# Regularized Mixed Variational-Like Inequalities

Authors: Muhammad Aslam Noor; Khalida Inayat Noor; Saira Zainab; Eisa Al-Said
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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 [1–3] 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 [4–7]  and  Noor et  al. [8–13].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 bifunctionη(·,·). 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, 4–34] 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

LetH 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 f:K→R and η(·,·):K×K→H be mappings. First of all, we recall the following well-known results and concepts; see [1–3, 21, 33].Definition 2.1.
Letu∈K. Then the set K is said to be invex at u with respect to η(·,·), if
(2.1)u+tη(v,u)∈K,∀u,v∈K,t∈[0,1].
K is said to be an invex set with respect to η(·,·), if K is invex at each u∈K. The invex set K is also called η-connected set. Clearly, every convex set is an invex set with η(v,u)=v-u,  for all u,v∈K, but the converse is not true; see [3, 33].From now onwards,Kis a nonempty closed and invex set inHwith respect toη(·,·), unless otherwise specified.Definition 2.2.
A functionf:K→R is said to be preinvex with respect to η(·,·), if
(2.2)f(u+tη(v,u))≤(1-t)f(u)+tf(v),∀u,v∈K,t∈[0,1].
Note that every convex function is a preinvex function, but the converse is not true; see [3, 33].Definition 2.3.
A functionf is said to be a strongly preinvex function on K with respect to the function η(·,·) with modulus μ, if
(2.3)f(u+tη(v,u))≤(1-t)f(u)+tf(v)-t(1-t)μ‖η(v,u)‖2,∀v,u∈K,t∈[0,1].
Clearly, a differentiable strongly preinvex functionf is a strongly invex function with constant μ>0, that is,
(2.4)f(v)-f(u)≥〈f′(u),η(v,u)〉+μ‖η(v,u)‖2,∀v,u∈K,
and the converse is also true under certain conditions.We remark that  ift=1, then  Definitions 2.2 and 2.3 reduce  to(2.5)f(u+η(v,u))≤f(v),∀u,v∈K.
One can easily show that  the minimum of the differentiable  preinvex function  on the invex set K  is equivalent to finding  u∈K such that   (2.6)〈f′(u),η(v,u)〉≥0,∀v∈K,
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.LetK be a nonempty closed and invex set in H. For given nonlinear operator T:K→H and a continuous function ϕ(·), we consider the problem of finding u∈K such that(2.7)〈Tu,η(v,u)〉+ϕ(v)-ϕ(u)≥0,∀v∈K,
which  is called the mixed variational-like inequalityintroduced and studied by [1]. It has been shown in [1–3] 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 ifη(v,u)=v-u, then the invex set K becomes the convex set K and problem (2.7) is equivalent to finding u∈K such that(2.8)〈Tu,v-u〉+ϕ(v)-ϕ(u)≥0,∀v∈K,
which is known as a mixed variational inequality. It has been shown [1–14, 17–35] 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).Ifϕ(·)=0,   then problem (2.7) is equivalent to finding  u∈K  such that   (2.9)〈Tu,η(v,u)〉≥0,∀v∈K,
which is known as the  variational-like inequality and has been studied extensively  in recent years.  For  η(v,u)=v-u, 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  [1–35] and  the references therein.Definition 2.4.
An operatorT:K→K is said to be(i)
η-Monotone, if and only if, 〈Tu,η(v,u)〉+〈Tv,η(u,v)〉≤0, for all  u,v∈K.(ii)
Partially relaxed stronglyη-monotone, if there exists a constant α>0 such that(2.10)〈Tu,η(v,u)〉+〈Tz,η(u,v)〉≤α‖η(z,u)‖2,∀u,v,z∈K.
Note that forz=v, partially relaxed strong η-monotonicity reduces to η-monotonicity of the operator T. For η(v,u)=v-u, 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 bifunctionη:K×K→H satisfies the condition
(2.11)η(u,v)=η(u,z)+η(z,v),∀u,v,z∈K.
In particular, it follows that η(u,u)=0  and
(2.12)η(u,v)+η(u,v)=0,∀u,v∈H.
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 givenu∈K satisfying  (2.7), we consider the problem of finding z∈K such that(3.1)〈Tw+E′(w)-E′(u),η(v,w)〉+ϕ(v)-ϕ(w)≥0,∀v∈K.
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 u∈K, consider the problem of finding a solution z∈K satisfying the auxiliary variational-like  inequality(3.2)ρn〈Tw+εnw+E′(w)-E′(u),η(v,w)〉+ρnϕ(v)-ρnϕ(w)≥0,∀v∈K,
where {ρn}n=1∞ be a sequence of positive real, and {εn}n=1∞ be a decreasing sequence of positive real such that εn→0 as n→∞. Clearly,  if  w=u and εn→0 as n→∞, then w is a solution of (2.7).Now, we consider the regularized version of (2.7). For a fixed but arbitrary n∈N and for εn>0, find uεn∈K such that(3.3)〈Tuεn+εnuεn,η(v,uεn)〉+ϕ(v)-ϕ(uεn)≥0,∀v∈K.Algorithm 3.1.
For a givenu0∈K, compute un+1∈K from the iterative scheme
(3.4)〈ρn(Tnun+1+εnun+1)+E′(un+1)-E′(un),η(v,un+1)〉+ρnϕ(v)-ρnϕ(un+1)≥0,∀v∈K,
where {ρn}n=1∞ be a sequence of positive real and {εn}n=1∞ be a decreasing sequence of positive reals such that εn→0 as n→∞.We now study the convergence analysis of Algorithm3.1.Theorem 3.2.
Let T  be a monotone operator.For the approximationTn of T, assume that there exists {δn} such that δn>0 such that
(3.5)‖Tn(u)-T(v)‖≤cδn(1+‖η(u,v)‖),∀u∈K,wherecisaconstant.
Also for the sequences {εn}, {δn}, and {ρn}, one has
(3.6)∑n=0∞δn2<∞,∑n=0∞(ρn2+δn2)<∞,∑n=0∞εnρn<∞,∑n=0∞αnρn<∞.
Then the approximate solution un+1 obtained fromAlgorithm 3.1converges to an exact solution u∈K satisfying (2.7).Proof.
Letuεn∈K satisfying the regularized mixed variational-like inequality (3.3).  Then replacing v by un+1 in (3.3), we have
(3.7)〈ρn(Tuεn+εnuεn),η(un+1,uεn)〉+ρnϕ(un+1)-ρnϕ(uεn)≥0.
Let un+1∈K be the approximate solution obtained from (3.4). Replacing v by uεn, we have
(3.8)〈ρn(Tnun+1+εnun+1)+E′(un+1)-E′(un),η(uεn,un+1)〉+ρnϕ(uεn)-ρnϕ(un+1)≥0.
For the sake of simplicity, we have T+εn=Fn and Fn+εn=F̃n in (3.7) and (3.8), respectively, and then adding the resultant inequalities, we have
(3.9)〈E′(un+1)-E′(un),η(uεn,un+1)〉≥〈ρnFnuεn-ρnF̃nun+1,η(uεn,un+1)〉.
We consider the Bregman function:
(3.10)B(u,w)=E(u)-E(w)-〈E′(u),η(w,u)〉≥μ‖η(w,u)‖2.
Now
(3.11)B(uεn-1,un)-B(uεn,un+1)=E(uεn-1)-E(un)-〈E′(un),η(uεn-1,un)〉-E(uεn)+E(un+1)+〈E′(un+1),η(uεn,un+1)〉=E(uεn-1)-E(uεn)+〈E′(un+1),η(uεn,un+1)〉+E(un+1)-E(un)-〈E′(un),η(uεn-1,un+1)〉-〈E′(un),η(un+1,un)〉≥E(uεn-1)-E(uεn)+〈E′(un+1),η(uεn,un+1)〉-〈E′(un),η(uεn-1,un+1)〉+μ‖η(un+1,un)‖2≥E(uεn-1)-E(uεn)+〈E′(un+1)-E′(un),η(uεn,un+1)〉+〈E′(un),η(uεn,uεn-1)〉+μ‖η(un+1,un)‖2≥E(uεn-1)-E(uεn)+ρn〈Fnuεn-F̃nun+1,η(uεn,un+1)〉+〈E′(un),η(uεn,uεn-1)〉+μ‖η(un+1,un)‖2≥μ‖η(un+1,un)‖2+μ‖η(uεn-1,uεn)‖2+〈E′(un)-E′(uεn),η(uεn,uεn-1)〉+ρn〈Fnuεn-F̃nun+1,η(uεn,un+1)〉.
SinceT is a monotone operator, Fn=T+εn is strongly monotone with constant (α+εn)=αn  (say), we have
(3.12)B(uεn-1,un)-B(uεn,un+1)≥μ‖η(un+1,un)‖2+μ‖η(uεn-1,uεn)‖2+〈E′(un)-E′(uεn),η(uεn,uεn-1)〉+ρnαn‖η(un,uεn)‖2-ρn〈F̃nun-Fnun+1,η(uεn,un+1)〉,
from which, we have
(3.13)B(uεn-1,un)-B(uεn,un+1)≥μ‖η(uεn-1,uεn)‖2+μ‖η(un+1,un)‖2+ρnαn‖η(uεn,un+1)‖2+τ1+τ2,
where
(3.14)τ1=〈E′(un)-E′(uεn),η(uεn,uεn-1)〉.
Using Lemma  2.1and Lipschitz continuity of operator E′, we have
(3.15)≥-β2ε22‖η(un,uεn)‖2-12ε2‖η(uεn-1,uεn)‖2.
Thus
(3.16)τ1≥-εnρn2‖η(un,uεn)‖2-β22εnρn‖η(uεn-1,uεn)‖2.
Solving for τ2, where
(3.17)τ2≥-ρn〈F̃nun+1-Fnun+1,η(uεn,un+1)〉≥-ε2ρn2‖F̃nun+1-Fnun‖2-ε2ρn2‖Fnun-Fnun+1‖2-ρn2ε2‖η(uεn,un+1)‖2.
Using (3.5), we obtain
(3.18)τ2≥-c2δn2ε2ρn2[1+‖η(un,un+1)‖]2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2,
where  we  have used the  Lipschitz continuity of Fn(=T+εn) with constant γn(=γ+εn).
Now using Assumption2.5, we have
(3.19)τ2≥-c2δn2ε2ρn2[1+‖η(un,uεn)+η(uεn,un+1)‖]2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2≥-c2δn2ε2ρn2[t+‖η(uεn,un+1)‖]2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2.=-c2δn2ε2ρnt2-c2δn2ε2ρn‖η(uεn,un+1)‖2,t≥1+‖η(un,uεn)‖-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2.
From (3.13), (3.16) and (3.19), we have
(3.20)B(uεn-1,un)-B(uεn,un+1)≥μ‖η(un+1,un)‖2+μ‖η(uεn-1,uεn)‖2+ρnαn‖η(uεn,un+1)‖2-εnρn2‖η(un+1,uεn)‖2-β22εnρn‖η(uεn-1,uεn)‖2-c2δn2ε2ρnt2-c2δn2ε2ρn‖η(uεn,un+1)‖2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2≥(μ-γn2)‖η(un+1,un)‖2+C1εnρn‖η(uεn-1,uεn)‖2+ρnαn‖η(un,uεn)‖2-εnρn2‖η(un,uεn)‖2-c2δn2t2γn-C2(δn2+ρn2)‖η(uεn,un+1)‖2.
Using conditions (3.6), we have
(3.21)B(uεn-1,un)-B(uεn,un+1)≥(μ-γn2)‖η(un+1,un)‖2.
If un+1=un, it is easily shown that un is a solution of the variational-like inequality (2.7).
Otherwise, the assumptionγn>2μ implies that B(uεn-1,un)-B(uεn,un+1) is nonnegative and we must have
(3.22)lim⁡n→∞‖η(un+1,un)‖=0.
From (3.22), it follows that the sequence {un} is bounded. Let û∈K be a cluster point of the sequence {un}and let the subsequence {uni} of this sequence converges to û∈K. Now essentially using the technique of Zhu and Marcotte [35], it can be shown that the entire sequence {un} converges to the cluster point  û∈K 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 givenu∈K, consider the problem of finding  z∈Ksuch that(3.23)〈Tu+E′(z)-E′(u),η(v,z)〉+ϕ(v)-ϕ(z)≥0,∀v∈K.
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 givenu∈K, consider the problem of finding a solution z∈K satisfying the auxiliary variational-like   inequality(3.24)ρn〈Tu+εnu+E′(z)-E′(u),η(v,z)〉+ρnϕ(v)-ρnϕ(z)≥0,∀v∈K,
where {ρn}n=1∞ be a sequence of positive reals, and {εn}n=1∞ be a decreasing sequence of positive reals such that εn→0 as n→∞.Note that ifz=u and εn→0 as n→∞, then z is a solution of (2.7).Algorithm 3.3.
For a givenu0∈K, compute un+1∈K from the iterative scheme
(3.25)〈ρn(Tnun+εnun)+E′(un+1)-E′(un),η(v,un+1)〉+ρnϕ(v)-ρnϕ(un+1)≥0,∀v∈K,n=0,1,2,…,
where {ρn}n=1∞ be a sequence of positive  and {εn}n=1∞ be a decreasing sequence of positive such that εn→0 as n→∞.Using the technique of Theorem3.2, one can prove the convergence of Algorithm 3.3. We include its proof for the sake of completeness.Theorem 3.4.
LetT 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 u∈K satisfying (2.7).Proof.
Letuεn∈K satisfying the regularized mixed variational-like  inequality (3.3), then replacing v by un+1, we have
(3.26)〈ρn(Tuεn+εnuεn),η(un+1,uεn)〉+ρnϕ(un+1)-ρnϕ(uεn)≥0.
Let un+1∈K be the approximate solution obtained from (3.25). Replacing v by uεn, we have
(3.27)〈ρn(Tnun+εnun)+E′(un+1)-E′(un),η(uεn,un+1)〉+ρnϕ(uεn)-ρnϕ(un+1)≥0.
For the sake of simplicity, we have T+εn=Fn and Tn+εn=F̃n in (3.26) and (3.27), respectively, and then adding the resultant inequalities, we have
(3.28)〈ρnT̃nun-ρnTnun+E′(un+1)-E′(un),η(uεn,un+1)〉≥0,
from which, we have
(3.29)〈E′(un+1)-E′(un),η(uεn,un+1)〉≥〈ρnTnuεn-ρnT̃nun,η(uεn,un+1)〉.
We consider the Bregman function:
(3.30)B(u,w)=E(u)-E(w)-〈E′(u),w-u〉.
Now, we investigate the difference. Using the strongly preinvexity  of E,  we have
(3.31)B(uεn-1,un)-B(uεn,un+1)=E(uεn-1)-E(un)-〈E′(un),η(uεn-1,un)〉-E(uεn)+E(un+1)+〈E′(un+1),η(uεn,un+1)〉≥E(uεn-1)-E(uεn)+〈E′(un+1),η(uεn,un+1)〉-〈E′(un),η(uεn-1,un+1)〉+μ‖η(un+1,un)‖2.
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
(3.32)B(uεn-1,un)-B(uεn,un+1)≥μ‖η(un+1,un)‖2+μ‖η(uεn-1,uεn)‖2+〈E′(un)-E′(uεn),η(uεn,uεn-1)〉+ρnαn‖η(un,uεn)‖2-ρn〈Tñun-Tnun,η(uεn,un+1)〉.
From which, we have
(3.33)B(uεn-1,un)-B(uεn,un+1)≥μ‖η(uεn-1,uεn)‖2+μ‖η(un+1,un)‖2+ρnαn‖η(un,uεn)‖2+τ1+τ2,
where
(3.34)τ1=〈E′(un)-E′(uεn),η(uεn,uεn-1)〉,=-〈E′(un)-E′(uεn),η(uεn-1,uεn)〉≥-ε22‖E′(un)-E′(uεn)‖2-12ε2‖η(uεn-1,uεn)‖2.
Using Lipschitz continuity of operator E′, we have
(3.35)τ1≥-β2ε22‖η(un,uεn)‖2-12ε2‖η(uεn-1,uεn)‖2.
Put ε=εnρn/β2, we have
(3.36)τ1≥-εnρn2‖η(un,uεn)‖2-β22εnρn‖η(uεn-1,uεn)‖2.
Solving for τ2, where
(3.37)τ2≥-ρn〈Tñun-Tnun,η(uεn,un+1)〉≥-ε2ρn2‖Tñun-Tnun‖2-ρn2ε2‖η(uεn,un+1)‖2≥-ε2ρn2‖Tñun-Tnun+1‖2-ε2ρn2‖Tnun+1-Tnun‖2-ρn2ε2‖η(uεn,un+1)‖2.
Using (3.5), we obtain
(3.38)τ2≥-c2δn2ε2ρn2[1+‖η(un,un+1)‖]2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2,
where we have used the  Lipschitz continuity of Tn(=T+εn) with constant γn(=γ+εn).
Now using Assumption2.5, we have, for any t≥1+∥η(un,uεn)∥,
(3.39)τ2≥-c2δn2ε2ρn2[1+‖η(un,uεn)+η(uεn,un+1)‖]2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2≥-c2δn2ε2ρn2[t+‖η(uεn,un+1)‖]2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2=-c2δn2ε2ρnt22-c2δn2ε2ρn2‖η(uεn,un+1)‖2+c2δn2ε2ρn2[-12‖t‖2-12‖η(uεn,un+1)‖2]-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2=-c2δn2ε2ρnt2-c2δn2ε2ρn‖η(uεn,un+1)‖2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2.
Combining all the results above, we have
(3.40)B(uεn-1,un)-B(uεn,un+1)≥μ‖η(un+1,un)‖2+μ‖η(un+1,un)‖2+ρnαn‖η(un,uεn)‖2-εnρn2‖η(un,uεn)‖2-β22εnρn‖η(uεn-1,uεn)‖2-c2δn2ε2ρnt2-c2δn2ε2ρn‖η(uεn,un+1)‖2-ε2ρnγn22‖η(un,un+1)‖2-ρn2ε2‖η(uεn,un+1)‖2.
Taking ε2=1/γnρn, we have
(3.41)B(uεn-1,un)-B(uεn,un+1)≥μ‖η(un+1,un)‖2+μ‖η(uεn-1,uεn)‖2+ρnαn‖η(un,uεn)‖2-εnρn2‖η(un,uεn)‖2-β22εnρn‖η(uεn-1,uεn)‖2-c2δn2t2γn-c2δn2γn‖η(uεn,un+1)‖2-γn2‖η(un,un+1)‖2-ρn2γn2‖η(uεn,un+1)‖2≥(μ-γn2)‖η(un+1,un)‖2+C1εnρn‖η(uεn-1,uεn)‖2+ρnαn‖η(un,uεn)‖2-εnρn2‖η(un,uεn)‖2-c2δn2t2γn-C2(δn2+ρn2)‖η(uεn,un+1)‖2.
Using conditions (3.6), we have
(3.42)B(uεn-1,un)-B(uεn,un+1)≥(μ-γn2)‖η(un+1,un)‖2.
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(uεn-1,un)-B(uεn,un+1) is nonnegative and we must have
(3.43)lim⁡n→∞‖η(un+1,un)‖=0.
From (3.43), it follows that the sequence {un} is bounded. Let û∈K be a cluster point of the sequence {un} and let the subsequence {uni} of this sequence converges to û∈K. Now essentially using the technique of Zhu and Marcotte [35], it can be shown that the entire sequence {un} converges to the cluster point  û∈K 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.



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*Source: 863450-2012-02-20.xml*