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Some remarks on regularized nonconvex variational inequalities
Journal of Inequalities and Applications volume 2013, Article number: 531 (2013)
Abstract
In this paper, we investigate and analyze the nonconvex variational inequalities introduced by Noor in (Optim. Lett. 3:411418, 2009) and (Comput. Math. Model. 21:97108, 2010) and prove that the algorithms and results in the above mentioned papers are not valid. To overcome the problems in the above cited papers, we introduce and consider a new class of variational inequalities, named regularized nonconvex variational inequalities, instead of the class of nonconvex variational inequalities introduced in the above mentioned papers. We also consider a class of nonconvex WienerHopf equations and establish the equivalence between the regularized nonconvex variational inequalities and the fixed point problems as well as the nonconvex WienerHopf equations. By using the obtained equivalence formulations, we prove the existence of a unique solution for the regularized nonconvex variational inequalities and propose some projection iterative schemes for solving the regularized nonconvex variational inequalities. We also study the convergence analysis of the suggested iterative schemes under some certain conditions.
MSC:47H05, 47J20, 49J40, 90C33.
1 Introduction
Variational inequality theory, 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, network and structural analysis, elasticity and optimization. Many research papers have been written lately, both on the theory and applications of this field. Important connections with main areas of pure and applied sciences have been made; see, for example, [2–4] and the references cited therein. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solution to important classes of problems; on the other hand, it also enables us to develop highly efficient and powerful new numerical methods to solve, for example, obstacle, unilateral, free, moving and complex equilibrium problems. One of the most interesting and important problems in variational inequality theory is the development of an efficient numerical method. There is a substantial number of numerical methods including projection method and its variant forms, WienerHopf (normal) equations, auxiliary principle, and descent framework for solving variational inequalities and complementarity problems. For the applications, physical formulations, numerical methods and other aspects of variational inequalities, see [1–22] and the references therein.
The projection method and its variant forms represent an important tool for finding an approximate solution of various types of variational and quasivariational inequalities, the origin of which can be traced back to Lions and Stampacchia [19]. The projectiontype methods were developed in 1970s and 1980s. The main idea in this technique is to establish the equivalence between the variational inequalities and the fixed point problems using the concept of projection. This alternative formulation enables us to suggest some iterative methods for computing an approximate solution.
It is worth mentioning that most of the results regarding the existence and iterative approximation of solutions to variational inequality problems have been investigated and considered so far to the case where the underlying set is a convex set. Recently, the concept of convex set has been generalized in many directions, which has potential and important applications in various fields. It is well known that the uniformly proxregular sets are nonconvex and include the convex sets as special cases. This class of uniformly proxregular sets has played an important part in many nonconvex applications such as optimization, dynamic systems and differential inclusions. For more details, see, for example, [9, 10, 13, 14, 21, 23, 24].
Very recently, Noor [25, 26] has introduced and considered a new class of variational inequalities, the socalled nonconvex variational inequalities (NVI) on the uniformly proxregular sets. He has also introduced a class of WienerHopf equations in [26]. The author has asserted that NVI (2.1) from [25, 26] is equivalent to the fixed point problem (3.1) from [25, 26] as well as the WienerHopf equation (2.5) from [26]. Then, he used the fixed point formulation (3.1) from [25, 26] and the equivalence formulations (4.1) and (4.2) from [26], and suggested some iterative schemes for solving NVI (2.1) from [25, 26]. He also studied the convergence analysis of the suggested iterative methods under certain conditions.
In this paper, we establish that the equivalence formulation (3.1), used by Noor in [25, 26], is not correct. That is, Lemma 3.1 in [25, 26], which is the main tool to suggest the algorithms and to prove the strong convergence of the sequences generated by the proposed iterative algorithms in [25, 26], is incorrect. Consequently, the algorithms and results in [25, 26] are not valid. To overcome these problems in [25, 26], we introduce and consider a new class of variational inequalities, termed the regularized nonconvex variational inequalities (RNVI), instead of the class of NVI (2.1) from [25, 26]. We also consider a class of nonconvex WienerHopf equations (NWHE) and establish the equivalence between RNVI and the fixed point problems as well as NWHE. By using the obtained equivalence formulations, we prove the existence of a unique solution for RNVI and propose some projection iterative schemes for solving RNVI. We also study the convergence analysis of the suggested iterative schemes under some certain conditions.
2 Preliminaries and basic results
Throughout this article, we let ℋ be a real Hilbert space which is equipped with an inner product \u3008\cdot ,\cdot \u3009 and the corresponding norm \parallel \cdot \parallel and K be a nonempty and closed subset of ℋ. We denote by {d}_{K}(\cdot ) or d(\cdot ,K) the usual distance function to the subset K, i.e., {d}_{K}(u)={inf}_{v\in K}\parallel uv\parallel. Let us recall the following wellknown definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis [12–14, 21].
Definition 2.1 Let u\in \mathcal{H} be a point not lying in K. A point v\in K is called a closest point or a projection of u onto K if {d}_{K}(u)=\parallel uv\parallel. The set of all such closest points is denoted by {P}_{K}(u), i.e.,
Definition 2.2 The proximal normal cone of K at a point u\in K is given by
Clarke et al. [13], in Proposition 1.1.5, give characterization of {N}_{K}^{P}(u) as follows.
Lemma 2.1 Let K be a nonempty closed subset in ℋ. Then \xi \in {N}_{K}^{P}(u) if and only if there exists a constant \alpha =\alpha (\xi ,u)>0 such that the following proximal normal inequality holds:
Definition 2.3 Let X be a real Banach space and f:X\to \mathbb{R} be Lipschitz with constant τ near a given point x\in X; that is, for some \epsilon >0, we have f(y)f(z)\le \tau \parallel yz\parallel for all y,z\in B(x;\epsilon ), where B(x;\epsilon ) denotes the open ball of radius \epsilon >0 and centered at x. The generalized directional derivative of f at x in the direction v, denoted as {f}^{\circ}(x;v), is defined as follows:
where y is a vector in X and t is a positive scalar.
The generalized directional derivative defined earlier can be used to develop a notion of tangency that does not require K to be smooth or convex.
Definition 2.4 The tangent cone {T}_{K}(x) to K at a point x in K is defined as follows:
Having defined a tangent cone, the likely candidate for the normal cone is the one obtained from {T}_{K}(x) by polarity. Accordingly, we define the normal cone of K at x by polarity with {T}_{K}(x) as follows:
In 1995, Clarke et al. [14] introduced and studied a new class of nonconvex sets, called proximally smooth sets; subsequently, Poliquin et al. in [21] investigated the aforementioned sets under the name of uniformly proxregular sets. These have been successfully used in many nonconvex applications in areas such as optimization, economic models, dynamical systems, differential inclusions, etc. For such applications, see [6–8, 10]. This class seems particularly well suited to overcome the difficulties which arise due to the nonconvexity assumptions on K. We take the following characterization proved in [14] as a definition of this class. We point out that the original definition was given in terms of the differentiability of the distance function, see [14].
Definition 2.5 For any r\in (0,+\mathrm{\infty}], a subset {K}_{r} of ℋ is called normalized uniformly proxregular (or uniformly rproxregular [21]) if every nonzero proximal normal to {K}_{r} can be realized by an rball. This means that for all \overline{x}\in {K}_{r} and all 0\ne \xi \in {N}_{{K}_{r}}^{P}(\overline{x}),
Obviously, the class of normalized uniformly proxregular sets is sufficiently large to include the class of convex sets, pconvex sets, {C}^{1,1} submanifolds (possibly with boundary) of ℋ, the images under a {C}^{1,1} diffeomorphism of convex sets and many other nonconvex sets, see [11, 14].
Lemma 2.2 [14]
A closed set K\subseteq \mathcal{H} is convex if and only if it is proximally smooth of radius r for every r>0.
If r=+\mathrm{\infty}, then in view of Definition 2.5 and Lemma 2.2, the uniform rproxregularity of {K}_{r} is equivalent to the convexity of {K}_{r}, which makes this class of great importance. For the case of r=+\mathrm{\infty}, we set {K}_{r}=K.
The following proposition summarizes some important consequences of the uniform proxregularity needed in the sequel. The proof of this result can be found in [14, 21].
Proposition 2.1 Let r>0 and {K}_{r} be a nonempty closed and uniformly rproxregular subset of ℋ. Set U(r)=\{u\in \mathcal{H}:{d}_{{K}_{r}}(u)<r\}. Then the following statements hold:

(a)
For all x\in U(r), {P}_{{K}_{r}}(x)\ne \mathrm{\varnothing};

(b)
For all {r}^{\prime}\in (0,r), {P}_{{K}_{r}} is Lipschitz continuous with constant \frac{r}{r{r}^{\prime}} on U({r}^{\prime})=\{u\in \mathcal{H}:{d}_{{K}_{r}}(u)<{r}^{\prime}\}.
In order to make clear the concept of rproxregular sets, we state the following concrete example: The union of two disjoint intervals [a,b] and [c,d] is rproxregular with r=\frac{cb}{2}, see [9, 13, 21]. The finite union of disjoint intervals is also rproxregular and r depends on the distances between the intervals.
3 Remarks on nonconvex variational inequalities
Let {K}_{r} be a uniformly rproxregular subset of ℋ. For a given nonlinear singlevalued operator T:\mathcal{H}\to \mathcal{H}, Noor [25, 26] introduced and considered the problem of finding u\in {K}_{r} such that
Meanwhile, by using Lemma 2.1, he asserted that problem (1) is equivalent to that of finding u\in {K}_{r} such that
where {N}_{{K}_{r}}^{P}(s) denotes the Pnormal cone to {K}_{r} at s in the sense of nonconvex analysis. However, this claim is not true in general.
Remark 3.1 Every solution of problem (1) is a solution of problem (2), but the converse is not necessarily true.
Proof Let {u}^{\ast}\in {K}_{r} be a solution of problem (1). Then we have
Inequality (3) implies that for all \alpha >0,
By using (4) and Lemma 2.1, it follows that
which leads to
The converse of the above statement does not hold in general. Indeed, suppose that inclusion (5) holds for some {u}^{\ast}\in {K}_{r}. Then, Lemma 2.1 implies that inequality (4) holds for some \alpha >0. However, by using inequality (4), we cannot deduce inequality (3). □
The following example illustrates that problem (4) does not imply problem (3).
Example 3.1 Let \mathcal{H}=\mathbb{R} and {K}_{r}=[0,\beta ]\cup [\gamma ,\delta ] be the union of two disjoint intervals [0,\beta ] and [\gamma ,\delta ], where 0<\beta <\gamma <\delta. Then {K}_{r} is an rproxregular set with r=\frac{\gamma \beta}{2}. Define T:\mathcal{H}\to \mathcal{H} by
where k\in \mathbb{R} and \theta <0 are arbitrary but fixed. Take {u}^{\ast}=\beta and let \alpha \ge \frac{\theta {e}^{k\beta}}{\gamma \beta} be arbitrary and fixed. Then, we have
If v\in [0,\beta ], then \beta \le v\beta \le 0 and
For v\in [\gamma ,\delta ], we have \gamma \beta \le v\beta \le \delta \beta and
The above facts guarantee that
Now, (6) and (7) imply that
However, it is obvious that \theta {e}^{k\beta}(v\beta )<0 for all v\in [\gamma ,\delta ], that is,
Hence, the inequality
cannot hold for all v\in {K}_{r}.
In Section 3 of [25, 26], the author claimed that problem (1) is equivalent to a fixed point problem.
Lemma 3.1 ([25, 26], Lemma 3.1)
u\in {K}_{r} is a solution of nonconvex variational inequality (1), if and only if u\in {K}_{r} satisfies the relation
where \rho >0 is a constant and {P}_{{K}_{r}} is the projection of ℋ onto the uniformly rproxregular set {K}_{r}.
Remark 3.2 By a careful reading, we found that there are two fatal errors in the proof of Lemma 3.1. Firstly, in view of Proposition 2.1, it should be pointed out that for any {r}^{\prime}\in (0,r), the projection of points in U({r}^{\prime})=\{u\in \mathcal{H}:{d}_{{K}_{r}}(u)<{r}^{\prime}\} onto the set {K}_{r} exists and is unique, that is, for any x\in U({r}^{\prime}), the set {P}_{{K}_{r}}(x) is nonempty and singleton. Equation (8) and Proposition 2.1 imply that the point u\rho T(u) should belong to U({r}^{\prime}) for some {r}^{\prime}\in (0,r). Unfortunately, it is not necessarily true. Indeed, equation (8) is not necessarily well defined. If u\in {K}_{r} and \rho <\frac{{r}^{\prime}}{1+\parallel Tu\parallel}, for some {r}^{\prime}\in (0,r), then we have
Therefore, u\rho Tu\in U({r}^{\prime}), that is, the set {P}_{{K}_{r}}(u\rho Tu) is nonempty and singleton. Hence, in the statement of Lemma 3.1, the constant ρ should be satisfied \rho <\frac{{r}^{\prime}}{1+\parallel Tu\parallel} for some {r}^{\prime}\in (0,r). Secondly, we note that the author [25, 26] used the nonconvex variational inclusion (2) as an equivalence formulation of the nonconvex variational inequality (1). However, in view of Remark 3.1 and Example 3.1, the two problems (1) and (2) are not necessarily equivalent.
Let the operator T be the same as in problem (1). Related to problem (1), Noor [26] considered the problem of finding z\in \mathcal{H} such that
where \rho >0 is a constant, {Q}_{{K}_{r}}=I{P}_{{K}_{r}} and I is the identity operator. Problem (9) is called the nonconvex WinerHopf equation (NWHE).
Noor [26] claimed that problem (9) is equivalent to problem (1).
Lemma 3.2 ([26], Lemma 4.1)
The nonconvex variational inequality (1) has a solution u\in {K}_{r} if and only if the nonconvex WienerHopf equation (9) has a solution z\in \mathcal{H}, satisfying
where \rho >0 is a constant.
By a careful reading, we discovered that Lemma 3.1 is the main tool to establish the statement of Lemma 3.2. As it is shown, the statement of Lemma 3.1 is not necessarily true. Consequently, the statement of Lemma 3.2 is not necessarily true.
Since Lemmas 3.1 and 3.2 are the main tools to suggest algorithms and to obtain the results in [25] and [26], in view of the above remarks, the results in [25, 26] and the papers where the same technique and method are used, are not valid.
4 Projection methods and convergence analysis
Instead of the nonconvex variational inequality (1), in this section, for a given nonlinear operator T:\mathcal{H}\to \mathcal{H}, we consider the problem of finding u\in {K}_{r} such that
Problem (11) is called the regularized nonconvex variational inequality (RNVI). We prove the equivalence between RNVI (11) and problem (2) as well as fixed point problem (8).
If r=\mathrm{\infty}, that is, {K}_{r}=K, the convex set in ℋ, then problem (11) collapses to the problem of finding u\in K such that
An inequality of type (12) is called the variational inequality, which was introduced and studied by Stampacchia [1] in 1964.
In the next proposition, the equivalence between nonconvex variational inclusion (2) and regularized nonconvex variational inequality (11) is established.
Proposition 4.1 If {K}_{r} is a uniformly proxregular set, then problem (11) is equivalent to problem (2).
Proof Let u\in {K}_{r} be a solution of problem (11). If Tu=0, because the vector zero always belongs to any normal cone, we have 0\in Tu+{N}_{{K}_{r}}^{P}(u). If Tu\ne 0, then for all v\in {K}_{r}, one has
Now, Lemma 2.1 implies that Tu\in {N}_{{K}_{r}}^{P}(u), and so
Conversely, if u\in {K}_{r} is a solution of problem (2), then Definition 2.5 guarantees that u\in {K}_{r} is a solution of problem (11). □
Problem (2) is called the nonconvex variational inclusion associated with RNVI (11). Now, by using the projection operator technique, we establish the equivalence between problem (11) and fixed point problem (8).
Lemma 4.1 Let T be the same as in problem (11). Then u\in {K}_{r} is a solution of problem (11) if and only if u satisfies equation (8), provided that \rho <\frac{{r}^{\prime}}{1+\parallel Tu\parallel} for some {r}^{\prime}\in (0,r).
Proof Let u\in {K}_{r} be a solution of problem (11). Since \rho <\frac{{r}^{\prime}}{1+\parallel Tu\parallel}, it follows that equation (8) is well defined. Then, by using Proposition 4.1, we have
where I is the identity operator and we have used the wellknown fact that {P}_{{K}_{r}}={(I+\rho {N}_{{K}_{r}}^{P})}^{1}. □
Definition 4.1 An operator T:\mathcal{H}\to \mathcal{H} is said to be:

(a)
monotone if and only if
\u3008TuTv,uv\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in \mathcal{H}; 
(b)
κstrongly monotone if and only if there exists a constant \kappa >0 such that
\u3008TuTv,uv\u3009\ge \kappa {\parallel uv\parallel}^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in \mathcal{H}; 
(c)
γLipschitz continuous if and only if there exists a constant \gamma >0 such that
\parallel TuTv\parallel \le \gamma \parallel uv\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}u,v\in \mathcal{H}.
In the next theorem, the existence and uniqueness of a solution for problem (11) are discussed.
Theorem 4.1 Let the operator T be the same as in problem (11) such that T is αstrongly monotone and βLipschitz continuous. If there exists a constant \rho >0 satisfying the following conditions:
and
where \delta =\frac{r}{r{r}^{\prime}}, then problem (11) admits a unique solution.
Proof Define F:{K}_{r}\to {K}_{r} by
By using condition (13), we can easily check that the mapping F is well defined. We establish that F is a contraction mapping. For this end, let u,v\in {K}_{r} be given. From (15) and Proposition 2.1, it follows that
where \delta =\frac{r}{r{r}^{\prime}}. Since the operator T is αstrongly monotone and βLipschitz continuous, we get
which leads to
Applying (16) and (17), we have
where
Condition (14) implies that \theta <1. From inequality (18), we infer that F is a contraction mapping. According to the Banach fixed point theorem, there exists a unique point {u}^{\ast}\in {K}_{r} such that F({u}^{\ast})={u}^{\ast}. It follows from (15) that {u}^{\ast}={P}_{{K}_{r}}[{u}^{\ast}\rho T{u}^{\ast}]. Now, Lemma 4.1 guarantees that {u}^{\ast}\in {K}_{r} is a solution of problem (11). This completes the proof. □
Noor [25] proposed the Mann iteration process for solving problem (1) as follows.
Algorithm 4.1 ([25], Algorithm 3.1)
For a given {u}_{0}\in \mathcal{H}, find an approximate solution {u}_{n+1} by the iterative scheme
where {\alpha}_{n}\in [0,1] for all n\ge 0.
Noor [26] suggested the following twostep and threestep iterative methods for solving problem (1).
Algorithm 4.2 ([26], Algorithm 3.4)
For a given {u}_{0}\in {K}_{r}, find an approximate solution {u}_{n+1} by the iterative schemes
where {\alpha}_{n},{\beta}_{n}\in [0,1], for all n\ge 0.
Algorithm 4.3 ([26], Algorithm 3.5)
For a given {u}_{0}\in \mathcal{H}, find an approximate solution {u}_{n+1} by the iterative schemes
where {\alpha}_{n},{\beta}_{n},{\gamma}_{n}\in [0,1], for all n\ge 0.
Remark 4.1 It should be pointed out that in the context of Algorithms 3.4 and 3.5 from [26], there are minor mistakes. In fact, in iterative processes (3.7) and (3.8) from Algorithm 3.4 in [26], {x}_{n} must be replaced by {u}_{n}, as we have done in Algorithm 4.2. Meanwhile, in the context of Algorithm 3.5 from [26], {u}_{n}\in \mathcal{H} must be replaced by {u}_{0}\in \mathcal{H}, as we have done in Algorithm 4.3.
By a careful reading, we found that Algorithms 4.14.3 do not work. Indeed, in a way similar to the argument of Remark 3.2, the points {u}_{n}\rho T{u}_{n}, {y}_{n}\rho T{y}_{n} and {w}_{n}\rho T{w}_{n} (n\ge 0) do not belong necessarily to U({r}^{\prime}).
By utilizing Lemma 4.1, we suggest and analyze the following explicit projection iterative methods for solving problem (11).
Algorithm 4.4 Let T be the same as in problem (11) and suppose further that \rho >0 is a constant satisfying condition (13). For an arbitrarily chosen initial point {u}_{0}\in {K}_{r}, compute the iterative sequence \{{u}_{n}\} in {K}_{r} in the following way:
Algorithm 4.5 Let T be the same as in problem (11), and let \rho >0 be a constant satisfying condition (13). For an arbitrarily chosen initial point {u}_{0}\in {K}_{r}, compute the iterative sequence \{{u}_{n}\} in {K}_{r} in the following way:
Algorithm 4.6 Let T be the same as in problem (11), and let \rho >0 be a constant satisfying condition (13). For an arbitrarily chosen initial point {u}_{0}\in {K}_{r}, compute the iterative sequence \{{u}_{n}\} in {K}_{r} in the following way:
We now study the convergence analysis of Algorithm 4.4 and this is the main motivation of our next result.
Theorem 4.2 Let the operator T be the same as in Theorem 4.1, and let all the conditions of Theorem 4.1 hold. Then the iterative sequence \{{u}_{n}\} generated by Algorithm 4.4 converges strongly to the unique solution of problem (11).
Proof Theorem 4.1 guarantees the existence of a unique solution u\in {K}_{r} for problem (11). Hence, Lemma 4.1 implies that u={P}_{{K}_{r}}[u\rho Tu]. Then, by using (20) and Proposition 2.1, we have
where θ is the same as in (19). Since \theta <1, it follows that the righthand side of the above inequality tends to zero as n\to \mathrm{\infty}, whence we deduce that {u}_{n}\to u as n\to \mathrm{\infty}. This completes the proof. □
In a similar way to the proof of Theorem 4.2, one can prove the strong convergence of the iterative sequence \{{u}_{n}\} generated by Algorithms 4.5 and 4.6.
5 WienerHopf equations technique
In this section, by utilizing Lemma 4.1 and the projection method, the equivalence between NWHE (9) and RNVI (11) is established. By using the obtained equivalence formulation, some iterative algorithms for solving RNVI (11) are suggested and analyzed. The convergence analysis of the proposed iterative algorithms under some certain conditions is studied.
Let the operator T be the same as in problem (11). Related to problem (11), we consider the problem of finding z\in \mathcal{H} satisfying (9).
If r=\mathrm{\infty}, then problem (9) is equivalent to finding z\in \mathcal{H} such that
where {Q}_{K}=I{P}_{K}. Equation (22) is the original WienerHopf equation mainly due to Shi [22].
Remark 5.1 It was shown that the WienerHopf equations had played an important and significant role in developing several numerical techniques for solving variational inequalities and related optimizations problems (see, for example, [18, 19, 22] and the references therein).
In the next lemma, the equivalence between RNVI (11) and NWHE (9) is proved.
Lemma 5.1 Let T be the same as in problem (11). Then u\in {K}_{r} is a solution of RNVI (11) if and only if NWHE (9) has a solution z\in \mathcal{H} satisfying (10), provided that \rho <\frac{{r}^{\prime}}{1+\parallel Tu\parallel} for some {r}^{\prime}\in (0,r).
Proof Let u\in {K}_{r} be a solution of problem (11). Since \rho <\frac{{r}^{\prime}}{1+\parallel Tu\parallel} for some {r}^{\prime}\in (0,r), it follows from Lemma 4.1 that
Taking z=u\rho Tu in (23), we have
By using (24) and the fact that z=u\rho Tu, we have
Obviously, the above equation is equivalent to
where {Q}_{{K}_{r}}=I{P}_{{K}_{r}}, that is, z\in \mathcal{H} is a solution of NWHE (9).
Conversely, if z\in \mathcal{H} is a solution of NWHE (9), satisfying
then Lemma 4.1 implies that u\in {K}_{r} is a solution of RNVI (11). This completes the proof. □
Noor [26] used the equivalence formulation between the two problems (1) and (9) and suggested the following iterative methods for solving problem (1).
Algorithm 5.1 ([26], Algorithm 4.1)
For a given {z}_{0}\in \mathcal{H}, compute {z}_{n+1} by the iterative schemes
where 0\le {\alpha}_{n}\le 1, for all n\ge 0 and {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}.
Algorithm 5.2 ([26], Algorithm 4.2)
For a given {z}_{0}\in \mathcal{H}, compute {z}_{n+1} by the iterative schemes
where 0\le {\alpha}_{n}\le 1, for all n\ge 0 and {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}.
Algorithm 5.3 ([26], Algorithm 4.3)
For a given {z}_{0}\in \mathcal{H}, compute {z}_{n+1} by the iterative scheme
where 0\le {\alpha}_{n}\le 1, for all n\ge 0 and {\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}.
Remark 5.2 As it is pointed out, the two problems (1) and (9) are not necessarily equivalent. Hence, the equivalence between problems (1) and (9) cannot be used for suggesting Algorithms 5.15.3 to approximate the solution of problem (1). Even without considering the mentioned fact, we note that Algorithms 5.15.3 do not work. Indeed, in a way similar to the argument of Remark 3.2, iterative scheme (25) is well defined provided that for each n\ge 0, the point {z}_{n} belongs to U({r}^{\prime}) for some {r}^{\prime}\in (0,r). Accordingly, {z}_{0} must be taken in U({r}^{\prime}) for some {r}^{\prime}\in (0,r). However, for a given {z}_{0}\in U({r}^{\prime}), iterative scheme (26) does not guarantee that {z}_{n}\in U({r}^{\prime}) for each n>0, because U({r}^{\prime}) is not necessarily convex.
The following example illustrates that for any given uniformly rproxregular set {K}_{r} in ℋ and {r}^{\prime}\in (0,r), the set U({r}^{\prime}) in ℋ is not necessarily convex.
Example 5.1 Let ℋ and {K}_{r} be the same as in Example 3.1. As has been mentioned in Example 3.1, {K}_{r} is an rproxregular set with r=\frac{\gamma \beta}{2}. Let {r}^{\prime}\in (0,r) be arbitrary but fixed. Then we have
which is clearly nonconvex.
By using NWHE (9) and Lemma 5.1, we get a fixed point formulation to construct a new projection iterative algorithm for solving RNVI (11).
By using (9) and (10), we have
This fixed point formulation enables us to construct the following iterative algorithm for solving RNVI (11).
Algorithm 5.4 Let the operator T be the same as in RNVI (11) and assume further that \rho >0 is a constant satisfying condition (13) for some {r}^{\prime}\in (0,r). For a given {z}_{0}\in U({r}^{\prime}), compute the iterative sequence \{{z}_{n}\} in U({r}^{\prime}) in the following way:
We now apply Lemma 5.1 and study the convergence analysis of Algorithm 5.4.
Theorem 5.1 Let the operator T be the same as in Theorem 4.1 and suppose that all the conditions of Theorem 4.1 hold. Assume further that \rho >0 is a constant satisfying conditions (13) and (14). Then there exists z\in \mathcal{H} such that z is a solution of problem (9) and the sequence \{{z}_{n}\} generated by Algorithm 5.4 converges strongly to z.
Proof Theorem 4.1 guarantees that RNVI (11) admits a unique solution u\in {K}_{r}. Hence Lemma 5.1 implies the existence of a unique point z\in U({r}^{\prime}) satisfying (10). By using (10), (27) and the assumptions, we have
From (10) and (27) and Proposition 2.1, it follows that
where δ is the same as in (14). Substituting (29) in (28), we have
where θ is the same as in (19). Applying (30), we deduce that
Condition (14) implies that \theta <1. Since \theta \in (0,1), it follows that the righthand side of the above inequality tends to zero as n\to \mathrm{\infty}, which implies that the sequence \{{z}_{n}\} generated by Algorithm 5.4 converges strongly to z. This completes the proof. □
6 Concluding remarks
In this paper, we have investigated and analyzed the nonconvex variational inequality (2.1) from [25, 26] and the WienerHopf equation (2.5) from [26]. We have proved that the problem (2.1) from [25, 26] is not equivalent to the fixed point problem (3.1) from [25, 26] and the WienerHopf equation (2.5) from [26]. That is, Lemma 3.1 in [25, 26] is incorrect. Lemma 3.1 in [25, 26] is the main key to suggest the Mann iteration processes in [25, 26] and to establish the strong convergence of the iterative sequences generated by the proposed algorithms in [25, 26]. Since Lemma 3.1 in [25, 26] is not valid, the algorithms and results in [25, 26] are also not valid. Indeed, we have pointed out that unlike the claim of the author in [25, 26], the existence and uniqueness of a solution for nonconvex variational inequality (1) ((2.1) in [25, 26]) up to now have not been established and still remain an open problem. We have also pointed out that the Mann iteration processes proposed by the author in [25, 26] for solving problem (1) do not work. To overcome these problems in [25, 26], we have introduced and considered the regularized nonconvex variational inequality (RNVI) (11) and the nonconvex WienerHopf equation (NWHE) (9). By using the projection operator technique, we have verified the equivalence between RNVI (11) and the fixed point problem (8) as well as NWHE (9). By using the obtained equivalences, we have proved the existence of a unique solution for RNVI (11) and suggested and analyzed some explicit projection iterative methods for solving RNVI (11). The convergence analysis of the proposed iterative schemes under some suitable conditions has also been studied. But the two following questions have not been replied and still remain open problems:

(1)
Can the existence of a solution for nonconvex variational inequality (1) be proved?

(2)
Can the Mann iteration process for solving nonconvex variational inequality (11) be presented?
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Acknowledgements
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by the Ministry of Education of the Republic of Korea (2013R1A1A2054617). The authors thank the anonymous referees for their constructive comments which contributed to the improvement of the present paper.
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Balooee, J., Kim, J.K. Some remarks on regularized nonconvex variational inequalities. J Inequal Appl 2013, 531 (2013). https://doi.org/10.1186/1029242X2013531
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DOI: https://doi.org/10.1186/1029242X2013531
Keywords
 regularized nonconvex variational inequalities
 nonconvex sets
 iterative algorithm
 proxregularity
 convergence analysis