A General System of Regularized Non-convex Variational Inequalities

The originally variational inequality problem introduced by Stampacchia [1] in the early sixties has a great impact and in influence in the development of almost all branches of pure and applied sciences and has witnessed an explosive growth in theoretical advances, algorithmic development. As a result of interaction between different branches of mathematical and engineering sciences, we now have a variety of techniques to suggest and analyze various algorithms for solving variational inequalities and related optimizations [2-6]. Verma [7-10] studied some systems of variational inequality with single valued mappings and suggest some iterative algorithms to compute approximate solutions of these systems in Hilbert spaces. Agarwal et al. [11] studied sensitivity analysis for a system of generalized nonlinear mixed quasi variational inclusions with single valued mappings. Several authors studied different kinds of systems of variational inequalities and suggested iterative algorithms to find the approximate solutions of the systems [12-15]. We remark that the results regarding the existence of solutions and iterative schemes for solving the system of variational inequalities and related problems are being considered in the setting of convex sets and the technique defined on the characteristics of the projection operator over convex a set which does not hold in general when the sets are non-convex. It is well known that the uniform prox regular sets are convex and include the convex set as special cases. Wen [16] considered a system of non-convex variational inequalities with different nonlinear operator and asserted that this system is equivalent to the fixed point problem and suggested an iterative algorithm for the system of non-convex variational inequalities. The convergence analysis of the proposed iterative algorithm under some certain assumption is also studied. In [17] point out the equivalence formulation used by Wen [16] is not correct. Inspired and motivated by the works of [1826], we introduced and studied a general systems of regularized nonconvex variational inequalities. By using the equivalence, we defined a projection iteration algorithm for solving GSRNVI. Further, we proved the existence and uniqueness of solutions of general system of regularized non-convex variational inequalities. The convergence analysis of the proposed iterative algorithm is also studied.


Introduction
The originally variational inequality problem introduced by Stampacchia [1] in the early sixties has a great impact and in influence in the development of almost all branches of pure and applied sciences and has witnessed an explosive growth in theoretical advances, algorithmic development. As a result of interaction between different branches of mathematical and engineering sciences, we now have a variety of techniques to suggest and analyze various algorithms for solving variational inequalities and related optimizations [2][3][4][5][6]. Verma [7][8][9][10] studied some systems of variational inequality with single valued mappings and suggest some iterative algorithms to compute approximate solutions of these systems in Hilbert spaces. Agarwal et al. [11] studied sensitivity analysis for a system of generalized nonlinear mixed quasi variational inclusions with single valued mappings. Several authors studied different kinds of systems of variational inequalities and suggested iterative algorithms to find the approximate solutions of the systems [12][13][14][15]. We remark that the results regarding the existence of solutions and iterative schemes for solving the system of variational inequalities and related problems are being considered in the setting of convex sets and the technique defined on the characteristics of the projection operator over convex a set which does not hold in general when the sets are non-convex. It is well known that the uniform prox regular sets are convex and include the convex set as special cases. Wen [16] considered a system of non-convex variational inequalities with different nonlinear operator and asserted that this system is equivalent to the fixed point problem and suggested an iterative algorithm for the system of non-convex variational inequalities. The convergence analysis of the proposed iterative algorithm under some certain assumption is also studied. In [17] point out the equivalence formulation used by Wen [16] is not correct. Inspired and motivated by the works of [18][19][20][21][22][23][24][25][26], we introduced and studied a general systems of regularized nonconvex variational inequalities. By using the equivalence, we defined a projection iteration algorithm for solving GSRNVI. Further, we proved the existence and uniqueness of solutions of general system of regularized non-convex variational inequalities. The convergence analysis of the proposed iterative algorithm is also studied.

Basic Foundation
Let H be a real Hilbert space endowed with norm ║ . ║and an inner product < . , . > respectively. Let Ω be nonempty closed subsets of H. We represent d Ω (.) or d(.;Ω) the distance function from a point to a set Ω that is where y is a vector in H and τ is a positive scalar.
The tangent cone to Ω at a point x € Ω, denoted by T Ω (x) is defined by where co(S) denotes the closure of the convex hull of S and is a closed convex cone and (x) P Q Ω is convex but may not be closed. Definition 2.6: [4] For a given t € (0, + ∞]; a subset t of H is called the normalized uniformly prox-regular (or uniformly t-prox-regular) if every nonzero proximal normal to Ω t can be realized by an t-ball.
That is for all Therefore, for all t x∈Ω and 0

Lemma 2.7 [23] A closed set
H Ω ⊆ is convex if and only if it is uniformly t-prox-regular for every t>0 Proposition 2.8 [25] Let t>0 and Ω t be a nonempty closed and uniformly t-prox -regular subset of H. Let U(t)={u € H : 0 ≤ d Ωt (u)<t} Then the following statements are hold: (a) for all x € U(t); P Ωt (x) ≠0; (b) for all t0 2 (0; t); Pt is Lipschitz continuous mapping with Q Ω is a closed set valued mapping, hence The infinite union of disjoint intervals is also uniformly t-prox-regular and t depends on the distance between the intervals.

Basic Remarks and Formulations
Let t be an uniformly Ω t prox-regular (nonconvex) set and g i : Ω t → Ω t be a given mapping for i = 1; 2; 3: For given mappings T 1 , T 2 , T 3 : Ω t → Ω t we consider the following problems of finding (x * , y * , z * ) € Ω t × Ω t × Ω t such that The problem (3.1) is called a general system of regularized non convex variational inequalities. We note that if T 1 =T 2 =T 3 =T : Ω t → Ω t is an univariant nonlinear operator, g i = I; i = 1; 2; 3 (the identity operator) and x*=y*=z*=u, then the system (3.1) reduces to the following classical variational inequalities defined on the nonconvex set Ω t find u € Ω t such that and (3.2) is equivalent to find u € Ω t such that Q Ω denotes the normal cone of Ω t at u over the non convex set. Lemma 3.1 (x * , y * , z * ) € Ω t × Ω t × Ω t is a solution set of problem (3.1) if and only if where P Ωt is the projection of H on to the uniformly t-prox-regular set Ω t . In the proof of Lemma 3.1, there occur three fatal errors. First in view of Proposition 2.8, for any t € (0, 1) the projection of points in onto the set Ω t exists and unique, that is for any x € U(t'), the set P Ωt (x) is nonempty and singleton. From the Lemma 3.1 and Proposition 2.6 the points g 1 (y * )r 1 T 1 (y * ; x * ); g 2 (z * ) -r 2 T 2 (z * , y * ) and g 3 (x * ) -r 3 T 3 (x * , z * ) should be in U(t') for some t' € (0, t) it is not necessary true, hence (3.4) are not necessarily , and , , and , for t' ε (0; t); then the equation (3.4) are well defined.
Secondly the following general system of regularized non convex variational inclusions is equivalence to the system (3.1): the system (3.1) is equivalent tothe following system: The system (3.1) is equivalent to the system (3.5) which is not true in general.
be the fixed arbitrary. Hence for all w ε Ω t

Proposition 3.3:
Let Ω t be an uniformly t-prox regular set. The system (3.9) is equivalent to the system (3.6).

Theorem 4.3:
Let the mappings T i , g i and r i ; i=1, 2, 3 be the same as in the system (3.1) such that g i (H) Ć Ω t : Let gi be the μ i -cocoercive with constant ξ i >0 and Lipschitz continuous mapping with constant μ i >0: Let T i be the relaxed (η i , v i )-cocoercive with respect to the first variable with constants η i , v i >0 and λ i -Lipschitz continuous mapping with constant λ i >0: If the constant r i for i=1; 2; 3: satisfy the following conditions: (1 (1 ) ) ) , then the system (3.9) admits a unique solutions.
Proof. Define , , : H H H φ ϕ ψ × → by We claim that ℑ is a contraction mapping. Indeed let (x,y), (x * ,y * ) ε H× H× H, g 1 (y) ε Ω t and 1 x y x y x g x P g y rT y x x P g y rT By µ 1 -cocoercive of g 1 and £ 1 -Lipschitz continuity of g 1 we have From (4.3)-(4.9) we have Where x y z i ∈ Ω ×Ω ×Ω = i=1; 2; 3 is a solution set of general system of regularized non-convex variational inequalities. This completes the proof. By using Lemma 3.4, we suggest the following explicit projection iterative method for solving the general system of regularize non-convex variational inequalities. Algorithm 4.4 Assume that mappings T i , g i and constant r i >0 for i=1; 2; 3 be the same as in the system (3.9) such that (H) i t g ⊆ Ω For arbitrary initial points x 0 , y 0 , z 0 ε H compute the sequences {x n },{y n } and {z n } in H in the following way: where {αn} is a sequence in [0,1] Assume that g i =I, i=1, 2, 3 is an identity mappings, then we have the following algorithm.

Algorithm 4.5:
Let mapping Ti and constant r i >0 for i=1; 2; 3 be the same as in the system (3.9). For arbitrary initial points x 0 , y 0 ,z 0 ε Ω t compute the sequences {x n }, {y n } and {z n } in Ω t in the following way: Again assume that g i =I, i=1, 2, 3 is an identity mappings, then we have the following algorithm. Algorithm 4.6 Let mapping Ti and constant r i >0 for i=1, 2, 3 be the same as in the system (3.9). For arbitrary initial points x 0 , y 0 , z 0 ε Ω t compute the sequences {x n }, {y n } and {z n } in Ω t in the following way:  then the iterative sequences {(x n , y n , z n )} generated by Algorithm 4.4 converges strongly to a unique solutions (x * , y * , z * ) of the system (3.9).