Some Existence Theorems for Nonconvex Variational Inequalities Problems

and Applied Analysis 3 the uniform r-prox-regularity C is equivalent to the convexity of C. Moreover, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class p-convex sets, C1,1 submanifolds possibly with boundary of H, the images under a C1,1 diffeomorphism of convex sets, and many other nonconvex sets; see 6, 8 . Now, let us state the following facts, which summarize some important consequences of the uniform prox-regularity. The proof of this result can be found in 7, 8 . Lemma 2.5. Let C be a nonempty closed subset of H, r ∈ 0, ∞ and set Cr : {x ∈ H;d x, C < r}. If C is uniformly r-uniformly prox-regular, then the following hold: 1 for all x ∈ Cr , projC x / ∅, 2 for all s ∈ 0, r , projC is Lipschitz continuous with constant r/ r − s on Cs, 3 the proximal normal cone is closed as a set-valued mapping. In this paper, we are interested in the following classes of nonlinear mappings. Definition 2.6. Amapping T : C → H is said to be a γ -strongly monotone if there exists a constant γ > 0 such that 〈 Tx − Ty, x − y ≥ γ∥x − y∥2, ∀x, y ∈ C, 2.5 b μ-Lipschitz if there exist a constants μ > 0 such that ‖Tx − Ty‖ ≤ μ‖x − y‖, ∀x, y ∈ C. 2.6 3. System of Nonconvex Variational Inequalities Involving Nonmonotone Mapping LetH be a real Hilbert space, and let C be a nonempty closed subset ofH. In this section, we will consider the following problem: find x∗, y∗ ∈ C such that y∗ − x∗ − ρTy∗ ∈ N C x∗ , x∗ − y∗ − ηTx∗ ∈ N C ( y∗ ) , 3.1 where ρ and η are fixed positive real numbers, C is a closed subset of H, and T : C → H is a mapping. The iterative algorithm for finding a solution of the problem 3.1 was considered by Moudafi 9 , when C is r-uniformly prox-regular and T is a strongly monotone mapping. He also remarked that two-step models 3.1 for nonlinear variational inequalities are relatively more challenging than the usual variational inequalities since it can be applied to problems arising, especially from complementarity problems, convex quadratic programming, and other variational problems. In this section, we will generalize such result by considering the conditions for existence solution of problem 3.1 when T is not necessary stronglymonotone. To do so, we will use the following algorithm as an important tool. 4 Abstract and Applied Analysis Algorithm 3.1. LetC be an r-uniformly prox-regular subset ofH. Assume that T : C → H is a nonlinear mapping. Letting x0 be an arbitrary point in C, we consider the following two-step projection method: yn projC [ xn − η Txn ] , xn 1 projC [ yn − ρ ( Tyn )] , 3.2 where ρ, η are positive reals number, which were appeared in problem 3.1 . Remark 3.2. The projection algorithm above has been introduced in the convex case, and its convergence was proved see 10 . Observe that 3.2 is well defined provided the projection on C is not empty. Our adaptation of the projection algorithm will be based on Lemma 2.5. Now we will prove the existence theorems of problem 3.1 , when C is a closed uniformly r-prox-regular. Moreover, from now on, the number r will be understood as a finite positive real number if not specified otherwise . This is because, as we know, if r ∞, then such a set C is nothing but the closed convex set. We start with an important remark. Remark 3.3. Let C be a uniformly r-prox-regular closed subset of H. Let T1, T2 : C → H be such that T1 is a μ1-Lipschitz continuous, γ -strongly monotone mapping and T2 is a μ2Lipschitz continuous mapping. If ξ r μ1 − γμ2 − √ μ1 − γμ2 2 − μ1 γ − μ2 2 /μ1, then for each s ∈ 0, ξ we have γts − μ2 > √( μ1 − μ2 )( ts − 1 ) , 3.3 where ts r/ r − s . It is worth to point out that, in Remark 3.3, we have to assume that μ2 < μ1. Thus, from now on, without loss of generality we will always assume that μ2 < μ1. Theorem 3.4. Let C be a uniformly r-prox-regular closed subset of a Hilbert space H, and let T : C → H be a nonlinear mapping. Let T1, T2 : C → H be such that T1 is a μ1-Lipschitz continuous and γ -strongly monotone mapping, T2 is a μ2-Lipschitz continuous mapping. If T T1 T2 and the following conditions are satisfied: a MδT C < ξ, where δT C sup{‖u − v‖;u, v ∈ T C }; b there exists s ∈ MδT C , ξ such that γts − μ2 ts ( μ1 − μ2 ) − ζ < ρ, η < min { γts − μ2 ts ( μ1 − μ2 ) ζ, 1 tsμ2 } , 3.4 whereM max{ρ, η}, ts r/ r − s , and ζ √ tsγ − μ2 2 − μ1 − μ2 ts − 1 /ts μ1 − μ2 . Then the problem 3.1 has a solution. Moreover, the sequence xn, yn which is generated by 3.2 strongly converges to a solution x∗, y∗ ∈ C × C of the problem 3.1 . Abstract and Applied Analysis 5 Proof. Firstly, by condition b , we can easily check that yn − ρTyn and xn − ηTxn belong to the set Cs, for all n 1, 2, 3, . . .. Thus, from Lemma 2.5 1 , we know that 3.2 is well defined. Consequently, from 3.2 and Lemma 2.5 2 , we haveand Applied Analysis 5 Proof. Firstly, by condition b , we can easily check that yn − ρTyn and xn − ηTxn belong to the set Cs, for all n 1, 2, 3, . . .. Thus, from Lemma 2.5 1 , we know that 3.2 is well defined. Consequently, from 3.2 and Lemma 2.5 2 , we have ‖xn 1 − xn‖ ‖projC ( yn − ρTyn ) − projC ( yn−1 − ρTyn−1 ‖ ≤ ts‖yn − yn−1 − ρ ( Tyn − Tyn−1 ‖ ≤ ts ‖yn − yn−1 − ρ ( T1yn − T1yn−1 ‖ ρ‖T2yn − T2yn−1‖ ] . 3.5 Since the mapping T1 is γ -strongly monotone and μ1-Lipschitz continuous, we obtain ∥ ∥yn − yn−1 − ρ ( T1yn − T1yn−1 )∥2 ∥ ∥yn − yn−1 ∥ ∥2 − 2ρ〈yn − yn−1, T1yn − T1yn−1〉 ρ2 ∥ ∥T1yn − T1yn−1 ∥ ∥2 ≤ ∥yn − yn−1 ∥∥2 − 2ργ‖yn − yn−1‖ ρμ1 ∥yn − yn−1 ∥∥2 ( 1 − 2ργ ρμ1 )∥ ∥yn − yn−1 ∥ ∥. 3.6 On the other hand, since T2 is μ2-Lipschitz continuous, we have ‖T2yn − T2yn−1‖ ≤ μ2‖yn − yn−1‖. 3.7 Thus, by 3.5 , 3.6 , and 3.7 , we obtain ‖xn 1 − xn‖ ≤ ts [ ρμ2 √ 1 − 2ργ ρμ1 ] ‖yn − yn−1‖. 3.8


Introduction
Variational inequalities theory, which was introduced by Stampacchia 1 , provides us with a simple, natural, general, and unified framework to study a wide class of problems arising in pure and applied sciences.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 solutions to important classes of problems.On the other hand, it also enables us to develop highly efficient and powerful new numerical methods for solving, for example, obstacle, unilateral, free, moving, and complex equilibrium problems.
It should be pointed out that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting; see 2-5 for examples.Moreover, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex.Notice that the convexity assumption, made by researchers, has been used for guaranteeing the well definedness of the proposed iterative algorithm which depends on the projection mapping.In fact, the convexity assumption may not require for the well definedness of the projection mapping because it may be well defined,

Preliminaries
Let H be a real Hilbert space whose inner product and norm are denoted by •, • and • , respectively.Let C be a nonempty closed subset of H.We denote by d C • the usual distance function to the subset C; that is, d C u inf v∈C u − v .Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis.
Let C be a subset of H.The proximal normal cone to C at x is given by The following characterization of N P C x can be found in 6 .
Lemma 2.3.Let C be a closed subset of a Hilbert space H.Then, Definition 2.4.For a given r ∈ 0, ∞ , a subset C of H is said to be uniformly prox-regular with respect to r if, for all x ∈ C and for all 0 / z ∈ N P C x , one has We make the convention 1/r 0 for r ∞.
It is well known that a closed subset of a Hilbert space is convex if and only if it is proximally smooth of radius r > 0. Thus, in view of Definition 2.4, for the case of r ∞, the uniform r-prox-regularity C is equivalent to the convexity of C.Moreover, it is clear that the class of uniformly prox-regular sets is sufficiently large to include the class p-convex 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 6, 8 .Now, let us state the following facts, which summarize some important consequences of the uniform prox-regularity.The proof of this result can be found in 7, 8 .Lemma 2.5.Let C be a nonempty closed subset of H, r ∈ 0, ∞ and set C r : {x ∈ H; d x, C < r}.If C is uniformly r-uniformly prox-regular, then the following hold: 3 the proximal normal cone is closed as a set-valued mapping.
In this paper, we are interested in the following classes of nonlinear mappings.Definition 2.6.A mapping T : C → H is said to be a γ -strongly monotone if there exists a constant γ > 0 such that

System of Nonconvex Variational Inequalities Involving Nonmonotone Mapping
Let H be a real Hilbert space, and let C be a nonempty closed subset of H.In this section, we will consider the following problem: find x * , y * ∈ C such that where ρ and η are fixed positive real numbers, C is a closed subset of H, and T : C → H is a mapping.
The iterative algorithm for finding a solution of the problem 3.1 was considered by Moudafi 9 , when C is r-uniformly prox-regular and T is a strongly monotone mapping.He also remarked that two-step models 3.1 for nonlinear variational inequalities are relatively more challenging than the usual variational inequalities since it can be applied to problems arising, especially from complementarity problems, convex quadratic programming, and other variational problems.In this section, we will generalize such result by considering the conditions for existence solution of problem 3.1 when T is not necessary strongly monotone.To do so, we will use the following algorithm as an important tool.Algorithm 3.1.Let C be an r-uniformly prox-regular subset of H. Assume that T : C → H is a nonlinear mapping.Letting x 0 be an arbitrary point in C, we consider the following two-step projection method: where ρ, η are positive reals number, which were appeared in problem 3.1 .
Remark 3.2.The projection algorithm above has been introduced in the convex case, and its convergence was proved see 10 .Observe that 3.2 is well defined provided the projection on C is not empty.Our adaptation of the projection algorithm will be based on Lemma 2.5.Now we will prove the existence theorems of problem 3.1 , when C is a closed uniformly r-prox-regular.Moreover, from now on, the number r will be understood as a finite positive real number if not specified otherwise .This is because, as we know, if r ∞, then such a set C is nothing but the closed convex set.
We start with an important remark.
, then for each s ∈ 0, ξ we have where t s r/ r − s .
It is worth to point out that, in Remark 3.3, we have to assume that μ 2 < μ 1 .Thus, from now on, without loss of generality we will always assume that μ 2 < μ 1 .
Theorem 3.4.Let C be a uniformly r-prox-regular closed subset of a Hilbert space H, and let T : C → H be a nonlinear mapping.Let T 1 , T 2 : C → H be such that T 1 is a μ 1 -Lipschitz continuous and γ -strongly monotone mapping, T 2 is a μ 2 -Lipschitz continuous mapping.If T T 1 T 2 and the following conditions are satisfied: where M ρ,η max{ρ, η}, t s r/ r − s , and Then the problem 3.1 has a solution.Moreover, the sequence x n , y n which is generated by 3.2 strongly converges to a solution x * , y * ∈ C × C of the problem 3.1 .
Proof.Firstly, by condition b , we can easily check that y n − ρTy n and x n − ηTx n belong to the set C s , for all n 1, 2, 3, . ... Thus, from Lemma 2.5 1 , we know that 3.2 is well defined.Consequently, from 3.2 and Lemma 2.5 2 , we have

3.5
Since the mapping T 1 is γ -strongly monotone and μ 1 -Lipschitz continuous, we obtain

3.6
On the other hand, since T 2 is μ 2 -Lipschitz continuous, we have
We claim that x * , y * ∈ C × C is a solution of the problem 3.1 .Indeed, by the definition of the proximal normal cone, from 3.2 , we have

3.13
By letting n → ∞, using the closedness property of the proximal cone together with the continuity of T, we have

3.14
This completes the proof.
Immediately, by setting T 2 0, we have the following result.
Theorem 3.5.Let C be a uniformly r-prox-regular closed subset of a Hilbert space H. Let T : C → H be a μ-Lipschitz continuous and γ -strongly monotone mapping.If the following conditions are satisfied: 15 and t s r/ r − s .Then the problem 3.1 has a solution.Moreover, the sequence x n , y n which is generated by 3.2 strongly converges to a solution x * , y * ∈ C × C of the problem 3.1 .
In view of proving Theorem 3.4, we can obtain the following result, which contains a recent result presented by Moudafi 9 as a special case.Theorem 3.6.Let C be a uniformly r-prox-regular closed subset of a Hilbert space H, and let T : C → H be a mapping.Let T 1 , T 2 : C → H be such that T 1 is a μ 1 -Lipschitz continuous and γstrongly monotone mapping, T 2 is a μ 2 -Lipschitz continuous mapping.If T T 1 T 2 and there exists s ∈ 0, ξ such that Ty n , ii By setting T 2 : 0, we see that Theorem

3.17
The problem 3.17 was introduced and studied by Verma 10 , when T is a strong monotone mapping.Hence, Theorem 3.4 extends and improves the results presented by Verma 10 .For further recent results related to the problem 3.17 , see also 2, 3, 5, 11-13 .

Further Results
By using the techniques as in Theorem 3.4, we can also obtain an existence theorem of the following problem: find x * ∈ C such that The problem of type 4.1 was studied by Noor 14 but in a finite dimension Hilbert space setting.In this section, we intend to consider the problem 4.1 in an infinite dimension Hilbert space.To do this, the following remark is useful.
We now close this section by proving an existence theorem to the problem 4.1 in a nonconvex infinite dimensional setting.Proof.Firstly, by using an elementary calculation, we know that the function h Moreover, we see that the net {t s } s∈ 0,r which is defined by t s : r/ r − s converges to 1 as s ↓ 0. Using these observations, together with the fact that h t ↓ γ/μ 2 as t ↓ 1, we can find s * ∈ 0, r r

Theorem 4 . 2 .
Let C be a uniformly r-prox-regular closed subset of a Hilbert space H, and let T : C → H be a γ -strongly monotone and μ-Lipschitz continuous mapping.If 0 < δ T C ≤ γr, then the problem 4.1 has a solution.
Clarke et al. 7 and Poliquin et al. 8 have introduced and studied a new class of nonconvex sets, which are called uniformly prox-regular sets.This class of uniformly prox-regular sets has played an important part in many nonconvex applications such as optimization, dynamic systems, and differential inclusions.
∀y ∈ C. 2.3 , y n was generated by 3.2 , then the sequence x n , y n strongly converges to a solution x * , y * ∈ C × C of the problem 3.1 .Remark 3.7.i An inspection of Theorem 3.6 shows that the sequences {Tx n } and {Ty n } are bounded. n 2 γ 2 − δ 2 T C / μ 2 r 2 − δ 2 T C such that μ 2 h t s * > γ.It is worth to notice that, from the choice of s * , we have γ/μ 2 − f t s * < s * /δ T C .Now, we choose a fixed positive real number ρ such that Next, let us start with an element x 0 ∈ C and use an induction process to obtain a sequence {x n } ⊂ C satisfying x n 1 proj C x n − ρTx n , ∀n 0, 1, 2, . . . .4.5 Note that, because of the choice of ρ, we can easily check that x n − ρTx n ∈ C s * for all n 1, 2, 3, . ... Following the proof of Theorem 3.4, we know that {x n } is a Cauchy sequence in C. If x n → x * as n → ∞, the closedness property of the proximal cone together with the continuity of T, from 4.5 , we see that x * is a solution of the problem 4.1 .This completes the proof.Remark 4.3.Theorems 3.4, 3.5, and 4.2 not only give the conditions for the existence solution of the problems 3.1 and 4.1 , respectively, but also provide the algorithm to find such solutions for any initial vector x 0 ∈ C.