𝑛+1𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝐸𝑢𝑛𝑢,𝜂𝜀𝑛1,𝑢𝑛+1𝜂𝑢+𝜇𝑛+1,𝑢𝑛2.(3.31) Since 𝑇 is partially relaxed strongly monotone with constant 𝛼>0, 𝑇𝑛=𝑇+𝜀𝑛 is partially relaxed strongly monotone with constant (𝛼+𝜀𝑛/4)=𝛼𝑛  (say), we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝜂𝑢𝜇𝑛+1,𝑢𝑛2𝜂𝑢+𝜇𝜀𝑛1,𝑢𝜀𝑛2+𝐸𝑢𝑛𝐸𝑢𝜀𝑛𝑢,𝜂𝜀𝑛,𝑢𝜀𝑛1+𝜌𝑛𝛼𝑛𝜂𝑢𝑛,𝑢𝜀𝑛2𝜌𝑛𝑇𝑛𝑢𝑛𝑇𝑛𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1.(3.32) From which, we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝜂𝑢𝜇𝜀𝑛1,𝑢𝜀𝑛2𝜂𝑢+𝜇𝑛+1,𝑢𝑛2+𝜌𝑛𝛼𝑛𝜂𝑢𝑛,𝑢𝜀𝑛2+𝜏1+𝜏2,(3.33) where 𝜏1=𝐸𝑢𝑛𝐸𝑢𝜀𝑛𝑢,𝜂𝜀𝑛,𝑢𝜀𝑛1,𝐸=𝑢𝑛𝐸𝑢𝜀𝑛𝑢,𝜂𝜀𝑛1,𝑢𝜀𝑛𝜀22𝐸𝑢𝑛𝐸𝑢𝜀𝑛212𝜀2𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2.(3.34) Using Lipschitz continuity of operator 𝐸, we have 𝜏1𝛽2𝜀22𝜂𝑢𝑛,𝑢𝜀𝑛212𝜀2𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2.(3.35) Put 𝜀=𝜀𝑛𝜌𝑛/𝛽2, we have 𝜏1𝜀𝑛𝜌𝑛2𝜂𝑢𝑛,𝑢𝜀𝑛2𝛽22𝜀𝑛𝜌𝑛𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2.(3.36) Solving for 𝜏2, where 𝜏2𝜌𝑛𝑇𝑛𝑢𝑛𝑇𝑛𝑢𝑛𝑢,𝜂𝜀𝑛,𝑢𝑛+1𝜀2𝜌𝑛2𝑇𝑛𝑢𝑛𝑇𝑛𝑢𝑛2𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛2𝑇𝑛𝑢𝑛𝑇𝑛𝑢𝑛+12𝜀2𝜌𝑛2𝑇𝑛𝑢𝑛+1𝑇𝑛𝑢𝑛2𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12.(3.37) Using (3.5), we obtain 𝜏2𝑐2𝛿2𝑛𝜀2𝜌𝑛2𝜂𝑢1+𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12,(3.38) where we have used the Lipschitz continuity of 𝑇𝑛(=𝑇+𝜀𝑛) with constant 𝛾𝑛(=𝛾+𝜀𝑛).
Now using Assumption 2.5, we have, for any 𝑡1+𝜂(𝑢𝑛,𝑢𝜀𝑛), 𝜏2𝑐2𝛿2𝑛𝜀2𝜌𝑛2𝜂𝑢1+𝑛,𝑢𝜀𝑛𝑢+𝜂𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12𝑐2𝛿2𝑛𝜀2𝜌𝑛2𝜂𝑢𝑡+𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12=𝑐2𝛿2𝑛𝜀2𝜌𝑛𝑡22𝑐2𝛿2𝑛𝜀2𝜌𝑛2𝜂𝑢𝜀𝑛,𝑢𝑛+12+𝑐2𝛿2𝑛𝜀2𝜌𝑛212𝑡212𝜂𝑢𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12=𝑐2𝛿2𝑛𝜀2𝜌𝑛𝑡2𝑐2𝛿2𝑛𝜀2𝜌𝑛𝜂𝑢𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12.(3.39) Combining all the results above, we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝜂𝑢𝜇𝑛+1,𝑢𝑛2𝜂𝑢+𝜇𝑛+1,𝑢𝑛2+𝜌𝑛𝛼𝑛𝜂𝑢𝑛,𝑢𝜀𝑛2𝜀𝑛𝜌𝑛2𝜂𝑢𝑛,𝑢𝜀𝑛2𝛽22𝜀𝑛𝜌𝑛𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2𝑐2𝛿2𝑛𝜀2𝜌𝑛𝑡2𝑐2𝛿2𝑛𝜀2𝜌𝑛𝜂𝑢𝜀𝑛,𝑢𝑛+12𝜀2𝜌𝑛𝛾2𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝜀2𝜂𝑢𝜀𝑛,𝑢𝑛+12.(3.40) Taking 𝜀2=1/𝛾𝑛𝜌𝑛, we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝜂𝑢𝜇𝑛+1,𝑢𝑛2𝜂𝑢+𝜇𝜀𝑛1,𝑢𝜀𝑛2+𝜌𝑛𝛼𝑛𝜂𝑢𝑛,𝑢𝜀𝑛2𝜀𝑛𝜌𝑛2𝜂𝑢𝑛,𝑢𝜀𝑛2𝛽22𝜀𝑛𝜌𝑛𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2𝑐2𝛿2𝑛𝑡2𝛾𝑛𝑐2𝛿2𝑛𝛾𝑛𝜂𝑢𝜀𝑛,𝑢𝑛+12𝛾𝑛2𝜂𝑢𝑛,𝑢𝑛+12𝜌𝑛2𝛾𝑛2𝜂𝑢𝜀𝑛,𝑢𝑛+12𝛾𝜇𝑛2𝜂𝑢𝑛+1,𝑢𝑛2+𝐶1𝜀𝑛𝜌𝑛𝜂𝑢𝜀𝑛1,𝑢𝜀𝑛2+𝜌𝑛𝛼𝑛𝜂𝑢𝑛,𝑢𝜀𝑛2𝜀𝑛𝜌𝑛2𝜂𝑢𝑛,𝑢𝜀𝑛2𝑐2𝛿2𝑛𝑡2𝛾𝑛𝐶2𝛿2𝑛+𝜌𝑛2𝜂𝑢𝜀𝑛,𝑢𝑛+12.(3.41) Using conditions (3.6), we have 𝐵𝑢𝜀𝑛1,𝑢𝑛𝑢𝐵𝜀𝑛,𝑢𝑛+1𝛾𝜇𝑛2𝜂𝑢𝑛+1,𝑢𝑛2.(3.42) If 𝑢𝑛+1=𝑢𝑛, then it can easily shown that 𝑢𝑛 is a solution of the variational-like inequality (2.7). Otherwise, the assumption 𝛾𝑛>2𝜇 implies that 𝐵(𝑢𝜀𝑛1,𝑢𝑛)𝐵(𝑢𝜀𝑛,𝑢𝑛+1) is nonnegative and we must have lim𝑛𝜂𝑢𝑛+1,𝑢𝑛=0.(3.43) From (3.43), it follows that the sequence {𝑢𝑛} is bounded. Let ̂𝑢𝐾 be a cluster point of the sequence {𝑢𝑛} and let the subsequence {𝑢𝑛𝑖} of this sequence converges to ̂𝑢𝐾. Now essentially using the technique of Zhu and Marcotte [35], it can be shown that the entire sequence {𝑢𝑛} converges to the cluster point ̂𝑢𝐾 satisfying the variational-like inequality (2.7).

4. Conclusion

In this paper, we have suggested and analyzed some new iterative methods for solving the regularized mixed variational-like inequalities. We have also discussed the convergence analysis of the suggested iterative methods under some suitable and weak conditions. Results proved in this are new and original ones. We hope to extend the idea and technique of this paper for solving invex equilibrium problems and this is the subject of another paper.

Acknowledgments

This research is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia. The authors are also grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan for providing the excellent research facilities.

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