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