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Article

Split Systems of Nonconvex Variational Inequalities and Fixed Point Problems on Uniformly Prox-Regular Sets

1
School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang 524048, China
2
Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania
3
The Key Laboratory of Intelligent Information and Big Data Processing of NingXia Province, North Minzu University, Yinchuan 750021, China
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(10), 1279; https://doi.org/10.3390/sym11101279
Submission received: 8 September 2019 / Revised: 30 September 2019 / Accepted: 8 October 2019 / Published: 12 October 2019
(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

Abstract

:
In this paper, we studied variational inequalities and fixed point problems in nonconvex cases. By the projection method over prox-regularity sets, the convergence of the suggested iterative scheme was established under some mild rules.

1. Introduction

Variational inequalities theory, introduced and improved by Stampacchia [1], has a tremendous potential in theoretical research and applied fields. For i { 1 , 2 , 3 , 4 } , given the operator S : H 1 H 1 and C i nonempty, closed, convex subsets of the Hilbert spaces H i , the variational inequality problem stated in [1] (in short, VIP) is to find v C 1 such that
S v , u v 0 , u C 1 ,
which helps us to understand a simple, unified, and efficient framework to research the actual problems arising in optimization, engineering, economy, and so on. More specifically, variational inequalities are an important tool for studying some equilibrium problems [2] and convex minimization problems [3]. Various types of equilibrium problems (e.g., Nash and dynamic traffic) can be modeled as VIP. Pang [4] showed that the VIP related to the equilibrium problem can be decomposed into a system of variational inequalities and discussed the convergence of the method of decomposition for a system of variational inequalities.
More specifically, let f : C 1 × C 2 H 1 and g : C 1 × C 2 H 2 be nonlinear bifunctions. The system of variational inequalities (SVI) (please, see [4,5]) is to find ( u , v ) C 1 × C 2 such that
f ( u , v ) , w 1 u 0 , w 1 C 1 , g ( u , v ) , w 2 v 0 , w 2 C 2 .
Using essentially the fixed point formulation and projection technique, many researchers [5,6,7,8,9,10,11] studied related iterative schemes for approximating the solutions to systems of variational inequalities. On the other hand, over the past three decades, there has been quite an activity in the development of powerful and highly efficient numerical methods to solve the VIP and its applications [12,13,14,15,16,17,18]. There is a substantial number of methods, including the linear approximation method [19,20], the auxiliary principle [21,22], the projection technique [9,11], and the descent framework [23]. For applications, numerical techniques and other aspects of variational inequalities and split problems, please see [19,20,24,25,26,27,28].
In 2012, Censor et al. [29] introduced the so called split variational inequality problem (SVIP), as follows. Let f : H 1 H 1 and g : H 2 H 2 be nonlinear operators and A be a bounded linear operator. Find v C 1 such that
f ( v ) , w 1 v 0 , w 1 C 1 ,
and such that u = A v C 2 solves
g ( u ) , w 2 u 0 , w 2 C 2 .
They also suggested some iterative algorithms for approximating the solutions to the SVIP. This problem is an important improvement of the VIP (1).
In 2016, Kazmi [30] proposed a system of split variational inequalities (SSVI), which is a generalization of the SVIP and the SVI, as follows. Let Φ : C 1 × C 2 H 1 , Ψ : C 1 × C 2 H 2 , ϕ : C 3 × C 4 H 3 , ψ : C 3 × C 4 H 4 be nonlinear bifunctions and A : H 1 H 3 and B : H 2 H 4 be bounded linear operators. The SSVI is to find ( x , y ) C 1 × C 2 such that
Φ ( x , y ) , w 1 x 0 , w 1 C 1 , Ψ ( x , y ) , w 2 y 0 , w 2 C 2 ,
and u = A x C 3 , v = B y C 4 solve
ϕ ( u , v ) , w 3 u 0 , w 3 C 3 , ψ ( u , v ) , w 4 v 0 , w 4 C 4 .
He proposed an iteration method for solving SSVI and proved that the sequence produced by the algorithm converges strongly to a solution of SSVI.
It is worth noticing that the results in [29,30] regarding the iterative schemes for approximating the solutions to variational inequalities are considered in underlying convex sets. In many practical cases, the existing results may not be applicable if the convexity assumption is not fulfilled. Thus, in this paper, we extend their results to split systems of nonconvex variational inequalities (SSNVI) in the context of uniformly prox-regular sets, which include the convex sets as special cases.

2. Preliminaries

Let H be a Hilbert space equipped with its inner product · , · and the norm · , please, see [9]. Assume that C is a nonempty, closed subset of H . Recall that the projection P r o j C from H onto C is defined by
P r o j C ( u ) : = { v C : u v = d i s t C ( u ) } ,
where d i s t C ( u ) = inf w C u w is the usual distance related to 2-norm from the point u to the set C.
Definition 1.
[31] Given v H , the proximal normal cone of C at v is given by
N C P ( v ) : = { τ H : v P r o j C ( v + α τ ) , α > 0 } .
Proposition 1.
[31] Let C be a nonempty, closed subset of H . Then τ N C P ( u ) if and only if there exists a constant α = α ( τ , u ) > 0 such that
τ , v u α v u 2 , v C .
We now give the definition of a uniformly l-prox-regular set.
Definition 2.
[32,33] A subset C l of H , l ( 0 , + ] , is said to be uniformly l-prox-regular if every nonzero proximal normal to C l can be realized by a l-ball, that is, for all u C l and all 0 τ N C l P ( u ) , one has
τ τ , v u 1 2 l v u 2 , v C l .
Obviously, the convex sets, p-convex sets [34], C 1 , 1 submanifolds [35], the images of C 1 , 1 diffeomorphism [36] are uniformly prox-regular sets. If we take l = , the convexity of C and the uniformly prox-regularity of C l are equivalent. For more details of uniformly prox-regular sets, please see [31,33,37].
Given an operator S, the nonconvex variational inequality problem
find v C l such that S v , u v 0 , u C l ,
was introduced by Bounkhel M. [38], and further studied in [37,39,40]. If C l = C , problem (2) and problem (1) are equivalent. We now give an example regarding the nonconvex case.
Example 1.
[37] Let u = ( x , y ) , v = ( t , z ) , and let S u = ( x , 1 y ) , and the set C be the union of two disjoint squares, A and B, having respectively, vertices at the points ( 0 , 1 ) , ( 2 , 1 ) , ( 2 , 3 ) , and ( 0 , 3 ) and at the points ( 4 , 1 ) , ( 5 , 2 ) , ( 4 , 3 ) , and ( 3 , 2 ) . The fact that C can be written in the form ( t , z ) R 2 : m a x { | t 1 | , | z 2 | } 1 { | t 4 | + | z 2 | 1 } shows that it is a uniformly prox-regular set in R 2 and the nonconvex variational inequality (2) has a solution on the square B.
Some properties of the uniformly l-prox-regular sets are given below.
Proposition 2.
[37] Let C l , l ( 0 , + ] , be a nonempty, closed, and uniformly l-prox-regular subset of H . Let U ( l ) = u H : d C l ( u ) < l . Then:
  • (i) For all u U ( r ) , P r o j C l ( u ) ;
  • (ii) For all l ( 0 , l ) , P r o j C l ( u ) is Lipschitz continuous with constant l l l on U ( l ) ;
  • (iii) The proximal normal cone N C l P ( u ) is closed as a set-valued mapping.
The next special operators are needed to develop our results.
Definition 3.
[40] For all u, v H , the operator S : H × H H is said to be:
  • (i) Monotone in the first variable if
    S ( u , · ) S ( v , · ) , u v 0 ;
  • (ii) α-strongly monotone in the first variable if α > 0 such that
    S ( u , · ) S ( v , · ) , u v α u v 2 ;
  • (iii) β-Lipschitz in the first variable if β > 0 such that
    S ( u , · ) S ( v , · ) β u v .
Definition 4.
[41] For all u, v H , the operator S : H H is said to be:
  • (i) ν-strongly monotone if ν > 0 such that
    S u S v , u v ν u v 2 ;
  • (ii) L-Lipschitz if L > 0 such that
    S u S v L u v ;
  • (iii) Uniformly L-Lipschitz if L > 0 such that
    S n u S n v L u v , n 1 ;
  • (iv) Generalized ( L , a ) -Lipschitz if L , a > 0 such that
    S u S v L ( u v + a ) .
Now, let us recall the class of the nearly Lipschitz operator, nearly nonexpansive operator, and nearly uniformly L-Lipschitz continuous operator briefly.
Definition 5.
[41] For all u, v H , the operator S : H H is said to be:
  • (i) Nearly Lipschitz with respect to { b n } [ 0 , ) with l i m n b n = 0 if k n > 0 such that
    S n u S n v k n ( u v + b n ) , n 1 .
The infimum of { k n } is called nearly Lipschitz constant and is denoted by
η ( S n ) = sup S n u S n v u v + b n : u v , u , v H .
A nearly Lipschitz operator S with respect to { ( b n , η ( S n ) ) } is said to be:
  • (ii) Nearly nonexpansive if η ( S n ) = 1 such that
    S n u S n v u v + b n , n 1 ;
  • (iii) Nearly asymptotically nonexpansive if η ( S n ) 1 for all n 1 such that lim n η ( S n ) = 1 ;
  • (iv) Nearly uniformly L-Lipschitz continuous if η ( S n ) L for all n 1 .
We need the following proposition in order to get the main result.
Proposition 3.
[41] For i { 1 , 2 } , let S i : C l C l be nearly uniformly L i -Lipschitz operators with respect to { b i , n } . Define a self-mapping S, S ( u , v ) = ( S 1 u , S 2 v ) for all ( u , v ) C l × C l . Then S = ( S 1 , S 2 ) : C l × C l C l × C l is a nearly uniformly max { L 1 , L 2 } -Lipschitz operator with respect to { b 1 , n + b 2 , n } . If ( u , v ) * = u + v , for any ( u , v ) , ( u , v ) C l × C l , we have
S n ( u , v ) S n ( u , v ) * = S 1 n u S 1 n u + S 2 n v S 2 n v L 1 u u + b 1 , n + L 2 v v + b 2 , n max { L 1 , L 2 } ( u , v ) ( u , v ) * + b 1 , n + b 2 , n , n 1 .
Example 2.
Let H = [ 0 , a ] , a ( 0 , 1 ] and an operator
S : H × H H × H , S ( u , v ) = ( S 1 u , S 2 v ) , ( u , v ) H × H ,
with S i ( x ) = λ i x , x [ 0 , a ) , 0 , x = a , for i = 1 , 2 , λ i ( 0 , 1 ) . Then S = ( S 1 , S 2 ) is a nearly uniformly max { λ 1 , λ 2 } -Lipschitz operator with respect to { λ 1 n 1 + λ 2 n 1 } .
Lemma 1.
[9] Let { a n } be a sequence of nonnegative real numbers and let { b n } be a sequence in [0,1] such that n = 1 b n = , { c n } R , c n 0 , n n 0 and lim n c n = 0 . If { a n } , { b n } and { c n } satisfy the property
a n + 1 ( 1 b n ) a n + b n c n , n n 0 ,
then lim n a n = 0 .
In the next sections, we are going to state and prove results regarding the existence and uniqueness of the solutions to a SSNVI, and also propose an iterative algorithm to determine the unique solution to SSNVI which is also a fixed point to some operators with suitable properties.

3. Split Systems of Nonconvex Variational Inequalities

In the section, we consider a SSNVI with several nonlinear operators.
For l, k > 0 , let C l H 1 be uniformly l-prox-regular and Q k H 2 be uniformly k-prox-regular. For i = 1 , 2 , consider the nonlinear operators Φ i : H 1 × H 1 H 1 , Ψ i : H 2 × H 2 H 2 , ϕ i : C l C l , and ψ i : Q k Q k . Let A and B be two bounded linear operators from H 1 to H 2 . The SSNVI is to find ( x , y ) C l × C l such that
Φ 1 ( y , x ) + x ϕ 1 ( y ) , ϕ 1 ( w 1 ) x + Φ 1 ( y , x ) + x ϕ 1 ( y ) 2 l ϕ 1 ( w 1 ) x 2 0 , w 1 C l : ϕ 1 ( w 1 ) C l , Φ 2 ( x , y ) + y ϕ 2 ( x ) , ϕ 2 ( w 1 ) y + Φ 2 ( x , y ) + y ϕ 2 ( x ) 2 l ϕ 2 ( w 1 ) y 2 0 , w 1 C l : ϕ 1 ( w 1 ) C l ,
and such that ( u , v ) Q k × Q k with u = A x , v = B y solves
Ψ 1 ( v , u ) + u ψ 1 ( v ) , ψ 1 ( w 2 ) u + Ψ 1 ( v , u ) + u ψ 1 ( v ) 2 k ψ 1 ( w 2 ) u 2 0 , w 2 Q k : ψ 1 ( w 2 ) Q k , Ψ 2 ( u , v ) + v ψ 2 ( u ) , ψ 2 ( w 2 ) v + Ψ 2 ( u , v ) + v ψ 2 ( u ) 2 k ψ 2 ( w 2 ) v 2 0 , w 2 Q k : ψ 2 ( w 2 ) Q k .
To study the existence of solutions to system (4), the following two lemmas are needed.
Lemma 2.
For i { 1 , 2 } , l > 0 , let Φ i : H 1 × H 1 H 1 and ϕ i : C l C l be nonlinear operators. Then system (4) and the following problem are equivalent:
find ( x , y ) C l × C l such that 0 Φ 1 ( y , x ) + x ϕ 1 ( y ) + N C l P ( x ) , 0 Φ 2 ( x , y ) + y ϕ 2 ( x ) + N C l P ( y ) .
Proof. 
Suppose that ( x , y ) C l × C l solves system (4).
If Φ 1 ( y , x ) + x ϕ 1 ( y ) = 0 , then:
0 Φ 1 ( y , x ) + x ϕ 1 ( y ) + N C l P ( x ) .
If Φ 1 ( y , x ) + x ϕ 1 ( y ) 0 , the following is always true
Φ 1 ( y , x ) + x ϕ 1 ( y ) , ϕ 1 ( w 1 ) x Φ 1 ( y , x ) + x ϕ 1 ( y ) 2 l ϕ 1 ( w 1 ) x .
By Definition 2 and Proposition 1, we have Φ 1 ( y , x ) + x ϕ 1 ( y ) N C l P ( x ) , and then
0 Φ 1 ( y , x ) + x ϕ 1 ( y ) + N C l P ( x ) .
Likewise,
0 Φ 2 ( x , y ) + y ϕ 2 ( x ) + N C l P ( y ) .
Conversely, if ( x , y ) C l × C l solves problem (6), Definition 2 guarantees that ( x , y ) C l × C l solves system (4).  □
We will obtain a uniqueness theorem for the solution to system (4) after verifying the equivalence between the fixed point formulation (7) and system (4).
Lemma 3.
For i { 1 , 2 } , l > 0 , let Φ i : H 1 × H 1 H 1 and ϕ i : C l C l be nonlinear operators. Suppose that max Φ 1 ( y , x ) , Φ 2 ( x , y ) < l , ( x , y ) C l × C l , and l ( 0 , l ) . Then ( x , y ) C l × C l solves system (4) if and only if
x = P r o j C l ϕ 1 ( y ) Φ 1 ( y , x ) , y = P r o j C l ϕ 2 ( x ) Φ 2 ( x , y ) ,
Proof. 
Suppose that ( x , y ) C l × C l solves system (4). By using ϕ 1 : C l C l and the projection operator technique, we have
d i s t C l ϕ 1 ( y ) Φ 1 ( y , x ) = inf w C l ϕ 1 ( y ) Φ 1 ( y , x ) w ϕ 1 ( y ) Φ 1 ( y , x ) ϕ 1 ( y ) < l .
From Proposition 2, we get ϕ 1 ( y ) Φ 1 ( y , x ) U ( l ) , and then the set P r o j C l ϕ 1 ( y ) Φ 1 ( y , x ) is a singleton. From Lemma 2,
0 Φ 1 ( y , x ) + x ϕ 1 ( y ) + N C l P ( x ) ,
that is,
ϕ 1 ( y ) Φ 1 ( y , x ) ( I + N C l P ) ( x ) .
Thus, we get x = P r o j C l ϕ 1 ( y ) Φ 1 ( y , x ) .
By the same way, we conclude that y = P r o j C l ϕ 2 ( x ) Φ 2 ( x , y ) . Thus, relations (7) are satisfied. It is easy to check the converse.  □
From Lemma 2, we find out the existence of a solution to system (4). By Lemma 3, system (4) admits a unique solution.
Theorem 1.
For i { 1 , 2 } , l > 0 , let Φ i : H 1 × H 1 H 1 be operators which satisfy the conditions from Lemma 3, and ϕ i : C l C l be nonlinear operators. Suppose that μ i , ν i , ζ i , θ i > 0 . Let the operators Φ i be μ i -Lipschitz and ν i -strongly monotone in the first variable and the operators ϕ i be ζ i -Lipschitz and θ i -strongly monotone. If the parameters satisfy
1 2 θ i + ζ i 2 0 ; 1 2 ν i + μ i 2 0 , i = 1 , 2 ; χ 1 + χ 2 < 1 ,
where χ i : = l l l 1 2 θ i + ζ i 2 + 1 2 ν i + μ i 2 , i = 1 , 2 , then system (4) admits a unique solution.
Proof. 
Define the operators φ 1 , φ 2 : H 1 × H 1 C l ,
φ 1 ( x , y ) = P r o j C l ϕ 1 ( y ) Φ 1 ( y , x ) , φ 2 ( x , y ) = P r o j C l ϕ 2 ( x ) Φ 2 ( x , y ) ,
for all ( x , y ) H 1 × H 1 . From Lemma 3, it is easy to check that relations (9) are satisfied. Define · * on H 1 × H 1 as in Proposition 3, that is
( u , v ) * = u + v , ( u , v ) H 1 × H 1 .
Clearly, ( H 1 × H 1 , · * ) is a normed space.
Define a self-mapping T : C l × C l C l × C l , T ( x , y ) = ( φ 1 ( x , y ) , φ 2 ( x , y ) ) for all ( x , y ) C l × C l .
Next, we prove that T is a contraction. Let ( x 1 , y 1 ) , ( x 2 , y 2 ) C l × C l . By Proposition 2, we have
φ 1 ( x 1 , y 1 ) φ 1 ( x 2 , y 2 ) = P C l ϕ 1 ( y 1 ) Φ 1 ( y 1 , x 1 ) P C l ϕ 1 ( y 2 ) Φ 1 ( y 2 , x 2 ) l l l ϕ 1 ( y 1 ) ϕ 1 ( y 2 ) Φ 1 ( y 1 , x 1 ) Φ 1 ( y 2 , x 2 ) l l l y 1 y 2 ϕ 1 ( y 1 ) ϕ 1 ( y 2 ) + y 1 y 2 Φ 1 ( y 1 , x 1 ) Φ 1 ( y 2 , x 2 ) .
In view of ϕ 1 , Φ 1 , for the first summand we have
y 1 y 2 ϕ 1 ( y 1 ) ϕ 1 ( y 2 ) 2 = y 1 y 2 2 2 ϕ 1 ( y 1 ) ϕ 1 ( y 2 ) , y 1 y 2 + ϕ 1 ( y 1 ) ϕ 1 ( y 2 ) 2 1 2 θ 1 + ζ 1 2 y 1 y 2 2 ,
and for the second summand
y 1 y 2 Φ 1 ( y 1 , x 1 ) Φ 1 ( y 2 , x 2 ) 2 = y 1 y 2 2 2 Φ 1 ( y 1 , x 1 ) Φ 1 ( y 2 , x 2 ) , y 1 y 2 + Φ 1 ( y 1 , x 1 ) Φ 1 ( y 2 , x 2 ) 2 1 ν 1 + μ 1 2 y 1 y 2 2 .
Thus,
φ 1 ( x 1 , y 1 ) φ 1 ( x 2 , y 2 ) l l l 1 2 θ 1 + ζ 1 2 + 1 2 ν 1 + μ 1 2 y 1 y 2 .
Similarly, we have
φ 2 ( x 1 , y 1 ) φ 2 ( x 2 , y 2 ) l l l 1 2 θ 2 + ζ 2 2 + 1 2 ν 2 + μ 2 2 x 1 x 2 .
Therefore, we have obtained
φ 1 ( x 1 , y 1 ) φ 1 ( x 2 , y 2 ) + φ 2 ( x 1 , y 1 ) φ 2 ( x 2 , y 2 ) χ 1 x 1 x 2 + χ 2 y 1 y 2 .
Finally, we rewrite the inequality above as
T ( x 1 , y 1 ) T ( x 2 , y 2 ) * κ ( x 1 , y 1 ) ( x 2 , y 2 ) * ,
where κ = max { χ 1 , χ 2 } . Since the parameters satisfy conditions (8), we get 0 κ < 1 . From inequality (10), it follows that the operator T is a contraction. Thus, there exists only one element ( x , y ) such that T ( x , y ) = ( x , y ) . Returning to relation (9), we have x = P r o j C l ϕ 1 ( y ) Φ 1 ( y , x ) and y = P r o j C l ϕ 2 ( x ) Φ 2 ( x , y ) . From Lemma 3, system (4) admits a unique solution.  □
The SSNVI is to find ( x , y ) C l × C l which solves system (4). Then its image ( u , v ) Q k × Q k has to solve system (5). So, Theorem 1 proved the validity of the existence and uniqueness theorem for the solution to SSNVI.

4. Iterative Algorithm

In this part, the set of solutions to SSNVI is denoted by y Ξ and the set of fixed points of S by F i x ( S ) . For any given ( x , y ) C l × C l , define S ( x , y ) = ( S 1 x , S 2 y ) as in Proposition 3. Notice that x F i x ( S 1 ) and y F i x ( S 2 ) if and only if ( x , y ) F i x ( S ) . If ( x * , y * ) Ξ F i x ( S ) , from Lemma 3, in relations (9) and for n 1 , we achieve
x * = S 1 n x * = P r o j C l ϕ 1 ( y * ) Φ 1 ( y * , x * ) = S 1 n P r o j C l ϕ 1 ( y * ) Φ 1 ( y * , x * ) , y * = S 2 n y * = P r o j C l ϕ 2 ( x * ) Φ 2 ( x * , y * ) = S 2 n P r o j C l ϕ 2 ( x * ) Φ 2 ( x * , y * ) .
We now construct the following iterative algorithm (12) by formulation (11) for approximating the unique common element of the set of fixed points of some nearly uniformly Lipschitz operators and the set of solution to SSNVI.
Theorem 2.
For i { 1 , 2 } , l , k > 0 , C l H 1 is a uniformly l-prox-regular and Q k H 2 is a uniformly k-prox-regular. Let Φ i , ϕ i be endowed with the same properties as in Theorem 1. Suppose ξ i , ϑ i , λ i , υ i > 0 . Let the operators Ψ i : H 2 × H 2 H 2 be ξ i -Lipschitz and ϑ i -strongly monotone in the first variable and the operators ψ i : Q k Q k be λ i -Lipschitz and υ i -strongly monotone. A and B are bounded linear operators from H 1 to H 2 , A * and B * are adjoint operators. Let S i : C l C l be two nearly uniformly L i -Lipschitz operators with respect to { σ i , n } , S = ( S 1 , S 2 ) be the same as in Proposition 3 with Ξ F i x ( S ) . Let the sequence { ( x n , y n ) } be computed as follows
f n = P r o j C l ( ϕ 1 ( y n ) Φ 1 ( y n , x n ) ) , g n = P r o j C l ( ϕ 2 ( x n ) Φ 2 ( x n , y n ) ) , h n = P r o j Q k ( ψ 1 ( B g n ) Ψ 1 ( B g n , A f n ) ) , z n = P r o j Q k ( ψ 2 ( A f n ) Ψ 2 ( A f n , B g n ) ) , x n + 1 = ( 1 ι n ) x n + ι n S 1 n P r o j C l ( f n + ς A * ( h n A f n ) ) , y n + 1 = ( 1 ι n ) y n + ι n S 2 n P r o j C l ( g n + ς B * ( z n B g n ) ) , n 1 ,
where { ι n } ( 0 , 1 ) with n = 1 ι n = . Suppose that l ( 0 , l ) , k ( 0 , k ) and
max Ψ 1 ( B g n , A f n ) , Ψ 1 ( B y * , A x * ) , Ψ 2 ( A f n , B g n ) , Ψ 2 ( A x * , B y * ) < k ,
ς < min 2 A 2 , 2 B 2 , r 1 + A * ( h n A f n ) , r 1 + B * ( z n B g n ) , n 1 .
Suppose that L = max L 1 , L 2 , w = l l l , M = max χ 1 + 2 χ 1 χ ¯ 2 , χ 2 + 2 χ 2 χ ¯ 1 < 1 with L w M < 1 , χ 1 and χ 2 are as in Theorem 1 and
χ ¯ 1 = k k k 1 2 υ 1 + λ 1 2 + 1 2 ϑ 1 + ξ 1 2 , χ ¯ 2 = k k k 1 2 υ 2 + λ 2 2 + 1 2 ϑ 2 + ξ 2 2 .
Then the sequence { ( x n , y n ) } computed by relation (12) converges strongly to an element of Ξ F i x ( S ) .
Proof. 
By Theorem 1, let ( x * , y * ) C l × C l be the solution to system (4). Therefore, ( x * , y * ) C l × C l is the unique solution to SSNVI. Let us take ( x * , y * ) Ξ F i x ( S ) . Since conditions (8) and (13) are satisfied respectively, then we obtain
x * = P r o j C l ϕ 1 ( y * ) Φ 1 ( y * , x * ) ,
y * = P r o j C l ϕ 2 ( x * ) Φ 2 ( x * , y * ) ,
A x * = P r o j Q k ψ 1 ( B y * ) Ψ 1 ( B y * , A x * ) ,
B y * = P r o j Q k ψ 2 ( A x * ) Ψ 2 ( A x * , B y * ) .
From the definition of Φ 1 , ϕ 1 , by relations (12), (8), (15) and Proposition 2, we have
f n x * l l l y n y * ( ϕ 1 ( y n ) ϕ 1 ( y * ) ) + y n y * ( Φ 1 ( y n , x n ) Φ 1 ( y * , x * ) ) χ 1 y n y * , n N .
In view of Φ 2 and ϕ 2 , from relations (12), (8), (16), and Proposition 2, we find
g n y * χ 2 x n x * , n N .
By looking into the definition of Ψ 1 and ψ 1 , from (12), (13), (17), and Proposition 2, we attain
h n A x * χ ¯ 1 B g n B y * , n N .
In light of Ψ 2 and ψ 2 , from (12), (13), (18) and Proposition 2, we conclude
z n B y * χ ¯ 2 A f n A x * , n N .
By relation (14), we get
f n x * ς A * ( A f n A x * ) 2 = f n x * 2 2 ς f n x * , A * ( A f n A x * ) + ς 2 A * ( A f n A x * ) 2 f n x * 2 ς 2 ς A 2 A f n A x * 2 f n x * 2 , n N .
Using (11), (12), (14), (19)–(21), (23), and Proposition 3, we obtain
x n + 1 x * ( 1 ι n ) x n x * + ι n S 1 n P r o j C l ( f n + ς A * ( h n A f n ) ) S 1 n P r o j C l ( x * + ς A * ( A x * A x * ) ) ( 1 ι n ) x n x * + ι n L 1 P r o j C l ( f n + ς A * ( h n A f n ) ) P r o j C l ( x * + ς A * ( A x * A x * ) ) + σ 1 , n ( 1 ι n ) x n x * + ι n L 1 w f n x * ς A * ( A f n A x * ) + ς A * ( h n A x * ) + ι n L 1 σ 1 , n ( 1 ι n ) x n x * + ι n L 1 w f n x * + ς A h n A x * + ι n L 1 σ 1 , n ( 1 ι n ) x n x * + ι n L 1 w χ 1 y n y * + 2 χ 2 χ ¯ 1 x n x * + ι n L 1 σ 1 , n , n N .
Likewise,
g n y * ς B * ( B g n B y * ) g n x * , n N .
By using relations (11), (12), (14), (19), (20), (22), (25), and Proposition 3, we have
y n + 1 y * ( 1 ι n ) y n y * + ι n L 2 w s x n x * + 2 χ 1 χ ¯ 2 u y n y * + ι n L 2 σ 2 , n .
It follows from (24) and (26) that
( x n + 1 , y n + 1 ) ( x * , y * ) * ( 1 ι n ) ( x n , y n ) ( x * , y * ) * + ι n L w M ( x n , y n ) ( x * , y * ) * + σ 1 , n + σ 2 , n = 1 ι n ( 1 L w M ) ( x n , y n ) ( x * , y * ) * + ι n ( 1 L w M ) ( σ 1 , n + σ 2 , n ) L 1 L w M , n N .
By applying Lemma 1 to relation (27), we achieve lim n x n = x * and lim n y n = y * . Thus, we conclude that the sequence { ( x n , y n ) } computed by (12) converges strongly to an element of Ξ F i x ( S ) .  □
We now have in view a special case of SSNVI, called the split systems of general variational inequalities (SSGVI), which is an improvement of SVIP in [29] and SSVI in [30].
Surely, if l = , the convexity of C and the uniformly prox-regularity of C l are equivalent. Thus, in a underlying convex set, the SSGVI is to find ( x , y ) C × C such that
Φ 1 ( y , x ) + x ϕ 1 ( y ) , ϕ 1 ( w 1 ) x , w 1 H 1 : ϕ 1 ( w 1 ) C , Φ 2 ( x , y ) + y ϕ 2 ( x ) , ϕ 2 ( w 1 ) y , w 1 H 1 : ϕ 2 ( w 1 ) C ,
and such that ( u , v ) Q × Q with u = A x , v = B y solves
Ψ 1 ( v , u ) + u ψ 1 ( v ) , ψ 1 ( w 2 ) u , w 2 H 2 : ψ 1 ( w 2 ) Q , Ψ 2 ( u , v ) + v ψ 2 ( u ) , ψ 2 ( w 2 ) v , w 2 H 2 : ψ 2 ( w 2 ) Q .
where C H 1 and Q H 2 are nonempty, closed, convex sets, Φ i , ϕ i , Ψ i , and ψ i ( i = 1 , 2 ) are the same as Theorem 2.
If l, k = , then the uniformly prox-regularity of C l , Q k collapse to convexity, respectively, that is to say C l = C , Q k = Q . Hence, we have the following corollary.
Corollary 1.
Let C H 1 and Q H 2 be nonempty, closed, convex sets. For i { 1 , 2 } , presume that Φ i , ϕ i , Ψ i ψ i , A, and B are the same as in Theorem 2. For each n 1 , let sequence { ( x n , y n ) } be computed as follows
f n = P r o j C ( ϕ 1 ( y n ) Φ 1 ( y n , x n ) ) , g n = P r o j C ( ϕ 2 ( x n ) Φ 2 ( x n , y n ) ) , h n = P r o j Q ( ψ 1 ( B g n ) Ψ 1 ( B g n , A f n ) ) , z n = P r o j Q ( ψ 2 ( A f n ) Ψ 2 ( A f n , B g n ) ) , x n + 1 = ( 1 ι n ) x n + ι n P r o j C ( f n + ς A * ( h n A f n ) ) , y n + 1 = ( 1 ι n ) y n + ι n P r o j C ( g n + ς B * ( z n B g n ) ) ,
where { ι n } ( 0 , 1 ) with n = 1 ι n = . Suppose that ς < min 2 A 2 , 2 B 2 and M = max χ 1 + 2 χ 1 χ ¯ 2 , χ 2 + 2 χ 2 χ ¯ 1 < 1 with
χ 1 = 1 2 θ 1 + ζ 1 2 + 1 2 ν 1 + μ 1 2 , χ 2 = 1 2 θ 2 + ζ 2 2 + 1 2 ν 2 + μ 2 2 ,
χ ¯ 1 = 1 2 υ 1 + λ 1 2 + 1 2 ϑ 1 + ξ 1 2 , χ ¯ 2 = 1 2 υ 2 + λ 2 2 + 1 2 ϑ 2 + ξ 2 2 .
Then the sequence { ( x n , y n ) } computed by relation (28) converges strongly to a solution to the SSGVI.

5. Numerical Example

Let H 1 = H 2 = R . Let x , y = x y , Φ 1 ( y , x ) = 7 6 y , Φ 2 ( x , y ) = 8 7 x , Ψ 1 ( y , x ) = 9 8 y , Ψ 2 ( x , y ) = 10 9 x for all x, y R . Let C = [ 0 , + ) and ϕ 1 , ϕ 2 : C C , ϕ 1 ( x ) = 3 2 x , ϕ 2 ( x ) = 4 3 x , respectively. Let Q = R and ψ 1 , ψ 2 : Q Q , ψ 1 ( x ) = 5 4 x , ψ 2 ( x ) = 6 5 x , respectively.
Clearly, ϕ 1 , ϕ 2 , ψ 1 , ψ 2 , Φ 1 , Φ 2 , Ψ 1 , Ψ 2 are 1-strongly monotone and 2-Lipschitzian. Let A x = 1 2 x and B x = 3 4 x from H 1 to H 2 , respectively. For ι n = 1 5 , ς = 1 . We now rewrite (28) as follows
f n = P r o j C 3 2 y n 7 6 y n , g n = P r o j C 4 3 x n 8 7 x n , h n = P r o j C 5 4 · 3 4 g n 9 8 · 3 4 g n , z n = P r o j C 6 5 · 1 2 f n 10 9 · 1 2 f n , x n + 1 = 4 5 x n + 1 5 P r o j C ( f n + 1 2 ( h n 1 2 f n ) ) , y n + 1 = 4 5 y n + 1 5 P r o j C ( g n + 3 4 ( z n 3 4 g n ) ) , n 1 .
For every n 1 , the operators and the parameters satisfy all conditions in Corollary 1. We find that the sequence { ( x n , y n ) } generated by relation (29) converges strongly to (0, 0).
Choosing initial values (10, 20), we see that Figure 1 demonstrates Corollary 1.

6. Conclusions

In this paper, we investigated the split system of nonconvex variational inequalities (SSNVI) in the context of uniformly prox-regular sets, which is an improvement of SSGVI, SSVI, and SVIP. By using an adequate formulation and the projection technique, we constructed an iterative algorithm for approximating the unique common solution to the set of fixed points of nearly uniformly Lipschitz operators and the set of solutions to SSNVI. The results of this paper are expected to be used as further study on numerical techniques.

Author Contributions

All authors participated in the conceptualization, validation, formal analysis, and investigation, as well as the writing of the original draft preparation, reviewing, and editing.

Funding

This work was supported by the Key Subject Program of Lingnan Normal University (Grant No. 1171518004), the Natural Science Foundation of Guangdong Province (2018A0303070012), and the Young Innovative Talents Project in Guangdong Universities (2017KQNCX125). Li-Jun Zhu was supported by the grants NXJG2017003, NXYLXK2017B09, and Advanced Intelligent Perception and Control Technology Innovative Team of NingXia.

Acknowledgments

The authors are grateful to the reviewers and to the editors for their valuable comments and suggestions which helped us improve the paper significantly.

Conflicts of Interest

The authors declare that they have no competing interests.

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Figure 1. The convergence of { ( x n , y n ) } with initial values (10, 20).
Figure 1. The convergence of { ( x n , y n ) } with initial values (10, 20).
Symmetry 11 01279 g001

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Chen, J.; Pitea, A.; Zhu, L.-J. Split Systems of Nonconvex Variational Inequalities and Fixed Point Problems on Uniformly Prox-Regular Sets. Symmetry 2019, 11, 1279. https://doi.org/10.3390/sym11101279

AMA Style

Chen J, Pitea A, Zhu L-J. Split Systems of Nonconvex Variational Inequalities and Fixed Point Problems on Uniformly Prox-Regular Sets. Symmetry. 2019; 11(10):1279. https://doi.org/10.3390/sym11101279

Chicago/Turabian Style

Chen, Jinzuo, Ariana Pitea, and Li-Jun Zhu. 2019. "Split Systems of Nonconvex Variational Inequalities and Fixed Point Problems on Uniformly Prox-Regular Sets" Symmetry 11, no. 10: 1279. https://doi.org/10.3390/sym11101279

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