Strong Convergence Results for Hierarchical Circularly Iterative Method about Hierarchical Circularly Optimization

An hierarchical circularly iterative method is introduced for solving a system of variational circularly inequalities with set of fixed points of strongly quasi-nonexpansive mapping problems in this paper. Under some suitable conditions, strong convergence results for the hierarchical circularly iterative sequence are proved in the setting of Hilbert spaces. Our scheme can be regarded as a more general variant of the algorithm proposed by Maingé.


Introduction
Variational inequalities were initially studied by Stampacchia [1] and ever since have been widely studied, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance.On the other hand, a number of mathematical programs and iterative algorithms have been developed to resolve complex real world problems.

Preliminaries and Lemma
2) A mapping is said to be : 3) A mapping is said to be quasinonexpansive if 4) A mapping is said to be strongly quasi-nonexpansive if

 
p Fix T  .5) (see [13]) A mapping is said to be Obviously, the above inequality is equivalent to and it is clear from the preceding definitions that every quasi-nonexpansive mapping is 0-demicontractive.Lemma 2.3 (see [14]) For and ) 1 For prove our result, we give the following lemma.
and the following properties are satisfied for all (sufficiently large) numbers sequence and .
In fact, .
  1 max : Lemma 2.5 ([11]) Assume that  is a sequence of nonnegative real numbers such that Then .

 0 
In [11], the existence and uniqueness of solutions of some related hierarchical optimization problems had been discussed.
Inspired by these results in the literature, a circularly iterative method in this paper is introduced for solving a system of variational inequalities with fixed-point set constraints.Under suitable conditions, strong convergence results are proved in the setting of Hilbert spaces.Our scheme can be regarded as a more general variant of the algorithm proposed by Maingé.The results presented in the paper improve and extend the corresponding results in [11] and other.

Main Results
First, we discuss the existence and uniqueness of solutions of some related hierarchical optimization problems.Theorem 3.1 Let be quasi-nonexpansive mappings and Then there exists a unique element .
Proof.The proof is a consequence of the well-known Banach's contraction principle but it is given here for the sake of completeness.It is known that both sets  are closed and convex, and hence the projections are well defined.It is clear that the mapping  1, 2, , is a contraction.Hence, there exists a unique element such that the following inequalities, .

 
This implies that and hence . This completes the proof. For mappings , suppose , , : we define the iterative sequences Recall that a mapping is demiclosed at zero iff and Tx . n We split the proof of Theorem 3.2 into the following lemmas.
Proof.Since i be strongly quasi-nonexpansive mappings, i f be contractions with the coefficient  .Then we have Similarly, we also have By induction, we have x are bounded.Consequently, the sequences are also Lemma 3.4 For each N n  , the following inequality holds: Similarly, we also have Lemma 3.5 If there exists a subsequence of Proof.In fact, we first consider the following assertion: by the iteration scheme (1), we have It follows from the boundedness of By using the same argument, we have

Lemma 2 . 4 (
[11]) Let   n  be a sequence of real numbers such that there exists a subsequence   Theorem 3.2 is completed. .